literature Cited ( 1 ) Bartlett, P. D., Dauben, H. J., J . A m . Chem. SOG.62, 1344 ( 1940). (2) Bennett, G. M., J . Chem. SOG.(London) 107, 357-8 (1915). ( 3 ) Bogomolov, N. A., Stepanenko, N. N., Z h . Fir. Khim. 26, 1664 (1952). ( 4 j Doucet, Y., Calmes-Perrault, F., Durand, M., Compt. Rend. 260(7), 1878-81 (1965). ( 5 ) Dreizler, H., Dendl, G., Z . Naturforsch. 19a, 512-14 (1964). ( 6 ) Goldschmidt, H., Udby, O., Z. Physik. Chem. 60,728 (1907). ( 7 ) Hartman, R. J., Borders, A. M., J . A m . Chem. Soc. 59, 2107-12 11937). ( 8 ) Hartman, R. J., Gassmann, A. G., Ibid., 62, 1559-60 (1940). ( 9 ) Hartman, R. J., Hoogsteen, H. M., Moede, J. A., Zbid., 66,1714-18 (1944). ( 10) Levanspiel, Octave, “Chemical Reaction Engineering,” p. 23, Wiley, New York, 1962. \ - - - - , -
\ - - -
I -
( 1 1 ) Zbid., p. 51. (12) Leyes, C. E., Othmer, D. F., Znd. Eng. Chem. 37, 968-77 (1945). (13) Rolfe, A. C., Hinshelwood, C. N., Trans. Faraday Soc. 30, 935-44 (1934). (14) Smith, H. A.,J . A m . Chem.Soc. 61,254-60 (1939). (15) Zbid., 62, 1136-40 (1940). (16) Smith, H. A., Burn, James, Zbid., 66, 1494-97 (1944). (17) Smith, H. A., Levenson, H. S., Zbid., 62, 2733-5 (1940). (18) Suter, C. M., Oberg, Elmer, Zbid., 56, 677-9 (1934). 119) Weast. R. C.. ed.. “Handbook of Chemistrv and Phvsics.” ’
46th ed.,’ p. E-50, Chemical Rubber Co., Cieveland, ’Ohib,
1965. (20) Williamson, A. T., Hinshelwood, C. N., Trans. Faraday Soc. 30,1145-9 (1934).
RECEIVED for review May 23, 1966 ACCEPTEDFebruary 4, 1967
MOMENTUM TRANSFER STUDIES IN EJECTORS Correlationsfor Single-Phase and Two-Phase @stems G.
s.
D A V I E S , ~A . K . M I T R A , A N D A . N . R O Y
Department of Chemical Engineering, Indian Institute of Technology, Kharagpur, India
The performance of an ejector in single-phase (air-air) and two-phase (air-liquid) systems has been studied with air as the motive fluid, and air and various liquids as the entrained fluid. Data for air-air system have been analyzed, using energy and momentum equations; the values of mass entrained calculated from the theoretical expression agreed with experimental results only in the limited range of small area ratios. Correlations based on the method of dimensional analysis have been developed for both air-air and air-liquid systems relating the mass ratio of entrained fluid to motive fluid in terms of Reynolds number of motive fluid, geometry of the ejector, and the physical properties of the fluid system. HE mechanism of jet flow has largely been exploited in Tejectors or jet pumps in which the momentum and kinetic energy of a high velocity fluid stream are used to entrain and pump a second fluid stream. Ejectors with steam o r compressed air as the motive fluid have found application in industrial operations for the creation of vacuum, exhausting corrosive fumes, pneumatic conveyor feeding, etc. Its incorporation especially in slurry-type chemical reactors holds considerable promise. By this technique the kinetic energy of the reactant motive gas can be utilized to maintain the solid catalyst particles in suspension, cause intense mixing between gas and fluid, and circulate the catalyst slurry through an external side tube, thereby obviating the difficulties normally encountered in conventional mechanically stirred reactors. Satisfactory operation of such a type of reactor employing an ejector as a pump for hydrocarbon synthesis has been reported ( 9 ) . The present investigation on the momentum transfer in ejectors has been carried out to obtain necessary data and correlation, with the ultimate object of designing and incorporating ejectors in slurry-type chemical reactors and allied process operations. In a multiphase system consisting of gas, liquid, and solid a large number of variables are involved which usually give rise
Present address, Indian Institute of Technology, Madras, India.
to problems of great complexity. Apart from the importance of fluid physical properties and solid particle characteristics, there are interrelated problems such as solubility, holdup, and slip. Hence, it is considered logical to carry out this investigation on momentum transfer in stages-first, using a singlephase system, where the motive and entrained fluids are gases, then a two-phase system where the motive fluid is gas and the entrained fluid is liquid, and finally a three-phase system where the motive fluid is gas and the entrained fluid is solid-liquid slurry. Studies carried out in single-phase systems with air as the motive and entrained fluid in the motive pressure range of 25 to 100 p.s.i.g. have been reported (8). This paper presents studies carried out in a single-phase (air-air) system in the low motive pressure range and in twophase (air-liquid) systems. Single-phase (Air-Air) System
The many analyses (7,4-7, 76, 78) that have been attempted in the design of ejectors, notably those of Keenan, Neumann, and Lustwerk (5, 6), Kastner and Spooner ( 4 ) , Smith (76), and Van der Lingen (78), have all been made using the equations of continuity, momentum, energy, and state. Most of them deal with constant area and constant pressure mixing. Nearly all the experimental investigations reported in the literature on the performance of ejectors relate to relatively high motive pressure, the only exception being the work of Kastner and Spooner. VOL. 6
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293
Design Consideration
The important design factors in the construction of an ejector system-the diffuser entrance, diffuser throat (mixing length) and diffuser outlet (tail piece), type of nozzle-have been discussed (8). The major dimensions are given in Tables I and 11. Other important variables that affect the performance are area ratio, projection ratio, pressure ratio, and mass ratio. Experimental Apparatus. The flow diagram of the experimental setup is shown in Figure 1. The ejector assembly is comprised of a spherical duct (2.5-inch diameter), a parallel mixing throat (0.5-inch diameter and 3.5-inch length), with a well-rounded entrance and 6.25-inch long divergent tail piece having a 10 O divergence angle (Figure 2). Procedure. Initially experiments were performed to determine the optimum projection ratio, which is defined as the ratio of the distance between the forcing nozzle and the commencement of the parallel throat of the diffuser to the diameter of diffuser throat, and a constant value of 1.9 was obtained for all the nozzles in the motive pressure range of 5 to 21 p.s.i.g. The nozzle was then fixed at the position of optimum projection ratio and purified compressed air a t the desired pressure was forced through the nozzle, which entrains air through side tube A . The flow rate of the motive air was metered by the rotameter, R, and of mixed fluid by the orifice meter, 0. The experimental data obtained for different nozzles a t motive pressures varying from 5 to 21 p.s.i.g. under steady conditions are given in Table 111. The critical area ratio a t which mass ratio is maximum is 197, which corresponds to a nozzle diameter of 0.90 mm. Data Analysis. T h e theoretical analysis of the present data was made by the method proposed by Smith (76). Based on energy and momentum equations, the final expression for the calculation of mass-entrained M e is given by
Table 1. Dimensions of Ejector Diameter of diffuser throat Length of diffuser throat Length of diffuser outlet Distance of nozzle outlet from diffuser throat Convergence angle of diffuser entrance Divergence angle of diffuser Diffuser exit diameter Table II. Nozzle No.
1 2 4 5
179 .O 91 .O 46.2 22.6
= 0
p3
=
- Po
1 m
For the calculation of M e , 0,and loss due to wall friction, F,, must be known. The value of fl has been calculated from experimental data (Table 111), by choosing a mean value of F, (76),defined as the loss of pressure due to wall fraction, equal to 0.60. The value of M e was calculated for the different nozzle area ratios using Equation 1 and compared with M
I
LEGENDS
COMPRESSOR
PRESSURE
ORIFICE
294
Apparatus for study of performance of ejector air-air system
I&EC PROCESS DESIGN A N D DEVELOPMENT
E
l
A
t
GAUGE
METER
C O NNEC T I N G
1 Figure 1.
(1)
where
I1
I1
loo 3 0
Dimensions of Nozzle Diameter, Area Ratio, d,, Mm. AR 247 .O 0.808
0.949 1.332 1.870 2.676
3
D
7 0 12.7 D 1.9 D Well rounded
PIECE
I
obtained experimentally (Figure 3). For small area ratios, the experimental M e agreed well with the calculated Me, but for higher area ratios the deviation was between 20 and 5070 of the experimental value. Thus this procedure based on energy and momentum equations applies in the limited range of small area ratios. Correlation of Mass Ratio with Other Variables Based on Dimensional Analysis. Dimensional analysis of the present system has been made to correlate the mass of entrained secondary fluid per unit time with other variables. This analysis involves the assumption of constant area mixing of two fluids, complete mixing within the parallel section of the diffuser, steady flow in the system, and negligible pressure loss a t the entry of the secondary fluid. On the basis of the above assumptions, mass of the secondary fluid entrained per unit time will evidently be influenced by the physical properties of motive fluid (air), p f and p f ; velocity of the motive fluid a t the nozzle tip, u1; dimensions of the ejector, d,, D,L’, and L ” , on one hand and on the other by the physical properties of the secondary fluid, p L , p L , and u L ; the diameter and the velocity of the secondary fluid a t the secondary entrance to the ejector, d’ and U L ,respectively. Therefore, if a theoretical equation exists for this problem, the mass entrained may be written in the following two general forms: Me = + [ P I , fir, d n , D,L’, L”1 (3) Me = $ ’ [ P L , P L ) UL, UL,d’, 91 (4) Applying dimensional analysis and solving in the usual way, the equation obtained from Equation 3 is
and from Equation 4 is Figure 2.
Nozzle No. 1
3
Nozzle Diameter, d,,, M m . 0.808
1.332
Area-Ratio, AR 247
91
Table 111. Typical Motive Motive Entrained Fluid, M , , Fluid ( M e f Lb./Hr. M,,,), Lb./Hr. 1.069 1.489 1.810 2.134 2.502
3.098 4.137 4.960 . . . ..
4
1.870
46
5
2.676
22.6
5.711 6,372 5.481 7.326 8 , 594 9.963 11.29 8.314 11.22 13.92 16.80 19.18
13.89 17.93 20.51 23.34 25.93 21.70 27.71 32.03 35.50 38.71 31.74 40.49 46.84 51.81 56.16 39.09 52.24 60.34 65.37 75.55 45.26 61.02 74.35 86.24 96.16
Experimental Results Motive Suction Pressure, Pressure,
fie,
P.S.I.G. 5 9 13 17 21
5 9 13 17 21 5 9 13 17 21 5 9 13 17 21 5 9 13 17 21
Po, Cm. of HzO
0.20 0.45 0.70 0.95 1 .oo 0.375 0.575 0.85 1.oo 1.15 0.80 1.30 1.55 1.65 2.45 0.90 1.60 2.30 3.10 3.90 0.975 2.10 3.20 4.45 5.40
Discharge Pressure Gage, Pa, Cm. of HzO 1.oo 1 .80 2.55 3.35 4.05 1.275 1.90 2.875 3.60 4.25 2.225 3.85 5.40 7.30 8.80 4.2 7.3 9.2 13.6 17.2 5.70 10.35 15.55 21.25 26.60
Ejector
P 0,003722 0.003841 0.003864 0.004535 0.004794 0,004963 0.004224 0.004429 0.005027 0.005126 0.006282 0.008791 0.008263 0.009782 0.010680 0.015820 0.015190 0.013670 0.018250 0.020030 0.020710 0.021250 0.022290 0.02809 0.03038
11.99 11.04 10.33 9.94 9.36 12.36 11.60 10.94 10.52 9.83 9.24 8.79 8.44 8.07 7.82 6.13 6.13 6.02 5.95 5.69 4.49 4.38 4.34 4.13 4.02 ~
VOL. 6
NO, 3 J U L Y 1 9 6 7
~~
295
I n the case of a gas-gas system, the effects of u and g are negligible and upon analyzing dimensionally the following relation is obtained
..
i
80
2j
c \
a
60
where d'Uopo/po is the Reynolds number (Re) of the secondary fluid at the entrance of the ejector. Since the Re of the secondary fluid is dependent upon the Re of the motive fluid and a linear relation in a log-log plot for different area ratio holds, Equation 10 simplifies to
tJ
a
9q
40
U
r" 20
"
20
I
I
I
40
60
eo
(EXPERIMENTAL
)
I
Correlation. For the evaluation of the Reynolds number of the motive fluid, U,, the velocity just after expansion from the nozzle is obtained on the basis of the isentropic expansion of the fluid as given by the equation:
100
LBS./HR.
Figure 3. Comparison of experimental with calculated values 5 4
and p I is calculated on the basis of the volume, after adiabatic expansion
3 2
pcucy = P l V l Y
4
c
.
x
2
The correlation obtained from the experimental data up to the value of the critical area ratio is given by
IO
:
5u '6 a a
5 4
z
3
