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Numerical Heat Transfer, Part A:
Applications: An International Journal of
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Numerical Investigation of Evaporation in
the Developing Region of Laminar Falling
Film Flow Under Constant Wall Heat Flux
Conditions
a
a
a
R. S. Maurya , S. V. Diwakar , T. Sundararajan & Sarit K. Das
a
a
Department of Mechanical Engineering, Indian Institute of
Technology Madras, Chennai, India
Published online: 21 Jul 2010.
To cite this article: R. S. Maurya , S. V. Diwakar , T. Sundararajan & Sarit K. Das (2010) Numerical
Investigation of Evaporation in the Developing Region of Laminar Falling Film Flow Under Constant
Wall Heat Flux Conditions, Numerical Heat Transfer, Part A: Applications: An International Journal of
Computation and Methodology, 58:1, 41-64, DOI: 10.1080/10407782.2010.490174
To link to this article: http://dx.doi.org/10.1080/10407782.2010.490174
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Numerical Heat Transfer, Part A, 58: 41–64, 2010
Copyright # Taylor & Francis Group, LLC
ISSN: 1040-7782 print=1521-0634 online
DOI: 10.1080/10407782.2010.490174
NUMERICAL INVESTIGATION OF EVAPORATION IN
THE DEVELOPING REGION OF LAMINAR FALLING
FILM FLOW UNDER CONSTANT WALL HEAT
FLUX CONDITIONS
Downloaded by [Eastern Michigan University] at 01:06 21 July 2013
R. S. Maurya, S. V. Diwakar, T. Sundararajan, and Sarit K. Das
Department of Mechanical Engineering, Indian Institute of Technology
Madras, Chennai, India
A finite-volume-based incompressible flow algorithm on Cartesian grid is presented for the
simulation of evaporation phenomena in a falling liquid film under low wall superheat conditions. The model employs the PLIC–VOF method to capture the free surface evolution,
and the continuum surface force (CSF) approximation to emulate the effects of interfacial
tension. The phase change process is represented through a source term in the interfacial
cells, which is evaluated from the normal temperature gradients on either side of the interface. In order to evaluate these discontinuous temperature gradients across the interface
accurately, a simple and efficient ghost fluid method has been implemented, which properly
takes into account the dynamic evolution of the interface. The overall numerical model,
including the phase change algorithm, has been validated against standard benchmark analytical results. Finally, the model is used to simulate the evaporating flow of a 2-D laminar,
developing film falling over an inclined plane surface, subjected to constant wall heat flux.
The results thus obtained, clearly illustrate the significance of inertial effects in the developing region of the falling film, which are generally neglected in the available analytical
models. It is also observed, that the evaporation of fluid commences only after the growing
thermal boundary layer reaches the interface, and the length of the nonevaporating section
reduces with the increase in wall heat flux value.
1. INTRODUCTION
Liquid film emerging from a distributor and flowing over a surface under the
action of gravity is commonly encountered in systems such as distillation columns,
evaporators, concentrators, food processors, and water desalinators. Generally,
liquid films undergo a development phase where the flow pattern changes continuously before a steady developed behavior is established. This developing region plays
a significant role in the optimal design of many chemical process equipments and,
hence, a proper understanding of its hydrodynamic and thermal behavior has a great
impact on the system development.
Received 27 October 2009; accepted 8 April 2010.
Address correspondence to Sarit K. Das, Heat Transfer and Thermal Power Laboratory,
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036,
India. E-mail: [email protected]
41
42
R. S. MAURYA ET AL.
NOMENCLATURE
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Cp
F(x, y, t)
Fv
g
hy
k
L
Lp
_
m
^
n
n
p
q
Re
Sc
Si
t
T
u
v
v
V
x, y
specific heat
color function
volume force
acceleration due to gravity
local heat transfer coefficient
thermal conductivity
latent heat of vaporization
length of phase
interfacial mass transfer rate per
unit area
unit normal
face normal
pressure
heat flux
film Reynolds number per unit
width
control volume surface
control volume interface
time
temperature
x-component of velocity vector
velocity vector
y-component of velocity vector
volume
coordinate directions
Le
h
b
d
/
m
q
r
s
Subscripts
1
0
av
e
i
l
lv
sat
v
w
nondimensional distance along plate
from the slit
angle of inclination of plane from
horizontal
ratio of film thickness to developed
film thickness (nondimensional film
thickness)
film thickness
fluid property
dynamic viscosity
mass density
surface tension
stress
developed region value
value at the inlet (slit)
average
evaporation
interface
liquid
ghost node of liquid region in the vapor
region
saturation condition
vapor
wall
Generally, the liquid to vapor phase transformation in a falling film can occur
via two modes. The first mode prevails in situations where a high degree of wall
superheat is available in the system that causes an onset of nucleation at the wall surface. Subsequently, vapor bubbles formed at the nucleation sites grow continuously
until the buoyancy and inertial forces make them depart and migrate to the liquid–
vapor interface. The other mode of phase change involves a pure evaporation process at the liquid–vapor interface, where no vapor bubble is formed at the heating
wall. Of these processes, nucleate boiling has a higher heat transfer coefficient as
compared to the pure evaporation process. Despite this fact, in several special industrial applications the evaporative mode of phase change is the preferred process than
the former. Fruit and vegetable juice evaporators are examples of such applications
where the low degree of wall superheat is preferred. This is mainly due to the sensitivity of juice to high temperature, which tends to degrade its taste and nutrients. So,
fruit-juice processing devices generally operate at a very low range of temperature
difference (3–10 C) depending on the heat transfer surface and the nature of the
liquid. Hence, quantifying the operational characteristics of such devices requires
a clear understanding of the behavior of falling film evaporation process, which is
the focal point here.
In the past, major work on modeling of hydrodynamics and the other transport
processes in evaporating falling film has been carried out through analytical
approaches, which mainly correspond to the fully developed region. A detailed review
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EVAPORATION OF LAMINAR FALLING FILM FLOW
43
of the available mathematical methods concerning the phase change process can be
found in the paper by Bankoff [1]. A fully developed evaporating liquid film flowing
along an inclined wall has been addressed by Murty and Sastri [2]. Many other works
[3–6] have also focused on the developed evaporating film region subjected to various
thermal and flow conditions. However, analyses pertaining to the fully developed
region alone do not provide a complete representation of falling film behavior,
as in actual systems the development region covers a significant portion. The characteristics of developing film region are governed by nonlinear interactions between
the flow, thermal, and phase change processes, which cannot be treated analytically
and hence, the focus needs to be shifted towards numerical simulations.
Several approaches have been introduced in the last two decades to address
the numerical challenges in phase change problems arising due to an evolving interface with discontinuous variation of properties across it. The first attempt in this
direction was by Welch [7], who simulated the two-dimensional film boiling process
using moving triangular grids, and this was further extended by Son and Dhir [8,
9]. Another landmark mathematical model for simulating the liquid–vapor phase
transition was presented by Juric and Tryggvason [10], to which further improvements in handling the interfacial conditions were suggested by Son [11]. A volume
of fluid (VOF) method-based adaptation of Juric and Traggvason’s [10] phase
change model was developed by Welch and Wilson [12], in which a mass source
term was introduced to emulate the phase change process. The capability of these
approaches has been demonstrated in droplet vaporization situations by Lacas et al.
[13]. Recent developments in the modeling techniques, include the energy of fluid
(EOF) method by Anghaie et al. [14] and the enthalpy-based formulation by
Shin and Juric [15]. In the current work, we choose an approach similar to that
of Welch and Wilson [12], where the VOF method has been used to track the
dynamics of interfacial fluid structures.
In view of the small mass transfer rates observed in the evaporation process,
the interface tracking process has to be very accurate since any numerically-induced
mass imbalance would hamper the effective characterization of phase change process. Hence, in this regard, the VOF method with its perfect volume conservation
characteristics has a definite edge over other multiphase tracking and capturing
techniques. However, even upon using the VOF method, special care is required
for the implementation of interfacial condition (say, the prescription of saturation
temperature) and the accurate estimation of mass source pertaining to the evaporation process. Particularly, with respect to the estimation of temperature gradients
on either side of the interface, the approach should be consistent with the drastic
change in fluid topology. Such sophistication is possible using the ghost node technique suggested by Fedkiw et al. [16]. Morgan [17] and Gibou et al. [18] have used
the ghost node technique coupled with level-set to implement the interfacial
condition and to capture the interfacial jump conditions accurately. The numerical
formulation for imposing accurate jumps in this method was further improved
and simplified by Tanguy et al. [19].
In the current work, a numerical model has been developed that couples the
VOF method and the ghost fluid technique to estimate the mass source arising out
of the evaporation process at the interface accurately. A simple and efficient way to
extrapolate information of ghost nodes on either side of the interface has been
44
R. S. MAURYA ET AL.
suggested. This coupled technique has been used to simulate the transient evaporating
characteristics in the developing region of a laminar liquid film. The simulations highlight the effects of wall heat flux on the temperature and velocity distributions across
the film, and the variation of wall temperature, film thickness, and evaporation rate.
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2. MATHEMATICAL MODEL
Consider a liquid film emerging from a 2-D slit and flowing over an inclined
plane wall under the action of gravity (Figure 1). The flow is assumed to be in the
laminar regime and a constant heat flux condition is imposed at the wall. As a result,
both the thermal and hydrodynamic boundary layers develop simultaneously until
they achieve asymptotic developed profiles. The behavior of the film in the developing region is fundamentally governed by gravitational, viscous, and surface tension
forces whose interactions determine the local features of development. In the case of
an evaporating liquid film, these forces render the hydrodynamics complex and
nonlinear, while the phase change introduces complexities in the associated heat
and mass transfer. The following assumptions are invoked to simplify the complex
interactions related to these processes.
The fluids are assumed to be incompressible and Newtonian with constant
thermophysical properties.
. Buoyancy effects are neglected in both phases.
.
Figure 1. Schematic diagram of evaporating falling liquid film.
EVAPORATION OF LAMINAR FALLING FILM FLOW
45
The interface is assumed to be at the saturation temperature corresponding to the
prevailing pressure. Hence, a constant surface tension value is employed for
the whole interface.
. The effects of kinetic energy, viscous dissipation, and surface tension work are
neglected in the energy equation.
. The gaseous medium is assumed to consist only of the vapor of liquid under
consideration.
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.
Subjected to these assumptions, the equations governing the laminar, incompressible flows of an evaporating liquid film and the ambient vapor medium can
be written for each control volume (computational cell) as
Mass balance
ZZ
Z
ðqv nÞdS þ ½qðv vi Þ n dS ¼ 0
ð1Þ
Sc
Si
Here, [/] indicates a jump in / across the interface.
Momentum balance
ZZZ
ZZ
ZZZ
q
qv dV þ
qðv nÞv dS ¼
ðqg rpÞ n dV
qt
V
Sc
V
ZZ
þ
ðs nÞdS
Sc
ZZZ
þ
F v n dV
ð2Þ
V
Energy balance
q
qt
ZZZ
V
qCp T dV þ
ZZ
qCp Tðv nÞdS ¼
Sc
ZZ
q n dS
ð3Þ
Sc
The second term of Eq. (1) corresponds to the interfacial mass source due to
the evaporation process, and it assumes significance only for interface cells where
the local thermal conditions favor the phase change process. The last term in
Eq. (2) represents the effects of surface tension, which is approximated as a
volumetric force in the vicinity of the interface, as discussed later.
The computational domain used in the current analysis of evaporating falling
film flow is shown in Figure 1. The boundaries of the domain are represented by
broken lines, and various conditions applied at these surfaces are listed below.
Boundary—A
A parabolic velocity profile is imposed at the inlet as follows.
" #
x
x 2
Inlet parabolic velocity; u ¼ 6uav ð4Þ
d0
d0
Boundary—B
qu
qv
qT
¼ 0;
¼ 0;
¼0
qy
qy
qy
and
p¼0
ð5Þ
46
R. S. MAURYA ET AL.
Boundary—C
qu
qv
qT
¼ 0;
¼ 0;
¼ 0 and
qx
qx
qx
p¼0
ð6Þ
qu
qv
qT
¼ 0;
¼ 0;
¼0
qy
qy
qy
p¼0
ð7Þ
Boundary—D
and
Boundary—E
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u ¼ 0;
v ¼ 0;
and
q ¼ constant
ð8Þ
At the interface
T ¼ Tsat (saturation temperature corresponding to the prevalent system
pressure)
3. SOLUTION ALGORITHM
In the current work, the unified governing equations (Eqs. (1)–(3)) corresponding to incompressible flow of the falling film and the vapor medium are discretized
using the finite-volume method. These discretized equations are marched explicitly
in time and proper pressure-velocity coupling is obtained by the use of SIMPLE
algorithm. The evolution of the liquid free surface is tracked with the help of a
density-based VOF method, and the associated interfacial tension is modeled as a
localized volumetric force based on the continuum surface model (CSF) approximation of Brackbill et al. [20]. Finally, modeling of the phase change process has
been carried out using the approach of Welch and Wilson [12], which employs a mass
source term to emulate the evaporation process at the interface. The details of these
models are discussed in the following subsections.
3.1. Density-Based VOF Approach to Model the Evaporation Process
A detailed review of the general VOF method can be found from the works of
Rider and Kothe [21] and Scardovelli and Zaleski [22]. In this method, the fluid
structures in the computational domain are identified with the help of a scalar
phase indicator called the liquid volume fraction, F. The basic relationship between
the F–value and the phase contained in the cell is given as
Cell with liquid phase F ¼ 1
Mixed cell (including interface)
Cell with gas phase F ¼ 0
0<F <1
ð9Þ
Here, the evolution of free surface in the domain is achieved by means of two
steps: namely, interface reconstruction and fluid advection. In the reconstruction
process, the 2-D interface segments are assumed to be straight lines (piecewise linear
interface calculation (PLIC)) in each cell whose approximate orientation and
EVAPORATION OF LAMINAR FALLING FILM FLOW
47
position are obtained through Youngs’ multidimensional stencil [23]. The fluid
advection process then geometrically estimates the liquid flux through the cell faces
and effectively tracks the dynamics of liquid structures by employing the volume
conservation equation given below.
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DF qF
¼
þ ðv rÞF ¼ 0
Dt
qt
ð10Þ
The essential features of the current VOF implementation and the procedure
involved in the evaluation of the surface tension force are essentially similar to those
used in the work concerning the hydrodynamics of falling liquid film by Maurya et al.
[6]. However, slight modifications in the conservation process have been incorporated here, since in situations involving change of phase, a procedure incorporating
the direct mass balance is more advisable than the volume conservation process
(Eq. (10)). Hence, a density-based approach has been employed to model the evolution of evaporating interface, where the fluxes obtained from the advection process
are used to evaluate the density directly through the mass conservation equation
given below.
q
qt
Z
qdV þ
Z
V
qv n dS ¼ 0
ð11Þ
Sc
With the new density field thus obtained, the volume fraction in each of the
cells is updated through the relation
F¼
q qv
ql qv
ð12Þ
3.2. Numerical Estimation of Phase Change Process
In a typical multiphase problem involving phase change process, additional
jump conditions pertaining to interfacial normal velocity and heat flux are encountered along with the general discontinuity of properties. Hence, the numerical formulation concerning the conservation of mass, momentum, and energy should be
adequately sophisticated to handle these jump conditions accurately. The discontinuity in the velocity component normal to the interface is given by,
_e
vv n vl n ¼ m
1
1
qv ql
ð13Þ
_ e Þ can be expressed in terms of the difference in normal
where the evaporation rate ðm
heat flux on either side of the interface, as
_e¼
m
ql qv
L
ð14Þ
48
R. S. MAURYA ET AL.
With the knowledge of the interface orientation and the segment length
obtained from the PLIC interface reconstruction procedure, the velocity jump evaluated through Eq. (13) can be used to estimate the mass source (second term on the
left-hand side of Eq. (1)) in an interfacial cell as
ðql qv Þ ^n 1
1
^ dS ¼ q
Mass source ¼ ½qðv vi Þ n
interface area
L
qv q l
ð15Þ
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In the above equation, the jump in normal heat flux can be written in terms of
the temperature gradients as
qT
qT
ðql qv Þ ^
n ¼ kv ðrTv ^
nÞ kl ðrT1 ^nÞ ¼ kv
kl
qn v
qn l
ð16Þ
From Eqs. (15) and (16), it is evident that the estimation of jump in heat
flux requires accurate computation of normal temperature gradients on either
side of the interface. Unfortunately, this information cannot be directly
extracted from the unified energy equation (Eq. (3)), which considers the two
phases as a single continuous medium with spatially varying thermophysical
properties. Also, due to the discontinuity of temperature gradient at the interface, the information used to estimate the normal gradients should be confined
to the respective phases and this involves preserving the identity of mesh points
with respect to the phases under consideration. Hence, the temperature gradients
evaluated in the individual phases can be extrapolated to the interface in order
to estimate the mass source term in Eq. (15). However, such an approach is
not trouble-free, particularly when the fluid structures are subjected to drastic
topological changes with respect to time. Figure 2 shows a typical interface
whose movement changes the association of certain nodes from the liquid
phase to vapor phase during the time step Dt, in a fixed Cartesian mesh. As a
result, the liquid temperature information that was associated with these nodes
becomes irrelevant, which in turn gives rise to an ambiguous situation when
these nodes are used to estimate the normal temperature gradient in the vapor
phase region. However, such an awkward situation can be successfully handled
using the ghost fluid method (GFM) proposed by Fedkiw et al. [16]. The GFM
implicitly treats the interfacial discontinuity by introducing ghost fluid nodes
in a thin band of Cartesian mesh points across the interface where the information pertaining to one fluid is extrapolated to the ghost fluid point on the
other side of the interface. This can be explained with the help of Figure 2a, where
the vapor and liquid ghost nodes derive information by extrapolation of values
from the actual vapor and liquid region at a particular instant. In the case of
drastic changes in fluid distributions, the nodes are self-contained with the information corresponding to the newly associated phase and this paves way for a
proper estimation of temperature gradients in the individual phases without
the influence of the complementary phase. The extrapolation procedure for the
nodal values and the estimation of temperature gradients are explained in
the following subsection.
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EVAPORATION OF LAMINAR FALLING FILM FLOW
49
Figure 2. (a) Ghost node and moving interface, and (b) ghost node construction and estimation of normal
temperature gradient.
3.3. Estimation of Ghost Node Values and Normal
Temperature Gradient
The current implementation of the ghost fluid method uses linear extrapolation
to estimate the values of temperature at the ghost nodes, which lie within the range
of max(2dx, 2dy) distance normal to the interface, where dx and dy are the cell
dimensions in the x- and y-directions, respectively. At each time step, the extrapolation process starts after the interface reconstruction procedure of VOF method,
which identifies the orientation and location of interface in each of the mixed cells
in the domain. The estimation of the ghost node value is then based on a nine cell
stencil, as shown in Figure 2b, where one reference node is chosen from each of
the phases. The selected reference nodes may be two of the extreme diagonal nodes
in the nine cell stencil, which are designated in Figure 2b, as fsw, fse, fne, or fnw corresponding to the farthest southwest, southeast, northeast, and northwest nodes,
respectively. For the illustrative fluid configuration shown in Figure 2b, node fsw
and fne can be chosen as reference points for the calculation of the liquid side
ðdT=dnÞjl and vapor side ðdT=dnÞjv temperature gradients.
Since the interface is assumed to be at saturated condition, the liquid side
normal temperature gradient can be written as
dT Tfsw Tsat
¼
dn l
dn
ð17Þ
where, dn is the normal distance from node fsw to the interface. Using the normal
temperature gradient thus obtained, the temperature value at the ghost liquid nodes
50
R. S. MAURYA ET AL.
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lying at a normal distance of dnl from the interface in the vapor region can be
evaluated as
dT
Tlv ¼ Tfsw þ
ðdn þ dnl Þ
ð18Þ
dn l
A similar approach can be applied on the vapor side to estimate the vapor phase
ghost node temperatures and the normal gradients which can be used to effectively
handle the ambiguous situations where grid points undergo a sudden change in their
phase association. Also, with the estimated temperature gradients ðdT=dnÞjl and
ðdT=dnÞjv , the source term in the mass conservation equation, Eq. (3), can be directly
calculated without any extra computational effort. Despite the use of linear extrapolation procedure, the current technique yields second order accuracy as evident from
the work of Gibou et al. [18]. Also, it is more convenient compared to the probe
method suggested by Udaykumar et al. [24].
3.4. Interfacial Velocity Conditions
Akin to the issues faced during the evaluation of temperature gradients, estimation of interfacial velocity also suffers due to the change in the association of grid
points with individual phases. In order to avoid this difficulty, the methodology
introduced by Nguyen et al. [25] has been used in the current work where a band
of ghost node velocities is defined on both sides of the interface. At every grid
location corresponding to the vapor phase, a liquid ghost velocity is obtained using
the evaporation rate, as follows.
!
1
1
1
1
_e
_e
ulv ¼ uv m
ð19Þ
ny
nx vlv ¼ vv m
qg ql
qv ql
Here, nx and ny are the direction cosines of the interface normal. Similarly, the vapor
phase ghost velocities can also be calculated at appropriate points.
The complete algorithm to model the flow of an evaporating liquid film can
now be summarized as follows.
1. Based on the initial fluid configuration for a time step, the approximate orientation and position of the PLIC interfaces are obtained using Youngs’ multidimensional stencil.
2. Using the known velocity field vn, the liquid flux through each cell face is estimated. This, in turn, is used to track the dynamics of fluid structures by estimating the density field at the new time level as
q
nþ1
dt
¼q þ
V
n
Z
qðvn nÞ dS
ð20Þ
Sc
3. The liquid volume fraction distribution (F) in the domain for the new time level is
calculated using Eq. (12) and, once again, the interfaces are reconstructed in all of
mixed cells.
EVAPORATION OF LAMINAR FALLING FILM FLOW
51
4. The various thermophysical and transport properties are evaluated in all of the
cells using the volume fraction F as weighting function.
For transport properties such as viscosity and thermal conductivity, the relation is
1
F ð1 F Þ
¼ þ
u ul
uv
ð21Þ
and for the other thermophysical properties, the relation is given as
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u ¼ ul F þ uv ð1 F Þ
ð22Þ
5. With the known interface positions in the mixed cells, liquid and vapor nodes are
properly identified in the domain and their vapor and liquid ghost node counterparts are created in the vicinity of the interface.
6. The temperature field in the domain is updated using the energy conservation
equation of the form
T
nþ1
k
2
¼ T þ dt v rT þ
r T
qCp
n
ð23Þ
Here, necessary care is taken to select either the original node or the ghost
node value depending upon the change in phase association of nodes over the
time step Dt.
7. The temperature gradients on either side of the interface are accurately evaluated
in all mixed cells and the resultant mass source term is calculated using Eq. (12).
8. Finally, the iterative solution procedure of the SIMPLE algorithm is used to
update the pressure and velocity fields. Here, using the body force obtained
from the CSF approximation and a guessed value of pressure, the momentum
equations are marched in time to arrive at the guessed velocity field. The resultant
mass imbalance corresponding to the guessed velocity field is used to obtain the
correct pressure value iteratively by solving the pressure correction equation.
However, with respect to the phase change process, the mass source term
evaluated in Eq. (15) is subtracted from the overall mass imbalance, and a revised
residue is used at each iteration.
4. VALIDATION OF THE EVAPORATIVE FLOW MODEL
In order to access the efficacy of the basic evaporation model, two standard
benchmark problems are considered here for which analytical solutions are readily
available from the work of Jamet and Duquennoy [26]. The first problem deals with
the phase change process in a saturated liquid layer caused because of external heat
supplied at the bottom boundary of the domain. The second test problem involves a
superheated liquid layer and a saturated vapor layer with no other source of heat.
Such a configuration forms a meta-stable system where the phase change process
is driven by the superheat available with the liquid. The results obtained for these
test cases using the current model are discussed in the remainder of this section.
52
R. S. MAURYA ET AL.
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4.1. One-Dimensional Liquid-Vapor Phase Change
Figure 3 shows the configuration of the first test problem with the corresponding boundary conditions. Here, the stratified liquid and vapor layers are initially at
rest (no flow) and at saturation temperature corresponding to the system pressure
(Patm). The bottom boundary of the domain is subjected to constant heat flux condition. Initially, the major part of the heat supplied is utilized in sensible heating of
the vapor layer during which the phase change process is sluggish or nonexistent.
Once temperature gradients are established in the vapor phase, the phase change
process at the interface slowly picks up and attains an asymptotic behavior. The
vapor mass formed due to the phase change process gently pushes the liquid layer
in the upward direction. Analytical estimates of this liquid velocity and the interfacial velocity can be obtained from the works of Jamet and Duquennoy [26]. In the
current simulations, the fluid properties are selected such that the Peclet number
in vapor phase is constrained to be much less than one (Pe51). This leads to the
dominance of conduction process in the vapor region, as considered by Jamet and
Duquennoy [26]. The computational domain chosen for the current analysis is of size
1 4 units and is discretized using a grid of 20 80 cells. The initial height of the
vapor column is chosen to be 3.5% of the vertical length in order to shorten the
initial transients in the simulation. Various nondimensional parameters and property
ratios used in the simulation are given below.
qref ¼ 1:0;
¼ 4:0;
Tsat ¼ 0:0;
L ¼ 1;
q ¼ 0:05;
ql =qv ¼ 2:0;
ml =mv ¼ 1:0;
Cpl =Cpv ¼ 1:0
kl =kv
ð24Þ
The temporal variations of interfacial and liquid region velocities are plotted in
Figures 4a and 4b. Evidently, the velocities exhibit an initial transient growth and an
Figure 3. One-dimensional phase change problem with imposed boundary heat flux.
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EVAPORATION OF LAMINAR FALLING FILM FLOW
53
Figure 4. Comparison of transient prediction with 1-D steady state results. (a) Interface velocity and
(b) liquid velocity.
asymptotic convergence to the steady state values obtained from the analytical
results [26].
4.2. Unsteady Phase Change of a Superheated Liquid
In the second test case, stratified fluid layers are also considered. The liquid
column here is at superheated condition and the bottom boundary is assumed to
be adiabatic. Despite the absence of an active heat source, the phase change process is triggered and sustained in this problem mainly by the superheat of the
liquid column. Akin to the previous problem, both the phases are initially at rest
and the vapor phase is at saturation temperature (Tsat). The liquid region in the
system is initialized to a uniform superheated temperature (Tsat þ DT), which
induces a sharp temperature jump at the interface. The uniform initial conditions
in the individual phases give rise to zero temperature gradients on either side of
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54
R. S. MAURYA ET AL.
the interface. However, any small perturbation to this meta-stable system initiates
the phase change process at the interface, and the sharp interfacial temperature
jump is smoothened by the diffusion process. The dynamics of this system has
been analytically and numerically studied by Welch & Wilson [12], who recommended the use of analytical solution at time t0 > 0 as the initial condition. In
line with this suggestion, the initial temperature perturbation in the current simulations is spread over a thickness of three grid points near the interface and the
profile is initialized using the detailed analytical solutions presented by Jamet and
Duquennoy [26]. Figure 5 shows the computational domain and the various
boundary conditions of the simulation. The computational domain chosen for
the simulation is of size 1 4 units and the domain is discretized into 20 80
cells. The various nondimensional parameters and fluid property ratios used in
the simulation are as follows.
qref ¼ 1:0;
¼ 4:0;
Tsat ¼ 0:0;
L ¼ 1;
DT ¼ 0:2;
ql =qv ¼ 2:0;
ml =mv ¼ 1:0;
Cpl =Cpv ¼ 1:0
kl =kv
ð25Þ
The evolution of thermal boundary layer in the liquid region is shown in
Figure 6a. The comparison between the predicted temperature profile and the
corresponding analytical result shows an excellent match. This is further corroborated by the comparison between numerical and analytical solutions in Figure 6b,
which show the variations of interface location and interface velocity with time.
Figure 5. One-dimensional phase change problem for initially superheated liquid.
55
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EVAPORATION OF LAMINAR FALLING FILM FLOW
Figure 6. (a) Evolution of temperature profile in liquid, and (b) Evolution of interface position.
5. RESULTS FOR EVAPORATING FALLING FILM
The thermal hydraulic analysis of an evaporating liquid film, falling over a
surface inclined at an angle of 60 to the horizontal (Figure 1) is now carried out.
_ =m (where m
_ is the film
In all simulations, the Reynolds number of the film Re ¼ 4m
flow rate per unit width) is maintained at a constant value of 5. The various parameters and thermophysical properties of the fluids considered are listed in Tables 1
and 2, respectively. A relatively smaller value of 100 kJ=kg has been assumed for the
latent heat of vaporization in order to reduce the computational requirement. Also,
56
R. S. MAURYA ET AL.
Table 1. Computational parameters
Parameters
Details
Computational domain size
Grid
Time step
Convergence criterion
0.048 m 0.012 m
80 40
Adaptive time, based on Courant
number criterion (C < 0.001)
Max. residue (R 1 1012)
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in order to prevent the onset of nucleate boiling, low wall heat fluxes (5 W=m2, 10 W=
m2, and 15 W=m2) are considered. The various nondimensional parameters
employed for the analysis of falling film behavior [5] are listed below.
T Tsat
ðqw d1 =kÞ
qw
Local heat transfer coefficient ¼
ðT Tw Þ
ð26Þ
Nondimensinal temperature ¼
Local Nusselt number ¼
ð27Þ
hy d1
k
ð28Þ
5.1. Grid Independence Study
The sensitivity of the predicted results with respect to the grid employed has
been investigated using meshes with 40 20, 60 30, 80 40, 100 50, and
160 80 cells. Relatively finer mesh has been employed in the y-direction, in view
of the steeper velocity gradients in that direction. Here, the fully developed film
thickness at Re ¼ 5 has been used as the parameter to check the sensitivity of the
predicted results on the computational grid. Table 3 shows the fully developed film
thickness obtained for various meshes mentioned above. It can be concluded that
80 40 is an optimum choice based on order of accuracy of result and, hence, it
has been used for all simulations presented in the current work.
5.2. Hydrodynamic Development of Falling Film
The hydrodynamic development of a non-evaporating falling film has been
analyzed first. The results obtained from these analyses have been compared with
the various experimental and analytical results available in the literature, which
further validates the capability of the current numerical model. The variation of film
Table 2. Thermophysical properties (selected) of fluids
Property
Liquid
Vapor
Density (kg=m3)
Dynamic viscosity (Ns=m2)
Thermal conductivity (W=mk)
Specific heat (J=kgK)
Surface tension (N=m)
800
0.16
1.0
2,000
1.2
5 105
0.1
1,000
0.072
EVAPORATION OF LAMINAR FALLING FILM FLOW
57
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Table 3. Sensitivity of results to grid size
Sl. no.
Grid size
Developed film thickness (mm)
1
2
3
4
5
40 20
60 30
80 40
100 50
160 80
3.3158
3.1295
3.0213
3.0132
3.0093
thickness (b) with nondimensional distance from the slit has been shown in Figure 7a.
The result obtained for b0 ¼ 1.80 using the current model shows a close agreement
with the experimental data of Fullford [27]. Figure 7b compares the variation of
developed film thickness with Re obtained from the present work with that of
Figure 7. Validations. (a) Comparison of developing film thickness (with experimental result), and (b)
developed film thickness variation with Reynolds number.
58
R. S. MAURYA ET AL.
Nusselt’s theory. The difference in the film thickness values predicted from these two
approaches increases with Re due to the increasing effect of interfacial shear, which
has not been accounted in Nusselt’s theory. Detailed investigations of falling film
hydrodynamics in the absence of evaporation for various ranges of flow parameters
can be found in the work of Maurya et al. [6].
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5.3. Thermal Development of Falling Film
5.3.1. Temporal evolution. Upon supply of heat to the inclined surface,
there is a simultaneous development of the thermal and hydrodynamic boundary
layers. The growth of the thermal boundary layer can be quantified through the normal temperature gradients at the wall. The transient evolution of wall temperature
gradient is presented in Figure 8 at a location of y ¼ 0.024 m from the flow inlet.
Due to the assumptions of pure conduction and nonaccelerating (negligible inertia)
film, Nusselt’s theory predicts a steady value of this temperature gradient at the given
Figure 8. Variation of wall temperature gradient with time.
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EVAPORATION OF LAMINAR FALLING FILM FLOW
59
wall location. However, in the case of an actual developing film such effects are
non-trivial and they manifest in the form of a transient decay of temperature
gradient which asymptotically approaches the value predicted by Nusselt’s theory.
In any case, the temperature gradients at the wall (and, hence, the heat transfer coefficient) are underestimated by Nusselt’s theory which ignores the inertial effects.
Similar features can also be observed from the transient evolution of temperature
profiles across the falling film shown in Figure 9a. The transient growth of the
parabolic temperature profiles is in stark contrast to the linear temperature profile
considered in Nusselt’s theory. Here, the interfacial temperature gradient
(at x ¼ 0.003 m) is initially zero and assumes significance only after the thermal
boundary layer transiently reaches up to the interface, thereby leading to the interfacial evaporation process. The variations of wall temperature for various supplied
heat fluxes are shown in Figure 9b. The figure shows that all the curves transiently
tend to attain an asymptotic state.
Figure 9. (a) Transient evolution of temperature profile at y ¼ 0.024 m, and (b) transient evolution of wall
temperature at mid plane y ¼ 0.024 m.
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R. S. MAURYA ET AL.
5.3.2. Spatial evolution. Figure 10 shows the spatial evolution of the nondimensional temperature profile across the film thickness at steady state. The profiles
clearly illustrate that the thermal boundary layer within the liquid film attains a fully
developed state at a distance of Le ¼ d=(d1Re) 1. The parabolic profile of temperature in the fully developed state clearly suggests the importance of the convective
(inertial) effects in the liquid film which were neglected in Nusselt’s approach. On
the whole, the liquid film can be ascribed to be thermally developing in the spatial
range of 0 Le 1. The variation of nondimensional wall temperature along the
flow direction at steady state is depicted in Figure 11 for various wall heat flux conditions. Once again, the temperature value slowly increases along the flow direction
and attains an asymptotic condition. This asymptotic condition represents a state of
thermal energy balance wherein most of the heat supplied at the wall is transferred to
interface for the phase change process. The spatial decay of the Nusselt number
along the flow direction is shown in Figure 12. The Nusselt number value at the fully
developed condition slightly deviates from the corresponding analytical result of Yih
and Lee [5]. This small deviation can be attributed to convective effects which have
been neglected in the analytical work.
5.4. Film Evaporation in Developing Region
The evaporative characteristics of a laminar film in the developing region can be
partially understood by examining the temperature profiles of Figure 10. In order to
further quantify the film evaporation process, the variation of dimensionless
evaporation rate (evaporating mass flux=flow rate at inlet) along the plate length is
depicted in Figure 13. It is obvious from this figure that the evaporation process is
Figure 10. Spatial variation of temperature across film thickness in developing region at wall heat
flux ¼ 5 W=m2.
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EVAPORATION OF LAMINAR FALLING FILM FLOW
Figure 11. Wall temperature along plate length.
Figure 12. Nusselt number along plate length.
Figure 13. Dimensionless evaporation rate in the developing region of film.
61
62
R. S. MAURYA ET AL.
nonexistent in the initial regions of the film where the thermal boundary layer has not
grown to the full extent of the local film thickness. After the temperature gradient
reaches up to the interface, the evaporation process slowly picks up and attains an
asymptotic developed condition. As expected, the evaporation rate amplifies with an
increase in the applied heat flux and the inception of evaporation process also occurs
at a distance closer to the inlet. For the conditions considered in the current work, the
evaporative heat flux (meL) is just a small fraction (6–8%) of the total heat supplied at
the solid wall and the remaining part is convected away with the falling film.
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5.5. Effect of Evaporation on Film Thickness
Figure 14a shows the effect of wall heat flux on the thickness of liquid film in
the developing region. As evident from the figure, the film thickness decays exponentially to a constant value mainly due to its acceleration along the sloping surface.
Figure 14. Effect of evaporation on film thickness.
EVAPORATION OF LAMINAR FALLING FILM FLOW
63
Such decay=growth behavior of the film is mainly governed by its inlet velocity and
the angle of surface inclination, and the effect of heat flux is relatively minor. The
magnified view shown in Figure 14b illustrates that the case with higher flux undergoes larger reduction in film thickness due to a higher evaporation rate.
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6. CONCLUSION
In the present work, the numerical study of an evaporating falling liquid film
flow subjected to constant wall heat flux has been carried out, including the effects
of flow development from the inlet and surface tension force at the interface. A
new methodology based on the ghost fluid method has been developed on a
Cartesian grid to accurately evaluate the discontinuous normal temperature
gradients on either side of the interface even in the case of dynamic flow evolution.
Using these gradients, a mass source term which emulates the phase change process
in the interface is evaluated. The present numerical model has been validated with
the analytical results of various benchmark test cases available in the literature.
Using the current numerical model, the characteristics of an evaporating falling
film in the developing region have been clearly brought out. The thermal boundary
layer within the evolving liquid film undergoes both transient and spatial growth
before a steady, developed temperature profile is established. Interestingly, the inertial effects significantly influence the growth rate of the thermal boundary layer,
particularly over the dimensionless distance of 0 Le 1. Surface evaporation does
not commence until the growing thermal boundary layer reaches the free surface of
the film. Once the effect of wall heating penetrates up to the interface, the evaporation process slowly picks up and attains an asymptotic steady value in the fully
developed region. With an increase in the wall heat flux, the steady state evaporation
rate increases and the inception point of evaporation also shifts towards the flow
inlet. The available analytical models such as Nusselt’s theory underestimate the wall
temperature gradients and the local heat transfer coefficient, due to the neglect of
inertial effects such as flow acceleration in the developing region of the film. The
present work clearly shows that such inertial effects could be significant even at a
low Reynolds number range of the film flow.
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