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 2 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. 4 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 5 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. 6 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. References Song, M., Deng, S., Dang, C., Mao, N., Wang, Z., 2018. Review on improvement for air source heat pump units during frosting and defrosting. Appl. Energy 211, 1150–1170. Lee, J., Lee, K.-S., 2018. The behavior of frost layer growth under conditions favorable for desublimation. Int. J. Heat Mass Transfer 120, 259–266. Sommers, A.D., Truster, N.L., Napora, A.C., Riechman, A.C., Caraballo, E.J., 2016. Densification of frost on hydrophilic and hydrophobic substrates –Examining the effect of surface wettability. Exp. Therm. Fluid Sci. 75, 25–34. 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