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. T h e 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 L a m b 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
and the mechanical energy equation :
T h e equations are further simplified for horizontal flow with negligible momentum or kinetic energy change to : PI - P2
=
Cd
(5)
P I - PJ = APT,
(6)
This is the only condition at which Cd and A P T pare equal. The most generally accepted method for predicting C, or APT, for two-phase flow is that proposed by Lockhart and Martinelli (27). The method is based upon data for flow in 0.62to I-inch horizontal pipes with a pressure range from atmospheric to SO p.s.i.g. It does not extrapolate well to larger diameters or higher pressure. Chenoweth and Martin ( 7 7) presented a method of correlation similar to that of Lockhart and Martinelli in that a ratio of the two-phase pressure drop to the pressure drop for the liquid is used. This applies to large as well as small pipe diameters and to high pressure. It is restricted to turbulent flow in horizontal pipe. Bertuzzi, Tek, and Poettman (7) proposed a correlation for horizontal flow that uses a two-phase f factor. The f factor is similar to the friction factor for single-phase flow with homogeneous flow assumed for calculating the velocity and the density to be used with the correlation. This correlation is applicable to all conditions of flow except when the gas mass flow rate per unit time is greater than that of the liquid. Hoogendoorn (78) has suggested individual correlations for the specific flow regimes. Pressure drop for horizontal annular flow has been considered by Aziz and Govier (3) and Wicks and Dukler (47). For vertical upward flow Galegar, Stovall, and Huntington (74) and Tek and Chan (36) have proposed friction factortype correlations for the combined potential energy and APT, terms. Hughmark and Pressburg (27) predicted APT, from the slip velocity and a combination of the physical properties and the mass velocity. Calvert and Williams (70), Anderson and Mantzouranis (2), and Collier and Heivitt (12) have analyzed pressure drop for annular flow. Prediction of pressure drop for gas-liquid flow depends upon prediction of the terms C, or APTp. For horizontal flow with negligible kinetic energy. the two terms have been shown to be identical. For vertical flow. the terms are not equal to each
other and are not equal to those for horizontal flow under identical flow conditions. Therefore, these terms must be considered as a function of pipe orientation as well as flow rates and physical properties of the phases. Correlation of the term APT, was selected for this work. The irreversible energy loss for single-phase fluid flow is designated as lost work and can be expressed in units of foot pounds force per pound mass. This lost work for single-phase flow is the product of the friction pressure drop and the fluid specific volume. By analogy the lost work for two-phase flow can be considered to be the product of the two-phase pressure drop, APTp. and the specific volume of the mixture in the pipe section. Thus a correlation can be used for the two-phase lost work if the t\vo-phase specific volume can be predicted. This is the correlation method used in this paper. In gas-liquid two-phase flow, the difference in the physical properties of the two phases causes the phases to flow at different average velocities in the pipe. Therefore. a knowledge of the mass flow rates per unit time and the specific volume of each phase does not define the specific volume of the mixture in the pipe. Hughmark (79) has recently presented a method for the determination of holdup in gas-liquid flow which is based upon the variable density single fluid model proposed by Bankoff (5). The holdup is the fraction of the pipe occupied by a phase and can be used to calculate the specific volume of the mixture in the pipe for the lost work correlation. Comparison of Two-Phase Lost Work for
Horizontal and Vertical Flow
The lost work terms for horizontal, vertical upward, and vertical downward cocurrent isothermal flow were calculated with APTp defined by Equation 2 and the two-phase density obtained from the holdup. Figures 1 through 4 show these lost work terms as a function of the liquid volume fraction, y L , and the Froude number. These are defined as:
+
WLDL
M~LZ~L W G ~ G
-Y L =
The lost work term as shown on the plots is modified by subtraction of the friction lost work for the total mass flowing at the
FrTp
-
20,000
10
t
01 0 0001
'
~
~
'
1
'
~ '
3~ ,
LLUL
'
0 01
0 001
01
I
I
LUU
10
YL
0.01 0.001
0 01
0.1 YL
Figure 1 . 316
Horizontal flow pressure drop
l&EC FUNDAMENTALS
Figure 2. Correlation for vertical upward flow pressure drop Frrp
>
1
0.11 0.01
I
I
I
I
I
I I I I I
1
I
I
\
I I l l
I
0.I YL
Figure 3. drop
Correlation for vertical upward flow pressure
0.I
0 001
, ,
1
I
0.01
0.1
!\,
YL
FrTp
