ARGUMENTS O F CONCENTRATION VARIABLES
= =
03
-
m
variable evaluated a t z variable evaluated a t L
= =
+
m
-m
SUBSCRIPTS ON C O N C E N T R A T I O N V A R I A B L E S
i, j , A , B
s
G
=
arbitrary solute species
= sodium chloride =
glycerol
SUPERSCRIPTS O N CONCENTRATION VARIABLES 0
* I
= value of variable taken a t feed conditions
equilibrium value = value of variable a t downstream end of a particular solute zone? or used to indicate differentiation with respect to x =
(5) Glueckauf. E.. Discussions Faradav SGC.7. 12 (1949) ( 6 ) Glueckauf. E.: J . Chem. SOC.1949, p. 3280. (7) Glueckauf, E., Coates, J. I., Ibid., 1947, p. 1315. (8) Griffin, R. P., Dranoff, J. S., A.I.Ch.E. J . 9, 283 (1963). (~, 9 ) Helfferich. F.. Anpew. Chem. (International Edition in English) ~, 1, 440 (1962). (10) Klamer. K.. van Heerden., C.., van Krevelen. D. LV.. Chem. ‘ Ens. Sci. 9, 10 (1958). (11) Klinkenberg, A., Sjenitzer, F., Ibid., 5 , 258 (1956). (12) Lapidus, Id., Rosen, J. B., Chem. Eng. Progr. Symp. Ser. 50, No. 14, 97 (1954). 1131 Liehtfoot. E. N.. Sanchez-Palma, R., Edwards. D. O., “Ne; Chernical Engineering Separation Techniques; I ’ H. Mi Schoen, ed., Interscience, New York, 1962. (14) Opler, A , , Hiester, N., “Tables for Predicting the Performance of Fixed Bed Ion Exchange and Similar Mass Transfer Processes.” Report, Stanford Research Institute, Pasadena, Calif., 1954. (15) Prielipp, G. E., Keller, H . W., J . Am. Oil Chemist$’ SGC.33, \
I
~
103 f19S6).
Literature Cited
(1) Asher, D. R., Simpson, D. W., J . Phys. Chem. 60, 518 (1956). (2) Coonev. D. O., Lightfoot. E. N., Ind. Eng. Chem. Fundamentals 4, 233 (1965). (3) Ekedahl. E., Hogfeldt, E., Sillen, L. G., Nature (London) 166, 723 (1950). (4) Gilliland, E R., Baddour, R. F.. Ind. Eng. Chem. 45, 330 (1953).
(16) k&ei:’J. B., J . Chem. Phys. 20, 387 (1952). (17) Shurts. E. L., White, R . R., A.I.Ch.E. J . 3, 183 (1957). (18) Vassiliou, B.. Dranoff. J. S., Ibid., 8. 248 (1962). ’ (19 ) Vermeulen, T., Aduan: Chem’. Eng.’ 2,’ (1958). (20) Vermeulen, T., Hiester, N., Chem. Eng. Progr. 48, 505 (1952). RECEIVED for review April 15, 1965 ACCEPTED August 16, 1965
TWO-PHASE FLOW IN A COILED TUBE Pressure Drop, Holdup, and Liquid Phase Axial Mixing G . R . R I P P E L , C. M . E I D T , J R . , A N D H . B. J O R D A N , J R . Esso ResearLh Laboratories, Humble Oil G3 Refining CG.,Baton Rouge, La.
Data on pressure drop, holdup, and axial liquid mixing are reported for two-phase concurrent flow in a ’/*-inch diameter spiral tube. The over-all objective of this study was to obtain scale-up data and correlations for a two-phase flow spiral reactor. Pressure drop and liquid holdup data have been compared to known correlations and a new correlation based on a gas drag coefficient is presented. Liquid holdups were measured by a “tracer” method in addition to the more conventional method of trapping. The tracer technique also permitted measurement of the axial mixing in the liquid phase.
number of articles on two-phase flow have appeared in recent years concerning the theory and correlation of data in horizontal and vertical upflow orientations. I n actual situations, few piping systems or isolated fluid contactors conform to either such simple geometry. I n this paper, data are presented on studies in a helical coil to complement the growing data bank of two-phase flow in the simpler flow orientations. T h e main variables which characterize two-phase flow are the increased pressure drop that occurs when a liquid is introduced into a tube with gas flow and the fractional pipe volume occupied by the flowing liquid. These two parameters are the minimum information needed for the fluid mechanical design of piping systems \L here two fluid phases flow simultaneously. Liquid entrainment and flow patterns are also important variables in two-phase flow phenomena. T h e problem of liquid phase residence time distribution in two-phase flow has not received so much attention as in single-phase pipe flow. However, the importance of such information for fluid contactors and reactors is generally recognized.
A
N INCREASING
Experimental Equipment and Procedure
Equipment. A diagram of the experimental equipment is shown in Figure 1. 32
l&EC PROCESS DESIGN AND DEVELOPMENT
TEST SECTION: 88.3’ OF 1/2” 0 D. TUBING
GAS SATURATOR
B
LIQUID
-
0
0 A 0
KATER
4
FREOK 12
AIR
I
HELIUM
1
0.00 IOU
IOUU
REYNOLDS NUMBER,
Re
Figure 2. Friction factor-Reynolds number plot for single-phase flow data
T h e two-phase flow !section consisted of a ','2-inch-diameter, 18-gage tubing coil \\round on a cylinder 8 inches in center diameter. T h e total length of this section bet\veen pressure measurement taps \vas 58.3 feet. About midlvay in this coil a glass tubing section of the same inside diameter a n d approximately 8 feet long \vas inserted for visual observation of flow patterns. Another glass section approximately G inches long a t the coil outlet housed a conductivity cell used to measure liquid phase residence time. Air o r other gases \cere introduced through suitable liquid saturators a t ambient temperature. Flow rates \vere controlled by rotameters adjusted by hand needle valves. Liquid rates could be controlled by either rotameters or a precision metering pump \Then very lo\v flolvs were required. Gas a n d liquid \ v u e fed simultaneously to the coil section through a smooth tubing tee with liquid on the straight tee run. Just upstream c,f the tee on the liquid line a small opening in the tubing \\.all was provided for injection of tracer solution. l'his openin:; \vas covered by a tight rubber diaphragm through Ivhich electrolytes such as KC1 or KhlnOb were injected in quantiiies of 0.5 to 1.0 cc. by means of a hypodermic needle \vith syringe plunger. Pressure d r o p measurements \ which have the following definitions:
AIR-WATER HELIIIM-WATER FREON 12-WATER A I R - 2-PROPANOL
1
LOCKHART-MARTINELLI LINE (IO)
h-
1(
-
I
I..(
-
.
0 1
LAMINAR
0
now
EXPERIMENTAL DATA, L I Q U I D I N S P I R A L L E D P I P E ( 7 )
TURBULENT FLOW
@
EXPERIMENTAL DATA, L I Q U I D IN STRAIGHT HORIZONTAL P I P E ( 8 )
0
T A Y L O R ' S THEORETICAL L I N E , L I Q U I D S (11,lZ)
0 01
1
10
100
lo5
104
1000
L I Q U I D REYNOLDS NUMBER
Figure 10.
*
Dispersion number in two-phase flow 1.6 X 10-5 sq. cm. per second
Sc = p / p D , D =
VOL. 5
NO. 1
JANUARY
1966
37
laminar and turbulent single-phase flo\v have been developed by Taylor ( 7 7 , 72) and others. T h e diffusion model as developed is strictly applicable to straight pipes. Bends. curved pipe, or marked end effects will result in average values for D. Nevertheless, \vork in curved pipes \\-ith single-phase flow shows the general usefulness of this type of model for residence time distribution. The tracer data from these studies resulted in the familiar type of C-curve (8). These data were fitted by a computer program which calculated the curve variance and a diffusion coefficient. The axial mixing Peclet number, Pe, Xvhich contains D , was also calculated for each experiment. T\vo typical tracer curves are presented in Figure 9. For correlation of axial mixing data, the longitudinal dispersion number, DiVd,, has been found more useful.
-1x - =L Pe
d,
D -V Lx
Conclusions
Two-phase pressure drop in a downward spiraled tube can be correlated by the Lockhart-Martinelli analysis derived from horizontal flo\\ data. Liquid holdups in the spiraled tube are less than the holdups obtained in horizontal pipe. The deviation from the horizontal data increases i \ i t h decreasing values of parameter X . l&EC
FOR ASSULARFLOW.
FOR BUBBLEAND SLUGF L O ~ .
FOR STRATIFIED FLO\\..
L - dl
Here, d , is some characteristic length associated with the local mixing process-Le., tube diameter for flow in pipes or particle size in packed beds. D / V d , varies from 0 for plug flow to m for complete backmixing. An increase in D,'Vd, corresponds to a rise in the degree of backmixing. T h e value of d , for use in tMo-phase flow has been more or ILSS arbitrarily defined as the thickness of the liquid holdup film assuming annular type flo~v. This value was chosen, rather than pipe diameter or some mixing length, because of the use of thickness for evaluating film Reynolds numbers. The longitudinal dispersion number is plotted as a function of liquid Reynolds numbers in Figure 10. For comparison, two sets of experimental data in single-phase liquid flow are indicated. Dispersion numbers, obtained by Levenspiel (8>9 ) in an analysis of all published data on mixing in straight pipes and pipe with a moderate number of bends, are shown in Figure 10 just above the theoretical line calculated by Taylor for smooth straight pipe. Experimental data for liquid flow in spiraled pipe taken from Kramers and M:esterterp ( 7 ) are indicated in the same region of Reynolds numbers as the present data. Only general comparisons can be made between the t\vophase and single-phase data. Dispersion in two-phase flow appears to be of the same order as for single-phase flow in spirals a t Reynolds numbers between 400 and 2000. However, the two-phase data do not indicate a maximum and tend to show slightly higher backmixing at the IoLver liquid Reynolds numbers. Relatively high gas to liquid rates in twophase flow contribute to scatter. In the case of very low liquid rates both entrainment and the bulk of liquid flow occurring in the laminar film a t the pipe wall produce the high dispersion numbers observed in the two experiments a t these conditions. At higher liquid Reynolds numbers the data tend to approach the single-phase dispersion numbers, indicating less importance for film flo~vand entrainment as axial mixing mechanisms. Also, at the higher liquid rates it is possible that secondary flow effects increase the radial dispersion, consequently reducing the axial dispersion.
38
Liquid properties affect holdup relationships to a greater extent than gas properties. Loiver holdups \vex obtained \vith 2-propanol than \vith \ \ ater. Pressure drop in t\\-o-phase flo\v can be correlated by a n v o phase drag coefficient. The correlations obtained in this study for a spiraled tubr in various flo\v patterns a r e :
PROCESS D E S I G N A N D DEVELOPMENT
Liquid axial dispersion in trio-phase f l o ~can be measured by tracer experiments. Dispersion numbers in t\\o-phase flo\\ tend to be higher than in single-phase flou a t lo\\ liquid rates but approach experimental single-phase data in spiraled pipe d t Re) nolds numbers of 400 to 2000.
Nomenclature
projected area for drag, sq. ft.
Ad
=
C D
= tracer concentration, Ib.;cu. ft.
do dt
=
F fd
G G,
H
=
= = =
= = =
L P
P
= = =
P, Q R
= = =
70
V
= =
X
=
8
=
P
= = =
iJ ad
longitudinal diffusion coefficient for mixing, sq. ft./ hr. pipe diameter, ft, characteristic distance for dispersion number, ft. drag force, Ib. drag coefficient acceleration of gravity. ft. sec.2 conversion factor, Ib. m ft. Ib. f set.* height. ft. length of pipe (axial), ft. pressure, Ib.;sq. f t . pressure, Ih.;'sq. ft. stagnation pressure, Ib.,'sq. ft. volumetric input flow rate, cu. ft.:hr. fraction of pipe volume occupied by flowing phase in tlvo-phase flow pipe radius, ft. linear velocity, ft.!sec. liquid fraction input flow = Q i / ( Q I Q,) time, hr. density, Ib.!cu. it. viscosity, Ib./ft.-hr. drag area, sq. ft. 'sq. ft. of inside pipe area I
+
DIMENSIONLESS GROUPS Re = dVp/p = Reynolds number = V L / D = axial Peclet number Pe = V2,'Gd = Froude number Fr D / V d , = longitudinal dispersion number X2 = two-phase flow parameter, ratio of gas to liquid pressure drops q2 = tivo-phase flow parameter, ratio of two-phase pressure drop to pressure drop for one phase fd = drag coefficient! dimensionless SUBSCRIPTS 0
g
1 gd
tp
based on superficial pipe dimensions gas phase liquid phase gas-phase drag = tlvo-phase
= = = =
literature Cited
(8) Levenspiel, O., “Chemical Reaction Engineering,” p. 264,
(1) Alves. G. E., Chem. Eng. Progr. 50 (9), 449 (1954). (2) Anderson, G. H.. Mantzouranis, B. G., Chem. Eng. Scz. 12, 109 (1960). (3) Calvert. S., L\.’illiams.B.. A.I.CI1.E. J . 1, 78, (1958). (4) Dean, \ V . R.. Phil. M a g . 4, 208 (1927). (5) f b i d . , 5 , 673 (1928). (6) Dengler. C. E.. Ph.D. thesis, Massachusetts Institute of Techn o l o q . 1952. (7) Kratncrs, H.. Lt’esterterp. K. R., “Elements of Chemical Kwctor Design and Operation,” p. 92, Chapman and Hall, S c w York, 1963.
(9) Levenspiel, O., f n d . Eng. Chem. 50, 343 (1958). (10) Lockhart, R. W.,Martinelli, R. C.. Chem. Eng. Proyr. 45,
IViley, New York, 1962. 39 (1949). (11) Taylor, G. I., Proc. Roy. SOC.(London) A219, 186 (1953) (12) Ibtd.: A223, 446 (1954). (13) LYhite, C. M., fbid., A123, 645 (1929) RECEIVED for review February 25. 1965 ACCEPTEDSeptember 7:1965 A.1.Ch.E. Regional Meeting, New Orleans, La., November 1964,
APPLICATION OF PENETRATION THEORY TO GAS ABSORPTION ON A SIEVE T R A Y R . K . S M I T H ’ A N D G .
B. W I L L S
Department of Chemical Engineering, Virginia PolytPchnic Institute, Blacksbur?. Va.
A method was developed for extending transient absorption measurements from simplified apparatus to operating, pilot-scale absorption equipment through measurement of the effective times of penetration and of the interfacial areas available for mass transfer in a 4-inch-diameter sieve tray column. The system studied was that of the absorption of carbon dioxide into dilute aqueous solutions of sodium hydroxide. The measured values of the times of penetration were about twice as large as previous estimates. The values of the interfacial areas determined were consistent with previous, independent estimates.
u
RECEST
years the penetration theory of Higbie ( 6 ) has
1’ found frequent use in the correlation of transient absorption
data ( I , .I, 6). This theory gives a n equation involving the time of penetration, t,:
I n a modification of Higbie’s work Danckwerts (3) suggested that each element of liquid was not exposed to the gas phase for a n equal period of time but that a Poisson distribution of ages of liquid exposures existed, and thus obtained a n equation involving a rate of surface renewal, s: ,-
One of the ultimate purposes in developing these characteristic models describing mass transfer was to facilitate the dcsign of process equipment. However, in the design of gas absorption apparatus exact theoretical analysis has been complicated by a n inability to evaluate the theoretical parameters, a? and either t , or .r. I t is then not the lack of adequate absorption models? but rather a lack of knowledge of values of the physical parameters describing the system which has prevented the application of these equations to equipment design. Given here is a method for determining these necessary parameters for a sieve tray column. While these methods were developed for a particular application, they are easily extended to a number of other chemical engineering systems. Present address, E. 1. du Pont de Nemours & Co.. Fabrics and Finishes Department. Experimental Station, IVilmington, Del.
Method of Analysis of Data
From the data of Danckwerts and Kennedy (3) it was observed (70), as expected, that the average rate of absorption of carbon dioxide into pure water was less than the average rate of absorption into dilute alkaline solutions. More interestingly, as the time of penetration approached zero, these rates of absorption approached the same limiting value. Thus, the ratio of the quantity of carbon dioxide absorbed into dilute N a O H to the quantity absorbed in pure water was observed to be a function of the time of penetration. This property together with conventional penetration theory formed the basis of the data analysis. Upon application of the penetration theory, given in Equation l , to gas absorption into two chemically different liquids under the same flow conditions (equal interfacial areas and times of penetration) the ratio of the average absorption rates reduces to the following equation :
(3) where R is the ratio of the average rates of absorption to the time of penetration for the absorption of carbon dioxide into dilute caustic and into water. This ratio was evaluated from the data of Danckwerts and Kennedy (Figure 1). T h e measurements upon which Figure 1 is based were made with simplified apparatus (a partially submerged, rotating drum) and under conditions such that the usual assumptions of diffusion into a stagnant, semi-infinite medium were applicable-namely, bulk concentration co constant, no internal VOL. 5
NO. 1
JANUARY
1966
39
