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Cited by 100Icing and De-icing TechnologiesRefrigeration and Air Conditioning TechnologiesHeat and Mass Transfer in Porous Media

Analysis of the frost growth on a flat plate by simple models of saturation and supersaturation

Lee, Y.B., Ro, S.T.·2005·Experimental Thermal and Fluid Science
DOI 10.1016/j.expthermflusci.2004.11.001
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Abstract

When humid air comes in contact with a cold surface maintained below the freezing temperature, heat and mass transfer occur and a frost layer begins to grow. Simple models for the growth of the frost layer on a flat plate are presented and they focus on diffusion and phase change inside the porous media. A simple model was built on the assumption that the water-vapor concentration at the frost surface is saturated, and that the gradient of vapor pressure is the same as the value obtained from the Clausius–Clapeyron equation. It was found that initial porosity of the frost layer affected the characteristics of frost growth. In order to solve the inconsistency of the diffusion resistance of water-vapor in the region of high porosity, the model was modified with the assumption that the concentration of water-vapor at the frost surface was in supersaturation. Two parameters controlling the densification and thickness growth of the frost layer were obtained.

AI snapshot

Written from the paper’s full text.

TL;DR

Two simple one-dimensional models are presented for frost growth on a flat plate, one assuming saturated water vapor at the frost surface and one assuming supersaturation. The supersaturation model introduces two parameters, j and l, that control vapor absorption and the split between densification and thickness growth, and it better explains diffusion resistance at high porosity.

Key findings

  • Initial frost porosity strongly affects predicted growth: lower initial density/higher porosity gives lower effective thermal conductivity, higher effective mass diffusivity, and a thicker frost layer.
  • The saturation model agreed adequately with experimental data, but the paper reports that no general fixed initial density could be found to match all data; calculations were therefore started after 60 min using experimental thickness and density as initial values.
  • In the supersaturation model, increasing j increases total mass flux, diffusive flux, deposition flux, and frost surface temperature; decreasing l increases diffusion resistance, causing more vapor to deposit on the surface and making the frost thicker and coarser.
  • The supersaturation model could explain the inconsistency of water-vapor diffusion resistance in the frost layer at high porosity, but single satisfying values of j and l for the whole test were difficult to obtain; the data between 60 and 120 min were used to match j and l.

Methods

  • Two simple models were developed for a flat plate: a saturation model assuming water-vapor concentration at the frost surface is saturated and vapor-pressure gradient follows the Clausius-Clapeyron relation, and a modified supersaturation model assuming water vapor at the frost surface is supersaturated.
  • The models solve one-dimensional mass and energy balances in the frost layer with supplementary equations for effective thermal conductivity, effective mass diffusivity, heat/mass transfer analogy, and water-vapor saturation pressure; frost surface temperature is found by trial and error.
  • Model results were compared with experiments in a test section 306 mm long and 164 mm high with a plate separation of 10 mm; representative conditions include w_a,in = 0.00300 kg/kgDA, T_w = 253.15 K, T_a,in = 278.15 K, Re_D = 2000, and x = 41 mm.

Limitations

  • The paper states that the modeling needs improvement; it ignores the Knudsen effect in microscale pores, possible nucleation by coagulation of water-vapor molecules due to supersaturation, and crystal growth inside the frost layer.
  • The paper states that j and l cannot be obtained without information on microscale phenomena and geometry, that single satisfying values for the whole test are difficult to obtain, and that l did not show a functional relation with porosity.
  • The text makes evident additional limitations: the model assumes one-dimensional transport, negligible convection and radiation inside the frost layer, quasi-steady growth, uniform porosity, equal gas and solid temperatures, constant total gas pressure, and heat/mass transfer correlations valid for surfaces without frost as approximations.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1016/j.expthermflusci.2004.11.001

Cites

20 of 20 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 11 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § 3. Results and discussion · p. 10

    Diffusion resistance factor that matches experimental data does not show a functional relation with porosity.

    “However, the diffusion resistance factor, which matches the experimental data, does not show the functional relation with the porosity.”
    Quote in the paperNot a substantial claim
  • Findingown data
    § 3. Results and discussion · p. 10

    Value of parameter j decreases with increasing porosity.

    “The value of parameter j decreases with increasing porosity.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 3. Results and discussion · p. 9

    Decrease of j results in an increase of saturation degree at the frost surface.

    “Decrease of j results in increase of saturation degree at the frost surface because diminishment of j implies that water-vapor, which is transported from free stream air to frost surface, cannot be absorbed into the layer.”
    Quote in the paperSubstantial claim
  • Findingown data
    § 3. Results and discussion · p. 6

    Initial thickness can be made to approach zero (about 2.0e-5 m) without causing problems in calculation results.

    “It was found that the initial thickness could be made to approach zero (ffi2.0 · 10 5 m) without causing problems in the results of the calculation.”
    Quote in the paperNot a substantial claim
  • Findingnone given
    § 3. Results and discussion · p. 9

    Difference between the saturation and supersaturation models increases as porosity increases.

    “Difference between the models increases as porosity increases.”
    Quote in the paperSubstantial claim
  • May combine two claims. 9
  • Classified as probably not a substantial claim. 8

  • May combine two claims.

    Findingown data
    § 3. Results and discussion · p. 6

    The model results agree adequately or well with experimental data.

    “The results from the calculation agree well with the experimental data.”
    • Also stated in: § 3. Results and discussion · p. 7 · plain
      “The results agreed with the experimental data adequately, even though a simple model was used.”
    • Also stated in: § 4. Conclusions · p. 11 · plain
      “The results from the model adequately agree with the experimental data.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 3. Results and discussion · p. 6

    As distance from the leading edge increases, mass transfer coefficient decreases and water-vapor flux to the frost layer diminishes, so the frost layer becomes thinner.

    “As distance from the leading edge increases, the mass transfer coefficient decreases and then water-vapor flux to the frost layer diminishes; therefore the frost layer becomes thinner.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given
    § 3. Results and discussion · p. 9

    The supersaturation model includes the effect of varying specific surface area, which decreases as porosity goes up and is an important parameter of the porous medium.

    “The supersaturation model also includes the effect of varying specific surface area, which decreases as porosity goes up and is important parameter of the porous medium.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Study design and conductnone given
    § 2.3.3. Analogy between heat and mass transfer on the frost surface · p. 5

    The heat and mass analogy equations (17)-(18) are valid for surfaces without frost and should be regarded as an approximation.

    “Eqs. (17) and (18), which are valid for surfaces without frost, should be regarded as approximation.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 3. Results and discussion · p. 6

    Sublimation density of ice crystal increases as crystal formation temperatures increase.

    “Fukuda and Inoue [6] and Miller and Young [22] found that the sublimation density of ice crystal increases as the crystal formation temperatures increase.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites Fukuda and Inoue [6]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites Miller and Young [22] Miller, 1979, A numerical simulation of ice crystal growth from the vapor phase, J. Atmos. Sci. source

  • May combine two claims.

    Interpretationreasoning
    § 3. Results and discussion · p. 6

    When effective thermal conductivity decreases, frost surface temperature increases and the amount of convective heat transfer decreases.

    “When the effective thermal conductivity decreases, the frost surface temperature increases and the amount of convective heat transfer decreases.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 3. Results and discussion · p. 6

    Initial value of the volumetric ratio of ice columns in the frost layer is a function of environmental parameters.

    “Moreover, according to the S ß ahin Õ s results [7], the initial value of the volumetric ratio of the ice columns in the frost layer is a function of environmental parameters.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Sahin [7] Şahin, 2000, Effective thermal conductivity of frost during the crystal growth period, Int. J. Heat Mass Transfer source

  • May combine two claims.

    Interpretationown data
    § 1. Introduction · p. 2

    The models are useful for predicting frost growth and analyzing microscale heat and mass transport within the frost layer.

    “The models are useful for predicting frost growth and analyzing microscale heat and mass transport within the frost layer.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation
    § 2.1. Water-vapor concentration in the frost layer · p. 4

    Water-vapor transport through a porous medium is mainly due to Fickian diffusion caused by the gradient of the vapor partial pressure.

    “Previous researchers assumed that the water-vapor at the frost surface is saturated with the surface temperature, and water-vapor transport through a porous medium is mainly due to Fickian diffusion caused by the gradient of the vapor partial pressure [4,11,13,14].”
    • Also stated in: § 3. Results and discussion · p. 7 · boosted
      “As presented, many have agreed that the water-vapor transport through a porous medium is mainly due to Fickian diffusion caused by the vapor partial pressure gradient and to water-vapor at the frost surface, which is saturated at the frost surface temperature.”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [4] Sherif, 1993, A semi-empirical transient method for modelling frost formation on a flat plate, Rev. Int. Froid source
    • cites [11] Le Gall, 1997, Modeling of frost growth and densification, Int. J. Heat Mass Transfer source
    • cites [13] Ismail, 1997, Frost growth around a cylinder in a wet air stream, Int. J. Refrig. source
    • cites [14] Lüer, 2000, Frost deposition in a parallel plate channel under laminar flow conditions, Int. J. Thermal Sci. source

  • Classified as probably not a substantial claim.

    Premisenone given
    § 1. Introduction · p. 2

    Growth characteristics of the frost layer cannot yet be accurately predicted and explained.

    “Despites a fairly large number of studies dealing with frost formation, growth characteristics of frost layer cannot yet be accurately predicted and explained.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3. Results and discussion · p. 6

    As initial porosity increases, effective thermal conductivity diminishes and effective mass diffusivity increases.

    “As initial porosity increases, effective thermal conductivity diminishes and effective mass diffusivity increases from Eqs. (14) and (16).”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Premisecitation
    § 3. Results and discussion · p. 7

    Water-vapor in the frost layer is supersaturated.

    “Fukada and Inoue [6] suggested that the water-vapor in the frost layer is supersaturated in their experimental research.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Fukada and Inoue [6] Fukada, 1999, Internal mass flux through frost, AIChE J. source

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § 3. Results and discussion · p. 7

    Vapor being transported across the thermal boundary layer undergoes homogeneous nucleation to the liquid phase, and vapor pressure is larger than the saturated value.

    “Epstein and Rosner [23] concluded that the vapor being transported across the thermal boundary layer is observed to undergo homogeneous nucleation to the liquid phase, and vapor pressure is larger than saturated value.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Epstein and Rosner [23] Epstein, 1970, Enhancement of diffusion-limited vaporization rates by condensation within the thermal boundary layer, Int. J. Heat Mass Transfer source

  • May combine two claims.

    Premisecitation
    § 3. Results and discussion · p. 7

    The Knudsen effect of the microsize pore in the frost layer and the possibility of nucleation of particles by coagulation of water-vapor molecules due to supersaturation reduce the mass diffusivity of water-vapor and affect the growth of the frost layer.

    “One is the Knudsen effect of the microsize pore in the frost layer, which is the space between microscale ice crystals. Second is the possibility of nucleation of the particles by the coagulation of the water-vapor molecule due to the supersaturation. Both of them reduce the mass diffusivity of the water-vapor and affect the growth of the frost layer.”
    Quote in the paperSubstantial claim
14 of 14

  • The authors state this as an assumption.

    Transport processes are one-dimensional in the frost layer.

    “(a) Transport processes are assumed to be one-dimensional in the frost layer.”

  • The authors state this as an assumption.

    Convection and radiation effects are negligible within the frost layer.

    “(b) Convection and radiation effects are negligible within the frost layer.”

  • The authors state this as an assumption.

    Frost growth is in quasi-steady state.

    “(c) Frost growth is in quasi-steady state.”

  • The authors state this as an assumption.

    Porosity of the frost layer is uniform in the frost layer.

    “(d) Porosity of the frost layer is uniform in the frost layer.”

  • The authors state this as an assumption.

    Temperature of the gaseous phase is the same as that of the solid phase.

    “(e) The temperature of the gaseous phase is the same as that of the solid phase.”

  • The authors state this as an assumption.

    Total pressure of the gaseous phase inside the frost layer is constant.

    “(f) And the total pressure of the gaseous phase inside the frost layer is constant.”

  • The authors state this as an assumption.

    Water-vapor concentration at the frost surface is saturated (simple model).

    “A simple model was built on the assumption that the water-vapor concentration at the frost surface is saturated, and that the gradient of vapor pressure is the same as the value obtained from the ClausiusClapeyron equation.”

  • The authors state this as an assumption.

    Gradient of vapor pressure is the same as the value obtained from the Clausius-Clapeyron equation.

    “A simple model was built on the assumption that the water-vapor concentration at the frost surface is saturated, and that the gradient of vapor pressure is the same as the value obtained from the ClausiusClapeyron equation.”

  • The authors state this as an assumption.

    Water-vapor concentration at the frost surface is in supersaturation (modified model).

    “In order to solve the inconsistency of the diffusion resistance of water-vapor in the region of high porosity, the model was modified with the assumption that the concentration of water-vapor at the frost surface was in supersaturation.”

  • The authors state this as an assumption.

    Rate of ablimation of water-vapor per unit area is proportional to the difference of water-vapor densities in air and on the ice crystal surface.

    “The rate of ablimation of water-vapor per unit area is assumed to be proportional to the difference of watervapor densities in air and on the ice crystal surface, and a symbol w is adapted as a coefficient for phase change.”

  • The authors state this as an assumption.

    The second term of Eq. (22) can be assumed as constant.

    “As assumed in Eq. (3), the second term of Eq. (22) can be assumed as constant.”

  • The authors state this as an assumption.

    Water-vapor densities on the interface between ice crystal and air are the same as the saturated value at the temperature of the location.

    “Here another assumption is used as boundary conditions at the frost surface, which is g = 1, and the base, which is g = 0; namely that water-vapor densities on the interface between ice crystal and air are same as the saturated value at the temperature of the location.”
12 of 12

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Deposition should be more homogeneous within the frost layer as shown by experimental observations (Lee and Ro, 2005) or the CFD results (Kim et al., 2015).

    “Deposition should be more homogeneous within the frost layer as shown by experimental observations (Lee and Ro, 2005) or the CFD results (Kim et al., 2015).”

    From Frosting modeling on a cold flat plate: Comparison of the different assumptions and impacts on frost growth predictions (2016) · cites this paper as Lee and Ro, 2005

    The Introduction (p. 2) reports experimental observations of an almost flat frost-density profile through the layer, and §2.1 (p. 3) uses that finding to assume uniform porosity. This is consistent with a more homogeneous layer, but Lee and Ro attribute the observation to Fukada and Inoue; their own experimental comparisons do not establish that deposition is homogeneous within the layer. (LLM, conf=0.82)

    Experimental — accuracy not yet measured

  • The conclusions of Lee and Ro on the effect of air moisture content on frost density are opposite to those of [50,54].

    “which was totally opposite to the conclusions of Lee and Ro [39,80]”

    From Frosting mechanism and behaviors on surfaces with simple geometries: A state-of-the-art literature review (2022) · cites this paper as [80]

    The Results and discussion (p. 7) say that air humidity affects the initial porosity and density of the frost layer, supporting an effect of air moisture content on frost density. Our copy does not state that Lee and Ro’s conclusion is opposite to those of [50,54], so it does not support the comparison. (LLM, conf=0.78)

    Experimental — accuracy not yet measured

  • In some studies, the average frost thickness on flat plates with a fixed surface temperature decreases with an increase in air temperature.

    “In some studies, the average frost thickness was decreased with an increase in air temperature [50,54,61,80].”

    From Frosting mechanism and behaviors on surfaces with simple geometries: A state-of-the-art literature review (2022) · cites this paper as [80]

    The Results and discussion (p. 7) examine frost-layer growth as air temperature varies, and the Fig. 8(c) caption specifies a fixed cooling-surface temperature. Our copy does not state that thickness decreases as air temperature rises; the figure is represented only by an image placeholder. The caption also specifies a plate location rather than an average thickness. (LLM, conf=0.76)

    Experimental — accuracy not yet measured

  • Lee and Ro [10] modeled the air gaps in the frost as cylindrical capillary tubes to model water vapor diffusion in the frost layer.

    “Lee and Ro [10] modeled the air gaps in the frost as cylindrical capillary tubes to model the water vapor diffusion in the frost layer.”

    From Numerical Simulation of Frosting on Fin-and-Tube Heat Exchanger Surfaces (2017) · cites this paper as [10]

  • The water vapor density gradient inside the frost layer near the plate surface was significantly smaller than that near the frost surface.

    “For water vapor density, its gradient inside the frost layer near the plate surface was significantly smaller than that near the frost surface [50,80,86,88,90,91].”

    From Frosting mechanism and behaviors on surfaces with simple geometries: A state-of-the-art literature review (2022) · cites this paper as [80]

Cites

Flagged first, then by citations

Research works printed but not linked: 5

  1. ASHRAE, ASHRAE Handbook Refrigeration 2002, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., Atlanta, USA (Chapter 6)
  2. Padki, 1989, A simple method for modeling the frost formation phenomenon in different geometries, ASHRAE Trans., 95, 1127
  3. Mao, 1992, Measurement and correlations of frost properties with airflow over a flat plate, ASHRAE Trans., 98, 65
  4. Brian, 1969, Cryogenic frost properties, Cryo. Technol., 5, 205
  5. ASHRAE, ASHRAE Handbook Fundamentals 2001, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., Atlanta, USA, 2001 (Chapter 6)

Cited by

Sorted by citations

Related

Authors
Lee, Y. B., & Ro, S. T.
Affiliations
  • LG (South Korea) (KR)
  • Seoul National University (KR)
Corresponding author
Ro, S.T.
Venue
Experimental Thermal and Fluid Science
Publisher
Elsevier BV
Published
2005-07
Volume
29
Issue
6
Pages
685-696
ISSN
0894-1777 (ISSN-L)
DOI
10.1016/j.expthermflusci.2004.11.001
OpenAlex
W2001794842
Semantic Scholar
3770b4c095a21f5e9c60e9dba5b40584095ae42d
Cited by
100
References
20
Citations by year
2026: 7 · 2025: 5 · 2024: 8 · 2023: 6 · 2022: 10 · 2021: 10 · 2020: 3 · 2019: 2 · 2018: 5 · 2017: 6 · 2016: 9 · 2015: 6 · 2014: 3 · 2013: 3 · 2012: 3
FWCI
0.76
Type
journal article
Language
en
Open access
closed
Licence
www.elsevier.com/tdm/userlicense/1.0/ (tdm)
Locations
Experimental Thermal and Fluid Science (publishedVersion)
Topics
  • Physical Sciences › Engineering › Aerospace Engineering › Icing and De-icing Technologies
  • Physical Sciences › Engineering › Mechanical Engineering › Refrigeration and Air Conditioning Technologies
  • Physical Sciences › Engineering › Computational Mechanics › Heat and Mass Transfer in Porous Media
Keywords
frost growth, frost layer, flat plate, saturation, supersaturation, water vapor diffusion, porous media, phase transitions

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