A semi-empirical transient method for modelling frost formation on a flat plate

A semi-empirical transient method for modelling frost formation on a flat plate S. A. Sherif Department of Mechanical Engineering, University of Florida, Gainesville, FL, 32611-2050, USA S. P. Raju Department of Mechanical Engineering, University of Washington, FU-10, Seattle, WA 98195, USA M. M. Padki Clean Energy Research Institute, University of Miami, Coral Gables, FL 33124, USA A. B. Chan Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 26 N o v e m b e r 1991," revised 4 M a y 1992

This paper models the frost formation process employing a semi-empirical transient model for a flat plate under forced convection conditions. This continuous differential model employs the Euler's method to solve for the temperature and thickness of the frost layer using existing correlations for the convective heat transfer coefficient and the Lewis analogy to compute both a mass transfer coefficient and an enthalpy transfer coefficient. The model is also based on empirical correlations for predicting the frost thermal conductivity and density. Model results are compared with existing experimental data and with numerical data of other investigators and are found to agree very well in the applicable temperature and humidity range of the frost density and conductivity correlations. (Keywords:frosting;frost; planesurface; forced convection;calculation)

M6thode transitoire semi-empirique pour la mod61isation de la formation de givre sur une plaque plate Dans l'article, on traite de la modklisation du processus de Jbrmation de givre qui utilise un modkle transitoire semi-empirique pour une plaque plate dans des conditions de convection forcke. Ce modkle diff~rentiel continu se fonde sur la mbthode de Euler pour r~soudre les caractkristiques de tempkrature et d'bpaisseur de la couche de givre en utilisant des corr~lations pour le coefficient de transfert de chaleur par convection et l'analogie de Lewis pour calculer un co£fficient de transJert de masse et un coefficient de tran~fert d'enthalpie. Le module se fonde ~;galement sur des correlations empiriques utiliskes pour prOvoir la conductivit~ thermique et la densitk du givre. On compare les rksultats du modOle avec les donn~es expOrimentales existantes et avec des donnkes num~riques obtenues par d'autres chercheurs, et on note une corrblation satisfaisante, dans la plage des temperatures et humid#ks courantes, pour la densitb et la conductivitb du givre.

(Mots cl6s: givrage; givre; surface plane; convection forc6e; calcul)

Whenever moist air comes in contact with a surface whose temperature is below both the dew point of water vapour in air and the freezing point, frost will form. Previous studies indicate that the heat transfer rate increases at the initial stages of deposition since the rough frost surface acts as a finned one. As the frost thickens, however, the insulating effect of the frost layer predominates resulting in a reduction in the heat transfer rate. The frost deposition process itself is complicated in part because the rate at which water vapour is diffused into the frost layer is dependent on the rate at which heat is transferred from the moist air stream to the frost layer. Other complicating factors include the time dependency of the frost properties and the time and spatial dependency of the frost surface temperature. Further complications arise, mainly due to the occurrence of repeated

*To whom correspondence should be addressed. 0140 7007/93/050321~)9 1993 Bunerworth Heinemann and IIR

cycles of melting and refreezing once the a i ~ f r o s t interface temperature reaches the freezing temperature. This results in structural changes in the frost layer that tend to increase both the frost density and thermal conductivity with time without a proportional increase in the frost thickness. This phenomenon is more likely to occur in high humidity and/or high temperature environments than low humidity and low temperature ones. In the latter case, the frost air interface temperature will rise to a steady value, below the melting point, such that the rate of frost deposition is balanced by the rate of sublimation from the frost. Previous studies on frost formation are numerous. A rather comprehensive list of references can be found in Padki et al.~. Studies pertaining to flat plate geometries under both free and forced convection conditions are given below, however. Free convection studies include those of Whitehurst 2,3, Whitehurst et al. 4. Barron and Han 5, Mulligan 6,7, Tajima et a l ) ~, G o o d m a n and Ken-

Rev. Int. Froid 1993 Vol 16 No 5

321

A semi-empirical transient method for modelling frost formation on a flat plate. S.A. Sherifet al.

Nomenclature

A Surface area, m C1 - C7 Constants defined by Equation (14) Mass ratio of frost melted to total frost Cmeh deposited c. Specific heat of dry air = 1.004 ( k J k g - l K i) Specific heat of frost = 2.261 (kJ kg-1 K - I ) C~m Specific heat of moist air = C v + W~Cp,~ (kJkg 1K-I) Cpm,a* Specific heat of moist air evaluated at t~ o r t v ( k J k g 1K 1) Specific heat of water vapour, C.v (kJ kg -1 K 1) Cpv,a~ Specific heat of water vapour evaluated at t, o r tp, (kJ kg-1 K - 1) Cpv,p D Coefficient of mass diffusion of water vapour in air (m -2 s -l) Ft(tp0 Function defined by Equation (23) h Average film coefficient of heat transfer (W m -2 K - ' ) Local film coefficient of heat transfer hx (W m-2 K ') Average enthalpy transfer coefficient hi (kg m -2 S-l) Local enthalpy transfer coefficient hix (kg m -" s-1) h, Average convective mass transfer coefficient (kg m -2 s -1) Local convective mass transfer coefficient hvx (kg m -2 S t) Enthalpy of moist air evaluated at t, (kJ kg ~) ip× Enthalpy of moist air evaluated at tpx (kJ kg -I) K Thermal conductivity of air ( W m ~ K ~) Kf Average thermal conductivity of frost ( W m -I K -I) Local thermal conductivity of frost xf~ (W m -1 K -I) L Latent heat of sublimation of frost (kJ kg-i) Lewis number = o(D Le Molecular weight of dry air -- 28.9 m~a Molecular weight of water vapour = 18 M~ Frost accumulation rate (kg m -2 s-~) my Total (barometric) pressure = 101320 Pa P~ Partial pressure of water vapour in air (Pa) P, Partial pressure of water vapour in air at Pvsat saturation (Pa) Pr Prandtl number = ].tCpm/K Q Total energy transfer rate (W)

nedy 12, Kennedy and G o o d m a n t3 and Yamakawa and OhtanP 4,t5. Forced convection to flat plates, on the other hand, was investigated by Coles and RuggerP 6, Glomb and WenzeW, Yonko ~8, Yonko and Sepsy tg, Hosoda and Uzuhashi z°, Trammell et al. 2Lz2, Jones and Parker 23, Seki et al. 24,2s, Abdel-Wahed et al. z6, Sherif et al. 27, and more recently by Padki et al.~ and Sami and Duong 2s. Review 322

Int. J. Refrig. 1993 Vo116 No 5

Re

Re~ St ¸

Sf, T ta [m

tmx

gp /px ts Va~

v~ w. w~x

Total energy transfer rate per unit area (W m -2) Average Reynolds number = U ~:)t/v Local Reynolds number = U~2"/v Average frost thickness (m) Local frost thickness (m) Absolute temperature (K) Dry-bulb temperature of air (°C) Average air-frost boundary layer temperature (°C) Local ai~-frost boundary layer temperature (°C) Average frost surPace temperature (°C) Local frost surface temperature (°C) Plate surface temperature (°C) Free-stream velocity (m s-t) Molar volume of dry air = 29.9 Molar volume of water vapour = 18:8 Humidity ratio of air evaluated at t. (kgv/kgdry ~ir) humidity ratio of air evaluated at tp~ (kg~/kgd~y air)

x

plate width (m) distance measured from leading edge of plate (m)

Greek syrnbols

2 /2 v p

Of

pf, pf~ P,c~r

Thermal diffusivity of air (m 2 s i) Plate length (m) Dynamic viscosity of air Kinematic viscosity of air (m 2 s ~) Density of air (kg m ~) Frost density (kg m ~) Frost density after melting and refreezing has started (kg m-3) Local frost density (kg m -3) Average density of ice = 897 kg m -~ Time (s)

Subscripts

a f m melt p px s sat t v x

Ambient air Frost Moist air Melt Frost-air interface Frost-air interface at distance x Solid surface Saturated vapour Total Vapour, mass transfer Evaluated at distance x from plate leading edge

studies were reported by O'Neal and Tree 29 and Kondepudi and O'NeaP °. Despite the relatively large number of articles dealing with the problem of frost formation, gaps and inconsistencies still exist. For example, investigators still disagree on the percentage of frost diffused in the porous frost layer versus that deposited on the top of the frost layer.

A semi-empirical transient method for modelling frost formation on a flat plate: S.A. Sherifet al. The frost diffused is more likely to increase the density of the frost sublayers without an appreciable increase in the frost thickness, whereas the reverse is true for the frost deposited on the upper frost layers. Also, it appears that little or no data exist on the development of the frost interface temperature with both time and position. As was shown by Padki et al?, most of the relevant frost parameters depend in whole or in part on the interface temperature. A reliable method to predict such a temperature should, therefore, prove useful in terms of developing a better overall understanding of the frost formation process. The model proposed by Padki et al. ~is one such method. In that model, a simple method of predicting the growth rate, as well as the frost thickness and surface temperature with both time and position was proposed. The method utilized known convective heat transfer correlations for different geometries along with the Lewis analogy to determine both a convective mass transfer coefficient and an enthalpy transfer coefficient. An iterative quasi-steady-state approach was adopted to compute the air-frost interface temperature, frost properties, the partial pressure of water vapour at the frost surface and the frost thickness. The technique was also provided with a capability for adjusting the frost density and thermal conductivity to account for changes in the frost layer structure as a result of the occurrence of a melting and refreezing phenomenon on the upper frost layers. Model results were shown to agree quite well with existing experimental data. This paper reports on a more refined method for modelling the frost deposition process on flat plates under forced convection employing a semi-empirical transient approach. This continuous differential model utilizes the Euler's method to solve for the temperature and thickness of the frost layer using existing correlations for the convective heat transfer coefficient and the Lewis analogy to compute both a mass transfer coefficient and an enthalpy transfer coefficient. The model is also based on empirical correlations for predicting the frost thermal conductivity and density. Model results are compared with existing experimental data and with models proposed by Yonko and Sepsy 19, Jones and Parker 23 and Sami and Duong 2s, and are found to agree very well in the applicable temperature and humidity range of the frost density and conductivity correlations. The model proposed is shown to be simple but surprisingly accurate and can be used as a basis for developing similar models for other simple geometries. Analysis

Figure 1 shows the physical model. Air at a specified temperature, t~, humidity, W~, and velocity, U~, flows over a solid surface whose temperature is ts. If ts is below the dew point temperature of water vapour in air, moisture will condense on the surface. If ts is also below freezing, frost will form. Initially, the frost surface temperature, tp, will equal that of the plate surface, ts. As the frost layer thickens, tp will increase and will generally assume a value between ts and ta. As tp increases, the humidity ratio of air in the neighbourhood of the frostair interface, Wp, will also increase resulting in a decrease in the water vapour concentration driving force and consequently in the frost deposition rate. In general, the water vapour transferred will partially diffuse into the

FREE AIR STREAM = Uoo

AIR-FROST BOUNDARYLAYER /

T~ C~

FLAT PLATE

Figure 1 P h y s i c a l m o d e l

Modblephysique

Figure 1

porous frost layer, thus contributing to an increase in the frost density, and partially deposits on the upper layers, thus contributing to an increase in the frost thickness. The following are the governing and constitutive equations.

Governing equations Energy balance. The energy transferred from the humid air stream will be partially conducted through and partially stored in the frost layer. This can be expressed by: Z

I

2

hi ( ia

--

ipO dx = I (Kr, dtpffdsfO dx

x-0

x=0

(1)

Z

+ (d/d0 ~ SfxPfxCpmtmxdx x--O

where the second term on the right-hand side represents the storage term which will be neglected in the present analysis (see Padki et al.~). The total energy transfer rate, Q, can be written as follows: Q = ~w [h, (t a

--

tpx) + L hvx (IV, - Wp0] dx

(2)

x=0

where the first and second quantities on the right-hand side represent the sensible and latent components of energy, respectively.

Mass balance. The water vapour transferred from the humid air stream will partially deposit as frost on the solid surface and partially diffuse into the porous frost sublayers, i.e.: A

R

2

j" hv (W, - Wpx)dx = S pr,dSr,/drdx + ~ Sa dpfffdrdx

x-o

x=o

x=o

(3)

where the first term on the right-hand side represents the

increase in the frost thickness, while the second term represents the water vapour diffused into the frost layer. The frost deposition rate, my, can thus be written as: 2

mv = j" [hvx (W~, -

Wp,). dx]A

(4)

x=O

Rev. Int. Froid 1993 V o 1 1 6 No 5

323

A semi-empirical transient method for modelling frost formation on a flat plate: S.A. Sherif et at Constitutive relations Air enthalpy. The average and local air enthalpies can be found from ASHRAE3~: i.

=

Cpt a + W,(L + Cp,,~t~)

15)

and iv~ = Cp/px + Wpx(L + Cpv,pxtpx)

(6)

Heat transjer coefficient. The local heat transfer coefficient on a flat plate in forced convection can be found from the following expression32: h~ = 0.322 Pr ~ Re~!6K/x

(7)

Mass transfer coefficient. The Lewis analogy requires that the heat and mass transfer coefficients be related by the following equation32: hvx = h x / G m

(8)

The above expression is based on a unity value for the Lewis number.

Enthalpy transjer coefficient. The enthalpy transfer coefficient is defined by the following equation: 2

Q = ~ [hix w(ia -- ip0]dx

(9)

"~'=0

It can easily be proved that this coefficient is numerically equal to the mass transfer coefficient (see Padki et al)), i.e.:

(io)

h~ = h,~ (numerically)

Mass d~ffusion coefficient. The mass diffusion coefficient, D, can be written in terms of the molecular weights and molar volumes of dry air and water vapour according to the following expression32: D = 0.04357

t,15[l/M~, + 1/Mv]°s/[p~(va, I-3 + V,I/3) 2]

(11) where P~ is expressed in pascals. The diffusion coefficient may also be made to depend on the frost surface temperature by using the air-frost boundary layer mean temperature instead of the ambient temperature.

Air humidity ratio. Assuming saturation conditions in the neighbourhood of the air-frost interface, the humidity ratio of air at that point can be expressed in terms of the saturation partial and total pressures as follows3': Wp~ = 0.62198

Pvs~t,px/(Pt

-

Pvsat,px)

(12)

The saturation partial pressure at the frost surface can be expressed as a function of the frost surface temperature according to the following3~: loge(P~,at,px) = C,/Tpx + C2 + C3Tv× + C4Tpx 2 + CsTpx~ -{- C6Tpx 4 -[- C 7 log~(Tp0

- I 0 0 ° to O°C.

Frost densio,. The frost density :generally depends on the environmental conditions and the plate surface temperature which, in turn, determine the structure of the frost layer. In order to simplify the frost density expression, however, Hayashi et al. 33 modelled the variation of the density as a sole function o f the frost surface temperature. This may be an over-simplification o f the problem, but it is not inaccurate. The justification for such a description is probably the fact that the frost surface temperature itself is a strong function of the environmental parameters, plate temperature and deposition time. Hayashi's expression is: p~. = 650 exp [0.227 tp]

(151

where p~ is in kg m- 3. The range of applicability of Equation (15) is - 2 5 ° C < tp < 0°C, airstream velocities between 2 and 6 m s- ~and an airstream humidity ratio of 0.0075 kg,/kgdry ,i,. Continuous deposition of frost will result in a contio nuous increase in the frost surface temperature, density, thermal conductivity and thickness. When the frost surface temperature reaches the melting point, the frost deposited will partially melt and seep through the pores of the porous frost layer, eventually reaching the metal surface, where it freezes into ice, and partially stay as frost (contributing to an increase in the frost thickness). Repeated occurrence of this phenomenon will gradually transform the bottom layer into ice and result in increasing both the density and the thermal conductivity of the deposited layer. Obviously, the bottom layer will stop being porous at some point and the melt will no longer penetrate all the way to the metal surface after very long deposition times'. When the frost surface temperature reaches the melting temperature, the frost density should be computed using the following equation: PI,,~ = [Sfpr ÷ CmeltASf~ce + (1 -- Cme|t)ASfpf] / (Sf + ASf)

(16) where Pr,,, is the new frost density alter one cycle of melting and refreezing has occurred. The constant, Cmdr, is merely a way of taking the aforementioned physical phenomenon into account. The exact numerical value of Cmeh is unknown, but is between zero and 1. This value should be determined using experimental data.

Frost thermal conductivity. Yonko and Sepsy m suggested the following correlation for the frost thermal conductivity: Kf~ = 0:024 2 4 8 + 0.000 723 It pf, + 0.000 001 183 pf~:: (17)

(13)

where Pfx is expressed in kg m 3and K~xis expressed in W m-~ K-~. The above equation is only valid for frost

where 324

CI = -5674.5359 C, = 6.392 524 7 C~ . . . . 0.967 784 3 x 10 : (4 .... 0.622 157 01 x 10 ~' (!4) ('~ -- 0.207 478 25 x 10 C,, := - 0 9 4 8 4 0 2 4 x 1 0 ~ ('3 ::: 4.163 501 9 The above equation is valid for the temperature range

Int. J. Refrig. 1993 Vo116 No 5

A semi-empirical transient method for modelling frost formation on a flat plate: S.'A. Sherifet al. densities less than 573 kg m -3. Equation (17) is the one used in this paper for prediction purposes. Other expressions for the thermal conductivity are those proposed by Sanders 34, Brian et al. 35,3~ and Marinyuk 37. The expressions developed by Sanders 34 and Marinyuk 37 are listed below: Kf = 0.001 202

pf0.963

(18)

Kr = 1.3 (Tp - L ) (0.156 exp [0.0137Tp] (19) - exp [0.0137Ts]) + 5.59 x 10-Spf. (exp[0.0214Tp] - exp[0.0214TJ)

Equation (18) should be applied under the following conditions: wall temperature, t~, from - 2 2 to I I°C; air temperatures, t,, from - 1 0 ° to 0°C; and air velocities, U~, from 4 to 9 m s- i. The maximum limit of applicability of Sanders' expression is 500 kg m-3. Equation (19) should be limited to densities up to 300 kg m ~, however. As suggested by Padki et al. ~, no attempt will be made to model the variability of the frost density or thermal conductivity analytically with either time or the frost surface temperature. Empirical equations developed by previous investigators for the frost density and thermal conductivity will be employed instead. Although this approach imposes some restrictions on the generality of the results obtained, it does not affect the generality of the methods employed and, if care is exercised in selecting the proper equations for the frost density and thermal conductivity, the method can be made to predict the relevant frost parameters to a high degree of accuracy. Numerical procedure

Equations (1) and (3) can be rewritten as follows after neglecting the storage term in Equation (1): (20)

and h,.x(W~ - Wp.) = Srxdp,-~/dr + pfxdSf~/dr

(21)

The second term on the right-hand side was identified as a diffusion term by Brian et a l ) 6. Differentiating the expression for the frost density, Equation (15), with respect to time, one gets: dpf~/dr = 147.55 exp [0.227 tp~] dtp~/dv = FfftpOdtpx/dr (22) where Ffftp0 = 147.55 exp[0.227 tp~]

(23)

Equation (21) can be rewritten in the following form: h,.~(W. - Wp.) = p,-.(dSfx/dtpO (dtp,./dr) + Sr~Ffftpx)dtvx/dr

(24)

Upon substitution from Equation (20) one gets: h,.x(iv. - Wp.) = dtp.~/dr[pr.Kfj{h~.(ia - ip0} + S,.~Ft(tvO]

dtp./dr = h~ (IV. - WpO/[p,.~K,-~/{h,~(i ~ - ip~} + S,xF,(tpO] (26) In a similar fashion, Equation (21) can be rewritten in the following form: h~(W. -

and

hi(i~ - ipOdx = (K~xdtp~/dSt-~)dx

which can be solved for dtp×/dv to yield:

(25)

Ww) = pf~dSr~/dr + Sr~Fl(tpO (dtp./dSrO (dSr~/dr)

(27)

which can be solved for d S a / d r after employing Equation (20):

dSf~/dr = h~ (W~ -

Ww)/{ptk + SrxF,(tvx ) hlx(i~ -

ipx)/Krx]

(28) Equations (26) and (28) were solved numerically to yield the frost thickness and surface temperatures as functions of time. Several integration schemes were tried with a range of integration steps from i/16th of a second to 1 s, and results were found to be fairly independent of the algorithm used or the time step employed. The sequence of numerical predictions is illustrated in Figure 2.

Results and discussion The method described in the previous section was applied to flat plates in forced convection, and comparisons with experimental data and other models were made whenever possible. Use of the empirical correlations for the frost density and thermal conductivity was made under conditions similar to the ones suggested by their author(s). Comparisons were made realizing that caution should be exercised in interpreting the results since these correlations are generally valid in a specified range. Most of the experimental data available in the literature are for average rather than local values. In the analysis given in the previous section, local quantities were the main focus. This was intended in order to make the analysis as general as possible. Derivation of average expressions is very straightforward, however, if an expression for the average heat transfer coefficient is employed and local variations in the frost properties and frost surface temperature are neglected. The development of the frost thickness as a function of time can be seen in Figures 3~10 for a host of conditions. In Figures 3 and 4, the air humidity ratio was changed after 60 min of frost deposition. A close examination of these figures indicates that the thickness profile experiences a sudden increase at the same time for all three models (present study, Jones and Parker 23 and Sami and Duong2~). As can be easily seen from Figures 3-7, the present model predicts the frost thickness to a very high degree of accuracy despite the simple nature of the algorithm employed. In most cases, the present model seems to pass through between Jones and Parker's and Sami and Duong's models. The latter model seems to be fairly representative of most of the data points, in part since it employs a molecular diffusion approach that accounts for the tortuosity phenomenon (defined as the pathRev. Int. Froid 1993 Vo116 No 5

325

A semi-empirical transient method for modelling frost formation on a flat plate: S.A. Sherif et a l 0,600 ,

!

tRead

t S ~ ta,W a

q,

Re

]

%-

0 , 5 0 0 ~/

i

0.400

tad Z

0.300

J - Porker d a t a 1 - Sami'e model 2 - Preunt model

t l

WO -- 0.0105

O

" Set initial conds. tp=ts, Sf=0

....... ~'~

0000

~~

i

~

0.0

20,0

I

i

t

t

40.0

60.0

80,0

100.0

120,0

TIME (MIN)

Figure 3 Variation of frost thickness with time Figure 3 Modification de l'dpaisseur du givre avec le temps

I Calculate:

T, - 255 K

o2oo~ °'1°°ii~~

Set simulation time Set time step

_1

60 < t < 120 mira

TO - 296 K I

i

- porkePS model

3

Wa -- 0.007 0 < t < 60 rain

tf, Pr, K,

I

h,

hv, hi,

Le, and D

l

Compute: Wp

! i L

0,420 T . . . . . . . . . . . . • - Parker data 0.360-t

'&'"f' I

ia , and i

p

1 - Sami'l

t

0,300--v 03 b~

1

model

2 -- present model 3 -- porker*s model

J

Wo = 0.0075

~"~"~?

0 < t < 60 mira

.::2o;;1: 6o

0.240-

,2o

3( 0.180-

Ts

~

Solve rate equations to obtain t and Sf P

F, 0

- 259 K

•

- 9100 ~

R

e A

~

~

i

/

3

1

0,1200,060 0.000

20.0

0.0

40.0

60:0

80.0

100.0

120,0

TIME (MIN)

Figure 4 Variation of frost thickness with time Figure 4 Modification de l'bpaisseur du givre avec le temps

No eFl

1.0100

o.go0 -

Figure 2 Flowchart Figure 2 Logiciel

• 1 2 3

--

Yonko & Sepsy d a t a Somi't model

%-

0.800

E o

0.700

Wo 1 0.0141

0,600

To Ts -

v

Pr e s e nt

model

parker's model 296 K 245 K

Re = 6400

length the molecules travel). The simplicity and generality of the model presented in this paper are positive features, however, perhaps making the model preferable in cases where simplicity of calculations and ease of applicability to other geometries are desirable: The effect of environmental parameters on the frost thickness can be seen in Figures 8-10. Figure 8 indicates that the Reynolds number has a rather small effect on the thickness, an observation that is consistent with previous frosting studies. This is not the case, however, for either the humidity ratio as indicated in Figure 9 or the air dry-

326

Int. J. Refrig. 1 9 9 3 V o 1 1 6 N o 5

0,5000.400 0.300O

0.200 0,1000.000

0.0

2(~.0

do

do

8~o

lO~O

12oo

TIME ( M I N )

Figure 5 Variation of frost thickness with time Figure 5 Modification de l'kpaisseur du givre avec le temps

1,oo

A semi-empirical transient method for modelling frost formation on a flat plate." S.A. Sherifet al.

0.500-

• - Yonko & I m p l y dora 1 - Sami's model

0,450-

2 -

Present

1.000 -I

--1

0.550 . . . . . . . . . . . . . . . . . . . . .

model

~

u

0.400 (/~

Wo - 0 , 0 1 0 6 To - 2 9 5 K

0.3500.300-

Z v 0 -r(--

Tt -

255 K

Re -

12000

0.800-

0.600 -

Wo -

0.0108

Re -

13900

1

-

To -

288 K

Ts ~ 2 4 5 K

2

-

To -

296 K

3

-

Ta -

303 K

Present

Z

model

0.2500.400 0.200-

/

0.150-

b. 0,I000.0500.000

'

0.0"

'

'

i . . . .

25.0

i . . . .

50.0

I

. . . .

I

75.0

. . . .

i

100.0

TIME

. . . .

i

125,0

. . . .

150.0

I

0.0

. . . .

i . . . .

i . . . .

25.0

50.0

i

75.0

. . . .

i . . . .

i . . . .

100.0

125.0

i .... 150.0

175.0

TIME (MIN)

(MIN)

Figure 9

Figure 6 Variation of frost thickness with time

Figure 9

Modification de l'#paisseur du givre avec le temps

Figure 6

0.000

0

175.

1.000 .

Effect of ambient temperature on the frost thickness Effet de la tempdrature ambiante sur l'kpaisseur du givre

.

.

.

.

.

.

.

.

1.000 . . . . . . . . . . . . . • - Yonko & sepsy data 0 . 8 0 0 -~

1 -

Somi'g

2 -

Present

3 -

Porker'S

Wo 0.600 -

model

&

model

0,600-

0.0108

(.3

Re "

13900

Present

~

model

To -

295 K

1

-

Wo - 0 . 0 1 0 5

Ts -

245 K

2

-

Wo -

0.0141

3

-

Wo -

0.0075

Z

To = 2 9 5 K TJ - 2 4 5 K

Z

Re = 1 3 9 0 0

0.800-

model

~

O.4OO-

0

i. 0.200 -

0,200

~

0.000

,

0,0

25.0

50.0

,

'

75.0

TIME

I . . . . 125,0

~ . . . .

100.0

0-000/ .... 0.0

i .... 25.0

i .... 50,0

i .... 75.0

i .... 100.0

i .... 125.0

i .... 150.0

175.0

150.0

TIME (MIN)

(MIN)

Figure 10 Effect of free stream humidity ratio on the frost thickness

Figure 7 Variation of frost thickness with time

F i g u r e 10

Modification de/'#paisseur du givre avec le temps

Figure 7

1.000-

0.800 -

v bJ Z

Wa -

0.0108

To -

295 K

I

-

Re -

Ts -

245 K

2

-

Re ~ 9 1 0 0

3

--

Re -

Present

0.600-

model

13900

5700

C)

I

T

0.400-

g b-

0.200-

0.000

0.0

. . . . . . . . 25.0

j .... 50.0

j .... 75.0

TIME

i .... 100.0

i .... 125.0

I .... 150.0

(MIN)

Figure 8

Effect o f R e y n o l d s n u m b e r o n t h e f r o s t t h i c k n e s s

Figure 8

Effet du nombre de Reynold~ sur l'#paisseur du givre

Effet de l'humidit# relative du courant d'air sur l'kpaisseur du

givre

175,0

bulb temperature, as indicated in Figure 10. (The reader should note that despite the relatively close proximity of the lines in Figures 9 and 10, the dependencies of the thickness on the ambient temperature and humidity are relatively large, since the ranges specified for those parameters in Figures 9 and 10 are not as big as the range of the Reynolds number specified in Figure 8.) Naturally, as the humidity ratio increases for the same temperature and velocity, the frost thickness will increase. On the other hand, as the air dry-bulb temperature increases for the same humidity ratio and velocity the frost thickness will decrease. This can only be explained by examining the corresponding change in the air-frost interface temperature (Figure 11). As the air dry-bulb temperature increases, the air-frost interface temperature will increase. This causes the air humidity ratio in the vicinity of the frost upper surface to increase, thus reducing the concentration driving force (the difference between the humidity ratio of the air in the free stream and that of the frost-air interface). This reduces the frost deposition rate and consequently the thickness (Figure 10). The relatively small dependency of the frost thickness on the Reynolds number (see Figure 8), on the other hand, can be Rev. Int. Froid 1993 Vo116 No 5

327

A semi-empirical transient method tor modelhng trost tormatton on a tlat plate. 5.A. 5hertt et ai

-s.oooj

i

~°'°°°T--~,=

Wa = 0 . 0 1 0 8 Re =

~. -io.ooo

ra

13900

T,, = 245 K

~

i

"~ :D

- ~ I ooo

=

1~9o~- ....................................................

i

295

!

is = 245

K K

~,----~=~

~

......

.........

"!

-

-14.000 16000

~ -l~°°°lf

~J Present

_22.ooo

b_

model

1P:";::°=

/I it

2 3 -

Present

model

]

-

WO =

2

""

Wa = 0 . 0 1 4 i

0.0108

~(~ -21.000 ~_

K

To-2~SK To-505K

3

We = 0 , 0 0 7 5

I -26.000

I . . . . 0.0

i . . . . 25.0

i . . . . 50.0

i . . . . 75,0

I . . . . 100,0

i . . . . 125.0

-26.000

i . . . . 150.0

175.0

-~

0.0

. . . .

Figure 11

Effect of ambient temperature on the frost temperature E(et de la temperature ambiante sur la temperature du givre

-7.000

0

WO = 0 . 0 1 0 8

-11.000-

To -

295

K

_ ~

_

_

.

~

-

~

m" ~

-15.000-

[1. ~

-19.000-

o ~

/// ,/

3-

~

= 5700

-23.000-

-27.000

. . . . 0.0

I . . . . . ' . . . . . . . . . . . 25.0 50.0 75.0 100.0

I . . . . 125.0

[ . . . . 150.0

75.0

TIME (MIN)

Figure 12 Effect of Reynolds number on the frost temperature Figure 12 Effet du nombre de Reynolds sur la tempkrature du givre

attributed to the following reason. As the Reynolds number increases, the convective heat and mass transfer coefficients increase, resulting in an increase in the frost deposition rate and consequently the frost thickness. However, increasing the Reynolds number also increases the frost-air interface temperature (Figure 12), which results in a reduction in the concentration (humidity ratio) driving force (as explained earlier) between the air stream and the frost surface. This causes the frost deposition rate and consequently the frost thickness to decrease with time. Apparently, the effect of the former phenomenon is more predominant than the latter, thus contributing to a slight overall increase in the frost thickness as the Reynolds number is increased. The reasoning presented above regarding the effect of the Reynolds number on the frost thickness profile with time can be used to explain the effect of the free stream humidity ratio on the frost thickness. For example, an increase in the free stream humidity ratio results in an increase in the frost-air interface temperature (Figure 13), which is always accompanied by an increase in the humidity ratio of the air in the neighbourhood of the 328

[ . . . . 50.0

I . . . . 75,0

I "r'-r'w 100.O

' I . . . . 125.0

I . . . . 150.0

175 0

TIME (MIN)

TIME (MIN)

Figure 11

t . . . . 25.0

Int. J. Refrig. 1993 V o 1 1 6 No 5

Figure 13 Effect of free stream humidi,ty ratio on the frost temperature Figure 13 I~\ffPttie l'humiditk relative du courant d'air sur la temp&ature du givre

frost interlace. This does not seem to reduce the concentration driving force as explained earlier since the increase in the free stream humidity ratio is inherently larger than the increase in the humidity ratio in the vicinity of the frost surface. Finally, it may be noted that the frost surface temperature experiences a relatively large increase at the initial deposition periods, which gradually tapers off afterwards. For lower humidity and temperature environments, the frost surface temperature takes longer to reach the melting temperature and, if the ambient temperature is below freezing, the frost surface temperature will never reach the melting temperature. This is not true, of course, in high humidity/high temperature environments, as explained earlier in this paper and as observed by Trammel et al. 22 and Yonko and Sepsy ~9 Conclusions

A simple method of computing the frost thickness and surface temperature with both time and position was presented using a semi-empirical transient model for a flat plate under forced convection conditions. This continuous differential model numerically solves for the temperature and thickness of the frost layer using existing correlations for the convective heat transfer coefficient and the Lewis analogy to compute both a mass transfer coefficient and an enthalpy transfer coefficient. The model is also based on empirical correlations for predicting the frost thermal conductivity and density. Model results are compared with existing experimental data and with other theoretical models and are found to agree very well in the applicable temperature and humidity range of the frost density and conductivity correlations. (A maximum of 30% difference between numerical predictions and experimental data is observed; Fc,r most of the time range, however, the model predictions were very close to the experimental data:) The simplicity and general nature of the model are obvious advantages that suggest trying similar models for other geometries in both free and forced convection frost deposition processes.

A semi-empirical transient method for modelling frost formation on a flat plate: S.A. Sherifet al. Acknowledgements This research was partially supported by a grant a w a r d e d t o t h e p r i m a r y a u t h o r in 1988 b y t h e U n i v e r s i t y of Miami's Research Council (Summer Awards for Natural Sciences and Engineering). Support from the Department of Mechanical Engineering at the University o f M i a m i is a l s o g r a t e f u l l y a c k n o w l e d g e d .

17

18

19

References 1 2 3

4 5 6 7

8

9

10 11

12

13 14 15 16

Padki, M.M., Sherif, S.A., Nelson, R.M. A simple method for modelling the frost formation phenomenon in different geometries ASHRAE Trans (1989), 95(2) 1127 1137 Whitehurst, C.A. Heat and mass transfer by free convection from humid air to a metal plane under frosting conditions A S H R A E J (1962), 5 (5), 8 8 4 7 Whitehurst, C.A. An investigation of heat and mass transfer by free convection from humid air to a metal plate under frosting conditions PhD dissertation, Mechanical Engineering Dept, Agricultural and Mechanical College of Texas (1962) Whitehurst, C.A., McGregor, O.W., Fontenot, J.E. Jr An analysis o f the turbulent free-convection boundary layer over a frosted surface Cryogen Technol (Jan/Feb) (1968) Barton, R.F., Han, L.S. Heat and mass transfer to a cryosurface in free convection J Heat Transfer (1965), 87 (4), 499 5O6 Mulligan, J.C. Properties of frost forming on a cold plate in still air PhD dissertation, Tulane University (1967) Mulligan, J.C. Model for heat transfer in frost and snow J Geophys Res (1968), 73 (8), 2663-2668 Tajima, O., Yamada, H., Kobayashi, Y., Mizutani, C. Frost formation on air coolers part I: natural convection for a cooled plate facing upwards Heat Transfer - Jap Res (1972), 1 (2), 3948 Tajima, O., Naito, E., Tsutsumi, Y., Yoshida, H. Frost formation on air coolers, part 2: natural convection for a flat plate facing downwards Heat Transfer Jap Res (1973), 3 (4), 55 66 Tajima O., Naito, E., Nakashima, K., Yamamoto, H. Frost formation on air coolers, part 3: natural convection for the cooled vertical plate Heat Transfer- Jap Res (1974), 3 (4), 55 66 Tajima, O., Naito, E., Goto, T., Segawa, S., Nishimura, K. Frost formation on air coolers, part 4: natural convection for two vertically opposed cooled plates Heat Transfer Jap Res (1975), 4 (3), 21 36 Goodman, J., Kennedy, L.A. Free convection frost formation on cool surfaces Proc 1972 Heat Transfer and Fluid Mechanics Institute (1972) Stanford University Press, Stanford, California, 338 352. Kennedy, L.A., Goodman, J. Free convection heat and mass transfer under conditions of frost deposition Int J Heat Mass TransJer (1974), 17 477 484 Yamakawa, N., Ohtani, S. Frost formation on air coolers part 1: natural convection for a cooled plate facing upward Heat Tran~/~,r Jap Res (1972), 1 (2), 1 10 Yamakawa, N., Ohtani, S. Frost formation on air coolers part II: natural convection for a cooled plate facing upward Heat TransJer Jap Res (1972), 1 (2), 75 82 Coles, W., Ruggeri, R. Experimental investiagation of sublima-

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tion of ice at subsonic and supersonic speeds and its relation to heat transfer National Advisory Committee for Aeronautics, TN 3104 (1954) Glomb, J.W., Wenzel, L.A. Forced convection frost deposition on a cold horizontal surface, AIChE Paper N. 25b Proc Am Inst Chem Eng 53rd National Meeting (1964) Yonko, J.D. An investigation of the thermal conductivity of frost while forming on a fiat horizontal plate M S thesis Ohio State University (1965) Yonko, J.D., Sepsy, C.F. An investigation of the thermal conductivity of frost while forming on a flat horizontal plate ASHRAE Trans (1967), 73, (Part II, Paper No 2043) 1.1.1-I. 1.10 Hosoda, T., Uzuhashi H. Effects of frost on the heat transfer coefficient Hitachi Rev (1967), 16 (6) Trammel, G.J., Catenbury, J., Killgore, E.M. Heat transfer from humid air to a horizontal fiat plate held at sub-freezing temperatures A S H R A E Trans (1967) 73, Part I, IV.3.1-1V.3.6 Trammel, G.J., Little, D.C., Killgore, E.M. A study of the frost formed on a fiat plate held at sub-freezing temperatures ASHRAE J (1968), 10, ( 7 ) 4 2 4 7 Jones, B.W., Parker, 3.D. Frost formation with varying environmental parameters J Heat Transfer (1975), 97 (2), 255 259 Seki, N., Fnkusako, S., Matsuo, K., Uemnra, S. Incipient phenomena of frost formation Trans Jap Soc Mech Eng B 50, 825 831 (in Japanese). Seki, N., Fukusako, S., Matsuo, K., Uemura, S. An analysis of incipient frost formation Warme- und Stoffubertragung (1985), 19,9 18 Abdel-Wahed, R.M., Hifnl, M.A., Sherif, S.A. Heat and mass transfer from a laminar humid air stream to a plate at subfreezing temperature lnt J Refrig (1984), 7 (1), 49 55 Sherif,S.A., Abdei-Wahed, R.M., Hifni, M.A. A mathematical model for the heat and mass transfer on a flat plate under frosting conditions A S M E Proc 1988 National Heat Transfer Conf(1988) HTD-96 3, 301 306 Sami, S.M., Duong, T. Mass and heat transfer during frost growth A S H R A E Trans (1989), 95 (Part I) O'Neal, D.L., Tree, D.R. A review of frost formation in simple geometries ASHRAE Trans (1985), 91 (Pt 2) Kondepudi, S.N., O'Neal, D.L., The effects of frost growth on extended surface heat exchanger performance: a review ASHRAE Trans (1987), 93 (Part 2) paper no 3070 ASHRAE, Handbook-1985fundamentals The American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc, Atlanta, GA (1989). Hohnan, J.P. Heat transJbr. Sixth ed. McGraw-Hill, New York (1986) Hayashi, Y., Aoki, A., Yuhara, H. Study of frost formation based on a theoretical model of the frost layer Heat Transfer Jap Res (1977), 6 (3), 79 94 Sanders, C.Th., The influence of frost formation and defrosting on the performance of air coolers, PhD dissertation, Technische Hogeschool, Delft, The Netherlands (1974) Brian, P.L.T., Reid, R.C., and Brazinsky, 1. Cryogenic frost properties Cryogen Technol (1969), 5 (5), 205- 212 Brian, P.L.T., Reid, R.C., Shah, Y.T. Frost deposition on cold surfaces I/EC Fundamentals" (1970), 9, (3), 375 380 Marinyuk, B.T. Heat and mass transfer under frosting conditions lnt J Refrig (1980) 3 (6), 366 368

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