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A New Model for Frost Growth Incorporating Droplet Condensation and Crystal Growth Phases.

BACKGROUND AND INTRODUCTION

When the surface temperature of an outdoor evaporator drops below the freezing temperature of water, frost forms on the evaporator. Frost increases the evaporator thermal resistance since it forms an insulating layer between the cold surface and the humid airstream, and it increases the air-side pressure drop due to airflow blockage.

One promising way of limiting frost growth on heat exchanger fins is to cover them with hydrophobic, hydrophilic, and hybrid or biphilic, surface coatings. This method was explored experimentally in several studies, but few researchers proposed analytical models for frost growth with different surface coating types. To the authors' best knowledge, the only frost models that exist for different surface types are composed of empirical correlations that are dependent on the water droplets' contact angle on the fin surfaces.

For the range of conditions typical of heat pump systems, frost formation consists of three primary phases: condensation and droplet growth, crystal growth, and growth of a homogeneous, porous frost layer (Hoke et al. 2000). The surface contact angle primarily affects the frost layer during the very beginning of frost growth, when supercooled water droplets condense, grow, and freeze on the surface. The next phase of frost growth, the crystal growth phase, is also affected by the contact angle because crystals grow on top of the frozen droplets, and thus the number of crystals present is dependent on the number, shape, and size of the frozen droplets. The frost layer growth period, which begins when the crystal growth phase ends, is least affected by surface coating types. It is important for frost models to incorporate all three of these phases in order to predict frost growth when various surface coatings are applied to the heat exchanger fins.

BRIEF LITERATURE REVIEW

The physics of droplet growth can be explained by dropwise condensation theory. This theory is old, but researchers are still refining and modifying it today so its use can be extended. In 1969, Graham pointed out that heat transfer through a single hemispherical droplet can be calculated by adding all the thermal resistances that exist between the vapor and the cold surface. Droplet growth was then calculated by assuming all the heat transfer through the droplet was latent heat that led to an increase in droplet mass and volume. Several researchers (such as Glicksmann and Hunt Jr. (1972), Wu and Maa (1976), Burnside and Hadi (1999), Abu-Orabi (1998), and Vemuri and Kim (2006)) calculated droplet growth this way in order to predict droplet size distribution and heat flux for surfaces. More recently, researchers modified the droplet growth model in order to account for contact angle and structure of the surface (Kim and Kim (2011), Sikarwar (2011), Miljkovic et al. (2012), and Rykaczewski (2012)).

Most researchers acknowledged the crystal growth phase of frost formation, as identified by Hayashi et al. (1977), but few studies focused on actually modeling this phase. Tao et al. (1993) and Sahin (1995) both presented models for crystal growth that were based on ice columns growing up from the cold surface. Their models were not validated for frost growth during the first six minutes, and their modeling results did not capture well the experimental data of the frost thicknesses at the beginning of frost periods. Other researchers used variations of the Tao et al. model for the crystal growth phase, but they do not seem to predict frost thickness well during this phase.

Few researchers coupled models for different frosting phases into one larger and more comprehensive model. Seki et al. (1985) presented a frost model, composed of droplet growth and frost layer growth phases, in which they calculated droplet growth and freezing and built a homogeneous frost layer on top of the frozen droplets. Tao et al. (1993) coupled their crystal growth model to a frost layer growth model, where the porous frost layer grows on top of the ice columns. Ismail et al. (1997) and Tahavvor and Yaghoubi (2009) also coupled variations of the Tao et al. model with frost layer growth models to predict overall frost growth. These models seemed not to match well the trends of the experimental data for very small values of frost thickness (i.e., for frost thickness less than 0.5 mm).

Based on the literature review, there is a need for developing a comprehensive frost model that incorporates all three phases of frost growth and that is experimentally validated for broader ranges of frosting conditions. Leaving out any of these phases could lead to an inaccurate prediction of the characteristics of frost growth as well as difficulty in adapting the model to surfaces with differing contact angles. In the current work, such a comprehensive model was developed and it is discussed in detail. First, the present model considers classical dropwise condensation theory for analyzing the very early stages of droplets growing on a subfreezing cold fin surface. Then the model switches to a crystal growth model, and it gradually transitions to porous frost layer growth mode during the frost operation period.

MODEL DEVELOPMENT

The newly developed model of the present work started from a dry cold horizontal flat plate surrounded by humid air and ran through three modes of simulation in order to build up frost on the top surface. The model was written in C++ language and the air and water properties were calculated by using CoolProp (Bell et al., 2014). A brief description of the model modes of simulation is given next.

Mode 1: Initial Condensation and Growth of Water Droplets

Thermal resistances, expressed as temperature differences, existed as a network through a droplet between the bulk air stream and the cold surface. The methods used in this work for calculating conduction resistance, interfacial resistance, and curvature resistance were all taken from Graham (1969) with updates to include varying contact angle, which were originally proposed by Kim and Kim (2011). Droplet growth models often consider resistances from the cold plate to a point directly above the droplet, but this approach does not take into account convective heat transfer from the surrounding fluid to the droplet. We therefore added a resistance for convection to the thermal resistance network. The temperature differences associated with each resistance used for this model are presented below.

Conduction Resistance:

[mathematical expression not reproducible] (1)

Interfacial Resistance:

[mathematical expression not reproducible] (2)

with

[mathematical expression not reproducible] (3)

Curvature Resistance:

[mathematical expression not reproducible] (4

Convection Resistance:

[mathematical expression not reproducible] (5)

The temperature difference between the droplet liquid surface, Ti , and the cold surface was set equal to the sum of all the temperature differences associated with the thermal resistances through the droplet itself (including all resistances except convection), and the resulting equation was then solved for the droplet heat transfer, Q. This process is demonstrated in Equations 6 and 7.

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

For this model, it was also assumed that all heat transfer through the droplet was latent heat transfer and went toward increasing droplet volume. This heat transfer related to droplet radial growth as shown below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE] (8)

An equation for radial growth rate was also needed. This equation was formulated by setting the Q from Equation 8 equal to the Q from Equation 7, then solving for the growth rate, dr/dt. The resulting final analytical expression is shown in Equation 9.

[mathematical expression not reproducible] (9)

Mode 2: Intermediate Frost Crystals Growth from the Iced Droplets

The next portion of the frost model covered the crystal growth phase, which started when the droplets froze and stopped growing in volume. For this part of the model, it was assumed that a single cylindrical crystal grows from the top of each frozen droplet (the assumption of one crystal per droplet was supported by the work of Cheikh and Jacobi, 2014). Similarly to the droplet growth model, crystal growth was governed by a thermal resistance network which included resistances for convection between the bulk air and the crystal, convection between the bulk air and the frozen droplet, and conduction through the crystal to the frozen droplet. It was assumed that all heat transfer by conduction through the crystal was latent heat and contributed to crystal growth, which led to the following equation for crystal radial growth:

[mathematical expression not reproducible] (10)

Crystal height was then calculated by assuming there was an aspect ratio, c, which directly related height to radius. Total frost height was calculated by adding crystal height to frozen droplet height.

Frost density was calculated during the crystal growth phase by taking a volume weighted average of the densities of the frozen droplets, the cylindrical crystals, and the air in the frost layer. For these calculations, the number of frozen droplets present was calculated from an assumed value of surface area coverage of frozen droplets; this value was based on information gathered from the literature. It was further assumed that all frozen droplets were the same size, which was assumed to be an average of the sizes of all frozen droplets. At first, frost density was high since frost thickness was low and mass of the frozen droplets was comparatively large, but as the crystals grew frost density began to drop. The transition point between the crystal growth phase and the frost layer growth phase occurred when the frost density reached a minimum value (Hayashi et al., 1977). While most frost layer growth models assumed initial values for frost thickness and density in order to begin the time step calculations, the current model used the predictions at the end of the crystal growth mode as inputs to the frost layer growth mode of the model.

Mode 3: Establishment of the First Layer of Frost and Subsequent Frost Layer Growth

There are many models available in the literature that are well capable of predicting frost layer growth, so it was unnecessary to develop a new one for this part of the model. In the present work, we started with the model proposed by Padhmanabhan (2011). In his model, Padhmanabhan treated the frost layer as a single uniform layer with average density and thermal conductivity. His model assumed that the total water vapor mass flux entering the frost layer split between a mass fraction penetrating the frost layer that augmented frost bulk density and a remaining mass fraction that stayed and deposited on top of the frost layer to increase frost thickness. These mass fluxes were used to calculate frost thickness and density at each time step. The main equations used to calculate the mass fluxes are given below.

Total Mass Flux:

Mass Flux to Increase Density:

[mathematical expression not reproducible] (12)

During the model preliminary validation, we noticed that Padhmanabhan's (2011) original model did not match the experimental data when using his embedded correlation for thermal conductivity. In the present work, we investigated six correlations of frost thermal conductivity present in the literature (Van Dusen 1929, Shin et al. 2003, O'Neal and Tree 1985, Yonko and Sepsy 1967, Ostin and Andersson 1991, and Oscarsson et al. 1990) and out of these six correlations, the correlation by Yonko and Sepsy (1967) performed best for almost every data set used in model validation of the present work. The Yonko and Sepsy correlation was used with our model from then on, both for this reason and because of the wide range of surface temperatures, relative humidities, and air velocities used as test conditions for the data from which it was developed. These test conditions are also similar to those for which the model will be used in the future.

Model Inputs

There were three types of inputs required for our newly developed model: test condition inputs, geometric inputs, and droplet characteristics inputs. The test condition inputs were air bulk temperature, pressure, bulk relative humidity, and average velocity, as well as cold plate surface temperature and contact angle. The geometric inputs were the length and width of the cold plate. The droplet characteristics inputs were initial droplet size (roughly equal to the minimum size a droplet must be to grow by condensation), time until droplets ceased growing and froze, plate area coverage by frozen droplets, and the aspect ratio of the cylindrical crystals growing from each individual iced droplet.

MODEL VALIDATION

Table 1 shows predictions of frost mass, thickness, density, and surface temperature compared to measured data from Piucco (2008) and highlights the agreement, given as percent errors and absolute temperature difference, between the predictions of the present model and the data in the literature. The model was able to predict frost mass well, with percent errors of less than one-half percent. It overpredicted one data point and underpredicted the other for frost thickness but the predicted values were still within 14% of the measured values. Frost surface temperature was predicted within the experimental uncertainty of [+ or -]1 K (1.8[degrees]R).

Figure 1 shows the model simulation results of frost thickness, frost density, frost surface temperature, and sensible, latent, and total heat fluxes compared to data from Lee et al. (2003). Frost thickness was predicted very closely for this case. There were close results for frost density and the heat fluxes as well, though with minor differences in the overall trends. The frost surface temperature predictions had a correct trend, but were found to be too high. These high surface temperatures in the simulation results could be due to the fact that our model used a simplified linear equation for temperature profile instead of a higher order equation. The average relative errors for the frost thickness, frost density, frost surface temperature, and total heat flux series were 4.6%, -1.23%, -2.7 K, and -4.3%, respectively, with maximum errors of 19.1%, -21.2%, -3.2 K, and -13.3%.

Figure 2 shows the comparison of model simulation results for frost thickness with measured data from Huang et al. (2010). Unlike the previous plots, where data was not available very early in the frosting process, this plot shows how the model handled the early stages of frost growth. The short, rather flat line at the very beginning of the process depicts the droplet growth stage. When the droplets froze, the model switched to mode 2, the crystal growth phase, which is represented by the nearly vertical line. When the model transitioned to mode 3, the frost layer growth mode, the curve flattened out again. The model was not able to accurately predict the slope of the frost thickness during the crystal growth phase, but it was able to predict frost thickness quite closely through the remainder of the test. The crystal growth model was used as a transitory mode of operation to connect modes 1 and 3 in terms of frost thickness and density while the instant at which the crystal growth model starts is assumed as the same as the droplet freezing time. This assumption seems reasonable for monodispersed droplets on a flat plate with no coalescence. While further work is still necessary to refine the crystal growth mode of the present model (i.e, mode 2), the proposed approach was able to provide accurate predictions of frost thickness and density at the beginning of the frost layer growth mode, that is, for the very first layer of porous frost established on the surface. This starting point enabled mode 3 of the model, the frost layer growth mode, to predict well the subsequent accumulation of frost on the plate.

All the above cases were for a flat horizontal cold plate in forced air convective flow. The model was also compared against data from Sommers et al. (2016) which was the only data set used for model verification in case of air natural convection on a cold flat plate. The model was not able to predict well Sommers et al. experimental data for frost density and could only predict frost thickness relatively well. Therefore, further investigations are needed to extend the present model to natural convection cases.

MODEL RESULTS

This section presents new simulation results of frost thickness and frost density for a flat horizontal cold plate in forced air convective flow and with variations of the plate surface temperature and air relative humidity.

Figure 3 shows frost thickness and frost density versus plate surface temperature. As intuitively expected, if the plate temperature decreased, the frost thickness increased and the frost density decreased. This is because more mass was deposited at colder plate temperatures, but with a less dense consistency. Even in the first ten minutes of frosting shown here, a temperature difference of 5[degrees]C produced large changes in both frost thickness and density. It is also worth noting that the same thickness trend exists for both the crystal growth and full growth phases of frost growth.

Figure 4 shows frost thickness and density versus air relative humidity. It can be seen that if relative humidity increased, frost thickness increased substantially but density decreased slightly. More mass was deposited for higher relative humidities, which explains the increase in frost thickness, but it was deposited with a slightly smaller density. Changes in relative humidity did not have a large effect on model predictions during the crystal growth period.

From these results, the newly developed frost model behaved as intuitively expected for both the full growth and crystal growth phases of frosting. In particular, the crystal growth mode of the present model seemed to capture well the main characteristics of the physical processes occurring during the early stages of frost formation.

CONCLUSIONS

This paper presented a new frost model that consists of three phases: droplet growth, crystal growth, and frost layer growth. The newly developed model started from a dry cold horizontal flat plate surrounded by humid air and ran through three simulation modes in order to build up frost on the top surface. The droplet growth mode used condensation theory to predict a radial growth rate. The crystal growth mode is fairly unique, and was based on a thermal resistance network applied to a cylindrical crystal growing on top of a frozen droplet. The frost layer growth mode was based on one available in the literature but with a modified correlation for frost thermal conductivity, which was selected after investigating six relevant correlations from the literature. The overall model was validated against four independent data sets from the literature, and matched experimental trends for all cases except the one with natural convection case. In addition, it was able to provide insight into how variations in plate surface temperature and humid air relative humidity affect frost growth characteristics, namely, frost layer thickness and frost density.

ACKNOWLEDGEMENTS

The authors would like to acknowledge and thank the National Science Foundation, Chemical, Bioengineering, Environmental, and Transport Systems Division for supporting the present work through Award No. 1604084.

NOMENCLATURE

c = crystal aspect ratio

[D.sub.AB] = binary diffusion coefficient of water vapor in air

[H.sub.cond] = latent heat of condensation

[H.sub.sub] = latent heat of sublimation

[h.sub.i] = interfacial heat transfer coefficient

[h.sub.m] = mass transfer coefficient

k = thermal conductivity

L = crystal height

m'' = mass flux

[p.sub.v] = vapor pressure of water

[sigma] = surface tension

Q = heat transfer

R = specific gas constant of water

r = droplet radius

[r.sub.c] = crystal radius

T = temperature

t = time

V = volume

[delta] = frost thickness

[theta] = contact angle

g = density

Subscripts

c = crystal

d = droplet

f = frost

fs = frost surface

i = interface

l = liquid

s = plate surface

v = water vapor

[infinity] = bulk air stream

REFERENCES

Abu-Orabi, M. 1997. Modeling of heat transfer in dropwise condensation. International Journal of Heat and Mass Transfer 41(1):81-87.

Bell, I. H., Wronski, J., Quoilin, S., and Lemort, V. 2014. Pure and pseudo-pure fluid thermophysical property evaluation and the open-source thermophysical property library CoolProp. Industrial & Engineering Chemistry Research 53 (6):2498-2508. www.coolprop.org.

Burnside, B. M., and Hadi, H. A. 1999. Digital computer simulation of dropwise condensation from equilibrium droplet to detectable size. International Journal of Heat and Mass Transfer 42(16):3137-3146.

Cheikh, A. E., and Jacobi, A. 2014. A mathematical model for frost growth and densification on flat surfaces. International Journal of Heat and Mass Transfer 77:604-611.

Glicksman, L. and Hunt Jr., A. 1972. Numerical simulation of dropwise condensation. International Journal of Heat and Mass Transfer 15(11):2251-2269.

Graham, C. 1969. The limiting heat transfer mechanisms of dropwise condensation. PhD dissertation, Massachusetts Institute of Technology.

Hayashi, Y., Aoki, A., Adachi, S., and Hori, K. 1977. Study of frost properties correlating with frost formation types. Journal of Heat Transfer 99:239-245.

Hoke, J. L., Georgiadis, J. G., and Jacobi, A. M. 2000. The interaction between the substrate and frost layer through condensate distribution. TR-177, Air Conditioning and Refrigeration Center, University of Illinois.

Huang, L., Liu, Z., Liu, Y., and Gou, Y. 2010. Experimental study of frost growth on a horizontal cold surface under forced convection. Journal of Mechanical Science and Technology 24(7):1523-1529.

Ismail, K. A. R., Salinas, C., and Goncalves, M. M. 1997. Frost growth around a cylinder in a wet air stream. International Journal of Refrigeration 20(2): 106-119.

Kim, S., and Kim, K. 2011. Dropwise condensation modeling suitable for superhydrophobic surfaces. Journal of Heat Transfer 133(8):081502.

Lee, K.-S., Jhee, S., and Yang, D.-K. 2003. Prediction of the frost formation on a cold flat surface. International Journal of Heat and Mass Transfer 46:3789-3796.

Miljkovic, N., Enright, R., Wang, and E. W. 2012. Effect of droplet morphology on growth dynamics and heat transfer during condensation on superhydrophobic nanostructured surfaces. ACS Nano 6(2): 1776-1785.

O'Neal, D. L., Tree, D. R., 1985. A review of frost formation in simple geometries. ASHRAE Transactions 91(2):267-282.

Oscarsson, S. P., Krakow, K. I., and Lin, S. 1990. Evaporator models for operation with dry, wet, and frosted finned surfaces: Part 1--Heat transfer and fluid flow theory. ASHRAE Transactions 96(1):373-380.

Ostin, R., and Andersson, S. 1991. Frost growth parameters in a forced air stream. International Journal of Heat and Mass Transfer 34(4-5):1009-1017.

Piucco, R. 2008. Theoretical and experimental analysis of frost formation on the inner walls of refrigerated cabinets. MSc Thesis, Federal University of Santa Catarina (in Portuguese).

Padhmanabhan, S. 2011. Study of frost growth on heat exchangers used as outdoor coils in air source heat pump systems. PhD Dissertation, Oklahoma State University.

Rykaczewski, K. 2012. Microdroplet growth mechanism during water condensation on superhydrophobic surfaces. Langmuir 28(20):7720-7729.

Sahin, A. Z. 1995. An analytical study of frost nucleation and growth during the crystal growth period. Heat and Mass Transfer 30(5):321-330.

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Sommers, A., Truster, N., Napora, A., Riechman, A., and Caraballo, E. 2016. Densification of frost on hydrophilic and hydrophobic substrates--Examining the effect of surface wettability. Experimental Thermal and Fluid Science 75:25-34.

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Van Dusen. 1929. International Critical Tables of Numerical Data, Physics, Chemistry and Technology.

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Ellyn Harges

Student Member ASHRAE

Lorenzo Cremaschi, PhD

Member ASHRAE

Ellyn Harges is a PhD student in the Department of Mechanical Engineering, Auburn University, Auburn, AL. Lorenzo Cremaschi, PhD is an associate professor in the Department of Mechanical Engineering, Auburn University, Auburn, AL.
Table 1. Error between model predictions and data from Piucco, 2008.
Ta = 22[degrees]C (72[degrees]F), RH = 50%, [T.sub.s] = -10[degrees]C
(14[degrees]F), V = 0.7 m/s (2.3 ft/s).

Time, h  Mass Error, %  Thickness Error, %  Density Error, %

1         0.474             13.6                -15.5
2        -0.251             -3.94                 3.63

Time, h  Frost SurfaceTemperature Error, K ([degrees]R)

1                   0.8 (1.44)
2                  -0.4 (-0.72)
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Author:Harges, Ellyn; Cremaschi, Lorenzo
Publication:ASHRAE Conference Papers
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Date:Jan 1, 2018
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