INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

INAUGURAL–DISSERTATION 

zur 

Erlangung der Doktorwürde 

der 

Naturwissenschaftlich–Mathematischen 

Gesamtfakultät 

der 

Ruprecht–Karls–Universität 

Heidelberg 

vorgelegt von 

M. Tech. Srikanth Reddy Gopireddy 

aus Nizambad, India 

Tag der mündlichen Prüfung: 06.12.2013

Numerical Simulation of Bi-component Droplet 

Evaporation and Dispersion in Spray and 

Spray Drying 

Gutachter: Prof. Dr. Eva Gutheil 

PD Dr. N. Dahmen

Dreams can never become a reality without hard work 

– Swamy Vivekananda

Abstract 

Spray drying is one of the most widely used drying techniques to convert liquid feed 

into a dry powder. The modeling of spray flows and spray drying has been studied 

for many years now, to determine the characteristics of the end products, e.g. particle 

size, shape, density or porosity. So far, the simulation of polymer or sugar solution 

spray drying has not been studied because drying behavior as well as properties are 

unknown. Previous studies concentrated on the systems of milk, salt solution, colloids 

or other materials for which the thermal and physical properties are well tabulated. 

The present study deals with the modeling and simulation of polyvinylpyrrolidone 

(PVP)/water and mannitol/water spray flows. PVP is a polymer, widely used as 

a pharmaceutical excipient, and mainly manufactured by BASF under several patented 

names, whereas mannitol is a sugar, which is used in dry powder inhalers and tablets. 

Experimental studies have shown that the powder properties of PVP and mannitol are 

significantly influenced by the drying conditions. The growing importance of PVP or 

mannitol powders and the inability of existing studies to predict the effect of drying 

conditions on the properties of the end product have prompted the development of a 

new reliable model and numerical techniques. 

Evaporating sprays have a continuous phase (gas) and a dispersed phase, which 

consists of droplets of various sizes that may evaporate, coalesce, or breakup, as well 

as have their own inertia and size-conditioned dynamics. A modeling approach which 

is more commonly used is the Lagrangian description of the dispersed liquid phase. 

This approach gives detailed information on the micro-level, but inclusion of droplet 

coalescence and breakup increase computational complexity. Moreover, the Lagrangian 

description coupled with the Eulerian equations for the gas phase, assuming a pointsource 

approximation of the spray, is computationally expensive. As an alternative to 

Lagrangian simulations, several Eulerian methods have been developed based on the 

Williams’ spray equation. The Euler – Euler methods are computationally efficient 

and independent of liquid mass loading in describing dense turbulent spray flows. 

The objective of this thesis is the modeling and simulation of spray flows and 

spray drying up to the onset of solid layer formation in an Euler – Euler framework. 

The behavior of droplet distribution under various drying conditions in bi-component 

evaporating spray flows is examined using, for the first time, direct quadrature method 

of moments (DQMOM) in two dimensions. In DQMOM, the droplet size and velocity 

distribution of the spray is modeled by approximating the number density function in 

terms of joint radius and velocity. Transport equations of DQMOM account for droplet 

evaporation, heating, drag, and droplet–droplet interactions. 

At first, an evaporating water spray in nitrogen is modeled in one dimension (ax- 

I

II 

ial direction). Earlier studies in spray flows neglected evaporation or considered it 

through a simplified model, which is addressed by implementing an advanced droplet 

evaporation model of Abramzon and Sirignano, whereas droplet motion and droplet 

coalescence are estimated through appropriate sub-models. The assumption of evaporative 

flux to be zero or computing it with weight ratio constraints was found to be 

unphysical, which is improved by estimating it using the maximum entropy formulation. 

The gas phase is not yet fully coupled to the DQMOM but its inlet properties are 

taken to compute forces acting on droplets and evaporation. The simulation results 

are compared with quadrature method of moments (QMOM) and with experiment at 

various cross sections. DQMOM shows better results than QMOM, and remarkable 

agreement with experiment. 

Next, water spray in air in two-dimensional, axisymmetric configuration is modeled 

by extending the one-dimensional DQMOM. The DQMOM results are compared with 

those of the discrete droplet model (DDM), which is an Euler – Lagrangian approach. 

Droplet coalescence is considered in DQMOM but neglected in DDM. The simulation 

results are validated with new experimental data. Overall, DQMOM shows a much 

better performance with respect to computational effort, even with the inclusion of 

droplet coalescence. 

Before extending DQMOM to model PVP/water spray flows, a single droplet evaporation 

and drying model is developed, because most of the evaporation models available 

in the literature are valid for salts, colloids or milk powder. The negligence of solid layer 

formation effects on the droplet heating and evaporation is addressed, and treatment 

of the liquid mixture as the ideal solution is improved by including the non-ideality 

effect. The PVP or mannitol in water droplet evaporation and solid layer formation 

are simulated, and the results are compared with new experimental data, which shows 

that the present model effectively captures the first three stages of evaporation and 

drying of a bi-component droplet. 

Finally, PVP/water spray flows in air are simulated using DQMOM including the 

developed bi-component evaporation model. Simulation results are compared with new 

experimental data at various cross sections and very good agreement is observed. 

In conclusion, water and PVP/water evaporating spray flows, and preliminary 

stages of PVP/water and mannitol/water spray drying, i.e., until solid layer formation, 

are successfully modeled and simulated, and show good agreement with experiment. 

Keywords: Sprays, PVP, Mannitol, DQMOM, Bi-component droplet

III 

Zusammenfassung 

Sprühtrocknung ist eines der am häufigsten eingesetzten Verfahren, um eine zugeführte 

Flüssigkeit in ein trockenes Pulver umzuwandeln. Die Modellierung von Sprühtrocknungsprozessen 

und des Sprays selbst wird seit vielen Jahren betrieben, um die Eigenschaften 

der Endprodukte, wie z.B. Partikelgröße, Form, Dichte oder Porosität, bestimmen 

zu können. Die Sprühtrocknung von Polymer- oder Zuckerlösungen wurde bisher 

noch nicht numerisch untersucht, da deren Trocknungsverhalten und Eigenschaften 

unbekannt sind. Bislang wurden nur Systeme mit Milch, Salzlösungen oder Kolloiden 

untersucht, deren thermische und physikalische Eigenschaften gut belegt sind. 

Die vorliegende Arbeit widmet sich der Modellierung und Simulation von Polyvinylpyrrolidon 

(PVP)/Wasser und Mannitol/Wasser-Sprays. PVP ist ein Polymer, weit 

verbreitet als pharmazeutisches Bindemittel und von der BASF unter verschiedenen 

patentierten Namen hergestellt, während Mannitol, ein Zucker, hauptsächlich in Trockenpulverinhalatoren 

und Tabletten verwendet wird. Experimentelle Studien haben gezeigt, 

dass die Eigenschaften von PVP- und Mannitol-Pulvern von den Trocknungsbedingungen 

signifikant beeinflusst werden. Die zunehmende Bedeutung von PVP- und 

Mannitol-Pulvern und das Fehlen geeigneter Methoden zur Bestimmung des Einflusses 

der Trocknungsbedingungen auf die Eigenschaften der Endprodukte haben die Entwicklung 

eines neuen zuverlässigen Modells sowie numerischer Methoden angeregt. 

Verdampfende Sprays bestehen aus einer kontinuierlichen Phase (Gas), und einer 

zerstäubten Phase, die aus Tropfen unterschiedlicher Größe besteht, die verdampfen, 

koaleszieren oder auch aufbrechen können, die aber auch ihre eigene Trägheit und 

größenabhängige Dynamik besitzen. Ein häufig verwendeter Modellierungsansatz ist 

die Beschreibung der zerstäubten, flüssigen Phase im Lagrangeschen Bezugssystem. 

Dieser Ansatz liefert detaillierte Informationen auf Mikroebene, aber Tropfen-Interaktionen 

wie Koaleszenz und Aufbrechen sind schwierig zu implementieren. Zudem 

ist der Lagrange-Ansatz, gekoppelt mit den Gleichungen der Gasphase im Eulerschen 

Bezugssystem unter Annahme der Punktquellen-Annäherung, zeitintensiv. Die Alternative 

zu Lagrange-Simulationen sind verschiedene Eulersche Methoden, die auf der 

Basis der Williams-Spraygleichung entwickelt wurden. Die Beschreibung von dichten 

turbulenten Sprayströmungen ist bei Verwendung dieser Euler – Euler Methoden zeiteffizient 

und unabhängig von der Massenladung der flüssigen Phase. 

Die Zielsetzung der vorliegenden Arbeit ist die Modellierung und Simulation der 

Sprühtrocknung bis zum Beginn der Partikelbildung im Eulerschen Bezugssystem. Zur 

Untersuchung des Verhaltens der Tropfenverteilung unter verschiedenen Trocknungsbedingungen 

wurde erstmals die Methode direct quadrature method of moments (DQ- 

MOM) zur Betrachtung der verdampfenden Zweikomponentensprays eingesetzt. In der

IV 

DQMOM wird die Tropfengrößen- und Geschwindigkeitsverteilung des Sprays modelliert, 

indem die Zahlendichtefunktion angenähert wird. Die Transportgleichungen der 

DQMOM berücksichtigen Tropfenverdampfung, Aufheizung, Widerstand und Tropfen- 

Tropfen-Interaktionen. 

Zuerst wird ein verdampfendes Wasserspray in Stickstoff in eindimensionaler Konfiguration, 

d.h. in axialer Richtung des Sprays, modelliert. Frühere Spraystudien vernachlässigten 

Verdampfungseffekte oder berücksichtigten diese durch ein vereinfachtes 

Modell. In dieser Arbeit wird die Tropfenverdampfung jedoch durch das Modell von 

Abramzon und Sirignano beschrieben, während Tropfenbewegung und -koaleszenz mit 

geeigneten Modellen abgeschätzt werden. Da die Vernachlässigung des Verdampfungsflusses 

oder seine Berechnung durch Einschränkungen des Gewichtsverhältnisses sich 

als unphysikalisch herausstellte, wurde der Fluss hier durch die Maximum-Entropie- 

Methode berechnet. Die Gasphase ist noch nicht vollständig an die DQMOM gekoppelt, 

stattdessen dienen die Gas-Einlaufbedingungen als Grundlage zur Berechnung 

der Kräfte, die auf Tropfen und Verdampfung wirken. Die Resultate der Simulationen 

werden mit der Quadratur-Momentenmethode (QMOM) und Experimenten an verschiedenen 

Querschnitten verglichen. Die DQMOM zeigt bessere Ergebnisse als die 

QMOM und auch erstaunliche Übereinstimmung mit dem Experiment. 

Als nächstes wird das Wasserspray in umgebender Luft in zweidimensionaler, axialsymmetrischer 

Konfiguration durch Erweiterung der eindimensionalen DQMOM modelliert. 

Die DQMOM-Resultate werden mit denen des diskreten Tropfenmodells (DDM), 

ein Euler – Lagrange Ansatz, verglichen. Tropfenkoaleszenz wird in der DQMOM 

berücksichtigt, in der DDM aber vernachlässigt. Die Simulationsergebnisse werden 

durch aktuelle experimentelle Daten validiert. Insgesamt zeigt die DQMOM deutlich 

bessere Recheneffizienz, sogar unter Einschluss der Tropfenkoaleszenz. 

Bevor die DQMOM auf PVP/Wasser-Sprays erweitert wird, wird ein Verdampfungsund 

Trocknungsmodell für einen Einzeltropfen entwickelt, da die meisten der in der Literatur 

bekannten Verdampfungsmodelle auf Salze, Kolloide oder Milchpulver angewendet 

werden. Das Modell berücksichtigt die Partikelbildung in Zusammenhang mit der 

Tropfenaufheizung und -verdampfung, und die Behandlung der flüssigen Mischung als 

ideale Lösung wird durch Einschluss nicht-idealer Effekte verbessert. Die Ergebnisse 

der Simulation dieses Modells werden mit aktuellen experimentellen Daten verglichen, 

und es kann gezeigt werden, dass das entwickelte Modell die ersten drei Phasen der 

Verdampfung und des Trocknens eines Zweikomponententropfen effektiv erfassen kann. 

Schließlich wird ein PVP/Wasser-Spray in umgebender Luft mittels DQMOM simuliert 

unter Anwendung des entwickelten Zweikomponentenverdampfungsmodells. Die 

Ergebnisse werden mit aktuellen experimentellen Daten an mehreren Querschnitten 

verglichen, und es konnte eine sehr gute Übereinstimmung festgestellt werden.

V 

Letztendlich können verdampfende Wasser- und PVP/Wasser-Sprays und die Frühphasen 

der Sprühtrocknung von PVP/Wasser- und Mannitol/Wasser-Tropfen, d.h. bis 

zum Einsetzen der Bildung einer festen Schicht, erfolgreich modelliert und simuliert 

werden, unter guter Übereinstimmung mit dem Experiment. 

Stichwörter: Sprays, PVP, Mannitol, DQMOM, Zweikomponententropfen

Contents 

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

I 

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

2. Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 

2.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 

2.2 Euler – Lagrangian Approach . . . . . . . . . . . . . . . . . . . . . . . 17 

2.2.1 Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 

2.2.2 Discrete Droplet Model (DDM) . . . . . . . . . . . . . . . . . . 19 

2.3 Euler – Euler Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 20 

2.3.1 Treatment of the Spray . . . . . . . . . . . . . . . . . . . . . . . 20 

2.3.2 NDF Transport Equation . . . . . . . . . . . . . . . . . . . . . 22 

2.3.3 Quadrature Method of Moments (QMOM) . . . . . . . . . . . . 23 

2.3.4 Direct Quadrature Method of Moments (DQMOM) . . . . . . . 25 

2.4 Single Droplet Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

2.4.1 Droplet Heating and Evaporation . . . . . . . . . . . . . . . . . 29 

2.4.1.1 Single Component Droplet . . . . . . . . . . . . . . . . 29 

2.4.1.2 Bi-component Droplet . . . . . . . . . . . . . . . . . . 32 

2.4.2 Droplet Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 

2.4.3 Droplet Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . 40 

2.4.4 Droplet Coalescence . . . . . . . . . . . . . . . . . . . . . . . . 41 

3. Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 

3.1 Finite Difference Method for Bi-component Droplet Evaporation and 

Solid Layer Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 

3.2 Spray Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 

3.2.1 Finite Volume Method for QMOM . . . . . . . . . . . . . . . . 47 

3.2.2 Finite Difference Scheme for DQMOM . . . . . . . . . . . . . . 49 

3.2.3 Wheeler Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 51 

3.3 Numerical Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

VIII 

Contents 

4. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 

4.1 One-dimensional Evaporating Water Spray in Nitrogen . . . . . . . . . 55 

4.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 56 

4.1.2 Initial Data Generation . . . . . . . . . . . . . . . . . . . . . . . 56 

4.1.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 58 

4.2 Two-dimensional Evaporating Water Spray in Air . . . . . . . . . . . . 66 

4.2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 66 

4.2.2 Initial Data Generation . . . . . . . . . . . . . . . . . . . . . . . 67 


4.3 Single Bi-component Droplet Evaporation and Solid Layer Formation . 76 

4.3.1 Vapor-Liquid Equilibrium . . . . . . . . . . . . . . . . . . . . . 77 

4.3.2 Non-ideal Liquid Mixture . . . . . . . . . . . . . . . . . . . . . 77 


4.4 Two-dimensional Evaporating PVP/Water Spray in Air . . . . . . . . . 92 

4.4.1 Experiment and Initial Data Generation . . . . . . . . . . . . . 93 


5. Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 99 

Appendix 103 

A. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 

B. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

List of Tables 

4.1 Initial weights and abscissas . . . . . . . . . . . . . . . . . . . . . . . . 57 

4.2 Droplet size distribution of water spray . . . . . . . . . . . . . . . . . . 61 

4.3 Experimental drying conditions . . . . . . . . . . . . . . . . . . . . . . 81 

4.4 Experiment vs simulation . . . . . . . . . . . . . . . . . . . . . . . . . 86

X 

List of Tables

List of Figures 

1.1 A schematic diagram of the spray drying process [3]. . . . . . . . . . . 2 

1.2 Chemical structure of polyvinylpyrrolidone (C 6 H 9 NO) n [8]. . . . . . . . 3 

1.3 SEM images of spray dried PVP [11]. . . . . . . . . . . . . . . . . . . . 4 

1.4 Chemical structure of mannitol (C 6 H 8 (OH) 6 ) [12]. . . . . . . . . . . . . 4 

2.1 Sketch of a pressure-atomized spray formation [69, 70]. . . . . . . . . . 11 

2.2 A typical log-normal distribution with different values of µ and σ. . . . 21 

2.3 Approximation of NDF in QMOM. . . . . . . . . . . . . . . . . . . . . 24 

2.4 NDF approximation in DQMOM. . . . . . . . . . . . . . . . . . . . . . 25 

2.5 Schematic diagram of stages in single droplet evaporation and drying. . 34 

2.6 Droplet breakup mechanisms based on Weber number [69, 178]. . . . . 41 

2.7 Droplet collision regimes: (a) bouncing, (b) coalescence [184]. . . . . . . 42 

2.8 Droplet collision regimes: (c) reflexive or crossing separation, (d) stretching 

separation [184]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 

3.1 Flowchart of the DQMOM computational code. . . . . . . . . . . . . . 52 

4.1 Photograph of the water spray formation. . . . . . . . . . . . . . . . . . 56 

4.2 Schematic diagram of spray with measurement positions. . . . . . . . . 56 

4.3 Experimental surface frequency distribution at cross section 0.14 m (left) 

and 0.54 m (right) away from the nozzle exit. . . . . . . . . . . . . . . 57 

4.4 Comparison of simulated and measured [199] droplet mass and temperature 

profiles for the evaporation of a pure water droplet. . . . . . . . . 58 

4.5 Homogeneous and inhomogeneous calculations of DQMOM. . . . . . . 59 

4.6 Comparison of QMOM and DQMOM results with experiment. . . . . . 60 

4.7 Droplet size distribution of water spray and at t = 0 s, and at t = 1 s 

with d 2 law evaporation rate. . . . . . . . . . . . . . . . . . . . . . . . 60 

4.8 Velocity profiles of three droplets with different initial radii and velocities 

under the influence of drag alone (left) and drag and gravity (right). . . 62 

4.9 Effect of liquid inflow rates on Sauter mean diameter computed with 

and without coalescence. . . . . . . . . . . . . . . . . . . . . . . . . . . 62

XII 


4.10 Profiles of Sauter mean diameter (left) and mean droplet diameter (right) 

computed with and without coalescence at surrounding gas temperatures 

of 293 K and 313 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

4.11 Profiles of specific surface area computed with and without coalescence 

at surrounding gas temperatures of 293 K and 313 K. . . . . . . . . . . 64 

4.12 Profiles of droplet number density computed with and without coalescence 

at surrounding gas temperatures of 293 K and 313 K. . . . . . . 65 

4.13 Schematic diagram of the experimental setup. . . . . . . . . . . . . . . 66 

4.14 Profile of effective cross-section area of the probe volume for measured 

droplet size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 

4.15 Experimental and DQMOM approximation of droplet number density 

for a water spray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 

4.16 Experimental and reconstructed NDF of 80 kg/h water spray at 64.5 mm 

from the center of the spray axis. . . . . . . . . . . . . . . . . . . . . . 69 

4.17 Experimental and reconstructed NDF of 121 kg/h water spray at 81 mm 

from the center of the spray axis. . . . . . . . . . . . . . . . . . . . . . 70 

4.18 Experimental and numerical profiles of the Sauter mean diameter of water 

spray with 80 kg/h liquid flow rate at the cross section of 0.12 m (left) 

and 0.16 m (right) distance from the nozzle exit. . . . . . . . . . . . . . 71 

4.19 Experimental and numerical profiles of the Sauter mean diameter at the 

cross section of 0.12 m distance from the nozzle exit for 120 kg/h. . . . 71 

4.20 Experimental and numerical profiles of the Sauter mean diameter at the 

cross section of 0.16 m distance from the nozzle exit for 120 kg/h. . . . 72 

4.21 Experimental and numerical profiles of the mean droplet diameter of water 



4.22 Experimental and numerical profiles of the mean droplet diameter of water 



4.23 Experimental and numerical profiles of the mean droplet velocity at the 

cross section of 0.12 m distance from the nozzle exit. . . . . . . . . . . 75 

4.24 Experimental and numerical profiles of the mean droplet velocity at the 


4.25 Numerical and experimental [214] results of water activity (a w ) in PVP/water 

solution at 73.0 ◦ C (left) and 94.5 ◦ C (right). . . . . . . . . . . . . . . . 78 

4.26 Numerical results of water activity (a w ) in mannitol/water solution at 

94.5 ◦ C and 160 ◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


XIII 

4.27 Effect of non-ideality on the vapor pressure of water at different temperatures. 

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 

4.28 Variation of vapor pressure of water with water mass fraction in PVP/water 

solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 

4.29 Experimental data of PVP [218] and mannitol [219] saturation solubility 

in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 

4.30 Effect of gas velocity on the evolution of mass and temperature of a 

mannitol/water droplet. . . . . . . . . . . . . . . . . . . . . . . . . . . 82 

4.31 Effect of elevated gas temperature on the surface area of a mannitol/water 

droplet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 

4.32 Effect of gas temperature on the temporal development of mannitol mass 

fraction inside the droplet at 0.5 s (left) and 0.9 s (right). . . . . . . . . 83 

4.33 Effect of gas velocity and temperature on porosity and final particle size 

of mannitol/water droplet. . . . . . . . . . . . . . . . . . . . . . . . . . 84 

4.34 SEM images of mannitol samples spray dried at 70 ◦ C (a), 100 ◦ C (b) 

and 90 ◦ C (c). Zoomed images of the surface structures of these particles 

at 70 ◦ C (d), 100 ◦ C (e) and 90 ◦ C (f) [225]. . . . . . . . . . . . . . . . . 85 

4.35 Effect of gas temperatures of 60 ◦ C and 95 ◦ C and relative humidity of 

1% R.H. (left) and 30% R.H. (right) on the droplet surface area. . . . . 86 

4.36 Time evolution of mannitol/water droplet surface area computed by 

present model and RMM. . . . . . . . . . . . . . . . . . . . . . . . . . 87 

4.37 Effect of initial droplet temperature on the evaporation rate of mannitol/water 

droplet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 

4.38 Effect of gas temperature on solid layer thickness inside the PVP/water 

droplet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 

4.39 Effect of gas temperature on the droplet mass and temperature. . . . . 89 

4.40 Temporal development of PVP mass fraction profiles inside the droplet 

subjected to dry air (left) and hot air with 5% R.H (right). . . . . . . . 90 

4.41 Time evolution of PVP/water droplet surface area predicted by present 

model and RMM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 

4.42 Effect of relative humidity on the water evaporation rate from the mannitol/water 

droplet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 

4.43 Effect of initial PVP mass fraction on the profiles of droplet radius and 

temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 

4.44 Photograph of the PVP/water spray formation with 112 kg/h liquid 

inflow rate in experiment [206]. . . . . . . . . . . . . . . . . . . . . . . 93 

4.45 Experimental and DQMOM approximation of droplet number density 

for PVP/water spray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

XIV 


4.46 Experimental and numerical profiles of the Sauter mean diameter (left) 

and mean droplet diameter (right) of PVP/water spray in air at the 


4.47 Experimental and numerical profiles of the Sauter mean diameter (left) 

and mean droplet diameter (right) of PVP/water spray in air at the 


4.48 Experimental and numerical profiles of the mean droplet velocity of 

PVP/water spray in air at the cross section of 0.12 m distance from 

the nozzle exit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 

4.49 Experimental and numerical profiles of the mean droplet velocity of 

PVP/water spray in air at the cross section of 0.16 m distance from 

the nozzle exit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

1. Introduction 

Drying has found applications in many areas of industry such as chemical, food, pharmaceutical, 

polymer, ceramics and mineral processing. The basic idea of drying is to 

remove liquid by evaporation from the material that has either dissolved or suspended 

solids to a dried powder. The objectives can be, to reduce the transportation costs, 

to increase the shelf life of a material, which is done for instance in the preservation 

of milk, tomato, etc. or because the material has better properties in dry form than 

when dissolved. 

Spray drying is one of the most widely used drying techniques and it is a process 

for converting a liquid feed into a powder by evaporating the solvent. The other drying 

techniques to produce powders mainly include freeze drying, supercritical drying and 

vacuum drying. Compared to other evaporation processes, spray drying has the great 

advantage that products can be dried without much loss of volatile or thermally unsteady 

or degradable compounds. These advantages are especially important in the production 

of pharmaceutical bulk materials such as polymers (e.g. polyvinylpyrrolidone), 

carbohydrates (e.g. mannitol) and food powders (e.g. milk and coffee powders) [1, 2]. 

Spray dryers are extensively used in industry, usually placed at the end-point of a 

process plant, which play an important role in the whole process, not because of their 

capital investment, size and operating costs but mainly because of their high energy 

efficiency and throughput production. 

Spray drying can be structured into several steps primarily consisting of the atomization 

of liquid feed into a population of poly-disperse droplets followed by the 

convection of droplets and gas, the evaporation of the liquid solvent from the droplets, 

droplet-droplet collisions, which might lead to coalescence, aggregation and eventually 

breakup. In some cases, chemical reactions are also involved, e.g. in the production 

of polymer via monomer polymerization using spray drying. These sub-processes are 

inter-dependent. The atomization leads to droplets with specific size distribution and 

kinetic energy, thus influencing the droplets convection and probability of collision. 

The drying gas in most of the cases is the ambient air heated to a desired temperature 

with controlled relative humidity and this hot air is either co-flow or counter-flow 

depending on the system requirement and the thermal sensibility of the material. A 

schematic diagram of the typical spray drying process in shown in Fig. 1.1 [3]. A review 

of available spray drying designs, process types and applications is given by Masters [1].

2 1. Introduction 

Fig. 1.1: A schematic diagram of the spray drying process [3]. 

Spray drying is a complex process and it is very difficult to predict the quality, 

properties and characteristics of the product such as particle size, density, porosity, 

flowability, shape, compressibility, etc. for the given drying conditions. The industrial 

practice to design is always based on the field experience and know-how followed by 

experimentation in pilot plant trials, which can be very expensive in case of rare materials 

[1, 4]. Problems associated with scale-up and hydrodynamics of the driers have 

resulted in limited success. The progress in computational techniques and computing 

prowess has given the advantage to develop a robust model of heat and mass transfer 

based on the equations of fluid flow within the spray chamber, which can predict the 

whole spray drying process i.e., from the liquid feed stock entering into the dryer to the 

end product, thereby resulting in cost effective and economic spray drier designs [1, 5]. 

The modeling of spray flows and the spray drying has been studied for many years 

now, which is done to predict the characteristics of the spray drying end products, 

e.g. particle size, shape, density or porosity. The previous studies mainly concern the 

systems of whole milk, salt solution, colloids or other materials for which the physical 

and thermal properties are well tabulated. The simulation of polymer or sugar solution 

spray drying is not studied so far because of unknown drying behavior of these materials 

and unavailability of properties. 

The present study deals with the modeling and simulation of polyvinylpyrrolidone 

(PVP)/water and mannitol/water spray flows, with an aim to predict the effect 

of drying conditions on the evolution of droplet properties. The synthesis, applications 

of PVP and mannitol, and the motivation to choose these systems are elaborated in 

the following paragraphs. 

PVP is a water-soluble polymer made from the monomer N-vinylpyrrolidone [6, 7].

3 

Fig. 1.2: Chemical structure of polyvinylpyrrolidone (C 6 H 9 NO) n [8]. 

It is a unique polymer providing a remarkable combination of properties that no other 

molecule is yet able to match. PVP offers a variety of properties, such as good initial 

tack, transparency, chemical and biological inertness, very low toxicity as well as high 

media compatibility and cross linkable flexibility. PVP was first synthesized by Prof. 

Walter Reppe and a patent was filed in 1939 for one of the most interesting derivatives 

of acetylene chemistry. Soluble PVP is obtained by free-radical polymerization 

of vinylpyrrolidone in water or 2-propanol, yielding the chain structure as shown in 

Fig. 1.2. There exists several grades of PVP, which are classified based on molecular 

weight, and spray drying technology is used in production of all types of PVPs [8]. 

PVP was initially used as a blood plasma substitute and later in a wide variety 

of applications in medicine, pharmacy, cosmetics and industrial production [9]. It is 

used as a binder in many pharmaceutical tablets; it simply passes through the body 

when taken orally. PVP binds to polar molecules exceptionally well, owing to its 

polarity. This has led to its application in coatings for photo-quality ink-jet papers 

and transparencies, as well as in inks for ink jet printers. PVP is also used in personal 

care products, such as shampoos and toothpastes, in paints, and adhesives that must 

be moistened, e.g. old-style postage stamps and envelopes [10]. It has also been used 

in contact lens solutions and in steel-quenching solutions. PVP is the basis of the 

early formulas for hair sprays and hair gels, and still continues to be a component of 

some [10]. As a food additive, PVP is a stabilizer and has E number E1201 [9]. In year 

2006, the total world wide production of PVP was 31,000 tonnes, out of which 47% was 

used in cosmetics and 27% in pharmaceuticals [9]. Figure 1.3 [11] shows the scanning 

electron microscope (SEM) images of PVP powder produced via spray drying. In the 

evaporation and drying of PVP dissolved in water droplet, the molecular entanglement 

prior to solid layer formation is observed, which is different from the colloids, silica 

or salt in water droplet where crust formation is found. It is observed that the spray 

drying yields hollow or solid particles with spherical or non-spherical shape but the 

initial droplet size, gas temperature and velocity and other drying conditions show 

enormous effect on the final powder characteristics such as flowability, particle density,


Fig. 1.3: SEM images of spray dried PVP [11]. 

porosity and shape [1]. To produce PVP powder with specific required properties 

is very important, for example, a uniform particle size distribution of PVP powder 

in pharmaceuticals not only helps in flowability of the powder but also improves the 

appealing of the final product. 

Similar to PVP, mannitol has several useful applications. Mannitol is a sugar 

alcohol and it is widely used as a carrier particle in tablets. Mannitol is commonly 

produced via the hydrogenation of fructose, which is formed from either starch or 

sucrose (common table sugar) [13]. Although starch is a cheaper source than sucrose, 

the transformation of starch is much more complicated. Hydrogenation of starch yields 

a syrup containing about 42% fructose, 52% dextrose, and 6% maltose [13]. Sucrose is 

simply hydrolyzed into an invert sugar syrup, which contains about 50% fructose. In 

both cases, the syrups are chromatographically purified to contain 90–95% fructose [13]. 

The fructose is then hydrogenated over a nickel catalyst into mixture of isomers sorbitol 

and mannitol with a typical yield of 50% sorbitol and 50% mannitol [13]. The chemical 

structure of mannitol is shown in Fig. 1.4 [12]. 

For many years active pharmaceutical ingredients (APIs) are delivered to the lung 

Fig. 1.4: Chemical structure of mannitol (C 6 H 8 (OH) 6 ) [12].

5 

via inhalation aerosols. Dry powder inhalers (DPI) are commonly used to achieve 

aerosols of a micronized solid API. To guarantee a reliable and constant dosing, the 

flow properties of the formulation are of high interest. Since the micronized API with 

a particle size of 1 µm - 5 µm [14] exhibits poor flowability, carriers consisting of larger 

particles are added to the formulation in order to carry the API particles on their 

surface. Due to the sufficiently large size of the carrier particles, the adhesive mixtures 

exhibit adequate flowability. In addition to the flowability the surface structure 

of the carrier is crucial to the formulation performance [15] and has to be controlled 

during development. In the last decade, mannitol was identified as a possible carrier 

for DPIs [16] and efforts were made to tailor the surface structure [17–19]. Spray 

dried mannitol particles in general have a spherical shape and can consist of two major 

polymorphs [17, 20]. Similar to PVP, recent studies [21, 22] of mannitol spray 

drying reveal that process parameters like droplet size, gas temperature and relative 

humidity exhibit strong correlation with the final powder characteristics. Compared to 

PVP/water, evaporation and drying of mannitol/water droplet leads to crust formation 

on the droplet surface prior to complete dried particle. 

The growing attention and wide applications of PVP and mannitol, as well as the 

effects of spray drying process on the final powder characteristics explains the particular 

interest towards these systems. The scarcity in thermal and physical properties 

of PVP and mannitol, and unknown behavior of evaporation and drying of 

PVP/water as well as mannitol/water droplets is a challenge for developing a mathematical 

model, numerical technique and validation. This project is part of the German 

Science Foundation (DFG) priority program ”SPP1423”, where spray flows and 

spray drying of PVP/water and mannitol/water are exclusively studied. This thesis 

deals with mathematical modeling and numerical simulation of both mono and bicomponent 

(PVP/water and mannitol/water) droplet evaporation and dispersion in 

sprays and spray drying with an objective to numerically investigate the effect of drying 

conditions such as gas temperature, gas velocity, relative humidity, initial droplet 

size and velocity distribution on the evolution of droplet properties, which will enable 

in better understanding the spray flows thereby helps in designing the spray dryer. 

The computational methods in the area of multiphase flow can primarily be categorized 

into two methods, (1) Lagrangian particle tracking method and (2) Euler – Euler 

or two continua/fluid methods. In both of these classical approaches, the continuous 

gas phase is modeled using the Navier – Stokes equations. Considering the presence 

of turbulence in the system, the Navier – Stokes equations can be solved on a fine 

computational mesh, which allows to capture all the macroscopic structures since all 

the considered length scales are considerably larger than the molecular length and time 

scales. Such a numerical resolution is defined as direct numerical simulation (DNS).


DNS solves all the characteristic scales of a turbulent flow and it requires no modeling 

of scales. DNS is implemented in a numerous flow problems for example particle dispersion 

[23, 24], turbulent reacting flows [25], spray flames [26], etc. A complete DNS 

of the spray drying is still not imaginary due to its high computational efforts [27, 28]. 

The alternative to DNS is the large eddy simulation (LES) [29–34] in which the 

large eddies are resolved and small eddies are modeled using a subgrid-scale model. 

This method requires spatial and temporal resolution of the scales in inertial subrange. 

The main disadvantage of LES method is that the accuracy of the flow field depends on 

the subgrid-model and filter size. Still, large computational time, and storage analysis 

of the huge data sets pose significant problems. 

The attractive approach to solve Navier – Stokes equations is the Reynolds-averaged 

Navier – Stokes (RANS) numerical simulation [35–37], where the instantaneous Navier – 

Stokes equations are averaged with respect to time whereby an instantaneous quantity 

is decomposed into its time-averaged and fluctuating quantities. RANS equations together 

with the turbulence closure model can be solved to resolve the turbulent flow and 

compute the mean flow field quantities [35–37]. There are several RANS turbulence 

closure models, which are extensively discussed by Pope [37], and the most notable 

turbulence models include k – ɛ and extended k – ɛ model [37]. 

The other methods like volume of fluid (VOF) [38] and lattice-Boltzmann (LB) [39] 

approaches also exist in the multi-phase flows to model the flow around the droplets 

or particles, and therefore the fluid flow can be fully resolved. These methods may 

be characterized under the DNS method for multi-phase flow problems to define the 

interfaces. 

In the Lagrangian particle tracking method, the droplets are injected into the gas 

and their trajectories are tracked by numerically evaluating the Lagrangian equations 

of motion. A typical spray consists of a large number of droplets and with limited 

computational resources, numerical parcels are implemented instead of droplets where 

each parcel contains of several number of droplets [35]. 

In Euler – Lagrangian approach, droplet–droplet interactions such as coalescence 

and breakup, which occur quite frequently in spray flows are difficult to account for due 

to computational complexity. The computational cost can be very expensive due to the 

large number of droplets needed to reach the statistical convergence, and computational 

cost is also dependent on mass loading of the dispersed phase. 

In the two-continua method, also known as Euler – Euler approach, a set of conservation 

equations is written for each phase, and the sets are coupled through their 

respective source terms. This was first proposed by Elghobashi and Abou-Arab [40] 

with the aim to establish a two-phase turbulence model. They derived a two-equation 

model based on the principle of k – ɛ model, and the Reynolds-averaged conservation

7 

equations are written in terms of volume fraction of each phase such that the sum of 

the volume fractions is unity. When the phases are equally distributed in the domain of 

interest with only moderate separation between the phases, the classical Euler – Euler 

approach is appropriate. 

Euler – Euler methods offer significant advantages over the Euler – Lagrange approach, 

e.g. the two-continua method is independent of disperse phase mass loading, 

and also the coupling between dispersed and carrier phase does not require averaging 

over the parcels unlike in Euler – Lagrangian. 

The merits of Euler – Euler methods play an important role when unsteady, turbulent 

gas-liquid flows with high dispersed phase mass loading are considered. Additionally, 

Euler – Euler methods can outperform the Euler – Lagrange in case of unsteady 

spray flows and the computational cost do not depend on the droplet mass loading. 

Most of the Euler – Euler methods in the field of spray flows are based on the description 

of dispersed phase as a number density function (NDF) and the evolution of 

this NDF due to physical processes of spray flows are described by the NDF transport 

equation, also known as population balance equation (PBE) [41]. This NDF transport 

equation is derived based on the kinetic equation [42] similar to the molecular kinetic 

theory, and it is known as general particle-dynamic equation in the field of aerosol 

science [43, 44]. 

There exists several Euler – Euler methods based on the kinetic equation such as 

Williams’ spray equation and are categorized mainly as multi-fluid methods [45–47] and 

moment based methods [48–54]. In the multi-fluid approach, the distribution function 

is discretized using a finite volume technique that yields conservation equations for mass 

and momentum of droplets in fixed size intervals called sections or fluids [46]. This 

approach has recently been extended to higher order of accuracy [55], but discretization 

of droplet size space is still a problem that needs to be addressed. On the contrary, 

moment based methods such as quadrature method of moments (QMOM) [51, 52, 56– 

58] or direct quadrature method of moments (DQMOM) [53, 54, 59] do not pose this 

problem and they are found to be efficient and robust in the poly-disperse multiphase 

flow problems. 

The scope of this work is modeling and simulation of mono and bi-component evaporating 

spray flows in an Euler – Euler framework. The focus is on the description of 

the characteristics of the spray flows and spray drying process, and the influence of the 

droplet size distribution on the droplet properties. In particular, spray inhomogeneity 

associated with the atomization process and its transport in the convective medium is 

not well understood. Subtle information is available about the particle formation and 

its influence on the properties of the resulting powder in spray drying. 

The present study aims to develop a comprehensive spray model, which can be


used to simulate the bi-component droplet evaporation and dispersion in spray flows 

and spray drying, and to predict the evolution of the droplet properties. In order 

to understand the behavior of droplet distribution under various drying conditions, 

the droplet size and velocity distribution of the spray is modeled using DQMOM. 

Transport equations of DQMOM should account for droplet evaporation, heating, drag 

and droplet–droplet interactions, which are calculated through appropriate sub-models. 

The systems of interest for the current study include water spray in nitrogen in one 

physical dimension, water and polyvinylpyrrolidone (PVP)/water spray in air in twodimensional, 

axisymmetric configuration. These systems were chosen for implementing 

and validating the numerical results. The experimental data for water spray in onedimensional 

configuration was provided by Dr. R. Wengeler, BASF, Ludwigshafen, 

whereas Prof. G. Brenn, TU Graz, Austria, provided the data for the two-dimensional 

water and PVP/water spray flows. The system of PVP or mannitol dissolved in water 

is also considered to verify numerical results of the single bi-component droplet 

evaporation and solid layer formation. 

Earlier studies based on DQMOM for spray flows either used simplified model for 

evaporation [60] or neglected the evaporation itself [61]. In the present study, the evaporation 

rate is computed using an advanced droplet evaporation model of Abramzon and 

Sirignano [62] for water spray, which accounts for variable liquid and film properties and 

includes the convective effects. For bi-component spray, the existing models to compute 

the evaporation rate neglect the non-ideality effect induced by non-evaporating 

component (e.g. PVP or mannitol) and ignore the solid layer resistance on the evaporation 

and droplet heating. In this study, for both PVP/water and mannitol/water 

droplet evaporation and solid layer formation, a mathematical model is formulated 

considering the non-ideality effect and solid layer resistance in the evaporation and 

heating. The evaporative flux, which is a point-wise quantity of the number density 

function of zero-droplet size, was either assumed to be zero [61, 63] or estimated with 

weight ratio constraints [60]. However, the later procedure was found to pose problems 

or to behave unphysical in multi-variate distributions [64, 65]. In the present study, 

evaporative flux is computed using the maximum entropy formulation [65–67]. 

The objectives of the current study are: modeling the evaporating water spray 

in nitrogen using DQMOM in one physical dimension and comparison of simulation 

results with QMOM and experiment, followed by extending the transport equations 

of DQMOM to two dimensions to simulate water spray in air, which is done for the 

first time, in axisymmetric configuration. The validation of DQMOM results with the 

discrete droplet (DDM) model [68], which is a well established Euler – Lagrangian 

technique, and with the new experimental data. In these configurations, the various 

physical processes due to gas–liquid and droplet–droplet interactions are accounted

9 

for through appropriate sub-models. In order to simulate PVP/water spray flows, 

existing evaporation model needs modifications to include the effects of non-ideality 

and solid layer formation on droplet heating and evaporation, so the present work aims 

to develop a mathematical model, which can predict the bi-component single droplet 

evaporation and solid layer formation prior to drying with prerequisites to account for 

non-ideality of liquid mixture and effect of solid layer resistance on droplet heating and 

evaporation. Final objective is to extend DQMOM to simulate PVP/water spray flows 

using the developed bi-component evaporation model and subsequent verification of 

the simulation results with that of the new experimental data. Complete spray drying 

is not yet simulated, which requires coupling of gas phase with the DQMOM and 

accounting for droplet temperature in transport equations of DQMOM. 

The dissertation is grouped into the following chapters. A review of the numerical 

simulation of sprays, governing equations of DDM and QMOM followed by the 

development of DQMOM and its transport equations are described in Chapter 2. The 

details of individual source terms of the spray flows such as bi-component droplet evaporation, 

droplet motion and droplet–droplet interactions and their equations are also 

elucidated in Chapter 2. In Chapter 3, numerical schemes to solve the single droplet 

evaporation and solid layer formation as well as DQMOM transport equations are explained 

along with the solution procedure of QMOM and Euler – Lagrangian approach 

DDM. A Wheeler algorithm to compute the initial data for DQMOM simulations and 

closure for QMOM unclosed moments is also given in Chapter 3. Chapter 4 presents 

the results and discussion starting from the water spray in nitrogen in one-dimension, 

followed by water spray in air in two-dimensional, axisymmetric configuration. The single 

droplet evaporation and solid layer formation model results are presented for both 

mannitol/water and PVP/water droplets. Finally, the results of PVP/water spray are 

presented. The conclusions and perspective future work are given in Chapter 5.

10 1. Introduction

2. Mathematical Modeling 

Spray constitutes of poly-disperse liquid droplets dispersed in gas medium. A typical 

sketch of a pressure-atomized spray breakup and its development is depicted in 

Fig. 2.1 [69, 70]. Here poly-disperse means that the properties of the disperse phase 

entities can be different for each entity. For example, evaporating sprays have a region 

near the nozzle where the liquid jet is not disperse, followed by a region after breakup 

of the primary jet that is composed of individual droplets having different properties 

such as size, velocity, temperature etc., which are defined as the poly-disperse 

droplets [41, 71]. To describe the poly-disperse characteristics of the spray flows, the 

mathematical modeling approach of Euler – Lagrangian and Euler – Euler framework 

is discussed, and the sub-models for the physical processes of sprays such as evaporation, 

forces, coalescence and breakup are elucidated in this chapter. Though the focus 

of this work is to model the spray flows using the direct quadrature method of moments 

(DQMOM), but the models like quadrature method of moments (QMOM) and 

discrete droplet model (DDM), which were used to compare and validate the DQMOM 

results, are also presented in this chapter. 

2.1 State of the Art 

The existing modeling approaches in the area of multiphase flows mainly include, (1) 

Lagrangian particle tracking method and (2) Euler – Euler or two continua/fluid meth- 

Fig. 2.1: Sketch of a pressure-atomized spray formation [69, 70].

12 2. Mathematical Modeling 

ods. In the Lagrangian particle tracking method, the gas phase behavior is typically 

predicted by solving the unsteady Reynolds-averaged Navier – Stokes equations with 

an appropriate turbulence model and sub-models for various source terms [35, 37, 72]. 

In this method, droplets are injected into the gas and their trajectories are tracked by 

numerically evaluating the Lagrangian equations of motion. 

A typical spray consists of a large number of droplets and with limited computational 

resources, numerical parcels are implemented instead of droplets where each 

parcel contains several number of droplets. The Euler–Lagrangian models are classified 

as locally homogeneous flow (LHF) method [73–75] and separated flow (SF) 

method [76–78]. 

The locally homogenous flow approximation of the LHF model for two-phase flow 

problems implies that the interphase transport rates are infinitely fast, so that both 

phases have same velocity and are in thermal equilibrium at each point of the flow [75]. 

This model neglects the slip effect between the liquid phase and gas phase. LHF 

approximation is the limiting case, which accurately represents spray with very small 

droplets [72]. 

Compared with the LHF model, the SF model has been used more widely in multiphase 

flow problems, because it provides the finite rate exchange of mass, momentum 

and energy between the phases [72]. The SF model assumes that each phase 

displays different properties and flows with different velocities, but the conservation 

equations are written only for the combined flow. In addition, the pressure across 

any given cross-section of a channel carrying a multiphase flow is assumed to be the 

same for both phases [72]. The SF models are further subdivided into discrete droplet 

model (DDM) [76–78], continuous droplet model (CDM) and continuous formulation 

model (CFM). The differences in these methods are explained by Faeth [72]. 

In DDM, the spray is divided into representative samples of discrete droplets whose 

motion and transport are tracked through the flow field, using a Lagrangian formulation. 

This procedure computes the liquid properties based on finite number of particles, 

called as parcels which are used to represent the entire spray [72, 76]. The gas phase is 

solved using Eulerian formulation, similar to the LHF method. The effect of droplets 

on the gas phase is considered by introducing appropriate source terms in the gas 

phase equations of motion. This type of formulation is is found to be convenient for 

considering a relatively complete representation of droplet transport processes [72]. 

The CDM was first introduced by Williams [79]. In this method, droplet properties 

are represented by a statistical distribution function defined in terms of droplet 

diameter, position, time, velocity, temperature, etc. [80]. Conservation principles yield 

a transport equation for the distribution function, which is solved along with the gas 

phase equations to deduce the properties of the spray [72, 79, 80]. Similar to DDM,

2.1. State of the Art 13 

the governing equations for the gas phase include appropriate source terms to compute 

the effects of droplets. 

The other important SF method for modeling sprays is the CFM, which employs 

a continuum formulation of the conservation equations for both phases [81, 82]. The 

motion of both droplets and gas are treated as interpenetrating continua. The work of 

Faeth [72] gives an extensive review of all the Euler – Lagrangian models. 

The Euler – Lagrangian approach is so far considered to be effective in many applications, 

which gives detailed information at the micro-level, however it has significant 

drawbacks as listed by Archambault [67]. For instance, inclusion of droplet–droplet 

interactions such as coalescence and breakup, which occur quite frequently in spray 

flows, increases the computational complexity. The computational cost could be very 

expensive due to the large number of droplets needed to reach the statistical convergence, 

and it may pose difficulties and numerical instabilities in coupling of Lagrangian 

description of dispersed phase with the Eulerian equations of the gas phase. The computational 

cost is also dependent on mass loading of the dispersed phase. According 

to Archambault [67], the vertices of the droplet trajectory and numerical grid of the 

gas phase never coincide, hence a sub-grid model is required in order to compute the 

exchange rate between the phases [83]. Grid independent solutions are quite difficult to 

obtain [84], which could be because of an insufficient number of droplets in a grid cell 

leading to a significant error as can be observed in the regions of high droplet number 

density. 

The study of Garcia et al. [85] and Riber et al. [86] describe and analyze the 

comparison of computational time between Euler – Euler and Euler – Lagrangian in 

homogeneous and non-homogeneous flows. 

There is a tremendous amount of literature available on the Eulerian – Lagrangian 

approaches in spray flows and spray drying [76, 87–93], and references therein. As the 

focus of the current work is about Euler – Euler approach to spray flows, this section 

presents the review of available literature in this area. 

A numerous Eulerian models have been recently developed where the disperse phase 

described based on a kinetic equation and continuum phase is resolved using Navier– 

Stokes equations. The basic idea in kinetic equation based Eulerian methods is that 

instead of solving the usual Euler equations for the dispersed phase, the evolution of the 

moment transform of the kinetic equation is solved, which resembles Navier – Stokes - 

like equation, and this equation is coupled to the continuum phase with the appropriate 

source terms. Such a kinetic equation is first derived by Williams [42], known as 

Williams’ spray equation which is analogous to Boltzmann’s equation of molecules [94, 

95]. The derivation of Williams’ spray equation is given by Archambault [67] and 

Ramakrishna [41]. This equation describes the temporal evolution of the probable


number of droplets within a range of droplet characteristics such as size, velocity, 

temperature and solute mole fraction within the droplet, which constitutes the phasespace 

at a spatial location. The solution of this equation coupled with the gas phase 

flow field equations provides the average properties of the spray, for example mean 

droplet diameter, Sauter mean droplet diameter, mean droplet velocity and many other 

statistical properties. 

Among the existing Eulerian models, the multi-fluid method allows the detailed 

description of poly-disperse droplet size and velocity through correlations. Such an 

approach has been shown to be derived from the Williams’ spray equation, Eq. (2.23), 

by Laurent and Massot [45] under the mono-kinetic spray assumption, which states 

that the velocity dispersion of the spray distribution function at a given time, spatial 

location and droplet size has to be zero. This assumption is important since it defines 

the validity limit of the multi-fluid model and also results in the ”pressure-less gas 

dynamics” structure of the transport equations for conservation of mass and momentum 

of droplets [96]. These conservation equations are derived by discretizing the droplet 

distribution using a finite volume technique in fixed size intervals called sections or 

fluids. 

This approach has been extended to higher order of accuracy [46], but discretization 

of droplet size phase-space is still a problem that needs to be addressed. The issues 

related to the mono-kinetic assumption have to be relaxed if the coalescence of droplets 

are to be considered, which is addressed by introducing a semi-kinetic equation and 

the results are presented for the evaporation and coalescence in spray flows [47, 97]. 

However, the validation in multi-dimensional configurations and the evaluation of the 

level of accuracy of such model versus the reference Lagrangian simulations as well 

as the related issue of a detailed study of the effective computational cost of the two 

approaches is not yet understood [98]. 

Next most notable method is the method of classes (CM) or discretized population 

balances (DPB) which is based on the discretization of droplet internal co-ordinates 

of the population balance equation [99–102] into a finite series of bins. The CM’s 

compute the mean properties of the population such as droplets or particles within 

these bins by solving the discretized population balance equation. CM’s are divided 

into two categories namely, (1) zero-order methods, and (2) higher order methods. In 

zero-order methods, the droplet size distribution (DSD) is considered to be constant 

in each class, and they are ”extremely stable”. Recently, Vanni [103] reviewed and 

compared the wide variety of zero-order CM’s. In higher order methods, the DSD 

is defined in a specific functional form for every section of discretization, and these 

methods are usually more accurate but less robust [101–103]. The CM’s present the 

main disadvantage of requiring a large number of classes to work with good accuracy,

2.1. State of the Art 15 

and if the final application of the solution is implementation in a computational fluid 

dynamics (CFD) code, then the solution has to be done in every cell of computational 

domain, resulting in a very high computational time and memory problems. 

On grounds of CM approach to solve the Williams’ spray equation, Tambour [104] 

discretized the size axis of the droplet size distribution into sections which are also 

known as bins to derive sectional equations. The droplet evaporation, collision and 

other physical processes are handled using the source terms among these bins. However, 

this method is found to be first order accurate with respect to the droplet size [105], 

thus resulting in a strong numerical diffusion when few sections are used. The problem 

of finding an appropriate way to improve the accuracy of this method and also 

minimizing the number of sections becomes critical, especially when considering industrial 

codes, which are intended to perform complex three-dimensional simulations. 

Later this approach is extended to higher oder by Dufour and Villedieu [55], but still 

requirement of high number of sections needs to be addressed. 

The other option is QMOM [51], which is based on the solution of moment transport 

equations of kinetic equation and the evaluation of unknown moments with the 

quadrature approximation. In this method, the distribution function is approximated 

with n-point Gaussian quadrature [51]. The moments are transported at every time 

and space step and the quadrature weights (number density), droplet radius, velocity 

and other phase-space variables, termed as abscissas, are computed using the 

product-difference (PD) algorithm of Gordon [106]. This method was first introduced 

by McGraw [51] in describing the aerosol dynamics to improve the method 

of moments (MOM) [107–113], and it is found to be a reliable method than MOM as 

the closure problem was observed with MOM [107, 114]. This method is proven to 

be promising in the problems of coagulation, aggregation and breakage [48–50], gasparticle 

flows [115]. One of the main limitations of QMOM is that since the dispersed 

phase is represented through the moments of the size distribution, the phase-average 

velocity of different phases must be used to solve the transport equations for the moments. 

Thus, in order to use this method in the context of sprays for which the inertia 

determines the dynamic behavior of the droplets, it is necessary to extend QMOM 

to handle cases where each droplet size is convected by its own velocity [53]. The 

efficiency and applicability of such methods [60, 116, 117] for moment inversion in 

multi-variate poly-disperse systems have remained a question of interest [115], which 

are characteristic in many technical applications. In the work of Marchisio and Fox [71], 

a comprehensive review of the existing moment methods is given. 

Recently CQMOM was introduced by Yuan and Fox [118], to address the issues 

related to moment inversion. CQMOM is a novel moment-inversion algorithm, which 

works even for multi-variate moments. One apparent disadvantage of CQMOM (as


well as other multi-variate moment-inversion algorithms [115, 119]), according to Yuan 

and Fox [118], is the existence of multiple permutations (i.e., the order of conditioning). 

In the context of multi-dimensional quadrature, there are a number of interesting open 

questions concerning CQMOM. For example, in order to have a realizable quadrature, 

the abscissas found from the conditional moments must lie in support of the distribution 

function. For one-dimensional distribution functions with compact supports, it is 

shown that the abscissas will always lie realizable [118, 120]. However, it still needs 

to be found under which conditions boundedness will hold for the abscissas found 

from CQMOM for a two-dimensional (or higher) distribution function with compact 

support [118]. For certain applications (e.g. turbulent reacting flows), guaranteed 

boundedness is critical because the source terms are only defined on the support of the 

distribution function. As per revelations made by Yuan and Fox [118], the other open 

question is whether further improvements in the moment-inversion algorithm are possible 

to increase the number of optimal moments controlled, perhaps up to the maximum 

number of degrees of freedom determined by the number of quadrature nodes [118]. 

In order to overcome these problems, DQMOM has turned out to be an attractive 

alternative to QMOM, which was introduced by Fan et al. [121], later extended and 

validated by Marchisio and Fox [53]. This approach is found to be a powerful method 

in the multiphase flow problems to include all the physical processes of interest. The 

principal physical processes that the droplets encounter during the spray flows are (1) 

transport in real space or convection, (2) droplet evaporation and drying, (3) acceleration 

or deceleration of droplets due to forces induced by the surrounding gas, and 

(4) coalescence and collision of droplets leading to poly-dispersity. In DQMOM [53], 

the transport equations of weights and abscissas are solved directly rather than the 

moment equations, which is done in QMOM, thus avoiding the ”moments” to ”weights 

and abscissas” conversion through moment-inversion algorithm, so the word ”direct” 

implies. DQMOM also allows each droplet to convect with its own velocity. 

The DQMOM is proven to be a robust method in the field of multiphase flows and 

it is applied to various research problems other than spray flows after its development 

by Marchisio and Fox [53]. DQMOM is adapted and validated for the coagulation and 

sintering of particles by extending to bi-variate population balance equations [54]. Recently, 

DQMOM has been applied in studying exhaust particle formation and evolution 

in the wake of ground vehicle [122], and DQMOM is compared with classes method for 

simulations of bubbly flows [123]. In latest studies, DQMOM is employed in modeling 

poly-disperse fluidized powders [124], and modeling of turbulent combustion [125]. It 

is also used in combination with micro-mixing model and compared with stochastic 

field method for treating turbulent reactions [59]. 

In the field of spray flows, DQMOM in combination with multi-fluid method is

2.2. Euler – Lagrangian Approach 17 

applied to study multi-component (fuel mixture) droplet evaporation [126]. Gumprich 

et al. [127, 128] analyzed the dense turbulent sprays using DQMOM, and DQMOM 

coupled with Eulerian multi-size moment model [129]. Madsen [61] extended DQMOM 

to include droplet coalescence in spray flows by neglecting the effects of evaporation, 

whereas Fox et al. [60] further improved DQMOM to model evaporating and coalescing 

spray flows but his study assumed simplified models for evaporation and coalescence. 

So far DQMOM has not been considered to treat the process of spray drying. 

In the present study, DQMOM is used to describe the disperse phase consisting of 

poly-disperse liquid droplets, whereas the gas phase is not yet resolved but its inlet 

flow properties are taken for computing the droplet motion and evaporation. In this 

work, the DQMOM is implemented in two dimensions, which is done for the first time, 

and applied to study bi-component evaporating spray flows. 

2.2 Euler – Lagrangian Approach 

In the Euler – Lagrangian approach, the mean field equations are used only for the 

continuous gas phase. The droplet properties are defined along the path lines followed 

by the droplet. The trajectories of droplets are tracked for each droplet group by using 

a set of equations that describe their physical transport in flow field. In the current 

study, the discrete droplet model is used to define the droplet phase whereas the gas 

phase is modeled using the Navier – Stokes equations. 

2.2.1 Gas Flow 

The Euler – Lagrangian model DDM includes Euler equations for the gas phase with 

source terms for the dilute spray, which is described in Lagrangian coordinates. The 

instantaneous Navier – Stokes equations in an axisymmetric, two-dimensional configuration 

with no swirl for a dilute spray yield [68, 130] 

∂(ρu i ) 

∂t 

+ ∂(ρu iu j ) 

∂x j 

∂ρ 

∂t + ∂(ρu j) 

= S l,1 , (2.1) 

∂x j 

= − ∂p 

∂x i 

+ ∂τ ij 

∂x j 

+ ρg i + S l,ui , (2.2) 

where ρ, u i and p are the density, velocity component and pressure of the gas flow, 

respectively. g i is the acceleration due to gravity and the quantities S l,1 and S l,ui are 

the source terms due to spray evaporation [130, 131]. τ ij is the viscous stress tensor 

given by 

( ∂ui 

τ ij = µ + ∂u j 

− 2 ∂x j ∂x i 3 

) 

∂u k 

δ ij , (2.3) 

∂x k


where δ is the tensorial Kronecker delta given by 

{ 

1 : i = j 

δ ij = 

0 : i ≠ j. 

(2.4) 

Neglecting the processes of radiation, friction heating, Dufour effect, and the viscous 

heating, the conservation equation of total stagnant enthalpy can be written as 

∂(ρh) 

∂t 

+ ∂(ρu jh) 

∂x j 

= ∂p 

∂t − ∂J d q,j 

∂x j 

− ∂J c q,j 

∂x j 

+ S l,h , (2.5) 

where h is the enthalpy of the gas flow and the terms on the right hand side (R.H.S) 

are the change rate of the pressure, the heat diffusion term, the heat conduction term 

and the source term due to spray evaporation, S l,h , respectively. The heat conduction 

term is expressed by the Fourier’s Law 

( 

) 

Jq,j c = −λ ∂T = λ¯Cp ∂h ∑N s 

∂Y α 

− h α , (2.6) 

∂x j ∂x j ∂x j 

where λ, T , ¯Cp are thermal conductivity, gas temperature and specific heat capacity, 

respectively. 

N s refers to the number of chemical species while h α and Y α are the 

enthalpy and mass fraction of species α. The heat diffusion term J d q,j is written as 

α=1 

∑N s 

Jq,j d = h α Jα 

m 

α=1 

∑N s 

Y α 

= − ρh s,α D α,M , (2.7) 

∂x j 

α=1 

where h s,α and D α,M are the specific sensible enthalpy of species α and diffusion coefficient 

of species α, respectively. Assuming a unity Lewis number, which is defined as 

the ratio of thermal diffusion to mass diffusion, (Le = k/(ρC p D), and equal diffusibility 

of all species, the total heat flux is 

J q = J c q,j + J d q,j = − λ¯Cp ( 

∂h 

∂x j 

− 

∂(ρh) 

∂t 

+ ∂(ρu jh) 

∂x j 

∑N s 

α=1 

) 

∂Y α 

h α − 

∂x j 

∑N s 

α=1 

The conservation equation of species mass can be written as 

∂(ρY α ) 

∂t 

ρh s,α D α,M 

Y α 

∂x j 

. (2.8) 

= ∂p 

∂t + ∂ ( ) 

∂h 

Γ h + S l,h . (2.9) 

∂x j ∂x j 

+ ∂(ρu jY α ) 

− ∂ ( ) 

∂Y α 

ρD α = S α + δ L,α S l,Yα , (2.10) 

∂x j ∂x j ∂x j 

where D α is the diffusion coefficient of species α while S α and S l,α are the source terms 

due to chemical reactions and spray evaporation, respectively. The mass fraction may 

be used to formulate mixture fraction. 

The advantage of an appropriately defined

2.2. Euler – Lagrangian Approach 19 

mixture fraction is that the source term S α will be zero. In the present work, for the 

water and polyvinylpyrrolidone (PVP)/water spray in air, the only possibility is to 

define the mixture fraction with reference to hydrogen as oxygen appears in both gas 

and liquid. A detailed study of different reference elements are given by Gutheil and 

Williams [132]. Thus the mass fraction Z A of element A, where A is either N or H or 

O, is defined as 

Z A = 

n∑ 

i=1 

a IA M A 

M I 

Y I , (2.11) 

where a IA is the mass of element A in molecule I and M A and M I are the molecular 

weights of element A and element I, respectively. Using this definition, mixture fraction 

can be defined as 

ξ = 

Z A − Z A,min 

Z A,max − Z A,min 

. (2.12) 

Multiplying Eq. (2.10) by a IAM A 

a lA M I 

and summing over total number of species under the 

assumption of equal diffusivity, the following conservation equation for mixture fraction 

is obtained 

∂(ρξ) 

∂t 

+ ∂(ρu iξ) 

∂x i 

= ∂ 

∂x i 

( 

Γ M 

∂ξ 

∂x i 

) 

+ S l,ξ , (2.13) 

where Γ M = ρD M is the mass diffusion coefficient of the mixture. 

Equations (2.1), (2.2), (2.9) and (2.13) are the instantaneous conservation equations 

of mass, momentum, energy and mixture fraction. These equations need to be 

averaged for application to turbulent flows, and the general averaging types include 

time-averaging, ensemble-averaging and Favre- or density-weighted averaging [37]. For 

turbulent compressible flows, a density-weighted averaging i.e., Favre-averaging for 

Navier – Stokes equations is useful, and for more details about this approach, see [35, 

72, 76, 77]. 

2.2.2 Discrete Droplet Model (DDM) 

The discrete droplet model (DDM) is a well established Euler – Lagrange approach 

for dilute sprays [72, 133, 134]. The droplet positions and velocities are captured 

using Lagrangian particle tracking method, thereby the source terms for the Eulerian 

equations of the gas phase are computed. The model captures the trajectories and 

dynamics of individual droplets, which are assumed to be parcels [35, 72, 76, 77]. A 

parcel refers to a collection of droplets, which are described by a set of properties, i.e., 

(x p,k , r p,k , v p,k , m p,k , T p,k , ∆V ij ), where x p,k is the position, r p,k is the radius, v p,k is the 

velocity, m p,k is the liquid mass and T p,k is the temperature of k th parcel in control 

volume ∆V ij . By tracking the trajectories of a system of parcels, the model captures


the flow properties, i.e., droplet dynamics, evaporation and heating, and these are 

described through sub-models. These sub-models for the different physical processes 

of interest are given in Subsection 2.4. 

The Lagrangian droplet equations are coupled to the gas phase through Eqs. (2.1) – 

(2.13), and the spray source terms are formulated using the particle-source-in-cell (PSIC) 

model [76, 130]. The system of gas and liquid phase equations is solved through a hybrid 

finite volume technique [130] with appropriate initial and boundary conditions. The 

DDM simulations were performed by Humza [68], so the details of initial and boundary 

conditions and numerical solution procedure of DDM are given in his work [68]. 

2.3 Euler – Euler Approach 

In the Euler – Euler approach, the disperse phase is generally described by a number 

density function (NDF). In order to understand the proposed modeling approach of 

DQMOM one needs to be familiar with the definition of the NDF, the NDF transport 

equation, and its underlying terms, which are explained in the next subsections. 

2.3.1 Treatment of the Spray 

The disperse phase constitutes of discrete droplets and each of these droplets can be 

identified by a number of properties known as coordinates. In general, the coordinates 

are categorized as internal and external. The external coordinates are spatial coordinates. 

Internal coordinates refer to the properties of the droplets such as droplet 

velocity, mass, volume (or surface area, size), and enthalpy (or temperature). The 

NDF contains the information about the population of droplets inside a control volume 

[28, 41]. 

Let us consider a population of droplets in a spray dispersed in a control volume 

located at the physical point x = (x 1 , x 2 , x 3 ) where the size of the control volume is 

dx = dx 1 dx 2 dx 3 . (2.14) 

Let ξ = (ξ 1 , ξ 2 , ..., ξ N ) be the internal-coordinate vector. The NDF n ξ (ξ; x, t) is defined 

as the probable number of droplets in the physical volume dx and in phase-space 

volume dξ, given by n ξ dxdξ. 

The NDF is an average quantity of the dispersed phase, and it has mathematical 

characteristics of an averaged function, i.e., it is smooth and differentiable with respect 

to time t, physical space dx and phase-space dξ. The number density of droplets 

contained in the phase-space volume dξ per unit volume of physical space is n ξ dξ. 

By integrating the NDF over all the possible internal-coordinates, different average

2.3. Euler – Euler Approach 21 

quantities of interest can be obtained. For instance, the total number density per unit 

physical volume N(x, t), which is defined as the total probable number of droplets in 

a unit physical volume, and it is also known as zeroth moment, M ξ (0), obtained as 

∫ 

N(x, t) = M ξ (0) = n ξ (ξ; x, t)dξ. (2.15) 

Similarly, the first moment defines the mean of the distribution, which is generally 

denoted by µ. The other moments which are common in probability and statistics are 

central moments [135], which are defined based on the distribution of droplets about 

the mean of the NDF. Thus the n th moment is given as 

∫ 

M(n) = (ξ − ¯ξ) n n ξ (ξ; x, t)dξ, (2.16) 

where the second moment (n = 2) is known as variance, generally denoted by σ 2 , and 

the third and fourth moments define the skewness and kurtosis of the distribution [135], 

respectively. Typical log-normal distribution with different values of mean, µ, and 

variance, σ 2 , are shown in Fig. 2.2. 

Likewise, when the NDF is defined with respect to more than one internal coordinate, 

the moments of NDF are known as multi-variate moments, which can be 

computed as 

∫ 

k 

M ξ (k) = ξ 1 k 

1 ξ 2 k 

2 ....ξ N N n ξ (ξ)dξ, (2.17) 

where k = (k 1 , k 2 , ...k N ) is a vector containing the order of the moments with respect to 

each component of ξ. In the description of spray flows, the NDF is in general described 

based on the droplet radius and velocities as the internal coordinates, ξ. 

1 

0.8 

µ = 0, σ = 0.5 

µ = 0, σ = 1.0 

µ = 1, σ = 0.5 

0.6 

n(ξ) 

0.4 

0.2 

0 

0 1 2 3 4 5 6 

ξ 

Fig. 2.2: A typical log-normal distribution with different values of µ and σ.


For the purpose of understanding the average properties of the spray dynamics 

from the simulations, the calculation procedure of mean droplet diameter and Sauter 

mean diameter is defined below. The considered NDF is defined based on the droplet 

diameter d. The mean droplet diameter, denoted as d 1,0 or d 10 , can be computed from 

the droplet diameter based NDF n d (d)as 

d 1,0 = 1 N 

∫ ∞ 

0 

dn d (d)dd, (2.18) 

where N is the total number density given by 

N = 

∫ ∞ 

0 

n d (d)dd. (2.19) 

Similarly, the Sauter mean diameter, d 3,2 , or simply d 32 , which is a very frequently 

used characteristic parameter especially in spray flows and spray drying, is given by 

∫ ∞ 

d 3 n 

0 d (d)dd 

d 3,2 = ∫ ∞ 

d 

0 2 n d (d)dd . (2.20) 

Any other average droplet diameter can be extracted by simply dividing the k + 1 th 

moment with k th moment, i.e., 

d k+1,k = 

∫ ∞ 

d k+1 n 

0 d (d)dd 

∫ ∞ 

d 

0 k n d (d)dd = M k+1 

. (2.21) 

M k 

Here, M k+1 and M k are the moments of droplet diameter based NDF. The equation of 

averaged droplet diameter changes with the definition of NDF, i.e., averaged droplet 

diameter is different in diameter based NDF from that of volume based NDF, and the 

relation between volume based NDF and diameter based NDF is given by 

n V (V ; x, t) = k V d 3 n d (d; x, t), (2.22) 

where k V 

is the volume shape factor. 

2.3.2 NDF Transport Equation 

The evolution of the NDF due to physical processes is, in general, written in terms of 

a transport equation known as population balance equation. This transport equation 

is a simple continuity equation written in terms of the NDF, and it can be derived 

based on the balance for droplets/dispersed entities in a fixed subregion of internal 

coordinates and physical space [41]. This type of equation is known by different names 

in different fields. In aerosol dynamics, it is known as particle-dynamics equation, 

and in evaporating spray flows it is known as Williams-Boltzmann equation or simply,


Williams’ spray equation, which was first introduced by Williams [42]. The derivation 

of such an equation is given by Archambault [67] and Ramakrishna [41]. 

In modeling the spray flows, the principal physical processes that must be accounted 

for are transport or convection, droplet evaporation, forces experienced by the droplets, 

and droplet–droplet interactions leading to poly-dispersity. Williams’ spray equation 

either accounts for these physical processes or it can be easily extended to include, 

and it has proven to be a useful starting point for testing novel methods for describing 

poly-disperse dense liquid sprays. The Williams’ spray equation [42], is given by 

∂f 

∂t + ∂(vf) 

∂x 

(Rf) 

= −∂ − ∂(Ff) 

∂r ∂v + Q f + Γ f . (2.23) 

Equation (2.23) describes the transport of the number density function f(r, v; x, t) 

in terms of time, t, and Euclidean space, x. In Eq. (2.23), first term at the left hand 

side (L.H.S) accounts for changes in the NDF with time and the second term includes 

the convective changes, v and F denote droplet velocity and total forces acting on the 

droplet per unit mass, respectively. The first term in the R.H.S includes the effect of 

evaporation, where R is the change in the droplet radius with time, i.e., R = dr/dt, and 

r is the droplet radius. The last two terms in Eq. (2.23) refer to the droplet–droplet 

interactions; Q f represents the increase in f with time due to droplet formation or 

destruction by processes such as nucleation or breakup, and Γ f denotes the rate of 

change in f due to droplet collisions. 

The next subsections present the governing equations of the liquid phase defined 

through QMOM and DQMOM, which are used to solve Williams’ spray equation. 

2.3.3 Quadrature Method of Moments (QMOM) 

Moment methods are an important class of approximate methods derived to solve 

kinetic equations, but require closure to truncate the moment set. In QMOM, closure 

is achieved by inverting a finite set of moments to reconstruct a point distribution 

from which all unclosed moments (e.g. 

spatial fluxes) can be related to the finite 

moment set [28, 51]. Figure 2.3 shows the typical quadrature approximation of the 

NDF in QMOM. The derivation of QMOM starts with the moment transformation 

of Williams’ spray equation, which is done in the current study with multi-variate 

moments of droplet radius and velocity denoted by M(k 1 , k 2 ; x, t) where k 1 and k 2 are 

the moment orders with respect to droplet radius and droplet velocity, respectively . 

The moment transformed Williams spray equation is given as [28], 

∂M(k 1 , k 2 ) 

∂t 

+ ∂M(k 1, k 2 + 1) 

∂x 

= 

∫ ∞ ∫ ∞ 

−∞ 

0 

r k 1 

v k 2 

[ 

− ∂ (Rf) − ∂(Ff) ] 

∂r ∂v + Q f + Γ f drdv. 

(2.24)


0.4 

Number density [(µm) -3 ] 

0.2 

-5 0 5 

Droplet diameter [µm] 

Fig. 2.3: Approximation of NDF in QMOM. 

In the evolution of every moment M(k 1 , k 2 ) one higher order moment, i.e., M(k 1 , k 2 +1) 

appears, see Eq. (2.24), which can be closed by using the product-difference algorithm 

(PD) [106]. The PD algorithm is quite efficient in a number of practical cases; 

however, it generally becomes less stable as the number of nodes, N, increases. It is 

difficult to predict a priori when this will occur, since it depends on the absolute values 

of the moments, but typically problems can be expected when N > 10 [71]. Another 

issue with PD algorithm is if the distributions with zero mean present, which can occur 

when the internal coordinate droplet velocity is ranging between positive and negative 

values, in this case the algorithm blows up due to the division by zero in the calculation 

of the coefficient matrix of the PD algorithm [71]. The alternative approach to the PD 

algorithm, which can handle cases with zero mean and remains stable even in the cases 

of N > 10, is the Wheeler algorithm proposed by Sack and Donovan [136]. The step 

by step procedure of implementing PD and Wheeler algorithms with corrections for 

moment realizability are given by Marchisio and Fox [71] with example calculations. 

The substitution of different sub-models for the droplet evaporation, total force 

acting on droplets, droplet breakup and collision in the R.H.S of the Eq. (2.24) and with 

the help of PD or Wheeler algorithm for the unknown moment terms, yields a closed 

transported moment equation, which can then be solved to find the change in moments 

with time and spatial location. As discussed in the literature review, see Section 2.1, 

this method lacks the ability to handle multi-variate moments as the moment-inversion 

algorithm gets cumbersome. The different sub-models to describe the various physical 

processes of interest are described in Subsection 2.4. The numerical solution for the 

QMOM moment transport equations with initial and boundary conditions, and closure


of the unknown moments applying the Wheeler algorithm are explained in Chapter 3. 

2.3.4 Direct Quadrature Method of Moments (DQMOM) 

In DQMOM, the NDF is approximated as sum of the Dirac-delta functions. Substitution 

of this assumed NDF in Williams’ spray equation yields transport equations in 

terms of the phase-space [60]. For the present study, a joint droplet radius-velocity 

number density function is considered, which is approximated in DQMOM as a sum 

of the product of weighted Dirac-delta functions [53] of radii and velocities [60], 

f(r, v) = 

N∑ 

w n δ(r − r n )δ(v − v n ), (2.25) 

n=1 

where w n and r n are chosen as N representative quantities of weights and radii, and v n 

are the corresponding velocities. Such an approximation with a three-node (N = 3) closure 

can be depicted as shown in Fig. 2.4. Application of DQMOM to Williams’ spray 

equation results in closed transport equations in terms of droplet weights or number 

densities, radii and velocities, which are written as 

∂w n 

∂t 

∂(w n ρ l r n ) 

∂t 

+ ∂(w nv n ) 

∂x 

+ ∂(w nρ l r n v n ) 

∂x 

= a n , (2.26) 

= ρ l b n , (2.27) 

and 

∂(w n ρ l r n v n ) 

+ ∂(w nρ l r n v n v n ) 

= ρ l c n , (2.28) 

∂t 

∂x 

where a n , b n and c n are the source terms that account for droplet evaporation, forces 

on droplet (drag, buoyancy, lift, basset and virtual mass effect and gravity etc.), coalescence 

and breakup. These Eqs. (2.26) – (2.28) form a set of coupled hyperbolic 

partial differential equations, which can be solved simultaneously by using appropriate 

initial and boundary conditions to find w n (x, t), r n (x, t) and v n (x, t), and thereby the 

evolution of droplet distribution function f can be computed. 

Fig. 2.4: NDF approximation in DQMOM.


The Eqs. (2.26) – (2.28) are closed by modeling the source terms, i.e., a n , b n and c n , 

using the physical models to account for effects of droplet evaporation, forces on droplet, 

coalescence and breakup. 

These source terms are calculated through the moment 

transformation of phase-space terms, which yields the following linear system 

∫ [ 

P k,l = r k v l − ∂(Rf) − ∂(Ff) ] 

∂r ∂v + Γ f + Q f drdv. (2.29) 

The exact form of the DQMOM linear system relies on the choice of moments, and it 

can be generated from 

∫ 

[ ∂f 

r k v l ∂t + ∂(vf) 

∂x 

] 

drdv = 

+ 

+ 

N∑ 

n=1 

N∑ 

n=1 

N∑ 

n=1 

(1 − k)r k nv l 1 

1,n v l 2 

2,n v l 3 

3,n a n 

(k − l 1 − l 2 − l 3 )r k−1 

n v l 1 

1,n v l 2 

2,n v l 3 

3,n b n 

r k nv l 1 

1,n v l 2 

2,n v l 3 

3,n (l 1 v −1 

1,nc 1,n + l 2 v −1 

2,nc 2,n + l 3 v −1 

3,nc 3,n ) 

+ δ k0 u l 1 

1 u l 3 

2 u l 3 

3 ψ, (2.30) 

where ψ is the evaporative flux, and u 1 , u 2 , and u 3 are three components of the gas 

velocity. The complete linear system is formed by combining Eqs. (2.29) and (2.30), 

which consists of 5N +1 unknowns, a n , b n , c n and ψ. To obtain a solution for this linear 

system, the moments are chosen in a way that the resulting coefficient matrix is nonsingular. 

Previous validation studies of DQMOM, and comparison of its performance 

with QMOM have demonstrated that by using two-node closure (N = 2) approximation 

for f is sufficient to track the lower order moments with small errors [48, 49, 54]. 

Increasing the number of nodes, N, to three (N = 3) have improved the results, and 

in general the evaporation and coalescence terms can be accurately approximated with 

N = 2–4 [48, 49, 54, 60]. In the present work, a three-node closure is used, i.e., N is 

set to be 3, and the corresponding moment set is chosen as [60, 137] k ∈ {1, ..., 2N}; 

l ∈ {0, 1}, where l is composed of three components l 1 , l 2 , and l 3 . The chosen set of 

k and l values conserves the mass and momentum of droplets, and these values are 

found to give non-singular source terms matrix [60]. Along with these moments set, 

the calculation of the source terms from the linear system requires the mathematical 

formulation for the evaporation, forces on droplet and droplet–droplet interactions, 

which enter as sub-models and these sub-models. 

discussed and mathematical formulation is given in Subsection 2.4. 

The sub-models are individually 

The evaporation term in Eq. (2.29) can be simplified by evaluating the integral on 

the R.H.S, which is given as 

∫ 

− r k v l ∂(Rf) 

∂r 

= −(r k v l Rf)| r=∞ 

r=0 + k 

∫ ∞ 

0 

r k−1 v l ∂ (Rf) dr. (2.31) 

∂r


With the assumption that the maximum droplet size is a finite value, the above equation 

can be further simplified as 

∫ 

− r k v l ∂(Rf) = δ k0 ψv l + k 

∂r 

∫ ∞ 

0 

r k−1 v l ∂ (Rf) dr, (2.32) 

∂r 

where δ k0 is the Kronecker delta, which is defined as δ k0 = 1 if k = 0 and δ k0 = 0 for 

any other k value. The quantity ψ = Rf(0) is the evaporative flux, which is a point 

wise quantity of the NDF representing the number of droplets having zero size. This 

quantity in DQMOM is computed by weight ratio constraints, which are introduced by 

Fox et al. [60] where ψ is treated as an additional variable along with a n , b n and c n ’s. 

These ratio constraints of weights, radii and velocities [60] are given by 

( ) ( ) 

D wn D rn 

= 0; 

= 0; (2.33) 

Dt w n+1 Dt r n+1 

where j is the index for three velocity components. 

( ) 

D vj,n 

= 0, (2.34) 

Dt v j,n+1 

Fox et al. [60] show that the 

estimation of evaporative flux via weight ratio constraints is found to give acceptable 

results in a stationary one-dimensional configuration. However, Fox et al. [60] suggests 

that this calculation procedure is found to pose problems in the case of complicated 

distribution functions [64]. 

In the current study, this is addressed by implementing the maximum entropy (ME) 

principle proposed by Mead and Papanicolaou [66] for water and PVP/water spray 

flow in air, which estimates the evaporative flux through reconstruction of the droplet 

distribution using its moments. 

The principle of maximum entropy in the problem of moments is that the distributions 

that satisfy the given moment set (also called as constraints), the most likely 

or least biased probability density function is the the one whose statistical entropy is 

a maximum. This formulation allows the determination of a number density function 

from the limited amount of information such as few known moments of a distribution 

[66]. The implementation of this method to compute ψ is explained by Massot et 

al. [98]. 

The ME method is first introduced by Mead and Papanicolaou [66] to compute 

a distribution for the given moment set based on the maximization of the following 

Shannon entropy from the information theory [66], 

H[f] ≡ − 

∫ rmax 

r min 

f(x) ln f(x)dx. (2.35) 

Mead and Papanicolaou have proven that there exists ME distribution satisfying the 

above entropy principle [66] for the case when the vector of moments M belongs to


the interior of the moment space M = {M(0), M(1), ..M(N)}. This is a standard constrained 

optimization problem where the constraints are to satisfy the given moments. 

In the ME method, following the moments satisfaction condition, below equation is 

the explicit representation of the ME approximation, 

f ME 

M (x) ≡ exp ( −Σ N j=0 ξ j x j) , (2.36) 

where the coefficients ξ 0 , ξ 1 ...ξ N are the Lagrange multipliers, and N is the number of 

moments. These coefficients are computed based on the condition of minimizing the 

following convex potential: 

∆ ≡ 

∫ rmax 

r min 

[ 

exp 

( 

−Σ 

N 

j=0 ξ j x j) − 1 ] dx + Σ N j=0 ξ j M(j). (2.37) 

The stationary points of Eq. (2.37) are given by ∂∆ 

∂ξ i 

equation 

∫ rmax 

≡ 0, which yields the following 

r min 

x i exp ( −Σ N j=0 ξ j x j) dx ≡ M(i). (2.38) 

The solution of the above equation gives ξ i , and substitution of these ξ i in Eq. (2.36) 

yields the required NDF. The above equation can be solved numerically using a Newton 

method, with the initial guess as ξ ≡ (− ln M(0)/(r min − r max )), 0, ...0), and updated 

ξ’s are estimated by 

Here H is the Hessian matrix defined by H i,j ≡ 

ξ + ≡ ξ − H −1 (M − 〈X〉 ξ 

). (2.39) 

∂∆ 

∂ξ i ∂ξ j 

≡ 〈x i+j 〉 for i, j ≡ 0, 1, ...N, and 

〈X〉 ξ 

≡ (〈x 0 〉 ξ 

, ... 〈 x N〉 ) is the vector of approximated moments, which are expressed 

ξ 

as 

〈〉 ∫ rmax 

x 

k ≡ x i exp ( −Σ N ξ j=0 ξ j x j) dx. (2.40) 

r min 

The numerical procedure to implement this approach is same as done by Mead and 

Papanicolaou [66] and Massot et al. [98], where a double-precision 24-point Gaussian 

quadrature method very efficiently produces the required accuracy for 〈 x k〉 ξ . More 

details about the derivation of this method and numerical solution procedure are given 

by Mead and Papanicolaou [66]. 

As the systems of interest in the present study are water spray in quiescent air or 

nitrogen as well as PVP/water spray in quiescent air, currently the gas phase is not 

fully coupled with DQMOM transport equations but its inlet properties taken from 

the experiment are used to compute the droplet motion and evaporation. 

2.4 Single Droplet Modeling 

This section presents the physical processes due to gas–liquid and droplet–droplet interactions, 

namely, droplet heating and evaporation, forces acting on the droplet, droplet

2.4. Single Droplet Modeling 29 

collisions and breakup, which enter as source terms in QMOM, DQMOM and DDM. 

2.4.1 Droplet Heating and Evaporation 

In spray flows, particularly in spray drying processes, the droplet evaporation can be 

critical because (1) it has direct effect on drying rate of droplets yielding powder, and 

(2) it influences the final powder characteristics. The evaporation process can be very 

complex under realistic spray drying conditions. Factors that increase the complexity 

of the evaporation models are (1) the multi-component character of the liquid solution, 

(2) the interaction between droplets in the turbulent gas environment, and (3) large 

differences in volatility of solutes. The study of single droplet heating and evaporation 

forms a basis for simulating complex spray flows. As stated before, few studies have 

been carried out for application of DQMOM on evaporating sprays [60, 126, 138]. However, 

these studies consider a simplified evaporation model to calculate the change in 

droplet size with time, i.e., either as a linear function of droplet volume or non-linear 

function of droplet volume, which is similar to the well established d 2 law. In the 

present study, an advanced droplet evaporation model of Abramzon and Sirignano [62] 

is used for the single component droplet evaporation, whereas, for the bi-component 

PVP/water droplet, the focus is to develop a mathematical model, which can predict 

the evaporation and drying of a single bi-component droplet, thereby include the developed 

model to study the PVP/water and mannitol/water droplet evaporation and 

solid layer formation. The following subsections present the mathematical models for 

mono- and bi-component droplet evaporation. 

2.4.1.1 Single Component Droplet 

The droplet evaporation is a complex process where simultaneous heat and mass transfer 

occurs leading to regressing droplet size. Fluid dynamics plays a major role when 

there is a relative motion between the droplets and the surrounding gas. The flow properties 

have a critical impact on the mass, momentum and energy exchanges between the 

gas and the droplets. Droplet evaporation was first studied by Langmuir [139] in 1918. 

Earlier studies reported that the droplet surface decreases constantly with time [140], 

famously known as d 2 law. After Langmuir [139], several studies were carried out in 

this area. Most notable works in droplet evaporation descriptions include the studies 

of Chigier [141], Clift et al. [142], Glassman [143], Lefebvre [144], and Williams [145]. 

A review of existing droplet evaporation models is given by Faeth [72], Law [146] and 

Sirignano [140]. 

The study of Abramzon and Sirignano [62] introduced a model for single component 

droplet evaporation, which includes the convection effects, droplet heating, and variable


liquid and film properties. 

The work of Sirignano [140] classified the single component droplet evaporation 

models into six types, and they are given in the order of complexity as, (1) constant 

droplet temperature model (also known as the d 2 law), (2) infinite liquid conductivity 

model (uniform but time dependent droplet temperature), (3) conduction limit (spherically 

symmetric transient droplet heating) model, (4) effective conductivity model, (5) 

vortex model of droplet heating, and (6) Navier – Stokes solution. There are various 

differences among these models, and some of these models are shown to be limits of 

another model [140]. 

Recent study of Sazhin [147] gives an overview of all the existing droplet evaporation 

models, particularly in the field of combustion studies. 

For evaporating water spray flow in air, the spatial gradients of the temperature 

within the droplet will not be significant when the evaporation conditions are room 

temperature and atmospheric pressure. Thus, in the current study, uniform but time 

dependent droplet temperature with convective effects can be used to predict the evaporation 

rate, and the droplet size regression. Therefore, for evaporating water sprays 

under room temperature and pressure conditions, the model of Abramzon and Sirignano 

[62] is implemented, which is a uniform temperature model that includes the convective 

effects, and considers the variable liquid and film properties. Here film means 

a thin layer across the droplet surface where the saturation of liquid vapor exists, and 

this vapor mass fraction is computed based on the vapor-liquid equilibrium. 

The rate of change of droplet mass with time due to convective evaporation and 

droplet heating in water spray is computed as [62] 

ṁ = 2πRρ f D f ˜Sh ln(1 + BM ), (2.41) 

where R is the droplet radius, ρ f is the density in the film, D f is the water diffusivity in 

the film, ˜Sh is the modified Sherwood number that accounts for the convective effects 

of droplet evaporation [62], given as 

˜Sh = 2 + Sh − 2 

B M 

(1 + B M ) ln(1 + B M ). (2.42) 

Here, Sh is the Sherwood number, which is defined as the ratio of convective mass 

transfer to the diffusion mass transport, and it is generally written in terms of the 

droplet Reynolds number, Re d , and Schmidt number, Sc, given by [62] 

Sh = 1 + (1 + Re d Sc) 1/3 f(Re d ). (2.43) 

The droplet Reynolds number is defined as the ratio of inertial forces to viscous forces, 

which is written as Re d = 2rρ g |u − v|/µ f . The Schmidt number is used to characterize


the fluid flows in which there is simultaneous momentum and mass diffusion, and 

it is defined as the ratio between momentum diffusion and mass diffusion, written 

as Sc = µ f /(ρ g D f ). 

In Eq. (2.43), the function f(Re d ) depends upon the droplet 

Reynolds number, and in case of low Reynolds number, it may be calculated as defined 

by Abramzon and Sirignano [62], with f(Re d ) = 1 for Re d ≤ 1 and f(Re d ) = Re d 

0.077 

for Re d ≤ 400. 

In Eq. (2.41), B M is the Spalding mass transfer number, expressed in terms of the 

mass fraction of vaporized liquid as, 

B M = Y s − Y ∞ 

1 − Y s 

. (2.44) 

Here Y s and Y ∞ are mass fractions of the water at the droplet surface and in the bulk 

of surrounding gas, respectively. 

Y s is computed from the vapor-liquid equilibrium 

through the vapor pressure of water, which is written as [148] 

Y s = 

M w 

M w + ¯M(¯p/p w − 1) . (2.45) 

The quantities M w and p w denote molar mass and vapor pressure of water while 

and ¯p represent molar mass and mean pressure of the surrounding gas, respectively. 

Although the initial temperatures of gas and the droplet are equal and are at 

room temperature, the droplet temperature is subject to change due to evaporation. 

Time evolution of droplet temperature for water spray is computed using the uniform 

temperature model [62], 

[ ] 

dT s 

mC pL 

dt = Q CpLf (T ∞ − T s ) 

L = ṁ 

− L V (T s ) , (2.46) 

B T 

where m is the droplet mass, Q L is the net heat transferred to the droplet per unit 

time, C pL and C pLf are the specific heat capacity of the liquid and in film, respectively, 

T s is the temperature at droplet surface, T ∞ is the temperature of the surrounding 

gas, and L V (T s ) is the temperature dependent latent heat of vaporization at T s . B T is 

the Spalding heat transfer number, which is calculated in terms of the mass transfer 

number using the relation [62] 

where the exponent φ is given by [62] 

B T = (1 + B M ) φ − 1, (2.47) 

φ = C pL 

˜Sh 1 

C pg Ñu Le . (2.48) 

Here C pg is the specific heat capacity of the gas, Le is the Lewis number, and 

¯M 

Ñu is 

the modified Nusselt number, which accounts for convective droplet heating, and it is


given by [62] 

Ñu = 2 + 

Nu 

. (2.49) 

(1 + B T ) 

−0.7 

The Nusselt number, Nu, defined as the ratio between convective heat transfer to 

conductive heat transfer and it is usually expressed in terms of the droplet Reynolds 

number, Re d , and Prandtl number, Pr, as 

Nu = 1 + (1 + Re d Pr) 1/3 f(Re d ). (2.50) 

The Prandtl number, Pr, is defined as the ratio of momentum diffusion rate to the 

thermal diffusion rate, written as, Pr = C pLf µ f /k f . 

2.4.1.2 Bi-component Droplet 

Many studies present the evaporation phenomena associated with pure and multicomponent 

droplet, but there is a lack of a mathematical model, which can predict the 

evaporation and drying behavior of a droplet containing a polymer or sugar dissolved 

in water because of the unknown physical behavior, unavailability of experimental 

results and complexity of the problem. The available literature in the area of single 

bi-component droplet evaporation and drying is reviewed in the following paragraphs 

followed by the development of new mathematical model to compute the evaporation 

and solid layer formation of a bi-component droplet. 

Charlesworth and Marshall [149] first investigated the process of single droplet 

evaporation and drying by measuring the change in droplet mass using the deflection 

of a thin, long glass filament. This study [149] also classifies different stages of droplet 

evaporation. Later, this experiment with some modifications is considered in many 

studies. The work of Sano and Keey [150] includes the drying behavior of colloidal 

material into a hollow sphere by considering the migration of solid matter towards the 

center of the droplet through the convection measurement inside the droplet, which is 

a challenge to experiment [151]. 

Most of the experiments concerning the droplet evaporation and drying available 

in literature are either related to salts [149, 152–154], milk powders [155, 156] or some 

other colloidal matter [150, 151, 157–159], but none deals with droplets of polymer 

or mannitol as a constituent. Previously developed models assume a uniform temperature 

gradient within the droplet [151, 152, 157], and neglect the effect of solid 

formation [152, 156, 157]. The study of Nesic and Vodnik [151] presents the kinetics 

of droplet evaporation to predict the drying characteristics of a colloidal silica droplet, 

where the crust formation on the surface occurring in this configuration is considered. 

The surface vapor concentration of the evaporating solvent is calculated using experimental 

material dependent factors, which are not available for every solution including


polymer and mannitol solutions in water. Moreover, in case of a polymer, molecular 

entanglement leads to solid layer formation. Nesic and Vodnik [151] use a more detailed 

description of various stages of droplet evaporation and drying. These stages, as 

described by Nesic and Vodnik [151], are that the droplet temperature initially rises 

to an equilibrium value and solvent evaporates continuously, which in turn increases 

the solute mass fraction within the droplet. When the solute mass fraction at the 

droplet surface rises to a critical value, then there starts a thin solid layer formation, 

and further drying leads to a dried particle. 

Farid [156] shows that the droplet evaporation and drying are controlled by thermaldiffusion 

rather than mass-diffusion as assumed by most of the earlier studies [149, 150, 

152]. In Farid’s model [156], the time taken for the formation of crust on a colloidal 

silica droplet is calculated using the energy balance, which does not account for solvent 

and solute composition changes, and the evaporation rate is computed using a 

simple relation without accounting for the variation in film and liquid properties. For 

droplets with suspended solids inside, the population balance approach is recently developed 

[158] to model the nucleation and growth of suspended solids inside an ideal 

binary liquid droplet with an assumption that there exist some nuclei of suspended 

solids initially. But this method cannot be applied in the present case of droplet 

with polymer or sugar, as solute is completely dissolved in water. Golman et al. [160] 

presents a model for the evaporation and drying of slurry droplets, which is an improvement 

over the receding interface model of Cheong et al [161] for slurry droplets, and 

the bi-component liquid mixture is treated as ideal. A detailed review of all existing 

theoretical models of evaporation and drying of single droplet containing dissolved and 

insoluble solids is given by Mezhericher et al. [162], and a review of evaporation models 

in the area of combustion is given by Sazhin [147]. 

Adhikari et al. [163] and Vehring et al. [164] give a review of the experimental 

studies in the area of single droplet evaporation and drying. Tsapis et al. [159] and 

Sugiyama et al. [165] have levitated droplets using Leidenfrost phenomenon on a concave 

hot plate whereas Yarin et al. [166] levitated droplets using an acoustic levitator. 

This technique was successfully used to study shell buckling during particle formation 

[165]. A drawback of this approach is that the flow field and temperature field 

in the vicinity of the droplet are different from those of a free flowing droplet in a 

spray dryer. A chain of mono-disperse free falling droplets has been used by several 

experimental groups to study heat and mass transfer, drying, and particle formation 

processes. El Golli et al. [154] measured salt droplet evaporation and compared their 

results with a theoretical model. A similar technique to study the effect of drying 

rates on particle formation was used by Alexander and King [167] and El-Sayed et 

al. [168]. Wallack et al. [169] compared measured evaporation rates with a numerical


Fig. 2.5: Schematic diagram of stages in single droplet evaporation and drying. 

model, achieving fairly good agreement. The droplet generators used in these studies 

produce a chain of closely spaced droplets, which leads to droplet–droplet interactions 

in processes that are limited by gas phase transport processes. 

The aim of the present work is to develop a mathematical model, which can be 

applied to predict the evaporation and drying characteristics of droplets of the polymer 

PVP dissolved in water and mannitol dissolved in water solution. Prerequisites 

of the method are, (1) accounting for the solid layer resistance in mass evaporation 

rate and energy calculation, and (2) treatment of the liquid mixture as non-ideal by 

computing the activity coefficient of the evaporating component. 

The problem under consideration is the evaporation and drying of an isolated single 

spherical droplet consisting of a binary mixture of a liquid and a dissolved solid material 

with low or zero vapor pressure. 

During the evaporation and drying of the bi-component droplet, the droplet undergoes 

four stages as explained by Nesic and Vodnik [151], which are depicted in Fig. 2.5. 

In the initial stage, the droplet temperature quickly rises to an equilibrium temperature, 

which is most often near to the wet bulb temperature for surrounding gas and 

humidity, with some solvent evaporation. 

In the second stage, the droplet starts to shrink as solvent evaporates causing the 

solute mass fraction to increase at the droplet surface; this leads to a slight raise in 

the droplet temperature (see Fig. 2.5). The increase in solute mass fraction at the 

droplet surface hinders further evaporation as the vapor pressure of the solvent at 

the surface drops. The third stage of drying starts when the solute mass fraction at 

the surface raises to a threshold value, which most often is equal to the saturation 

solubility of the solute in the solvent, whereupon the crust formation starts for salts, 

sugar and colloidal material. In the case of polymers, molecular entanglement and


gradual increase in concentration lead to solid layer formation at the droplet surface. 

In the latter case, the solid layer thickens and develops into the droplet interior as 

shown in Fig. 2.5, and a rapid fall in evaporation rate is observed. 

In this period, 

the heat penetrated into the liquid is used for heating the droplet, which causes the 

droplet temperature to rise rapidly. Further drying behavior of droplet depends on the 

vapor diffusivity through the solid layer. In the final stage of drying, boiling followed 

by particle drying, eventually leading to dried product formation, takes place. 

The different assumptions in developing this mathematical model include the following: 

1. The droplet remains spherical in shape throughout the evaporation with spherical 

symmetry. 

2. Solubility of gas in liquid is negligible. 

3. Gas phase is in a quasi-steady state. 

4. No influence of chemical reactions occurs within and outside the droplet. 

5. No heat transfer due to radiation. 

6. No mass diffusion by temperature and pressure gradients. 

7. No change in droplet radius once the solid layer formation starts. 

8. Internal circulation of water and capillary effects are negligible. 

The problem of evaporation and drying of a single droplet can be well defined using 

the species mass diffusion and heat conduction equations in spherical coordinates. The 

diffusion equation for the substance i in the droplet, formulated in terms of mass 

fraction Y i , reads 

∂Y i 

∂t = D [ ( 

12 ∂ 

r 2 ∂Y )] 

i 

, (2.51) 

r 2 ∂r ∂r 

where D 12 is the binary diffusion coefficient in the liquid, r is the radial coordinate 

within the droplet radius, and t stands for time. In this equation i = 1 denotes the 

solvent (water) and i = 2 denotes solute (PVP or mannitol). Initially, the droplet is a 

homogenous mixture, Y i = Y i0 at t = 0 s. At the droplet center, r = 0 m, the regularity 

condition must be satisfied at any time, ∂Y i /∂r = 0. The boundary condition at the 

droplet surface must account for the change in droplet size, 

∂Y i 

−D 12 

∂r − Y ∂R 

i 

∂t = 

ṁi 

Aρ l 

(2.52) 

at r = R(t). Here ṁ i is the mass evaporation rate of substance i across the droplet 

surface, R(t) and A(t) are time dependent droplet radius and surface area, respectively,


and ρ l is the liquid density. 

ṁ i is zero for non-evaporating solute (PVP or mannitol), 

i = 2. The diffusion process described through Eq. (2.51) provides the mass fraction 

profiles inside the droplet. In order to close this equation, the evaporation rate from the 

droplet surface, ṁ i is needed, which appears in Eq. (2.52). This rate of evaporation is 

determined based on Sherwood analogy of Abramzon and Sirignano’s model [62], and 

in the present study, it is used in the extended form for a bi-component liquid mixture 

as modified by Brenn et al. [170], 

ṁ i = 2πR i ρ f D f ˜Sh ln(1 + BM,i ), (2.53) 

where R i is volume equivalent partial radius of component i, based on its corresponding 

volume fraction, computed as R i = R(V i /V ) 1/3 , ˜Sh is the modified Sherwood number 

defined by Eq. (2.42), which accounts for the effect of convective droplet evaporation 

[62], D f is water vapor diffusivity in film, and ρ f is the density in the film. B M,i is 

the Spalding mass transfer number for component i, and it is calculated as [62, 171], 

B M,i = Y i,s − Y i,∞ 

1 − Y i,s 

, (2.54) 

where Y i,s and Y i,∞ are the mass fractions of evaporating component i at the droplet 

surface and in the bulk of the gas, respectively. Nesic and Vodnik [151] implemented 

a similar approach, but they do not account for the volume fraction based radius in 

the calculation of the evaporation rate, i.e., droplet radius R is used instead of R i 

in computing ṁ i . The evaporation rate retardation due to solid layer resistance may 

be considered through modification of Eq. (2.53) by extending the work of Nesic and 

Vodnik [151] to yield 

ṁ = 

∑ N 

i=1 2πR iρ f D f ˜Sh ln(1 + BM,i ) 

1 + ˜ShD 

, (2.55) 

f δ/[2D s (R − δ)] 

where ṁ is the total evaporation rate, δ is the solid layer thickness at the droplet surface 

and D s is the diffusivity of vapor in the solid layer. Since the solute vapor pressure is 

low or zero and the droplet’s solute evaporation rate is zero or very small, negligence 

of the volume correction (using R in the place of R i ) may lead to an artificial increase 

in evaporation rate. In the present situation, the summation in Eq. (2.55) is only over 

component 1, because the solute (PVP or mannitol) does not evaporate, but for the 

sake of generality, the summation is kept. 

During the initial and second stage, δ equals zero. But once the solute mass fraction 

at the droplet surface reaches a threshold value, which is most often near saturation 

solubility level, there is initiation of solid layer. This solid layer on the droplet surface 

offers significant resistance to evaporation and is evident from the second term in the 

denominator of Eq. (2.55) [172]. The effect of capillary force on water vapor diffusion


through solid layer due to pressure difference in pores is not considered, and it is the 

scope of the future study. Moreover, the influence of internal circulation within the 

droplet is neglected, which can be modeled by a correction for the diffusion coefficient 

rather than adding a convection term [151]. 

The heat conduction equation, describing the conductive heat transfer within the 

droplet, is written as 

∂T 

∂t = α [ ( ∂ 

r 2 ∂T )] 

, (2.56) 

r 2 ∂r ∂r 

where T is the liquid temperature and α denotes the thermal diffusivity. The above 

equation is solved with the following initial and boundary conditions: At t = 0 s, the 

droplet is at uniform temperature, T = T 0 . 

At the droplet center, r = 0 m, zero 

gradient condition prevails at any time, ∂T /∂r = 0. The energy balance at the droplet 

surface is given through the boundary condition, 

∂T 

k l 

∂r = h(T ∂R 

g − T s ) + L V (T s )ρ l 

∂t 

(2.57) 

at r = R, where R is the droplet radius. In Eq. (2.57), T s denotes droplet surface 

temperature, T g stands for gas temperature in the bulk, k l is the liquid thermal conductivity, 

h is the convective heat transfer coefficient, and L V (T s ) is the latent heat of 

vaporization at the surface temperature, T s . 

In this work, first Eq. (2.56) is solved numerically with initial and boundary condition 

as defined above using a finite difference method. It is observed that the gradient 

in droplet temperature from the center to the droplet surface is very small as the computed 

Biot number, which is a measure of heat transfer resistances within and outside 

the droplet, (Bi = h/k s R = k g /(2k s )Nu), always remains below 0.5. Therefore, in the 

remaining simulations, uniform temperature within the droplet is assumed, which is a 

valid assumption as per the revelations made by Mezhericher et al. [173]. The droplet 

temperature continuously changes due to heat transfer from ambient gas to the binary 

liquid droplet, and it is computed using the energy balance across the droplet, which 

gives the net heat transferred into the droplet [62], as 

[ ] 

dT s 

mC pL 

dt = Q CpLf (T g − T s ) 

L = ṁ 

− L V (T s ) , (2.58) 

B T 

where m is the total droplet mass, m = Σ N i=1m i , C pL , C pLf are the specific heat capacity 

of liquid and in the film, respectively and B T is the Spalding heat transfer number. 

This equation can be used to calculate the time evolution of droplet temperature. Here, 

the heat transfer number, B T , is calculated in terms of mass transfer number defined 

by Eq. (2.47). 

Equation (2.58) needs modification in order to account for the solid layer formation 

at droplet surface, and this is achieved through the equation written in terms of the


solid layer thickness, δ, as 

where 

mC pL 

dT s 

dt = Q L + ṁL V (T s ) 

1 + Ñu k gfδ/[2k s (R − δ)] − ṁL V (T s ), (2.59) 

Ñu is the modified Nusselt number defined by Eq. (2.49), which accounts for 

the effect of convective droplet heating [62], k s and k gf are the thermal conductivity of 

the solid layer and in the film, respectively, and Q L is the net heat transferred to the 

droplet [62], given by Eq. (2.58). Similar to Eq. (2.55), the second term in denominator 

inside the bracket of Eq. (2.59) denotes the resistance due to solid formation at the 

droplet surface, and its effect becomes significant only when the solid layer thickness, 

δ, is positive. The difference between heat transfer and mass transfer resistance is 

that the ratio of diffusion coefficients D f /D s is larger than the ratio k g /k s [151], which 

implies that resistance to mass transfer due to solid layer formation is higher than the 

heat transfer. 

In the present study, simulations are also carried out with rapid mixing model (RMM) 

which is a simple model based on the assumption that the liquid mixture inside the 

droplet is always homogeneous (no spatial gradients of mass fraction within the droplet) 

and infinity conductivity within the droplet, thus the droplet is at uniform temperature 

at every time. In this work, the RMM is extended to account for solid layer resistance 

on the droplet evaporation rate and heating, thus the governing equations in the RMM 

are Eq. (2.55) and Eq. (2.59), and the time evolution of solute (i = 2) mass fraction is 

calculated based on the simple mass balance of solute and solvent mass fraction within 

droplet, given as 

Y 2,RMM = Y 02 − m 02 

m − ṁ , (2.60) 

where Y 02 and m 02 are the initial solute mass fraction and solute mass within the 

droplet, respectively. 

2.4.2 Droplet Motion 

The dynamics of liquid droplets in sprays is the basic physical process that needs to 

be computed for the coupling of gas–liquid phases due to its strong dependance on 

the flow of surrounding gas. The droplet velocity v at position x can be computed as 

following 

v = dx 

dt . (2.61) 

The acceleration of droplets due to different forces acting on droplets can be written 

as [174] 

dv 

dt = ΣF + ρ ( 

g Du 

2ρ l Dt − dv ) 

, (2.62) 

dt


where the first term in R.H.S includes all the forces such as aerodynamic drag, gravity, 

Basset, lift, and buoyancy etc. 

and the second term in R.H.S is the added mass 

D 

force [174]. In Eq. (2.62), is the substantial or material derivative. 

Dt 

The force experienced by the droplets due to difference in velocities of droplets 

and surrounding gas is known as drag force, F d . The droplet velocity evolution by 

interactive drag induced by the surrounding gas, and gravity per unit droplet mass is 

commuted using the following relation, which describes droplet motion [175] 

F d = 3 8 

1 ρ g 

(u − v)|u − v|C D + g, (2.63) 

r ρ l 

where ρ g and u are the density and velocity of the surrounding gas, respectively, while 

ρ l , C D and g are liquid density, drag coefficient, and gravitational acceleration, respectively. 

The dependencies of the drag force are confined to the droplet radius, droplet 

shape, droplet density, ρ l , relative velocity between gas and droplet, u − v, gas density, 

ρ g , kinematic viscosity of the gas, η g , and surface tension, σ d . 

The drag coefficient, C D , is calculated as a function of the droplet Reynolds number, 

Re d = 2rρ g |u − v|/µ f , where µ f is the mean dynamic viscosity in the film, as [176] 

{ 

24 

(1+ Re 

C D = 

1 d 6 Re0.687 d ) if Re d


surrounding gas on the droplets. Buoyancy force is equal to the weight of the displaced 

gas due to droplet motion. 

The added mass force, defined by second term in the right hand side of Eq. (2.62), 

accounts for the acceleration of the gas due to the droplet motion. When a droplet 

accelerates in gas, it implies an acceleration of the surrounding gas at the expense of 

the force exerted by the droplet. Since the added mass force depends on the fluid 

density, it is often neglected for droplets much denser than the gas [177]. In this work, 

the ratio between liquid density and gas density is about 10 3 , so the effect of added 

mass can be neglected. 

The unsteady behavior of the droplet, buoyancy effects, compressibility of the gas, 

rotation effects, the fluid motion within the droplet or other subtle forces are not 

considered. It can be shown [174] that terms originating from these phenomena are 

negligible for large ratios of droplet to gas densities, and for low droplet Mach numbers, 

Ma = |u − v|/c < 0.03, where c is the speed of sound in the gas. 

2.4.3 Droplet Breakup 

Liquid drops generated from the primary breakup of the liquid sheet, moving in the 

surrounding gas may undergo further breakup or disintegration under certain conditions, 

leading to formation of smaller droplets. This phenomenon is called as droplet 

breakup or secondary atomization. The exact mechanisms of the droplet breakup is 

not yet completely understood as there are many uncertainties in the quantitative description 

of the process. The relative motion between a droplet and the surrounding 

gas causes a non-uniform distribution of pressure and shear stress on the droplet surface, 

which results in deformation of the droplet and cause it to disintegrate when 

they overcome the opposing force of surface tension. The newly formed droplets may 

still undergo further breakup until surface tension force of the newly formed droplet 

is higher than the external forces. The work of Pilch and Erdmann [178] explained 

the various regimes of breakup, which are depicted in Fig. 2.6. Faeth et al. [70] and 

Faeth [72] give an overview of existing mechanisms of droplet breakup. 

According to Faeth et al. [70], the breakup regime transitions are mainly functions 

of the gas Weber number, We g , and the Ohnesorge number, Oh. The Weber number 

is defined as the ratio between the drag force to surface tension force, written as 

We g = 2rρ g|u − v| 2 

, (2.67) 

σ 

where σ is the surface tension and r is the droplet radius. The Ohnesorge number, 

represents the ratio of viscous forces to inertial and surface tension forces, given as 

Oh = 

µ l 

√ 2ρl rσ , (2.68)


(a) Vibrational breakup, We g ≈ 12 (b) Bag breakup, We g < 20 

(c) Bag / streamer breakup, We g < 50 (d) Stripping breakup, We g < 100 

(e) Catastrophic breakup, We g > 100 

Fig. 2.6: Droplet breakup mechanisms based on Weber number [69, 178]. 

where µ l is the liquid viscosity. The existing breakup models developed based on 

the various mechanisms include, wave breakup (WB) model [179], Taylor analogy 

breakup (TAB) model [180], enhanced Taylor analogy breakup (ETAB) model [181], 

Rayleigh-Taylor instability (RTI) model [182], and droplet deformation and breakup (DDB) 

model [183]. Madsen [61] extended DQMOM to include droplet coalescence and 

breakup in spray flows by neglecting the effects of evaporation. In the present study, 

the focus is on the influence of droplet coalescence, evaporation and drag on droplet 

characteristics, and the study concerns the spray at a distance after the atomization, 

which may not breakup further, the droplet breakup is currently neglected. 

2.4.4 Droplet Coalescence 

The droplets in spray flows when come close enough, they interact with each other 

leading to collision of droplets. The collision dynamics of liquid droplets is important 

in the evolution of spray flows as they can significantly effect the spray characteristics 

such as droplet size and velocity distribution, and in turn influence the final powder 

characteristics in spray drying process.


Fig. 2.7: Droplet collision regimes: (a) bouncing, (b) coalescence [184]. 

The outcome of the droplet collision is mostly dependent on the size, mass, surface 

tension, and velocity of colliding droplets. The collision outcome is classified into four 

different types: bounce, coalescence, reflexive separation and stretching separation. 

This classification is depicted in Figs. 2.7 and 2.8 [184]. Reflexive separation which 

occurs in head on and near-head on collision of droplets from miscible liquids does not 

exist for the immiscible liquids [185], and the collision mechanism in immiscible liquids 

is identified by Planchette and Brenn, and termed it as crossing separation [185]. 

The spray models developed to account for the droplet–droplet interactions mostly 

assume that there are only two possibilities of collision outcome: the droplets rebound 

without any change in droplet size or they coalesce to give a single droplet [186]. These 

models are only applicable to the study of two droplets colliding with each other but 

Fig. 2.8: Droplet collision regimes: (c) reflexive or crossing separation, (d) stretching 

separation [184].


not to the spray itself as the extension of these models to dense spray flows is much 

more complex [184], because of individual droplet tracking requirement. Implementation 

of droplet coalescence models needs tracking of individual droplets as done in 

Euler – Lagrangian simulations. In case of Euler – Euler models, droplet distribution 

is computed but the individual droplets are not tracked. Hylkema and Villedieu [187] 

developed a droplet collision model based on the droplet distribution, which can be 

implemented in Euler–Euler methods. In the current study, as the spray flow is modeled 

using Eulerian approach where the global droplet distribution is computed, so the 

droplet collisions are taken into account as described by Hylkema and Villedieu [187] 

and Laurent [46]. To emphasize upon coalescence only, standard assumptions [46] for 

droplet coalescence have been employed. These assumptions imply that each binary 

collision either leads to coalescence (E c = 1) or rebound (E c = 0), and conservation 

of mass and momentum before and after the collision [46] is assured. In addition, the 

mean collision time is assumed to be smaller than the inter-collision time. Thus, the 

coalescence function can be written in terms of the flux of newly formed droplet, Q + c 

and flux of the vanishing droplets, Q − c , given by [187] 

Γ f = Q + c + Q − c , (2.69) 

where Q + c and Q − c are calculated as 

Q − c 

∫ ∞ 

= − 

−∞ 

∫ ∞ 

0 

f(t, x; r, v)f(t, x; r 1 , v 1 ) × B(|v − v 1 |)dr 1 dv 1 , (2.70) 

Q + c = 

∫ ∞ ∫ ∞ 

−∞ 

where B(|v 1 − v 2 |) is given by, 

0 

1 

2 f(t, x; r 1, v 1 )f(t, x; r 2 , v 2 ) × B(|v 1 − v 2 |)dr 1 dv 1 , (2.71) 

B(|v 1 − v 2 |) = π(r 1 + r 2 ) 2 |v 2 − v 1 |E c , (2.72) 

and B(|v − v 1 |) is defined accordingly. In the above equations, (r, v) refer to postcollision 

properties, which are related to pre-collision properties (r 1 , v 1 ) and (r 2 , v 2 ) 

through the relations [46, 187] 

v = r3 1v 1 + r2v 3 2 

, (2.73) 

r1 3 + r2 

3 

r 3 = r1 3 + r2. 3 (2.74) 

The collision efficiency is computed following the work of O’Rourke [186], which is 

written as 

E c = 

K 2 

(K + 1/2) 2 , (2.75)


where K is given as [186] 

K = 2 ρ l |v 1 − v 2 |r2 

2 . (2.76) 

9 µ g r 1 

For DQMOM, substitution of all these sub-models defined in Subsection 2.4 into the 

Eq. (2.29) results in a linear system, which in turn is substituted into Eq. (2.30) to 

yield in a linear system of equations. 

The solution of this linear system gives the 

various source terms, i.e., a n , b n and c n that appear in DQMOM transport Eqs. (2.26), 

(2.27) and (2.28), which then constitutes a closed system. The numerical solution 

procedure to solve these equations with initial and boundary conditions is explained 

in Chapter 3. Droplet–droplet interactions are currently neglected in DDM because of 

the computational complexity involved if Lagrangian models are used. 

In summary, the new implementations of the present study include implementation 

of an advanced droplet evaporation model for water sprays in DQMOM, a new mathematical 

model development for the polymer or sugar dissolved in water droplets evaporation 

and solid layer formation at the droplet surface, and improvement of the evaporative 

flux calculation with maximum entropy formulation. Extension of DQMOM 

to simulate two-dimensional system, and implementation of developed bi-component 

evaporation model in DQMOM.

3. Numerical Methods 

In the numerical simulation, the governing equations are discretized and solved by 

computer programs where appropriate numerical algorithms are required. An ideal 

numerical algorithm should 

• be linearly stable for all cases of interest; 

• ensure the positivity property when appropriate; 

• be reasonably accurate; 

• be computationally efficient. 

There are several numerical methods available for the fluid mechanics. The methods 

ranging from the most discrete (or particulate) in nature to the most continuous (or 

global) include: 

• particle methods 

• characteristic methods 

• Lagrangian finite difference/finite volume method 

• Eulerian finite difference/finite volume method 

• finite element methods 

• spectral methods 

Each method has advantages and disadvantages, consequently has the preferable applications. 

Usually, it is difficult or inefficient for a stand-alone method to simulate 

a complex system. Hybrid method, which is like a bootstrapping process, combines 

the advantages of the multiple methods and minimizes their disadvantages. The disadvantage 

of hybrid method is that the consistency problem is more serious. Special 

strategies are needed to keep consistency between the multiple methods. 

In this chapter the numerical methods employed to solve the single droplet evaporation 

and drying equations, QMOM and DQMOM transport equations are explained. 

In this work, the DDM computations performed by Humza [68] are used to validate the 

DQMOM results for water spray in air in two-dimensional, axisymmetric configuration. 

Hence, the numerical details of the DDM simulations can be referred to Humza [68].

46 3. Numerical Methods 

3.1 Finite Difference Method for Bi-component 

Droplet Evaporation and Solid Layer 

Formation 

The partial differential equation, Eq. (2.51), with initial and boundary conditions is 

solved numerically at every time and spatial location within the droplet using second 

order explicit finite difference method, given as 

Y j+1 

i 

∆t 

− Y j 

i 

[ 

r 

2 

= D i+1 (Y j 

i+1 − Y j 

i ) − r2 i−1(Y j 

i − Y j 

i−1 )] 

12 , (3.1) 

ri 2∆r2 where r i+1 = r i + ∆r, r i−1 = r i − ∆r, and i and j are the spatial location within the 

droplet and time step indices, respectively. Equation (3.1) can be simplified to yield 

the following equation 

Y j+1 

i 

= Y j 

i 

+ D 12∆t [ 

r 

2 

ri 2 i+1 (Y j 

∆r2 i+1 − Y j 

i ) − r2 i−1(Y j 

i − Y j 

i−1 )] . (3.2) 

The initial condition to compute Eq. (3.2) is provided as a Dirichlet condition, i.e., 

Y = Y i0 at every location inside the droplet at t = 0. A Neumann boundary condition 

is applied at the center of the droplet, i.e., ∂Y i /∂r = 0 at r = 0, which implies the 

radial symmetry within the droplet. A Robin boundary condition is employed at the 

droplet surface, and it is given by Eq. (2.52). 

The energy Eq. (2.59) is an ordinary differential equation, solved using Runge-Kutta 

4 th order method. The droplet is discretized into equal distant grid points at any given 

time. As the droplet size decreases with time thereby the grid size changes because 

grid points are fixed, thus a moving grid problem is solved, and grid independency of 

the numerical method is tested using different grid sizes with the number of grid points 

varying from 10 to 100. The value of 50 grid nodes is found to perform well. 

The numerical stability of the method is tested using various time steps following 

the Courant-Friedrichs-Lewy (CFL) condition [188]. The CFL condition defines the 

limiting criteria for the numerical grid size when the time step and fluid velocity are 

known, and it is defined as 

C = u∆t 

∆x ≤ C max, (3.3) 

where C is the dimensionless number known as Courant number, u is the velocity, ∆t 

and ∆x are the time step and grid size, respectively. C max is the maximum possible 

Courant number to get a stable numerical solution, and it is generally taken as any 

positive value lower than or equal to 0.5 [188]. The step-by-step procedure of Abramzon 

and Sirignano [62] is applied to calculate the mass evaporation rate given by Eq. (2.55). 

Numerical simulations of pure water, mannitol dissolved in water droplet evaporation

3.2. Spray Modeling 47 

and solid layer formation is done to test the implementation of this algorithm and the 

numerical results are compared with experimental data. The results are presented in 

Chapter 4. 

3.2 Spray Modeling 

3.2.1 Finite Volume Method for QMOM 

In the present study, QMOM is implemented with a three-node (three weights or number 

densities, three droplet radii, and three droplet velocities) closure approximation 

of the NDF, which requires a total of nine moments of the NDF to compute the initial 

data of droplet radii and velocities and corresponding weights. The transport equations 

are generated by selecting k 1 ∈ {0, 1, 2, 3} and k 2 ∈ {0, 1} in Eq. (2.24), which 

is equivalent to three-node closure. The choice of three-node closure with the mentioned 

values of k 1 and k 2 is proven to be accurate in previous studies [48, 49, 51]. The 

substitution of k 1 and k 2 values results in the following equations: 

∂M(0, 0) 

∂t 

+ 

∂M(0, 1) 

∂x 

= 

∫ ∞ ∫ ∞ 

−∞ 

0 

[ 

− ∂ (Rf) − ∂(Ff) ] 

∂r ∂v + Q f + Γ f drdv, (3.4) 

∂M(1, 0) 

∂t 

∂M(0, 1) 

∂t 

∂M(1, 1) 

∂t 

∂M(2, 1) 

∂t 

+ 

+ 

+ 

+ 

∂M(1, 1) 

∂x 

∂M(0, 2) 

∂x 

∂M(1, 2) 

∂x 

∂M(2, 2) 

∂x 

= 

= 

= 

= 

∫ ∞ ∫ ∞ 

−∞ 

0 

∫ ∞ ∫ ∞ 

−∞ 

0 

∫ ∞ ∫ ∞ 

−∞ 

0 

∫ ∞ ∫ ∞ 

−∞ 

0 

[ 

r − ∂ (Rf) − ∂(Ff) ] 

∂r ∂v + Q f + Γ f drdv, (3.5) 

[ 

v − ∂ (Rf) − ∂(Ff) ] 

∂r ∂v + Q f + Γ f drdv, (3.6) 

[ 

rv − ∂ (Rf) − ∂(Ff) ] 

∂r ∂v + Q f + Γ f drdv, (3.7) 

[ 

r 2 v − ∂ (Rf) − ∂(Ff) ] 

∂r ∂v + Q f + Γ f drdv (3.8) 

∂M(3, 1) 

∂t 

+ 

∂M(3, 2) 

∂x 

= 

∫ ∞ ∫ ∞ 

−∞ 

0 

[ 

r 3 v − ∂ (Rf) − ∂(Ff) ] 

∂r ∂v + Q f + Γ f drdv. (3.9) 

The M(0, 2), M(1, 2), M(2, 2) and M(3, 2) fall away from the selected moment set 

defined by k 1 and k 2 values, and these four unclosed moments are computed in terms 

of the weights and abscissas:


M(0, 2) = w 1 v 2 1 + w 2 v 2 2 + w 3 v 2 3, (3.10) 

M(1, 2) = w 1 r 1 v 2 1 + w 2 r 2 v 2 2 + w 3 r 3 v 2 3, (3.11) 

M(2, 2) = w 1 r 2 1v 2 1 + w 2 r 2 2v 2 2 + w 3 r 2 3v 2 3, (3.12) 

M(3, 2) = w 1 r 3 1v 2 1 + w 2 r 3 2v 2 2 + w 3 r 3 3v 2 3. (3.13) 

These weights and abscissas are computed using the Wheeler algorithm (see Subsection 

3.2.3). Similarly, if any of the terms on the right hand side of these Eqs. (3.4) – (3.9) 

contain unknown moments, they will be closed in the analogous manner. 

To solve Eqs. (3.4) – (3.9) a numerical scheme based on a kinetic transport scheme 

to evaluate the spatial fluxes [96, 189] can be employed. A first-order, explicit, finite 

volume scheme for these equations can be written for the set of moments 

as 

M = 

M n+1 

[ 

M(0, 0), M(1, 0), M(0, 1), M(1, 1), M(2, 1), M(3, 1)] T 

(3.14) 

i = Mi n − ∆t [ 

] 

G(Mi n , M n 

∆x 

i+1) − G(Mi−1, n Mi n ) + ∆tSi n (3.15) 

where n is the time step, i is the grid node, S is the right hand side estimate of 

Eqs. (3.4)– (3.9), and G is the flux function. Using the velocity abscissas, the movement 

of the quadrature node from left to right or right to left is determined. The flux function 

at any time step is expressed as [28] 

G(M i , M i+1 ) = H + (M i ) + H − (M i+1 ) (3.16) 

where 

⎛ ⎞ 

⎛ ⎞ 

⎛ ⎞ 

1 

1 

1 

r 1 

r 2 

r 3 

H − (M) = w 1 min(v 1 , 0) 

v 1 

r 1 v + w 2 min(v 2 , 0) 

v 2 

1 

r 2 v + w 3 min(v 3 , 0) 

v 3 

2 

r 3 v , 

3 

⎜ 

⎝r1v 2 ⎟ 

⎜ 

1 ⎠ 

⎝r2v 2 ⎟ 

⎜ 

2 ⎠ 

⎝r3v 2 ⎟ 

3 ⎠ 

r1v 3 1 r2v 3 2 r3v 3 3 

(3.17) 

and


⎛ ⎞ 

⎛ ⎞ 

⎛ ⎞ 

1 

1 

1 

r 1 

r 2 

r 3 

H + (M) = w 1 max(v 1 , 0) 

v 1 

r 1 v + w 2 max(v 2 , 0) 

v 2 

1 

r 2 v + w 3 max(v 3 , 0) 

v 3 

2 

r 3 v . 

3 

⎜ 

⎝r1v 2 ⎟ 

⎜ 

1 ⎠ 

⎝r2v 2 ⎟ 

⎜ 

2 ⎠ 

⎝r3v 2 ⎟ 

3 ⎠ 

r1v 3 1 r2v 3 2 r3v 3 3 

(3.18) 

Higher-order flux schemes can also be developed to control numerical diffusion [190]. 

However, the key characteristics of the flux function is that the quadrature method 

provides a realizable set of weights and abscissas at every grid node that can be used 

to determine the node velocities. In the present study, only the steady state solution 

of Eqs. (3.4) – (3.9) is needed due to the fact that experimental data provides only the 

time averaged droplet properties. The steady form of Eqs. (3.4) – (3.9) is solved using 

Runge-Kutta 4 th order method. The QMOM simulations are carried out only for onedimensional 

water spray in nitrogen in order to compare and validate DQMOM results, 

and the initial data for the QMOM simulations are generated from the experimental 

data by calculating the above moment set, and the initial data generation procedure 

is outlined in Chapter 4. 

3.2.2 Finite Difference Scheme for DQMOM 

A generalized model for three-dimensional physical space has been discussed for application 

to evaporating sprays [191]. At first, DQMOM is applied to study the steady 

spray flows in one physical dimension, i.e., in the axial direction x. Thus, inhomogeneous 

formulation also known as steady state form of DQMOM transport Eqs. (2.26) – 

(2.28) can be rewritten as below by neglecting the terms containing time, t, 

where 

∂U n 

∂x = S n, (3.19) 

U n ∈ {w n v n , w n ρ l r n v n , w n ρ l r n v n v n }, 

S n ∈ {a n , ρ l b n , ρ l c n }. 

Similarly, the homogeneous formulations of DQMOM transport Eqs. (2.26) – (2.28) 

can be rewritten as following by ignoring the spatial terms, 

∂U n 

∂t 

= S n , (3.20)


where 

U n ∈ {w n , w n ρ l r n , w n ρ l r n v n }, 

S n ∈ {a n , ρ l b n , ρ l c n }. 

To solve these equations, the choice of numerical scheme is important because this 

array of equations are strongly coupled. Different finite difference schemes with varying 

order of accuracy are tested. 

It has been shown [191] that Runge-Kutta 4 th order 

method can accurately solve the system of inhomogeneous equations represented by 

Eq. (3.19) and proven to be computationally efficient for DQMOM in one-dimensional 

physical space [191]. In the current study, the NDF is approximated by a three-node 

closure in DQMOM, which is proven to be accurate in previous studies [48, 49]. The 

three-node approximation of NDF implies that a total of nine coupled equations, which 

are generated by substituting n = {1, 2, 3}, in Eq. (3.19). These equations are solved 

to find the evolution of NDF, which is achieved by discretizing and estimating these 

equations with Runge-Kutta 4 th order method. At every spatial location within the 

geometry, the source terms are computed through the models proposed in Chapter 2. 

The flowchart shown in Fig. 3.1 outlines the step by step procedure of the computational 

code. 

The previous studies concerning DQMOM in spray flows, transport equations are 

never solved in two-dimensional configuration but only in one dimension. In this study, 

the DQMOM transport equations are solved in two-dimensional (axial and radial direction) 

geometrical configuration for water and PVP/water sprays in air, by implementing 

a finite difference numerical scheme. At each axial and radial location, the 

coupled steady state transport equations of DQMOM are solved and the source terms 

such as droplet heating, evaporation rate, total forces acting on droplet and droplet coalescence 

are computed from the weights and abscissas available from the initial values 

at first iteration and from the last computed value in the next iterations. The steady 

form of the DQMOM transport Eqs. (2.26) – (2.28) in two dimensions can be written 

as 

where 

∂U n 

∂x + ∂E n 

∂z = S n, (3.21) 

U n ∈ {w n v n , w n ρ l r n v n , w n ρ l r n v n u n , w n ρ l r n v n v n }, 

E n ∈ {w n u n , w n ρ l r n u n , w n ρ l r n u n u n , w n ρ l r n u n v n }. 

(3.22) 

In Eq. (3.21), x is the axial direction, z is the radial direction, and the corresponding 

velocities are v and u, respectively. 

To keep the computational efficiency, ease of


application and numerical accuracy, a second order explicit finite difference scheme is 

applied to solve steady state form of Eqs. (2.26) – (2.28) [192], which are represented 

by Eq. (3.21). Thus the solution formula may be written as [193] 

U j+1 

n,i 

= U j n,i − ∆x [ 

1.5E 

j 

i 

∆z 

− 2Ej i−1 + ] 0.5Ej i−2 + ∆xS 

j 

i , (3.23) 

where i and j are grid nodes in radial and axial directions, respectively. 

The above formulation is applied to an equidistant rectangular grid, where the size 

of each grid cell is 1.5 × 10 −3 m in radial direction and 1.0 × 10 −4 m in axial 

direction, resulting in a maximum of 80 × 1000 grid nodes. The initial data to start 

simulations in both the configurations, i.e, one and two-dimensional cases is generated 

from the experimental data provided by Dr. R. Wengeler, BASF Ludwigshafen (onedimensional 

water spray in nitrogen) and Prof. G. Brenn, TU Graz (two-dimensional 

water and PVP/water spray in air) using Wheeler algorithm (see Subsection 3.2.3). The 

experimental data closest to the nozzle exit is taken for generating the initial data and 

the procedure for calculating this initial data from experiment is explained in Chapter 

4 along with brief description about the experimental setup. The boundary conditions 

in solving DQMOM include (1) if droplets hit the axis of symmetry, they are reflected, 

and (2) Neumann boundary is applied for the lateral sides of the computational domain 

and exit plane. The experimental data available at other cross sections away from the 

nozzle exit is used to validate the simulation results. The flowchart of the computational 

code is illustrated in Fig. 3.1. 

3.2.3 Wheeler Algorithm 

The Wheeler algorithm developed by Sack and Donovan [136], requires 2N +1 moments 

to compute N weights (number density) and N abscissas (droplet radii or velocities). 

The moment set is represented as M = [M(0), M(1), ...M(2N + 1)] T . This algorithm 

is used to generate the initial data in DQMOM whereas in QMOM it is used to compute 

the unknown moments. The first step in Wheeler algorithm is to compute the 

coefficients π α based on these 2N + 1 moments of the distribution function n(ξ), given 

as 

π α+1 (ξ) = ξπ α (ξ). (3.24) 

The above recursive relation has the properties of π −1 (ξ) = 0 and π 0 (ξ) = 1. Here, α 

is a subset of number of moments 2N + 1, i.e., α ∈ 0, 1, 2..N − 1. From these coefficients 

π α (ξ), a symmetric tridiagonal matrix is computed through some intermediate 

quantities: 

∫ 

σ α,β = n(ξ)π α (ξ)π β (ξ)dξ, (3.25)


Initialization 

(Generate weights and abscissas from the 

moments through Wheeler algorithm) 

Grid generation 

Compute droplet evaporation, Eq. (2.55) 

Compute droplet heating, Eq. (2.59) 

Compute drag force, Eq. (2.63) 

Compute coalescence, Eq. (2.69) 

Solve source terms of spray, Eq. (2.30) 

Update weights, abscissas, 

Eqs. (2.26) – (2.28) 

No 

Is the 

final measurement 

position 

reached? 

Stop 

Yes 

Fig. 3.1: Flowchart of the DQMOM computational code.

3.3. Numerical Performance 53 

where β ∈ α, α + 1, ...2N − α − 1. These quantities, σ α,β , are calculated by initializing 

and a 0 = M(1)/M(0), b 0 = 0. The recurrence relation is 

σ −1,α = 0, (3.26) 

σ 0,α = M(α), (3.27) 

σ α,β = σ α−1,β+1 − a α−2 σ α−1,β − b β−1 σ α−2,β , (3.28) 

where the tridiagonal matrix components are given as 

a α = σ α,α+1 

− σ α−1,α 

, (3.29) 

σ α,α σ α−1,α−1 

σ α,α 

b α = . (3.30) 

σ α−1,α−1 

Here, the values of a α are the diagonal elements and b α are the upper and lower diagonal 

elements of the symmetric tridiagonal matrix. The eigenvalues of this matrix are the 

abscissas (droplet radii, velocities) where as the corresponding eigenvectors are the 

weights (number densities). More details about derivation of this algorithm is given 

by Gautschi [194], and example calculations are given by Marchisio and Fox [71]. 

3.3 Numerical Performance 

For the DDM computations, which are carried out Humza [68], a hybrid finite volume 

method based on the SIMPLER (Semi-Implicit Method for Pressure-Linked Equations 

- Revised) algorithm [68, 195] is used to solve the mean conservation equation of the 

gas flow, and a Lagrangian stochastic droplet parcel method is used for the spray flow. 

The initial and boundary conditions are generated from the experimental data. A 

non-equidistant rectangular numerical grid is used, which is finer in the region near 

the nozzle exit with a total of 78 × 101 grid nodes. The numerical time step for 

the governing gas phase equations is controlled by applying the CFL condition [188]. 

The solution algorithm and numerical details of the DDM calculation are given by 

Humza [68]. 

The DQMOM simulations are carried out on a PC with two Intel dual core 2.2 GHz 

processors having 8 GB RAM. The DDM is simulated on a PC having an AMD quad 

Opteron 1.8 GHz processor with 64 GB RAM [68]. The latter PC had several jobs 

running simultaneously, so that the available RAM on both the PCs is about identical. 

All simulations are run on a single processor. The computations for DQMOM and DDM 

take about one hour and three days, respectively. Thus, the DQMOM computations 

show a much better performance with respect to the computational cost.

54 3. Numerical Methods

4. Results and Discussion 

Results presented in this chapter are categorized into four sections based on the model 

development and implementation, which are discussed successively: At first, onedimensional 

water spray in nitrogen results are given, followed by the results of twodimensional 

evaporating water spray flows in air in axisymmetric configuration. Later, 

single bi-component droplet evaporation and solid layer formation results are discussed, 

and finally, results of PVP/water spray in air in an axisymmetric configuration are presented. 

4.1 One-dimensional Evaporating Water Spray in 

Nitrogen 

A spray can be generated by pumping the liquid through a nozzle that facilitates 

dispersion of liquid into a spray. Nozzles are mainly used to distribute a liquid over an 

area thereby liquid surface area is increased. There are three types of nozzles normally 

used, which include spinning disk nozzle, single-fluid or centrifugal pressure nozzle, 

twin-fluid nozzle [196]. The spinning disk nozzles are also known as rotary atomizers. 

The single-fluid nozzles include pressure-swirl nozzle, plain-orifice nozzle, hollow cone 

nozzle, etc., whereas the twin-fluid nozzles can be internal-mix or external-mix two-fluid 

atomizers [196]. The present study concerns the simulation of water spray generated 

using a hollow cone nozzle, which is single fluid nozzle. However, the model presented 

in this work is equally applicable to other type of nozzles as the current work focuses 

on the simulation of spray after the primary breakup. 

Evaporating sprays are of special interest as those occur not only in many industrial 

applications but also constitute the defining physical phenomena in spray drying 

process. Therefore, having models validated for evaporating sprays motivate their application 

in simulations of spray drying. A water spray injected through a hollow cone 

Delavan SDX-SE-90 nozzle in a vertical spray chamber and carried by nitrogen is simulated 

by DQMOM and the results are compared with the QMOM, and validated with 

the experiment.

56 4. Results and Discussion 

Fig. 4.1: Photograph of the water 

spray formation. 

Fig. 4.2: Schematic diagram of spray 

with measurement positions. 

4.1.1 Experimental Setup 

Experiments have been carried out by Dr. R. Wengeler at BASF, Ludwigshafen, where 

a water spray is injected into a cylindrical spray chamber. The carrier gas is nitrogen 

at room temperature. Three different experiments are conducted by keeping the 

spray inflow rate at 80, 150 and 200 kg/h while the gas volumetric flow rate is fixed 

at 200 Nm 3 /h. The droplet size distribution is recorded at sections of 0.14, 0.54, 

and 0.84 m distance from the nozzle exit using laser Doppler anemometry (LDA). 

Measurements at 0.14 m are taken as a starting point for initial data generation for 

computations. Figure 4.1 shows the photograph of water spray formation in experiment, 

and the schematic representation of spray with dimensions and measurement 

positions in experiment is shown in Fig. 4.2. The spray column has a diameter of 1 m. 

The present simulations concern the experimental data generated using the Delavan 

nozzle SDX-SE-90 with an internal diameter of 2 mm and an outer diameter of 12 mm 

at the nozzle throat and 16 mm at the top. 

4.1.2 Initial Data Generation 

The experimental data provide the cumulative volume frequency of different droplet 

sizes. These volume frequencies are converted into surface frequencies by dividing the 

individual volume frequency with the corresponding diameter. Figure 4.3 shows the 

surface frequencies at the distance of 0.14 m (left) and 0.54 m (right) from the nozzle 

exit, respectively, as obtained from the experimental data. At 0.14 m distance away 

from the nozzle, there is a higher number of small-sized droplets shown in the left side 

of Fig. 4.3, whereas at 0.54 m distance, an increased number of larger size droplets 

is found, see right part of Fig. 4.3. The droplet velocities are not measured in the

4.1. One-dimensional Evaporating Water Spray in Nitrogen 57 

Surface frequency [µm 1 ] 

0.12 

1 

0.1 

0.08 

0.06 

0.04 

0.02 

0 

0 

10 1 10 2 10 3 


0.8 

0.6 

0.4 

0.2 

Cumulative surface frequency [µm 1 ] 

Surface frequency [µm 1 ] 

0.12 

1 

0.1 

0.08 

0.06 

0.04 

0.02 

0 

0 

10 1 10 2 10 3 


0.8 

0.6 

0.4 

0.2 

Cumulative surface frequency [µm 1 ] 

Fig. 4.3: Experimental surface frequency distribution at cross section 0.14 m (left) and 

0.54 m (right) away from the nozzle exit. 

experiments, and they are calculated using the relation 

√ 

ρ l − ρ g 

v = 1.74 gd (4.1) 

ρ g 

given by Stieß [197], where d is droplet diameter, ρ l and ρ g are the liquid and gas 

densities, respectively, and g is the acceleration due to gravity. This relation is the 

estimate of the terminal velocity of the droplets [197], and it is proven to give accurate 

value of the droplet velocity [191, 198]. 

The moment sets are calculated by means 

of these droplet radius, velocity and surface frequency, which are used as initial data 

for QMOM whereas for DQMOM these moments are in turn used to calculate the 

weights (representing surface frequencies), radii and velocities through the Wheeler 

algorithm [136] as explained in Chapter 3. These data (weights, radii and velocities) 

are then used as initial data to start the computations. 

Tab. 4.1 lists these initial 

values with three-node approximation for 80 kg/h and 150 kg/h water inflow rate. 

Tab. 4.1: Initial weights and abscissas 

Liquid flow rate [kg/h] Weights [(µm) −1 ] Radii [µm] Velocities [m/s] 

80 0.638 24.424 1.09 

0.276 86.432 1.94 

0.086 143.29 2.76 

150 0.733 21.739 1.03 

0.223 79.706 1.84 

0.044 128.350 2.72


4.1.3 Results and Discussion 

In this subsection, the simulation results of DQMOM in one dimension are presented, 

and compared with experimental data and with the simpler model QMOM. Computations 

are carried out considering different ambient gas temperatures, i.e. 293 K, 

as in experiments, and 313 K, and different inflow rates of liquid as in experiments 

in order to investigate the effect of evaporation and drag force along with gravity on 

droplet characteristics and spray dynamics. The gas inflow rate remains fixed in accordance 

with experiments. Although the surrounding gas velocity is fixed as 0.078 m/s 

in experiments, the simulations are performed using different velocities of surrounding 

gas in order to analyze the effect of drag force on the droplet dynamics. Also, the 

cases of spray with and without coalescence are compared to analyze the influence of 

coalescence on droplet distribution. 

First, the implementation of the droplet evaporation model is tested for a single pure 

water droplet. The numerical predictions and experimental results of water droplet 

mass and temperature are shown in Fig. 4.4. The experimental data are taken from 

Werner [199], and refer to a 6 µl water droplet evaporation in air at 40 ◦ C, 3.75% 

relative humidity (R.H.) and flow velocity of 0.3 m/s. The droplet mass continuously 

decreases due to water evaporation, and initially there is no significant increase in 

droplet temperature. When the droplet mass reduces to a negligible value (less than 

5% of its initial mass), temperature raises quickly to the gas temperature, and there is 

good agreement between the simulation and the experiment. 

6 

60 

Droplet mass [mg] 

5 

4 

3 

2 

Mass [Simulation] 

Mass [Experiment] 

T [Simulation] 

T [Experiment] 

50 

40 

30 

20 

Droplet temperature [°C] 

1 

10 

0 

0 100 200 300 400 500 600 700 

0 

800 

Time [s] 

Fig. 4.4: Comparison of simulated and measured [199] droplet mass and temperature 

profiles for the evaporation of a pure water droplet.


3E+08 

Number density [m 3 ] 

2.8E+08 

2.6E+08 

2.4E+08 

2.2E+08 

2E+08 

1.8E+08 

1.6E+08 

1.4E+08 

1.2E+08 

Inhomogeneous 

Homogeneous 

Experiment 

1E+08 

8E+07 

6E+07 

0.14 0.28 0.42 0.56 0.7 0.84 

Position [m] 

0 1 2 3 4 5 

Time [s] 

Fig. 4.5: Homogeneous and inhomogeneous calculations of DQMOM. 

Before carrying out simulations with an inhomogeneous system of DQMOM transport 

Eq. (3.19), homogenous formulations of these equations, given by Eq. (3.20), are 

simulated and the results are compared with inhomogeneous computations. Fig. 4.5 

shows the computed and experimental profiles of number density at different cross sections 

of the spray chamber with homogeneous and inhomogeneous system of DQMOM 

equations (see Eq. (3.19)) for 80 kg/h water spray in nitrogen flowing with 0.078 m/s 

velocity at 293 K. In the homogeneous results, the time axis of the model is matched 

to experimental position through the droplet velocity. The number density decreases 

along the spray axis due to droplet evaporation, and the predictions with inhomogeneous 

formulation captures the physics of the spray more accurately [198, 200]. Thus, 

the present work includes the numerical solutions of the inhomogeneous linear system, 

which are formed through application of DQMOM in one physical dimension (axial 

direction). 

Figure 4.6 displays the results of Sauter mean diameter showing the comparison of 

QMOM and DQMOM, where DQMOM results are shown for both lower and higher 

Reynolds number. Here, the liquid flow rate is 80 kg/h. It can be seen that the 

QMOM results strongly deviate from the experiment whereas DQMOM improves the 

results of QMOM significantly even for lower droplet Reynolds numbers, and with 

higher Reynolds number, which is the case in this simulations, the agreement between 

DQMOM and experiment is very good [191]. A general intuitive question could be 

”why the Sauter mean diameter increases in spite of evaporation in simulations as well 

as the experiment?”. The answer is elaborated with an example by considering the 

droplet size distribution of a water spray shown in Fig. 4.7, which displays the droplet


Sauter mean diameter [µm] 

140 

135 

130 

125 

120 

115 

110 

DQMOM (Re>1) 

DQMOM (Re


Tab. 4.2: Droplet size distribution of water spray 

d(t = 0 s) [µm] f [(µm) −1 ] d(t = 1 s) [µm] d(t = 50 s) [µm] ] 

10.643 0.409 0.000 0.000 

12.913 4.091 0.000 0.000 

15.588 11.434 0.000 0.000 

18.745 11.762 0.000 0.000 

22.468 8.472 0.000 0.000 

26.858 7.112 0.000 0.000 

32.037 7.093 5.138 0.000 

38.145 7.481 21.332 0.000 

45.350 7.304 32.506 0.000 

53.848 6.609 43.584 0.000 

63.870 5.439 55.492 0.000 

75.692 4.491 68.770 0.000 

89.635 4.073 83.872 0.000 

106.080 4.319 101.257 0.000 

125.478 4.574 121.427 0.000 

148.355 3.579 144.946 0.000 

175.337 1.490 172.462 0.000 

207.163 0.255 204.735 0.000 

244.697 0.015 242.646 99.381 

The k value is usually estimated from the material properties such as density, diffusivity, 

etc., and in general, it has a value in the range of 10 −7 to 10 −11 . Just for the sake of 

explanation, k is assumed to be 1.0E-09 m 2 /s. The change in the droplet diameter is 

computed at t = 1 s using d 2 law is given in Tab. 4.2, see the third column. Comparing 

the values of droplet diameter at t = 0 s and t = 1 s, it shows that the droplet diameter 

decreases and lower size droplets vanish. Using these data, the computed d 32 at t = 

1 s is 117.126 µm, which shows an increase from initial value. This increase continues 

till certain evaporation time (see last column in Tab. 4.2), whereupon the Sauter mean 

diameter starts to decrease because most of the smaller size droplets vanish and only 

few droplets have finite size. The Sauter mean diameter of this distribution decreases 

to 99.381 µm after 50 s of d 2 law evaporation rate (see the last column in Tab. 4.2). 

Figure 4.8 shows the plots of droplet velocities subjected to only drag force (left), 

and drag force with gravity (right) for three different sized droplets, respectively. In 

case of only drag caused by the surrounding gas with initial velocity of 0.078 m/s (experimental 

value), the velocity decreases at first due to drag force and later the droplets


3 

3 

Veloctiy [m/s] 

2.5 

2 

1.5 

1 

u[1] = 1.09 m/s, r[1] = 24 µm 

u[2] = 1.94 m/s, r[2] = 86 µm 

u[3] = 2.76 m/s, r[3] = 143 µm 

Velocity [m/s] 

2.5 

2 

1.5 

1 

u[1] = 1.09 m/s, r[1] = 24 µm 

u[2] = 1.94 m/s, r[2] = 86 µm 

u[3] = 2.76 m/s, r[3] = 143 µm 

0.5 

0.5 

0 

0 0.2 0.4 0.6 0.8 1 

Time [s] 

0 

0 1 2 3 4 5 

Time [s] 

Fig. 4.8: Velocity profiles of three droplets with different initial radii and velocities 

under the influence of drag alone (left) and drag and gravity (right). 

follow the streamlines of the gas after reaching a steady value, cf. left side of Fig. 4.8. 

On the other hand, when the droplets encounter gravity in addition to drag force applied 

by the surrounding gas, the droplet velocity initially decreases due to drag and 

then increases linearly due to gravity as seen in right part of Fig. 4.8. In a previous 

study [198, 200], it has been shown that the moderate droplet evaporation under the 

present conditions does not significantly influence droplet velocity. 

140 

130 

Experiment - 80 kg/h 

Evaporation - 80 kg/h 

Evaporation and Coalescence - 80 kg/h 

Experiment - 150 kg/h 

Evaporation - 150 kg/h 

Evaporation and Coalescence - 150 kg/h 


120 

110 

100 

90 

80 

70 

0.14 0.28 0.42 0.56 0.7 0.84 

Position [m] 

Fig. 4.9: Effect of liquid inflow rates on Sauter mean diameter computed with and 

without coalescence.


The variations in Sauter mean diameter with axial position of the spray for two different 

liquid inflow rates of 80 kg/h and 150 kg/h are shown in Fig. 4.9. The results for 

80 kg/h show an increasing Sauter mean diameter with evaporation. Inclusion of coalescence 

in addition to evaporation leads to excellent agreement between computational 

and experimental results. On the contrary, the computational results for 150 kg/h at 

x = 0.54 m seem to be deviating far away from the experimental data. The observed 

deviation is due to inconsistency in experimental data, which is evident from the fact 

that the experimental flow rate does not match the prescribed value of 150 kg/h at 

0.54 m. Therefore, the results from 80 kg/h will be discussed for the remaining part of 

this section. 

Figure 4.10 displays the profiles of the Sauter mean diameter (left) and mean droplet 

diameter (right) of water spray subjected to evaporation at 293 K and 313 K temperatures 

of surrounding gas as well as with and without coalescence. As expected, Sauter 

mean diameter increases substantially with evaporation that causes the decrease and 

eventual loss of small size droplets. Higher temperature imposes a rise in evaporation, 

which considerably accelerates the rate of increase of Sauter mean diameter. A comparison 

with experimental data reveals the importance of modeling the droplet coalescence, 

which not only improves the simulation results but also has excellent agreement with 

experiment (see left side of Fig. 4.10). 

Similar to Sauter mean diameter, the mean droplet diameter is an important physical 

quantity for several applications such as particle size analysis of powder sampling 

in food and pharmaceutical industries [201]. Mean droplet diameter of a number density 

based distribution can be computed using the Eq. (2.18). Since very small size 

160 

150 

Experiment - 293 K 

Evaporation - 293 K 

Evaporation and Coalescence - 293 K 



90 

85 







140 

130 

120 

110 

D 10 

[µm] 

80 

75 

70 

65 

60 

55 

100 

0.14 0.28 0.42 0.56 0.7 0.84 

Position [m] 

50 

0.14 0.28 0.42 0.56 0.7 0.84 

Position [m] 

Fig. 4.10: Profiles of Sauter mean diameter (left) and mean droplet diameter (right) 

computed with and without coalescence at surrounding gas temperatures of 

293 K and 313 K.


droplets may completely evaporate, leading to decreased total droplet number at any 

cross section, cf. Fig. 4.12, the mean value of droplet diameters increases. Therefore, 

as the droplets move and start to evaporate and vanish completely, the mean droplet 

diameter of the spray starts increasing at cross sections away from the nozzle although 

individual droplet diameters decrease (see right part of Fig. 4.10). This observation 

is in agreement with the behavior of the Sauter mean diameter shown in left side of 

Fig. 4.10. Coalescence causes an increase of droplet diameter as anticipated. 

Figure 4.11 shows the results of the droplet specific surface area. The specific 

surface area is an important parameter, which is used particularly to characterize 

powder materials, and it is defined as the ratio of total surface area of the individual 

droplets/particles to the total volume [201]. It can be seen that the specific surface 

area decreases as a result of evaporation, which leads to decrease in number density 

of droplets. This is evident from the behavior of specific surface area at a higher 

temperature. A comparison with experimental data confirms the role of coalescence in 

improving the results as observed in case of the Sauter mean diameter and the mean 

droplet diameter displayed in Fig. 4.10. 

Figure 4.12 shows the plots of total droplet number density in axial direction. Since 

the geometric configuration considered for the numerical solution is one-dimensional, 

the integral value of droplet number density over the corresponding cross sections 

is displayed. It can be seen that evaporation causes the droplet number density to 

decrease as the spray develops. This decrease is much pronounced at the higher temperature 

(313 K) due to enhanced evaporation. It is worthwhile to note that inclusion 

6.0E+04 






Specific surface area [m 2 /m 3 ] 

5.5E+04 

5.0E+04 

4.5E+04 

4.0E+04 

3.5E+04 

0.14 0.28 0.42 0.56 0.7 0.84 

Position [m] 

Fig. 4.11: Profiles of specific surface area computed with and without coalescence at 

surrounding gas temperatures of 293 K and 313 K.


3E+08 

2.5E+08 






Number density [m -3 ] 

2E+08 

1.5E+08 

1E+08 

5E+07 

0.14 0.28 0.42 0.56 0.7 0.84 

Position [m] 

Fig. 4.12: Profiles of droplet number density computed with and without coalescence 

at surrounding gas temperatures of 293 K and 313 K. 

of coalescence affects the calculation of droplet number density significantly as it can 

be inferred through comparison of the numerical results with experimental data. This 

may be understood by the fact that only coalescence is considered in the present work 

and processes of breakup, reflexive and stretching separation along with formation of 

satellite droplets is neglected in the present simulations, which leads to a lower droplet 

number density at any given position. This may be improved by including a more 

advanced droplet–droplet interaction model [184]. Moreover, in these computations 

the evaporating flux at zero droplet size is computed through the ratio constraints of 

weights, radii and velocities given by Eqs. (2.33)– (2.34), which are derived based on 

a smooth and continuous density function [60]. This approach is prone to errors and 

may be rectified by implementing an maximum entropy model [66] explained in Section 

2.3.4, which is done in the case of two-dimensional water and PVP/water in air 

spray flows. 

The successful implementation of DQMOM in studying the one-dimensional water 

spray flow in nitrogen and the good agreement with experimental data has led to the 

extension of DQMOM to two dimensions in order to model the evaporating water 

spray in air in two-dimensional configuration. The DQMOM extension is outlined in 

Section 2.3.4. The next section presents the results of two-dimensional water spray in 

air in axisymmetric configuration.


4.2 Two-dimensional Evaporating Water Spray in 

Air 

A water spray injected into air through a hollow cone Delavan SDX-SE-90 nozzle in 

a vertical spray chamber, is modeled by DQMOM and DDM. The one-dimensional 

transport equations of DQMOM [191] are extended to two-dimensional to model the 

spray flow in axisymmetric configuration [202, 203]. The starting data for the simulations 

are taken from experimental data, where the experiments are conducted by the 

group of Prof. G. Brenn at TU Graz, Austria. The experimental setup is explained in 

the next section. The generation of initial data is discussed in the following section. 

The simulation results of DQMOM are compared with the results of DDM and both 

these model results are validated with the experiment [202, 203]. 

4.2.1 Experimental Setup 

A series of experiments is carried out at TU Graz by the group of Prof. G. Brenn where 

a water spray in air is studied for different liquid mass inflow rates. The droplet sizes 

and velocities are recorded at various cross sections for different liquid inflow rates 

using phase Doppler anemometry (PDA) [204]. The present simulations concern the 

experimental data generated using a Delavan nozzle SDX-SE-90 having an internal 

diameter of 0.002 m, an outer diameter of 0.012 m at the nozzle throat and 0.016 m 

at the top, for liquid inflow rates of 80 kg/h and 120 kg/h. A water spray is injected 

into a cylindrical spray chamber of diameter 1 m. The carrier gas is air at room 

temperature and atmospheric pressure. Measurements are recorded at cross sections of 

0.08 m, 0.12 m and 0.16 m. Figure 4.13 illustrates the schematic of the experimental 

Fig. 4.13: Schematic diagram of the experimental setup.

4.2. Two-dimensional Evaporating Water Spray in Air 67 

setup. The data at 0.08 m are taken as starting point for initial data generation for 

computations, and results are compared at later cross sections [205]. 

4.2.2 Initial Data Generation 

The experimental data at the closest position to the nozzle is used to generate initial 

data for the numerical computations of DQMOM. The nearest experimental position 

is 0.08 m from the nozzle, where the measurements are available at radial positions 

separated by 1.5 × 10 −3 m distance. The PDA data at every radial position consists 

of droplet radius, velocities in axial and radial directions, and the time elapsed for 

each measurement, which gives the total time carried out over a period. These data 

are grouped into 100 droplet size classes [206]. The effective cross sectional area of 

the probe volume is computed, which is done to eliminate errors in measuring volume 

due to nonlinearity in phase/diameter relationship in large size droplets because of the 

nonuniform beam intensity [207]. The result of the calculation for a water flow rate of 

80 kg/h, at a position of 0.066 m from the center is shown in Fig. 4.14. The trajectory 

length exhibits strong fluctuations, and fluctuations increase with the droplet size. 

Furthermore, the number of droplets in the size classes for the larger diameters is 

typically much lower than in the smaller size classes. Therefore, the properties such 

as droplet trajectory lengths through the probe volume are statistically unreliable for 

drops with sizes greater than a certain threshold value [206, 207]. In particular, the 

decrease of the effective probe volume size with increasing droplet size such as from 

0.45 

Effective crosssection area [mm 2 ] 

0.4 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

x 

x 

x x x xx x x 

xxx 

x x x 

x x x x 

Experiment 

Corrected 

Trend line 

x 

x 

x x 

x 

x x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 

x x x 

x 

x 

x x 

x 

x x x x x 

x x 

0 

0 100 200 300 400 

Drop size class [µm] 

Fig. 4.14: Profile of effective cross-section area of the probe volume for measured droplet 

size.


200 µm as shown in Fig. 4.14 is invalid as the effective cross-sectional area should 

increase with droplet size [207]. The effective cross-section area is therefore calculated 

using a linear trend line from a threshold diameter. In the first step, the linear trend 

line is calculated using a linear regression scheme based on the data in the droplet size 

classes up to 60% of the maximum droplet size. 

In the second step, for all droplet size classes larger than 40% of the maximum 

droplet size class for this experimental position, the values of the effective cross sectional 

area are obtained as values of the linear trend line. Therefore, there is an 

overlap of the size class ranges used for computing the trend line and those whose 

probe volume cross-section areas are calculated using the trend line. Once the effective 

cross-sectional area probe volume is corrected, the number density is corrected 

correspondingly. Then, the moment sets of droplet size and velocities are computed, 

which in turn are used to calculate the initial weights (number densities), radii and 

velocities using the Wheeler algorithm [136]. In the present study, the spray distribution 

is approximated by a three-node closure, which is proven to be accurate in 

previous studies [48, 49, 191, 202]. The three-node approximation of NDF implies that 

the required number of moments is 12 (3 each: weights, droplet radii, axial velocities, 

radial velocities). The same procedure is followed at every radial position for the crosssection 

of 0.08 m. Figure 4.15 shows the experimental distribution of droplets and 

DQMOM approximation at 0.066 m from the center of the spray for 80 kg/h water 

flow rate. The problem of negative moments is handled by employing the adaptive 

Wheeler algorithm [208]. 

Number density [ (µm) 1 ] 

0.05 

0.04 

0.03 

0.02 

0.01 

Experiment 

DQMOM 

0 

0 50 100 150 

Droplet radius [ µm ] 

Fig. 4.15: Experimental and DQMOM approximation of droplet number density for a 

water spray.


The DDM simulation are carried out only for water spray in two-dimensional configuration 

by Humza [68]. In his work, the same experimental data is used to generate 

a system of parcels for DDM, where the properties of the k th parcel are denoted by 

(x k , r k , u k , v k , m k ) for the present two-dimensional configuration. The liquid mass of 

k th parcel is computed assuming the spherical symmetry of the droplets, i.e., 

m k = 

N∑ 

i=1 

4 

3 πρ lr 3 i , (4.3) 

where N refers to the number of droplets in the parcel. The number of parcels for the 

inflow rate of 80 kg/h is 3,704 and for 120 kg/h, it is 3,464. A non-equidistant rectangular 

grid with 7,878 grid points (78 in radial and 101 in axial direction, respectively) 

is used [68]. 


At first the implementation of maximum entropy (ME) method for the calculation of 

evaporative flux is studied. Fig. 4.16 shows the computed NDF at radial distance of 

64.5 mm from the center and 0.08 m axial distance from nozzle for 80 kg/h liquid flow 

rate using ME method and its comparison with experiment, where a good agreement 

between the ME approximated NDF and experiment can be found. The evaporative 

flux computed using the weight ratio constraints, which are defined by Eqs. (2.33)– 

(2.34), for this position is found to be 0.39, where as with ME method it is 0.022 and 

0.05 

0.04 

Flow rate: 80 kg/h 

Axial position: 80 mm 

Radial position: 64.5 mm 

ME reconstruction 

Experiment 

NDF [(µm) 1 ] 

0.03 

0.02 

0.01 

0 

0 25 50 75 100 125 150 175 

Droplet radius [µm] 

Fig. 4.16: Experimental and reconstructed NDF of 80 kg/h water spray at 64.5 mm 

from the center of the spray axis.


0.03 

Flow rate: 121 kg/h 

Axial position: 80 mm 

Radial position: 81 mm 

NDF [(µm) -1 ] 

0.02 

0.01 

ME reconstruction 

Experiment 

0 

0 25 50 75 100 125 150 175 


Fig. 4.17: Experimental and reconstructed NDF of 121 kg/h water spray at 81 mm from 

the center of the spray axis. 

in experiment the same is about 0.03 (see Fig. 4.16). Similar observation is made with 

higher liquid flow rate as well, Fig. 4.17 shows the experimental and reconstructed 

NDF for water spray of 121 kg/h at 81 mm radial position from the center and 0.08 m 

away from the nozzle. Thus, the ME approach improves the DQMOM evaporative flux 

calculation procedure and it has excellent agreement with the experiment. Therefore, 

in the current study, the ME method is used for the ψ calculation in two-dimensional 

evaporating water and PVP/water spray flows. 

In numerical simulation of water spray in two-dimensional configuration, average 

droplet properties such as mean droplet diameter, Sauter mean diameter and mean 

droplet velocity are computed using both the methods, i.e., DQMOM and DDM, and 

the simulation results are compared with the experiment at the cross sections of 0.12 m 

and 0.16 m away from the nozzle exit. Figure 4.18 shows the computed and experimental 

profiles of the Sauter mean diameter at cross sections of 0.12 m (left) and 

0.16 m (right) downstream to the nozzle orifice for 80 kg/h. The DDM simulation 

results match quite well with the experiment at the center of the spray at 0.12 m 

away from the nozzle exit, but slightly underpredicts towards the periphery of the 

spray whereas good agreement is observed at 0.16 m cross section between DDM and 

experiment. 

The DQMOM simulation results are in good agreement with experiment at 0.12 m 

downstream the nozzle exit, and it is closer to the experimental data at higher radial 

distance as well. Further downstream, at 0.16 m from the nozzle orifice (see right part 

of the Fig. 4.18), the DQMOM simulations reveal some scattering near the centerline,


180 

180 


160 

140 

120 

100 

80 

60 

40 

DQMOM 

DDM 

Experiment 


160 

140 

120 

100 

80 

60 

40 

DQMOM 

DDM 

Experiment 

20 

20 

0 

-100 -50 0 50 100 

Radial Position [mm] 

0 

-100 -50 0 50 100 

Radial position [mm] 

Fig. 4.18: Experimental and numerical profiles of the Sauter mean diameter of water 

spray with 80 kg/h liquid flow rate at the cross section of 0.12 m (left) and 

0.16 m (right) distance from the nozzle exit. 

while at higher radial distances, they underpredict the experimental results. This 

discrepancy may be the result of numerical scheme, which employs an explicit finite 

difference method to solve the transport equations of DQMOM; the results can be 

improved by implementing an implicit method. The post-processing of experimental 

data, which is explained in Subsection 4.2.2, may be the reason of the deviation, too. 

For an elevated liquid inflow rate of 120 kg/h, the computed and experimental 

180 

160 

DQMOM 

DDM 

Experiment 


140 

120 

100 

80 

60 

40 

20 

0 

-120 -80 -40 0 40 80 120 


Fig. 4.19: Experimental and numerical profiles of the Sauter mean diameter at the cross 

section of 0.12 m distance from the nozzle exit for 120 kg/h.


160 

140 

DQMOM 

DDM 

Experiment 


120 

100 

80 

60 

40 

20 

0 

-120 -80 -40 0 40 80 120 


Fig. 4.20: Experimental and numerical profiles of the Sauter mean diameter at the cross 

section of 0.16 m distance from the nozzle exit for 120 kg/h. 

profiles of Sauter mean diameter at cross sections of 0.12 m and 0.16 m away from 

the nozzle exit are shown in Figs. 4.19 and 4.20. An increased liquid flow rate causes 

a somewhat decreased droplet size: For a given liquid, increased mass flow rate leads 

to higher pressure drop in the atomizer, which decreases liquid sheet size and breakup 

length to yield smaller particles as can be seen when compared with Fig. 4.18. 

At the cross section of 0.12 m, it can be seen that DQMOM performs better than the 

DDM results as DDM overpredicts the experimental values. The scattering behavior 

of DQMOM simulation results near the centerline is observed in this case, too. As the 

droplets move to the next cross section, a decrease in large size droplets is evident, 

which is predicted by both DQMOM and DDM. The results show that the DQMOM 

shows better agreement with experiment, while DDM predicts somewhat higher values 

than the experiment at corresponding radial positions [205]. 

The overall shape of a hollow cone spray is captured quite nicely by both methods, 

although some deviations are observed, particularly in DQMOM as compared to 

experimental profile. This is possibly due to the post-processing of the experimental 

data as explained in Subsection 4.2.2, which is done to correct the number frequency 

at every measuring position to rule out the fluctuations in the effective cross sectional 

area of the measuring volume for the larger droplet sizes [207]. This correction of experimental 

data is position dependent, whereas DQMOM and DDM results account 

for these corrections for the initial condition but not at positions further downstream. 

Another reason for the discrepancies in the DQMOM results may be due to the fact 

that the spray equations are not yet fully coupled to the gas phase.


Comparing the maximum values of the Sauter mean diameter at the two cross 

sections displayed in Figs. 4.18, 4.19 and 4.20, a decrease in large size droplets is 

observed as the droplets move away from the nozzle. Even though the process of 

evaporation is considered in the present models, the major reason for the decrease in 

droplet size may be attributed to the influence of drag force applied by the surrounding 

gas, because significant evaporation may not occur at the present room temperature 

condition. This decrease is more evident in the large droplet size region, where the 

dynamic interaction of droplet with surrounding gas dominates, as observed in profiles 

of mean droplet velocity (see Figs. 4.23 and 4.24). 

Besides the Sauter mean diameter, in many technical applications such as particle 

size analysis of powder sampling in food and pharmaceutical industries, the mean 

droplet diameter is an important physical quantity [201]. Radial profiles of the mean 

droplet diameter compared with experiment are shown in Fig. 4.21 for 80 kg/h at 

0.12 m (left) and 0.16 m (right) distance away from nozzle. DDM results are in very 

good agreement with the experiment. A slight decrease in the mean droplet diameter 

is observed as the droplets move away from nozzle indicating some mass transfer from 

liquid to gas, which is attributable to gas–liquid interactions. The DQMOM results 

are in excellent agreement with experiment at the cross section of 0.12 m near the 

centerline, and DQMOM results show better agreement than DDM results (see left part 

of Fig. 4.21). At 75 mm radial position, the DQMOM results are below experimental 

values, which may stem from the explicit finite difference technique. At the cross 

section of 0.16 m, a good agreement is observed between DQMOM and experiment 

near the axis of symmetry, even though some scattering behavior is found (see right 

100 

100 

Mean droplet diameter [µm] 

80 

60 

40 

20 

DQMOM 

DDM 

Experiment 


80 

60 

40 

20 

DQMOM 

DDM 

Experiment 

0 

-100 -50 0 50 100 


0 

-100 -50 0 50 100 

Radial Position [mm] 

Fig. 4.21: Experimental and numerical profiles of the mean droplet diameter of water 


0.16 m (right) distance from the nozzle exit.


120 

DQMOM 

DDM 

Experiment 

80 

DQMOM 

DDM 

Experiment 

100 


80 

60 

40 


60 

40 

20 

20 

0 

-120 -80 -40 0 40 80 120 


0 

-120 -80 -40 0 40 80 120 


Fig. 4.22: Experimental and numerical profiles of the mean droplet diameter of water 


0.16 m (right) distance from the nozzle exit. 

side of Fig. 4.21). 

In Fig. 4.21, deviations from the experiment occur in the large droplet size region, 

which is due to the fact that the numerical technique captures the distribution function 

globally, and there could be some local discrepancies as well. This may be improved 

by solving the gas phase equations for DQMOM, which is not yet done in the present 

study, where the inlet gas flow properties are used to calculate the source terms for 

transport equations for DQMOM [209]. 

Figure 4.22 shows the computed and experimental profiles of the mean droplet 

diameter at cross sections of 0.12 m (left) and 0.16 m (right) away from the nozzle exit 

for liquid inflow rate of 120 kg/h. Similar to Sauter mean diameter, elevated liquid 

flow rate leads to somewhat decreased droplet size (compare Fig. 4.22 with 4.21) . At 

0.12 m away from the nozzle exit, both DDM and DQMOM agree well with each other 

near the centerline, where they show relatively higher values than the experiment. At 

the radial positions away from the centerline, DQMOM is in good agreement with the 

experiment, and it is better than the DDM results. As the droplets move away from 

the nozzle exit, a decrease in size can be observed at the cross section of 0.16 m away 

from the the nozzle exit (see right part of the Fig. 4.22), which is similar to the case 

of liquid flow rate of 80 kg/h. Near the centerline at 0.16 m away from the nozzle 

exit, both DQMOM and DDM show the same behavior and predict slightly higher 

values than experiment. At higher radial positions, DDM values are higher compared 

to DQMOM and experiment, whereas DQMOM coincides with the experimental data. 

In Figs. 4.23 and 4.24, the radial profiles of mean droplet velocity are displayed 

at different cross sections. It can be seen that the droplet velocity is higher for larger


Mean droplet velocity [m/s] 

6 

5 

4 

3 

DQMOM 

DDM 

Experiment 

2 

-100 -50 0 50 100 


Fig. 4.23: Experimental and numerical profiles of the mean droplet velocity at the cross 

section of 0.12 m distance from the nozzle exit. 

droplets as anticipated. Interestingly, the small size droplets near the axis of symmetry 

also move at a higher velocity as observed in the experiment and thus causing the 

velocity profile bimodal, which is predicted quite nicely by both models. 

A closer look reveals that the width of the jet is captured by the DQMOM, whereas 

the DDM predicts somewhat broader profiles with a lower maximum value at the centerline. 

At the spray edge, a judgement of the numerical methods is difficult, since the 


6 

5 

4 

3 

DQMOM 

DDM 

Experiment 

2 

-100 -50 0 50 100 


Fig. 4.24: Experimental and numerical profiles of the mean droplet velocity at the cross 

section of 0.16 m distance from the nozzle exit.


experimental data are somewhat spread at 0.12 m from the nozzle exit. At 0.16 m, the 

slopes of the numerical results deviate from the experimental data, particularly in large 

size droplets region where the effective cross sectional area shows strong fluctuations 

in experiment as shown in Fig. 4.14. This implies that the post-processing of experimental 

data plays an important role in the corrections of number density and thereby 

the droplet properties [207]. Comparing the velocity profiles at the two different cross 

sections, it is seen that the velocity decreases as droplets move away from the nozzle. 

This is because the droplets are strongly decelerated by the dynamic interaction with 

the surrounding gas. The gas around the spray stagnates and is driven into motion 

only due to the spray entrainment. The gas motion driven by the spray arises at the 

expense that the droplet loses momentum. 

The droplet properties are predicted quite well by the present simulations, which 

confirms their applicability for spray flows. There are some deviations between simulation 

and experimental results, which are attributable to the post-processing of the 

experimental data as mentioned before. In case of DDM, neglecting droplet–droplet 

interactions may need reconsideration. For DQMOM, the improved numerical scheme 

and the simultaneous solution of the gas phase equations may improve the simulation 

results. 

Based on these simulation results and comparison with the experiment, it can be 

concluded that the DQMOM is a robust method, which can predict the spray flows 

accurately. This led to the implementation of DQMOM to study bi-component evaporating 

spray, i.e., PVP/water spray flow in two dimensions. In order to perform 

simulations of PVP/water spray flows, the predictability and efficiency of developed bicomponent 

droplet evaporation and solid layer formation model (see Subsection 2.4.1.2) 

needs to be verified under different drying conditions. The next section presents the 

numerical simulation of single bi-component droplet evaporation and solid layer development, 

and comparison of simulation results with experiment. 

4.3 Single Bi-component Droplet Evaporation and 

Solid Layer Formation 

The model presented in Subsection 2.4.1.2 to predict the evaporation and solid layer 

formation for PVP/water droplet and mannitol/water droplet is simulated with different 

conditions such as initial solute mass fraction, gas temperature and velocity, 

relative humidity, initial droplet size etc. In the next subsections, the vapor-liquid 

equilibrium calculation followed by non-ideality effect caused by the solute (PVP or 

mannitol) presence on the droplet heating and evaporation rate is explained. Finally, 

the single droplet evaporation and solid layer development results are presented.

4.3. Single Bi-component Droplet Evaporation and Solid Layer Formation 77 

4.3.1 Vapor-Liquid Equilibrium 

The vapor-liquid equilibrium for the evaporating component i is needed for the calculation 

of Spalding’s mass transfer number, B M,1 , cf. Eq. (2.54), for both PVP/water 

and mannitol/water droplet evaporation and solid layer development cases, in which 

the mass fraction, Y 1,s , of the evaporating component appears; this mass fraction is 

calculated through the mole fraction, X i , i = 1, of the evaporating component water, 

in terms of the activity coefficient 

X i = p vap,i 

p m 

γ i X L,i , (4.4) 

where p vap,1 is the vapor pressure of pure water and p m is the total mixture pressure, 

which is equal to the ambient gas pressure, and in the present study it equals the 

atmospheric pressure. Here, X L,i is the mole fraction of evaporating component i in 

liquid phase, and γ i is the activity coefficient of evaporating component i, which is 

calculated through equation given as, 

γ i = a wY L,i 

X L,i 

. (4.5) 

Here a w is the water activity, Y L,i with i = 1, is the water mass fraction within the 

droplet. The calculation of water activity coefficient is described in next subsection. 

4.3.2 Non-ideal Liquid Mixture 

The presence of polymer or mannitol with water leads to non-ideal liquid behavior, 

which must be accounted for in calculating the mole fraction of water vapor at the 

droplet surface. In this work, the liquid mixture is treated as non-ideal by determining 

the influence of individual components on each other through their activity coefficients. 

The universal functional activity coefficient (UNIFAC) method is the accurate and 

most extensively used procedure [210], which estimates the activity coefficient as a 

sum of combinatorial and residual terms. This method, however, cannot be applied for 

polymer solutions as they have significant difference in accessible volume for a molecule 

in the solution [211]. 

The work of Oishi and Prausnitz [211] extended UNIFAC method to account for 

such differences in accessible volume by introducing a free-volume term, which enabled 

the UNIFAC approach to be applied to polymer solution systems. However, it is 

proven that their model fails for aqueous polymer systems because of the inadequacy 

of its free-volume term [212]. In the current study, the activity coefficient of water 

in PVP/water solution is computed using the UNIFAC-van der Waals-Free Volume 

method known as UNIFAC-vdW-FV method [212], which accounts for the free-volume


1 

1 

0.8 

0.8 

Water activity [] 

0.6 

0.4 

Temperature = 73.0°C 

Water activity [-] 

0.6 

0.4 

Temperature = 94.5°C 

0.2 

UNIFAC 

UNIFACvdWFV 

Experiment 

0.2 

UNIFAC 

UNIFAC-vdW-FV 

Experiment 

0 

0 0.05 0.1 0.15 0.2 0.25 

Water mass fraction [] 

0 

0 0.05 0.1 0.15 0.2 0.25 

Water mass fraction [-] 

Fig. 4.25: Numerical and experimental [214] results of water activity (a w ) in PVP/water 

solution at 73.0 ◦ C (left) and 94.5 ◦ C (right). 

effect in aqueous polymer solutions. In case of mannitol/water droplet evaporation 

study, the activity coefficient of water is calculated using the analytical solution of 

groups (ASOG) contribution method [213], as it is proven to perform better than the 

UNIFAC method [213]. 

Before implementation of the UNIFAC-vdW-FV method into the current PVP/water 

droplet code, it has been verified by comparing the water activity (a w ) computed using 

the UNIFAC method [210]. Results from these two methods are compared with 

1 

0.96 

T = 94.5°C 

T = 160°C 

Water activity [] 

0.92 

0.88 

0.84 

0 0.1 0.2 0.3 0.4 0.5 0.6 

Mannitol mass fraction [] 

Fig. 4.26: Numerical results of water activity (a w ) in mannitol/water solution at 94.5 ◦ C 

and 160 ◦ C.


experimental results [214]. Molecular properties data such as van der Waals volume 

and radii for PVP are taken from Bondi [215], Danner and High [216], and the interaction 

parameters of individual molecules required in the UNIFAC-vdW-FV method 

are taken from Daubert and Danner [217]. In Fig. 4.25, variation of the weight based 

water activity with water mass fraction in PVP/water solution is exemplarily shown 

at a temperature of 73.0 ◦ C (left) and 94.5 ◦ C (right), respectively. 

The results reveal that the UNIFAC-vdW-FV method improves the UNIFAC method 

results, and the UNIFAC-vdW-FV predictions are in excellent agreement with the experimental 

data. Therefore, in the current study, the UNIFAC-vdW-FV method is 

implemented to compute water activity in PVP/water solution. 

The change in water activity with mannitol mass fraction in mannitol/water solutions 

at a temperature of 94.5 ◦ C and at 160 ◦ C computed using ASOG method is 

displayed in Fig. 4.26. The results show that the water activity in mannitol/water solution 

decreases not only with increased mannitol mass fraction but also with increased 

liquid temperature. 

The effect of non-ideality through activity coefficient on the reduction of vapor 

pressure of water in PVP/water solution for different mass fractions of PVP dissolved in 

water at different temperatures is shown in Fig. 4.27. For the sake of comparison, ideal 

condition is also shown where the activity coefficient always remains at unity so that the 

vapor pressure is independent of solute mass fraction. It can be clearly observed that 

the liquid mixture strongly deviates from ideal behavior and the deviation increases 

with the increasing PVP mass fraction in water. 

Figure 4.28 shows the effect of non-ideality through PVP presence in PVP/water 

1 

0.8 

Ideal 

Y PVP 

= 0.2 (nonideal) 

Y PVP 

= 0.3 (nonideal) 

1 

0.8 

T = 50°C [P s 

= 0.12 bar] 

T = 100°C [P s 

= 1 bar] 

Vapor pressure [bar] 

0.6 

0.4 

Vapor pressure [bar] 

0.6 

0.4 

0.2 

0.2 

0 

0 20 40 60 80 100 

Temperature [K] 

0 

0 0.1 0.2 0.3 0.4 

Water mass fraction [] 

Fig. 4.27: Effect of non-ideality on the vapor 

pressure of water at different 

temperatures. 

Fig. 4.28: Variation of vapor pressure of 

water with water mass fraction 

in PVP/water solution.


solution on the water vapor pressure at 50 ◦ C and 100 ◦ C of liquid temperature. The 

vapor pressure of pure water at 50 ◦ C is about 0.12 bar whereas at 100 ◦ C it is 1 bar. 

An increase in water mass fraction increases the vapor pressure and it equals the pure 

water pressure when the water mass fraction is above 0.4. Thus, it infers that the 

role of water activity coefficient is important when the water mass fraction within the 

droplet falls below 0.4 at 50 ◦ C and 100 ◦ C, which occurs in the present simulations. 


The simulation of evaporation and solid layer development of single droplet containing 

PVP or mannitol in water is carried out under various drying conditions such as 

surrounding gas temperature, gas velocity, and relative humidity to investigate their 

effect on drying characteristics. The effect of the initial solute (PVP or mannitol) mass 

fraction on the final particle characteristics is also studied. The droplet is assumed to 

be spherical during the entire evaporation and drying process. The simulations are 

also carried out with rapid mixing model (RMM), which is a simple model based on 

the assumptions that the liquid mixture inside the droplet is always homogeneous and 

infinity conductivity within the droplet thus the droplet is at uniform temperature at 

every time. The governing equations of RMM are presented in Subsection 2.4.1.2. 

The thermal properties of PVP and mannitol are taken from Dakroury et al. [220], 

and mass diffusivity of PVP in water is obtained from Metaxiotou and Nychas [221], 

whereas the mass diffusivity of mannitol in water is taken from Grigoriev and Mey- 

0.7 

0.6 

Solute mass fraction [] 

0.5 

0.4 

0.3 

PVP 

Mannitol 

0.2 

0.1 

0 20 40 60 80 100 

Temperature [°C] 

Fig. 4.29: Experimental data of PVP [218] and mannitol [219] saturation solubility in 

water.


Tab. 4.3: Experimental drying conditions 

Drying condition 

Values 

Initial solute mass fraction 0.075, 0.05 and 0.15 

Initial droplet radius 70 µm 

Initial droplet temperature 

Gas temperature 

Gas velocity 

20 and 70 ◦ C 

60, 67, 95, 100, 160 and 210 ◦ C 

0.05, 0.65 and 10 m/s 

Relative humidity (R.H.) 0.5 1.0, 2.0 and 30% 

likhov [222]. The critical temperature and pressure of PVP and mannitol are taken 

from Daubert and Danner [217]. The vapor diffusion coefficient through the solid layer 

of PVP or mannitol, D s , and solid thermal conductivity, k s are not available in literature, 

therefore, they are computed similar to the work of Nesic and Vodnik [151]. The 

physical and thermal properties in the film are estimated at the reference composition 

using the 1/3 rule [223]. The PVP/water and mannitol/water solution physical and 

thermal properties are computed with the standard rules of mixing. The variation 

of saturation solubility of PVP in water and mannitol in water with temperature is 

taken from measurements [218, 219], and it is shown in Fig. 4.29. The solid layer at 

the droplet surface is presumed to develop when the PVP mass fraction at the droplet 

surface reaches 20% above its saturation solubility limit, and in the case of mannitol, 

it is assumed that the crust and solid layer formation begins when the mannitol mass 

fraction reaches 0.9, which is much higher than the saturation solubility, in order to 

avoid re-dissolution of solid layer with increased temperature as it shows large variation 

of solubility with temperature, see Fig. 4.29. 

The numerical results presented refer to a droplet of initial radius 70 µm at 20 ◦ C 

containing 0.15 PVP or mannitol initial mass fraction subjected to air with 0.5% relative 

humidity (R.H.) flowing at 0.65 m/s with 100 ◦ C initial gas temperature [172]. 

The various drying conditions for numerical simulations taken from the experimental 

study of Littringer et al. [21] and Sedelmayer et al. [224], are listed in Tab. 4.3 and 

numerical results are compared with available experimental data [21, 224]. 

Figure 4.30 shows the change in mannitol/water droplet mass and temperature 

with time for the above conditions and for increased initial gas velocity (U g = 10 m/s). 

Initially, there is no significant increase in droplet temperature, and droplet mass reduces 

due to continuous water evaporation. After an initial heating period, the droplet 

temperature rises very quickly indicating the formation of solid layer whereupon the 

rate of evaporation is reduced due to added resistance coming from solid layer, which is 

reflected in the droplet mass profile. The higher gas flow rate increases convection and


Droplet mass [µg] 

1.6 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

Mass [ U g 

= 0.65 m/s ] 

Mass [ U g 

= 10 m/s ] 

T [ U g 

= 0.65 m/s ] 

T [ U g 

= 10 m/s ] 

120 

100 

80 

60 

40 


0.2 

0 0.5 1 1.5 2 

20 

2.5 

Time [s] 

Fig. 4.30: Effect of gas velocity on the evolution of mass and temperature of a mannitol/water 

droplet. 

thereby the water evaporation, hence there is quicker development of the solid layer. 

The solid layer forms in about 1.7 s with U g = 0.65 m/s, whereas with U g = 10 m/s, the 

solid layer forms in about 0.75 s. A closer look reveals that there is higher droplet mass 

at any given time after solid layer formation when compared with lower gas velocity 

situation, which means that increased gas velocity would lead to larger particle and 

the porosity, defined as the ratio of the volume occupied by water at the instance of 

1 

0.9 

0.8 

U g 

= 0.65 m/s 

T g 

= 67°C 

T g 

= 100°C 

T g 

= 160°C 

(d/d 0 

) 2 [-] 

0.7 

0.6 

0.5 

0.4 

0.3 

0 0.5 1 1.5 2 2.5 3 3.5 

Time [s] 

Fig. 4.31: Effect of elevated gas temperature on the surface area of a mannitol/water 

droplet.


solid layer formation over that of the whole particle volume, would be higher in case 

of increased gas velocity (see Fig. 4.33). 

Figure 4.31 shows the effect of initial gas temperature on the temporal change of 

the dimensionless surface area of a mannitol/water droplet. Elevated gas temperature 

leads to higher energy transfer from the gas to the droplet, and thereby, an increase in 

the rate of droplet evaporation and drying. The surface area continuously decreases due 

to water evaporation until the beginning of solid layer formation whereupon particle 

size remains constant, which is reflected in Fig. 4.31. 

The higher the gas temperature the quicker the time taken for the solid layer formation: 

In case of 67 ◦ C the solid layer develops in about 2.9 s and with 100 ◦ C the solid 

layer forms in 1.7 s, whereas with 160 ◦ C, the same is observed in about 0.9 s. There 

is larger surface area at the time of solid layer formation with higher gas temperature, 

which means that elevated gas temperature would give larger particles towards the end 

of the drying process (see Fig. 4.33). 

The effect of gas temperature on the development of mannitol mass fraction profiles 

inside the droplet of initial radius 70 µm subjected to dry air with 0.5% R.H., flowing at 

0.65 m/s with temperatures of 67, 100 and 160 ◦ C is shown in Fig. 4.32 at 0.5 s (left) 

and at 0.9 s (right), respectively. Initially, the droplet interior has a homogenous 

mannitol mass fraction distribution of 0.15 (not shown here) and with time, there is 

development of mannitol mass fraction gradients inside the droplet, and the droplet 

size reduces due to continuous water evaporation. For 100 ◦ C initial gas temperature, 

the droplet radius is 62 µm at 0.5 s whereas at 0.9 s it reduces to 56 µm, which is 

seen in Fig. 4.32, respectively. The increased initial gas temperature yields higher mass 

fraction gradients inside the droplet mainly due to the decreased activity coefficient of 

0.35 

Time = 0.5 s 

0.8 


0.3 

0.25 

0.2 

T g 

= 67°C 

T g 

= 100°C 

T g 

= 160°C 


0.7 

0.6 

0.5 

0.4 

0.3 

Time = 0.9 s 

T g 

= 67°C 

T g 

= 100°C 

T g 

= 160°C 

0.15 

0 10 20 30 40 50 60 70 

Radial position inside the droplet [µm] 

0.2 

0 10 20 30 40 50 60 70 


Fig. 4.32: Effect of gas temperature on the temporal development of mannitol mass 

fraction inside the droplet at 0.5 s (left) and 0.9 s (right).


Porosity [-] 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

Porosity [ U g 

= 0.65 m/s ] 

Porosity [ U g 

= 10 m/s ] 

Radius [ U g 

= 0.65 m/s ] 

Radius [ U g 

= 10 m/s ] 

50 

49 

48 

47 

46 

45 

44 

Particle radius [µm] 

0.2 

43 

0.1 

42 

60 90 120 150 180 210 240 

Gas temperature [°C] 

Fig. 4.33: Effect of gas velocity and temperature on porosity and final particle size of 

mannitol/water droplet. 

water in mannitol (see Fig. 4.26) and enhanced heat transfer. The activity coefficient 

decreases not only with increase in temperature but also with mannitol mass fraction, 

which is quite clearly seen at later times, i.e., 0.9 s shown in right part of Fig. 4.32. 

The effect of elevated gas velocity and temperature on the final particle porosity 

and particle radius is shown in Fig. 4.33. The porosity increases with higher gas temperature 

and velocity because of quicker solid layer formation, thereby yielding larger 

particles. The computed porosity of mannitol particle with 160 ◦ C gas temperature 

and 0.65 m/s gas velocity is 0.39 and the corresponding value in experiment is found 

to be 0.41 [21]. The final particle radius is reported as 42 µm in experiments [21], 

which can be compared to the corresponding computed value of 44.8 µm, showing a 

very good agreement. 

In experiments [21], it is reported that increased gas temperature leads to less 

porous particle with shriveled or non-spherical shape as seen in Fig. 4.34, which shows 

the the scanning electron microscope (SEM) images of the mannitol samples spray dried 

under different drying temperatures. For the outlet temperature of 70 ◦ C, a spherical 

mannitol particle is obtained, see Fig. 4.34(a). For the higher outlet temperature 

of 100 ◦ C the particle shape changes from spherical to a ’raisin like’ structure, cf. 

Fig. 4.34(b), that may occur due to inflation of a drying shell and this transition is 

observed at 90 ◦ C, see Fig. 4.34(c). Higher temperatures lead to faster evaporation 

of the water, leading to less time to form a stable structure on the droplets surface. 

From the high porosity in combination with cuts of spray dried mannitol particles in 

previous studies [21], the formation of a particle with an outer shell is evident [225].


Fig. 4.34: SEM images of mannitol samples spray dried at 70 ◦ C (a), 100 ◦ C (b) and 

90 ◦ C (c). Zoomed images of the surface structures of these particles at 

70 ◦ C (d), 100 ◦ C (e) and 90 ◦ C (f) [225]. 

This shell formation is in good accordance with the simulations performed so far. 

The particle surface consists of small, needle shaped structures in case of low drying 

temperatures as shown in Fig. 4.34(d), and smaller, non-needle shaped structures for 

higher drying temperatures, cf. Fig. 4.34(e), and the shift from needle shape to nonneedle 

structures is seen in Fig. 4.34(f). The increased gas temperature not only effects 

the final particle shape but also internal structure [225]. In computations, the change 

in particle shape is not accounted for, and it is assumed to be spherical throughout the 

evaporation and drying period, therefore, the present numerical results show increase 

in porosity with temperature as anticipated, see Fig. 4.33. This behavior will change 

when the final drying step is added to the present model, and if non-spherical particle 

formation will be considered. 

Figure 4.35 shows the effect of gas temperatures of 60 ◦ C and 95 ◦ C and relative 

humidity of 1% R.H. (left) and 30% R.H. (right), respectively, on the droplet surface 

area and comparison with experimental data. The experiments are carried out by 

Sedelmayer et al. [224] at the University of Hamburg in an acoustic levitator. The 

simulation results show excellent agreement with the experiment. The droplet surface 

area continuously decreases due to water evaporation until a critical value where the 

solid layer formation starts, which is quite nicely predicted by the simulation. Increased 

temperature increases the evaporation rate and thereby quicker solid layer formation 

as seen in left part of Fig. 4.35, whereas increased humidity increases the solid layer 

formation time, i.e., at 60 ◦ C at 1% R.H. the solid layer forms in about 65 s and with


1 

0.8 

R.H. = 1% 

Simulation [ T g 

= 60°C ] 

Experiment [ T g 

= 60°C ] 


= 95°C ] 


= 95°C ] 

1 

0.8 

R.H. = 30% 


= 60°C ] 


= 60°C ] 


= 95°C ] 


= 95°C ] 

(d/330 µm) 2 [-] 

0.6 

0.4 

(d/360 µm) 2 [-] 

0.6 

0.4 

0.2 

0.2 

0 

0 10 20 30 40 50 60 70 80 

Time [s] 

0 

0 50 100 150 200 250 

Time [s] 

Fig. 4.35: Effect of gas temperatures of 60 ◦ C and 95 ◦ C and relative humidity of 

1% R.H. (left) and 30% R.H. (right) on the droplet surface area. 

30% R.H. the same observed in about 205 s, see right part of Fig. 4.35. The profiles of 

the normalized droplet surface, (d/d 0 ) 2 , shown in the Fig. 4.35, reveal that the droplet 

evaporation rate prior to solid layer formation in the present case deviates from the 

linear decrease with time as would be expected from the classical d 2 law, where a 

constant evaporation constant is assumed. 

These experiments are carried out with different initial droplet radius for every 

experiment, and Tab. 4.4 gives the initial droplet radii (R 0 ) and particle size at the 

time of solid layer formation (t s ) in every experiment and its corresponding computed 

value from simulation. 

The comparison between rapid mixing model (RMM) and the present model is given 

in Fig. 4.36, which shows the time evolution of mannitol/water droplet surface area for 

initial droplet radius of 70 µm at 20 ◦ C temperature and subjected to hot air of 160 ◦ C 

with 0.5% R.H. and flowing at 0.65 m/s. Even though there is little difference between 

RMM and the present approach during the initial time period, however, in the later 

Tab. 4.4: Experiment vs simulation 

T g R.H. R 0 Particle radius at t s [µm] 

[ ◦ C] [%] [µm] Simulation Experiment 

60 1.0 330 145.1 147.9 

30.0 360 155.2 155.1 

95 1.0 215 95.0 109.4 

30.0 280 122.5 121.4


1 

0.9 

0.8 

T g 

= 160°C, U g 

= 0.65 m/s 

RMM 

Present model 

(d/d 0 

) 2 [] 

0.7 

0.6 

0.5 

0.4 

0.3 

0 0.2 0.4 0.6 0.8 1 1.2 

Time [s] 

Fig. 4.36: Time evolution of mannitol/water droplet surface area computed by present 

model and RMM. 

time period RMM overpredicts the decrease in droplet surface and thereby the time of 

the solid layer formation caused by the fact that the assumption of homogeneous liquid 

mixture within the droplet. This assumption leads to more water to be evaporated, 

which increases the solute mass fraction to the critical value so that the formation of 

solid layer begins. 

The effect of initial droplet temperatures of 20 ◦ C and 70 ◦ C on the evaporation 

1.6 

T g 

= 160°C 

120 


1.4 

1.2 

1 

0.8 

0.6 

0.4 

Mass [ T 0 

= 20°C ] 

T [ T 0 

= 20°C ] 

Mass [ T 0 

= 70°C ] 

T [ T 0 

= 70°C ] 

100 

80 

60 

40 


0.2 

0 0.2 0.4 0.6 0.8 1 

20 

1.2 

Time [s] 

Fig. 4.37: Effect of initial droplet temperature on the evaporation rate of mannitol/water 

droplet.


Solid layer thickness [µm] 

45 

40 

35 

30 

25 

20 

15 

10 

U g 

= 0.65 m/s 

T g 

= 100°C 

T g 

= 160°C 

5 

0 

0.5 0.75 1 1.25 1.5 1.75 2 2.25 

Time [s] 

Fig. 4.38: Effect of gas temperature on solid layer thickness inside the PVP/water 

droplet. 

rate is shown in Fig. 4.37. In both the cases, the initial droplet radius is 70 µm, and 

it is subjected to hot air flowing at 0.65 m/s with 160 ◦ C. The droplet with 20 ◦ C 

initial temperature quickly raises to an equilibrium temperature, which is most often 

equal to the wet bulb temperature, whereupon no significant rise in temperature is 

found. Whereas with 70 ◦ C, the wet bulb temperature for the gas temperature of 

160 ◦ C and 0.5% R.H., is lower than the initial droplet temperature (70 ◦ C), so the 

droplet temperature decreases until it equals the wet bulb temperature, and remains 

almost constant in further development. Similarly the droplet evaporation rate is 

higher in this initial period, and it is reflected in the reduction of droplet mass as seen 

in Fig. 4.37. In the later time period, the final particle temperature is same, and it is 

equal to 105 ◦ C. 

Similar trends are observed for PVP/water evaporation and solid layer formation. 

The effect of elevated gas temperature on the temporal development of solid layer 

thickness in PVP/water droplet is shown in Fig. 4.38 for the same conditions that are 

studied for mannitol/water. Increased gas temperature of T g = 160 ◦ C leads to higher 

energy transfer and earlier molecular entanglements of PVP and solid layer formation, 

with 100 ◦ C the solid layer forms in about 1.4 s whereas with 160 ◦ C, the same is 

observed in 0.7 s, see Fig. 4.38. 

Comparison of PVP/water droplet evaporation and solid layer formation with that 

of mannitol/water under the same drying conditions reveals that the solid layer forms 

quicker in case of PVP/water (in about 1.5 s with 100 ◦ C, see Fig. 4.38) than mannitol/water 

(about 1.7 s with 100 ◦ C, see Fig. 4.30). This is due to the fact that the


1.5 

Mass [ T g 

= 100°C ] 

Mass [ T g 

= 160°C ] 

T [ T g 

= 100°C ] 

T [ T g 

= 160°C ] 

100 


1.25 

1 

0.75 

0.5 

80 

60 

40 


0.25 

0 0.5 1 1.5 2 

20 

2.5 

Time [s] 

Fig. 4.39: Effect of gas temperature on the droplet mass and temperature. 

required solute mass fraction for initiation of the solid layer formation is less in case of 

PVP (about 0.78 at 100 ◦ C, see Fig. 4.29) compared to mannitol, which is fixed to 0.9. 

Figure 4.39 shows the effect of gas temperature on the temporal evolution of 

PVP/water droplet mass and temperature when the droplet is subjected to 100 ◦ C 

and 160 ◦ C gas temperatures. Elevated temperature leads to higher energy transfer 

from the gas to the droplet, and thereby, an increase in the rate of droplet evaporation 

and drying, which is reflected in Fig. 4.39. The higher the gas temperature the quicker 

the time taken to see molecular entanglement leading to solid layer formation: in case 

of 160 ◦ C, the solid layer develops in about 0.7 s whereas with 100 ◦ C , the same is 

observed in about 1.5 s, which is in agreement with Fig. 4.38. This means that an 

increase in gas temperature would give larger particles towards the end of the drying 

process. 

Figure 4.40 shows the temporal development of PVP mass fraction profiles inside 

the PVP/water droplet of initial radius 70 µm subjected to hot air flowing at 0.65 m/s 

with 100 ◦ C temperature and no humidity, i.e., dry air (left) and with 5% R.H. (right), 

respectively. Initially, the droplet has a homogenous PVP mass fraction of 0.15 and 

with time the droplet size decreases, and there is development of PVP mass fraction 

gradients inside the droplet due to continuous water evaporation, which can be seen at 

later times in both the figures. The PVP mass fraction at the droplet surface reaches 

the value of 0.78 in about 1.4 s with dry air, as seen left side of Fig. 4.40, which is 

equivalent to 20% above the saturation solubility whereas the same is achieved after 

1.8 s with 5% R.H., see right part of Fig. 4.40. This indicates that the increase in 

humidity prolongs the drying period because of the reduced driving force for water


0.8 

0.8 

Time 

PVP mass fraction [] 

0.7 

0.6 

0.5 

0.4 

0.3 

Time 

0 s 

0.5 s 

1.0 s 

1.4 s 

PVP mass fraction [] 

0.7 

0.6 

0.5 

0.4 

0.3 

0 s 

0.5 s 

1.0 s 

1.5 s 

1.8 s 

0.2 

0.2 

0.1 

0 10 20 30 40 50 60 70 


0.1 

0 10 20 30 40 50 60 70 


Fig. 4.40: Temporal development of PVP mass fraction profiles inside the droplet subjected 

to dry air (left) and hot air with 5% R.H (right). 

evaporation, cf. Eq. (2.54). It is also observed that there is a lower PVP mass fraction 

gradient within the droplet for 5% R.H., cf. Fig. 4.40, when compared with dry air 

before solid layer develops, implying that humidity leads to smaller size particles with 

less porosity. Thus, it appears that the relative humidity plays a major role in the 

mass fraction gradients development within the droplet. 

Figure 4.41 shows the comparison of present model predictions of PVP/water 

droplet surface and that of RMM. The behavior is similar to the revelations made in 

1 

0.9 

T g 

= 100°C, U g 

= 0.65 m/s 

RMM 

Present model 

0.8 

(d/d 0 

) 2 [] 

0.7 

0.6 

0.5 

0.4 

0.3 

0 0.3 0.6 0.9 1.2 1.5 1.8 

Time [s] 

Fig. 4.41: Time evolution of PVP/water droplet surface area predicted by present model 

and RMM.


Evaporated water mass [µg] 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

T g 

= 100°C, U g 

= 0.65 m/s 

R.H. = 0.5% 

R.H. = 2.0% 

0 

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 

Time [s] 

Fig. 4.42: Effect of relative humidity on the water evaporation rate from the mannitol/water 

droplet. 

mannitol/water droplet case, i.e., the RMM overpredicts the decrease in droplet surface 

area, and thereby the time required for solid layer formation due to the assumption 

of homogeneous liquid mixture within the droplet, which delays the formation of the 

solid layer. 

Figure 4.42 shows the effect of relative humidity on the water evaporation rate from 

a mannitol/water droplet subjected to air with 0.5% and 2.0% R.H. For low relative 

humidity, the mass fraction of water vapor in the bulk of the air, Y 1,∞ , cf. Eq. (2.54), 

is decreased, leading to a higher driving mass transfer rate, and this would eventually 

cause faster water evaporation, and thereby somewhat quicker solid layer formation. 

With 0.5% R.H. the solid layer develops in about 1.7 s whereas with 2% R.H., the same 

is observed in 1.9 s. 

Figure 4.43 shows the effect of modification of initial PVP mass fraction on the evolution 

of droplet radius and temperature for 0.075 and 0.15 PVP initial mass fractions. 

All other conditions remain fixed. Less initial PVP mass fraction implies that there 

is more water to evaporate leading to smaller size particle with longer drying time. 

With an initial PVP mass fraction of 0.15, the droplet radius reduces to 46.4 µm in 

about 1.4 s whereas with 0.075 PVP initial mass fraction, the droplet radius decreases 

to 38.5 µm in about 1.7 s before the solid layer formation begins, which is indicated 

by the quick rise in droplet temperature reaching the same value in both cases as 

seen in Fig. 4.43, showing that initial mass fraction of PVP does not affect the final 

temperature of the particle. 

Though the final drying step is not yet added to this model, the results presented


70 

65 

Radius [ Y PVP 

= 0.075 ] 

Radius [ Y PVP 

= 0.15 ] 

T [ Y PVP 

= 0.075 ] 

T [ Y PVP 

= 0.15 ] 

100 


60 

55 

50 

45 

80 

60 

40 


40 

35 

0 0.5 1 1.5 2 

20 

2.5 

Time [s] 

Fig. 4.43: Effect of initial PVP mass fraction on the profiles of droplet radius and temperature. 

here for PVP/water and mannitol/water single droplet evaporation and solid layer formation 

is very promising. The first three stages of bi-component droplet evaporation 

and drying, i.e., till the solid layer development on the droplet surface is effectively 

predicted by present model and the comparison of the numerical results with the experiment 

exhibits very good agreement. Thus, this model is included in DQMOM 

mathematical formulation in order to simulate bi-component PVP/water spray flows 

in an axisymmetric, two-dimensional configuration and the results of this system are 

presented in the next section. 

4.4 Two-dimensional Evaporating PVP/Water 

Spray in Air 

This section presents the numerical and experimental results of bi-component evaporating 

spray flows. Though the developed model can be applied to simulate PVP/water 

and mannitol/water spray flows, but here only the results of PVP/water spray flows 

are presented as the initial data with respect to mannitol/water spray is not available. 

The experimental setup and initial data generation to start numerical simulations are 

presented in the next subsection followed by the results and discussion.

4.4. Two-dimensional Evaporating PVP/Water Spray in Air 93 

Fig. 4.44: Photograph of the PVP/water spray formation with 112 kg/h liquid inflow 

rate in experiment [206]. 

4.4.1 Experiment and Initial Data Generation 

The PVP/water spray in air experiments have been carried out by the group of 

Prof. G. Brenn by spraying a solution of 20% PVP and 80% water (by mass) through 

the Delavan nozzle SDX-SE-90 at room temperature. The droplet sizes and velocities 

are recorded at the cross sections of 0.08, 0.12 and 0.16 m away from nozzle orifice 

using PDA, which provides both droplet size and velocity distributions, similar to the 

water spray in air measurements. The liquid mass flow rate of these experiments is 

112 kg/h and other conditions of the experiment such as gas velocity, gas temperature 

and pressure are same as the water spray in air. Figure 4.44 displays the PVP/water 

spray formation in experiment [206]. To generate the initial data for simulating the 

PVP/water spray in air, the same procedure as outlined in two-dimensional water spray 

in air is followed here. Figure 4.45 shows the experimental droplet size distribution 

and the corresponding DQMOM approximation at the radial position 0.036 m from 

the spray axis and 0.08 m downstream of the nozzle orifice. 


PVP/water spray flow in air is modeled using the DQMOM where the bi-component 

droplet evaporation of PVP/water droplets [172, 225, 226] are accounted through the 

single droplet evaporation and solid formation model presented in Chapter 2 and the 

results are discussed in Subsection 4.3.3. For droplet motion, droplet coalescence, the


0.15 

0.12 

Experiment 

DQMOM 

NDF [(µm) 1 ] 

0.09 

0.06 

0.03 

0 

0 20 40 60 80 100 120 


Fig. 4.45: Experimental and DQMOM approximation of droplet number density for 

PVP/water spray. 

same sub-models as employed in water spray are applied here, see Subsection 2.4. 

In Fig. 4.46, computed and experimental profiles of Sauter mean diameter (left) and 

mean droplet diameter (right) of PVP/water spray for a mass inflow rate of 112 kg/h 

at 0.12 m away from the nozzle exit are shown. Similar to the water spray, the spray 

distribution assumes a hollow-cone shape, and it is nicely predicted by DQMOM. In 

both the figures, a closer look reveals that across all the radial positions, the DQMOM 


140 

120 

100 

80 

60 

40 

DQMOM 

Experiment 


80 

70 

60 

50 

40 

30 

20 

10 

DQMOM 

Experiment 

20 

-75 -60 -45 -30 -15 0 15 30 45 60 75 


0 

-75 -60 -45 -30 -15 0 15 30 45 60 75 


Fig. 4.46: Experimental and numerical profiles of the Sauter mean diameter (left) and 

mean droplet diameter (right) of PVP/water spray in air at the cross section 

of 0.12 m distance from the nozzle exit.



140 

120 

100 

80 

60 

40 

DQMOM 

Experiment 


100 

80 

60 

40 

20 

DQMOM 

Experiment 

20 

-100 -80 -60 -40 -20 0 20 40 60 80 100 


0 

-100 -80 -60 -40 -20 0 20 40 60 80 100 


Fig. 4.47: Experimental and numerical profiles of the Sauter mean diameter (left) and 

mean droplet diameter (right) of PVP/water spray in air at the cross section 

of 0.16 m distance from the nozzle exit. 

underpredicts the experimental results and towards the periphery of the spray some 

deviation is observed particularly in profiles of Sauter mean diameter as compared to 

the experiment. This can possibly be explained by the fact that the DQMOM predicts 

the global droplet distribution, but there could be local discrepancies induced by the 

gas phase, which is not resolved in the present study. Coupling of DQMOM with the 

gas phase would eventually improve the simulation results. 

Figure 4.47 displays the Sauter mean diameter (left) and mean droplet diameter 

(right) at further downstream the nozzle exit, i.e., at the cross section 0.16 m. 

Comparing the maxima in Fig. 4.46 and 4.47 reveals that there is an increase in the 

Sauter mean diameter and mean droplet diameter, which is converse to the the water 

spray where decrease in droplet size is found. At a given temperature, the evaporation 

rate of water from pure water droplets is higher than from the droplets containing PVP 

dissolved in water due to the non-ideality effect (see Fig. 4.25). An analysis of droplet 

coalescence reveals that it occurs 1.5 times more often in PVP/water spray compared 

to water spray, which also contributes to an increased droplet size in the PVP/water 

spray. The elevated viscosity of PVP leading to higher viscous PVP/water droplets 

compared to pure water droplets influences the droplet coalescence. The present model 

is suitable to capture these effects, and a good agreement between the experiment and 

simulation is found [209]. 

The mean droplet velocity of PVP/water spray with 112 kg/h liquid inflow rate at 

0.12 m away from the nozzle is shown in Fig. 4.48. Increased liquid flow rate leads to 

higher droplet velocity (compare Figs. 4.23 and 4.48), which increases the chances of


23 

21 

DQMOM 

Experiment 


19 

17 

15 

13 

11 

9 

7 

-75 -60 -45 -30 -15 0 15 30 45 60 75 


Fig. 4.48: Experimental and numerical profiles of the mean droplet velocity of 

PVP/water spray in air at the cross section of 0.12 m distance from the 

nozzle exit. 

collision. The smaller size droplets that lie closer to centerline of the spray are moving 

at higher velocity than the larger size droplets, which is in quite contrast with that of 

water spray (see Fig. 4.23) where both the larger and smaller size droplets move with 

higher velocity. This may be because initially the gas around the spray is stagnant 

and the droplets decelerate by aerodynamic drag. The surrounding gas acquires the 

momentum lost by the droplets, and this creates a flow field in which gas is continually 

entrained into the spray. As the entrained gas enters the spray, it drags small liquid 

drops at the outer regions of the spray inward, and the momentum lost by the droplets 

at the periphery of the spray is larger than the ones that lie closer to the axis of 

symmetry, which explains the smaller velocity of larger droplets [227, 228]. 

Further downstream of the nozzle exit, i.e., at the cross section of 0.16 m away 

from the nozzle exit, the retardation of the droplet velocity in large size droplet region 

is observed (see Fig. 4.49) similar to water spray as this effect is dependent on initial 

liquid flow rate, where low liquid flow rate leads to larger droplets, which take more 

time to follow the streamlines of the gas than the smaller size droplets [209, 225]. 

The simulation results are in good agreement with the experiment, particularly in 

smaller size droplets region whereas towards spray edge there is deviation, which can 

be attributed to the post-processing of the experimental data and the non-resolved gas 

phase. 

Concerning the differences in evaporation characteristics for water and PVP/water 

droplet evaporation in air, it is found that for a given liquid flow rate and axial po-


23 DQMOM 

Experiment 

21 


19 

17 

15 

13 

11 

9 

7 

-80 -60 -40 -20 0 20 40 60 80 


Fig. 4.49: Experimental and numerical profiles of the mean droplet velocity of 

PVP/water spray in air at the cross section of 0.16 m distance from the 

nozzle exit. 

sition from the nozzle exit, droplet size is larger in water spray (120 kg/h) than the 

PVP/water spray (112 kg/h), compare Fig. 4.19 with left side of Fig. 4.46, which is 

because of the high viscosity of PVP/water solution. Moreover, at a given temperature, 

the evaporation rate from pure water droplet is higher than from PVP/water droplet 

due to the non-ideality effect caused by polymer presence.

98 4. Results and Discussion

5. Conclusions and Future Work 

The objective of the present work is modeling and simulation of polymer or sugar solution 

spray drying until the solid layer formation at the droplet surface, and dispersion 

in bi-component evaporating spray flows in an Eulerian framework. 

In order to understand the behavior of droplet distribution under various drying 

conditions, the direct quadrature method of moments (DQMOM) is implemented, for 

the first time, in two dimensions to study the bi-component evaporating spray flows. In 

DQMOM, the droplet size and velocity distribution of the spray is modeled by approximating 

the number density function in terms of joint radius and velocity. The DQMOM 

has been extended to accommodate gas–liquid interactions such as convective droplet 

evaporation, drag force and gravity as well as droplet–droplet interactions by including 

coalescence. The effect of these physical processes on the evolution of droplet size 

distribution and kinetic properties is analyzed and validated with the experiments. 

The DQMOM simulation results are also compared with the quadrature method of 

moments (QMOM) in one-dimensional configuration whereas in two-dimensional axisymmetric 

configuration DQMOM is compared with discrete droplet model (DDM), 

which is a well known Euler – Lagrangian approach. 

First, evaporating water spray in nitrogen is modeled using DQMOM in one physical 

dimension, and the simulation results are compared with QMOM. The water evaporation 

is accounted through convective evaporation model of Abramzon and Sirignano, 

which accounts for variable liquid and film properties. The drag and droplet coalescence 

are included through appropriate sub-models. The gas phase is not yet fully coupled 

with DQMOM but its inlet flow properties are used to compute droplet evaporation 

and drag. The initial data to start simulations is generated from the experimental data, 

which were provided by Dr. R. Wengeler, BASF Ludwigshafen. The simulation results 

are validated with experiment at various cross sections. The influence of individual 

physical processes is analyzed. It is demonstrated that the model reflects the evaporation 

to have a pronounced effect on the parameters pertaining to droplet size. More 

importantly, when evaporation is considered in combination with droplet coalescence, 

the numerical results are improved significantly and show excellent agreement with 

experiments. The droplet velocity is largely influenced by the drag force and gravity. 

Based on the successful of implementation of DQMOM, it is then extended to model 

evaporating water in air in two-dimensional, axisymmetric configuration. The same

100 5. Conclusions and Future Work 

system is also modeled using DDM. In DQMOM, the source terms are computed same 

as done in the one-dimensional case, and the evolution of droplet size and velocity distributions 

are analyzed with both DDM and DQMOM. Droplet collisions are included 

in DQMOM by modeling the droplet coalescence. The DDM does not include droplet 

collisions due to computational complexity such as redistribution of droplet classes 

and increased computational effort. For initialization and validation of the simulation 

results, experimental data is used, which was provided by Prof. G. Brenn TU Graz, 

measured using PDA. The experimental data contains droplet size and velocity in axial 

and radial direction and this data is post-processed in order to eliminate errors in the 

large size droplets region. The experimental data at the cross section closest to the 

nozzle exit are used for the generation of initial conditions for the simulations, and the 

numerical results of DQMOM are compared with experimental data at the later cross 

sections, and with DDM. 

Overall, both the methods i.e., DQMOM and DDM show good agreement with 

the experiment. Some deviations between DQMOM and experiment are observed that 

might result from the present DQMOM formulation, which is not yet fully coupled 

with the gas phase equations. Concerning the experimental data, a post-processing of 

the raw data has been performed in order to correct the number density of large size 

droplets with respect to the effective cross section area, leading to different correction 

factors for different axial positions in experimental data away from the nozzle exit, 

which may also lead to discrepancy between numerical and experimental results. The 

DDM performs somewhat better in the periphery of the spray and DQMOM near the 

centerline. However, DQMOM shows an excellent numerical performance, and droplet 

coalescence is included with relative ease compared to DDM. Therefore, the DQMOM 

is further extended to simulate PVP/water sprays in air. 

Before simulating the PVP/water spray in air using DQMOM, a model to describe 

the bi-component droplet evaporation and solid layer formation is developed. The 

system under consideration is governed by the continuity (diffusion) and energy equations. 

Brenn’s model is modified to include the resistance from the solid layer, and 

this extended formulation is used to compute the evaporation rate of water from the 

bi-component droplet. The temperature inside the droplet appears to be uniform, and 

the change in droplet temperature due to heat exchange between the droplet surface 

and the surrounding gas is calculated with similar modifications used for mass evaporation 

rate to account for the resistance from the solid layer. The variable physical 

and thermal properties and the volume fraction based radius are introduced based on 

Brenn’s model. The predictability and efficiency of the developed single droplet model 

is first verified by simulating PVP/water and mannitol/water droplets. The liquid 

mixture is treated as non-ideal with the activity coefficient calculation using the im-

101 

proved UNIFAC-vdW-FV method for PVP in water and ASOG contribution method 

for mannitol/water solution. 

The effect of various drying conditions on the evolution of single droplet characteristics 

is analyzed and the results are compared with experimental data. These drying 

conditions include the effect of gas temperature, gas velocity, initial solute (PVP or 

mannitol) mass fraction and relative humidity, which are found to have significant 

effect on the evaporation and drying characteristics of PVP/water as well as mannitol/water 

droplet. The study reveals that an increase in gas velocity and temperature 

cause earlier formation of the solid layer and faster drying, leading to larger particles 

with higher porosity. Humidity in air leads to smaller size particles with less porosity. 

The lower initial solute (PVP or mannitol) mass fraction implies that there is more 

water to evaporate resulting in smaller size particle with longer drying time. The variation 

of the activity coefficient of water with PVP mass fraction is much higher than 

that of mannitol, causing a stronger retardation of water evaporation rate from the 

PVP/water droplet compared to mannitol/water, which results in a faster solid layer 

formation for mannitol/water droplets. The present model successfully predicts the 

first three stages of droplet evaporation and drying, i.e., until solid layer formation at 

the droplet surface, which can readily be incorporated into an overall model of spray 

drying. 

The developed bi-component single droplet evaporation and drying model is then 

included in the DQMOM to simulate evaporating PVP/water spray flows in air in 

two-dimensional, axisymmetric configuration. Even though the developed spray model 

is equally applicable to simulate PVP/water and mannitol/water spray flows but only 

computations of PVP/water spray flows are performed as mannitol/water spray flow 

experimental data is not available. The physical processes of the spray such as drag 

and coalescence are included through the appropriate sub-models. Numerical results 

are compared with experimental data at different cross sections, and results are found 

to be in good agreement with experiment. Some deviations between DQMOM and 

experiment in case of mean droplet diameter are observed that might have originated 

from the present DQMOM formulation, which is not yet fully coupled with the gas 

phase equations. Moreover, the employed numerical technique uses an explicit finite 

difference method to solve the DQMOM transport equations – an implicit scheme may 

lead to considerable improvement. Additionally, the Schiller–Naumann correlation for 

drag coefficient may need revision. In conclusion, mono- and bi-component evaporating 

spray flows and preliminary stages of spray drying are successfully modeled and 

simulated using DQMOM. 

During spray drying, elevated gas temperature enhances droplet heating and evaporation 

thus advanced formulation of DQMOM is required in order to model spray

102 5. Conclusions and Future Work 

drying. Even though the current DQMOM method includes the droplet radius and 

velocity as internal variables, inclusion of droplet temperature and analogous terms 

in the Williams’ spray equation will enable modeling of spray drying. The gas phase 

equations should be resolved and coupled to the DQMOM transport equations, which 

would eventually enable the model to predict the spray drying process, i.e., from the 

atomized liquid droplets to the dried end product. 

Concerning the bi-component single droplet evaporation and drying, the developed 

model needs further extension to account for the capillary force, internal circulation, 

shriveling effect or disorientation of particle shape towards the final stages of drying as 

observed in experiment. This would further improve the model predictability of solid 

layer formation, porosity within the solid layer and final particle size.

Appendix

A. Nomenclature

106 A. Nomenclature 

Symbol Unit Description 

a w - Water activity 

a α , b α - Variables in Wheeler algorithm 

a n 

Source term in DQMOM 

A(t) m 2 Time dependent droplet surface area 

B M - Spalding mass transfer number 

B M,i - Spalding mass transfer number of coefficient i 

B T - Spalding heat transfer number 

Bi - Biot number 

b n 


c m/s Speed of the sound in gas medium 

C D - Drag coefficient in spray model 

C pg J/(kg K) Specific heat capacity of gas 

C pL J/(kg K) Specific heat capacity of liquid 

C pLf J/(kg K) Specific heat capacity in the film 

c n 


c 1,n , c 2,n , c 3,n 

Source terms in DQMOM 

d m Droplet diameter 

D 12 m 2 /s Binary diffusion coefficient of liquid mixture 

d 1,0 , d 10 m Mean droplet diameter 

d 3,2 , d 32 m Sauter mean diameter 

d k+1,k m General definition of mean diameter 

D f m 2 /s Mass diffusion coefficient in the film 

D s m 2 /s Mass diffusion coefficient of vapor in solid layer 

E c - Efficiency of the droplet coalescence 

F m/s 2 Total force per unit mass on droplets 

F h m/s 2 History term or Basset force per unit mass 

F L m/s 2 Lift force per unit mass 

f 

Droplet distribution function 

f m −3 Number density function 

fM ME(x) 

m−3 Maximum entropy method approximation of number density 

function 

g m/s 2 Acceleration due to gravity 

G 

Moment flux term defined in finite volume method 

h W/(m 2 K) Convective heat transfer coefficient 

h J/kg Total specific enthalpy 

H - Hessian matrix 

H[f] 

Shannon entropy

107 

J c q J/(m 2 s) Heat flux due to thermal conductivity 

J d q J/(m 2 s) Heat flux due to molecular mass diffusion 

k m 2 /s 2 Turbulent kinetic energy 

k m 2 /s Evaporation constant in d 2 law 

k gf W/(m K) Thermal conductivity in the film 

k l W/(m K) Thermal conductivity of the liquid 

k s W/(m K) Thermal conductivity of the solid layer 

L V (T s ) J/kg Latent heat of vaporization at T s 

Le - Lewis number 

m kg Droplet mass 

m p,k m Droplet mass in k th parcel 

ṁ kg/s Total evaporation rate 

M 

Ma - Mach number 

M k 

Moment set defined in QMOM 

k th moment 

M w kg/mol Water molecular weight 

¯M kg/mol Mean molecular weight in film 

n d (d) m −3 Number density function based on the droplet diameter d 

N m −3 Total number density 

N - Number of nodes in DQMOM 

Ñu - Modified Nusselt number 

Nu - Nusselt number 

Oh - Ohnesorge number 

p Pa Pressure 

P k,l 

Phase-space transform defined in DQMOM 

p m Pa Total pressure in the film 

p vap,i Pa Vapor pressure of component i 

Pr - Prandtl number 

Q L J/s Net heat transferred to the droplet 

Q f 

r m Radial coordinate 

Rate of change in f due to droplet coalescence 

r n m Approximated droplet radius of n th node in DQMOM 

r p,k m Droplet radius in k th parcel 

Re - Reynolds number 

Re d - Droplet Reynolds number 

R 0 m Initial droplet radius 

R i m Volume fraction based droplet radius 

R m Droplet radius 

R = dr 

dt 

m/s Rate of change in droplet radius due to evaporation


S n 

S g 

S l 

Vector of source terms 

Source term due to gas phase 

Source term due to liquid phase 

S α kg/s Chemical production rate of species α in mass 

Sc - Schmidt number 

˜Sh - Modified Sherwood number 

Sh - Sherwood number 

T K Droplet temperature 

T g K Gas temperature 

T s K Droplet surface temperature 

T ∞ K Temperature in the bulk of the gas 

T p,k K Droplet temperature in k th parcel 

t s Time 

t s s Time taken for initiation of solid layer 

u m/s Gas velocity in physical space 

V m/s Gas velocity in sample space 

v m/s Droplet velocity in physical space 

U g m/s Gas velocity 

u x m/s Axial component of gas velocity 

u r m/s Radial component of gas velocity 

v x m/s Axial component of droplet velocity 

v r m/s Radial component of droplet velocity 

V m 3 Droplet volume 

V i m 3 Volume of component i within the droplet 

We g - Gas Weber number 

〈 

x 

k 〉 Approximated k th moment 

X i - Mole fraction of species i 

X L,i - Mole fraction of species i in the liquid 

x m Geometrical coordinates 

Y i - Mass fraction of component i inside the droplet 

Y s - Mass fraction of vapor at the droplet surface 

Y ∞ 

Mass fraction of vapor in the bulk of the gas 

Y i,s - Mass fraction of species i at the droplet surface 

Y i,∞ - Mass fraction of species i in the bulk of the gas 

γ i - Activity coefficient of component i 

Γ f 

Droplet coalescence function 

Γ h kg/(m s) Thermal diffusion coefficient 

Γ h,eff kg/(m s) Effective thermal diffusion coefficient 

Γ k,eff kg/(m s) Effective exchange coefficient for k

109 

Γ ɛ,eff kg/(m s) Effective exchange coefficient for ɛ 

Γ M,eff kg/(m s) Effective mean mass diffusion coefficient of the mixture 

∆t s Time step 

∆x m Spatial step 

δ k0 - Kronecker delta 

ɛ m 2 /s 3 Dissipation rate of turbulent kinetic energy 

λ W/(m K) Thermal conductivity 

µ kg/(m s) Dynamic viscosity 

µ eff kg/(m s) Effective viscosity coefficient 

µ f kg/(m s) Dynamic viscosity in the film 

µ l kg/(m s) Laminar viscosity coefficient 

µ t kg/(m s) Turbulent viscosity coefficient 

ξ - Set of internal coordinate 

π α - Variable in Wheeler algorithm 

ρ kg/m 3 Mass density 

ρ g kg/m 3 Gas density 

ρ l kg/m 3 Liquid density 

σ α,β 

Variable defined in Wheeler algorithm 

σ ɛ - Effective Schmidt number for ɛ 

φ - Relates the Spalding mass and heat transfer coefficients 

χ s −1 Dissipation rate of mixture fraction 

ψ - Evaporative flux 

Ω g s −1 Gas vorticity 

Subscripts and Superscripts 

Symbol Quantity 

d Droplet 

f Film 

g Gas 

l Liquid 

m Mixture 

n Index for the number of node 

p Parcel 

s Surface 

w Water 

〈〉 Moment


List of Abbreviations 

Abbreviation 

Meaning 

CDM 

CFD 

CFL 

CFM 

CM 

CQMOM 

DDB 

DDM 

DNS 

DQMOM 

DSD 

ETAB 

LB 

LDA 

LES 

LHF 

ME 

MOM 

NDF 

PBE 

PD 

PDA 

PSIC 

PVP 

QBSM 

QMOM 

RANS 

RMM 

RTI 

SF 

TAB 

VOF 

WB 

continuous droplet model 

computational fluid dynamics 

Courant-Friedrichs-Lewy condition 

continuous formulation model 

method of classes 

conditional quadrature method of moments 

droplet deformation and breakup 

discrete droplet model 

direct numerical simulation 

direct quadrature method of moments 

droplet size distribution 

enhanced Taylor analogy breakup model 

lattice-Boltzmann method 

laser Doppler anemometry 

large eddy simulation 

local homogeneous flow model 

maximum entropy method 

method of moments 

number density function 

population balance equation 

product-difference algorithm 

phase Doppler anemometry 

particle-source-in-cell method 

polyvinylpyrrolidone 

quadrature based sectional method 

quadrature method of moments 

Reynolds-average Navier – Stokes equations 

rapid mixing model 

Rayleigh-Taylor instability 

separated flow model 

Taylor analogy breakup model 

volume of fluid method 

wave breakup model

B. Acknowledgements 

The research work was carried out at Interdisciplinary Center for Scientific Computing 

(IWR), University of Heidelberg, and this study is funded by German Science 

Foundation (DFG) through the priority program ”SPP1423”. 

First and foremost I would to like to express my deep sense of gratitude to my 

supervisor, Prof. E. Gutheil, who constantly motivated, taught, encouraged, supported 

and patiently guided me throughout this work. She offered me this work while I was 

about to finish my Master studies, and I thank her for bestowing enormous confidence 

on me. She perseveringly corrected the research abstracts, papers, this thesis, and her 

style of correction given me an opportunity to be familiar with scientific writing. She 

helped me a lot in day to day activities as well. Her endless kindness and incomparable 

support cannot be thanked adequately here. 

I am thankful to my co-supervisor, PD Dr. N. Dahmen, for his guidance and support 

in my research, and for correcting this thesis. 

I would like to thank E. Wimmer, Prof. G. Brenn (TU Graz), Dr. R. Wengeler (BASF 

Ludwigshafen) for providing the experimental data. Special thanks to Prof. G. Brenn 

for extending his support and comments in understanding and correction of experimental 

data. Special thanks to Prof. N. Urbanetz for the fruitful discussions on mannitol/water 

system. I would like thank all the members of the DFG ”SPP 1423” for 

their support. 

I express my sincere thanks to all my colleagues, L. Cao, H. Großhans, R. M. Humza, 

Y. Hu, H. Olguin, M. Trunk, X. G. Cui, D. Urzica and E. Vogel. Thanks to N. Wenzel 

for his help in experimental data handling. Special thanks to H. Großhans for proof 

reading this thesis and translating the abstract in German. I sincerely thank E. Vogel 

for her kind help in all administrative matters, which made my life easier in Germany, 

and thanks to her also for the corrections of German abstract. 

I thank my family members, especially, my mother (Sulochana), beloved brother 

(Surender), and sister (Swarupa) for their constant support and endless love. I wish 

to express special thanks to my wife (Tanuja), who encouraged me in every hard hour, 

and corrected this thesis with humongous patience. 

Last but not least, the financial support of DFG through ”SPP1423” and HGS- 

MathComp is gratefully acknowledged.

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