International Journal of Heat and Fluid Flow 89 (2021) 108807
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow
journal homepage: www.elsevier.com/locate/ijhff
A CFD model of frost formation based on dynamic meshes technique via
secondary development of ANSYS fluent
Yonghua You a, b, c, d, *, Sheng Wang a, Wei Lv a, Yuanyuan Chen a, Ulrich Gross b
a
State Key Lab. of Refractories and Metallurgy, Wuhan University of Science and Technology, Wuhan 430081, China
Institute of Thermal Engineering, TU Bergakademie Freiberg, Gustav-Zeuner-Str. 7, Freiberg 09599, Germany
c
National-provincial Joint Engineering Research Center of High Temperature Materials and Lining Technology, Wuhan University of Science and Technology, Wuhan
430081, China
d
Key Lab. for Ferrous Metallurgy and Resources Utilization of Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China
b
A R T I C L E I N F O
A B S T R A C T
Keywords:
Frost formation
Dynamic mesh technique
Heat and mass transfer
Frost profile evolution
Secondary development
ANSYS fluent
To simulate the non-uniform frost growth in flow direction for humid air flowing through a freezing channel, a
2D numerical frosting model based on dynamic meshes technique is developed in the current work via the
secondary development of commercial ANSYS Fluent. The computation domain consists of both frost layer and
humid air regions, and the local heat and vapor fluxes at the surface of frost layer are determined by numerical
temperature and vapor fraction fields in the humid air region rather than by empirical correlations. The frost
layer is treated as a growing packed bed with heat and mass transfer dominated by molecular diffusion, where
local absorption coefficient of vapor desublimation and local vapor fraction are both determined by solving the
pseudo steady vapor diffusion equation with a source term theoretically. The interface of frost layer and humid
air regions is treated as two walls for the iteration of its temperature, of which the humid air side is specified with
the temperature equal to the frost-side counterpart and the frost side takes the heat flux including the extra latent
heat caused by vapor deposit. User-defined functions are compiled to implement the above treatments to ANSYS
Fluent. Frosting experiments in the literature are simulated with the current model for validation. How the
profile of frost layer evolves with time in the frosting process is explored. The contours and profiles of velocity,
temperature and vapor fraction are presented to discuss the effects of heat and mass transfer on frost formation.
Numerical results demonstrate that the proposed CFD model can predict the frost growth and densification with a
relative deviation less than 5% compared with experiments. Besides, the computation load of current model is
small due to no solution of complex multiphase flow. In addition, dynamic meshes help current model to capture
the interface of frost layer and humid air regions accurately.
1. Introduction
Frost formation occurs due to vapor desublimation when humid air is
exposed to a freezing surface. This phenomenon is frequently observed
in heat exchangers of heat pump or cryogenics and results in the
decrease of heat transfer rate and the increment of power consumption
(Song et al., 2018; Lee and Lee, 2018; Sommers et al., 2016, 2018). Frost
formation is involved in complex heat and mass transfer together with
the growth and densification of the frost layer. Experiments have
demonstrated that the frost layer takes a porous structure and its for­
mation consists of three stages, i.e., crystal growth period, dry frost
growth and wet frost growth periods (Şahin, 1994). Besides, empirical
density, thermal conductivity and thickness of frost layer under different
operation conditions were put forward (Şahin, 1994; Hayashi et al.,
1977; Schneider, 1978).
Numerical simulation was adopted by scholars to study frost for­
mation processes (Léoni et al., 2016; Cheng and Cheng, 2001; Şahin,
1995). Léoniet al. (Léoni et al., 2016) reviewed several numerical
frosting models and compared five correlations with experiments for
frost thickness. With the assumptions that the convection in porous frost
layer could be ignored and vapor desublimation rate was proportional to
vapor density, Lee et al. (Lee et al., 1997) and Hermes et al. (Hermes
et al., 2009) developed one-dimensional (1D) mathematical models of
frost formation for humid air flowing over a cold surface. In their
models, the vapor diffusion and heat conduction equations of frost layer
with respective source terms were numerically solved, and empirical
* Corresponding author at: State Key Lab. of Refractories and Metallurgy, Wuhan University of Science and Technology, Wuhan 430081, China.
E-mail address: hust_yyh@163.com (Y. You).
https://doi.org/10.1016/j.ijheatfluidflow.2021.108807
Received 8 August 2020; Received in revised form 5 November 2020; Accepted 12 March 2021
Available online 16 April 2021
0142-727X/© 2021 Elsevier Inc. All rights reserved.
Y. You et al.
International Journal of Heat and Fluid Flow 89 (2021) 108807
correlations were utilized to determine the heat and vapor fluxes from
bulk humid air. By assuming that the temperature took a linear distri­
bution in frost layer and the variation of supercooling degree was pro­
portional to that of wall supersaturation degree, Hermes et al. (Hermes,
2012) simplified the thermal boundary condition of 1D model and ob­
tained an algebraic expression for the temporal evolution of frost layer
thickness. These 1D models have the advantage of small computation
load. However, frost growth is usually non-uniform along flow direction
and can greatly change channel geometry. Thus, researchers developed
two-dimensional (2D) or three-dimensional (3D) frosting models (Lee
et al., 2003; Bartrons et al., 2018, 2019; Armengol et al., 2016; Kim
et al., 2015). For instances, Lee et al. (Lee et al., 2003) presented a 2D
model of frost formation where the conservation equations of frost layer
and bulk humid air are coupled. Their model was validated by the ex­
periments of humid air flowing through a cold channel. However, some
significant information, like velocity, temperature and vapor fraction
fields, etc., wasn’t reported. Bartrons et al. (Bartrons et al., 2018) con­
structed a 2D frosting model of dynamic grids in their software Ter­
moFluids and correction steps were adopted for the frost growth. It is
noted that only the frost layer was modeled in their dynamic model and
empirical correlations were adopted for the heat and vapor fluxes at the
surface. To improve the prediction precision, Bartrons et al. (Bartrons
et al., 2019) developed a frost formation model of static grids by treating
the humid air flow and frost layer as a porous medium with different
porosities. In this model, the working medium was divided into three
states, i.e., “all humid air”, “all ice” and “partial ice”, and the equations
based on density and porosity were respectively adopted for the cells of
“all humid air” and “partial ice”.
For the past three decades, commercial CFD softwares were widely
applied to simulate various flow and heat/mass transfer processes (Luo
et al., 2016; Guo et al., 2015; Yang et al., 2014; You et al., 2015; Yan
et al., 2019; Cui et al., 2011), and several numerical studies of frost
formation based on multi-phase model have been reported, where the
humid air and ice particles were respectively set as the primary and
secondary phases (Wu et al., 2016, 2017; Afrasiabian et al., 2018). In
more details, based on the analysis of Gibbs free energy, Wu et al. (Wu
et al., 2016, 2017) proposed a dimensionless model of vapor desbuli­
mation rate and applied it into the Euler multi-phase model of ANSYS
Fluent to simulate the frost formation. It is noted that static grids were
adopted in their model and an empirical inequality constraint was used
to regulate the growth of frost layer. Afrasiabian et al. (Afrasiabian et al.,
2018) split the velocity and supersaturation terms of the inequality
constraint in their 3D numerical simulation of a plate fin evaporator.
In the current work, a 2D numerical frosting model based on dy­
namic meshes technique will be set up via the secondary development of
commercial ANSYS Fluent (Fluent, 2013). The computation domain
consists of both frost layer and humid air flow regions. The frost layer is
treated as a growing packed bed with the heat and mass transfer
dominated by molecular diffusion. User-Defined Functions (UDFs) will
be compiled for the vapor desublimation in the frost layer and the vapor
deposit at the surface of frost layer, together with for the meshes
deformation of whole computation domain. The frosting experiments
performed by Şahin (Şahin, 1994) will be simulated by the current
model for validation, and the temporal evolution of frost layer profile
will be explored. In addition, contours and profiles of velocity, tem­
perature and vapor fraction will be presented for discussions. With the
proposed CFD model, it is expected that the non-uniform frost growth in
the flow direction can be predicted at a small computation load due to
no solution of complex multi-phase flow. Besides, the sliding meshes
technique is expected to help current model to capture the interface
between frost layer and humid air accurately.
2. Numerical model
2.1. Governing equations
Fig. 1 depicts the schematic of a convective frost formation process,
where the thickness of frost layer increases with the humid air flowing
over a freezing wall. To capture the frost growth accurately, the dynamic
meshes technique is applied in the present simulation. With the
assumption of incompressible laminar flow, the heat and mass transfer
processes in the humid air flow region, where no vapor desublimation
occurs, are governed by the following conservation equations of conti­
nuity, momentum, energy and vapor fraction, (Fluent, 2013; You et al.,
2019)
∫
∫
(
)
d
(1)
ρdV + ρ u − ug dA = 0
dt V
∂V
d
dt
d
dt
d
dt
∫
∫
∫
)
(
ρudV +
ρu u − ug dA =
∂V
V
∫
∫
ρh u − ug dA =
∂V
V
∫
∫
(2)
k∇TdA
(3)
∫
)
(
ρhdV +
( − p + μ∇u)dA
∂V
ρYV dV +
∂V
(
)
∫
ρYV u − ug dA =
∂V
V
D∇YV dA
∂V
(4)
where ∂V represents the boundary of the control volume V; ρ, h, T
and YV stand for the mixture density, specific enthalpy, temperature and
vapor fraction, respectively, while u and ug refer to the fluid and grid
velocities, respectively.
With the meshes deformation taken into consideration, the time
derivative terms in the above equations are calculated by
∫
d
(ρφV)n+1 − (ρφV)n
ρφdV =
(5)
dt V
δt
where φ is the general variable and the cell volume at (n + 1)th time
level is calculated by V(n+1) = V(n) + dV/dt × δt.
The physical process in the frost layer, involved in the phase change
of the vapor of humid air in the porous structure (ice particles), is quite
complicated. To facilitate the numerical study, it is assumed in the
modeling that the humid air and ice particles are in the locally thermal
equilibrium and the heat and mass transports are dominated by the
molecular diffusion. Besides, the vapor desublimation rate is assumed to
be proportional to local vapor fraction (YV) and calculated by (Lee et al.,
1997)
SY = K⋅α⋅ρ⋅YV
(6)
where α is the porosity of frost layer and K denotes the absorption
coefficient of vapor desublimation.
With the assumption that K is constant at the cross section, the K and
YV in the frost layer can be determined by theoretically solving pseudo
steady one-dimensional diffusion equation with the source term of vapor
desublimation, i.e., (Lee et al., 1997)
(
) ⎤2
⎡
ρV,sat Tf |yf
1
( ) ⎦
K = D⋅⎣ cosh− 1
(7)
yf
ρV,sat Tp
(
)
(√̅̅̅̅̅̅̅̅̅̅ )
ρV,sat Tp
YV =
⋅cosh
K/D ⋅y
ρha
(8)
Here cosh and cosh− 1 stand for the hyperbolic cosine and inverse
hyperbolic cosine functions, respectively; yf and y refer to the thickness
of frost layer and the displacement from plate, respectively. ρha and ρV,sat
represent the densities of humid air and saturated vapor.
The porosity (α) of frost layer decreases with the advancement of the
vapor desublimation and is calculated by
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Y. You et al.
International Journal of Heat and Fluid Flow 89 (2021) 108807
Fig. 1. Frosting schematic of humid air flowing through a freezing channel (Bartrons et al., 2018).
∂α
= − SY /ρice
∂t
[
( )2/3 ]( )0.11
− 1000)Pr
d
Pr
√̅̅
3 × 103 ⩽Re⩽1 × 106
1+
Nu =
L
Pr
ξ
w
2/3
1 + 12.7 8(Pr − 1)
(9)
ξ
(Re
8
The humid air and ice particles of each control volume have the same
temperature in the frost layer, which can be determined by numerically
solving the following energy equation.
∫
∫
∫
]
∂ [
αρha hha + (1 − α)ρf hf dV =
keff ∇TdA + SY LdV
(10)
∂t V
∂V
V
(15)
where d and L stand for characteristic diameter and channel length,
while ξ refers to the friction factor, ξ=(1.82 logRe-1.64)-2.
2.3. Computation domain, grids and boundary conditions
Here the last term on the right hand side denotes the energy source
due to vapor desublimation and L is the latent heat of vapor desu­
blimation. Vapor condensation can also occur when local temperature is
greater than 0℃ and then the L takes the value of vapor condensation.
The keff is the effective thermal conductivity of frost layer, which de­
pends on frost layer density and follows below empirical function (Lee
et al., 1997);
)
(
keff = 0.132 + 3.13 × 10− 4 × ρf + 1.6 × 10− 7 × ρ2f 50⩽ρf ⩽400 kg⋅m− 3
The test section of Ref. (Şahin, 1994) is adopted as current compu­
tation domain (refer to Fig. 1) and the adiabatic segments in front of and
behind the freezing plate are not modeled. The computation domain
consists of the frost layer and humid air regions. Both regions deform
with the progress of frost growth. The test channel has a width much
larger than its height and thus 2D model is adopted to decrease the
computation load. The inlet takes the fully-developed momentum
boundary condition together with uniform temperature and vapor
fraction ones, while the outlet takes the atmospheric pressure. The
interface between frost layer and humid air regions is treated as two
walls, of which the humid air side is specified with the temperature
equal to the counterpart of the frost side, while the frost side takes the
heat flux calculated by Eq. (13). With this method, the temperature
continuity through interface is obtained and the latent heat released at
the interface is appropriately considered in the simulation. As for the
species boundary condition at the interface, it is assumed that the vapor
fraction takes the saturation value of the temperature there. The quad­
rilateral meshes are generated with a constant streamwise size, and the
grids of humid air region are refined near the interface. During the
simulation, the displacements of grids at the interface are computed via
Eq. (12) and the meshes deformation of whole domain is determined
according to the interface displacements. At the start of frosting simu­
lation, the frost layer is very thin (10 μm) and then its thickness increases
with the progress of vapor desublimation. The strategy of varying cell
size rather than that of adding or decreasing cell amount is adopted for
meshes deformation. Initially, the meshes in the frost layer are quite thin
and Fig. 2 illustrates the deformed grids of a typical case after running
for 20 min.
(11)
It is noted that at the interface of frost layer and humid air regions,
the heat and species fluxes on the humid air side are calculated by the
numerical gradients of temperature and vapor fraction, rather than by
empirical convection heat and mass transfer correlations in the current
work.
2.2. Energy and species conservations at the interface
Here it is assumed that one part of the vapor from humid air region
diffuses into porous frost layer to densify it through desublimation,
while the rest desublimates at the surface of frost layer and increases its
thickness. Therefore, the increment rate of frost thickness expressed by
δyf can be determined by
(
)
⃒
⃒
1
dY ⃒
dYV ⃒⃒
δyf =
ρD V ⃒⃒
− αρD
(12)
(1 − α)ρice
dy y=y+
dy ⃒y=y−
f
f
Here the superscripts + and – stand for the sides of humid air and
frost layer, respectively.
As the vapor deposit at the interface of frost layer and humid air
regions could release latent heat, the frost-side heat flux is calculated by
⃒
dT ⃒
qT |y=y− = k ⃒⃒
+ Lδyf (1 − α)ρice
(13)
f
dy y=y+
2.4. Solution procedure and computation scheme
The commercial CFD package ANSYS Fluent is quite flexible in
simulating complex flow and heat transfer processes because it provides
many types of macros with which users can compile User-Defined
Functions (UDFs) for their particular modeling needs. In the current
work, the 2D double-precision version of this package is adopted for the
present numerical simulation. The species transportation model is acti­
vated for the humid air, and the convection heat and mass transfer in the
humid air flow region is simulated based on numerically solving the
Navier-Stokes equations. The frost layer region is treated as a porous
medium with ice particles as the skeleton. In this region, the fluid is set
with a zero velocity and an energy source of vapor desublimation, and
its vapor fraction is specified with the values calculated by Eq. (8) by
using
the
model-specific
DEFINE_PROFILE
macro.
The
f
The current simulation is involved in the channel flow with the Re
number of 3700. To cover the effect of slight turbulence at the cost of no
notable increase of computation load, a correction factor is figured out
for the thermal conductivity and diffusivity of the bulk humid air by
comparing the following empirical correlations of laminar flow and
transitional turbulence flow (Taler, 2016);
Nu = 1.86⋅(Re⋅Pr)1/3
( )1/3 ( )1/4
d
u
Re⩽2400
L
μw
(14)
3
Y. You et al.
International Journal of Heat and Fluid Flow 89 (2021) 108807
Fig. 2. Computation domain, meshes and boundary conditions after running for 20 min.
DEFINE_PROFILE macro is also compiled to specify the temperature,
heat flux and vapor fraction at the interface of frost layer and humid air
regions. The dynamic meshes model is activated to capture the meshes
deformation. The general purpose macro of DEFINE_ADJUST is used to
determine the current frost growth by comparing the vapor fluxes on the
two sides of the interface. It is also compiled to compute the absorption
coefficient of vapor desublimation and the porosity variation of frost
layer at the current time step. With the frost growth obtained by the
general purpose macro, the dynamic meshes macro of DEFINE_­
GRID_MOTION is executed to regulate the grids motion of the frost layer
and humid air regions.
Implicit segregate solver is selected and the time-dependent terms
are discretized with the Euler scheme. SIMPLE algorithm is used for the
decoupling of pressure and momentum. The second order upwind
scheme is applied for the discretizations of momentum and energy
terms. The solution procedure of current numerical simulation is
depicted in Fig. 3 for readers’ references, where treq and δt refer to the
required frosting duration and time step, respectively. It is seen that
before solving the governing equations for a new moment, the DEFIN­
E_GRID_MOTION macro will be executed to regulate the grid motion and
update the frost layer thickness.
During the computation, the inlet velocity, temperature and vapor
fraction of humid air are respectively equal to experimental measure­
ments. The surface of cold plate takes the experimental temperature as
well and its vapor fraction is equal to the saturation value. Monitors are
set for the temporal variations of average thickness and porosity of frost
layer. The relative residual below 1e-4 is adopted as the convergence
criteria of all governing equations except for the energy equation, which
takes the threshold residual of 1e-8. The effects of cell amount and time
step are studied. The finer grids facilitate a better precision; meanwhile
they could increase computation load and worsen numerical stability.
After the comprehensive balance, the grid system in the final compu­
tation takes about 4 K cells (～1K cells for frost layer and ～3K cells for
humid air region), and the time step is variable with the maximum value
of 0.1 s. With the above setting, it takes about 5 h to run a typical case by
using the current computer with the Intel(R) Core(TM) i7 CPU (four
cores).
3. Numerical results and discussions
The whole computation domain, including both the frost layer and
humid air regions, has the length and height of 300 and 12.7 mm,
respectively. The original frost layer is specified with a small thickness, i.
e., yf = 10 μm. The inlet humid air, with the temperature (Tin) of 286 K
and the vapor fraction (YV,in) of 0.007 or 0.0057, takes the parabolic
velocity profile. The mean inlet velocity varies in such a way that the
whole mass flowrate keeps constant during the simulation. The surface
of freezing plate takes two temperatures (Tp), i.e., Tp = 248 or 258 K. In
brief, four typical cases of frost formation are studied with different
combinations of Tp, YV,in and Re number, as depicted in Table 1. These
values are consistent with the experimental counterparts (Şahin, 1994).
For the first several millimeters’ freezing plate, the frost grows at a
large rate during the first several minutes’ running because the bound­
ary layer is quite thin. The booming frost near the inlet can induce
Fig. 3. Solution procedure of current frosting simulation.
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Y. You et al.
International Journal of Heat and Fluid Flow 89 (2021) 108807
Table 1
Working parameters of experimental cases used in current study.
Equation
inlet temp.
(Tin), K
inlet vapor
fraction (YV,in)
plate temp.
(Tp), K
Re number
Case I
Case II
Case III
Case Ⅳ
286
0.007
0.007
～0.007
0.0057
248
248
258
248
2400
3700
3700
～3700
backward flow and weaken the stability of current numerical model. On
the other hand, the sharp frost peak can level off due to air blow. With
the consideration that the freezing plate has the length of several hun­
dred millimeters, the first 4 mm long frost layer is assumed to take the
same growth rate as that 4 mm away from the inlet. With this treatment,
the current numerical simulation runs quite smoothly while maintains
an acceptable accuracy. As for the frost layer with its temperature over
273.15 K, it is assumed that vapor condensation rather than desu­
blimation occurs and static liquid water is generated.
3.1. Temporal variations of mean thickness and density of frost layer
Frosting experiments with working parameters listed in Tab. 1
(Şahin, 1994) are simulated with current model. The predicted temporal
variations of mean thicknesses of frost layer are depicted in Fig. 4(a)
with continuous lines of different patterns. For the convenience of
comparison, Fig. 4(a) presents the experimental measurements with
discrete solid square or other marks.
From the curves in Fig. 4(a), it is seen that as the humid air flows
through the channel with a freezing surface, the averaged frost layer
thickness (yf ) increases with time monotonously and the increment rate
decreases gradually for all the cases. Besides, the yf increases with the
increment of inlet velocity (i.e., Re number) and vapor fraction (YV,in), or
with the decrement of plate temperature (Tp). In more details, for the
Case II, i.e., under the condition of YV,in = 0.007, Tp = 248 K and Re =
3700, the numerical yf is equal to 3.54 mm at the moment of 60 min. If
the Tp is increased to 258 K (Case III) or the YV,in is decreased to 0.057
(Case IV), the numerical yf can decrease by ～49.6% or 29.7%, respec­
tively. Comparing the above curves with corresponding discrete marks
in Fig. 4(a), it is seen that numerical predictions match well with
experimental measurements and average relative deviation is ～4.7%.
The temporal variations of mean frost density are numerically
simulated for the above cases and the results are depicted in Fig. 4(b)
with continuous lines of different patterns. Fig. 4(b) presents the
experimental data with disperse marks as well for comparison. From the
curves in Fig. 4(b), it is seen that the mean frost density takes a similar
variation trend for all the cases, i.e., it increases very quickly at the first
10～20 min, and then the growth rate decreases and takes an approxi­
mately linear profile. This variation trend is consistent with the report in
Ref. (Lee et al., 2003). Comparing the curves and marks in Fig. 4(b), it is
clear that the numerical simulations match the experiments quite well,
and the averaged relative deviation is ～ 4.3%.
Scrutinizing the curves and corresponding marks in Fig. 4(a) and (b),
it is observed that the present model can over-predict the frost thickness
and density for those cases that frost for a long time(greater than80 min)
and have a thick frost layer. This phenomenon could be partly related to
the vapor condensation at the surface of frost layer. The current model
does consider the vapor condensation when the surface of frost layer has
the temperature over 0℃ that can occur after a long time of frosting.
However, when the amount of liquid water is large at the surface, it will
flow downstream and creep into the porous structure, which can go
against the frost growth. On the other hand, the air flowrate is kept
constant in the current work despite the channel becomes gradually
narrower due to the frosting, thus the heat and mass transfer between
bulk air and frost layer can be overrated, as a result, the predicted vapor
desublimation for the case with a thick frost layer (frosting for a long
Fig. 4. Comparisons of frost growth and densification between current nu­
merical model and experiments in the literature (Şahin, 1994). (a) Comparison
of frost layer thickness; (b) Comparison of frost layer density.
time) can be more than the practical value.
The above cases considered for validation cover different inlet tem­
peratures and vapor fractions along with different plate temperatures,
and the temporal variations of frost thickness and density obtained by
current model are both compared with the experimental counterparts.
Thus it can be concluded that the current CFD model could predict the
frost formation processes with a reasonable precision.
3.2. Evolution of frost layer profile
Fig. 5(a) depicts the frost layer profiles of Case II at eight different
moments, i.e., t = 0, 2.5, 5, 10, 20, 30, 40 and 60 min. As is mentioned
above, the first 4 mm long frost layer takes the same thickness for all the
moments. It is clear from Fig. 5(a) that the local thickness (yf) of frost
layer increases with the frost formation in progress. Besides, the yf varies
notably in the flow direction and different variation trends can be seen at
different frosting stages. In more details, the upstream frost layer grows
at a larger rate and its thickness takes the downward profile for the first
～20 min. After that, the growth rate near the inlet drops abruptly to a
small value and the yf takes the convex variation trend longitudinally,
whose peak becomes more notable with time and shifts downstream. As
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Y. You et al.
International Journal of Heat and Fluid Flow 89 (2021) 108807
Based on the above factors, it is expected that a peak appears in the frost
layer profile after frosting long enough.
3.3. Discussions based on contours and profiles
It is well known that the convection heat and mass transfer in the
channel plays a significant role in the frosting process, thus Fig. 6(a), (b)
and (c) are presented for the numerical contours of velocity, tempera­
ture and vapor fraction after frosting for 30 min, respectively. It is seen
from Fig. 6(b) and (c) that notable thermal and vapor fraction boundary
layers are generated in the bulk humid air region, which act as the main
resistances of heat and mass transfer from bulk humid air to the surface
of frost layer. Meanwhile, the temperature and vapor fraction of bulk
humid air are observed to decrease in the flow direction, which results in
the downstream having a smaller heat and mass transfer flux. Similar to
the bulk humid air, the surface of frost layer has a dropping temperature
in the flow direction (refer to Fig. 5(b)), which determines the vapor
fraction there based on the saturation assumption. On the other hand, it
is seen in Fig. 6(a) that the channel becomes narrower and the velocity
increases with the frost growth, which facilitates the convection heat
and mass transfer in the channel.
The heat and mass transfer in frost layer is quite significant for the
frost growth and densification as well. It is noted that the constant
temperature and zero vapor flux boundary conditions are specified at
the freezing plate in the present simulation. As the vapor from bulk air
diffuses through frost layer with some vapor desublimating on ice par­
ticles, it is expected that vapor flux can be smaller at the section closer to
the freezing plate and thus the vapor fraction increases along the
Fig. 5. Temporal variations of local thickness and surface temperature of frost
layer for Case II. (a) Local thickness of frost layer; (b) Surface temperature of
frost layer.
liquid water has a much larger density than porous frost and vapor
condensation at the surface of frost layer can generate a much slower
frost growth than that of vapor desublimation, the above-mentioned
transition of yf profile from the downward variation trend to convex
counterpart could be related to the local occurrence of vapor conden­
sation. To confirm this assumption, Fig. 5(b) presents the temperature
profile of the surface of frost layer at different frosting moments. It is
seen from Fig. 5(b) that when the frosting time is limited, the surface
temperature is below the freezing point and takes the dropping profile in
the flow direction. After running for over ～20 min, the surface tem­
perature of frost layer near the inlet increases to 273.15 K, which con­
firms that the vapor is transformed into liquid water rather than porous
ice particles there. With the consideration that the convection heat
transfer coefficient near the inlet drops quickly, the frost layer further
from the inlet can grow to a larger thickness before its surface temper­
ature reaches the condensation point. Thus, the downstream frost layer
could have some chances to take the thickness exceeding the counter­
part near the inlet. As for the frost layer far from the inlet, where the
vapor fraction of bulk humid air is small due to upstream vapor desu­
blimation, the vapor deposits slowly and the local thickness is small.
Fig. 6. Contours of velocity, temperature and vapor fraction at the moment of
30 min. (a) Velocity magnitude; (b) Temperature; (c) Vapor fraction.
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Y. You et al.
International Journal of Heat and Fluid Flow 89 (2021) 108807
thickness (or y) direction with an acceleration. On the other hand, in the
frost layer, the latent heat released due to vapor desublimation need be
transported to the freezing plate, thus the section closer to the freezing
plate has a higher heat flux and the frost temperature is expected to
increase with a deceleration in the y direction. Fig. 7(a) and (b) depicts
the numerical cross-sectional profiles of temperature and vapor fraction
at various moments, respectively. Scrutinizing the bottom left segments
of the curves in Fig. 7(a) and (b), which depict the temperature and
vapor fraction distributions in the frost layer, it is observed that the frost
temperature takes the convex profile while the vapor fraction profile is
concave, consistent with aforementioned expectations. Besides, as the
frost has a much greater thermal conductivity than humid air, one can
seen in Fig. 7(a) that the temperature gradient takes a great variation at
the surface of frost layer.
Generally, a thicker frost layer is expected to have a higher surface
temperature and thus a greater surface vapor fraction, which results in a
smaller vapor flux from the bulk humid air. Besides, a greater thickness
and a larger vapor fraction, which facilitates more vapor desublimation
in the frost layer (refer to vapor desublimation model of Eq. (6)) and
smaller vapor deposit at the surface of frost layer. The deduction accords
with the curves in Fig. 7(b), where the thickness of frost layer increases
with the frosting in progress, while its increment rate decreases.
4. Conclusions
In the current work, a 2D numerical frosting model is developed with
ANSYS Fluent for humid air flowing in a freezing channel. Its compu­
tation domain consists of both frost layer and bulk humid air regions.
The dynamic meshes technique is utilized to track the frost growth.
User-defined functions are compiled by using the macros of DEFIN­
E_ADJUST, DEFINE_PROFILE and DEFINE_GRID_MOTION, etc. for the
internal vapor desublimation and surface vapor deposit of frost layer,
and the meshes deformation of computation domain. Frosting experi­
ments in the literature are simulated with the current model for vali­
dation. Besides, the temporal evolution is explored for the profile of frost
layer. In addition, the contours and profiles of velocity, temperature and
vapor fraction are presented for discussions. Numerical results demon­
strate that
1) The current frosting model built via secondary development of
ANSYS Fluent can predict the temporal frost growth and densifica­
tion with a reasonable precision (relative deviations against experi­
ments smaller than 5%), and the uneven profile of frost layer is
captured accurately with the adoption of dynamic meshes. Besides,
the present numerical simulation has a small computation load due
to no solution of complex multi-phase flow.
2) The frost formation depends on the joint contributions of flow, heat
and mass transfer in the frost layer and bulk humid air regions. The
local thickness of frost layer increases with frost formation and the
upstream grows at a greater rate in the early stage After a sufficiently
long running, water condensation occurs at the surface near the inlet
and the frost layer takes the convex profile.
Fig. 7. Distributions of temperature and vapor fraction at the section with a
streamwise displacement of 15 mm after running for 10, 20, 30, 40 and 60 min.
(a) Temperature distributions; (b) Distributions of vapor fraction.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgements
This work was financially supported by the Natural Science Foun­
dation of China (NSFC No. 51804234) and China Scholarships Council
(CSC No. 201808420319)..
The current work presents a computationally cheap and accurate
CFD frosting model that can be referred for the optimal design and
performance improvement of heat transfer devices involved in frost
formation.
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CRediT authorship contribution statement
Yonghua You: Conceptualization, Investigation, Validation, Writing
- original draft. Sheng Wang: Investigation, Software, Validation. Wei
Lv: Data curation, Formal analysis. Yuanyuan Chen: Investigation,
Project administration, Funding acquisition. Ulrich Gross: Conceptu­
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