Frost Deposition on Cold Surfaces P. 1. Thibaut Brian, Robert C. Reid, and Yatish T. Shah' Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, 31ass. 02139
Water frosts were deposited on a liquid nitrogen-cooled plate from a humid air stream. The plate formed the bottom of a high-aspect wind tunnel. Reynolds numbers were varied from 3770 to 15,800, gas-phase humidities from dew points of 14" to 58"F, and gas temperature from 34" to 93°F. Frost densities and thermal conductivities were determined and the heat and mass fluxes measured. The data were correlated b y a simple analytical model which emphasized the internal diffusion processes within the frost. Density measurements were also made within the thin frost layer. These measurements indicated that within the accuracy of the method used there are no significant density gradients in the frost. This result, though not thoroughly explained, i s believed due to a combination of internal diffusion and nucleation with particle transfer by thermal diffusion forces.
PROCESSES
in which heat is transferred to a refrigerated surface with the simultaneous deposition of a frost layer are important in gas coolers, refrigerators, regenerators, freeze-out purification of gases, cryopumping, and the storage of cryogenic liquids. Frost will form on a cold surface having a temperature below the dew point of the condensable component, if that temperature is also less than the freezing point. I n general, for small temperature differences between the bulk gas and the frost surface, the condensable component will be transported to the wall as a vapor by molecular and turbulent diffusional processes. If these temperatures are known, heat and mass fluxes can be predicted with the usual transport correlations. If, on the other hand, there are large temperature differences between the gas and wall, the condensable component may freeze out or fog in the vicinity of the wall. Particle impingement or trapping of the particles on the surface then becomes the dominant mechanism of frost deposition. This type of deposition is, however, usually of short durationthat is, after a short while, the frost deposited has insulated the cold surface and, in effect, has imposed a thermal resistance which results in a sufficiently high frost surface temperature so that a diffusional mechanism of gas-phase mass transport predominates. As additional frost deposits upon this cold surface and further insulates the surface, the rate of heat transfer from the warm gas to the cold surface decreases. After an initial transient period (in which the heat flux decreases), numerous investigators have observed that the heat transfer rate levels off and becomes essentially constant with time, even though frost continues to accumulate on the surface. Reid et al. (1966) concluded that this "steady-state" heat transfer rate could be explained only by postulating that the frost layer densifies with time. During densification, the frost thermal conductivity increases very substantially, presumably because of greater interparticle contact. T h e fact that frost densification just counterbalances the increasing mass of frost so as to keep the frost thermal resistance constant with time is surely no coincidence; the great sensitivity of the densification rate to the frost surface temperature results in a high-gain feedback effect which keeps the surface temperature almost constant with time (Brian et al., 1969). The mechanism of densification 1
Present address, University of Pittsburgh, Pittsburgh, Pa.
13214
is still not completely clear, and various mathematical models have been proposed. These are discussed below. The experimental studies reported in this paper concentrated on the properties of water frost deposited from a turbulent, humid air stream onto a liquid nitrogen-cooled surface. Measurements were made of the frost density and thermal conductivity as a function of time for various values of the gas-phase Reynolds number, humidity, and gas temperature. I n the analytical study various models of the frosting process were proposed, and time variations of the frost properties were calculated and compared with the experimental values. A summary of the literature dealing with similar frosting studies is given elsewhere (Biguria, 1968; Biguria and Wenzel, 1970; Brian et al., 1969). Experimental
The experimental equipment and operational techniques were similar to those described by Brazinsky (Brazinsky, 1967; Brian et al., 1969). The experimental setup is schematically described in Figure 1. Humid air passed over a liquid nitrogen-cooled flat copper plate inserted in the bottom of a high-aspect-ratio rectangular wind tunnel. Frost deposited on the plate and, as the thickness of the frost increased, the copper plate was lowered so as to keep the frost surface always flush with the bottom of the wind tunnel. T h e independent parameters varied were Reynolds number (3770 to 15,800), gas phase humidity (dew points of 14' to 58"F), gas temperature (34' to 93"F), and time of deposition (up to 180 minutes). During a run, temperatures within the frost and a t the frost surface were measured, as were the heat flux and frost thickness. At the end of a run, a sample of the frost was cut out and weight determined; from the volume and weight of the sample, the average density was determined. By making a number of runs with all independent variables constant, but for different lengths of time, the average frost density could be determined as a function of deposition time. Further details are given elsewhere (Shah, 1968). The average thermal conductivity of the frost was calculated from heat flux, surface temperature, and thickness measurements; the local thermal conductivity within the frost was calculated from the heat flux and temperature gradients within the frost. Temperature gradients were obtained by differentiating experimental temperature-distance profiles such as shown in Figure 2. The data shown in this figure were obtained by means of thermocouples placed a t known Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
375
,
Figure 1 .
Flowsheet of equipment
IO'
Experimental Conditions Re =15,800;Humidity=Gas dew point at 30°F; Frosting time = 79 minutes
LL
-
A
0
L
.c \
o
m
v
9.
1
1
1
I
I
1
C
/
-1
.-
-
0
-
-
I
I
I
I
I
0.05
,
,
I
I
I
I
I
0.10
,
0.15
0) V
F
10.0
0)
V
&i
wLn
C
0
$
0
0 Tg =93OF, T,=
+ 0
0
0)
@ T g = 7PF, Ts = @ Tg =50°F, Ts = @ T, =34OF, Ts =
I
Distance from Cold Wall (inches)
Figure 2.
1
c
E -160
f
0
I
-
3 I-
!?
I
System Conditions Frost i n g 75OF 8OF Frosting 58F 5OF Frosting 34F 2OF Frosting 82OF - I I05 F Non-Frosting Theoretical Predictions under Non -Frosting Conditions
o
"F + -80
1
I
Experimental Data : Gas Cold Surface Temperature Temperature 93OF I5OF
End of run temperature profile within frost
15OF 8OF
5OF 2OF
distances from the cold plate. For each set of system conditions, the temperature profiles were obtained for the runs with different deposition times. These profiles indicated that during the transient state of the frosting process, the temperature a t any depth within the frost decreases with an increase in the deposition time (Shah, 1968). For a few system conditions, the temperature profiles were obtained for two identical repeat runs. These profiles were found to be in agreement within 10% deviation in the temperature values.
Figure 3. Heat transfer coefficient vs. Reynolds number (at various gas temperatures) under frosting and nonfrosting conditions
Typical experimental results are shown below, when a comparison with analytically predicted values is made. It can be noted in Figure 3, however, that using the measured frost surface temperature and experimental heat fluxes, calculated gas-phase heat transfer coefficients agreed well (within 10%) with those predicted from the theoretical equation of Sleicher and Tribus (1957) for heat transfer in the entrance region. The theory assumes a nonfrosted flat plate, whereas in the experiment there is obviously an effect of a rough surface. Other experiments, not described here, wherein dry nitrogen gas was passed over the nonfrosted copper test plate yielded heat transfer coefficients within 3% of those predicted using the Sleicher and Tribus method noted above.
Gas-phase mass transfer coefficients were calculated using the Lewis analogy (Eckert and Drake, 1955) and the theoretical values for the heat transfer coefficient are shown in Figure 3. This technique led to the predicted values of the mass transfer coefficient shown in Figure 4. Experimental values of this coefficient were obtained from the mass deposition rate data and the water vapor partial pressure in the bulk gas and at the frost surface, the latter being assumed to be the vapor pressure of ice at the frost surface temperature. There is some Bcatter, which is believed to be due to slight inaccuracies in the surface temperature measurements, but the agreement between theory and experiment is satisfactory.
376
Ind. Eng. C h h . Fundam., Vol. 9, No. 3, 1970
I .o
2x10~
I
I
I
I
I
I
1
I
io4
I
1
4x11
Reynolds Number
Simple Analytical Model of Frosting Process
A microscopic, detailed examination of the frosting process would reveal it to be extremely complex. The movement of water vapor to the frost surface, if temperatures in the gas boundary layer are not too low, is believed to be similar to other gas-phase transport processes. While not simple, such processes are describable and amenable to predictive calculations. Heat transfer and deposition rates will, however, be a function of the frost surface temperature and therefore of the properties of the frost layer. The frost is not a simple thermal resistance. The thickness increases with deposition time, the density increases (whether or not deposition occurs) as it matures, and the resulting average thermal conductivity is a function of time as well as deposition conditions. As noted earlier, however, all studies of heat transfer across a frost layer have shown that the heat flux soon becomes almost constant with time, even though the thickness continues to increase. With constant gas-phase conditions, this would lead one to predict that during the period of quasi-steady heat flux, the frost surface temperature also becomes essentially time-independent. This assertion was shown to be true in earlier work (Brian et al., 1969) and also in the present studies. A very simple analytical model may be formulated during this quasi-steady-state heat transfer portion of a frosting process. The development of the model is summarized elsewhere (Brian et al., 1969). A key assumption in this model is that water vapor diffuses away from the frost-gas surface into the colder frost interior by virtue of the equilibrium partial pressure gradient corresponding to the temperature gradient existing a t the surface. I n some "undefined" way, this water penetrates the frost to produce a frost of uniform density, though this density is time-variant. This assumption is examined in detail later and some experiments are described to support the model. The three basic equations which must be solved are as follows:
Table 1. Empirical Correlation of Frost Thermal Conductivity with Average Frost Density and Temperature Equation
IC,3.875 x 10-6~1.441+ 4.08 x 10-9 - 0.025)Ta.0S5
+
- P , ) A ~ T= ICf9s(dT/dz)s= - T,)/6 i , V S
Kg(pg -
PA)
=
Pi(ds/d)
+ 6(dp,/de)
= 8.49 X 10-'7T5~44 6.86 X (fit - 0.025)T4.84
p, in g/cma; IC, in Btu/hr-ft-OIt; T i n OR.
I
=
I
I
I I I I I
I
I
I
lxperimental Data : Gas Tempe rat u re 0 93OF 0 75°F A
0
-
Cold Surface Te mp e r atu re + 15OF + 8OF 5OoF t 5OF 34OF + 2OF Theoretical Predictions from Lewis Analogy
i 0
@ Tg=93OF; Ts=15OF @ Tg=75OF, T s = 8OF
O
(1)
-PVA[(I - P , / P ~ ~ ~ ) / ~ I ( ~ T(3) /~&
I n addition, one must have, for water frost, a relationship of the form k , = f(T, p,). For water frost deposited on a flat plate, such a correlation was developed from the present experimental data (Shah, 1968) and the data of Brazinsky (Brazinsky, 1967; Brian et al., 1969). Expressed analytically, this correlation yields the equations shown in Table I. Also, to solve Equations 1 to 3, initial values of 6 and pf are required. To test the applicability of the model, initial values of 6 and p , were chosen equal to the first experimental values taken in a frosting test. The equations were then solved by finite difference techniques, using a digital computer to predict the time variation of frost thickness, total mass deposited, average frost density, and heat flux. The predicted values are shown and compared with experimental values in Figures 5 and 6 for several widely differing deposition conditions. The agreement is excellent. The conclusion one can draw from comparisons such as those illustrated in Figures 5 and 6 is that, given only initial values of p , and 6, with a
Tg =5OoF, Ts = 5°F
@ T, = 34OF, Ts= 2?
'"2x
(2)
Internal Diffusion
(dp,/W
T
+ k f = 6.58 X 10-6T1.272 + 8.17 X 10-6p,T'.74 IC,
Mass Balance
N
> 0.025 < 460 p / > 0.025 T > 460 < 0.025 T < 460
= (PI
Energy Balance 9 = h ( T , - T,)
Conditions
103
IO
3 ~ 1 0 ~
- 0.40 G
System Conditions R e = 14,700 HumidityZGas dew point at 29.2"F
0.30
"'
Tg= 7 O o F
e :0.10
.t 0.20
I C
-
- 0
- 0
v
0.15 0.14
> 0,13 0
N -
1,000
*
A .-
/
t
A'
0 Y
LL
$
0.04 40
80 120 Time (rnins.)
Time (rnins.)
Figure 5. Experimental facts and theoretical predictions of the effect of gas temperature on time variation of frost properties and heat and mass fluxes
0
A
--
Re = 5 7 9 0 ; humidity = gas dew point a t 27.2 OF; T, = 3 4 Re = 5 1 0 0 ; humidity = gas dew point a t 27.4 OF; T, = 9 3 Theoretical curve
1
20
60 100 140 Time (inins.)
0
--
I
I
I
I
1
0.041
Variable Density Model
0
40
i
80 120 160 Time (mins.)
Re = 8 4 3 0 ; humidity = gas dew point a t 29.8 O F ; T, = 52 OF Re = 8000; humidity = gas dew paint a t 30.4 OF; T , = 7 6 O F Theoretical predictions of simple model
-
For a given set of frosting conditions, the choice of p , a t 0 = 0 was found (Shah, 19:s) to be important in predicting iiumerical values of p , and 6 a t later times, but the rate of change of p , and 6 a t long times was nearly independent of the
choice of the initial average frost density. This finding, while interesting, still does not aid one in an a priori estimation of sonie starting values of 6 and p , for the analytical model. It is possible, however, that the problem of obtaining starting values of 6 and p , might be resolved by either empirically 378
I
O F
OF
Figure 6. Experimental facts and theoretical predictions of the effect of gas temperature on variation of frost properties and heat and mass fluxes with time
A
3 0 mins.
correlating p , with the independent system parameters a t zero time or correlating both p , and 6 with independent system parameters a t any arbitrary time if the experimental data are available.
t T.
2
0,05- einitioi
Ind. Eng. Chem. Fundam., Val. 9, No. 3, 1 9 7 0
Even though the simple model presented above accurately predicts the time variation in heat flux and frost properties, one might question the assumption that the frost density is constant throughout the layer while being time-dependent. I n fact, the simple model disregards completely any processes occurring within the frost. As the key concept of the simple niodel was that water vapor diffused from the frost surface int,o the interior by a n equilibrium partial pressure gradient, it seemed logical to extend this assunipt,ion to include the entire frost layert,hat is, one might postulate that the time-variant frost densification results solely from subliination processes between frost layers at different temperatures. At each point within the frost there is assumed to be equilibrium between the solid and water vapor. With this concept as a basis, one cannot assume that the frost density is uniform wit'h depth. I n fact, solutions of the partial differential equat'ioiis describing heat and mass transport within the frost will lead to predictions of the density'and temperature profiles as well as the thickness and heat flux a t different, times. These equations, together with the details of their numerical solution, are given elsewhere (Shah, 1968). An example of the results of the numerical solution is shown in Figure 7 . The results in Figure 7 are computed for a hypothetical run a t a Reynolds number of 14,700, a gas humidity corre-
-
1
-
-
i
T
T
-
I
Curves Generoted by Simple Model
o A
Curves Generoted b y Variable D e n s i t y
S
o
800 ''I7-
30
0,20 I
VI
40 45 Time ( m i n s )
X c la) V Q n)
.-
c ln-
0,14
0
0.16 -
50
-1 m 0.15 0.14
35
2
L 40 45 Time ( m i n s . )
A 50
30
35
~ 40
45
A
$ c.10-
f .
vr
0,O7 ~
Experimental Conditions Re = 15,800 H u m i d i t y = G a s dew point at Tg = 7 l 0 F
0.13 -n 0.12 0.11 -
1 0,08
l
l
l
i
l
~
Average Density of the Entire Frost Layer Density of the Bottom 0.0625" Thick Frost Layer Density of the Bottom 0.12 5 " T h i c k Frost Loyer
c
H u m i d i t y = G a s dew p o i n t a t 29.2OF
1
l
29.8v
U
bIy
. 30
--
l
1 :.
Svstem Conditions
v)
0
35
l
i
~
/
9
4
:/:
0.09 0.08 0.07 -
.
-
-
-
om Time (min. ) F i g u r e 9.
Density distribution within frost as function of t i m e
~
~
~
50
Time (mins.) F i g u r e 8. C o m p a r i s o n of t h e o r e t i c a l p r e d i c t i o n s of s i m p l e and v a r i a b l e density models
sponding to a dew point of 29.2'F, and a gas temperature of 70'F. Results for other conditions were qualitatively similar. For the results in Figure 7 , the calculation was started a t a time of 30 minutes with a frost layer 0.14 inch thick of uniform density equal to 0.076 gram per cc. This layer was then exposed to the 70°F gas with a dew point of 29.2'F. As additional frost deposited, internal diffusion of water vapor caused densification of the frost layer, but to an appreciable extent only in the outer regions of the layer. Apparently, in the inner regions of the frost layer the temperature is so low and the ice vapor pressure so small that frost densification occurs a t a n insignificant rate. Therefore, as shown in Figure 7 , large variations of density with position were predkted. Over a short period of time, this model predicted heat and mass fluxes and frost thicknesses in good agreement with the simple model and with experimental data. Figure 8 indicates the type of agreement for one set of conditioiis. B u t the two niodels disagree strongly in the predicted density distribution within the frost. The only previous experiinental data on density gradients within a frost were taken by Braeinsky (Braeinsky 1967; Brian et al., 1969) who found no variation of density with depth. As these data were few and the result was somewhat unexpected, additional experimental confirmation was also sought in this work. Experimental Measurements of Density Gradients in a Frost layer
A series of frosting runs was made a t a gas-phase Reynolds number of 15,800, a humidity corresponding t o a 30'F dew point, arid a gas temperature of 71'F. These conditions closely approximate those chosen for the theoretical results in Figure 7 . The deposition times varied from 26 to 144 minutes; corresponding to these times the frost thickness a t
the end of the run varied between 0.091 aiid 0.256 inch. T o sample the frost at the end of each run, a coring tube was constructed by gluing together 0.0625-inch polystyrene rings. When a frost saniple was removed with this tube, the 0.0625iuch rings (with frost) could easily be separated with a cold razor blade in a -20'F cold room. The density of each segment was determined by measuring the frost volume and mass. The results of these tests are shown in Figure 9. Though the density varied for runs of different length, the frost density on the bottom 0.0625-inch layer was always essentially the same as the average density for the entire layer. With the help of repeat runs, the data shown in Figure 9 were estimated to be accurate within a inaxinium error of 10%. These results and others of n similar nature force one to reject a n internal densification model based on the transport of water b y vapor-phase diffusion driven by a n equilibrium partial pressure gradient, a t least in the colder regions of the frost interior. Indeed, vapor-phase diffusion is unlikely even with a rionequilibrium partial pressure profile. Using the experimentally determined rate of densification of t'he frost layer for a given run, the partial pressure profile required to produce a uniform densification of the frost layer by vaporphase diffusion can be calculated. Such calculations (Shah, 1968) resulted in water vapor partial pressure profiles such as that shown in Figure 10. The slope of the required partial pressure profiles must, at each point within the frost, be equal to the rat,e of densification of the frost layer between t'he point in question and the cold wall. Thus the slope a t the cold wall is zero, and the slope a t the frost surface is that required to densify the entire frost layer. R u t while the slope of the required profile is fixed a t every point', the absolute pressure level is not. Thus the required profile is represented by a family of curves obtained by sliding the curve in Figure 10 up or down. B u t since negative partial pressures must. surely be excluded, the curve presented in Figure 10 represents the lowest possible curve in the family of partial pressure profiles which could account for a uniform frost densification a t the experimentally observed rate. Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
379
"1
* e o
S Rn:I,yo d ':i;tn ies = 14,700 H u m i d i t y = G a s dew point T4= 7 O o F
I .o 3.0
0 .-+ 0
a
A
5.0 4.0
3 .O
2 .o I .o
0 0
0.04 0.08 0.12 0.16 0.20 Distance f r o m Cold W a l l (inches)
Figure 10. Required pressure profile compared to equilibrium profile within frost
Shown for comparison in Figure 10 is the profile of the ice vapor pressure in equilibrium with the local frost temperature. At the frost surface, the slope of the equilibrium profile is nearly equal to the slope of the required profile because the former agrees with the over-all rate of densification of the frost layer. But the required profile lies substantially above the equilibrium profile, indicating supersaturation ratios over 1000 within the frost layer. Even a t the frost surface, a supersaturation ratio of about 5 is indicated. Such high supersaturation ratios would appear to be unattainable in such circumstances. But even more convincing is the fact that the required partial pressure a t the frost surface, 14 m m of Hg, is substantially higher than the water vapor partial pressure in the bulk gas stream, which was only 6.3 mm of Hg. This contradiction indicates that vapor-phase diffusion cannot account for a uniform densification of the frost layer. Therefore while the success of the simple model indicates that water is transported from the frost surface to the interior by vapor diffusion driven by an equilibrium partial pressure gradient, the transport of water to the cold regions deeper within the frost would appear t o occur by other mechanisms. Water Transport as Nuclei
Since the densification of the cold regions of the frost layer cannot be explained by vapor diffusion, it is postulated that very small ice particles are nucleated in the vapor within the frost and that these nuclei are transported to the cold regions by thermal diffusion. Thus the water transport into the frost is visualized as occurring by vapor-phase diffusion in the warm region near the frost surface but by thermal diffusion of nuclei in the cold region. Intermediate between these two would lie a region in which both vapor-phase diffusion and thermal diffusion of ncclei might be important. Brock (1962) has presented a theory for the movement of colloidal-sized solid particles in a gas with a temperature 380
Ind. Eng. Chem. Pundam., Vol. 9, No. 3, 1970
gradient. Assuming an ice particle 50 A in diameter, Brock's theory has been used t o calculate (Shah, 1968) the thermal diffusion velocity corresponding to the temperature gradient at various points within the frost, as given in Figure 2. The calculated velocity varies from about 40 feet per hour near the frost surface to about 170 feet per hour near the cold wall. The water flux required to explain frost densification is of the order of 0.05 lb,/hi ft2 near the fiost surface, but it decreases to zero at the cold wall. I n order to get a flux of 0.02 lb,/hr ft2with a particle velocity of 100 feet per hour, the concentration of particles would have to be 2 X lb,/ft3, which corresponds to a volume fraction of 4 X 10-6. This does not appear to be an unreasonable nuclei concentration. While nucleation within the frost appears likely and while Brock's theory predicts thermal diffusion velocities high enough to transport the nuclei within the fiost, there ale a t this time no experimental observations to corroborate the postulate that theimal diffusion of nuclei is the mechanism of densification of the cold regions of the frost. Furthermore, it is by no means clear how this water transport mechanism would result in a densification rate which is uniform over the frost layer. But no other mechanism of densification of the cold layers appears likely, and so this postulate is currently being pursued. Nomenclature
D
= molecular diffusivity of water vapor in air, ft2/hr
h
= =
AH
k
K, M AT p
kT X
heat transfer coefficient, Btu/hr-ft2-OF heat of sublimation (negative) of water vapor, Btu/lb, = thermal conductivity, Btu/hr-ft-OF = mass transfer coefficient, ft/hr = molecular weight of water, lb,/lb mole = flux of water vapor to surface, lb,/hr-ft2 = partial pressure of water vapor, mm of Hg = heat flux, Btu/hr-ft2 = gas constant, lbi-ft3/ftz-lb mole-OF = temperature, O F = distance from the cold wall, ft
GREEKLETTERS
e
time, hr frost density, lb,/ft3 P(T,) DLlf2AHpg/R2Tsar,lb,/'F-ft-hr 7 = tortuosity 6 = frost thickness, inches = = =
P
SUBSCRIPTS f = frost gas phase
9
=
ice
= solid ice
S W
= =
frost surface wall
A bar over a symbol indicates an integrated average value. Literature Cited
Biguria, Gabriel, Ph.D. thesis, Lehigh University, Bethlehem, Pa., 1968. Biguria, Gabriel, Wenxel, L. A,, IND.ENG.CHEM.FUNDAM. 9, i 2 9 (i970). ' Brazinsky, I., Sc.D. thesis, Massachusetts Icstitute of Technology, Cambridge, hIass., 1967. Brian, P. L. T., Reid, R. C., Braainsky, I., Cryog. Technol. 5 , 205 11969). Brock, J. R., J . Colloid Sci. 17,768 (1962). Eckert, E. R . G., Drake, R. I f . , ('Heat and Mass Transfer," 2nd ed., p. 7, McGraw-Hill, New York, 1955. Reid, R. C., Brian, P. L.T., Weber, M.E., A.I.Ch.E. J . 1 2 , 1190 (1966).
Shah, Y. T., Sc.D. thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1968. Sleicher, C. A,, Tribus, X., Trans. ASME 79, 789, 1957. RECEIVED for review July 7, 1969 ACCEPTEDJune 1, 1970
