March 1953
INDUSTRIAL AND ENGINEERING CHEMISTRY
This means that a product containing a valuable constituent is to be concentrated from 80 to 99.9% purity, but that a loss of 25% of the valuable component is accepted. I n this case
M + N 5.5
Exact Equation 1 1 Equation 17
6.1) 5.2
The ( M - N ) values were well represented by Equation 1 2 over the entire range of variables. For high Rz and low R, this equation simplifies to Equation 18. The x values were adequately represented by Equation 13 over the entire range shown in Figure 3.
R
= amount of a certain component in product A divided by
S
= solvent ratio (throughput of solvent A divided by through-
amount in product B
put of solvent B ) z = reduced extraction factor Z A , XB, X F = mole, weight, or volume fraction of component 1 in product A , product B, and feed, respectively K = ratio of partition coefficients, -* p Ki
>> = - electric heaters which were located on the exterior of a section of the conduit, IC, of Figure I and upstream from the c,alibrated orifice meter, I,. This orifice meter was used to measure the rate of flow of the natural gas.
Folsom and Ferguson ( 8 ) investigated the mixing of liquids in jets and compared their effectiveness with propeller-type mixers. Corrsin and Uberoi (6) studied the momentum and thermal transfer associated with a heated jet of air and made a number of contributions to the analysis of such phenomena. The turbulent mixing of two-dimensional streams in supersonic Row has been reported by Bershader and Pai (2). Forstall and Shapiro (9) . . reviewed the literature associated with jets and presented some experimental data concerning the material and momentum transfers in coaxial streams. This review of the applicable literature was sufficiently complete to render further consideratioii in this paper unnecessary. I ' e As a supplement t o the work discussed and as a part of an investigaSchematic Arrangement of Working Section Figure 3. tion of combustion of natural gas and air, an experimental study was The differential pressure at the orifice meter was det'ermined by made of macroscopic diffusion of these components under turbumeans of a kerosene-in-glass manometer used in conjunction with lent flow conditions. a cathetometer. The Row rate of the natural gas was controlled to within 0.5y0 of a predetermined value by a manually operated Equipment throttle valve in the supply line, The equipment for the control The equipment employed in this study provided a horizont'al of temperaturepermitted both the and natura] gas to be annular air stream of circular section into which a coaxial stream maintained within o.lo F. of a predetermined value. ~h~ rate of natural gas was admitted. Provisions were made to deterof flow of air was controlled to u.ithin 0,5% of the desired value mine the velocity and composition of the stream as functions of by varying the speed of the motor driving the centrifugal blower, radius a t nine stations downstream from thepoint of initial mixing, A , in Figure 1. Sfter the fluid passed through the working section, J , the gas was ignited outyide the building a t the burner, A schematic diagram of the arrangement of the apparatus is M. presented in Figure 1. A centrifugal blower, A , drew air through
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1953
The arrangement of the working section is shown in Figure 3. Nine sampling ports were located a t distances varying from 3 to 100 inches downstream from the point of initial mixing. These ports were provided with removable plugs which were flush with the interior surface of the copper working section. Each of these plugs could be removed and replaced by a pitot tube assembly. The pitot tube was arranged so as to permit deter-
659
in-glass manometer. After cooling t o room temperature, the change in weight of the bulb was determined on an analytical balance with an uncertainty of 0.3 mg. Over the range of composition of interest in this study the weights of the gas samples in the bulbs a t nearly atmospherio pressure varied from 400 to 600 mg. The specific weight of the samples was determined with an uncertainty of approximately 0.3%. I n establishing the composition from the specific weight determinations, the deviation of the natural gas-air mixtures from ideal solutions (IS)was not significant. The influence of humidity of the air upon the specific weight of the mixtures was taken into account.
Procedure and Results The composition distribution and the flow pattern were studied a t gross velocities of 2 5 , 50, and 100 feet per second. The streams of air and natural gas were maintained a t the same gross velocity and temperature. The velocity a t each of the nine stations was determined a t intervals of 0.1 inch across the working section,
BEL
Figure
4.
Variation of Velocity with Radial Distance from A x i s of Working Section
mination of the impact pressure as a function of radius a t each station. This pressure was measured with a kerosene-in-glass manometer used in conjunction with a cathetometer or with a micromancimeter ( 4 ) . The latter instrument was used when the pressure differential was less than 2 . 5 inches of kerosene. The pitot tube was also employed to withdraw samples from the working section. The composition of the air-natural gas mixture was established from measurements of the specific weight of the sample withdrawn. The specific weights were determined by me of weighing bulb techniques. The samples were withdrawn a t such a rate that the velocity of the gas entering the pitot tube was equal t o that of the flowing stream. The glass weighing bulbs used in the specific weight measurements were immersed in an agitated air bath while the sample was introduced. The temperature of the air bath was controlled within 0.1' a t 100' F. The pressure in the weighing bulbs was measured relative to that of the atmosphere by a kerosene-
P a2
BELOW t -R A D I A L
Figure 6.
WSTANCE
FROM
AXIS
INCHES
-
ABoVL
Variation of Composition with Radial Distance from A x i s
Specific weight determinations were made upon samples taken
at the same radial positions a t which the velocity was measured when the gas stream contained more than 1 mole per cent natural gas. Representative velocity and composition profiles are presented in Figures 4, 5 , and 6, and the detailed experimental data are available (16). The marked asymmetry resulting from horizontal flow in a gravitational field is evident from Figure 6 which has been presented for a gross velocity of 25 feet per second in order to show the effectin greater detail.
Eddy Viscosity and Eddy Diffusivity The determination of the eddy diffusivity from the experimental data which have been obtained requires the simultaneous solution of the following two partial differential equations:
I/>
1
a" I X 4L
DLTAN'E
F R O N P O N i OF
I
I eo
I
*D
Figure 5.
j
08 ~
CAS
hJLCTON
80 INCHES
Variation of Composition with A x i a l Distance from Point of Initial M i x i n g
(5)
Equation 4 results from a combination of Equation 1 with a material balance for one of the components in the mixing region, Equation 5 is a n expression of continuity in steady flow. For
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
660
Vol. 45, No. 3
the case in which the flow is symmetrical about the axis of the flow channel, Equations 4 and 5 may be combined to yield the following expression: I
This equation, which neglects diffusion in the direction of g i o d ~ motion as represented by $ d . # , may be solved by appropriate graphical integrations and differentiations. An approximate analytical solution of Equation G inag be obtained on the basis of a number of bimplifying assumptions which in some cases differ materially from the actual behavior. The required assumptions may be stated in the following w a ~ :
% ''
SWAQE
FEET X 1 0 3
Figure 8. Comparison of Experimental and Predicted Compositions at Gross Velocity of 5 0 Feet per Second
In this solution the boundary conditions are described in the following u'ay: U.her1
when ?J
=
0,
"" = 1
for (0
5T5
b)
no = 0
for ( b
5r 5
a)
0.8
$
a
9
0.6
E2
0.4
I e
The first equality of Equation 8 in dependent upon perfect gas and ideal solution behavior for a const'ant temperature and pressure. The solutions to Lquation 8 that were obtained by taking a = 0.159 foot, and b
=
0.0392 foot, and given values of
? L!'
U
are
shown hy the solid lines in Figulos 7, 8, and 9. In these figures
w
nois plotted as a function o* f'?! c"
d
The nitwmwnents which were niadc at each of the three grow velocities were employed as iiidicated below in conjunction with
02 ~
0.8
0
0
ID
0.25
0.50
&Y
0.75
1.03
1.25
SOVARE FEET X I 0 2
Figure 7. Comparison of Experimental and Predicted Compositions a t Gross Velocity of 100 Feet per Second
08 v)
4 i
It is implicit in the above that the specific weight must be constant in order that the radial velocity component be zero. such circumstances, Equation 4 reduces to
Under
CL 3
s
06
z
e 0
04
Ct L
(7)
w 0
=
02
The following solution of this partial differential equation niay be obtained (5)if
u,. - 1s treated as constant:
$d,v
Y +c
SQUARE
FEET x i 0 3
Figure 9. Comparison of Experimental and Predicted Compositions at Gross Velocity of 25 Feet per Second
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1953
Equation 8 to obtain average values of the eddy diffusivity .throughout the working section of the duct. I n Figure 7 the experimental values of the composition which were determined at a gross velocity of 100 feet per second are superimposkd upon the solution of Equation 8. By an iterative procedure it was found that the best agreement between this equation and the experimental values was obtained by taking $d,v
3 equal to
w
*
1.55 X
feet, which corresponds to an average
value of f d in the mixing region of approximately 0.016 square foot per second, The points shown in the figure are the average of experimental values obtained above and below the center line of the axis of flow. Similar comparisons were made for each of the lower velocities. In order to compensate the effect of gravity on the flowing stream, the axis of symmetry was chosen to correspond to the locus of maximum gas concentration. Figures 8 and 9 show the experimental results superimposed upon a solution of Equation 8. In these instances it was necessary to employ interpolated data in order to average values obtained upon either side of the axis of maximum concentration of natural gas. The closest agreement between the solution of Equation 8 and experiment was obtained with values of
9 of 1.9 X lo-* and 2.3
X
feet a t gross
velocities of 50 and 25 feet pc~rsecond, respectively. The agreement between Equation 8 and the experimental data under these conditions is depicted in Figures 8 and 9.
661
the preparation of these data are acknowledged. reviewed the manuscript.
W. N. Lacey
Nomenclature
a
=
radius of working section, 0.159 f t . b = radius of natural gas conduit, 0.0392 ft. C = concentration, pounds of component/cubic foot D = molecular diffusion coefficient, square feet/second d = differential operator e = base of natural logarithms g = acceleration of gravity, feet/square second J = Bessel function of the first kind 6’1= , weight rate of diffusion per unit area, pound/sscond/square foot n = mole fraction r = radial distance from axis of working section, feet
U
= gross velocity
u y
= point velocity, feet/second = axial distance in working section from point of initial mix-
($ f
= roots of the equation
a = €d
‘d
e, Em
feet/second
ing, feet
sc = total Schmidt number
Bi
TU&),
5 Ed
J1(flia)= 0
partial differential operator eddy diffusivity, square feet/second
= total diffusivity, square feet/second = eddy viscosity, square feet/second = total viscosity, square feet/second
v u 8
= kinematic viscosity, square feet/second = specific weight, pounds/cubic foot = angular displacement about axis of working section
ity or Reynolds number as is shown in Figure 10. The varia-
7
=
U with gross velocity is similal to that of the analogous ratio for uniform flow. All values of total
Subscripts g refers to r refers to y refers to 8 refers to 0 refers to 1 refers to
The data indicate a small variation of ?Lr with the gross vdoc-
U
tion in the quantity
$
shear stress/unit area parallel to flow, pounds/square foot natural gas radial direction axial direction angular direction Bessel function of zero order Bessel function of first order
literature Cited
*
2.2 5
I
0 X
$
2.00
W (r
1.75
GROSS
VELOCITY
FEET P E R
SECOND
Figure 10. Effect of Velocity upon Total Diffusivity
diffusivity considered are based upon the assumption that they are independent of the radial position in the channel. The flow in the mixing section was too complex to permit an accurate evaluation of the point values of the eddy transport properties.
Acknowledgment This experimental work was made possible by the financial support of the United States Air Force through the Jet Propulsion Laboratory, Pasadena, Calif. The assistance of Theodore Wadsley and Forrest Gilmore in the accumulation of the experimental data and the aid of Robert Sears and David Wilford in
( 1 ) Bahkmeteff, B. A., “Mechanics of Turbulent Flow,” Princeton. Princeton University Press, 1941. (2) Bershader, D., and Pai, S. I . , J . AppE. Phys., 17, 399 (1950). (3) Carslaw, H. S., “Mathematical Theory of the Conduation of Heat in S o l i d s , ” ~118, . New York, Dover Publication, 1945. (4) Corcoran, W. H., Page, F., Jr., Schlinger, W. G., and Sage, R . H., IND. ENG.CHEM.,44, 410 (19.52). (5) Corrsin, S., J . Aeronauf. Sci., 18, 417-23 (1951). (6) Corrsin, S., and Uberoi, M. S., Natl. Advisory Comm. Aeronaut., Tech. Note 1865 (1949). ( 7 ) Dryden, H. L., IND.ENG.CHEM.,31, 416 (1939). (8) Folsom, R. G., and Ferguson, C. K., Trans. Am. Soc. Mcch. Engrs., 7 1 , 7 3 (1949). (9) Forstall, W., Jr., and Shapiro, A. H., J. Appl. Mechanics, 17, 399 (1950). (10) Kalinske, A. A., and Pien, C. L., Ibid., 36, 220 (1944). (11) K&rm&n,Th. von, J . Roy. Aeronaut. Soc., 41, 1109 (1937). (12) K&rmBn,Th. yon, Trans. Am. Soc. Mech. Engrs., 6 1 , 7 0 5 (1939). (13) Lewis, G. N., Proc. Am. Acad. Arts. Sei., 43, 259 (1907). (14) Liepmann, H. W., and Laufer, J., Natl. Advisory Comm. Akeronaut., Tech. Pub. 1257 (1947). (15) Murphree, E. V., IND.ENG.CHEM.,24, 727 (1932). (16) Schlinger, W. G., and Sage, B. H., American Documentation Institute, Washington, D. C., Doc. 3263 (1951). (17) Sherwood, T. K., IND. ENG.CHEM.,42, 2077 (1950). (18) Sherwood, T. K., and Woertz, 33. B., Trans. A m . Inst. Chem. Engrs., 35, 517 (1939). (19) Towle, W. L., and Sherwood, T. K., IND. ENG.CREM.,31, 457 (1939). (20) Willis, J. B., Australian Council Aeronaut., Rept. ACA-19 (Oct. 19, 1945). RECEIVED for review March 5, 1951. ACCEPTED October 30, 1962. For detailed tables supplementary to this article order Document 3263 from the American Documentation Institute, Library of Congress,Washington 25, D. C., remitting $1.00 for microfilm which yields images 1 inch high on standard 35-mm. motion picture film or $1.00 for photocopies 6 by 8 inches which are readable without optical aid.
