18) Kramers. H.. Alberda. G.. Chem. Enp. Sci.2. 173 (1953) (9j Laan, E.’T. ;an der, Zbid.; 7, 187-(1?58). (10) Latinen, G. A., Stockton, F. D., A.1.Ch.E. Natl. Meeting,
Subscripts
B C
i
= = = = = =
k
=
L
= =
D F
i
0
% X
J
u
0
= = = = = =
I
back-flow model value continuous phase value diffusion model value feed-end value phase i stagej index in summation outlet-end value over-all plates or stages; particle (in d,) total; two-phase phase X phase Y transient behavior value single-stage value
Superscripts
* 1
= equilibrium value = feed value
literature Cited (1) Aris, R., Amundsen, N. R., A.I.Ch.E. J . 3, 280 (1957). (2) Colburn. A. P., Ind. Eng. Chem. 28, 526 (1936). (3) Danckwerts, P. V., Chem. Eng. Sci.2, 1 (1953). Nagata, S., Chem. Eng. (Japan) 23, 146 (1959). (4) Eguchi, W., (5) Einstein, H. A., Dissertation, Eidgenossische Technische Hochschule, Zurich, Switzerland, 1937. (6) Fan, L. T., Bailie, R. C., Chem. Eng. Sci. 13, 63 (1960). (7) Hill. F. B., Brookhaven National Laboratory, unpublished work. 1957.
\
,
St. Paul, Minn., September 1959. (11) Levenspiel, O., 2nd. Eng. Chem. 51, 1431 (1959). (12) McHenry, K. LV., Jr., Wilhelm, R. H., A.I.Ch.E.J. 3, 83 (1957). (13) McMullen, .A. K., Miyauchi, T., Vermeulen: T., U. S. At. Energy Comm. Rept. UCRL-3911-Suppl., 1958. (14) Madden. A. J., Darnerell, G. L.. A.I.Ch.E.J. 8 , 233 (1962). (15) Mattern. R. V., Bilous? O.? Piret, E. L., Ibid., 3, 497 (1937). (16) Miyauchi, T.: D. Eng. Dissertation, Univ. of Tokyo, Tokyo. Japan, 1956. (17) Miyauchi, T., Vermeulen, T., 2nd. Eng. Chtm. Fundamentals 2, 113 (1963). (18) Oya, H., M.S. Thesis in Chem. Eng., Univ. of Tokyo, Tokyo. Japan, 1960. (19) Rifai, M. N. E., “Dispersion Phenomena in Laminar Flow through Porous Media,” Ph.D. Dissertation, Univ. of California, Berkelev. Calif.. 1956. (20) Sege. G., TVoodfield, F. LV., Chem. Eng. Progr. Symp. Ser. No. 13, 50, 39 (1954). (21) Sherwood, T. K., Jenny, F. J., Ind. En?. Chem. 27,265 (1935). (22) Sleicher, C. A , , Jr..A.I.Ch.E.J. 5 , 145 (1959). (23) Ibid.. 6, 529 (1960). 124) Vandenem. J. H.. U. S. At. Enerw y , Comm. Reut. UCRL‘ 8733. 1960. ’ (25) Vkrrneulen, T.. IVilliams, G. M., Langlois, G. E., Chem. Eng. P r o p . 51, 85 (1955). (26) Yagi, S., Miyauchi, T., Chem. Eng. (Japan) 19, 507 (1955). RECEIVED for review November 12, 1959 RESUBMITTED March 1, 1963 ACCEPTEDJune 28, 1963
AXIAL DISPERSION IN A SPRAY-TYPE EXTRACTION TOWER D. E. HAZLEBECK A N D C. J . C E A N K O P L I S The Ohio State University, Columbus 70, Ohio
Axial dispersion coefficients were obtained in a spray-type liquid-liquid extraction tower with water as the continuous and methyl isobutyl ketone as the discontinuous phase. The dispersion coefficient, DL,was obtained b y introducing a step function of KCI solution into the continuous phase inlet nozzle. The breakthrough or F curve was measured a t the outlet point of the continuous phase b y an electrical conductivity and probe. DL values varied directly as the velocity of the continuous phase to the 0.45 power (Uw0.45) were much greater than those found for packed beds of spheres. For packed beds, D L varied approximately as Uw~’.O. The Peclet number varied from 0.008 to 0.023, and these values were abcut 1 /10 as large as those for packed beds. Brutvan using glass beads as the discontinuous phase in a spray tower obtained DL values approximately twice those of the present work. DL values calculated using Taylor’s equation for laminar flow in an open tube were approximately 10 times the DL values for the continuous phase in this investigation.
liquid-liquid extraction spray towers have by many investigators and experimental mass transfer coefficients obtained. An important factor in understanding the fundamental mass transfer mechanisms in such a tower is the effect of axial dispersion o r the residencetime-distribution of the continuous phase. Several mathematical methods have been derived by Sleicher (78), Epstein ( 6 ) , and Miyauchi (76) for the effect of axial dispersion on the mass transfer coefficients and extraction efficiency in extraction towers and packed beds. In these final equations the experimental value of the axial dispersion coefficient, D,, is needed for the continuous phase. I t is often assumed that the axial dispersion in the discontinuous phase of a n extraction tower is negligible. This assumption of plug flow is more reasonable if the drops of the discontinuous phase are uniform in size and do not collide, and the continuous OUNTERCURRENT
C been studied
310
I&EC FUNDAMENTALS
phase velocity profile is flat. No experimental data are available o n the axial dispersion in the discontinuous phase. Many workers (7, 77, 73, 74, 20) have measured so-called “end effects” in spray-type extraction towers, and postulated that these were caused by recirculation of the continuous phase. Brutvan ( 7 ) measured the D, of the continuous phase in a spray tower. H e used glass beads as the discontinuous phase by dropping them into the continuous water phase. Increasing the dispersed phase flow rate increased D ,slightly. His values for Peclet number were considerably lower than those found for packed beds ( 2 , 3, 5, 75). H e measured D, only in a short center portion of the column and not over the entire continuous phase from inlet to outlet. I n the present work, D, values were obtained in a spray-type tower for the continuous phase water. The water flowed downward and the dispersed or discontinuous phase of methyl isobu-
tyl ketone flowed upward. The water and ketone rates were varied. D, was obtained by introducing a step function of dilute KC1 into the continuous phase inlet nozzle. The concentration a t the outlet of the continuous water phase was measured by a n electrical conductivity probe, and the breakthrough or F curve was determined. The D, data obtained were correlated and compared to those of Brutvan ( 7 ) and others for packed beds (2, 3, 5, 75,79). Theory
Two mathematical models have generally been used to calculate D ,from experimental data obtained by introducing a step concentration function into a flowing liquid: the diffusional and Einstein statistical models. The basic equations for the diffusional model have been derived by Danckwerts ( 4 ) . His starting point is the following basic differential equation where the longitudinal dispersion coefficient. D,,uniquely characterizes the mixing process:
I t is assumed that the phenomena of axial mixing can be described by the same equations as those for molecular diffusion (4. 72) and that this mixing is proportional to the concentration gradient with D,>constant. The main experimental limitation is that the bed length must be long enough for the diffusion assumptions to be true (2). This model assumes a uniform radial concentration in the continuous phase which has been shobvn to be approximately true (2, 20). Making the substitution, Z = X - U@,the boundary conditions are
co c(o,e)= c0/2 e 2 o C(2,O) =
(2) (3)
Equation 3 states that ' j 2the tracer material is ahead of the point (Lle)/tV = 1.O and '/z behind. Equation 1 was solved for the case where the step function was generated by shutting off the tracer flow.
c
-
co
=
a
[I
(+)I
+ erf x- Le 2dDdi
(4)
Equation 4 was differentiated and evaluated a t the reduced time of e = 1.0
Hence by measuring the slope of the curve of C,C, us. ( v O ) / (tLr) a t 8 = 1.0, D, can be calculated by Equation 5. This point a t = 1.0 is the period of time to pass one void volume of fluid through the vessel in plug flow. The basic equations for Einstein's statistical model have been summarized in detail by Cairns and Prausnitz ( Z ) ? where axial mixing is assumed to be a result of a series of motion and rest phases. Their final equation is
e
where l o is the Bessel function of zero order 2nd first kind of imaginary argument. T h e main advantage of this statistical model is that it works well for a very short bed. At long bed lengths as shown below, the diffusion model and statistical model give similar results.
Figure 1. A'. A. B. C. D. E.
F. G, H. J. K.
I. M. N.
P.
Process flow diagram
Storage of dispersed phose ketone Storage of continuous water phase Overflow and constont head tank Storage of tracer solution Throttle valve to overflow tank Control valve for water flow Control valve for ketone flow Valves Microswitch Recorder Adjustable loop for interface control Storage bottle Tared bottle for timed flow measurement Conductivity probe
For large values of ,V > 50, Equation 6 reduces to Equation 5 of the diffusional model. At N = 50 the difference is 0.25y0 between the D, values obtained from Equations 6 and 5; N = 5, approximately 2.5Ye, N = 2.5, approximately 5%. In the present work, 14 out of 16 runs had IV values above 3, the majority above 5. Hence, it appears that the diffusional model in Equation 5 is sufficiently accurate to represent the residence-time-distribution curve end D, for this work. Experimental Methods
Apparatus. The arrangement of the apparatus and the process flow diagram are shown in Figure 1. T h e equipment was modified from that used by Vogt and Geankoplis (20). The tracer solution of O.l.V KCI solution was stored in C (Figure 1) and entered the main water flow a t the inlet nozzle. Its flow was controlled with valve G and was less than 1% of the continuous phase flow rate. To make a run, the main flows of water saturated with ketone and ketone saturated with water were set. T h e KCl tracer flow was adjusted to obtain a reading of about 6000 ohms on recorder K for the conductivity probe. P. When steady state was reached after about 1 hour, the step function was introduced by stopping the tracer solution flov. This was done by closing valve H. which zutomatically tripped a microswitch. J . T h r microswitch introduced a zero time pip on the recorder. The breakthrough curve was then recorded for 15 minutes to 1 hour. depending on the type of run. T h e time V O L . 2 NO. 4 N O V E M B E R 1 9 6 3
311
required for the step function to enter the column proper from the nozzle after closing the valve was 0.3 to 2.0 seconds, depending on the flow rate. This correction \vzs applied to the time scale of the breakthrough curve. It was small, since the time for 6 = 1.O was 250 to 1000 seconds for all runs. The tower length between the top of the dispersed phase nozzle and the tip of the continuous phase nozzle where the dispersed phase coalesced \*;as 36.9 inches. In Figure 1 the KCl tracer entered through valve H and then through a side arm to the inlet nozzle: the same nozzle used by Vogt and Geankoplis (20). The constriction in the nozzle induced rapid mixing of the tracer solution and the main continuous phase water f l o ~ r . The dispersed phzse inlet nozzle is similar to that used by Vogt and Geankoplis (20). The number of open tips used varied from 4 to 10, depending on the ketone rate. This procedure kept the velocity in the tips constant and gave the same bubble size a t different flow rates. Unused tips were plugged with small stoppers. The holdup or void fraction of the ketone-dispersed phase in the column was determined as follows. At steady state both the ketone and water lines were closed simultaneously. The increase in interface level of ketone \vas equal to the dispersed phase holdup of ketone. Analytical. The electrical conductivity probe shown in Figure 2 was placed a t the exact end or outlet of the continuous phase as shown in Figure 1. It consisted of two parallel platinum plates lvith copper lead wires sealed with epoxy resin in a '/'s-inch-diameter stainless steel tube. The resistance across the conductivity probe was recorded by a Leeds and Northrup conductivity recorder with I-second response time. The probe was calibrated a t different temperatures using standard solutions of KCl saturated with ketone. The experimental calibration curves were obtained a t various KC1 concentrations and temperatures and the cross plot of these curves is given in Figure 3.
FOIL
LATINUM
COPPER WIRE
SOLDER
EPOXY R E S I N CELLULOID
Figure 2.
Electrical conductivity probe
Experimental Data and Calculations
K C L CONCENTRATION,(GM.-MOLE / L I T E R )
x
lo4
Cross plot of electrical conductivity Calibration
Figure 3.
0.06 0.05
0.04 Y
w 0.03 0.02
0.0 I 0
0
50
100
150
200
250
300
VK , C . C . / M I N .
Figure 4. rates 312
Dispersed phase holdup at various flow
l&EC FUNDAMENTALS
The experimental data for the holdup or void fraction of the dispersed phase were determined. The velocities of the ketone and water phases were varied over the same ranges used in the D , runs. The actual experimental data were first plotted as C~ us. V,, for constant values of V K . A cross plot of these curves was then made (Figure 4). The data appear to be consistent and extrapolate as straight lines to zero-zero. As expected, the continuous phase flow rate has only a small effect on the ketone holdup. The data essentially agree with those of Johnson and Bliss (9). The bubble diameter was calculated as 0.34 cm. by using the correlation of Hayworth and Treybal ( 8 ) and the physical property data of Keith and Hixson (70). This size was constant throughout all the runs. since the number of tips was varied directly as the volumetric ketone flow rate, giving a velocity in the tips of 10 cm. per second. Actual drop size data are not available, but Hayworth and Treybal ( 8 ) show that drop size is uniform in the velocity range. Others (70) also indicate this, Visual observation of the tower showed essentially no coalescence of drops. The experimental data for D, for the 16 runs are given in Table I. The ketone and water rates were varied. To calculate D , the values of C'C, were plotted against the dimensionless number 8 = (v,$e//~,V), where volume V was 985 cc. At the point where 8 = 1.0, the slope was determined (run 5, Figure 5). L'sing the experimentally determined value of D , = 4.75 sq. cm. per second and Equation 4, the breakthrough curve was calculated (Figure 5). The calculated points check the experimental residence-time-distribution curve reasonably well. Hence, the use of Equation 5 for the diffusion model appears justified within the limits of the experimental error.
Table 1. Run
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
c. 26.5 26 6 27.0 26 2 26 0 28.3 28.5 27 1 26 -~ n . 26.4 27.2 27.2 26.8 24 5 25 0 26 5
vw 96.9 98.3 45.3 144.5 166.3 49.4 9:. 0 107.0 195.7 248.3 48.0 138.6 250.8 247,9 248 1 46.4
VK 241 8 238 2 222 4 246 3 253 7 95 9 94 1 96 1 97 6 95 3 140 4 152 0 154 4 221 0 242 6 221 3
€W
0 9583 0 9589 0 9633 0 9558 0 9539 0 9846 0 9841 0 9835 0 981’ 0 9802 0 9766 0 9732 0 969’ 0 95’3 0 9568 0 9638
Experimental Data for
63 58 00 25 11 30 80 63 83
9: 93 40 44 55
5’ 13
(7)
These values of D , at a given velocity are severalfold greater than the values for packed beds (75). Both spray towers and packed beds have a continuous phase and a dispersed phase. However, a big difference is the 0.04 void fraction in the spray tower as compared to 0.50 for a packed bed. Hence! in any comparisons this must be borne in mind. The data cover the range of values from 3 to 18. The lrue laminar region for flow past a sphere is in the region a t a
1
NR,
.VRe”
6.28 6.40 2.98 9.28 10 66 3,34 6.61 7.06 12.58 16.13 3.18 9.18 16.41 15.35 15.64 3.01
6.56 6.68 3.10 9.70 11.17 3.39 6.72 7.18 12.80 16.45 3.26 9.44 16 90 16.03 16 33 3.12
.VPe
0 0 0 0 0 0 0 0 0
3.87 4.01 3.18 4.27 4.75 2.44 3.77 4.25 5.24 6.09 3 09 4.74 5,93 6.51 6.36 3.41
0 0 0 0 0 0 0
0146 0144 0083 0199 0206 0115 0147 0144 0214 0234 0089 0169 0245 0223 0231 0080
?;&I
66.2 67.4 31.4 97.8 112.1 33.1 69.6 74.3 132.2 169.9 33.5 96.8 172 8 161 7 164 6 31.7
Reynolds number less than 1.0. However, Carberry and Bretton (3) show that the laminar region for packed beds extends to a Reynolds number of about 20, and Streeter (79) shows that this region goes u p to about 20 for pressure drop in unconsolidated sands. Perry (77) shows that this region extends to approximately 80 for pressure drop in packed beds. Hence, the present data appear to be in the region where laminar flow is just ending. It should be iemembered that the void fraction of the spheres in the present work was about 0.04, while that of packed beds was about 0.50. Others (2, 3. 5, 75) found that D , for packed beds of spheres was proportional to the velocity to the 0.96 to 1.00 power in the Reynolds number region below 100. Streeter (79) shows that for beds of unconsolidated sands D, was proportional to Lrn,l :1 for the Reynolds number region below 10. ‘The value of 0.45 as the exponent o n the velocity for the present work is about ‘/2 of the range ofvalues of 0.96 to 1.I7for packed beds of spheres and sands. I n Figure 7 the data are plotted as AVpeLS. -YRe”, The plot shows that the Peclet number for this \+ark is not constant but is proportional to (,VHe”)O 55. in marked contrast to the data for packed beds (2, 3. 5, 75, 79) v here the Peclet number is approximately constant. The values of the Peclet number in the present work range
In Figure 6 D , is plotted against thc interstitial velocity of the continuous phase. It is apparent that D,,is independent of the dispersed ketone flow rate in the range of ketone flows of 95 to 240 cc. per minute. This nondependence o n the flow rate of ketone is somelvhat expected, since the bubble size is constant, which makes the actual velocity of the bubbles constant a t all flow rates of ketone. Also, the value of e,,. does not vary much and is approximately 0.96 for this range of velocities. The maximum deviation of all the data from the best line is -2O%, and the mean deviation is +5.4%. \vithin the expected error in measuring D,. The data for D,,were plotted o n log-log scales 1s. the velocity. The equation for the best line through the data is (Lrw)0.45
167 170 0778 250 288 0830 163 180 330 419 0812 236 428 428 432 0796
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Discussion
D L = 9.00
DL
CN
CrK
9 9 10 9 9 10 9 9 8 7 9 9 8 8 8 10
DL
’
.
1
’
I
I
‘
I
’
-
0 EXPERIMENTAL
CALCULATED
-
-
-
-
I
I
I
I
I
I
--- I
I
1
-
-01 . 1
VOL. 2 NO. 4 NOVEMBER 1 9 6 3
313
from 0.008 to 0.023. Over the same Reynolds number range, the values for packed beds are much greater and r m g e from 0.25 to 0.9. Hence, the present values are about as large as those for packed beds. The Peclet numbers for beds of unconsolidated sands (79) vary from 0.06 to 0.08. Brutvan's data ( 7 ) are plotted in Figure 7 for comparison with the present work. His experimental values of the Peclet numbers scatter considerably and are estimated to be about '/z as large as those of the present work at the same Reynolds number values. The slope of the line through his data is approximately the same as that of this work. Brutvan found a slight increase of D, with an increase in the dispersed phase flow. H e used a spray tower 1.5 inches in diameter and 59.5 inches in length with 0.40-cm. spherical glass beads as the dispersed phase. His dispersed phase flow was downward and not upward as in this work. Since the beads were rigid, they did not become distorted nor undulate in shape as drops do in passing through a fluid. No coalescence of drops occurred at the end of the tower as in the present work. Brutvan measured D, experimentally only in a short section 0.5 foot long in the center of the tower and not over the entire column length. Hence, his data are not really representative of the data expected from a liquid-liquid spray tower. Because of these differences it is not surprising that the two lines in Figure 7 are not the same. A calculation was made using Taylor's (72) equation for D, in a n open tube in laminar flow. These values of D, were about 10 times the experimental values for this work given in Figure 6. Evidently the dispersed ketone phase greatly reduces the parabolic velocity profile of the continuous phase. This reduction in turn reduces the Taylor D,,which is caused primarily by this parabolic velocity profile. The axial dispersion coefficient in spray towers is undoubt-dly caused by a combination of several factors. One factor is a result of some turbulent eddy mixing in the axial direction; another is caused by the main velocity profile which is closer to that for turbulent than for laminar flow. However, a considerable thickness of the laminar sublayer probably still exists, so molecular diffusion would be of some significance. In Figure 6 only a
0.20
' '
I l l
'
'K T H I S WORK 0 240 0 150 95 BR U T V A N
d
v)
a-J
UW,CM./ SEC. Figure 6.
~
0.06 -
z
limited range of large values of V , were covered and the data show no effect of this flow rate on D,. As V , drops to very small values or to zero, a large effect on D, would be expected, since Taylor's equation for D, in a n open tube would be expected to be valid. Nomenclature
C C,
= tracer concentration, gram mole/liter
= tracer concentration a t inlet to tower? gram mole/liter
DL
= axial dispersion coefficient, sq. cm./sec.
d, dl
= diameter of dispersed phase sphere, cm. = tower diameter, cm.
I
I
I
i l l 1
'tl
38
J5 fi
237
A; II
0
-
/' 9
0.02-
-
a0 ' 0
' d ? / 0.01 7 0.008 -/ -
I&EC FUNDAMENTALS
I
h -
Ill
-
31 4
I
r*
0.04-
0.006-
Effect of velocity on DL
x
0.10 0.08 -
W Q
I
I
\
3
I
1
I l l 1
I
I
-
I I l l
I
I
F
zo
=
literature Cited
L/ L O
= Bessel function of zero order and first kind
of imaginary
argument
L
-v
= length of test section, cm.
dimensionless axial distance from tracer sourcc = C-X(’DL iVPe = c;i-d,,i DL = d, Cw’p;’p =
= dtUw-‘p,’p
c‘
dDrn-p,’p
interstitial velocity, cm.,/sec. interstitial velocity of ketone phase, cm./sec. velocity of water based o n empty tube, cm.)’sec. interstitial velocity of water phase, cm./sec. flow rate of ketone, cc./min. floiv rate of water? cc.,’min. volume of test section of tower, cc. flow rate, cc./sec. flow rate of water. cc. ’sec. axial distince from p i n t of tracer injection to point of observation, cm. ‘I‘ - c-0 volume fraction o r holdup of phase volume fraction o r holdup of ketone phase volume fraction o r holdup of water phase time, sec. reduced time = U & ’ E ~ V viscosity. gram cm. sec. density, gram, ‘cc.
(1) Brutvan, D. R., Ph.D. dissertation, Rensselaer Polytechnic Institute, Troy, N.Y . , 1958. (2) Cairns. E. J.: Prausnitz, J. M., Chem. Eng. Sci. 12, 20 (1960). 13) Carbrrrv. J. J.. Bretton. R. H.. A.I.Ch.E. J . 4. 367 11958). (4j Danckiderts, P: V., Chem. Eng.’Sci.2, 1 (1953): (5) Ebach. E. A , , \Vhite, R. R.. A.I.Ch.E. J . 4, 161 (1958). (6) Epstein, Norman, Can. J . Chern. Eng. 36, 210 (1958). 17) Gier. T. E.. Hougen. J. O., Znd. E w . Chem. 45. 1362 (1953). (8) Hay&-orth.’C.B.,-Treybal, R . E., Ib;’d., 42, 1174 (1950). (9) Johnson, H. F., Bliss, Harding, Trans. A.I.Ch.E. 42, 331 (1946). (IO) Keith, F. I V . , Hixson, A. N.. Ind. Eng. Chem. 47,258 (1955). (11) Kreager. R. M.?Geankoplis, C. J., Ibid., 45, 2156 (1953). (12) Levenspiel, Octave, Ibid.! 50, 343 (1958). (13) Licht, W., Conway. 3. B., Ibzd., 42, 1151 (1950). (14) Licht, LV., Pansing, T.V. F., Ibid., 45, 1885 (1953). (15) Liles, A. W.: Geankoplis, C. J., A.I.Ch.E. J . 6, 591 (1960). (16) Miyauchi, Terukatsu, V. S. At. Energy Commission, Rept. UCRL-3911 (1957). (17) Perry, J. H., “Chemical Engineers’ Handbook,” McGraivHill, New York, 1950. (18) Sleicher. C. A.. Jr.. A.Z.Ch.E. J . 5 . 145 11959). (19j Streeter, V. L.’, “Handbook of Fiuid Dynamics,” McGrawHill, New York, 1961. (20)JVogt, H. J., Geankoplis, C. J., Ind. Eng. Chem. 46, 1763 (1954). ~
~
~
~
~~~
\
I
RECEIVED for review August 17, 1962 ACCEPTED July 12. 1963 Financial assistance in the form of an Ohio State University fellowship is acknowledged.
P R E S S U R E DROP IN H O R I Z O N T A L A N D V E R T I C A L COCURRENT GAS-LIQUID FLOW G . A.
HU GHM A R K
,
Ethyl Carp., Baton Rouge, La.
A new “lumped type” pressure drop correlation for gas-liquid flow utilizes a lost work term from an energy balance derived equation for gas-liquid flow. Experimental data for horizontal, vertical upward, and vertical downward flow show that the lost work term i s a function of the pipe orientation. The average absolute deviation between experimenral and calculated pressure drop data i s 19% for horizontal flow in 0.5- to 1 0-inch pipe and 15% for vertical upward flow in equivalent diameters to 2*/2 inches. The correlation i s applicable to all gas-liquid flow regimes. REDICTION of two-phase gas-liquid pressure drop in pipe P h a s been a subject of interest for many years. An appreciable amount of experimental data has been obtained for isothermal two-phase flow with the greatest emphasis on horizontal and vertical upward flow. Many correlations have been proposed for these data. The correlations apply with reasonable accuracy to the data on which they are based, but are limited in some form of application. Either they d o not extrapolate to data other than those on which they are based or they are valid only for specific flow conditions. S o n e of the correlations apply with reasonable accuracy to all of the experimental data. Lamb and it’hite (26) recently presented derivations of momentum and mechanical energy equations to show their relation to pressure drop correlations in two-phase flow. T h e derived momentum equation can be rearranged to the form suggested by Martinelli and Nelson (28) :
PI
- PZ = cd
f
(PLRL $. P&G)(HZ 46
- H I )f
If a linear pressure change Tvith length is assumed for the twophase section, the mechanical energy equation derived by Lamb and White and by Vohr (38) can be reduced to the form suggested by Hughmark and Pressburg (27).
v,&)f WG(v& - v&) (2) 2gJW~t7~ W G ~ G )
wL(vi2 -
+
It is apparent that C, and APT, are not identical. Cd can be considered as a drag coefficient applied to the pipe wall, which has a positive sign in most flow conditions but is negative for conditions of vertical upward flow in which the net liquid flow a t the pipe wall is downward. A P l p represents the irreversible energy loss through the two-phase section and must be positive for all flow conditions. The major part of the isothermal two-phase data has been obtained at conditions in which the momentum or kinetic energy change is negligible. When this term is negligible, the momentum equation gives: PI
- PZ = c d
$.
’
Se
( P L R L-I- ~ G R G ) (-HH~ I )
V O L . 2 NO. 4 N O V E M B E R 1 9 6 3
(3) 315
