FACTORS INFLUENCING THE RATE OF HEAT CONDUCTION IN FREEZEIDRYING THOMAS A. TRIEBES’ AND C. JUDSON K I N G University of California, Berkeley, Calif.
A steady-state device was used to investigate rates of heat conduction through freeze-dried turkey breast meat. The apparent thermal conductivities of the specimens increase with increasing pressure and increasing thermal conductivity of the residual gas. There is no evident effect of temperature level; however, increasing the relative humidity of the residual gas does serve to increase the apparent thermal conductivity. There is a marked effect of grain orientation on the heat conduction rate. The variation in conductivity between like samples is also reported. The experimental heat conduction data are interpreted in terms of structural features of the dried meat.
involves the removal of water from a frozen substance by means of sublimation. As a piece of frozen material is freeze dried, an outer, dried region develops surrounding a central, frozen core. The heat of sublimation necessary for water vapor generation is usually supplied to the frozen core by conduction through the layer of material which has already been dried. Recent reviews of the interaction of heat and mass transfer processes to be expected during freezedrying ( 7 7 , 76, 27) indicate that freeze-drying as carried out commercially should be rate-limited during much of the drying cycle by resistances to heat and mass transfer within the pieces being dried. Furthermore, conditions are most often such that the drying is heat transfer controlled-i.e., an increase in thermal conductivity of the outer dried layer is much more efficient in accelerating drying rates than is an increase in vapor permeability and/or diffusivity. These predictions are confirmed by the recent experimental measurements of Sandall, King, and Wilke (78). For these reasons, there is considerable importance attached to a knowledge and understanding of the environmental and structural factors determining the rate of heat conduction in freeze-dried substances. At present and in the foreseeable future, foodstuffs will provide the primary application for freeze-drying, although several other uses are undergoing active research. Among foodstuffs, poultry meat carries high interest as a future large scale product. Turkey meat (pectoralis superficialis, the major breast muscle) was chosen for this work because of its importance, because of the large sampie size available, and because of the opportunity to study the effect of grain orientation. Harper and El Sahrigi (4, 5 ) have reported the variation of effective thermal conductivity with pressure level for specimens of beef and several fruits in the presence of various residual gases and applied a semitheoretical model to their data. Kessler ( 8 ) utilized the radiation-dependent model of Krischer and Esdorn (70) to explain his experimental conductivities for freeze-dried apple and other materials. Saravacos and Pilsworth (79) measured heat conduction in several low density model food gels. Lusk, Karel, and Goldblith (73) and Ehlers, Hackenberg, and Oetjen (7) were also interested in the thermal conductivities of freeze-dried foods, but obtained data under less controlled conditions. REEZE-DRYING
Theoretical Analysis
T h e porosity of freeze-dried meats is 0.65 to 0.75, with the 1
430
Present address, Dow-Corning Corp., Midland, Mich. I & E C PROCESS D E S I G N A N D DEVELOPMENT
result that much of the piece volume is occupied by the surrounding gas phase. The conductivities are low enough so that the contribution of the gas to total heat conduction can be highly significant. Comprehensive reviews of conduction processes in heterogeneous media have been given by Woodside and Messmer (25), Meredith and Tobias (75), and Gorring and Churchill ( 3 ) . In the terminology of the latter authors, the present system represents a case of “continuous pairs.” Because of the difficulty of characterizing such systems mathematically, most approaches for the interpretation of data have been based upon concepts of unit cells involving combinations of series and parallel arrangements of the two phases. I n the present work, the conductivity of the gas phase is considered first, and then the interaction of conduction in the gas and solid is analyzed. The contribution of the gas phase to the heat conduction process should be dependent upon the pressure level. For a n unconfined gas a t moderate pressures, kinetic theory predicts that the thermal conductivity is independent of pressure. When gas is confined in a solid matrix, the gas thermal conductivity should still be independent of pressure as long as the mean free path is small compared with the pore spacings or void sizes of the matrix; as a consequence, the apparent thermal conductivity of the solid-gas matrix should be independent of pressure. As the pressure is reduced, with the mean free path becoming comparable to and greater than the pore spacings, the gas thermal conductivity and hence the conductivity of the matrix should begin to drop continuously. At very low pressures, the gas should provide a negligible contribution to the total heat conduction; thus a lower asymptotic thermal conductivity, due to the solid alone, should be reached a t very low pressures. Harper (4) has summarized several past theoretical approaches to the dependence of gas thermal conductivity in porous media on pressure. Interestingly, all of these approaches lead to an expression of the form
Simpler theories (9, 22) predict that C is independent of pressure. For a gas confined between parallel plane walls a distance 6 apart, Kistler (9) predicts for conduction normal to the planes that
c = PAo T2 ( *2 )*
Equations 6 and 7 contain four constants, two of which may be set by the conditions that
Simple, rigid-sphere kinetic theory predicts that
RT = 1/2nu2
PN
(3)
hence PA0 should be independent of pressure. A more elaborate approach by Kennard (7) for conduction between parallel plates a distance d apart yields two different expressions. For high pressures (slip region for flow)
while for low pressures (free molecule region)
CL = PA0
*
(+)-
10 2
6
?+I (2).
Combining Equations 4 and 5, one finds that the ratio of Ch to Cl is 0.4 y / N p , , which for air is 0.77. Taking the Eucken relationship (2) between h r p , and y, one finds that the ratio of C h to C I is (9 y - 5)/10, which reduces to unity only for a monatomic gas. Figure 1 shows a unit cell model of the sort proposed by Wyllie and Southwick (26) for heterogeneous media in connection with studies of electrical conduction. T h e model has been applied to heat conduction by Woodside and Messmer (25) and Harper and El Sahrigi (5). The cell depicts both series and parallel conduction paths, each in parallel with one another. Previous use of this model has generally been based on the assumption that there is zero lateral conductivityLe., that the heat flux lines in the cell are perfectly vertical, with no bending. The expression for the over-all conductivity then becomes ax
y = d x + l - d
+b+cx
(6)
where y = k / k s and x = k,/k,. Harper and El Sahrigi (5) have endeavored to generalize this model by including effectiveness factors to account for interactions between the three adjoining conduction paths. Unfortunately these effectiveness factors must be dependent upon x; consequently, a complex, complete mathematical solution for the geometry would be required to use them reliably. A simpler approach is to consider the other limiting extreme, the case of infinite lateral conductivity. In that situation, the plane a distance 1 - d above the bottom must be isothermal and the over-all conductivity is given by 1 -=
y
d 1 - d cx+a+b+(c+a)x+b
a+
(8)
b f c = 1
b +ad = 1
-
(9)
E
A third constant may be set by matching the low pressure conductivity. For the remaining constant in Equation 6, Woodside and Messmer (25) employed the "electrical resistance formation factor" of a bed of nonconducting solids, the ratio of the resistivity of the porous material fully saturated with an electrolyte to the resistivity of the electrolyte itself. Harper and El Sahrigi (5) found empirically for various food samples that (dk/dk,),y=o = (dy/dx),,o = 1, and used this as the remaining boundary condition for Equation 6. Apparatus and Procedure
By passing a steady state heat flux of known magnitude through a turkey sample and a piece of Lucite in series, two independent sample conductivities were calculated :
T h e experimental apparatus is shown in Figure 2. The test sample, the reference Lucite, and the test heater were surrounded by a guard ring of identical construction. T h e heaters consisted of a matrix of 30-gage Chromel-X heating wire enclosed in a brass block and insulated from the brass by thin mica sheets and Sauereisen cement. The temperature of all points just under each heater could be made equal by adjusting individual heater currents. T h e cold plate, also constructed of brass, was cooled by passing refrigerated alcohol through a 5 X 5 X inch baffled recess located in its center,
I
I
GUARD AREA
I
I
(7)
P
I
2
I lNCHES
INSULATION
HEATER LEADS DlFF
TC LEADS
SECTION A-A
Figure 1. Unit cell for Wyllie-Southwick model; direction of heat flux is vertical
Figure 2. VOL. 5
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No temperature fluctuations were observed with the thermocouple located just above the cold plate during any datataking period. Temperature differences across sample and Lucite were noted by using differential iron-constantan thermocouples, insulated wires, 0.004 inch in diameter, butt-welded electrically. Thermocouples were placed in grooves etched in the surface of the plastic and held in place with a thin coat of Lucite cement. Sample thermocouples were sandwiched between the sample and a thin piece of soft gum rubber, the conductivity of which is about 0.1 B.t.u./hour sq. ft. (' F./ft.). Thermocouples for measuring the absolute temperatures of various points in the thermal pile were constructed and installed similarly. The circuit for the differential thermocouples included the junctions on either side of the sample or the Lucite. The circuit for absolute temperatures included a controlled cold junction. The gum rubber provided good contact at all points between the heaters and cold plate. Experiments run initially without the rubber gave nonreproducible results, and conductivities by power input seldom matched those obtained by reference to Lucite. With the gum rubber promoting good contact, results did not vary when the pressure applied to the top of the thermal pile was doubled from 6 to 12 pounds. A glass bell, sealed to the cold plate by a ground glass joint with rubber and grease seals, enclosed the entire thermal pile. Electrical leads into the sample chamber were through tungsten-to-glass seals. Thus, the environment around the sample could be maintained a t any pressure from 0 to 1 atm. by bleeding various gases into the evacuated chamber. Dry cylinder helium, carbon dioxide, and Freon-I2 were used along with water vapor and dry air. Water vapor was throttled into the chamber after boil-off from a vessel of distilled water a t ambient temperature. Dry air (dew point equal to -80' F.) was supplied by drawing room air through a bed of indicating Drierite. Chamber pressure was measured with a variety of gages including a 5-micron to 1.5-mm. of Hg McLeod gage, a 0- to 2.5-mm. of Hg type 521 N R C thermocouple gage, a 0- to 20mm. of Hg Model SP-1s Hastings-Raydist thermocouple gage, a multirange-type 530 N R C Alphatron, and a L-tube mercury manometer; the calibrations of all gages were checked against a large Type GM-100A CVC McLeod gage in the range 1 micron to 10 mm. of Hg. Gages were used in the indicated ranges with no preference being given to any one in particular. Allowance was made for the change in calibration from gas to gas. The largest piece of turkey of uniform thickness, '/4 inch nominal, which could be obtained from the breast muscle was about 1 X 2 inches. Therefore, the '/4 X 2 X 2 inch samples were prepared by spot gluing two to four pieces of turkey together. Edges were trimmed by cutting with a razor blade and sanding. Random micrometer readings of the thickness of finished samples showed the maximum deviation from average thickness to be about 0.010 inch, or 4% of the thickness. The mean deviation was on the order of 1 to 2y0, however. The turkey guard ring was constructed similarly. The thermal conductivity of Lucite was measured independently by following the centerline temperature change during transient cooling of a large Lucite cylinder, a method which has been described elsewhere (5,23,24). The length-todiameter ratio of the cylinder was IO, and the Biot modulus was maintained above 50. Detailed results are given elsewhere (20). The indicated thermal conductivity was 0.118 B.t.u./ hour sq. ft. (' F./ft.), a t 0 to 2 5 O C., agreeing to within 2% 432
I & E C PROCESS D E S I G N A N D DEVELOPMENT
with the experimental values reported by Harper and El Sahrigi ( 5 ) and Masamune and Smith (74). All Lucite reference samples were machined from the large cylinder to tolerances of about 0.001 inch. Care was exercised to orient all pieces of Lucite such that the heat flux would pass normal to the former cylinder axis, as this was the direction of heat flow in determining the conductivity of the Lucite. The plastic guard ring was a composite of four smaller pieces held together with spots of Lucite cement. The nominal thickness of Lucite was chosen as '/z inch to achieve comparable temperature differences across Lucite and turkey. Turkey conductivity dependence on residual gas pressure, nature of residual gas, grain orientation, absolute temperature level, and meat moisture content were all studied. The conductivity of a consolidated polystyrene foam was also obtained for comparison. A run was continued until timeinvariant temperatures were obtained. Turkey samples were prepared by cooking, slicing, freezing, and freeze-drying. For samples 1 and 2, the freezing was accomplished by placing the samples in a room held a t -30' F. Most of the remaining samples were frozen by radiation to the walls of a Dewar flask surrounded by liquid nitrogen. The drying process did not seem to alter the physical characteristics of the samples appreciably. Results and Discussion
Pressure, Nature of Gas Phase, and Grain Orientation. Figure 3 shows apparent thermal conductivities as a function of pressure for various residual gases. These results were obtained for sample 1, where the heat flux was in the direction of the grain. Figure 4 presents similar results for sample 2, where the heat flux was normal to the grain direction. These figures include points measured at 40' =k 25' F. For clarity only, the conductivities obtained by reference to Lucite are shown. The conductivities calculated from the heater power input agreed with these values to within 10% in general. but showed slightly more scatter. A complete tabulation of all recorded data is available (20). Figures 3 and 4 both clearly show a transition between a low pressure plateau and a high pressure plateau, as predicted. In Figure 5, the apparent conductivities a t the high pressure asymptote are plotted against the thermal conductivities of the various gases; the common low pressure asymptotes are also included (zero gas conductivity). Thermal conductivities of helium, air, carbon dioxide and Freon-12 a t 40' F. are taken as 0.081, 0.0142, 0.0090, and 0.0050 B.t.u./hour sq. ft. ('F./ft.), respectively.
i
0I O 1
-
'SAMPLE 1
HEL'UM
PARALLEL ORlENTdTlON
DIOXIDE WYLLiE AND SOUTHWICK
fk",ON
/
A+
0
I
I l l 1
I
! I l l
~
I l l 1
i
I
1 1 1 1
I
I
I
!
+
Figure 4. all gases
HCl.IUK
SAMPLE 2
NORM4L O R I E N T A T I O N
~*
Effective conductivity vs. pressure; sample 2-
Figure 5. Effective conductivity vs. aas - .Dhase conductivity (0 and 1 atm. values), k in B.t:u.jhr. sq. ft. ("F./ft.)
Figure 5 shows that the residual gas contributes substantially to the apparent conductivity, even when the gas conductivity is considerably less than the over-all conductivity. The behavior implies an appreciable "short-circulating'' of tortuous solid conduction paths, resulting from heat flux across short gas phase bridges. This effect must be brought into any quantitative analysis of the data. T h e unit cell model of Figure 1 and Equations 6 and 7 is capable of accounting for such behavior if d is near unity. Figure 5 shows that the initial slope is considerably greater than unity for the present samples 1 and 2; hence, the fourth boundary condition employed by Harper and El Sahrigi cannot be used. T h e curves shown in Figure 5 are based on Equation 6 with the remaining constant obtained in both cases by fitting to the experimental result for air a t atmospheric pressure. Equation 7, with the remaining constant determined in the same manner, produces curves which are virtually indistinguishable from those derived from Equation 6 and shown in Figure 5. This result suggests that the prediction for a finite lateral conductivity may also have the same shape. T h e constants determined for Equation 6 are a = 0.207, b = 0.100, c = 0.693, d = 0.969 for sample 1 and a = 0.311, b = 0.0515, c = 0.638, d = 0.799 for sample 2. T h e constants determined for Equation 7 are a = 0.310, b = 0.0086, c = 0.681, d = 0.940 for sample 1 and a = 0.447, b = 0.0207, c = 0.532, d = 0.625 for sample 2. I t was necessary to employ a porosity in Equation 9 for determination of the constants. Microscopic examination of samples of freeze-dried poultry meat indicated a porosity on the
order of 65%. An Aminco-Winslow mercury intrusion porosimeter was employed to measure pore size distributions (see below) ; the results for samples 1 and 2 indicated a porosity of 68 to 72% due to pores 200 microns in diameter and less. Gross measurements with a micrometer of total volumes of the samples employed in the penetrometer indicated porosities of about 74%. T h e slopes of the curves in Figure 5 must also equal the porosity a t the pointy = x (k = k , = ks). This conclusion follows from Equations 6 and 7 and from the fact that total parallel (a = 0) and total series ( b = c = 0) models both have a slope equal to porosity a t this point. These two extremes must necessarily bracket the data. Extrapolation of the data indicates a preference for e = 0.70 if k, is to have the same value for both samples. A porosity of 70% was used in deriving values of the four constants in Equations 6 and 7. T h e true thermal conductivity of the solid phase, k,, is equal to the apparent conductivity a t the point where k = k,. From Figure 5 k, is estimated as 0.165 B.t.u./hour sq. ft. (" F./ft.). This result compares with k, = 0.15 derived by Harper and El Sahrigi ( 5 )for pear, apple, and beef. The conductivity of a consolidated polystyrene foam was measured in the same apparatus, yielding a low pressure asymptote of 0.0150 B.t.u./hour sq. ft. (" F./ft) and a high pressure asymptote with air of 0.0225 B.t.u./hour sq. ft. (" F./ft.). The difference between these values is on the order of ekQ (considerably less than found for turkey meat) indicating the absence of any appreciable gas phase short-circuiting effect. This result is in qualitative agreement with previous results on similar materials. Following Equations 2, 4, and 5, the pressure range over which the transition between asymptotes occurs in Figures 3 and 4 should be related to the pore spacings of the dried meat. Figure 6 shows pore size distribution curves obtained for central cuts of samples 1 and 2 by means of the mercury intrusion porosimeter. T h e pore diameter has been related to the mercury pressure by postulating circular pores and a contact angle of 140". Since pores are not circular, the reported "diameter" would be that of a circle having the same ratio of perimeter to cross-section area as does the pore. Duplicate portions of sample 1 were analyzed, as shown. For both samples 95% of the pore volume with access diameters between 0.04 and 250 microns lies in the access size range of 2 to 100 microns. The size ranges shown in Figure 6 account for a porosity of 68 to 72%; hence, one must conclude that most of the gas phase which influences conduction is included in the range of Figure 6. T h e curves shown in Figures 3 and 4 are obtained from Equation 6, coupled with Equation 1. A constant value of C PORE D I A M E T E R
i microns]
100 80
60 40 0 A
0 LL
c z
" W
SAMPLE 1
n SAMPLE 2
20
W
0 05
I
Figure 6.
IO MERCURY PRESSURE
1000 (psi01
10.003
Pore size distribution measurements VOL. 5
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was employed for each gas and each sample, and was obtained by fitting the inflection point of each curve. T h e use of a constant value of Cis an oversimplification on a t least two accounts: Since there is a range of pore spacings presrnt, one would expect the transition between asymptotes to cover a greater pressure range than indicated by Equation 1. Also, Equations 4 and 5 indicate that C should be less a t high pressure than a t low pressure, thus reducing the pressure transition range in comparison to Equation 1 with a constant value of C. In reality, the two factors seem to be unimportant or to offset one another to a large extent, and the curves in Figures 3 and 4 derived for constant C match the data well. Values of C derived from Figures 3 and 4 are reported in Table I. Values of C derived from Equations 2, 4, and 5 are also given. In these equations, a: was taken equal to unity. Since Figures 3 , 4 , and 6 all show a high degree of symmetry on semilogarithmic coordinates, one might suspect that the diameter allowing access to half the pore volume would be an appropriate value for the pore spacing. Empirically, the diameter allowing access to 40% of the pore volume below 250 microns has been used in the predictive equations to allow for cases where the pores deviate markedly from a circular cross-section and for situations where a large size void is given access only through narrow pores. T h e dimensions obtained from the mercury intrusion method are indicative of the narrowest section of a pore-i.e., the "access" diameter. The predicted ratios of C for helium to C for air fall in the range of 1.7 to 2.4. T h e experimentally observed ratios are approximately 8. From measurements of pear, apple, and beef, Harper and El Sahrigi ( 5 )report ratios of C for helium and nitrogen in the range of 3.0 to 4.6. One rationalization of this result could be that the accommodation coefficient, CY, for helium is substantially less than that for air. This would be possible since air is composed of diatomic molecules with internal degrees of freedom, whereas helium is monatomic. Taking a substantially less than unity for helium would serve to increase all predicted values of C for helium without changing the values predicted for air. The experimental values of C for sample 2 are 4.6 times greater than those for sample 1, although the ratio of effective pore spacings derived from Figure 6 (and hence the ratio of predicted values of C) is only 1.7. The difference is probably explainable in terms of the anisotropic grain structure of the meat. In the case of normal grain orientation, the highest heat flux density is likely to be along the minimum spacing between adjacent fiber cylinders; while in the case of parallel grain orientation the primary gas phase heat flux is probably along pores where the wall spacing for molecule collisions is substantially greater. Applying this reasoning to the results for air with sample 2 in Table I and taking a: = 1, one would conclude that the minimum fiber spacing is on the order of the access diameter for 40 to 75y0 of the pore volume (8 to 16 microns). From the results with air in sample 1, one would conclude that the effective dimension for transport along pores
is on the order of 80 to 170 microns. This is the access diameter for less than 6% of the total pore volume, suggesting that a substantial amount of the voidage can be reached only through finer pores. This interpretation would also be in qualitative accord with microscopic observations of meat samples. The use of Equations 6 and 7 for data analysis implies that conduction is the only significant form of heat transport in the samples. For substances with pore spacings of the order of magnitude indicated for turkey meat it may easiIy be shown that natural convection is unimportant. Radiation in porous media has been examined by Larkin and Churchill (72); the importance of this effect decreases with increasing bulk density and decreasing pore spacings. By extrapolation of their results or by use of the Stefan-Boltzmann equation, it may be shown (20) that radiation should provide a contribution of less than 10% in the extreme. Piece-to-Piece Variation. Figure 7 shows measured thermal conductivities for 16 different samples where the heat flux was parallel to the grain and for two different samples where the heat flux was normal to the grain. The samples denoted by circles and by X were frozen by placement in a room held at -30' F. The remainder were frozen by radiation to the walls of a Dewar cooled by liquid Nz. The piece-to-piece scatter is considerably greater than the scatter for any one piece. Extrapolations to atmospheric pressure for the various parallel grain samples show that the mean conductivity is 0.0542 B.t.u./hour sq. ft. ('F./ft.). The standard deviation is 0.0066, or 12%. Extrapolations to 100 microns pressure for the parallel grain samples give a mean k of 0.0311 B.t.u./hour sq. ft. (' F./ft.), with a standard deviation of 0.0076, or 24%. One would expect that there could be a dependence of k or C upon freezing conditions; however, this is not clearly discernible from the data. Temperature Level. Thermal conductivities in the presence of air were measured at temperatures ranging from 15' to $120' F. For sample 1 in the high pressure region, k was 0.053 B.t.u./hour sq. ft. (' F./ft.) a t -15' to +15O F., 0.052 at 30 to 60° F., and 0.053 a t 90 to 120' F. For sample 1 in the low pressure range, k was 0.018, 0.017, and 0.019 in the same three ranges, respectively. For sample 2 at low pressure, k was 0.0089, 0.0085, and 0.0090 in the same three temperature ranges. No discernible temperature dependence exists. I t was also confirmed experimentally that k for Lucite does not vary significantly in entire temperature range considered. Saravacos and Pilsworth (79) report that the apparent atmospheric pressure thermal conductivity of freeze-dried starch gel increases with increasing temperature. The porosity of this material was 9370, however, with the result that the atmospheric pressure conductivity is much closer to that for the pure
-
.
ob a
v r a 4 PI D
011GRAIN P A R A L L E L
a + GRAIN N O R M A L
a
0.05
Table 1. Sample No.
Values of C for Use in Equation 1 (Mm. of Ha) Predicteda Gas Exptl. Eq. 2 Eq. 4 Eq. 5
13.5
13.5
m
;0.02 0.0 1
13 ._
23 a Assuming ct = 7, and 6 = minimum access diameter for 40y0 of pore volume (31 microns for sample 7 and 18 microns for sample 2).
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Figure 7.
Piece-to-piece scatter
gas (0.0228 us. 0.0158 for air a t 106” F.) and reflects the contribution of air much more strongly. Their results correspond to k proportional to TO.7 for dry samples; whereas the thermal conductivity of air increases as To.86. Moisture Content. When water vapor was employed as the residual gas with sample 1, the over-all conductivity was frequently greater than would be predicted from Figure 3 on the basis of the known thermal conductivity of water vapor. T h e results for water vapor with sample 1 are shown in Figure 8. The parameter is relative humidity, defined as the ratio of the pressure (pure water vapor) to the vapor pressure of water a t the temperature under consideration. The nature of the apparatus caused temperature to vary appreciably across the sample ; consequently, the relative humidity varied across the sample by a factor of 1.3 to 2.5. Assuming that the conductivity is linear in relative humidity, the appropriate average relative humidity is the logarithmic mean of the extreme values. The humidities reported in Figure 8 are derived on this basis. If the moisture sorption curve reported by Salwin (77) for chicken is extrapolated to lower temperatures and applied to turkey, the adsorbed moisture would be 5.5% a t 3070 relative humidity. If it is assumed that adsorbed water has the thermal conductivity of pure water [0.35 B.t.u./hour sq. ft. (” F./ ft.)] and that the adsorbed water contributes in proportion to its volume fraction, the increase in over-all conductivity a t 30% relative humidity would be 0.007 B.t.u./hour sq. ft. (” F./ft.). This is qualitatively similar to the increase observed a t water vapor pressures above 10 mm. of Hg. Saravacos and Pilsworth (79) report higher conductivities for high porosity freeze-dried gels when equilibrated with a relative humidity of 52% (12 to 15% adsorbed moisture) than for dry samples. The increases in thermal conductivity ranged from 0.0005 to 0.004 B.t.u./hour sq. ft. (” F./’ft.) (2 to 12Yo). This is again roughly the order of magnitude increase given by the above prediction method.
substantial piece-to-piece variation in thermal conductivity (a standard deviation of 12 to 24%). This will cause a n important piece-to-piece variation in drying rate, even when pieces of the exact same size are compared. Allowance must be made for adsorbed moisture and gradients in adsorbed moisture in predicting thermal conductivities and other physical properties. Ac knowledgment
This study was carried out as part of U. S. Department of Agriculture Research & Service Contract, No. 12-14-100-771 3 (74). Some of the data were obtained by S. K. Fung and 0.C . Sandall. Some of the preliminary design of the apparatus was carried out by V. Errunza. Microscopic observations were made by R. A. Cardenas. T h e authors are grateful for the helpful advice of C. R . Wilke, R. D. Gunn, and 0. C. Sandall. Nomenclature
constants in Equations 6 and 7 cross-sectional area, normal to heat flux, sq. ft. constant in Equation 1, mm. of Hg heat capacity a t constant pressure, B.t.u./lb. F. thermal conductivity, B.t.u./hr. sq. ft. (” F./ft.) Avogadro’s number, 6.025 X loz3 Prandtl group, C p p / k o pressure, mm. of H g heat flux, B.t.u./hr. sq. ft. gas constant, 62,370 mm. of Hg, cc./g. mole ” K. temperature, ” K. temperature difference across specimen, ” F. k,/ka thickness of specimen in direction of heat flow, ft.
klk, thermal accommodation coefficient ratio of heat capacity a t constant pressure to heat capacity a t constant volume pore spacing porosity mean free path of molecules, cm. viscosity, Ib./ft. hr. gas molecule effective diameter, cm.
Process Implications
T h e processing ramifications of the thermal conductivity behavior will be discussed in more detail in subsequent publications; however, several points are worth noting a t this time. Freeze-dried meats are anisotropic, with the thermal conductivity parallel to the grain being greater than that normal to the grain. This factor must be taken into account along with vapor permeation anisotropy in predicting drying rates of individual pieces. A residual gas pressure of 0.1 to 1 mm. of H g can cause a substantial increase in the conductivity parallel to the grain without being sufficient to increase the conductivity in the normal direction appreciably. There is a
f
OO’[
002
1
g = gas phase L = Lucite s = solid phase T = turkey 0 = unconfinedgas Literature Cited
(1) Ehlers, H., Hackenberg, V., Oetjen, G. W., Trans. Nad. Vacuum Symp. 8, 1069 (1961). ( 2 ) Eucken, A., Physik. Z. 14, 324 (1913). ( 3 ) Gorrin R. L., Churchill, S. W., Chem. Eng. Progr. 57 ( 7 ) , 53 (19617; (4) HarDer. J. C.. A.Z.Ch.E. J . 8. 298 (1962). (5) Harper; J. C.’, El Sahrigi, A. F., Ind. Eng. Chem. Fundamentals 3, 318 (1964). ( 6 ) Hock, L., Kessler, R. H., Chem. Ber. 86, 1166 (1953). ( 7 ) Kennard, E. H., “Kinetic Theory of Gases,” Chaps. 4, 8, McGraw-Hill, New York, 1939. (8) Kessler, H. G., Chem. Zng.-Tech. 34 ( 3 ) , 163 (1962). (9) Kistler, S. S.,J . Phys. Chem. 39, 79 (1935). (10) Krischer, O., Esdorn, H., Forsch. Gebiete Zngenieurw. 22, 8 (1956). (11) Lambert, J. B., Marshall, W. R., Jr., in “Freeze-Drying 01 Foods,” F. R. Fisher, ed., .. pp. 105-132, Natl. Acad. of Sciences, Washington, 1962. (12) Larkin, B. K., Churchill, S. W., A.Z.Ch.E. J . 5 , 467 (1959). (13) Lusk G.. Karel,. M... Goldblith, S. A.. Food Technol. 18 (10). . .. 181 (1964): (14) Masamune, S., Smith, J. M., J . Chem. Eng. Data 8 , 54 (1963). (15) Meredith, R. E., Tobias, C. W., in “Advances in Electrochemistry and Electrochemical Engineering,” C. W. Tobias, ed., Vol. 2, pp. 15-48, Interscience, New York, 1962. (16) Rowe. T. W. G.. in “AsDects Theorioues et Industriels de la
Figure 8. Water vapor-effective conductivity vs. gas phase pressure a t various relative humidities VOL. 5
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J. Abbl. Phvs. 52.16RR 11961i
slty 01 tialltomla, Herkele (21) van Arsdel, \.
..
Avi Publ. Go., Westport, Conn., 1963. (22) Verschaor, J. D., Greebler,
P., Trow. ASME 74, 961 (1952).
RECEIVED for review April 4, 1966 ACCEPTED August 1, 1966
(23) Williams, I., Ind. Eng. Chcm. 15, 154 (1923).
A LARGE SPHERICAL VESSEL FOR
COMBUSTION RESEARCH COMBU: A L P H O N S E B A R T K O W I A K A N D JOSEPH M. K U C H T A Explosiucs Research Cenfcr, Unifcd SfofcrUeportment of the Interior, Bureau of Mines, Pittsburgh, Pa.
Design and construction features ore given for a 12-foot diameter spherical steel vessel which hos been added to U. S. Bureau of Mines facilities at Bruceton, Pa., for use in combustion reseorch. The vessel i s constructed of A-21 28 Firebox quality steel which hos high tensile strength and resistance to brittle fracture. The maximum design of the vessel is 300 p.s.i.g. It i s equipped with both inner and outer doors which permit entry of personnel for installation o f instrumentation, and make possible experiments at elevated, atmospheric, and reduced pressures. The relatively large size of the vessel makes it particularly suitable for ignition, flammability, and detonobility studies with minimum wall effects. It can b e used for experiments with solid explosives as well as vapors and gases.
on the flammability and detonability of combustible gaseous mixtures is essential for safety in the mining and chemical industries as well as in various civilian and military facilities where combustible gases or vapors may be encountered. The U. s. Bureau of Mines and other research organizations have obtained such information for many combustible materials. Until recently, most of the Bureau studies have been made in cylindrical vessels up to 12 inches in diameter and in spherical vessels up to 24 inches in diameter (4, 5 ) . These vessels are suitable for many combustion studies, but they are less adequate for conducting limit-offlammability experiments or basic studies on the growth and propagation of explosions in which the vessel walls influence the flame propagation. Accordingly, the Bureau of Mines has had a 12-foot diameter spherical steel vessel fabricated for use in this type ofresearch (Figure 1 ) .
I
NFORMATION
Figure 1. 436
Although data are available on the limits of flammability and detonability for many combustible gases and vapors, the results reported by different workers are not always in agreement. For example, Linnett and Simpson (7) note that lower limit values reported for methane-air mixtures range from 5.0 to 6.3 and upper limit valves from 12.8 to 15.0 volume yo; similarly, for ethylene-air mixtures, lower limit values vary between 2.72 and 3.45 and upper limit values between 13.7 and 34.0%. Such variations are due largely to vessel size effects and are more pronounced a t reduced pressures; however, Coward and Jones (3) and Zabetakis (72) have shown that, if the vessel diameter is sufficiently large, the lower and upper limits of most hydrocarbon-oxygen-nitrogen mixtures do not vary greatly with decreasing pressures. Such size effects are also observed in flammability experiments with many halogenated hydrocarbons and with materials which are thermally
Test facility showing 12-foot diometer spherical steel vessel
I h E C P R O C E S S D E S I G N A N D DEVELOPMENT
