Measuremt Measurement and Correlation of Water Frost Thermal Cc Conductivity and Density Gabriel Biguria' and Leonard A. Wenzel Department of Chemical Engineering, Lehigh University, Bethlehem, Pa. 18016
Water frost densities and thermal conductivities were measured in the temperature range of -20' to -145OF when the frost condensed from air under forced convection conditions on a flat plate in a test section of a wind tunnel. Correlations are presented to predict these frost properties and a structural model based on cinephotomicrographic studies is suggested to explain the observed behavior of the frost crystal deposit.
FROST
deposition is a common phenomenon in the cryogenic industries, aerospace technology, meteorology, some of the process industries, and other specialized areas (Pingry and Engdahl, 1963; Whitehurst, 1966). To predict heat and mass transfer flnxes under frost formation conditions requires the solution to equations modeling the unsteady-state deposition of a constituent from a moving multicomponent gas stream to a cryosurface. To solve this moving boundary problem one requires values of the frost thermal conductivity and its density. A comprehensive Iit,eratnre survey (Biguria, 1968) indicated a need to understand better the effects of variables which affect frost properties as well as t o obtain heat flux data under frost formation conditions between 0" and -300'F. I n this paper, we report results of a n investigation conducted to measure these frost properties and to formulate a structural model for the two-phase crystalline deposit. In analyzing the data, crystal growth rate arguments were needed, particularly in efforts to formulate a structural model for the observed deposit behavior and to explain a n nnexpected independency of frost height with length. For this reason cinephotomicrographic studies of crystal growth and heterogeneous nucleation on the plate surface were necessary.
steam t o enter the heating section of the wind tunnel. The test plate is made of two pieces of 0.016-inch and one of 0.091inch brass. The 0.091-inch piece has channels through which the coolant passes. One of the design requirements is that the maximum temperature difference between any two points on the plate should be less than 5'F. For the heat fluxes expected in this investigation, and with Freons as coolants, the test plate should be able to stand flow rates of 15 pounds per minute. Various unsuccessful designs were tried, but one in which the coolant entered through a labyrinth to the front of the plate and left through the side a t the rear next to the entrance of the coolant fina.lly met the requirements. The plate sits in the tunnel test section on a Plexiglas support which has a leading edge 4.06 inches long. The thickness of the support is 5/8 inch a t the front of the bra.ss plate, and its total length is 16.9 inches. The test plate cooling unit is able to maintain a constant plate temperature for about 1 to 4 hours. Freons circulate from a n insulated tank to a heat exchanger where they are cooled by liquid nitrogen. From the heat exchanger, the coolant passes through the turbine flowmeter and then t o the test plate. Measurements and Instrumentation
Description of Experimental Apparatus
The cryogenic test plate is situated in the test section of a wind tunnel in which velocities from 5 t o 50 feet per second can be obtained. The air stream which enters the 25-inch-long test section through an 8 by 8 inch opening is followed by a 22inch section where heaters are located. The flow then passes through a blower and is recirculated back to the test section through a number of screens, regulating fins, and a convergent section. The test section of the wind tunnel (Figure 1) is made of 3/8-iuch-thick Plexiglas and has a removable top and a side wall which opeus t o p e r m i t easy access to the test plate. An aluminum plate located in the lower part of the test section can be raised or lowered t o control the pressure gradient. The top can also be raised or lowered. Measurements of velocity and intensity of turbulence a t various cross sections of the test section showed that, for a zero pressure gradient, the velocity wa$ constant to within ~ 0 . 1foot per second. The intensity of turbulence, for a stream at 31.7 feet per second, was 0.36% 7 0.01%. The heaters were designed so that they could supply 3000 Btu per hour. Air temperature is automatically controlled to 10.5'F. Humidity is controlled by manually allowing 8-psig ~1
Present address, The International Nickel Co., Research
Stations, Port Colhorne, Ontario, Canada
The measurement of frost thermal conductivity is complicated by the dynamic nature of the process, the numerous variables which affect conductivity, the need to do i n situ measurements, and the sensitive nature of the crystals.
Figure 1.
Close view of test section
VOL 9 NO. 1 FEBRUARY 1970
l&EC FUNDAMENTALS
129
~~
Frost Density Data Obtained 1 Inch from Metal leading Edge
Table 1.
H, = 0.0065 Boundary Layer, lb/Ft3 Untripped Tripped
T, T,
=
20
=
-140
U, U, U,
= 41.1 = 8.8 = 41.1
T s = -4
U, = 8.8
T , = 20
U, U,
T p = -20
T s = -4
= 41.1
8.8 U, = 41.1 U, = 8.8 =
To measure conductivity, it is assumed that the temperature profile across the deposit is linear a t any time, although this is exactly true only a t steady state and constant conductivity. From an energy balance around the plate one finds:
-
= 41
pu,
qheatleak
kf (Ts - Tp)A
= -
(1)
h
The average or effective conductivity is calculated from Equation 1. qzDJT,, and h are variables which change with time. To calculate q t the following equation is used: qt
=
mcp(Tout- Tin)
(2)
I n addition to the variables already mentioned a continuous record of air velocity, humidity, and temperature must be kept. Intensity of turbulence of the air stream, acceleration or deceleration of the main stream, and the boundary layer regime have to be determined a t frequent intervals. The temperature difference between the inlet and outlet coolant streams is measured with a 16-function thermopile to &O.Ol°F. Average plate surface temperature a t the platefrost interface is calculated from the outlet coolant teniperature. An optical radiometer-thermometer is used to measure frost surface temperature at the frost-air interface to i l ° F without any physical contact between radiometer and frost surface. Frost height is measured with a cathetometer. Turbulence intensity and air velocity measurement3 are made with a hot-wire anemometer. Cinephotomicrographic movies are taken with a microscope and cine tube attachment coupled to a 16-mm reflex movie camera.
TURBULENT EXPERIMENTAL
1.0-
IT G'
0.8-
------d
\Wl 06-
[r
r-n,
04-
Z
LAMINAR WITH LEADING EDGE CORRECTION ( X - 0 5 ) -
W
02LAMINAR NO LEADING EDGE CORRECTION ( X - O 5, -
1
I
0
Figure 2. position 130
2 4 6 8 LONGITUDINAL DISTANCE (INCHES) FROM METAL PLATE LEADING EDGE Frost density distribution along longitudinal
I&EC FUNDAMENTALS
VOL.
9 NO. 1 FEBRUARY 1970
H, = 0.0135 Boundary layer, lb/Ft3 Untripped Tripped
6.34 5.42 1.87 1.72
5.60 3.79 1.82 1.27
3.90 3.22 1.'io 1.46
3.96 2.85 1.86 1.40
10.6 8.14 2.65 1.86
9.50 7.81 2.20 1.41
9.14 3.90 2.50 1.68
6.43 3.34 2.10 1.40
The recorded data were fed to computer programs which performed the calculations. Frost Density Experimental Design and Correlation
A Z5 factorial experimental design was used initially to investigate the effects of five variables and their interactions on frost density. The variables studied and their high and low values were: low
Air stream velocity, U r n ,ft/sec Bir stream humidity, If,, lb HsO/lb dry air Plate temperature, T,, O F Boundary layer (untripped-artificially tripped) Frost surface temperature, O F
High
8.8
41.1
0.0065 -20
0.0135 - 140
-1 -4
+I 20
The frost samples were collected a t 1, 4, and 8 inches from the metal leading edge for every run. N o s t runs were repeated to estimate the reproducibility. The results of the factorial experimental design are shown in Table I. Standard deviation of the reproducibility tests was 75.5%. The factorial experiment was analyzed statistically by the Yates algorithm method using a computer program. Instead of using the frost' surface temperature, a nondiniensional temperature defined by
Ts - T , T m - TP
r p = ___
would better correlate the frost density. The Yates algorithm showed that if one designates the boundary layer tripping effect as the smallest significant effect, then the air velocity, air humidity, plate temperature, diiiiensionless frost surface temperature, and three second-order interactions were also significant. The significant second-order internctioiis were: velocity with plate temperature, huriiidity with frost surface dimensionless temperature, and velocity with frost surface dimensioiiless temperature. The statistical experinierital design also showed a distribution of densities with longitudinal distance. The frost height was independent of position for all runs except for the small distance of about l / 4 inch a t the metal edge where the deposit height increased smoothly from zero to the height characteristic of the deposit away from the leading edge. Figure 2 shows the effect of longitudinal position on the frost density for six laminar and six tripped boundary layer runs. Because there is a smooth lc:i,ding edge where deposition does not occur, a correction to the mass transfer coefficients
as given b y Kreith (1963) was used. The observed density distribution and the over-all average frost density ratio appear to be slightly higher than predicted. To estimate the over-all average frost density, the frost was assumed to be distributed according to the corrected leading edge equation and the average density was computed by integrating over the whole plate.
p=--- P1x+J f(x)
=
( L - 2.0) x-"5/ 1 -
[
- pi(0.612)
(3)
(;)0,75]0'33
(4)
The necessary numerical integration was performed with the computer. The result, which indicates that p I / p l f = 0.612, is 6.1% lon-er than the measured average ratio of about, 0.65 for laminar flow. Some factors that may explain the surprisingly flat density distribution are: The deposited molecules diffuse on the metal surface from regions of high to low concentration, tending to make the downstream densities higher than expected. The leading edge correct'ion to the mass transfer coefficient is not descriptive of the real correction. The roughness of the deposit induces turbulence which modifies the mass transfer coefficient', so that it becomes less dependent on position. If nucleation occurs within the boundary layer, some of the particles may be carried by the air stream and deposited at a downstream position. The effective frost surface temperature a t the air interface is not, constant with length. The diffusion on the plate surface and the leading edge correction are probably not as significant as the effect of the surface roughness on the transfer coefficients and the effect of a nonconstant effective frost surface temperature on the distribution of rate of deposition. These last two effects may be significant during the initial stage of deposition, when the deposit, typically grows about 2 nim in 5 or 10 minutes. According to Kraenier (Schlichting, 1960), whose study excluded mass or heat transfer, a wire which is used t'o promote early transition is fully effective if
(5)
V
When heat or mas5 is transferred from the stream to the plate, the stability of the boundary layer is increased (Schlichting, 1960). I n the present investigation when tripping occurred, the signal of the hot-wire anemometer was characteristic of high frequency random turbulence and the frost surface suffered a remarkable change. The criterion for turbulence induction, which was deduced from the investigation, hac a larger value of the constant than does Equation 5 , as would be expected. The critical height can be predicted by Cmhcr,t V
2 1200
hadm
L 7-
+
+
lnp,(n) = -11.952096 0.024217706 * T, 35.549849 * H, - 0.03553795 * U, 1.2062987 * IO-' * T, * L', - 0.038382644 * BL 13.160559 * 71s 0.021328733 * T, * r f s - 81.955 * H , * r f s lnf(x) (8)
+ +
+
i l
- 4-
m
Tp =-I47
'
/
To=-39-
i
(6)
and froqt surface temperature ranged from -15' to +23"F when tripping occurred. Even if the d e p o d is not tripped, however, roughness can affect the transport coefficients. The admissible roughness in a turbulent boundary beyond which a surface is not hydrodynamically smooth can be calculated from 100 v
second it is 0.51 mm. During the first 5 or 10 minutes, before the deposit meshes into a more uniform composite phase, the effect of roughness may be significant. I n the laminar boundary layer, the effect of surface roughness is to advance transition (Schlichting, 1960). Hence, the transport coefficients in the laminar boundary layer are also affected by the surface roughness. The nonuniform effective frost surface temperature is probably another important reason why the frost density deposition profiles are flatter than those predicted. The effective frost surface temperature rose faster in the upstream than in the downstream regions of the plate, the discrepancy being as much as 15°F between the front and the back of the plate after depositing for 2 or 3 minutes. The temperature difference became progressively smaller as the deposit grew, and after about 5 minutes the temperature was constant from the front to the rear of the plate. The metal plate surface temperature was never allowed to have a temperature difference of more than 4"F, so the observed frost surface temperature variation with length in the initial stages of deposition must be explained on the basis of the density distribution of critical clusters and crystal growth. These topics will be discussed in a later article. The Yates algorithm analysis on the 2 j factorial experimental design showed that the five main variables had significant effects. These effects were then studied to find the type of relation between the dependent and independent variables. Figures 3 to 6 show the various effects. It was found that the density varied exponentially with the independent variables. A multilinear regression analysis program was used to determine the best functioiinl form. The following equation waS considered to be an adequate representation of the experimental data obtained from 50 observations. The standard deviation of the correlation was ~ 0 . 1 4 5lb,/ft3, which is less than the standard deviation for experimental data. The most probable error expected in the density measurements is T i % .
(7)
ba
This equation showy that a t a bulk velocity of 40 feet per second, the admissible height is 0.103 m m and a t 8 feet per
I 0L 02
L
L
03 04 DIMENSIONLESS
-
0.5
J
06
-
07
U
08
09
FROST SURFACE TEMPERATURE (TfsI
Figure 3. Frost density variation with dimensionless frost surface temperature VOL. 9 NO. 1 FEBRUARY 1970
l&EC FUNDAMENTALS
131
-9
9 t 8 .
I
Hm=O.O1OO Ib H20/lb dry air
Variable
H,
urn
T*
FG
h
m + \
Trn
7 !
!
6
1
IO
0.0136 lb HzO/lb dry air 43 feet per second -20°F 2 . 5 x 105 88°F
FLOW
If one neglects the effects of surface diffusion, frost surface roughness, and fog formation, the expected density distribution functions are: Expected laminar, no leading edge correction:
20 30 40 5 0 FREE STREAM VELOCITY
60
( f t /set) Effect of stream velocity on frost density
Figure 4.
High Value
0,005 8.1 - 145 5 . io x 104 78
H 2 0 / l b dry air
0
0
Conditions
This equation was obtained from data within the range of conditions shown in Table 11. The experimentally determined density distribution functions were: L‘4MISAR
-H,=001361b
Table II.
Low Value
f(z) =
(11)
2-0,5
Expected laminar, with leading edge correction:
‘O 7 -
-
-
- 8
Y?-?
cc
Expected turbulent:
\
I
26
-
-
t.
f(.)
t
cn 2 4 W n -
Um=41.1 f t / s e c
0‘ 440 Figure
dry air
H,=0.0134 Ib H,O/lb
T, = 2 0 ° F i
BL=-1 X
= I inch I
420
I
I
I
I
Theoretical Models to Relate Frost Thermal Conductivity to Its Density
I
400 380 360 340 320 PLATE TEMPERATURE (OR)
5. Effect of plate temperature
of the effects. The cinephotoniicrographic studies helped to explain the observed behavior of the frost density. Crystal growth rate and heterogeneous nucleation play very important roles.
on frost density
Heat can be transferred in a two-phase composite material in paths that are in series or in parallel. These two paths and the distribution of phases serve as the limiting bounds in the prediction of the thermal conductivity of the composite. Brailsford and Major (1964) give the following relation for the minimum and niaxiniuni composite conductivity: 1
__=-
K,,,
1 - P kz
and
K,,,
I
=
(1 - P)kz
FREE STREAM HUMIDITY ( H,XI03 Ib H,O/lb dry air) Figure 6.
Effect of free stream humidity on frost density
l&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
+ P-ki (resistances in series)
(14)
+ Pkl
(15)
(resistances in parallel)
Various equations have been proposed for syqtems with other distributions of the phases than those assumed in Equations 14 and 15. The Russel equation was developed for porous media where the solid is the continuous phase and there is a distribution of cubical pores arranged in a simple cubic lattice (Perry, 1963). The Russel equation is:
K - -
132
(13)
= 2-0
No attempt will be made a t this point to explain the nature
kz
+1+ 1 - P2’3+ P
(LY)P2’3
( L Y ) ( P ”-~ P )
with LY
= kl/k?
P2‘3
(16)
According to Laubitz (Perry, 1963), when Equation 16 is applied to powders, where the fluid phase is CoxitinuouS, the right-hand side must be doubled. Maxwell arid Rayleigh (Brailsford and hlajor, 1964) derived an equation for the electrical conductivity of a two-phase medium \%-hereuniform spheres of one material were randomly distributed in the second material. The AIaxwell and Rayleigh equation for the caw of fluid pores distributed in a continuous solid is:
K -
=
(1
+
- 2 P (iTa)/(1 P
[ea])) (17)
k2
The same type of 3laxwell-Rayleigh distribution of phases but for the case of continuous fluid phase gives: -
ks
=
{: +
!!! k?
2 P (1
- :))/is
-P
(1
-
I):
(18)
Brailkford and Major (1964) solved the problem of a random two-phase assembly with regions of both single phases in the correct proportions, embedded in a random mixture of the same two phases which has a conductivity equal to the average conductivity. The equation should be a good predictor for the case in which there is no spatial continuity of one phase.
K = { ( 3 P - l)kl
+ (38 - l ) k J + ( [ ( 3 P - 1)ki + (38
The frost densities formed under forced convection conditions over plates a t subfreezing temperatures have densities between 0.08 and 15 lb,/ft3 before they reach their quasisteady-state condition. The values of the frost densities are lower than the densities of packed snow or aged frost. Thus, from the point of view of the present investigation, the range of densities covered by snow studies is useful only near the quasi-steady state. In Figure 7 are presented experimental curves for packed snow and some of the theoretical curves evaluated a t 32°F. The effective air thernial conductivit'y given by Equation 29 was used. The Devaux and Van Dusen curves give good bounds for most of the experimental snow data a t densities larger than about 15 1b,/ft3 (Coles, 1954). The frost thermal conductivity curve obtained in this investigation for a plate temperature of -20'F is seen t o fall about 20 to 30% higher than the extrapolated values of snow thermal conductivity. The theoretical models which come closer t o the experimental data are the ones which assume that a random mixture exists (Equation 19), and the Woodside model (Equation 20) which assumes a cubic lattice distribution of spherical particles. Figure 8 shows the effect of the average temperature on the conductivities calculated from the Woodside and the random mixture model. Past investigators who have tried a single correlating curve between frost density and thermal conductivity have, a t times, not taken into account the effect of average temper'a t w e .
- 1)k2]* + 8 klkJ1/Z]/4
(19)
n'oodqide (1958) derived the equation to determine the thermal conductivity of a cubic lattice of uniform solid spherical particles in a gas. Heat transfer by radiation and by convection was neglected. His result is:
where
and is valid for 0 5 S 5 0.5236. Koodside used Equatlorl 20 to calculate the snow thermal conductivity a t an average temperature of 0°C. He took into account the contribution to the air conductivity in the snow pores caused by the diffusion of water vapor from a cold to a warm region. Krisher (Woodside, 19%) gives the following expression for the effective thermal conductivity of air inside the pores of a material whose pore walls are wetted:
Thermal Conductivity Results
A preliminary investigation in which the authors determined four sets of thermal conductivities a t four average temperatures showed that a t very low densities the theoretical models could not correlate frost density and thermal conductivity. .A full scale investigation of the main variables was then undertaken. The variables htudied were: free stream velocity, free stream humidity, boundary layer regime, plate temperature, and time.
0.28
I
I
I
I
/
WOODSIDE
(EQ 201
e
016
L
c \
2 012
VAN DUSEN
rn
v
The diffusion coefficient obtained by Krisher is:
cc
008 P 4ATM
Tp
004Roodside's calculations showed that the predicted snow thermal conductivities could be lower by as niuch as 47.6% if the water vapor diffusion was neglected. He also found that the predictions calculated from Equation 20 fell within the broad lines of experimental data when the water vapor diffusion was considered. The density range was between 6.4 and 31 lbn,/ft3.
0
7
I
0
OTHIS INVESTIGATION U=,41 .THIS INVESTIGATIDN U,=16
5
IO 15 FROST DENSITY
=+3Z0 F
I F T / SEC 5 FT/SEC
20
25
I
30
( 1 b, / f t 3 )
Figure 7. Frost thermal conductivity as a function of frost density according to different structural models Comparison with experimental lines of Schrapp, Devaux, and Van Duren VOL. 9 NO. 1 FEBRUARY 1970
l&EC FUNDAMENTALS
133
I
0 25 030
!+
I
I
I
E !
t
U-2 I O f t / s e c THE EFFECTIVE AIR T H E R M A L CONDUCTIVI T Y WAS USED, EQ 2 9
RANDOM-MIXTUREEQ i 9 , T = - 5 O o F
L 0 2 0 r
RANDOM MIXTURE
c
.
Lc
\
0 I 5 - -WOODSIDE __
c
m
EQ 2 0
u
'\
-+-
,
0 IO-
Le
0.05EO 19 ,T=-IOO"F I
I
0
I
I
20
25
1
IO
5
15
FROST DENSITY (IbM/ft3) Figure 8. density
Frost thermal conductivity as a function of frost Effect of averoge temperature
-0 . 3
Figures 9 and 10 show the typical raw data as a funct'ion of time. From the instantaneous frost height, corrected heat flux, frost surface temperature, and plate temperature, one can calculate the instantaneous average thermal conductivity. Equation 1 is used. Figure 11 shows that the ininimum heat flux is not always at the quasi-steady state for runs where tripping of the boundary layer occurs. Figure 10 indicates that, for low velocity runs, the measured frost surface temperature took as long a s 3 minutes to drop to its lowest value before it inrreased with time. I t was thought that perhaps it was because the frost layer was not thick enough, but this does not appear to be the case. As can be seen from Figure 9, the layer may have a value of about 1 mm after 3 minutes. According to Hsii (1963), the emissivity of frost is 0.985 for surfaces as thin as 0.004 inch. Since the radiometer thermometer needs no calibration for emissivities so close to 1.0, and its response is in the order of seconds, the initial drop in frost surface temperature cannot be explained by radiometer error. A patch on the plate was painted with black paint which has an emissivity very close to 1. The radiometer, when placed on top of this patch, again showed the initial drop in frost surface temperature for the low vclocity runs. The explanation of the initial frost surface ternperat,ure decrease a t low velocity probably lies in the complex relation between relative rate of increase of height and increase in frost thermal conductivity-that is, rn
2.5
''or H,=OOlOJIb
Y
5 ,-
E
1.51
9/%
Tp :-20'
1 HpO/lb dry
R d
if the deposit's resistance (Rd) approaches zero, the frost surface temperature approaches the plate temperature, and if the resistance becomes very luge, the frost surface ternperature approaches the melting point. The initial frost surface teniperature drop a t low velocities implies that the resistance decreases to a niiniinurn value and then increases. This may happen if initially the relative iiicrease in frost thermal conductivity is higher than the relative increase in height.
l-Uw:41.1fl/,sc
:UN230 TURBULENCE TR AKS FROM HERE CFI
0.5
l
0 Figure 9.
0 R U N 112 A R U N 128 0 R U N 130
.
1
4
+ hc
F
l
n
10
20 30 4045 TIME (Minutes)
1
.
1
*
,
l
-1
Variaton of frost height with time U4:,l
I fl/sec dry air
H =0103IbH O/lb
Eoo/
\BOUNDARY LAYER BECAME TLRBULENT
LBOUNDPRY
3
v
T ~ Z - Z O O F . BL:-I
400-
LAYER BECAME
1 1 1
TUQBULENT
1 RUN 0 RUN a RUN A RUN
112 127 128
129
i
I 1
0
IO 20 30 40 TIME (Minutes)
50
Figure 10. Frost surface dimensionless temperature variation with time 134
I&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
U,=886fl/sec
2 00;
0
1
IO
20
,
I
1
30
40
53
TIME (Minutes) Figure 1 1 . Total corrected heat flux (4.) to frost surface as a function of time
01 I t 0 10-
-k-
1
"' "I
008-
I: 0 0 6 -
U= ,16
c
-k 0 0 5 -
\ 3
m
i
006-
48 f t / s e c
L
-
P
0.071
004-
c
U,-8@6f'/sec
r
I
Y
002 t
H
i
=COIO3 Ib H p O / l b d r y o r
TF =-EO" F
BL = - I
0
5
15
IO
20
25
30
35
~
I
A
RUN 145 H,=00136 Ib H20/lb d a
1
40
TIME (Minutes)
Figure 1 2. with time
Typical frost thermal conductivity variation Tp =-83" F
001 -
U,
5 =148X105
0.1 I 0. IO-
LL
0
t 0.08 -
Figure 14. conductivity
0 L 4-
-;
/
m 004: L
Figure 13. conductivity
Tp
:
=-ZOO
0 , --I
F
;
20 30 40 TIME (Minutes) Effect of free stream humidity on frost thermal
:
- 008 A
\
002/
0
IO
1
006-
1
:236ft/sec
BL = - I
H ~00136Ib H O / l b d r y oir t IO m 1 n u t e s ~ Tp =-150' F BL=tI
15 20 25 30 35 45 45 FREE STREAM VELOCITY ( f t/sec) Effect of free stream velocity on frost thermal IO
!-007;
I : 15minuter--
c
.
-t=iOminutes
1 005; 2
041 - 0003; I
=U ,
L
Y
236 ft/sec
H,=001361bH20/Ib
002-
d r y air
BL = + I
001;
Figure 12 shows the typical change of frost thermal conductivity with time a t various temperatures and velocities. From Figure 13 one infers that the frost thermal conductivity increases linearly with air velocity for runs a t the same humidity, plate temperature, boundary layer regime, and time. Increased free stream humidity appears to decrease the frost thermal conductivity a t low temperatures and to increase it a t higher temperatures if the runs are at the same plate temperature, air velocity, boundary layer regime, and time. Figure 14 shows the effect of stream humidity. The boundary layer regime seems to have an interaction with plate temperature in its effect on the frost thermal conductivity. The effect of plate temperature is shown in Figure 15. The curves of Figure 15, when plotted on log-log paper, indicate that the frost thermal conductivity probably varies as Tpl,afor observations a t the same time, air stream velocity, humidity, and boundary layer regime. observed in the low humidity runs ( H , = 0.0115 lb HaO/lb dry air) a t plate temperature lower than about -100"F, that patches of frost were blowi away if the air bulk velocity exceeded a,bout 18 feet per second. However, if the stream humidity was increased to 0.0136 pound of H20per pound of dry air, the frost did not flake away even a t temperatures of -145'F and air bulk velocities of 41.6 feet per second. A multilinear regression analysis program was used to correlate the frost thermal conductivity with the five main independent variables and their significant second-order
I
1 250
300
350
PLATE TEMPERATURE
400 I
i I
( O R )
Figure 15. Variation of frost thermal conductivity with plate temperature
interactions. Simple least-squares analysis for each iiidividual run showed that the frost thermal conductivity could be represented by a quartic polynomial with time. The following equation was considered to be an adequate representation of the experimental average frost thermal conductivity data within the range of conditions of Table 11.
* T p 1 . 3+ * I;, + 6.2047771 * * t - 8.9475394 * * t2 + 1.0182528 * lo-' * t 3 + 2.6084586 * lo-* * t 4 4.2023418 * lo-* * H , * T , + 0.11349924 * t * H , + 1.0859212 * IO-' * BL + 2.1232614 * lo-' * 0'- * t 2.6856724 * IO-; * BL * T , (24)
if =
+ 1.0342876 * 18.007637 * I€, + 3.5719847 *
-0.23376438
The standard deviation of the correlation was = ~ 2 5 % . Reproducibility tests on four runs showed that the most probable error in reproducibility of frost thermal conductivity was r 1 2 % . The greatest error i n reproducibility occurs in the first 4 minutes of the run. The expected most probable error in the thermal conductivity measurements is =F 9%. VOL. 9 NO. 1 FEBRUARY 1970
l&EC FUNDAMENTALS
135
Substitution of the experimentally determined frost distribution,f(x), in Equation 25 gives: LAMIKAR FLOW
r,
k,(x) = (1.40)
(26)
2-0.270hf
TURBULENT FLOW k&)
Figure 16.
Schematic of initial crystal growth
Explanation of Observations on Growth
- 0.08 +
Lc
< 0.06 L
3
c
m y-
0.04
Y
b
0
H,=O.l
03 i b H 2 0 / l b d r y a i r
T = -
2 2 "F
2 4 6 8 1 AVERAGE DENSITY (Ib,/ft3)
0
Figure 17. Frost thermal conductivity as a function of frost density at low densities
>Urn,
=. f, I
'
1
I
0
\-Urn2
J
13
3 F R O S T D E N S I T Y (lb,
/it3)
Figure 1 8. Observed frost thermal conductivity variation with frost density
Equation 24 represents the average frost thermal conductivity over the entire length of the 11.06-inch-long plate. If the height is constant with length as has been found in this investigation, and the frost surface temperature is independent of position, from Equation 1 one concludes that the frost thermal conductivity should vary with position as the heat transfer coefficient varies with position. The arguments that apply to the frost density distribution also apply to the frost thermal conductivity distribution. If the frost thermal conductivity is distributed in the same manner as the frost density, the following equation gives the relation between average measured thermal conductivity and the expected point thermal conductivity:
(25)
136
1.11 x-O.07%,
l&EC FUNDAMENTALS
(27)
The observations discussed up to now were supplemented with cinephotomicrographic studies which helped to determine the morphology of the frost crystals and the structural changes of the deposit as it was growing.
0.I O
Y
=
VOL. 9 NO. 1 FEBRUARY 1970
of Deposit
One of the first observations that must be explained is the independence of deposit height with length. The cinephotomicrographic studies show clearly that the deposit initially is like a forest of trees as depicted in Figure 16. The mechanism of frost formation is, therefore, through clusters and one does not observe the growth of a homogeneous layer of ice. The air can circulate freely around the ice dendrites which, during the early stages of deposition, are relatively far apart. Once a critical cluster grows into a crystal, whether a t the front or end of the plate, the crystal grows a t about the same rate a t any position and, hence, the deposit appears to have a height independent of plate length. The distribution of critical clusters is not constant and decreases from the front region to the rear of the plat'e. The observed increases in frost density and thermal conductivity with a n increase in air velocity are caused by the decrease in growth rate in the linear dimension and the increase in nucleation and cluster formation caused by the higher velocity. The deposit tends to form more crystals and branches, with a consequent decrease in the porosit'y. The lower values of frost density and thermal conductivity a t low temperatures are caused by the decrease in the value of the thermal conductivity of air, the constituent material that cont,rols the thermal conductivity a t low densities, and the faster rate of linear crystal growth a t the lower plate temperatures. The thermal conductivity of ice increases as the temperature decreases, and this effect becomes important a t very high frost densities (Figure 8). The tendency of a higher humidity is to lower the frost density and t,hethermal conductivity by increasing the crystal growth rate and decreasing the critical cluster density, but the effect appears to have a temperature interaction. The models proposed to calculate snow t,hermal conductivities predict low values of the frost thermal conductivity. The same is true of the extrapolated curves of snow thermal conductivity as a function of its density a t low density. Figure 17 shows that a t the beginning of deposition the frost t,hermal conductivity rises from a value close to the air thermal conductivity a t the plate temperature (shown in Figure 12 as a function of time). The conductivity then drops to a minimum and again rises. For the region up to the minimum, and possibly somewhat beyond, the frost thermal conductivity is also a function of the air velocity. Thus, in the very low density region, when the crystals are beginning to branch, the convective heat transfer by eddies within the air phase is a significant contributor to the frost thermal conductivity. On the basis of the experimental evidence, one can explain the three regions shown in Figure 18 by the following model for the frost' thermal conductivity.
Region I. Assume that in the initial stage (I), heat is transferred only in parallel by the two phases (Figures 16, 17, and 18).Hence, =
keffrro,t
(keffair)
P
+ (1 -
(28)
where keff,ir
=
(kair
+k
vapor diffusion
+
kradiation
+
koonvective) turbulent eddies
(29)
I n Region I, a t very low densities, p f < 1.3 lb,/ft3, the air phase heat transfer is much larger bhan the ice phase heat transfer. Furthermore, in this initial rough deposit, full of ice trees and air spaces, the convective eddy diffusion probably controls. Thus, one would expect the frost thermal conductivity to rise with a n increase in frost density.
1.3 < p f
< 3 lbm/ft3
Region 11. I n this region, t,he cinephot,omicrographic studies showed that the deposit begins to mesh and the branches of the crystal dendrites begin to cross each other. Thus, one would predict that the convective eddy diffusion contribution should become smaller and eventually nonsignificant when the deposit consists of a close-knit mesh of dendrites. T h e frost thermal conductivity should drop until its density increases to the point where the ice phase and air phase heat transfer can no longer be considered to be simple parallel heat transfer. Region 111. When the deposit reaches densities of about 3 lb,/ft3, the thermal conductivity again starts to rise and continues to rise a t higher densities. A calculation from Equation 28 a t a frost density of 3 lb,/ft3 indicates that the contribution of the ice phase, if it were in a completely parallel independent path, would be about 0.066 Btu/hr ft O F . Thus, one can no longer say bhat the air phase alone controls the effective frost thermal conduct'ivity. Because a t densities higher than about 3 lb,/ft3 the deposit becomes a mesh of dendrites which enclose air pockets, the model based on the assumption of completely parallel arrangement of the two phases breaks down and one must search for a model such as that of Woodside (Equation 20) or the random mixt'ure model (Equat'ion 19). The Woodside model seems to be adequate if the contribution of the convective eddy diffusion is included in the effective air conductivit'y. With this correction, the Woodside model should reproduce the experiment'al data for frost densities higher than about 3 lb,/ft3 and less than 15 lb,/ft3. i l t higher densities the Woodside equation may give slightly lower values than those expected from snow data. The frost thermal conductivity curve given by the random mixture model (Equation 19) is very close to the experimental data if the cont,ributions of convective eddy diffusion and vapor diffusion in the effective air thermal conductivity are considered. This random mixture model is the best now available. The convective eddy diffusion contribution was determined from a comparison of the experimental frost thermal conductivity data with the values predicted by the random mixture model. The radiation contribution was neglected and the effective thermal conductivity was assumed to be given by Equation 29. By a trial and error procedure the following relation was obtained for the convective eddy diffusion contribution : kconveetive turbulent eddies
=
(0.001) LTm Btu/hr ft
O F
(30)
the downstream regions. The reason for this small frost surface temperature difference with length in the initial stages of growth is that the number of critical clusters varies from the front to the end of the plate. The higher concentration of crystals in the front reduces the area of plate surface seen b y the air stream. As the crystals mesh, the air stream begins to see a frost surface temperature which is more characteristic of the tips of the crystals and less of the plate surface. When this happens, the variation in frost surface temperature with length disappears, because the frost crystals seem to have the same temperature a t the air interface a t all positions along the plate. Summary
A correlation is presented (Equation 8) to estimate frost densities to F7Yob. A similar correlation for the thermal conductivity is given by Equation 24, which has a probable accuracy of r 2 5 % . Cinephotomicrographic studies, made to help formulate a structural model for the growing deposit, showed that the frost deposit grows according to a critical cluster mechanism. Once a critical cluster forms, the crystal grows a t about the same rate independent of its position along the cooled surface. The critical cluster density is higher in the front than in the rear regions of the plate. The effects of gas velocity, plate temperature, boundary layer regime, free stream humidity, and time have been partially explained. A structural model 1s suggested based on a random mixture of the phases (Equation 19) to calculate the effective frost thermal conductivity as a function of the conductivities of the constituents and the frost density. ;Z correction is given to account for the contribution of turbulent eddies in the gas phase to the effective air thermal conductivity. Acknowledgment
The authors thank Air Products and Chemicals, Inc., for the fellowship which made this work possible. Nomenclature
surface area, ft2 boundary layer tripped or untripped, +1 or - 1 = diffusion coefficient of water vapor inside wetted pores, cm2/sec = humidity of free stream, lb H,O/lb dry air = thermal conductivitv of two-ohase comDosite. Btu/hr ft O F = cooled plate length or total length including Plexiglas leading edge, ft = porosity or volume fraction of fluid phase = total pressure, g/cm2 = vapor pressure of water vapor, g/cni2 = deposit thermal resistance = average Reynolds number, Re = L C J V = gas constant, g-cm/g = volume fraction of solid phase = temperature within frost deposit, "K = melting point temperature of water, OR = plate average temperature, OR = frost surface temperature a t frost-air interface,
A BL D
= =
H, K
L P
Pi Pa,
R d
Re
R
s
T
T, T P
T.
O R
Tin Tout
One last effect must be explained. A slight variation of the effective frost surface temperature with distance from the leading edge mas cbserved. I n the upstream regions of the plate, the frost surface temperature rises faster than in
urn
C_p
hc
h
coolant inlet temperature, OR coolant outlet temperature, OR = free stream velocity, ft/sec = coolant heat capacity, Btu/lb, O F = average convective heat transfer coefficient, Btu/hr ft2 O F = frost height, ft or mm
= =
VOL. 9 NO. 1 FEBRUARY 1970
I&EC FUNDAMENTALS
137
admissible roughness height in a turbulent boundary layer beyond which a surface is not hydrodynamically smooth critical height. When critical height is exceeded, boundary layer is tripped thermal conductivity of fluid phase, Btu/hr ft O F thermal conductivity of solid phase, Btu/hr ft O F molecular air thermal conductivity, Btu/hr ft O F effective air thermal conductivity, Btu/hr ft O F effective frost thermal conductivity, Btu/hr ft O F convective eddy contribution to effective air thermal conductivity, Btu/hr ft O F radiation contribution to effective air thermal conductivity, Btu/hr ft O F contribution to air thermal conductivity by diffusion of water vapor through pores total heat transferred to plate, Btu/hr total heat transferred through frost deposit only, Btu/hr heat transferred to plate not through frost deposit time, minutes distance in direction of air flow, inches length of Plexiglas leading edge, inches GREEKLETTERS = ratio of thermal conductivities, kl/kz a! = latent heat of vaporization of water, cal/g AHv v = air kinematic viscosity, ftz/sec = frost density at 1 inch from leading edge, lb,/ft3 P1 f = average frost density, lb,/ft3 Pf
In
of frost density dimensionless temperature parameter,
= natural logarithm
pf
=
711
T. - T,
T~~
=
literature Cited
Biguria, G., “The Moving Boundary Problem with Frost Deposition to a Flat Plate at Subfreezing Temperatures and Forced Convection Conditions, Measurement and Correlation of Water Frost Properties,” Ph.D. thesis, Chemical Engineering Department, Lehigh University, May 1968. Brailsford, A. D., Major, K. G., Brit.J . A p p l . Phys. 15, No. 3, 313 (1964). -___
_
_
j
Coles, W. D.’, NACA Tech. Note, No. 3134 (1954). Devaux, J., ilnn. Phys. 20, 10,5-67 (1933). Hsu,,S. T., “Engineering Heat Transfer,” p. 585, Van Nostrand, Princeton, N. J., 1963. Kreith. F.. “Princioles of Heat Transfer.” , v. . 293. International Textbook Co.. Siranton. Pa.. 1963. Perry, J. H., “Chemical Engineer’s Handbook,” 4th ed., p. 3-224, hIcGraw-Hill, New York, 1963.(( Pingry, J. R., Engdahl, R. B., Surface-Moisture Phenomena under Icing Conditions,” Paper 17, Proceedings of International Symposium on Humidity and Moisture, Washington, D. C., May 20-23, 1963. Schlichting, H., “Boundary Layer Theory,” Series in Mechanical Engineering, 4th ed., p. 562, RIcGraw-Hill, ?Jew York, 1960. Whitehurst, C. A., ASHRAE J . 8, No. 10, 50 (1966) Woodside, W., Can. J . Phys. 36,815 (1958). RECEIVED for review February 26, 1969. ACCEPTED October 29, 1969.
Role of Coupling in Nonisothermal Diffusion Gas Absorption George B. Delancey’ and S. H. Chiang Chemical Engineering Department, University of Pittsburgh, Pittsburgh, Pa. 15213
An analysis is developed for multicomponent gas absorption accompanied by the heat effect of solution. The results are used to illustrate the interactions of the coupled fluxes in the ammonia-water system and in a hypothetical quinary system.
COUPLED
DIFFUSION has recently been subjected to both experimental (Cullinan and Toor, 1965; Cussler and Dunlop, 1966; Cussler and Lightfoot, 1965; Dunlop, 1965) and theoretical (Cullinan, 1965; Cussler and Lightfoot, 1963; Stewart and Prober, 1964; Toor, 1964) studies. hlost of these works have been devoted to isothermal and nonreactive systems, although some work has been reported for these other cases (DeLancey, 1967; DeLancey and Chiang, 1968; Hudson, 1967; Toor, 1965). Thermodynamic coupling provides additional modes of transport in the processes of heat and mass transfer that are not considered by conventional mechanisms. Nonisothermal multicomponent systems, and thus coupled phenomena, are 1 Present address, Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Hoboken, N. J.
07030 138
I&EC FUNDAMENTALS
VOL. 9 NO. 1 FEBRUARY 1970
prevalent in chemical processes throughout the biological and industrial areas. As these processes are of prime scientific and economic importance, the intrinsic role of coupling must be thoroughly understood. The purpose of the present study is to provide some insight into the possible extent to which thermal and mass transfer phenomena may interfere and to illustrate the behavior of the interactions. This is accomplished by studying the characteristics of a simple physical model of a nonisothermal process in which the coupled transport mechanisms are represented. The selection of a particular physical model is not a restriction, since the interactions that occur in any coupled transport problem are universal. A particular situation merely limits or enhances the extent of the interactions so that analysis of certain special cases can lead to a better understanding of the role of coupling in general.
