INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1951
1229
If the benzene is to be cooled from 110” to 70” I?. by passing / ? P = Reynoldq nutnhri = ( D V p ) / p , dimensionless with consistent units cold water through the shell, an average bulk temperature of = average bulk temperature o,f liquid a t which physical 90’ F. is again used, with the A’, Y coordinates of 6.0, 8.3. 1 3 ~ ~ t the previous method, h = 282 from the chart as compared with properties are determined, F. V = average linear velocity of liquid, feet per second h = 290 by numerical methods. p = absolute visrosity of liquid, centipoises = density of liquid, grams per ml. ACKNOWLEDGMENT = density of liquid, pounds per cubic foot The aid of C. W. Hurley, 111, A. F. Bentz, €I. F. Wyatt, and kO.6 0 8 0 4 ~0.6(p’)0.8(cp’)0.4 +(t)ic = = 0.0650 F. S. Glessner in collecting data on the physical properties of ,,o.~ these liquids is gratefully acknowledged.
-:;&
NOMENCLATURE
of liquid, joules/(grani)( O (7.) of liquid, B.t.u./(lh.)(’ F.) I> = inside diameter of pipe, inches h = film, $o$Iicient of heat transfer, B.t.u./(hour)(sq. foot) = specific heat cp’ = specific heat
cp ~
k *
t.)
= thermal conductivity of liquid, B.t,.u./(hour)(sq. foot)
( F. per foot) Pr = Prandtl number = (c,p)/Ic, dimensionless with consistent units
-
LITERATURE CITED (1) Dittus, F. W., and Boelter, L. M. K., l i m a . CuW. ( B e r k e l e y ) Pubs. Eng., 2, 443 (1930). (2) MoAdams, W. H., “Heat Transmission,” 2nd ed., p. lG8, Sew
York, McGraw-Hill Book Co., 1942. (3) Ryant. C. J., Jr., IND.ENG.CHEM.,35,1187 (1943). RECEIVED Augost 28, 1950.
Thermal Conductivity of Granular Beds Filled with Compressed Gases
Engjrnediring Process development I
JOSEPH L. WEININGER’ McGlLL UNIVERSITY, MONTREAL, CANADA
WILLIAM G. SCHNEIDER NATIONAL RESEARCH COUNCIL, OTTAWA, CANADA
*
x
T h e present measurements were undertaken to obtain information on the heat transfer in granular beds filled with gas, and more particularly to investigate the possibility of deriving from such measurements the pressure coefficient of thermal conductivity of the gas contained in the powder. Measurements of thermal conductivity on granular beds of aluminum oxide and powdered borosilicate glass when filled with helium, hydrogen, or nitrogen at varying pressures yielded linear relations between thermal conductivity and pressure; carbon dioxide in the same beds gave rise to a parabolic relation. Except for the measurements with helium and hydrogen, the measured values of the thermal conductivity were much greater than would be predicted from calculated values of the pressure coefficient of thermal conductivity of the gas in question. This behavior is attributed to increasing gas adsorption on the solid granules with increasing pressure. While the system employed in the present measurements effectively eliminates heat transfer by convection in the compressed gas, because of a possible gas adsorption (particularly the heavier gases) it ,does not appear very promising as a method for measuring the pressure coefficient of thermal conductivity of a gas. It may, however, be useful for measuring the thermal conductivity of liquids where the convection problem is also encountered. Finally, the gas-granule systems used i n the present meaeurements bear a close resemblance to granular catalyst beds used industrially.
T
HIS report concerns the experimental measurement of the
thermal conductivity of granular aluminum oxide or borcsilicate glass, when filled with gases at varying pressures u p t o 100 atmospheres. Although such measurements yield valuable information regarding the heat transfer in granule-gas or powdergas systems, i t appeared worth while also t o investigate the possibility of deriving from the data the pressure coefficient of thermal conductivity of the gas itself. It is extremely difficult t o measure the pressure coefficient of gas conductivity by the ordinary methods, owing t o the experimental difficulty of completely eliminating convection at high gas pressures. T o date, the only reported measurements are those of Varhaftik (16) and Keyes (10). I n the present work a modification of the Schleiermacher hot wire &ell (12)was used; the cell space was filled with the granular material, into which the gases t o be studied were compressed. With this arrangement the space available t o the gas is broken u p into a large number of minute cells, and convection is effectively eliminated. This method gave reliable measurements of heat transfer in the granule-gas systems, but attempts t o derive the pressure coefficient of conduction for the individual gases from t h e data were not too successful. With possibly two exceptions, the derived pressure coefficient was too high. This behavior was attributed t o increasing gas adsorption on the granules with increasing pressure. The systems investigated here are in many respeots similar t o a reacting system employing a granular c a t s lyst bed. Some thermal conductivity measurements on gas-filled powPresent address, University of North Carolina, Chapel Hill, N. C.
1230
INDUSTRIAL AND ENGINEERING CHEMISTRY
ders u p to one atmosphere pressure have been reported by a group of A4ustralian workers ( 1 , 7 ) . They showed that, using different gases for the same powder, the measured conductivities, although greater than those of the gases alone, were still in the same ratio as the gas conductivities (1). The conductivities fell off at a low pressure, converging a t a higher pressure (but still below atmospheric pressure), to a more or less constant value regardless of the grain size of the powder (?).
passed. These included the plat,inum heating wire, L, which was found on a grooved borosilicate glass tube, C, potential leads, A;, and two copper-constantan thermocouples, K. The latter, on passing through the borosilicate glass tube inside t,he >Ionel cylinder, were silver-soldered, respectively, at, the cent,er and the bottom of t,he Monel tube. Thus, an end loss of heat nould have been registered by a difference in the temperature readings of these thermocouples. The potent,ial leads cut off a central portion of the heating wire, t’his region constituting t,he part of the cell ovei which the measurements were made. The dimensions of the cell io 11077 : Steel cylinder Outside diameter Inside diaiiieter i\Ionel cylinder, outside dimnerei. Distance between potential leads Total volume of cell
T
Vol. 43, No. 5
3 493 C I I I . 2.210 ?In. 0.9416
riii
15,135 c n i . 110 06 c c .
The thermocouple and heater lead \virea w r e brought out of the cell by being passed through two neoprene washers which were clamped in a block a t the top of the pressure vessel. A gas inlet both a t the t’op and bottom of the cell provided for immediate pressure equalization throughout the cell. The pressure was measured by a large Bowdon gage x-ith a sensitivity of =t0.10 atmosphere. It was calibrakd against,a free piston gage.
Figure 1. Thermal Conductivity Cell
In the present investigation calcined, crystalline, alumina grain (supplied by the iiorton Co.) and ground borosilicate glass were used, respectively, as the aggregate material in conjunct,ion with the following gases-helium, hydrogen, nitrogen, and carbon dioxide. Heat transfer measurements were made on these systems with a cell suitable for measurements up to 100 atmospheres and in vacuum. CONDUCTIVITY APPARATUS
Figures 1 and 2 show t,he conductivity cell. I t consisted of two major parts-an outer steel cylinder, A , and an inner Monel cylinder, B , xvhich t’ook the place of the coaxial vire of the conventional cell. The outer cylinder was bored and honed to a snicoth finish. I t terminated a t the top in a flange and r a s closed a t the bottom by a hexagonal pressure nut. Annealed copper gaskets, H , x-ere used to make the pressure seals. The inner cylinder which is shown in detail in Figure 2 was accurately centered \Tithin the outer one. This \vas accomplished by tx-o steel plugs, E and F , soldered to the inner cylinder. To ensure proper alignment,,the parts were held in a special jig for the soldering. The cylinder was thermally insulated a t each end by glass which, in turn, was sealed tmoa small Kovar t,ubr, D , through which the electric leads
Figure 2.
Detail of Inner Cylinder
The heating current to the cell v a s supplied by a bank of tn-elve Edison cells and was accurately measured potentiometrically with the aid of a calibrated resistor. The electromotive force of the potential leads was measured on the same instrument while the thermocouples were measured separately with a Leeds & iiorthrup K-2 potentiometer. Evidence for the absence of convection in the gas was obtained by carrying out experiments n i t h different amounts of heat input As the conductivity, calculated from heat transfer data, did not varv, it was ooncludrd that convection TT as absent.
May 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
1231
where A' is the conductivity of the solid-gas system at pressure P; TABLE I. SUMMARY OF THERMAL CONDUCT~VITY MEASUREMENTS a refers t o the intercept on t h e conductivity axis ("zero-pressure AT HIGHPRESSURES AT 30 " C. conductivity of the system") of the conductivity-pressure plot. Intercept The slope of the plot, m, is equivalent t o ( b A ' / d P ) p . These a (bX'I3P)T con:tant 104 X Cal. lo7 X Cal. parameters, shown in Table I, were derived from the experimental No. of Cm. Sec. Atm. Cm. Atm.-l X Expt. data by a least squares fit. C. Sec. C. 104 Detn. Packing Cas O
Heliuni
AlnOa, grain .UzOs,, grain Borosilicate dilzOa, grain AlzOa, grain Borosilicate AlzOa, grain AlsOs, grain Borosilicate . 1 1 2 0 8 , grain &Os. grain Borosilicate AlzOa, grain AlzOa, grain Borosilicate
Hydrogen Sitrogen
-
coz T'acuuin
20.76 20.70 9.54 23.31 23.02 10.28
3.76 3.76 1.49 8.35 7.88 0.79
11.0 11.0 4.34 20.2 18.2 1.91
10 4 12 3 4 6
16
6.44
36 glass 16 36 glass
6.29 4.38
7.88 7.88 3.48
136.1 136.1 60.10
8 4 6
1G 36 glass 16 3B glass
4.66 4.28 3.17 1.44 0.954 0,871
16
36 glass
FL
Throughout this work the bath temperature was kept a t 30" 0.02" C. The thermostat was similar in design to that described by Beattie ( 2 ) .
*
CONDUCTIVITY MEASUREMENTS
The thermal conductivity was obtained from the experimental data (the heat input and the temperature gradient of the cell) by the equation:
where X,
The results shown were corrected to a mean temperature of 30' C., and a correction was also applied for the heat loss across the wall of the outer cylinder of the pressure vessel. Unlike the other gases, carbon dioxide showed a parabolic relation, a linearity of the conductivity with pressure being observed only up t o about a pressure of 20 atmospheres. These results (Figures 4 and 5 ) illustrate the behavior, under these conditions, for a gas close t o the critical point. The pressure during the measurements was in all cases kept below the saturation pressure to avoid condensation of the gas in the packing (critical pressure = 72.9 atmospheres, critical temperature = 31.0' (3.). At low pressures the conductivity for all gases fell off rapidly with a further decrease in pressure. I n a n ordinary gas system the region of pressure, where this phenomenon occurs, is of the order of magnitude of a few millimeters of mercury, whereas in the gas-powder system molecular conduction becomes evident a t pressures up to approximately 19 atmospheres, the pressure depending on the nature of the gas and the particle size of the packing. This is due to the very minute gas cavities possible in the region of contact of two neighboring powder grains, the length of these cavities in the direction of the temperature gradient being of the same order of magnitude as the mean free path of the molecules in the compressed gas.
= mean thermal conductivity of gas
q = quantity of heat put into cell per cm. height of
cylinder
T I , TZ = temperatures of respective cylinders DI, D Z = diameters of respective cylinders Two different grain sizes of the alumina aggregate were used. The one denoted by "grain size 16" had an average particle diameter of 1.088 mm.; the other, "grain size 36," had a n average particle diameter of 0.4797 mm. The single borosilicate glass sample used had grains of a n average diameter of 0.5404 mm. Figures 3,4, and 5 show the results of conductivity measurements on the various gases. Only one or two series of experiments are shown for each gas t o indicate the scatter of the experimental points. The average of several such determinations for each cell filling was used in calculating the thermal conductivity data summarized in Table '1.
0
IO
20
30 PRESSURE
Figure 4.
40 IN
50
60
70
SO
ATMOSPHERES
Conductivity of Alumina Grain with Nitrogen and Carbon Dioxide us. Pressure
The vacuum measurements on the different packings were performed in order to estimate the contribution of radiation and the conduction of the solid packing t o the over-all heat transfer. ANALYSIS OF RESULTS
20 0
I 0
0
20
1
I
30
4 0
PRESSURE
Figure 3.
IN
I
I
50
60
70
80
&TMOSPnERES
Conductivity of Alumina Grain with Hydrogen and Helium us. Pressure
Within the accuracy of the present measurements, the conductivity was linear with pressure for all gases except carbon dioxide, and ran be expressed by: A' = a
+ 7nP
(2)
I n assuming heat transfer by molecular'collision (thermal conduction) in a gas-granule system to be the only cause of a heat gradient one excludes both convection and radiation as other possible causes of heat transport. If, furthermore, no surface phenomenon, such as adsorption or lack of accommodation, is taking place a t the solid-gas boundaries, there remain the conduction through the solid (within each particle and at contact points or faces) and conduction by the gas (within each void space). Evidence regarding the absence of convection was obtained by performing experiments with varying heat input. Similarly, radiation could be considered t o be negligible. This was shown by the small value of the apparent conductivity of the system in
INDUSTRIAL AND ENGINEERING CHEMISTRY
1232
Vol. 43, No. 5
so that the number 01 collisions of a gas molecule with other gas molecules is large relative to the number of collisions wit,h the surface of ihe powder grains. An alternative method of analyzing the data was suggested by Woolley ( 1 7 ) . This will L)P referred to as the “calibration method,” since it involves the use of the “zero-pressure’’ heat conductivities of the system t o derive foi each gas a “cell constant.“
5.0
SLOPE 11ETHOI)
According to our basic assumption, tho slope of the thermal conductivity us’. pressui’c plots lor the powder-gas systems is directly proportional t o the rate of increase of thermal conduotivitg of the gafi wit’h pressurr. JYriting
4 3.0
I
0
10
20
30
40
PRESSURE
Figure 5.
50 IN
70
60
ATMOSPHERES
Conductivity of Borosilicate Glass
x
YO
80
:= A0
x
c’t’
(3)
where is the thermal c:onductivit,y ol the gas alone at any pressure I ] , and io is the thermal conductivity of the same gas a t one atmosphcic, then the slope c’ will be identical with the slopo m of Equation 2. Dividing Equation 3 bv i o a more convc’nicnt form is obtained :
cs. I’ressurr
vacuum as well as by a theoretical calculation according to Smoluchowski (14). The simplest and most plausible nicchanism for the hcut trimsfer in the powder-gas system involves the assumption that the heat transferred by the solid is not pressure dependent, w,iereas the heat transferred by the gas is pressure dependent owing to the increase of the thermal conductivity of the gas with increasing density. ( a n additional condition’ to be specified is that t,hc: increase in the heat transferred by the gas as a result of iricreasccl density be small compared t o the total heat transferretl; ot,hei,wise, if this increase is large the heat transferred by t,hc powdo. grains will be measurably altered.) Accordingly, the slope of the plot of thermal conductivity of the powder gas system against pressure should be a measure of the rate of increase of the thermal conductivity of the gas with pressure. The data presented here were analyzed on this basis, and for brevity it will be referred to simply as the “slope method.” When applied to a powder-gas system it is necessary t h a t the powder grains be sufficiently large
X’XO = 1 \\
htw c
=
+ (c’/X,)P = 1 + cP
(4)
c’, Xi, = vz
Values used for Xo, the thermal conductivity a t atmospheric: pressure and 30” C., are as follorvs: He-
liuin Conductivity, oal./cm. Reference
sec.
’C.
X 104
3.43 (9)
Hydrogen
SitroKen
COZ
4.13
0.579 (16)
0.350
(9)
(8:
Table I1 lists values of XlOo/Xo for helium, hydrogen, and nitrogen, where Xloo is the thermal conductivity of the gas a t 100 atmospheres. For the alumina packing a single value is shown for each gas and represents the average of the rrsuks for the two grain sizes given in Table I. CALIBRATION METHOD
For this method the zero-pressure conductivities for each powder-gas system (obtained by extrapolation of the thermal conductivity cs. pressure plots) were plotted as abwissas, and the known conductivities of the pure gases at atmospheric pressure were plotted as ordinates. Curves were then drawn in as shown in Figure 6, and were taken as calibration curves to convert mearured thermal conductivities of the powder-gas systems undei varying gas pressures into true gas conductivities. Table I1 shows the values of iloo/Xo obtained in this manner. It will be noted that the calibration curve for each powder (Figure 6 ) contains five points, one for each of the four yaseq measured, and the fifth point representing the vacuum measurement on the powdcr packing.
5 c
DISCUSSION OF RESULTS
ZERO P R E S S U R E CONDUCTIVITY OF THE P O W D E R - G A S S Y S T E M [ 1 0 4 x cal/cm.
Figure 6.
Calibration Curve
wc
‘c]
Comparison of the results obtained by applying the two methods of analysis (Table 11) shows that for helium and hydrogen with ground borosilicate glass packing the t,u-o met,hods are i n good agreement,; for all other systems the slope method gives much higher values of the ratio Xloo/xo. Both methods of analysi~ may be expected to be valid only if appreciable adsorption of thc gas on the powder is absent. If gas adsorption on the powder increases with increasing pressure, the slope method will yield too high a value for the pressure coefficient of thermal conductivity
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1951
of the gas. Under these conditions additional heat transfer in the system may occur as a result of two-dimensional thermal conduction of the adsorbed gas on the surface of the grains, and also of improved contact areas of the powder grains themselves as-a result of adsorption. The calibration method likewise gives erroneous results if appreciable adsorption occurs, since the calibration curve was derived from the experimental low pressure data, where presumably adsorption is not appreciable and, hence, cannot be used t o convert the experimental data a t high pressures where gas adsorption may be quite considerable.
1233
second virial coefficient of helium by Schneider and Duffie (15). Considering only the first-order term of the power series of Equation 6, the‘ variation in the conductivity with density is given by
+ 0.575bpX
(7)
- 1 - T(dZ/dT)p
(8)
X/Xo = 1
where bpX
=
Z
z = PV RT OF
THERM AL CONDUCT] VITY RATIO, XlQO/hO
Alumina packing Slope method Calibration method Borosilicate glass packing Slope method Calibration method Equation 6 Enskog equation (65)
Varhaftik
Heliuin
Hydrogen
Nitrogen
1 110
2 361
1 031
1.182 1.057
1 043 1 040
1.019
1 601
1 0230
(16)
Keyes (10)
1.016 1 ,0298 1,0473
....
1 181 1 157 1 152 1.226 1 178 1 204
In terms of the second virial coefficient BT and the Amagat density D,,2 becomes Z =
x = EC%q (5) a constant of proportionality ( = 2.5 for monatomic gases) C, = specific heat a t constant volume 7 = viscosity
where *
L
e =
This is the relation most commonly used in estimating the thermal conductivity of gases at higher pressures when C, and q are known as functions of pressure. For the present calculations the specific heat data of Deming and Shupe ( 4 )and Michels (11)were used for hydrogen and nitrogen, respectively. For the viscosity of both gases Gibson’s data (6)were used. The Enskog equation ( 5 ) for the thermal conductivity as a function of density is given by: X/Xo = b p
1
{bpx.
+ 56 + 0.7574 b p X . . . . II
Here the conductivity ratio X / ~ Ois given in terms of b p and a power series in ( b p X ) ,where b is tbe actual volume of the gas molecules, p the density of the gas, and X a factor allowing for multiple encounters of gas molecules which become important at higher densities. The values of the term bpX are tabulated in the literature for hydrogen (18) and nitrogen (6) but had t o be calculated for helium. Use was made of a recent determination of the
= 1
+ BTD,
(9)
D , was evaluated directly from the virial equation by D a
In view of the above considerations it appears reasonable to interpret the results of Table I1 as follows: For helium and hydrogen with ground borosilicate glass gas adsorption isabsentorisnegligibly small, and the reported values of Xloo/Xo should be close to the true values; for the other systems appreciable gas adsorption occurs and the measured values of X I O O / ~ Oare, on the basis of the slope method of analysis a t least, too high. According t o this interpretation, one would predict from the results of Table I1 that alumina was a bptter adsorbent for the gases listed than borosilicate glass. This is borne out by low pressure adsorption isotherm measurements (5). Likewise, as expected, nitrogen would be more strongly adsorbed than helium and hydrogen and would give rise to a larger slope of the thermal conductivity us. pressure plot. For comparison with the present measurements Table I1 includes some values of the ratio X~OO/XO calculated theoretically, and for nitrogen, some experimental measurements reported in the literature. According t o the kinetic theory, the thermal conductivity of a gas is related t o the viscosity and heat capacity by the relation
PV RT
- AT
P
+ BTP
There are no previous measurements of the pressure coefficient of conductivity of helium and hydrogen for comparison. Values reported for nitrogen by Varhaftik ( 1 6 ) and by K e y s (10) are listed in Table 11. The conclusion t o be reached on the basis of the above analysis is that the present method of measuring the pressure coefficient of thermal conductivity of gases is not particularly satisfactory and has a rather limited application. Thus, while it may be suitable for hydrogen and helium, and these only at higher temperatures, it is likely t o lead t o erroneous results when used for heavier gases owing to gas adsorption on the granules of the solid. The method is, however, an effective way of eliminating convection and may find useful application in the measurement of the thermal conductivity of liquids. Thus by first using the granulefilled cell t o measure the thermal conductivity of a liquid, whose conductivity is accurately known, the thermal conductivity of any other liquid can be measured. LITERATURE CITED (1) Aberdeen, J., and Laby, T. H., Proc. R o y . SOC.( L o n d o n ) , A113, 459 (1926). (2) Beattie, J. A . , Rev. Sci. Instruments, 2, 458 (1931). (3) Davis, R. T., DeWitt, T. W., and Emmett, P. H., J . P h y s . & Colloid Chem., 51, 1232 (1947). (4) Deming, W. E., and Shupe, L. E., P h y s . Rev., 37, 638 (1931). (5) Enskog, D., Kg1. Svenska Vetenskapsakad. Handl., 63, No. 4 (1921). (6) Gibson, R. O., dissertation, University of Amsterdam, 1933. (7) Gregory, M., and Archer, C. T., P h i l . Mag., 13, 30 (1933). (8) Kannuluik, W. G., and Law, P. G., Proc. R o y . Soc. Victoria, 58, 142 (1947). (9) Kannuluik, W. G., and Martin, L. H., Proc. R o y . SOC.( L o n d o n ) , A114,496 (1934). (10) Keyes, E. G., and Sandell, D. J., Jr., T r a n s . Am. SOC.Engrs., 72, 767 (1950). ( 1 1 ) Miohels, A,,’Wouters, H., and deBoer, J., P u b . V a n der W a a l s Fund., 37 (1934); 51 (1936). (12) Sohleiermacher,A., Ann., 34, 623 (1888). (13) Sohneider, W. G., and Duffie, .T. A., J . Chem. P h y s . , 17, 751 (1949). (14) Smoluchowski, M., B u l l . Acad. Cracow, 5A, 548 (1911). (15) Varhaftik, H. R . , and Parfenov. J..J. Tech. P h y s . ( U . S . S . R . ) , 8, 189 (1938). (16) Varhaftik, H. B., and Timrot, D. L., Izvesttla VTI, Sept. 9 , 1935; J . Tech. P h y s . (U.S.S.R.), 9, 63, 461 (1939). (17) Woolley,’H.W., private communication. (18) Woolley, H. W., Scott, R. E., and Brickwedde, F. G., J . Research Nntl. B u r . Standards, 41, 378 (1948). RECEIVED May 12, 1950. Presented before the Division of Physical and Inorganic Chemistry a t the 117th Meeting of the AMERICAN CHEarxcaL SOCIETY, Detroit, Mich.
