Modeling of Hydrogen-Peroxide
Films for Aseptic Processing Applications
A Parameter Study on Hydrogen-Peroxide Exposure
to Internal Surfaces of mm-wide Slits
Master’s thesis in Innovative and Sustainable Chemical Engineering

JESPER ERIKSSON

Department of Mechanics and Maritime Sciences
CHALMERS UNIVERSITY OF TECHNOLOGY
Gothenburg, Sweden 2019





Master’s thesis 2019:26

Modeling of Hydrogen-Peroxide Films
for Aseptic Processing Applications

A Parameter Study on Hydrogen-Peroxide Exposure
to Internal Surfaces of mm-wide Slits

JESPER ERIKSSON

Department of Mechanics and Maritime Sciences
Division of Fluid Dynamics

Chalmers University of Technology
Gothenburg, Sweden 2019



Modeling of Hydrogen-Peroxide Films for Aseptic Processing Applica-
tions
A Parameter Study on Hydrogen-Peroxide Exposure to Internal Surfaces of mm-
wide Slits
JESPER ERIKSSON

© JESPER ERIKSSON, 2019.

Supervisors: Per Abrahamsson (ÅF), Mårten Regnér (Tetra Pak) and Prof. Srdjan
Sasic (Chalmers)
Examiner: Srdjan Sasic, Mechanics and Maritime Science

Master’s Thesis 2019:26
Department of Mechanics and Maritime Science
Division of Fluid Dynamics
Chalmers University of Technology
SE-412 96 Gothenburg
Telephone +46 (0)31-772 1000

Cover: Fluid dynamic simulation of a food packaging machine.
The Tetra Pak® A3-speed Machine. Retrieved 03-06-2019 from:
https://www.tetrapak.com/packaging/tetra-pak-a3speed

Typeset in LATEX
Printed by Chalmers Reproservice
Gothenburg, Sweden 2019

iv



Modeling of Hydrogen-Peroxide Films for Aseptic Processing Applications
A Parameter Study on Hydrogen-Peroxide Exposure to Internal Surfaces of mm-
wide Slits
JESPER ERIKSSON
Department of Mechanics and Maritime Sciences
Division of Fluid Dynamics
Chalmers University of Technology

Abstract
Disinfection methods are used within many fields to keep equipment and environ-
ments free from harmful biological agents such as fungi, bacteria, viruses and spore
forms. Within the food and drug industry, aseptic processing refers to a process
chain in which sterilized products are filled into sterilized containers. Aseptic pro-
cessing brings benefits such as improved quality and increased shelf life of a product,
which in turn leads to less waste due to the expiration of products. The benefits of
aseptic processing introduce a demand for the development of effective disinfection
methods.

A common chemical disinfection method is the usage of hydrogen-peroxide to kill
bacteria. One way to perform hydrogen-peroxide disinfection is to spray a vapor
containing air, water and hydrogen-peroxide which is allowed to condense on cold
surfaces forming a liquid film of water and hydrogen-peroxide which conducts the
desired bacteria killing. It is important that all surfaces have been exposed to a
hydrogen-peroxide film to ensure a rapid disinfection. This project aims to inves-
tigate the process of hydrogen-peroxide film condensation to evaluate the degree
of liquid film coverage and gas hydrogen-peroxide exposure within mm-wide slits.
The investigation was carried out using Computational Fluid Dynamics (CFD). The
process of hydrogen-peroxide gas disinfection is well understood, but the process of
liquid film formation and dynamics is more complex and many challenges remain
within this field. In particular, there are no previous studies investigating film for-
mation within small slits.

A parameter study was performed to evaluate how the characteristic width of a slit
h and the external gas flow velocity v influence the degree of liquid film coverage
on the inside surfaces of a slit. The results suggest that the critical value of h as a
function of v to ensure full surface film coverage follow hcrit(v) = 6.44v−0.2838−1.84,
which is valid within the range v = [0.5, 2.0]. Expressed in terms of the dimension-
less mass transfer Péclet number Peh, the criteria for full surface film coverage is
according to the results Peh > 97. The results remove the need of having a high
spatial resolution around mm-narrow spaces. Instead the developed macro-scale
model can be used to predict the degree of hydrogen-peroxide exposure within the
slit only with the knowledge of gas velocity external to the slit.

Keywords: hydrogen-peroxide, aseptic, disinfection, film, condensation, model,
dropwise, CFD, slit.

v





Acknowledgements
First of all I am thankful to the department of Technical Analysis, Fluid Dynamics,
at ÅF for hosting me so well during the course of my Master’s Thesis. I’m par-
ticularly thankful to all colleagues on the department for always being so helpful
and patient when answering my questions. Special thanks to my ÅF supervisor Per
Abrahamsson for always being there to discuss relevant issues and direction of the
project.

I am very grateful to Professor Srdjan Sasic at the Department of Applied Mechan-
ics and Maritime Science, Division of Fluid Mechanics, for being my examiner and
supervisor from Chalmers. It has contributed with a lot of motivation to have such
an enthusiastic and skilled Professor as a lecturer during the course of Multiphase
Flows and during my Master’s Thesis project.

A big thanks to Tetra Pak for having me as a Master’s Thesis worker. Thank you
for letting me work independently and for your guidance both regarding the thesis
work and practically at your site in Lund. In particular, thank you Mårten Regnér
for being my Tetra Pak supervisor.

To my family who have supported me all the way through my study period. To ev-
eryone I have met along the way during this five year period of studies at Chalmers
University, some of them who will be friends for life. For all the memorable mo-
ments and group study sessions in the Chemistry building. For all afternoon trips
to Gunillas Baguetter. A big thank you. A special thank you to my dear friend
Alejandro Gracia Tabuenca for your support, friendship and letting me practice the
thesis presentation for you.

Thank you to CD-adapco for providing me with STAR-CCM+ power-on-demand
licence hours.

Jesper Eriksson, Gothenburg, June 2019

vii





Contents

List of Figures xi

List of Tables xiii

1 Introduction 1
1.1 Aseptic Processing within Food Packaging Industry . . . . . . . . . . 1
1.2 Physical Description of Thin Liquid Films and Available Models . . . 2
1.3 Problem Description and Aim . . . . . . . . . . . . . . . . . . . . . . 3

2 Theory 5
2.1 General Equations of CFD . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Gas Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Assumptions of Gas Modeling . . . . . . . . . . . . . . . . . . 6
2.2.2 Governing Gas Equations . . . . . . . . . . . . . . . . . . . . 6
2.2.3 Degree of Saturation . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.4 Concentration Boundary Layer . . . . . . . . . . . . . . . . . 8
2.2.5 Non-Dimensional Analysis of Gas Slit Flow . . . . . . . . . . . 9

2.3 Film Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Thin Film Modeling . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Assumptions of Liquid Film Modeling . . . . . . . . . . . . . 12
2.3.3 Governing Film Equations . . . . . . . . . . . . . . . . . . . . 12
2.3.4 Fluid Film Modeling in STAR-CCM+ . . . . . . . . . . . . . 15

2.3.4.1 Film Motion . . . . . . . . . . . . . . . . . . . . . . 15
2.3.4.2 Surface Tension . . . . . . . . . . . . . . . . . . . . . 15
2.3.4.3 Evaporation and Condensation . . . . . . . . . . . . 16

2.4 Solid Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Wettability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.1.1 Filmwise Condensation: The Nusselt Model . . . . . 19
2.4.1.2 Dropwise Condensation . . . . . . . . . . . . . . . . 19

2.4.2 Topography of a Solid Surface . . . . . . . . . . . . . . . . . . 20

3 Methods 21
3.1 Geometrical Representation of a Slit . . . . . . . . . . . . . . . . . . 21
3.2 Meshing of the Slit Representation . . . . . . . . . . . . . . . . . . . 22
3.3 Simulations of Film Build-up and Film Drying . . . . . . . . . . . . . 22

3.3.1 Assumptions Employed for the Simulations . . . . . . . . . . . 24
3.3.2 Judging Time-Step Convergence During Simulations . . . . . . 24

ix



Contents

3.3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.4 Description of the Various Simulation Cases . . . . . . . . . . 25

3.4 Design Manager . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Film Surface Coverage . . . . . . . . . . . . . . . . . . . . . . 26
3.5.2 Péclet Number Setup . . . . . . . . . . . . . . . . . . . . . . . 27
3.5.3 Time-Extrapolation of Slit Film Coverage . . . . . . . . . . . 27

3.6 Description of the Experimental Validation Methods . . . . . . . . . 28

4 Results and Discussion 29
4.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Sensitivity Analysis of Total Simulation Time and Mesh Configuration 30

4.2.1 Time Evolution of the Film . . . . . . . . . . . . . . . . . . . 30
4.2.1.1 Extrapolation of Film Coverage for Increased Accu-

racy - Validation . . . . . . . . . . . . . . . . . . . . 32
4.2.2 Grid-Independence . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Flow Characteristics Inside and in Vicinity of Slits . . . . . . . . . . . 33
4.4 Final Surface Film Coverage Fraction . . . . . . . . . . . . . . . . . . 36

4.4.1 Critical Parameter Conditions for Complete Slit Film Coverage 37
4.5 Slit Surface Exposure to Gas H2O2 . . . . . . . . . . . . . . . . . . . 40
4.6 Improvements and Suggestions for Future Studies . . . . . . . . . . . 41

5 Conclusion 43

References 45

A Appendix 1 I

x



List of Figures

2.1 Different behaviours of a droplet on solid surfaces. The equilibrium
contact angle θ increases with increasing hydrophobicity of the solid
material. The contact angle θa ≈ 135◦ corresponds to a hydropho-
bic solid material, while θs ≈ 20◦ corresponds to a hydrophilic solid
material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Figure illustrating the geometry produced in ANSA. The inlet and
outlet surfaces are hidden to make the plate visible. . . . . . . . . . . 21

3.2 Illustration of the mesh in the region around a 2-mm wide slit. The
bottom figure illustrates a more detailed view of the cells close to the
solid-fluid interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1 Illustration of the evolution of condensate on inside walls of a food
packaging machine. The early stages of condensation is character-
ized by very small droplets with a large mean free distance between
droplets. The middle stage of condensation has a wide range of
droplet sizes. The drying stage to the right is characterized by a
lower amount of large droplets. . . . . . . . . . . . . . . . . . . . . . 30

4.2 The film time evolution of four different cases, during the first fifteen
seconds of film build-up. The purple line is extended in graph 4.4,
and illustrated with film coverage scenes between t = 1 and t = 5 in
Figure 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 The film time evolution of the case h=3 mm and v=1 m/s, during
the first five seconds. The black line at t=5 s indicates the position
of the film front at t=4 s. An extension of the film coverage evolution
is shown in Figure 4.4 up to t = 32 s. . . . . . . . . . . . . . . . . . . 31

4.4 Time evolution of film build-up between t=5 s and t=32.5 s, for the
case of h=3 mm and v=1 m/s. . . . . . . . . . . . . . . . . . . . . . . 32

4.5 Comparison between simulation data and fitted curve for cases [h,v]
= [2,0.5] and [h,v] = [2,1.0]. The fitted curves was regressed based
on the time-span t = 1-4 s. . . . . . . . . . . . . . . . . . . . . . . . . 33

4.6 An illustrative example of liquid film distribution and gas ppm levels
of H2O2 after t = 4 s film build-up. The parameter values are h = 3
mm and v = 0.5 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.7 An illustrative example of liquid film distribution and gas ppm levels
of H2O2 after t = 4 s film build-up. The parameter values are h = 7
mm and v = 0.5 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xi



List of Figures

4.8 An illustration of the velocity field around a h = 3 mm slit. The bulk
velocity is v = 0.5 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.9 An illustration of the Turbulent Kinetic Energy field around a h = 3
mm slit. The bulk velocity is v = 0.5 m/s. . . . . . . . . . . . . . . . 36

4.10 The slit surface film coverage α at t = 4 s as a function of slit width
h, for different iso-velocities. . . . . . . . . . . . . . . . . . . . . . . . 37

4.11 The slit surface film coverage α at t = 8 s as a function of slit width
h, for different iso-velocities. The solution has been obtained by time-
extrapolation in the interval t = 4-8 s. . . . . . . . . . . . . . . . . . 38

4.12 The critical slit widths, as to ensure 100 %, 75 % and 50 % slit surface
film coverage, as a function of external gas flow velocity v. . . . . . . 38

4.13 The slit surface film coverage fraction as plotted against the dimen-
sionless mass transfer Péclet-number. The blue circles represent the
raw simulation data and the orange line is the least square fit. . . . . 39

4.14 Illustration of the elevated concentration levels of gas H2O2 in the
slit, during drying stage. The parameter values are h = 6 mm and
v = 0.5 m/s. The snapshot is taken after 12 seconds of simulation,
four seconds film build-up followed by eight seconds drying. . . . . . . 40

4.15 Illustration of the elevated concentration levels of gas H2O2 in the
slit, during drying stage. The parameter values are h = 2 mm and
v = 0.5 m/s. The snapshot is taken, similarly to Figure 4.14, eight
seconds after initiation of drying phase. . . . . . . . . . . . . . . . . . 41

A.1 The surface film coverage fraction α at t=2.5 s as a function of slit
width h for different iso-velocities. . . . . . . . . . . . . . . . . . . . . II

A.2 Comparison of simulation results at t = 4 s and the results obtained
extrapolating to t = 8 s. . . . . . . . . . . . . . . . . . . . . . . . . . II

A.3 Illustration of the pressure drop over a h = 3 mm slit. There is a
total pressure drop of approximately ∆P = 0.1 pa. . . . . . . . . . . III

A.4 Sideview of the geometrical representation of a mm-wide slit in the
case of h = 2 mm. An illustration of the parameters v and h. The
liquid film results are based on the underside wall of the plate. . . . . IV

A.5 Sideview/overview of the geometrical representation of a mm-wide
slit in the case of h = 2 mm. . . . . . . . . . . . . . . . . . . . . . . . IV

xii



List of Tables

3.1 Inlet boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Boundary specification of the walls. . . . . . . . . . . . . . . . . . . . 25

4.1 Proportion of surface film coverage αhv=α(h, v) at t=2.5 s using two
different mesh setups, one with three million cells and one with five
million cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A.1 All designs (h and v parameter combinations) for which the parameter
study was performed. . . . . . . . . . . . . . . . . . . . . . . . . . . . III

A.2 Péclet number data, for designs where α < 1, which was used to
produce the slit film coverage model given in Equation 4.4 and Figure
4.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV

xiii



List of Tables

xiv



1
Introduction

The employment of disinfection methods is widely used as a means to kill pathogenic
biological agents and to maintain environments sanitary [1]. Disinfection applica-
tions are used within the healthcare, drug and food industry where sanitary re-
quirements are strict [2][3]. Every year the bacteria Clostridium Botulinum, which
produces one of the most potent toxins known [4], is the cause of severe cases of
botulism, which is a potentially fatal disease [5]. One of the transmission routes
for botulism infection is through conserved food [5], which introduces strict re-
quirements on disinfection during food packaging. Furthermore, ensuring sterile
conditions during food and drug packaging contributes to less waste as a results of
increased shelf life of products. In several food packaging machines used by Tetra
Pak [6], disinfection of food containers and internal surfaces of the machines is per-
formed by introducing a gas mixture containing air, hydrogen-peroxide and water
into the machine. The gas mixture is allowed to condense on the food containers
and internal machine walls. After condensing, the machine is heated and vented
with air in order to evaporate the liquid deposited on walls. Water is preferentially
evaporated due to its higher volatility [7][8], leaving a liquid with high concentration
of hydrogen-peroxide which conducts the desired bacteria killing.

1.1 Aseptic Processing within Food Packaging In-
dustry

Aseptic processing within food packaging refers to a process chain in which all in-
volved steps are treated under sterile conditions [9]. A sterile product is filled into
containers that are produced and sealed under sterile conditions. A major benefit of
aseptic processing is the increased shelf life of a food product [9]. The existence of
biological agents implies a risk of biological reproduction [10], which highlights the
importance of disinfecting all internal surfaces of the machine and food containers.
For food packaging machines that utilize hydrogen-peroxide condensation, it is im-
portant to expose all surfaces to a film with high concentration of hydrogen-peroxide.
However, not only the liquid film of hydrogen-peroxide performs disinfection. Given
that a solid surface has been, during a long enough time-period, exposed to a gas
containing a high enough concentration of hydrogen-peroxide, the surface is disin-
fected completely [11].

With background of the importance of complete disinfection, the problem formula-
tion of this study arise in the investigation of hydrogen-peroxide disinfection charac-

1



1. Introduction

teristics in small confined spaces. More specifically, slits with a width of a few mm
will be investigated since they appear in food packaging machines. Although these
slits vary in their geometrical configuration, they are unified in that they have two
or more surfaces of the machine that lie close to each other with a small space in
between. It has been observed that for glas-glas slits, a characteristic slit width of
h=2 mm strongly hinders film formation while a h=10 mm slit enables a complete
film to be formed over both internal surfaces.

Machines that are employed for aseptic processing typically involves a wide range of
relevant length scales. For instance, from a macroscopic point of view, a length scale
of m is relevant for the process machine as a whole, whereas the smallest relevant
length scale at µm is relevant for a detailed description of the liquid film configu-
ration. This wide range of intimately coupled length scale induce both theoretical
and numerical modeling challenges.

1.2 Physical Description of Thin Liquid Films and
Available Models

The wide range of physical phenomena that govern formation and appearance of a
film is complex and this is also reflected in the wide range of different behaviour
and configuration a film can show [12]. Film dynamics are both governed by the
properties of the solid material at which the condensation takes place and the fluid
dynamics, including the multiphase interaction between gas and liquid film [13].
To mention an example of this, the configuration of condensed liquid on a solid
surface may vary significantly depending on the degree of hydrophobicity of the
surface [14]. In the case of a hydrophobic surface, a dropwise liquid configuration
(partially wetted surface) will form, whereas a more homogeneous film is formed in
the case of a hydrophilic surface [14]. The process of dropwise condensation and
its properties have been investigated in numerous studies [15][16][17], where it has
been demonstrated how dropwise films form and behave as condensation proceeds.
The surface topography of a solid is an additional property that will influence the
process of condensation on walls [18]. A rough surface will exhibit a different con-
densation behaviour than a smooth surface. This can be modeled by including a
parameter for the surface roughness of a material, as done by Mu et al. [18] and
Wenzel [19] among others. Surface roughness has implications on the surface density
of nucleation sites, where primary nucleation is allowed to occur which initiates the
condensation process [18]. Finally, modeling of liquid film systems requires a proper
understanding of the surface tension interplay between phases, since it plays a major
role for the mechanical behaviour of thin liquid films [20].

To study and analyze the process of disinfection within mm-wide slits, Computa-
tional Fluid Dynamics (CFD) is used through the software Simcenter STAR-CCM+
[21]. One benefit of using a CFD approach is that experiments do not have to be
physically performed. Instead, experiments can be performed electronically to spare
not only time, but also material- and financial resources. However, a major short-

2



1. Introduction

coming of employing a CFD method is that errors to the solution are introduced,
partly as a result of the implementation of the finite volume method [22], partly
as a result of introduced models which simplify reality. The governing model the-
ory implemented in STAR-CCM+ is studied with the purpose of validating results
and to be able to suggest alternative approaches for the case of film condensation
within mm-wide slits. The Fluid Film Model of STAR-CCM+ is employed, which
is a model developed to describe the dynamic and formation of thin layers of liquids
[23]. The versatility of the Fluid Film Model, and thin film models in general, is
limited because of the various assumptions made during its derivation. Therefore,
thin film assumptions are discussed in particular detail.

1.3 Problem Description and Aim
The project aims to enhance the understanding of the disinfection process in and
around slits of mm width. More specifically, the problem will be analyzed on basis
of the width of a slit. With the background of observations described in 1.1, the
range of interest for the parameter h is 2-10 mm. Together with the velocity v of a
gas flowing past a slit, a parameter study is performed with the aim to find critical
conditions of h and v as to ensure that the internal surfaces of a slit are disinfected.
A model is developed, describing the liquid film exposure and gas hydrogen-peroxide
levels within slits of mm-width as a function of parameters h and v. A number of
questions are proposed and the aim is to be able to answer the questions based on
the simulations.

• What does the flow structure look like inside and in vicinity to the slits?

• How well is the disinfection accomplished in slits of different mm widths?

• What are the requirements to form a film with full surface coverage inside the
slits, in terms of the studied parameters h and v?

• How valid are the results? What are the indications that simulation results
do not accurately describe reality?

• What can an alternative model look like, describing the exposure to hydrogen-
peroxide in mm-wide slits?

3



1. Introduction

4



2
Theory

To be able to understand and describe the process of disinfection within mm-wide
slits, it is necessary to include theory from a range of fields. First a review of
basic CFD equations is presented in Section 2.1, these are transport equations of
momentum, mass and heat. Some elements of these fundamental equations are
of particular interest for the studied case and they are presented further, such as
modeling of phase transition and surface tension. Furthermore, in Section 2.2, gas
modeling and its assumptions are presented. In Section 2.3, theory of thin film
modeling is presented and connected to both the studied case and to the Fluid Film
Model formulation in STAR-CCM+. Lastly, properties of solid materials that are
relevant for the liquid film formation will be discussed in Section 2.4. The reader is
encouraged to be observant of when the described theory regards models that are
self-derived, review of existing literature or is describing the theory implemented in
the software of STAR-CCM+.

2.1 General Equations of CFD
Most commonly CFD rely on what is called the finite-volume method, which is a
framework that is based on the principle of local conservation of physical quantities
[22]. More specifically, the domain of interest is split into small sub-volumes, allow-
ing a set of partial differential equations to be transformed into algebraic equations
for each volume which describe the conservation of mass, energy and momentum
[22]. By doing this, an error is introduced to the solution since the algebraic equa-
tions are approximations. The general equations that are solved in CFD can be
derived from the principle of conservation by setting up a micro control volume and
simply balancing the input, output and source terms. The transport equation of
momentum reads

∂ui
∂t

+ uj
∂ui
∂xj

= −1
ρ

∂P

∂xi
+ 1
ρ

∂τji
∂xj

+ gi + SMOM (2.1)

where for a newtonian fluid

τji = µ

(
∂ui
∂xj

+ ∂uj
∂xi

)
(2.2)

From left to right the terms represent accumulation of velocity, convection, motion
induced due to pressure gradient, shear stress due to a strain rate and lastly gravity.

5



2. Theory

Transport equations of mass and energy can be written in a similar fashion to the
momentum equation.

∂ρ

∂t
+ ∂ρuj

∂xj
= SM (2.3)

and

∂ρcpT

∂t
= −uj

∂ρcpT

∂xj
+ λ

∂2T

∂xj∂xj
+ SE (2.4)

The source terms can SM and SE can represent for example addition of mass and
energy due to phase transitions. Equations 2.1, 2.3 and 2.4 constitutes the general
equations solved in CFD. Note the similarities between the equations, including
terms for accumulation, convection and diffusion. Determining which elements of
these equations that are of highest importance is case-dependent. In order to op-
timize the model in terms of balance between accuracy and speed, it is up to the
CFD-engineer to include vital parts and exclude redundant parts of the equations.

2.2 Gas Modeling
By introducing a set of assumptions that are valid for the gas phase, the general
equations presented in Section 2.1 can be simplified to obtain a set of equations that
govern the gas. Assumptions of the gas flow are presented in Section 2.2.1 and they
are followed up with resulting 3D governing gas equations in Section 2.2.2.

2.2.1 Assumptions of Gas Modeling
The influence of gravity is neglected because of the absence of strong density gradi-
ents, moreover there is no preferential direction of gravity because of the diversity
of possible slit configurations. It is also assumed that condensation induce no effect
on the momentum of the liquid film. It is further assumed that isobaric conditions
apply. As a result, pressure gradient terms and gravity terms are excluded from
Equation 2.1. Finally, the gas bulk is assumed to obey the ideal gas law.

2.2.2 Governing Gas Equations
By employing the assumptions presented in Section 2.2.1, the general momentum
balance equation (2.1) is reduced to

∂ui
∂t

+ uj
∂ui
∂xj

= 1
ρ

∂τji
∂xj

(2.5)

Note that the source term SMOM also is excluded, which results from neglecting the
influence of condensation on momentum transfer. The general continuity equation
can be modified for the gas bulk conditions by modeling the source term SM . The
source term will be active at locations where phase transition occur. With the

6



2. Theory

assumption that condensation does not occur in the gas bulk, Equation 2.3 can be
written as

∂ρ

∂t
+ ∂ρuj

∂xj
= Ṁ vapδ(r − rint) (2.6)

where δ is the kronecker delta function, activating the source term at locations
where a liquid-gas interface is present. The condensation rate of the right hand side
of Equation 2.6 can be described for a species A by

Ṁ vap
A = −kcMA(CA∞ − CAs) = −kcMA

Ptot
RT

(xA∞ − xAs) (2.7)

where kc is the convective mass transfer coefficient for A. In order to close equation
2.7, kc has to be modeled. Many empirical models exist that describe the dimension-
less Sherwood number for various geometries, which is the ratio between molecular
mass-transfer resistance and convective mass transfer resistance [24]

ShL = kcL

DAB

(2.8)

where L is a characteristic length of the system. The models describing Sh is empir-
ically derived and, for forced convection, they are functions of the Reynolds number
and the Schmidt number

ShL = f(ReL, Sc) (2.9)
A wide range of correlations exist to model Sh for different geometrical constructions.
For the example of laminar flow over flat plates, the average Sherwood number ShL
over a plate of length L is given by

ShL = 0.664Re1/2
L Sc1/3 (2.10)

which is valid for ReL < 2 · 105. Similar correlations exist for the turbulent range
[24]. The saturation pressure psatA of Equation 2.7 is solved through Antoine rela-
tions, which are semi-empirical relations describing a certain species temperature
dependence on its saturation pressure. For water within the temperature range of
273 K to 647 K, the saturation pressure is described by

lg[psatH2O] = A− B

C + T
(2.11)

where psatA is in bar and A=11.89 , B=3962.56, C=-39.89. For hydrogen-peroxide,
the same equation apply with different coefficients. However, since the mixture of
water and hydrogen-peroxide is non-ideal, the need to include activity coefficients in
the model is introduced. The activity coefficients for water and hydrogen-peroxide,
γH2O and γH2O2 , are modeled by

γH2O = exp

(1− xH2O)2

1.987T

(
0.97T−1017+85(1−4xH2O)+13(1−2xH2O)(1−6xH2O)

)
(2.12)

7



2. Theory

γH2O2 = exp

(1− xH2O2)2

1.987T

(
0.97T−1017+85(4xH2O2−1)+13(2xH2O2−1)(6xH2O2−1)

)
(2.13)

Lastly, the energy equation for the gas bulk can be written as

∂ρcpT

∂t
= −uj

∂ρcpT

∂xj
+ SE (2.14)

Heat conduction excluded due to the low heat conduction of gases. The source
term is assumed to be constituted of formation of latent heat and convection across
liquid-gas interface

SE =
(
Ṁ vap∆Hvap + h(Ts − T∞)

)
δ(r − rint) (2.15)

Analogous to the convective mass transfer correlation of Equation 2.10, models exist
for the Nusselt number which in heat transfer corresponds to the Sherwood number
of mass transfer. Equations 2.5, 2.6 and 2.14 constitutes the gas flow customized
governing equations. However, the essence of modeling this process is the two-way
coupling between gas and liquid film. The two-way coupling is expressed through
source terms. Therefore, many of the gas source terms will appear as liquid film
source terms with reversed sign in Section 2.3.3.

2.2.3 Degree of Saturation
The amount of condensable species in the condensing gas is one of the crucial pa-
rameters in the process since it is one part of what constitutes the driving force for
film condensation. In a gas, concentration of a certain species can be measured by
its partial pressure. Together with the saturation pressure of the same species, the
relative saturation of a species A is defined as

φA = pA
psatA (T ) (2.16)

where pA is the partial pressure of species A and the saturation pressure of species A
psatA is a function of temperature as described in Equation 2.11. By using Equation
2.16, the temperature and partial pressure of a species can be clumped into one
single parameter. A benefit of using relative saturation is that it provides a more
intuitive description of how high the gas concentration of a certain species is in
relation to its saturation concentration at the actual temperature.

2.2.4 Concentration Boundary Layer
To make the formation of a liquid film on surfaces within a slit possible, the first
basic requirement is that condensing species enter the slit. With a gas that is
to a large extent drained on water and hydrogen-peroxide, film condensing is not
possible because the species that are supposed to condense do not reach their dew-
point because of their low saturation. The answers to why gas would not bring

8



2. Theory

condensable species into the slit are found within the analysis of boundary layer
development. It is generally distinguished between boundary layers of momentum,
mass and energy. A boundary layer of momentum refers to the velocity profile that
is formed for flow close to a wall. Because of the connection between momentum
transport and the transport of energy and mass, the boundary layer for momentum is
generally used as a basis to analyze the boundary layers of mass and energy through
the Schmidt (Sc) and Prandtl (Pr) numbers, respectively [24]. Blasius’s solution
for the laminar boundary layer on a flat plate demonstrates how the thickness of
the hydrodynamic boundary layer increases with length in the flow direction. The
derived boundary layer thickness is

d = 5
√
νx

v∞
(2.17)

where the flow starts at x=0 and v∞ is the bulk velocity. To analyze the degree of
which condensing species enter the slit, it is necessary to couple the hydrodynamic
boundary layer to the boundary layer of a species concentration. For a fluid with
a Schmidt number other than unity, the concentration boundary layer is related to
the hydrodynamic boundary layer by

d

dc
= Sc1/3 (2.18)

By combining Equations 2.17 and 2.18, the concentration boundary layer thickness
is given as

dc = 5Sc−1/3
√
νx

v∞
(2.19)

Further, a measure of the slit width h relative to the concentration boundary layer
width can be defined, describing the ratio between the two

ĥ = h

dc
= 0.2hSc1/3

√
v∞
νx

(2.20)

If ĥ � 1, expressing that the slit width is much larger than the concentration
boundary layer width, it can safely be assumed that the concentration boundary
layer does not affect the intake of condensable species into the slit space. If ĥ ≈ 1, h
and dc are of comparable size and an influence on the intake of condensable species
can be expected.

2.2.5 Non-Dimensional Analysis of Gas Slit Flow
The gas flow within a slit of width h can be characterized in a dimensionless manner
by the mass Péclet number

Pe = RehSc = uh

D
(2.21)

which describes the ratio between convective transport and diffusion transport of
mass. The velocity u is the mean velocity over the width of the slit. Usually in engi-
neering applications, the Péclet number is very high which enables easier modeling

9



2. Theory

because the effect of diffusion may be neglected. However, what is characteristic
for the situation of gas flowing between two narrowly placed plates is that there is
always a wall present close to the gas. Close to walls, it is rather viscous forces that
dominate as opposed to inertial forces.

2.3 Film Modeling
Liquid films are frequently appearing in industrial application and in everyday life.
For instance, liquid films are treated in rainwater management, in heat exchangers
and lubrication[25][26][27]. Because of the wide range of application areas, litera-
ture on liquid film theory is rich [20][28][29][30]. However, liquid films may, in many
systems, show up complex behaviour and many challenges in film modeling remains
[28]. The detailed morphology of a film formed from condensation depends on a
range of factors, including physiochemical properties of the condensing fluid, wall
temperature and wall surface roughness [13].

Governing equations that formulate the film transport are often based on a set of
assumptions. The purpose is to make the solutions procedure easier and to elimi-
nate redundant computational resources, at a low cost of accuracy. A short review
of assumptions, which are both frequently used within film modeling and of inter-
est in this particular process, will be presented in Section 2.3.1. Furthermore, the
precise assumptions which the film models derived in this study are based upon are
presented in Section 2.3.2. In section 2.3.3, the assumptions are followed up with
the resulting governing film equations. Finally, in Section 2.3.4 the fluid film theory
used in the software of STAR-CCM+ is outlined.

2.3.1 Thin Film Modeling
The thin film assumption has been used by several authors [28][29], it is usually
comprised by a number of assumptions which will be discussed in this section. Even
though the exact formulation of thin film assumptions may vary, they are all based
on the same general idea of applying appropriate simplifications to describe the mo-
tion of a thin layer of liquid on a solid surface. Most frequently, the boundary layer
approximation is applied, meaning that spatial gradients of dependent variables are
negligible in a surface tangential direction, compared to those in a surface normal
direction [29].

A set of assumptions were set up by Bai and Gosman [29] to treat the film modeling
with a thin film approach. Although they developed a model for impinging sprays,
which is not the focus of this study, some of the assumptions for the film are of more
general character and will here be described. The following set of assumptions are
made by Bai and Gosman:

1. The film is thin enough to be laminar. It is also thin enough for the boundary
layer approximation to be valid and for the inertial forces to be negligible.

10



2. Theory

2. The liquid film is locally subjected to a total normal pressure of

pL = pd + pg + pσ (2.22)

where the pd and pg is related to impingement of droplets and will be overseen in this
study. The capillary pressure pσ arises from surface tension and interface curvature
according to the Young-Laplace equation

pσ = −σ
(

1
R1

+ 1
R2

)
≈ −σ∇2δf (2.23)

where R1 and R2 are the principal curvature radii of the film interface and δf is
the local film thickness. The derivation of the approximation in the right hand side
of Equation 2.23 relies on an assumption that the curvature of the film interface is
negligible, which is usually true under the thin film assumption [29].

3. Film motion is caused by surface gradients of the total pressure pL, shear at the
film interface, and gravity.

4. Energy flux due to radiance and species gradients (Dufour effect) is omitted.
Similarly, pressure and thermal diffusion flux in the species equation are omitted.

5. The gaseous mixture behaves like an ideal gas.

6. A normalized film velocity profile is assumed, constituted of a third degree poly-
nomial in height over the wall and with a no-slip boundary condition

u

uδ
= a1η + a2η

2 + a3η
3 (2.24)

where η=z/δ is non-dimensional height over the wall and uδ is the velocity com-
ponent in x-direction at the interface. The same expression, but with different
coefficients, is valid for the y-direction velocity profile. Directions x and y consti-
tutes the surface tangential plane. To derive the explicit form of the velocity profile
of Equation 2.24, phenomena that are assumed to govern the film motion, i.e those
pointed out in assumption number 3, must be combined with the governing film
momentum equations.

O’Rourke and Amsden [28] have used even more relaxed assumptions. It is explicitly
mentioned that the assumption of a film thickness much smaller than the radii of
curvature of the walls is adopted. An obvious limitation of adopting the thin film
framework is its incapability to handle surfaces with strong curvature. As opposed
to Bai and Gosman, O’Rourke and Amsden have assumed that the effect of gravity
on the film is negligible and that the film velocity vary linearly with height above the
wall. It is further assumed that the interface pressure gradient terms are negligible.
Lastly, the computation of gas velocity close to the film is treated as if the gas flow
along a solid wall. An approximate upper bound value of the film thickness of δ=100
µm is given, for which the thin film assumption can be assumed valid. The upper

11



2. Theory

bound film thickness provides a perception of acceptable film thickness, for which
the thin film assumption hold, and will be used as a benchmark in this study. It can
be noted that in the studies brought up in this section, even though they are unified
in that they use thin film modeling, quite different assumptions are adopted.

2.3.2 Assumptions of Liquid Film Modeling
One central assumption of the study is that the existence of bulk condensate is over-
seen. In reality, the steam is saturated to the degree that nucleation occurs in the
gas bulk, forming small droplets which moves along the gas. However, this study
investigates solely the condensation occurring directly on walls and film. Therefore
this assumption includes also omitting the influence of droplet impingement on film
motion and build-up. It is also assumed that the liquid layer can be treated as a film
rather than very small droplet deposited close to each other, which is an assumption
made based on the pre-condition of using and evaluating existing STAR-CCM+
models.

Phenomena that are assumed to govern the fluid film momentum include gas shear
stress, wall shear stress resulting from a no-slip condition and surface tension forces.
Inertial forces are assumed to not influence the film motion, this assumption is vali-
dated by the usage of a microscope to inspect the film behaviour, which shows that
the droplets forming the film are static. Along with the assumption of static film
follows that inertial terms of the governing equations and the unsteady term of the
momentum equation may be neglected. Finally, the energy transfer is assumed to
consist of three parts, convective heat transfer from the gas, conduction to the wall
and addition of energy due to condensation.

2.3.3 Governing Film Equations
In accordance with the assumptions described in Section 2.3.2, governing equations
are set up which are specific for the case of thin film flow accompanied by condensa-
tion from surrounding gas. According to the thin film assumptions, diffusion terms
are treated in wall normal direction and the equations are transformed into its 2D
counterpart which is done by integration in surface normal direction over the entire
film thickness. For sake of clarity, the governing equations will now be expressed
explicitly in the directions x, y and z. The velocities in surface tangential directions
x and y, are denoted u and v respectively. Surface normal direction is z and its
corresponding velocity is denoted w. By conducting the film integration, the 2D
film continuity equation is obtained as

∂δfρf
∂t

= Smδ (2.25)

where δf is the local film thickness. The source term Smδ of Equation 2.25 is only due
to condensation, which may be modeled by considering the diffusivity of condensing
species and their respective deviation from saturation concentration at the current
temperature. Doing so, Equation 2.25 becomes

12



2. Theory

∂δfρf
∂t

=
N∑
i

ρDi
Yi − Yi,sat(Tg)

δf/2
(2.26)

where i are indices for condensing species and N is the number of condensing species.
The mass film governing equation is governed by the driving force of difference be-
tween the condensing species concentration at the film interface and their respective
saturation concentrations. The saturation concentration is obtained by Antoine re-
lations and activity coefficients as described in Section 2.2.2. The change of film
species composition xj may then also be calculated by

∂xj
∂t

=
ρDj

Yj−Yj,sat(Tg)
δf/2

δfρf +∑N
i ρDi

Yi−Yi,sat(Tg)
δf/2

(2.27)

The film integrated momentum transport equations becomes

∂δfρfu

∂t
= −∂δfpL

∂x
+ τx (2.28)

∂δfρfv

∂t
= −∂δfpL

∂y
+ τy (2.29)

Terms contributing to momentum transfer are split into pressure based (surface
gradients of pressure normal to wall) and stress based (forces acting tangential to
wall). The pressure pL is assumed to consist of only the surface tension normal
pressure pσ, which in turn is expressed in Equation 2.23, according to

pL = pσ ≈ −σ∇2δ (2.30)
which is valid for small curvature of the film interface. Instead of calculating the
instantaneous shear force at every location, as expressed in Equation 2.1, the stresses
can under the thin film approach be divided into separate contributions from gas
shear, wall shear and tangential surface tension force according to

τ = τg + τw + τθ (2.31)
These terms have to be modelled separately. In [30], the assumption that τg � τw
is done, enabling τg to be neglected. However, τw can be modelled by using a
mean film velocity ūf derived from an assumed velocity profile, which enables the
expression

τw = µf (T̄f )
ūf
δf/2

(2.32)

to be set up, describing a finite difference approximation of the shear stress asserted
by the wall on the film. The expression of Equation 2.32 is based on the boundary-
layer approximation, meaning that in the wall normal direction only the diffusion
term is considered. In expression 2.32 a non-moving wall with a no-slip boundary
condition is assumed. It is also assumed that the dynamic viscosity taken at the
mean film temperature is representative for the film as a whole. The surface tension
shear stress τθ arise at the location of a contact line, i.e a line separating a dry

13



2. Theory

and a wet region. For the case of a uniform film, which is the case of the STAR-
CCM+ formulation of contact line force described in Section 2.3.4, the contact
line force is simply present at the different locations of transition from wet area
to dry area. However, in the case of a dropwise configuration, contact lines are
on a macroscale present everywhere on the dropwise film since every micro-droplet
brings about a small circle constituting the contact line. The contact lines around
droplets are important for the film configuration of dropwise films. If there is a high
surface density of contact line forces and other forces on droplets are negligible in
comparison, the result is a static dropwise film because the contact line is only pulled
in towards the center of droplets until it is balanced by the pressure gradient force.
By including all of the submodels to the momentum equation, and by excluding the
transient momentum term, the finalized momentum equations are written as

µf (T̄f )
ūfx
δf/2

= −∂δfσ∇
2δf

∂x
+ σ(1− cos(θ))∂Γ

∂x
(2.33)

µf (T̄f )
ūfy
δf/2

= −∂δfσ∇
2δf

∂y
+ σ(1− cos(θ))∂Γ

∂y
(2.34)

where Γ is a film indicator function that is Γ=1 in wet regions and Γ=0 in dry
regions, it is described in more detail in Section 3.5.1. Because of the exclusion of
the transient momentum terms, Equations 2.33 and 2.34 express a situation where
pressure gradient force, viscous shear force from the wall and contact line forces bal-
ance each other resulting in a static film. Lastly, if the energy equation is integrated
over the film height

∂ρfcpδfTf
∂t

= Seδ (2.35)

Similarly to shear stresses being divided into different contributions, Seδ can be
divided into energy that is transferred from gas to film, energy transferred from wall
to film, and energy arising from phase transition.

Seδ = Seg + Sew + Sept (2.36)

It is necessary to, in similar fashion to the velocity profile, assume a temperature
profile in order to model these contributions, since each of these contributions depend
on temperature. In [28] it is assumed that the temperature profile is piece-wise linear,
varying linearly from Tw to Tf from the wall to the middle of the film, and from Tf

to Tg from the middle of the film to the gas interface. The heat source term due to
phase transition can simply be achieved through

Sept = Ṁ vap(∆Hvap + Cp(Tg − Tf )) (2.37)

considering both latent heat due to phase transition and the sensible heat to cool
the condensate to the mean film temperature. The convective source terms can be
written as

Seg = −k(Tf )
Tf − Tg
δf/2

(2.38)

14



2. Theory

Sew = −k(Tf )
Tf − Tw
δf/2

(2.39)

for the gas contribution and wall contribution, respectively. The temperature of gas
Tg and temperature of wall Tw are evaluated at the center of their respective first cell
layer outside of the liquid film. Arguably, the convective heat sources of Equations
2.38 and 2.39 could be neglected against the heat source term due to condensation in
Equation 2.37. Combining equations 2.35 with the source term equations 2.37-2.39
yields the final energy equation

∂ρfcpδfTf
∂t

= Ṁ vap(∆Hvap + c̄p(Tg − Tf ))− k(Tf )
Tf − Tg
δf/2

− k(Tf )
Tf − Tw
δf/2

(2.40)

To summarize the film modeling, Equations 2.26, 2.33, 2.34 and 2.40 constitute the
derived thin film model of this study.

2.3.4 Fluid Film Modeling in STAR-CCM+
The Fluid Film Model in Simcenter STAR-CCM+ is a tool applied to represent
the build-up and transport of a thin layer of liquid on a solid surface. The model
accounts for a range of different phenomena responsible for the mass, heat and
momentum transfer both within the film itself and over the liquid-gas interface. For
the present application, the most essential utilities of the Fluid Film Model include
the formulation of wall condensation and the formulation of surface tension effects.
The governing film equations that are solved in STAR-CCM+ are based on thin
film modeling [23].

2.3.4.1 Film Motion

In STAR-CCM+, the velocity profile of the film is assumed to be a polynomial of
second order in height over the wall. The way in which the coefficients of the velocity
profile are determined by the following constraints [23]:

1. No-slip boundary condition at the wall.

2. At the gas side, fluid velocity is fixed to gas velocity to ensure continuity of
velocity.

3. It is satisfied that the stress term plus the pressure term are equal on both sides
of the gas-liquid interface.

2.3.4.2 Surface Tension

The surface tension formulation that is employed in the Fluid Film Model is based
partly on a surface normal contribution pσ, similar to that of Equation 2.23, however
it contains an additional empirical parameter a

pσ = −aσ
(

1
R1

+ 1
R2

)
≈ −aσ∇2δ (2.41)

15



2. Theory

Surface tension force in the wall tangential direction appears at the transition from
a dry region to a wet region. The line separating a wet and a dry region is called a
contact line. At the contact line, the tangential surface tension force is calculated
according to

τσ = bσ(1− cos(θ))∇Γ (2.42)
where θ, which is described more thoroughly in Section 2.4.1, describes the apparent
contact angle as a fundamental parameter of the wetting behaviour of the film. The
parameters a and b are empirical parameters which are used to account for discrep-
ancies between theory and reality and it can be fitted with respect to experimental
results, however the default values are a=b=1. As seen by the definition of Γ in
Section 3.5.1, the factor ∇Γ is simply activating the contact line force at locations
where the fluid film thickness pass the critical value δf=0.1 µm.

2.3.4.3 Evaporation and Condensation

In the Fluid Film Model, the solution procedure of condensation is treated separately
below and above the saturation temperature. Below saturation temperature, the
rate of condensation is controlled partly by hydrodynamic effects - vapor needs to be
delivered to or carried away from the surface, partly by thermal effects - heat needs
to be conducted away from the interface to balance the latent heat of evaporation.
Practically, the following set of equations are treated iteratively through the secant
method to solve for the interfacial temperature Ts

f(Ts) = kg
dT

dy

∣∣∣∣∣
gs

− kl
dT

dy

∣∣∣∣∣
ls

− Q̇v(Ts) (2.43)

Q̇v =
Nv∑
i

∆Hvap
i ṁv,i (2.44)

ṁv,i = Yiṁv − ρDi
dYi
dy

(2.45)

ṁv = −
∑Nv
j ρDi

dYi

dy

1−∑Nv
j Yj

(2.46)

However, at saturation conditions the hydrodynamic procedure described above is
cut short and the interfacial temperature is set to Ts = Tsat. A shortcoming of
the method prescribed in Equations 2.43 - 2.46 is the treatment of condensation
on perfectly dry walls. In STAR-CCM+, the heat transfer coefficient of the film is
generally given by

hf = 2kf
δf

(2.47)

and consequently the following approximation hold

kl
dT

dy

∣∣∣∣∣
ls

≈ 2kf
δf

(Tl,c − Ts) (2.48)

16



2. Theory

which is based on the assumption of piecewise linear temperature profile. The tem-
perature Tl,c is the film liquid temperature at the cell center. Note from Equation
2.47 that, for vanishing film thickness, the heat transfer coefficient tends to infinity,
expressing that heat is transferred towards the wall infinitely fast. The limitations
of the heat transfer model presented so far can be explained partly by the absence
of physics describing how the condensation is modeled. Condensation at dry walls
occurs as earlier described not in a flat film but in a dropwise mode. During early
stages of conensation, when a fully wetted film has not been formed, the interfacial
contact area between vapor and liquid is small and may be the limiting factor for
condensation and evaporation rates. To overcome this issue, STAR-CCM+ uses a
simplified drop-wise approach, which is based on modifying only the interfacial area
between vapor and film. With the assumption that droplets form a 90◦ apparent
contact angle and that all droplets are identical with a radius R, the relation be-
tween droplet radius, mean film thickness δf and nucleation site density N (m−2) is
expressed through

R =
(

3δf
2πN

)1/3

(2.49)

Further it is assumed that the evaporation rates as calculated by Equation 2.45 and
2.46 are still valid. However the interfacial area between liquid and vapor is modified
with aid of the factor

fA = 2πNR2
eff (2.50)

where Reff is evaluated as max(R,Rmin), where Rmin is a user specified value for the
minimum nuclei radius. An expression for the minimum nuclei radius Rmin, that
will not spontaneously disappear due to its instability, is presented in Section 2.4.2.
The factor fA is an expression of the area available for condensation, i.e the area
wetted by a liquid. When fA < 1, expressing that only a fraction of the solid surface
is covered by liquid, it is used as a multiplier for the condensation rates. Not only
Rmin, but also nucleation site density N is a user specified value. Theory regarding
density of nucleation sites is related to solid materials and it is presented in Section
2.4.2.

2.4 Solid Material Properties
The wide range of different solid material properties is one of the main reasons to why
understanding and modeling a liquid film is complex. There are two main factors of
a solid material that govern the behaviour of a liquid exposed on its surface. The
first influential factor is whether the material is hydrophobic or hydrophilic [30].
If the surface is hydrophilic, a uniformly distributed film is allowed to be formed,
which can be characterized by that the liquid layer on the surface is transparent.
However, with increasing hydrophobicity of a surface comes an increased tendency
that the liquid instead takes on a dropwise configuration [16]. Recall the behaviour of
liquid deposited on different surfaces illustrated in Figure 2.1. The second important
solid material characteristic is surface topography. The surface structure of a solid

17



2. Theory

material influence the primary nucleation process in that it affects the surface density
of nucleation sites [18]. Both outlined factors are however coupled to each other and
great progress has been made in recent years in understanding the relation between
surface topography and the degree of hydrophobicity [31].

2.4.1 Wettability

Wettability refers to the degree to which the surface of a solid is allowed to be wetted
by a liquid [32]. The contact angle, θ in Figure 2.1, is a fundamental parameter used
to describe the wettability of a surface in a given liquid-solid system [33]. There
are several different definitions of a contact angle. An equilibrium contact angle can
be derived through the Young-Laplace equation 2.23 by assuming the ideal case of
homogeneous pure liquid on a perfectly flat, rigid, and smooth substrate without any
impurity or heterogeneity [34]. However this is hardly the case for any real system.
Advancing contact angle and receding contact angle are the maximum and minimum
contact angles that are possibly attained for a surface [35]. The two latter contact
angles are referred to as being dynamic. Taking the difference between advancing
and receding contact angle gives the contact angle hysteresis

∆θ = θa − θr (2.51)

which has become a popular topic due to the interest in superhydrophobic and self-
cleaning surfaces [36]. Lastly, the apparent contact angle is defined as the angle
made up by the apparent solid surface and the tangent to the liquid-fluid interface
[37]. In other words the apparent contact angle is the contact angle observed on a
macro-scale, overseeing the micro-scale roughness of the solid surface. However, on
a length scale comparable to the surface roughness, there are intrinsic contact angles
that often differ from the apparent contact angle. The large range of length scales
between intrinsic and apparent contact angles results in tough modeling challenges.
If a computational grid is fine enough to resolve the surface roughness, the compu-
tational resources needed to calculate the liquid motion over the surface will quickly
get too large. This limitation describes why macro-scale models such as the Fluid
Film Model and the results it brings should be examined with caution. Conclusively,
it is challenging to introduce models that have a good balance between accurately
predicting the liquid behaviour at walls and ensuring that the simulations are not
too computationally expensive.

Figure 2.1 provides an illustration of a fundamental wetting parameter, the equi-
librium contact angle θ. A low value of θ corresponds to a liquid on a hydrophilic
surface, whereas a high value corresponds to a liquid on a hydrophobic surface.

18



2. Theory

Figure 2.1: Different behaviours of a droplet on solid surfaces. The equilibrium
contact angle θ increases with increasing hydrophobicity of the solid material. The
contact angle θa ≈ 135◦ corresponds to a hydrophobic solid material, while θs ≈ 20◦
corresponds to a hydrophilic solid material.

2.4.1.1 Filmwise Condensation: The Nusselt Model

Condensation in a filmwise mode has been studied for a long time, starting 1916
with nusselt’s solution to condensation on vertical walls. Nusselt derived a model for
the heat transfer coefficient h of filmwise condensation of an ideal solution, without
presenting the derivation, h according to Nusselt is

h =
ρLgk3(ρL − ρv)[hfg + 3

8cpL(Tsat − Tw)]
4µ(Tsat − Tw)

1/4

(2.52)

where h includes heat transfer due to both latent and sensible heat. The latent
heat term hfg should be evaluated at the saturation temperature and the liquid
properties should be evaluated at the film temperature. The Nusselt model has
provided a base for the development of film condensation models and it has allowed
improved models to be derived over the years, reducing the number of assumptions
leading up to the solution of Equation 2.52 [24].

2.4.1.2 Dropwise Condensation

When condensation occurs on a hydrophobic surface, it appears in a dropwise mode
[15]. Primary nucleation is then only allowed on certain discrete nucleation sites.
The density of such sites is related to the surface topography and it is further de-
scribed in Section 2.4.2. Dropwise condensation differ from filmwise condensation in
that it has an order of magnitude larger heat transfer coefficient [16]. The progress
within studies of heat transfer due to dropwise condensation of steam has been re-
viewed by Rose [15] and it is described how the results, that were straggling for a
long time, in recent decades agree well with each other. Rose compile these results
and expresses a heat transfer model for dropwise condensation that capture the re-
sults of studies that are mutually consistent.

Factors influencing drop growth and coalescence include surface orientation and com-
position, vapor and surface temperatures, gas concentration of condensing species,
and vapor flow rate [38]. The dynamics of a dropwise film have been demonstrated
by Rose [16] where it is observed how the film build-up occurs, primary nucleation

19



2. Theory

takes place and as droplets grow they start coalescing alongside with the ongoing pri-
mary nucleation. Rose concluded that the ratio between largest and smallest drops
may amount to as much as 106 [16]. Leach et al [38] have studied the dropwise
condensation on hydrophobic surfaces by direct observation and simulations and
discovered that the droplets size distribution is very insensitive to initial droplet
radius. This insensitivity is explained by the critical role that coalescence play in
determining the size distribution. The observed drop growth kinetics suggest that
the smallest drop grows principally by diffusion of water adsorbed on the substrate
to the drop perimeter. On the other hand, larger drops (d>50µm) grow principally
by direct deposition from the vapor onto the drop surface.

2.4.2 Topography of a Solid Surface
The wettability of a solid material is coupled to the structure of its surface. Studies
have been done to investigate the influence of surface topography on nanometer scale
on density of nucleation sites [18]. It was shown that surface topography has a great
influence on nucleation density sites. The density of nucleation sites increases with
the fractal dimensions of the surface [18]. Rose [16] proposed a formula to calculate
the nucleation site density

Ns = 0.037
r2
min

(2.53)

which takes into account the minimum nuclei radius. Mu et al [18] modified Equation
2.53 according to experiments to also consider the influence of surface topography
on nucleation site density. An equation, including the influence of fractal dimension
D, was formulated to improve equation 2.53

N ′s = 0.01 + 2.019 · 10−21e15.62D

r2
min

(2.54)

which is valid for fractal dimensions D within the range 2.628-2.812. The minimum
nuclei radius included in Equations 2.53 and 2.54 may also be modeled. A derivation
[39] starting with gibbs free energy to obtain the critical equilibrium radius yields

rmin = 2σVl
RTln[pv/psat(T )] (2.55)

where Vl is the molar specific volume of the liquid.

20



3
Methods

The methods used to study this process include a literature study on both film
modeling and the interaction between liquid and solid. Furthermore, a geometry
that represent the problem description was produced, followed with suitable mesh-
ing and model set-up. Simulations were performed for shorter film build-up cases
and longer cases including both film build-up and drying. Used models were vali-
dated partly with literature, partly with experimental observation of the film using
a microscope. Used boundary conditions were obtained by analyzing the results of
previously performed simulations of a food packaging machine called A3-speed.

3.1 Geometrical Representation of a Slit
Geometries that represent the occurrence of different types of narrow gaps were
produced using Computer Aided-Design (CAD) through the pre-processing software
ANSA [40]. The geometry consists of a cuboidal containing a plate located close to
one of the sides, which is illustrated in Figure 3.1

Figure 3.1: Figure illustrating the geometry produced in ANSA. The inlet and
outlet surfaces are hidden to make the plate visible.

The dimensions of the box are 24x30x7 cm. The inlet and outlet are specified so that
the gas flows in a positive x-direction. Of the remaining four faces of the cuboid,

21



3. Methods

three are defined as slip-walls so that there is no influence of a boundary layer from
these sides, this is done because these sides are not actually walls, but just an outline
of the investigated domain. Through defining these slip-walls, the need to include a
fine spatial resolution close to the walls are eliminated. The remaining face is there
to represent a physical wall, it has a solid plate located parallel to its surface with a
narrow width. The dimensions of the plate are 10x10x0.5 cm. The variation of this
geometry consists of altering the parameter h which represents the distance of free
space between plate and wall.

3.2 Meshing of the Slit Representation

With the motivation of having a geometry consisting of only perpendicular angles,
a trimmed cell mesher was chosen, which aims to produce rectangular cells. When
adjusting meshing control parameters, the goal is to produce a mesh that gradually
refines the mesh as distance to a solid surface decreases. At the solid surface it is
desired to have a fine spatial resolution in order to obtain a detailed description of
the film formation. To achieve this goal, the important mesh controls include the
targeted cell size at solid surfaces, specification of prism layers and the growth rate
at which cells grow as function of distance from a solid surface.

Initially, a mesh consisting of around one million cells was produced. It was observed
that the resolution at solid interfaces was coarse and the solution was not grid-
independent. The mesh was refined to 3.16 million cells. Once more, the grid-
independence was investigated by refining the mesh from 3.16 million cells to around
5.00 million cells, whereupon a sample of three different cases (v = 1 m/s, h = 2,3,4
mm) were chosen to compare the results obtained when using the two different
meshes. Mesh refinement was performed by changing the so called Base Size. Many
meshing parameters are set in relation to the Base Size, so an alteration of this
parameter changes a range of relevant meshing parameters, for example the bulk
cell size, surface cell size and the total thickness of boundary layer (near wall) cells.
This refinement method ensures that the mesh refinement is performed in interesting
areas, such as at solid surfaces, and not only in the gas bulk. The results of the
grid-independence analysis are presented in Section 4.2.2. The mesh consisting of
3.16 million cells is illustrated in Figure 3.2, in the particularly interesting region,
around the 2-mm wide slit.

3.3 Simulations of Film Build-up and Film Dry-
ing

Simulations were performed with a few key considerations in mind; quality of mesh,
judging convergence, used models and boundary conditions. In this section, a de-
scription is provided including assumptions, the procedure of judging convergence,
choice of boundary conditions and a description of performed analyses.

22



3. Methods

Figure 3.2: Illustration of the mesh in the region around a 2-mm wide slit. The
bottom figure illustrates a more detailed view of the cells close to the solid-fluid
interface.

23



3. Methods

3.3.1 Assumptions Employed for the Simulations
One of the central assumptions for the gas phase regards the velocity specification
at the inlet boundary. The gas will be seen as if it flows from only one direction. In
reality, many slits are located so that the gas would be able to flow into the slit from
more than one side, i.e at locations where the velocity magnitude and direction are
not completely uniform. However, the effect of multiple inlets is overseen and it is
assumed that flow from one direction is sufficient to describe the actual slits and to
enhance the general understanding of film formation within slits.

It is assumed that no condensation occurs in the gas bulk, i.e there are no nucleation
of droplets occurring and there are no droplets dispersed in the gas. It has been
observed by experiment in the actual process that small drops are formed in the
bulk which form a fog. In other words this study treats only the wall condensation
phenomena as opposed to the possible influence of droplet impingement. However
it can be motivated that the influence of impinging droplets is not large because the
droplets are very small, as seen from videos of the process, which implies that they
follow the gas stream line well.

3.3.2 Judging Time-Step Convergence During Simulations
The method by which convergence is judged in STAR-CCM+ is based on setting up
stopping criteria that define when the solution has converged. Stopping criteria for
transient simulations include the specification of inner iteration convergence criteria,
which makes up the definition of when a time step has converged and calculation of
the next time step can proceed. The convergence of time-step were based on four
different reports; Average Fluid Film Thickness, Maximum Fluid Film Thickness,
Average Evaporation Rate and Average Velocity (average velocity of a cross-section
in the flow direction). Numerically the stopping criteria for these reports are, based
on the last four iterations, defined by the maximum deviation of two iterations
normalized by the mean value of that report during these four iterations. The time-
step is converged if none of the normalized deviation exceeds 1e-3.

3.3.3 Boundary Conditions
Inlet boundary conditions were chosen based on available simulation results of a food
packaging machine. The investigated range of inlet velocities is of central importance
since it is a part of the parameter study. To determine which range of velocities that
is of interest, available simulation results revealed that common velocities through-
out the machine are in the range of 0.5 m/s to 2 m/s. The inlet turbulence conditions
were specified through values of turbulent kinetic energy and turbulent dissipation
rate, k and ε, which were also determined from available simulation results. Inlet
values of k=0.1 m2/s2 and ε=1 m2/s3 were used. Inlet boundary condition for the
concentration of species were simply determined by trying to achieve a desired degree
of relative saturation according to the theory outlined in Section 2.16. A relative
saturation of around φ=90 %, for both water and hydrogen-peroxide, was used as

24



3. Methods

inlet condition. Together with a gas inlet temperature of Tin=50 ◦C, the species inlet
mole fraction of water and hydrogen-peroxide were determined to be xH2O=0.043
and xH2O2=0.0037 respectively. The inlet boundary conditions are summarized in
Table 3.1.

Table 3.1: Inlet boundary conditions.

Mole Fraction T (◦C) v (m/s) k (m2/s2) ε (m2/s3)
xH2O=0.043, xH2O2=0.0037 50 0.5-2 0.1 1

Besides from the inlet and outlet boundary conditions, for the three boundaries that
border to the ambient machine conditions, slip-walls were chosen to reduce the influ-
ence of these walls on the studied domain. Recall that the slip-wall boundaries are
not physical walls but just there to limit the domain under investigation. The phys-
ical wall and the plate were both defined with no-slip shear specification. Thermal
specification for the physical wall were specified with room temperature outside of
the wall and a convective heat transport coefficient of h=10 Wm−2K−1. The initial
temperature of the plate and wall were set to T=25 ◦C. Specification methods of
the wall-boundaries are summarized in Table 3.2.

Table 3.2: Boundary specification of the walls.

Boundary Shear Stress Thermal
Machine Ambient Boundaries Slip Adiabatic
Physical Wall Boundary No-slip Convection, room temperature

3.3.4 Description of the Various Simulation Cases
The main analysis was conducted based on the alteration of slit width and gas ve-
locity. As outlined in the introduction, the interesting domain for h is between 2
and 10 mm, and as described in Section 3.3.3, the investigated domain of the gas ve-
locity is between 0.5 m/s and 2 m/s. A design table was produced in STAR-CCM+
Design Manager, which includes all parameter combinations for which simulations
were run. The slit width domain was discretized as h = [2, 3, 4, 5, 6, 7, 8, 9] mm
and the gas flow velocity domain was discretized as v = [0.5, 1.0, 1.5, 2.0] m/s. For
all different combinations of h and v, film build-up cases were run where the gas
mixture consisting of hydrogen-peroxide, water and air was used as inlet condition
for a short period of time. Initially, the film build-up phase was simulated during
2.5 seconds. However, it was realised that the rate of film build-up is still relatively
fast at t = 2.5 s, so the film build-up simulation time was increased to t = 4 s. The
time-dependency of film build-up is thoroughly presented in the result section and
in Appendix I. After 4 seconds of building up the liquid film, the film build-up rate
has decreased significantly and the film coverage was considered stable. Finally, the
results were mainly based on the final surface fraction of film coverage on the inside
walls of the slit, as a function of h and v.

25



3. Methods

In addition to film build-up analysis, simulations of full cycles including both film
build-up phase and drying phase were conducted. For these cycles, simulations of 20
seconds were performed, of which the first eight seconds constitute the film build-up
phase and the remaining eight seconds constitute the drying phase. For the drying
phase, air of Tin=60 ◦C and xAir=1 was used as inlet condition. The objective
with full cycle simulations were to investigate the film evaporation process and to
investigate to which degree walls are exposed to vapor H2O2.

3.4 Design Manager
To perform the parameter study, the STAR-CCM+ integrated tool Design Manager
was used to set up the various parameter designs. Continuously spaced data points
within the velocity range 0.5-2.0 m/s was chosen with increments of 0.5 m/s and
similarly for the slit width range 2-9 mm increments of 1 mm were chosen, i.e the
total number of parameter combinations for the design study is 8x4. Between each
of the 32 different design cases, the procedure of altering the geometry, remeshing
and running the simulation has to be performed. To skip the inconvenience of
doing this manually, a parameter determining the slit width were set up, which
work in conjunction with Design Manager. Each time a design case was finished,
the geometry and remeshing was automatically performed and the simulation was
initiated for the sequent design of the design table.

3.5 Post-Processing
Post-Processing of the simulations include analyzing how far the film reaches into
the slit. The results are also demonstrated in the form of plots showing the fraction
of film surface coverage as functions of parameters h and v, after a certain time
period. The latter method is a way to compress the information of the results,
enabling the dependency on h and v to be more easily visualized in one plot. The
corresponding dependency on the Péclet number is also presented together with the
critical Péclet number value for which the slit is completely disinfected.

3.5.1 Film Surface Coverage
For the purpose of being able to analyze the degree of film coverage on surfaces, it
is necessary to set up a film criteria. A critical fluid film thickness was chosen so
that when the film thickness is larger than this value, it is denoted a wet region,
when the film thickness is smaller than this value, it is considered a dry region. By
fitting the critical film thickness value to observations, which is described in Section
3.6, a critical film thickness of δcrit= 0.1 µm was chosen. Note that this the same
value as for where the contact line was defined in Section 2.3.4.2. However, using
the criteria, a film indicator function is defined

Γ(δf ) =

1, if δf ≥ δcrit

0, if δf < δcrit

26



3. Methods

To monitor the specific surface area exposed to a liquid film, the film indicator
function was first set up as a field function. Further, a surface average report was
set up, reporting the average value of the film indicator function on the underside of
the plate of Figure 3.1, the average corresponds to how large fraction of the chosen
surface that is covered. Mathematically the surface fraction that is covered by film
is expressed through

α =
∑
i Γ(δf > δcrit) · ai +∑

j Γ(δf < δcrit) · aj
Atot

=
∑
i ai∑

i ai +∑
j aj

(3.1)

where ai are the surface areas of cells with a fluid film thickness larger than the
critical value of δcrit=0.1 µm and aj are the surface areas of cells with a fluid film
thickness smaller than the critical value of δcrit=0.1 µm.

3.5.2 Péclet Number Setup

The Péclet number was obtained by reporting the mean velocity of gas within the
slit, which was then used in the definition of the Péclet number given in Section
2.2.5. The correlation between film coverage within the slit and the Péclet number
was analyzed. Furthermore, the correlation between gas slit flow velocity and the
parameters slit width h and gas bulk velocity v was analyzed, which allows the
results to be expressed in gas bulk flow velocity if desired.

3.5.3 Time-Extrapolation of Slit Film Coverage

Because of the long time it takes for the film build-up to reach a relatively stable
point, i.e when the time derivative of film coverage fraction has decreased signifi-
cantly, a workaround was developed making it possible to extrapolate the fluid film
coverage to a later stage. Even though the film build-up cases ran for only 4 s each,
the solution can be extrapolated to t = 8 s without adding a significant error. It
should be mentioned that a large fraction of the film has already been formed at t =
4 s, but the remaining time period typically contributes with a few additional per-
centage points of film coverage fraction. To perform this extrapolation, the Curve
Fitting Toolbox of MATLAB may be employed to fit the surface film coverage α as
a function of time for each set-up of the parameters h and v. All cases were fitted
with a power-equation of the form

αhv(t) = atb + c (3.2)

where the constants a, b and c are given by the Curve Fitting Toolbox for each
given h and v. Furthermore, different set-ups of h and v give different constants to
this equation, so the constants could be fitted according to a=a(h,v), b=b(h,v) and
c=c(h,v) to provide one equation including both parameters.

27



3. Methods

3.6 Description of the Experimental Validation
Methods

The film appearance on the inner walls of a food packaging machine was investigated
using a small microscope. The microscope was attached with duct tape on the
outside of a window of the machine, pointing in to the machine, capturing the
film on the inside of the glass during film build-up and evaporation. Videos with
different magnification were recorded, however the best results were achieved with
a magnification of around 700x, which gave an appropriate resolution of the film
configuration. Snapshots were taken both during recording and afterwards using
the recorded video.

28



4
Results and Discussion

First of all, findings of the experiments using a microscope are presented to illus-
trate the film configuration on a µm level during different stages of the process. The
results of the sensitivity analysis of total simulation time and mesh configuration
is presented. Furthermore, the main results of the study are presented providing a
detailed description of how the film forms and develops in early stages of conden-
sation within mm-sized slits. The results of slit liquid film coverage is presented as
functions of the parameters h and v. Critical conditions for the parameters h and v,
and in dimensionless form with the Péclet number, are given as to ensure full film
surface coverage during film build-up stage. Finally, to complement the description
of liquid film disinfection, a description of the wall surface exposure to gas H2O2
is given. Finally, based on the experience and findings of this study, improvements
and suggestions for future studies are given.

4.1 Experimental results
Results from the experimental investigation is presented in the form of snapshots
taken from one of the microscope videos showing the process of film build-up. Figure
4.1 shows three snapshot taken from a video showing the film build-up process, at
different stages of the process. The left snapshot shows the early stage of condensa-
tion, which is characterized by very small droplets with a large mean free distance
between droplets. As condensation proceed, droplets grow to a size large enough to
cover the entire surface and for coalescence between droplets appear with high fre-
quency. The middle snapshot of Figure 4.1 is taken when the film build-up process
has proceeded for a while. The drying stage starts between the middle and right
snapshot. Since tiny droplets have a large ratio between interfacial area and droplet
volume, tiny droplets evaporate first. Consequently, the evaporation stage is char-
acterized by having fewer, but larger, drops. The snapshot to the right of Figure
4.1 shows the beginning of the drying stage. Snapshots of the film formation video
makes it possible to evaluate features such as droplet size distribution and droplet
number density at different time levels of the process, which could be implemented
as Rmin and N in Equations 2.49 and 2.50. It can be seen from the middle snap-
shot of Figure 4.1 that the largest droplets are 50-100 µm and the smallest visible
droplets are around 5 µm. However, it can not be concluded that droplets smaller
than 5 µm do not exist. Based on the findings of Rose [16], finding a maximum
droplet size ratio of 106, it is reasonable to not discard the existence of droplets
small enough to not be resolved with this magnification. Interesting to note from

29



4. Results and Discussion

Figure 4.1: Illustration of the evolution of condensate on inside walls of a food
packaging machine. The early stages of condensation is characterized by very small
droplets with a large mean free distance between droplets. The middle stage of
condensation has a wide range of droplet sizes. The drying stage to the right is
characterized by a lower amount of large droplets.

Figure 4.1 is the presence of nucleation free zones in a radius around large droplets
and small spots of dirt, where no primary condensation occurs. The phenomena of
nucleation free zones is reported by other authors, such as R. N. Leach et al [38].

4.2 Sensitivity Analysis of Total Simulation Time
and Mesh Configuration

As a part of result validation, two sensitivity analyses were performed, one which
was based on total simulation time, one which was a traditional grid-independence
check. The purpose of the sensitivity analysis is to evaluate how the uncertainty of
achieved results can be allocated to these two input parameters. The simulation time
analysis is presented in Section 4.2.1 and the grid-independence check is presented
in Figure 4.2.2.

4.2.1 Time Evolution of the Film
The analysis of time evolution of the film gives insight in when it is possible to stop
the simulation without affecting the final results to a large extent. Since the film
propagation clearly slows down after a few seconds of simulation, too long simula-
tions are not desired because it takes a lot of time while only affecting the results
marginally. Too short simulations are not desired because it will have an consid-
erable affect on resulting liquid film. However, it is important that the simulation
time reflects a time-scale that is relevant for the actual process. The evolution of
film propagation was captured during fifteen second runs. A sample of four cases
([h,v] = [2,0.5], [2,1.0], [3,1.0], [6,0.5]) not reaching α = 1 were chosen, running film
build-up for fifteen seconds each. The time dependency of slit film coverage for the
four chosen cases is shown in Figure 4.2. The four cases of Figure 4.2 illustrate
clearly in which cases the film coverage uncertainty is large for low simulation times
such as t = 4 s. Notice that for both cases of h = 2 mm, the rate of slit film build-
up has clearly diminished and is nearly zero. On the contrary, for the two upper
curves, the rate of film build-up is still high after t = 4 s, suggesting that certain

30



4. Results and Discussion

Figure 4.2: The film time evolution of four different cases, during the first fifteen
seconds of film build-up. The purple line is extended in graph 4.4, and illustrated
with film coverage scenes between t = 1 and t = 5 in Figure 4.3.

Figure 4.3: The film time evolution of the case h=3 mm and v=1 m/s, during the
first five seconds. The black line at t=5 s indicates the position of the film front at
t=4 s. An extension of the film coverage evolution is shown in Figure 4.4 up to t =
32 s.

combinations of h and v have a larger time scale.

Additionally, one case of the cases shown in Figure 4.2, h = 3 mm and v = 1, was run
for a longer time. As an illustration of film time evolution, snapshots every second
for the case of h = 3 mm and v = 1 m/s is shown in Figure 4.3. It is clearly demon-
strated that the rate of film propagation slows down quickly in the early stages of
simulation. In particular, the brief change in film coverage between t = 4 s and t =
5 s is demonstrated by the black line furthest to the right of the figure. In Figure
4.4, an extension to the film formation shown in Figure 4.3 is shown, where the time
period t = 5 s and t = 32 s is presented. The purpose of the long run was to confirm
the presented declining behaviour of slit film coverage and to investigate the trend
after a long time.

31



4. Results and Discussion

Figure 4.4: Time evolution of film build-up between t=5 s and t=32.5 s, for the
case of h=3 mm and v=1 m/s.

The graph of Figure 4.4 shows that the slit film surface coverage is constantly in-
creasing, but with a declining rate. For instance, the change of slit film coverage,
in percantage points, during the time period t = 25-30 s is ∆α = 1.1 %, as op-
posed to ∆α = 39.0 % during the time period t = 0-5 s. The drawn conclusion of
the simulation time dependency in this section is that it is a good approximation
to simulate for four seconds for cases showing a lower final slit film coverage. For
cases showing a larger final slit film coverage (e.g α > 0.5), there is an error to the
solution when using four second simulations. Four second simulations demonstrate
the general trends of this process. However, it should be kept in mind that the short
simulation time of four seconds provide a source of error to the final solution, as
demonstrated for example in the graph of Figures 4.2 and 4.4.

4.2.1.1 Extrapolation of Film Coverage for Increased Accuracy - Vali-
dation

According to the method described in Section 3.5.3, the solution of slit film coverage
was extrapolated to t = 8 s. A validation of the fitted curves was performed. A
comparison between simulation data from two of the fifteen second cases and their
respective fitted curves (fit based on the time-span t = 1-4 s) is shown in Figure
4.5. The accuracy of the fitted curve for v = 1.0 m/s is extraordinary and follows
the simulation data well up to t = 12 s. There is a lower accuracy for the case
of v = 0.5 m/s, however it is accurate enough to provide a great improvement
to the solution compared to simulation of t = 4 s. The conclusion is that the
results of the extrapolation are good enough to be used as a measure to accurately
obtain the solution at a later stage, for all designs, and at the same time strongly
reduce simulation time. Considering that the simulation speed is approximately 3
hours/simulation second, it is a result of great importance.

32



4. Results and Discussion

Figure 4.5: Comparison between simulation data and fitted curve for cases [h,v] =
[2,0.5] and [h,v] = [2,1.0]. The fitted curves was regressed based on the time-span t
= 1-4 s.

4.2.2 Grid-Independence
The results of simulations using a five million cell grid were almost identical with
the results when using three million cells. The surface film coverage at t=2.5 s for
different slit widths h is presented in table 4.1 for the two different mesh setups.

Table 4.1: Proportion of surface film coverage αhv=α(h, v) at t=2.5 s using two
different mesh setups, one with three million cells and one with five million cells.

Case αhv 3 million cells (%) 5 million cells (%) %-point deviation
α21 10.09 10.09 0.00
α31 28.22 27.95 0.27
α41 51.84 51.63 0.21

From the table it can be noticed that the film coverage fraction is identical for h=2
mm, while differing 0.27 and 0.21 percentage points for h=3 mm and h=4 mm,
respectively. The cell size in boundary layer is accordingly refined as described in
the method Section 3.2. Since the results from using three million cells and five
million cells are close to identical, the solution obtained using a mesh setup of three
million cells is considered grid-independent.

4.3 Flow Characteristics Inside and in Vicinity of
Slits

For all of the design cases presented in Table A.1, results exist in the form of snap-
shots showing; 1. the hydrogen-peroxide gas ppm levels within the slit and liquid

33



4. Results and Discussion

film distribution, 2. the liquid film distribution within the slit and 3. liquid film
distribution on the wall neighbouring the plate. The frequency of snapshots are 10
snapshots per second. Due to the high number of snapshots captured, only a few
illustrative examples will be presented in this section. For the case of h = 3 mm
and v = 0.5 m/s, the hydrogen-peroxide gas ppm levels and liquid film distribution
within the slit are illustrated in Figure 4.6. It can be seen from Figure 4.6 that
the concentration front of hydrogen-peroxide permeates a small distance into the
slit before it diminishes to levels below 100 ppm. The reason for the rapid draining
of H2O2 once entering the slit is that the liquid film on both slit walls provide a
H2O2 mass sink. The point where hydrogen-peroxide is drained corresponds well
with the point of how far the liquid film reaches into the slit. Furthermore, there is
only a limited amount of H2O2 entering the slit, the amount which enters the slit
is reduced mainly by two factors; the reduced entrance area available for the gas to
enter narrow slits, compared to wider slits, and the concentration boundary layer
that forms along the wall before entering. As opposed to a liquid film neighbouring
a bulk gas flow, which always provides fresh H2O2 to the film due to turbulence and
the large availability of bulk H2O2, the provision of H2O2 is throttled within a slit.
In the next section it is presented how far into the slit the liquid film reaches as
function of the parameters h and v.

Figure 4.6: An illustrative example of liquid film distribution and gas ppm levels
of H2O2 after t = 4 s film build-up. The parameter values are h = 3 mm and v =
0.5 m/s.

34



4. Results and Discussion

Figure 4.7: An illustrative example of liquid film distribution and gas ppm levels
of H2O2 after t = 4 s film build-up. The parameter values are h = 7 mm and v =
0.5 m/s.

For h = 7 mm, corresponding gas H2O2 concentration and film distribution is shown
in Figure 4.7. The large difference between narrow and wider slits are apparent when
comparing Figure 4.6 and Figure 4.7. In the case of h = 7 mm, the accessibility of
high H2O2 levels over the entire slit is reflected in the complete liquid film coverage.
Levels of 2000 ppm H2O2 reach as far as half of the slit length, for h = 7 mm, while
not entering the slit at all for h = 3 mm. Interesting to note from Figure 4.7 is that
the concentration profile within the slit is not fully developed even at the end of the
slit, after t = 4 s. It would be interesting to also include the influence of the slit
length L for future studies.

The hydrodynamic behaviour of gas around the slit is illustrated by the velocity
field in Figure 4.8 and by turbulent kinetic energy in Figure 4.9. It is clear that,
within the slit, the velocity is much lower than the bulk velocity and that there is an
absence of turbulent kinetic energy. The turbulence within a slit can be particularly
interesting to analyse when it comes to the transition from laminar to turbulent
flows, since turbulent flows generate greater mixing which will influence the liquid
film condensation and evaporation. However for the lower range of h parameters
in this study, the Reynolds number is not close to the value where transition to
turbulent flow occurs.

35



4. Results and Discussion

Figure 4.8: An illustration of the velocity field around a h = 3 mm slit. The bulk
velocity is v = 0.5 m/s.

Figure 4.9: An illustration of the Turbulent Kinetic Energy field around a h = 3
mm slit. The bulk velocity is v = 0.5 m/s.

4.4 Final Surface Film Coverage Fraction
The most fundamental way to present the simulation results is to explicitly describe
the dependency of the parameters h and v on the slit film coverage fraction α. Fig-
ure 4.10 shows the slit film coverage data, at t = 4 s, obtained from the design
study as function of slit width, in the form of iso-velocities. It is apparent from
Figure 4.10 that there is a strong gas velocity dependency, especially for low gas
velocities. Consequently, it can also be noted that the slit width at which the slit
surface are fully covered with a liquid film differs considerably between gas velocities.

After extrapolation, the result at t = 8 s is shown in Figure 4.11. It is apparent
that the solution is different, and according to the analysis in the previous chapter,

36



4. Results and Discussion

Figure 4.10: The slit surface film coverage α at t = 4 s as a function of slit width
h, for different iso-velocities.

more accurate. A comparison between Figure 4.10 and Figure 4.11 is presented in
Appendix I, Figure A.2.

4.4.1 Critical Parameter Conditions for Complete Slit Film
Coverage

By interpolating the various iso-velocities of Figure 4.11 onto α = 1, a critical slit
width hcrit is obtained that describe the criteria of h to ensure full slit film coverage.
A plot of the critical slit widths hcrit to achieve 100 %, 75 % and 50 % slit surface
film coverage, as a function of v, is shown in Figure 4.12. By fitting a power function
to the data points of α = 1 in Figure 4.12, an expression describing the critical slit
width is obtained as

hcrit = avb + c (4.1)
where a = 6.44, b = -0.2838 and c = -1.84. Equation 4.1 and Figure 4.12 describe
the criteria in h and v explicitly, however in a more compressed form the criteria
can be expressed in the dimensionless mass transfer Péclet number

Peh = hvh
DAB

(4.2)

Note that for the Péclet number, the velocity is not the gas bulk velocity v, but the
mean gas flow velocity within the slit vh. Mean gas flow within slits are presented
in comparison to the gas bulk velocities in Table A.2. However the function of slit
mean velocity as function of h and v was fitted to a second degree polynomial

37



4. Results and Discussion

Figure 4.11: The slit surface film coverage α at t = 8 s as a function of slit width
h, for different iso-velocities. The solution has been obtained by time-extrapolation
in the interval t = 4-8 s.

Figure 4.12: The critical slit widths, as to ensure 100 %, 75 % and 50 % slit surface
film coverage, as a function of external gas flow velocity v.

38



4. Results and Discussion

Figure 4.13: The slit surface film coverage fraction as plotted against the dimen-
sionless mass transfer Péclet-number. The blue circles represent the raw simulation
data and the orange line is the least square fit.

vh(h, v) = 0.0705− 0.4039h− 0.2433v + 0.0012h2 + 0.1376hv + 0.0576v2 (4.3)

The slit surface film coverage α is plotted against Peh in Figure 4.13, both as data
points and a least square fitted power function. The power function of Figure 4.13
is described according to

α = 0.03491Peh0.733 (4.4)

if Peh < 97

α = 1 (4.5)

if Peh > 97. From the fitted model of Equation 4.4, a critical parameter condition
can be extracted as Peh = 97. However, there is reason to believe that the criteria
of Peh = 97 is slightly over-predicted, because the actual process uses a film build-
up phase that lasts longer than eight seconds. Simulating for a longer time would
increase the coverage fractions α and consequently lower the requirements on the
governing Péclet number. To complete the model of Equation 4.4, Equation 4.3
describing vh(h, v) can be inserted so that the model is expressed explicitly in h and
v.

39



4. Results and Discussion

Figure 4.14: Illustration of the elevated concentration levels of gas H2O2 in the
slit, during drying stage. The parameter values are h = 6 mm and v = 0.5 m/s. The
snapshot is taken after 12 seconds of simulation, four seconds film build-up followed
by eight seconds drying.

4.5 Slit Surface Exposure to Gas H2O2

The results of the liquid film analysis in the previous section showed that not all
parameter combinations of h and v resulted in a complete liquid film coverage within
the slit. When an area is not covered by a liquid film, the area must instead be ex-
posed to high enough gas concentration levels of H2O2. The result of the H2O2 gas
exposure investigation shows that the slit gas concentration levels of H2O2 undergoes
a strong increment during the course of the drying phase. Figure 4.14 shows the ppm
levels of H2O2 within the slit, after totally 12 seconds simulated, four second film
build-up and eight seconds drying. Interesting from Figure 4.14 is the completely
reversed H2O2 concentration profile as compared to during film build-up, which can
be visualized by comparing Figure 4.7 with Figure 4.14. In other words there is a
tendency that areas not covered by liquid films are, in some cases, complemented
through gas sterilization in an effective way. The results indicate that the liquid
films, both within the slit and on the wall neighbouring the slit, provides a source of
H2O2 to the gas during the drying phase. In the case of evaporation, there is a re-
versed concentration boundary layer, compared to that during condensation, where
the concentration of H2O2 is largest close to the wall. The interesting phenomena
of liquid film evaporation next to the slit (to the right of Figure 4.14), followed by
the entrance of gas into the slit due to advection transport, can provide an effective
mechanism to allow high ppm levels of gas H2O2 to enter narrow slits.
On the other hand, the drying mechanism for very small slits, e.g h = 2 mm, does not
follow the same pattern as for the case shown in Figure 4.14. In terms of liquid film
behaviour, the difference comparing narrow and wider gaps (e.g 2 mm and 6 mm)
is mainly shown in how far into the slit the liquid film reaches. The spread of liquid
film will in turn affect the evaporation behaviour during the drying phase and thus
also concentration levels of H2O2 gas. Figure 4.15 shows the concentration profile of
H2O2, eight seconds after the initiation of drying phase. It is shown from Figure 4.15
that the beginning of the slit provide a source of H2O2 due to an evaporating liquid
film. It is also illustrated in Figure 4.15 that the concentration boundary layer,

40



4. Results and Discussion

Figure 4.15: Illustration of the elevated concentration levels of gas H2O2 in the
slit, during drying stage. The parameter values are h = 2 mm and v = 0.5 m/s. The
snapshot is taken, similarly to Figure 4.14, eight seconds after initiation of drying
phase.

beside the slit, during the drying stage causes H2O2 to blow into the slit. With a
narrow gap such as 2 mm, the H2O2 introduced to the slit easily gets concentrated
due to the very limited space. After a while, the gas velocity in the slit is carrying
the H2O2 concentrated gas through the slit in parallel with ongoing evaporation.

4.6 Improvements and Suggestions for Future Stud-
ies

In this section, suggestions and necessary improvements for future studies will be
presented based on the experience and results of this study.

First of all, the model from which the distribution and spreading of the dropwise liq-
uid film is calculated should be validated against experimental results to a greater
extent. In particular, the validity of the employed surface tension model, with a
contact line force at δf = 0.1 µ