Heat and mass transfer under frosting conditions

B. T. Marinyuk

Transport de chaleur et de masse dans des conditions de givrage

On a #tudi# exp#rimenta/ement /'influence de la formation de givre sur /e transport de chaleur entre un cy/indre d'essai et son environnement gazeux. Les pnnc/paux param#tres examines dans cet article sont les suivants. flux thermique total, coefficient de

transport de cha/eur par convection en r#gime permanent et masse de givre adh#rant au cy/indre d'essai. L 'article met/ 'accent sur /a conductivit# thermique du givre. Les r#su/tats indiquent que /es m#canismes de diffusion du transport d'eau I'int#rieur de/a couche de givre provoquent une augmentation dans /e temps de/a densit# du givre et de la conductivit# thermique. La conduct/vit6 therm/que du givre est fonct/on de/a temperature locale et de/a densit# moyenne. Cet article pr#sente une #quation qu/ peut @tre utilis#e par les ing#nieurs concevant des syst#mes ~ basse temperature avec des surfaces non/solVes.

The ef fect of frost format ion on heat t ransfer between a test cyl inder and its gaseous envi ronment was studied experimental ly . The main parameters discussed in the paper are: the total heat flux, the steady-state convect ive heat t ransfer coeff ic ient , and the mass of frost adhering to the test cylinder. The emphasis of the paper is on the thermal conduct iv i ty of frost. The data indicate that

the dif fusion mechanisms of moisture t ransfer wi th in the frost layer causes the frost density and thermal conduct iv i ty to increase wi th t ime. Frost thermal con- duct iv i ty is a funct ion of the local tempera- ture and average density. The paper presents an equat ion which can be used by designers of low tempera ture systems wi th uninsulated surfaces.

Frost forms on a refrigerated surface when the temperature is below both the dew point of the gas and the freezing point of the vapour. A frost layer usually forms on air coolers, on evaporators of different types, on freezers, and on uninsulated short cryogenic ducts. The characteristics of the heat transfer process between the low temperature surface and its gaseous environments are therefore of great importance in the design of such apparatus. The analysis of the frosting process presents considerable difficulties not only due to the complex deposition mechanisms, but also due to the scarcity of good experimental data. The data available are primarily conerned with forced convection and with the temperature range used in commercial refrigeration equipment. 1.2.3 Much less is known about this problem when associated with natural convection and when the surface temperatures are very low.

An analysis and correlation of experimental data for natural convection cooling over the temperature range 93 233 K are presented in this paper.

The author is the Assistant Professor and lecturer of the Department of Cryogenics, Moscow Institute of Chemical Engineering. Paper received July 1980.

01 40 7007/80/060366-03502.00 © 1980 IPC Business Press Ltd and IIR 366

E x p e r i m e n t a l

The test element was a thin-walled cylinder of 64 mm diameter and 100 mm high mounted vertically on top of a 1 0 I Dewar vessel. To prevent heat leakage from the surroundings it was insulated top and bottom with polyurethane foam.

The liquid refrigerant was supplied continuously to the test element by pressure transfer from the Dewar vessel. The level of the liquid refrigerant in the test element was continuously indicated by a float- indication device installed within it. Both the Dewar vessel and the test element were housed within a double-walled chamber having the dimensions of a 1 m cube, its temperature being automatically controlled within the range 0 to -40°C by a three- stage refrigeration machine and an electric heater.

To create the proper conditions for natural convection mechanism, heat was transferred to the working volume through the wall. The double- walled construction of the chamber prevented undesired inward leaking of air. These measures controlled the air temperature to 4-0.3 K.

Revue Internationale du Froid

Low pressure steam jets were installed in the chamber to make up the vapour entrapped by the test element. The air temperatures and humidity were measured continuously by means of an aspiration psychrometer with dry and wet bulbs. The test surface temperature was measured by means of copper-constantan thermocouples 0.2 mm in diameter. Similar thermocouples were used to obtain data on the temperature profiles within the frost layer (Fig. 1 ). Visual observations were performed through multipane windows by a telescope.

The total heat transferred from the surroundings to the test element is the sum of heat fluxes delivered to the frost surface by convection Qc, mass transfer Qm, and radiation QR. This can be expressed in the following equation:

Q T = O c + Q M + O R (1)

The total heat was obtained by measuring the evaporation rate of the test liquid by a gas meter, and applying the following equation:

O T = R R V (2)

where RR is a latent heat of vaporization of the liquid refrigerant, Jm -a (related to vapour volume at test conditions) and V is volume flow rate of the boil-off gas measured by the gas meter, m 3 s -1.

Values of the total heat flux measured in experiments for a variety of natural convection conditions in air showed that the heat flux started at a maximum value, but after a short period of time decreased to an approximately constant value during the quasi-steady state period. The decrease of the heat flux at the start of the experiment may be attributed to an increase of frost layer thermal resistivity. The main factors which affect heat transfer fluxes with natural convection are the ambient air conditions and the metal surface temperature. The heat transferred by radiation was calculated by

OR= CsF[ (L /1 00) 4 -- (Ts/1 00) 4] (3)

Fig. 1 The general v iew of the frost surface wi th thermocouples

Fig. 1 Vue g#n#ra/e de/a surface du givre avec /es thermocoup/es

40

5 0 - - 5

2 £ - - 4

O I 2 :5 4 5 Time, h

Fig. 2 Accumula t ion of frost massvs time for a variety of gravi ty convect ion condi t ions. 1 - - t a i r = 2 2 ° C , ( = 3 0 . 1 % , Twal l=233K; 2 - - ta i r=26°C, ( = 6 4 % , Twau=233K; 3 - - ta i r=26°C, ( = 6 4 % , TwaH=93K; 4 - - tair= 32°C, ( = 5 4 % , Twall=93K; 5 - - tair= 32°C, ( = 76%, Twau= 93K

Fig. 2 Accumu/ation de/a masse de givre en fonction du temps pour diverses conditions de convection par gravit& 1 - tair=22°C, ~=30.1%, Tparoi=233 K," 2- ta i r=26°C, ~=64%; Tparoi=233 K, 3 - tair = 26°C, ~ = 64%, T p a r o i = 93 K; 4 - tair = 32°C, ~ = 54%, Tparo i = 93K; 5-tai r = 32°C, ~ = 76%; Tparo i = 93K

where C is emissivity constant, 5.77 x 10 -8 W m -2 K-4; s is frost surface emissivity 4 0.92: T a is air temperature, K; T s is frost surface temperature, K; and F is surface heat and mass transfer, m 2. The quantity of heat transferred as latent heat is given by the equation:

OM=G x Rf ... (4)

where G is mass of frost adhering to the test element, kg s -1 (obtained by scraping off and weighing) and Rf is latent heat of condensation and freezing of water vapour, J kg -1.

Fig. 2 shows the increase of frost mass with time. The effect of humidity is seen by comparing the results of tests four and five. Each of these tests was run at approximately the same chamber and metal surface temperatures but at different humidities. The higher humidity resulted in a higher frost deposition rate. The convective portion of the total heat flux can be found from the equation:

Oc=OT--OM--OR (5)

On the other hand, it can be found from Newton's law of convective heat transfer,

Oc=~Fz~T (6)

The results of correlation in terms of Nusselt, Grashof and Prandtl numbers are shown in Fig. 3. A line can be drawn through the plotted data, represented by the equation:

Nu =0.730(CrPr) °25

This is in agreement with the data presented by Barron and Han. 5

(7)

Volume 3 Num6ro 6 Novembre 1980 367

According to the procedure of correlating the experimental data, frost thermal conductivities estimation was based on the local thermal conductivity defined by the equation:

~-f= ( O T - - O i n ) / F c31~ t× Wm ~K 1 (8)

where 8t/Sx is the differential of temperature with respect to x, (defined by graphical differentiation of temperature curve); Q,n is heat leakage from the surroundings, J, and F is heat transfer surface, m 2.

Temperature distribution curves and calculated local thermal conductivities considered with previous data on mass transfer yielded an empirical correlation between local thermal conductivities, average frost densities and temperatures within the layer. The resulting correlation was:

)~f=0.0904 exp(0.0137T- 3.74)

[1 +4 .64x10-3~exp(0 .00775T-2 .12) ] (9)

where T is local temperature, K, and ~ is average frost density.

In spite of the empirical nature of (9), its structure revealed some physical justification. It is known that local thermal conductivity consists of two parts, the first is conductivity of the frost structure, the second is diffusion of moisture within the frost layer. The latter depends on the diffusion coefficient and the gradient of the water vapour pressure with temperature, which is expressed by an exponential function. To find an average value of the thermal conductivity with respect to temperature it is necessary to integrate (9) by the formula:

4C

) , f = ( T 2 __ T1 ) - 1 1 2 f d T

T1

(10)

50

20

J

o j..j o4

I I I I I I I I ,o 2 3 4 5 o ~ 8 9 , ~

Grl~"

Fig. 3 Heat transfer coefficients in terms of Nusselt. Grashof and Prandtl numbers, Nu=0.730 (GrPr) °25

Fig. 3 Coefficients de transport de chaleur en fonction des nombres de Nusselt, de Grashof et de Prandt/. Nu = 0.730(GrPr)°25

where T 2 and T~ are temperatures of the frost surface and the wall, K. The result is:

2,-- 1.3(T 2 - f 1 ) -1 [0.1 56{exp (0.01 37T2)

-exp(0 .0137T 1)}+5.59 x 10 -5 x

~{exp(0.O214T2) - exp(0.0214T1)}] (11)

Equation (11) predicts average values of frost thermal conductivities for the density range 60-300 kgm 3

It follows from (11) that different values of thermal conductivity may be associated with the same frost density due to the temperature conditions of frost formation.

In the literature one finds attempts to draw a single curve on the conductivity versus density graph, in cases where the difference between them can exeed 100%. So when correlating experimental data on thermal conductivities, it is necessary to take into account the temperature of frost formation.

C o n c l u s i o n s

The following conclusions can be drawn:

The whole 3rocess of frost formation can be divided into two stages. During the first unsteady stage, all the parameters such as convective coefficients, surface and frost temperatures change rapidly with time. During the second quasi-steady state period, the same parameters remain approximately constant.

In constrast to forced convection, frost formation under natural convection conditions does not increase heat transfer convective coefficient in comparison with the dry state,

Frost thermal conductivity is strongly dependent on the formation temperature.

R e f e r e n c e s

1 Yanve l , B. K. Investigation of heat and mass transfer coefficients along plate under frosting conditions. Kho/odi/naya Tekhnika, 12 (1 968) 1 3 17

2 Hosoda, T., Uzuhashi , H. Effects of frost on the heat transfer coefficient, Hitachi Review 16 6 (1967) 254-259

3 Ga tch i l ov , T. S., I vanova, V. S. Characteristics of the frost formed on the surface of the finned air coolers. Report presented by B1 Commission at XV International Congress of Refrigeration, Venice (September 1 979)

4 Ho l ten , D. C. Heat and mass transfer to uninsulated oxygen containers. Cryogenic Engineering 6 (1961 ) 499

5 Bar ron , R. F., Han, L. S. Heat and mass transfer to a cryogenic surface in free convect ion J Heat Transfer 87 4 (1965) 499

3 6 8 I n t e r n a t i o n a l J o u r n a l of R e f r i g e r a t i o n