ENGINEERING AND PROCESS DEVELOPMENT literature Cited Smoothed Values of Relative Conductivity
Table IV.
.oh .rl. Y+
12.00 14.00 16.00 18.00 20.00 22.00 24.00 26.00 25.00
40.00
42.00 44.00 46.00 48.00 50.00 Standard deviations b
Averageb
2.26 2.73 3.25 3 83 4.48 5.20
2.04 2.48 2.97 3.54 4.13 4.76
2.15 2.59 3.11 3.68 4.32 4.08
9.70 10.50 11.27 12.04 12.85 13.66 14.53 15.38
10.66 11.56 12.46 13.36 14.26 15.10 15.95 16.77
fi.00
32.00 34.00 36.00 38.00
a
Upper waI1a
6.85 7.72 8 65 9.59 10.58 11.60 12.62 13.64 14.66 15.66 16.70 17.75 18.70
30 00
\1
Lower walla
1.165
0.327
...
...
0.5664 0.608b
Based upon present measurements. Based upon present and earlier data.
7 = absolute viscosity, pound second/square foot
= thermometric conductivity, square feet/second = kinematic viscosity, square feet/second u = specific weight, pounds/cubic foot K
Y
T
T~
= shear, pounds/square foot
= shear a t wall, poundsjsquare foot
Dimensionless parameters Pr, = molecular Prandtl number Y/K Pr - = total Prandtl number srn/~,, Pr, = eddy Prandtl number ern/+ Re = Reynolds number BYOY/Y u + = velocity parameter (Equation 9) y + = distance parameter (Equation 8) Subscript
w = wall
Bakhmeteff, B. A., “The Mechanics of Turbulent Flow,” Princeton, N. J., Princeton University Press, 1941. Boelter, L. M . K., Illartinelli, R. C., and Jonassen, F., Trans. Am. SOC. Mech. Engrs., 63, 447-55 (1941). Brough, H . W., Schlinger, W. G., and Sage, B. H., 1x0. ESG. CHEM.,43,2442-6 (1951). Connell, W. R.. Schlinger, W. G . , and Sage, B. H., Washington, D . C., Am. Doc. Inst., Doc. 3657 (1952). Corcoran, W. H., Page, F., Jr., Schlinger, W. G., and Sage, B. H., IND.ENQ.C H E W ,44, 410-19 (1952). Dunn, L.G., Powell, W. G., and Seifert, H. S.,“Heat-Transfer Studies Relating to Rocket Power-Plant Development,” Third Anglo-American Aeronautical Conference 1951, Published by The Royal Aeronautical Society. Hirschfelder, J. O., Bird, R. B . , and Spotz, E. L.,Trans. Ana. Soc. Mech. Engrs., 71, 921-37 (1949). Hsu, N. T . , Kendall, B. H., Schlinger, W. G., and Sage, B. R., Washington, D . C., Am. Doc. Inst., Doc. 3906 (1952). K&rm&n,Th. yon, Trans. Am, Soc. illech. Engrs., 61, 703-10 (1939). Page, F., Jr., Corcoran, W. H., Schlinger, W. G . , and Sage, B. H., Washington, D . C., Am. Doc. Inst., Doc.3293 (1950). Page, F., Jr., Corcoran, W. H., Schlinger, W. G., and Sage, B . H., IND.ENQ.CHEM.,44, 419-24 (1952). Page, F., Jr., Schlinger, W. G., Breaux, D . K., and Sage, B. H., Washington, D . C., Am. Doc. Inst., Doc. 3294 (1951). Page, F., Jr., Schlinger, W. G., Breaux, D . Ii)(Sec.) 2.55 X 10-2
108 lo7 lo7
1.583 X 2.44 X 4.33 X 5.94 X 3.53 X 2.01 X 2.35 X 4.01 X 1.295 X 2.05 X
lo7
lOQ 10’ 107 10s
108 lo9
100
Friction
Factor,
f
10-8 10-8 10-8 10-8
10-a 10-8 IO-’ 10-2 10-2
lo-*
Reynolds h-io.. Re
with needle valves. The gaq leaving the sample was saturated with water and was metered with a calibrated wet-test meter or by a soap film moving in a buret and timed with a stop watch The temperature of the gas Ptream just before and after the sample was measured by means of calibrated copper-constantan thermocouples mounted in a samp!e holder by mean? of thermocouple glands. Nearly all the hamples were removed from the sample holder and reversed. No change in the flow characteristics was observed with the change in the flow direction. Typical flow data appear inSTable 11. The flow rate and pressure measurements are accurate within 501, except for the data taken on very low permeability samples for which the flow rates were extremely small. In addition, since considerable variations in the propertie8 of different samples taken from the same bulk sample were found the results should be interpreted with due regard to the variable nature of the materials being considered. Twellty-four sandstone, limestone, and dolomite samplcs were analyzed. Their characteristics are given in Table 111. The photomicrographs of the exposed faces of six of these sample9 are given as Figure 3. Each of the sandstones. with the exception of the Spraberry sandstone, had a well-defined quartz grain structure. The Spraberry sandstone had very fine grains with a great deal of cementing material and had a more dense appearance than did the others. The fractured faces of the brown dolomite samples revealed an irregular crystalline structure as did the limeCYLINDRICAL CONSOLIDATED SAMPLE - LUCITE MOUNT O-RING GASKET
GAS
FLOW
Figure 2. Details of Sample Holder for Flow Measurements
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 10
ENGINEERING AND PROCESS DEVELOPMENT
c
Wilcox Sandstone
No. 1
Spraberry Sandstone No. I 2
Figure 3.
Bromide Sandstono No. 4
Bromide Sandstone No. 6
Brown Dolomite No. 20
Canyon Reef Limestone No. 24
Photomicrographs of Consolidated Samples ( X 5 )
stone. The limestone samplr, however, was ground flat for t h e photograph t o show the nature of the internal porosity more clearly. Samples from 0.65 t o 1.31 inches long and about 0.72 inch in diameter were used for the flow measurements. Somewhat longer samples were used for the other measurements of the properties of the samples, since the ends of the samples were fractured off after mounting in plastic prior t o the flow measure-
Table 111. F:
I
Sample Type
.
BromideQ
Burbanka Spraberry' Outcropa
Torpedo0 Brown dolomite
15.5 17.4 16.6 36.2 36.2 62.5 86.8 30.0 39.1 34.8 53.8 60.3 14.4 14.4 12.1
0.175 0.158 0.160 0.110 0.110 0.030 0.022 0.123 0.150 0.163 0.080 0.084 0.206 0.206 0.227
413.4 9.8
0.228 0.143 0.160 0.125 0.095 0.095 0.094 0.061 0.087
.. .. ..
64.7 70.2 68.5 142.5 123.0
CanyonReef limestone a Sandstone samples.
October 1953
Flow Data Correlated on Friction Factor Plot
Summarized Characteristics of the Consolidated Samples Studied X, K (Perme-
ResisFractivity tional Factor Porosity
Wilcox"
ments. The permeability of the samples varied from 0.01 t o 500 millidarcies. Porosities of from 2 t o 23% were found. Electrical resistivity factors of 12 t o 140 were observed. The pore size distribution of four types of samples was obtained for the larger pores and ranged from 6 X to 2 X foot. The flow measurements were made with ases having molecular weights of 4.0 (He), 16.04 (CH,), 28.2 (N$, and29.0 (air). This selection of gases provided viscosities ranging between about 0.01 and 0.02 cp. (4, 16, 19,21).
a,Feet-*
4.90 3.77 6.35 4.fi0 3.92 1.015 1.97 6.30 8.60 2.65 9.75 8.53 3.00 4.50 2.22
X 10'1 X 101' X 1011 X 10'8 X 10'8 X 10'6 X 10" X 1012 X10'2 X 10'2 X 101' X 1016
x
X X 1.850 X X 2.00 2.40 x 1.7.5 X 9.20 x 1.25 X 4.56 X 7.70 X 9.10 X
1012
10'2 10'1 101' 10'2
ioi* 1018 1018
1018 10"
1014
lot*
8 , Feet-1 8.23 3.09 9.95 2.80 1.99 9.00 1.96 4.40 1.31 3.47 2.00 1.74 7.10 8.20 5.90 4.80 6.25 1.55 3.22 7.25 8.24 2.68 1.033 9.05
X 10' X 107 X 10' X 109
X
lo9
X 10'2 X lo'* X 108 XlOg, X 108 X 10" X 10'8
x
X X X X
x
108
108 108 108 108 108
X 100
x
1010
X X X X
109 10" 1012 1010
ability), Millidarcies 192 250 148 2.0 2.4 0.093 0.048 15.0 11.0
36.5
0.097 0.011 31.3 20.9 424 508 47.0 39.2 5.4 1.02 7.5 0.207 0.122 1.03
DE Feet
&
0.5) LE/LB 4.60 X 10-6 1.64 5.42 X 10-5 1.66 4.08 X 10-1 1.63 7.10 X 10-8 1.99 7.68 X l o - * 1.99 1.36 1.98 X 10-0 1.68 X 10-0 1.38 1.74 X 10-5 1.92 1.70X10-' 2.41 2.90 X 10-6 2.38 1.88 X 10-8 2.07 6.71 X 10-7 2.25 1.75 x io-' 1.72 1.43 X 10-6 1.72 5.90 X 10-1 1.66 (]E1
6.80 X 3.99 X 10-s 10-0
1.74 2.66
6.7o'xio-e 1.90 X 10-8 3.10 X 10-6 3.44 X 10-8 9 . 3 0 X 10-6
2.48 2.58 2.64 2.69 3.26
.....
... ...
]E2
1.26 3.48 1.33 1.05 1.38 0.00512 0.053 1.95 0.901 1.62 0.119 0.00441 0.310 0.328 8.11 10.0 1.19
*..
o.'iii 0.390 0,0714 0.0677 0.176
INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 4 shows typical flow data for three Wilcox sandstones plotted in the form described by Equation 22 to fall on a straight line. T h e quantities and @ can be Obtained directly from this plot as its intercept and slope. Summarized values of a and p are given in Table 111. T h e low p e r m e a b i l i t y samples all showed deviations on this type of straight-line data plot a t low mean pressures because of molecular streaming. Figure 5 shows typical flow data on low permeability bromide sandstone for which molecular streaming was s factor. Molecular streaming, however,
2149
ENGINEERING AND PROCESS DEVELOPMENT 2.0x1012
'
'
'
'
NO 3. K
s
148 MILLIDARC+'
I
I 0
"
'
I , , 5,000 c+
,
0
Figure 4.
,. .
,
FT'I 10,000
Flow Data Plot to Obtain Sandstones
CY
I5,WO
and p-Wilcox
X = Air H = NB 0 = Chc A = He ,
80x1013
,
,
,
10-6 /
I0-4
I 0-5 DE- FEET
Figure 6. kr Correlation for Sandstones
Y
t:
0 = Wilcox Nos. 1-3
6.0~1013
= Bromide Nos. 4-8 W = Burbonk Nos. 9-10 Spraberry Nos. 11-12 Outcrop Nos. 13-1 4 3 Torpedo Nos. 15-1 6
u l &O
a
-I E+ -a*N
-
I N
%- y i
*'
.'
I I
0
,
,
,
.
5000
l0,OOO
15,000
are derived. Data taken a t low mean pressures affected by molecular streaming have been omitted from this plot.
Yp,Ff'
Figure 5.
Flow Data Plot to Obtain
CY
and @-Bromide
Sandstones X = Air
W =
NP A = He
was not under study in this u-ork. and its effects were avoided by extrapolating the data taken a t higher mean pressures. Similar flow data were taken on a total of 24 consolidated porous samples. The values of k2 given in Table 111 were computed from the experimentally determined values of C Y , p, F , and X and plotted versus DE with the type of material as a parameter as in Figures 6 and 7. These correlations of k 2 show some scattering of the data. This is undoubtedly due to the sampling and experimental errors and t o the arbitrary assumption of kl = 0.5. The average deviation of the values of kz for all of the samples excluding the bromide sandstones was 12.4%. The various bromide sandstones differed considerably in their characteristics as is apparent from the photomicrographs in Figure 3 and, consequently, a good correlation would not be expected for them. The conventional means of presenting fluid flow data is a dimensionless friction factor-Reynolds number plot. Such a plot is given a s Figure 8 for the flow data of this investigation and that of Green and Duwez (13) for porous metals. Three expressions are given for the friction factor and Reynolds number. The first forms are due to Green and Duwez and are based on CY and 8, the empirical flow constants. These would be used in correlating flow data. The second set of forms off and R e may be used if fluid flow data are not available and flow rates must be predicted. Knowledge of the permeability, porosity, and electrical resistivity factor with Figures 6 and 7 giving kz values is sufficient t o predict the entire range of viscous and turbulent flow. The h a 1 forms off and Re are the basic theoretical friction factor and Reynolds number from which the other two forms
2150
Relationship to Other Methods I s Shown
Previous applications of the Kozeny equation to flow problems have required the estimation of the increased length of the path through an irregular bed. The use of Equation 7 enables one to measure this property of the bed directly. Table I11 lists the
--,'I
'
" '
1
'"'1
Figure 7. kl Correlation for Limestones and Dolomites
0 = Brown dolomites Nos. 17, 20-22 0 = Canyon Reef limestone Nos. 23-24
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 10
ENGINEERING AND PROCESS DEVELOPMENT measured values of L E / L Bfor the samples studied. The LEILB ratios vary from 1.36 t o 3.26. Since this ratio is squared in the laminar flow equation, it is evident that it is of considerable importance in any fluid flow correlation Heretofore, the evaluation of the Kozeny constant has depended on an assumed value of LEILB. With a method of measuring LE:/LB directly now available it is possible to calculate the Kozeny constant accurately for any porous media whose permeability, porosity, surface area, and electrical resistivity factor are known. The theory developed here permits an analysis of the F j and F R terms ~ presented by Brownell, Dombrowski, and Dickey (6).
stant ( c ) is due to the fact that two different reference points on the friction factor plot have been used. Summary
Procedures for predicting and correlating turbulent flow through consolidated porous media are shown using the permeability, porosity, and electrical resistivity factor of the solid and the properties of the fluid. A method of evaluating the actual length of the path through a porous medium by means of an electrical resistivity measurement is shown. A procedure for evaluating the Kozeny constant more accurately than previously possible is shown. The relationship of this work to that of Brownwell, Rozeny, Green and Duwez, and others is given. Acknowledgment
The constant (cl) relates the particle diameter to the effective diameter of the flow path.
DE
i=
(~(DB)
This work was carried out under the Soeony-Vacuum fellowship in chemical engineering a t the University of Michigan. The samples studied and the pure methane were provided by the Phillips Petroleum Co.
(25)
These values of F j and Fxe should not be interpreted as varying with X-2 and X-1 when the porosity of a porous medium is altered, because actual changes in the porosity will also change the values of LE/’LBand kz in the usual case, thus complicating the nature of the porosity function. The presence of the con-
Nomenclature c, c1 = dimensionless proportionality factors
DE = effective diameter of pore structure, feet D, = particle diameter of unconsolidated particles, feet F = electrical resistivity factor: electrical resistance of
10,000
WlLCOX SANDSTONES, NO. 1-3 BROMIDE SANDSTONES, NO. 4 - 8 A BURBANK SANDSTONES, NO. 9 -10 SPRABERRY SANDSTONES, NO. I I- 12 0 OUTCROP SANDSTONES, NO. 13-14 @‘TORPEDO SANDSTONES, NO. 15-16 0 BROWN DOLOMITES, NO. 17-22 * CANYON REEF LIMESTONES, NO. 23-24 0 GREEN h DUWEZ POROUS METALS +
(u
boo0
11
-
m
.
“
100
5
ct e It -I-
IO
Figure 8,
October 1953
Friction Factor-Reynolds No. Plot
INDUSTRIAL AND ENGINEERING CHEMISTRY
2151
ENGINEERING AND PROCESS DEVELOPMENT sample saturated with brine/resistance of quantitg of brine of same size and shape as bulk volume of sample I;i = Brownell’s friction factor factor PR. = Brownell’s Reynolds nunihcr factor f = friction factor gc = conversion factor, 32.17 pounds mass X foot/pounds force X square second G, = superficial mass velocity, pounds mass/square foot x second 120 = Kozeny constant, dimensionless 121 = dimensionless, geometrical constant in laminar term of flow equation kz = dimensionless, geometrical constant in turbulence term of flow equation LB = over-all length of porous media, feet L E = effective length of flow path, feet m = hydraulic radius, feet M = molecular weight of a gas P or Pj = pressure, pounds force/square foot absolute R = gas constant: 1543 pound: force X cubic feet/squar& foot X pound mole R. T = absolute temperature, R. Y, = superficial velocity, feetjsecond V E = effective velocity in pore structures, feet/second X = fractional porosity z = compressibility factor of a gas a = coefficient of laminar term of flow equation, feet-’ P = coefficient of turbulence term of flow equation, feet-‘ 6 = dimensionless constant relating hydraulic radius and effective diameter E = roughness dimension of a pipe, feet p = viscosity, pounds mass/(foot)(second) p = fluid density, pounds mass/cubic feet
5
literature Cited
(4) Bicher, L. B., and Kats, D. L., Trans. A m . Inst. Mining M d , Rngrs., 155,246 (1944). (5) Brownell, L. E., Dombrowski, H. S., and Dickey, C. A., Chem. Eng.Progr., 46,415 (1950). (6) Brownell, L. E., and Kata, D. L.. Ibid.,43, 537 (1947). (7) Carlson, A. J., and Eastman, J. E., A m . Inst. il4ining M e t . Engrs., Tech. Publs., 1196 (,\lay1940). ( 8 ) Carman, P. C., Trans. I n s t . Chena. Enyrs. (London), 15, 150 (1937). (9) Ibid.,16, 168 (1938). (IO) Dallavalle, J. XI., “AIicromeritics,” Kew York, Pitman Publishing Co., 1943. (11) Ergun, S., Chem. Eng. Progr., 48, 89 (1952). (12) Fancher, G. H., and Len%, J. A., IXD.ENR. CHEM.,25, 1139 (1933). (13) Green, L., and Duwea, P., J . AppZ. N e c h o n i c s , 18, 39 (1951). (14) Grunberp, L., and Nissan, A. €I.,J . I n s t . Petroleum Technol., 29, 192 (1943). (15) Handbook of Chemist1.g arid Physics, 24th ed., Cleveland, Ohio, Chem. Rubber Publishing Co.. 1941. (16) International Critical Tables, Vol.-V, p. 2 , Sew I-ork, LicGIawHill Book Co., 1929. (17) Kozeny, J., Ber. Wien. Aknd., 136a. 271 f l 9 2 7 ) . (18) Martin, J. J., Ph.D. thesis, Carnegie Institute of Technulogy, 1948. (19) Nichels, A., and Gibson, R. O., Proc. Rou. Soc. (London), A134, 288 (1931). (20) Muskat, M., “Physical Principlesof Oil Pioduction,” Sew Yoi k, McGraw-Hill Book Co.. 1949. (21) Perry, J. H., “Chemical Engineers’ Handbook,” 3rd ed., Scw York, McGraw-HillBook Co., 1950. (22) Sullivan, R. R., and Hertel, K. I>., “Advances in Colloid Science,” Vol. I, New Pork, Interscience Publishers, 1942. (23) Wyllie, M. R. J., and Rose, W. D., J . PetroZeum Technol., 2, 105 (1950). (24) Wyllie, bl. R. J., and Spangler, 11. B., Gulf Research a n d De-
velopment Co., Research Project 4-G-l, Rept. 15 (i\Iaich
(1) Archie, G . E., Trans. Am. Inst. Mining M e t . Etagrs., 146, 54 (1942). (2) Bakhmeteff, B. A., and Feodoroff, N. V., Trans. A m . Soc. Mech. E’ngrs., 59, A-97 (1937).
19,1951). (25) Young, J. TV., and Pot, R., Petroleum Engr., 22B, 50 (.Janualy 1950).
(3) Barrer, R. E., “Diffusion in and through Solids,” London, Cambridge University Press, 1941.
RErEIvm) for review June 30, 19.52.
ACCEPTED August 8 , 1953.
Radiation-Conduction Correction for eratwre Measurements in Mot Gases W. E. WEST, JR.,
AND
J. W. WESTWATER
University o f Illinois, Urbana, 111.
A
C C C R h T E measurement of the temperature of a hot g a ~i q a problem beset with difficulties. Consider the common case of a hot gas flowing through a duct. A number of possible temperature measuring devices could be used, but objections ran be cited for each. If some sort of radiation meter is sighted through a peep-lioic, it “sees” the duct walls as well as the gas. The durt nalls cannot be a t the gas temperature unless the outside of the duct is provided with perfect insulation. Usually the duct is losing heat to the surroundings, and the duct temperature and gas temperature are different. Thus the pyrometer may give readings that are in error, possibly by several hundred degrees (6). One radiation scheme does give good accuracy; this is the line-reversal method which consists essentially of measuring the radiation emitted by a salt suspcnded in the hot gas. This method is suitable for flames only and is so inconvenient to use that it is rarely seen in industry. If a thermometer or similar probe device is inserted into the gas stream, it will have a radiation error, because the probe can see
2152
the cool duct ~aalls. Thtw \\ill I N > :i contluction error also (tlie so-ralled fin effect) cawed i)y the, flair of heat along the probe to the duct u-all. Two methods of attacking the p i oblem are possible. Onr consists of trying different design&of temperature measuring ilc~vices until one is found that produce- a dirert reading of the true gas temperature. A large body of litciature deals Hith thii appioach. Some of the methods use shiolci
