I
ARPAD
G. NEMET
Escher Wyss Ltd., Zurich, Switzerland
M a x i m u m Liquid Transport.
..
Flow of Gas-liquid Mixtures in Vertical Tubes Relationships between pipe length and diameter, as well as material characteristics, are used to calculate maximum liquid transport SoLumov O F PROBLEMS involving gasliquid flow are often complicated theoretically and cannot be treated with classical methods of fluid mechanics. Yet empirical knowledge based on experiments is not complete and no generally valid laws can be established. For horizontal flow of gas-liquid mixtures, a significant step forward has been made. by which pressure drop can be calculated with an accuracy of i l 5 % (5. 14). However, vertical gas-liquid flow is in some respects even more complicated because introduction of the gas has a much greater influence. For fully developed vertical flow above the point where gas is introduced, correlations have been developed (4,9, 7 7 ) from which pressure loss and flow patterns can be determined. I n this report, maximum liquid transfer in vertical flow is discussed. Total pressure drop includes also losses in the mixing chamber. Data obtained a t the Technical University of Budapest. Budapest, Hungary, are used, as well as other data which have been published (6, 19-21).
experimentally, that movement of rising bubbles which fills the whole cross section of the tube is characterized by the constant Froude's number. This seems to contradict another report ( 8 ) ; ho\$ever, this is not true when capillary forces (75) are considered. The investigations mentioned in the previous paragraph are based mostly on relevant physical constants of the liquids ( 7 7 ) . \Vhen the liquid is in movement, velocities can be superimposed, because the influence of small turbulence in the main flow is practically negligible (75). T h e effect resulting from movement of singular bubbles has only a qualitative importance in the work described here, because when liquid transfer is maximum, orderly flo~vof certain types of bubbles is not possible (below). M'hen introduced in large quantities, the gas rises mainly in the form of big cavities which fill the whole cross section of the tube. At maximum liquid transfer, this form of flow probably exists in nearly all practical cases.
Dimensional Considerations
Flow of gas-liquid mixtures can be described by the dimensional equation: F ~ ( Q IQ o,,
PI,
PO,
PZ,
~3
D ,L k , A p ,
g) = 0
(1)
For further examination, the number of variables can be reduced. Because the maximum amount of liquid delivered is the principal concern. delikery rate can be written as Qz
F n - i ( Q Q . f c , .P L
1
(2)
To have the maximum value of Qz for a variable quantity of gas (keeping other variables constant), the first partial derivative can be equated to zero :
and QinIax
=
Fn-? ( P I . P ~ . ,, , . . )
Density of the gas is assumed negligible compared with that of the liquid-i.e., it is assumed sufficient to characterize
Values of Vertical Gas-liquid Flow .As gas is introduced into an open liquid column (Figure 1) the fluid level rises, and when it spills over the top, fluid flow begins (I). As more gas is introduced, liquid flow increases and reackes an optimum mixing rat? (11). If the quantity of gas is increased still further, the maximum rate of liquid delivery is reached (111). After that point, liquid flow decreases until it becomes practically zero. Also, as gas is introduced various types of flow occur (3, 8 ) . For small quantities of gas, movement of singular bubbles occurs and ample empirical or theoretical methods are available for investigating the movement of such bubbles ( 2 ) . I t has been shown (72, 77) that viscosity of the liquid becomes less important as deformation of the bubbles increases, but that surface tension becomes more important. Also, it has been shown (7j, both theoretically and
At maximum liquid transport, the gas rises mainly in the form of big cavities which
fill the whole cross section of the tube VOL. 53, NO. 2
FEBRUARY 1961
151
columns (13), the complete set of dimensionless products is
where Ap' represents pressure loss which may be indicated by
Pressure coefficient Froude's number 1Veber.s number ,
MIXTURE
This is similar to homogenous flow ( V , is assumed constant along the length. A/). The mean local density of the mixture is ( p , = 0) I)
Reynolds numhrr
LIQUID L E M L
Thus, the characteristic equation for liquid transfer of gas-liquid flow in vertical tubes is
pn'
=
r;
and so
7 maximum -
0 (4)
The pressure drop for unit length is given by
Viscous flow in long vertical tubes should be distinguished from turbulent flow in short tubes. For long tubes. friction is most important. but in short tubes, density ratio, surface tension. and gas introduction are important. Although some questions remain unanswered-eg., definition and the boundaries of regions for the two kinds of flow-general relations for the two cases can be found. For turbulent AJM. in long pipes (78: 22): between the t\vo regions, given formulas are invalid.
Decrease in density of the mixture leads to higher frictional losses. T h e prmsure drop as a function of p m p2 has an extremum. which is always a minimum. If we suppose that ,f depends on P , ~ p, 2 onlv slightly. the minimum occurs a t the density ratio
@(TI,
CHAMBER
'
-6
These symbols are used in the equations
TP.
x3? ~
1
L, I D : k l D , P,I'P[)
=
and will be given by the flow by the average local density of the mixture and liquid. Moreover, experience has shown that viscosity of the gas has no appreciable influence. By using these assumptions, the characteristic equation can be written as Pn-2
(Qlrnax,
pi, ~ r n ,
5,
D,L, k . JP, 9) = 0
(3)
.4ccording to the theorem of Buckingham, the phenomenon can be described \+ith seven dimensionless products. Three of them can be selected directlv. the geometrical ratios, L I D and k D, and the density ratio, p n p r . To test a dimension matrix of seven
Maximum liquid Transport with Viscous Flow i n l o n g Tubes
Compressed air-oil lifts M hich are used in mines or deep wells belong in this range. Here the tubes are long with generally small submergence and friction has a predominant influence; effect of gas introduction is negligible.
Based on this rough estimate, an upper limit for fluid delivered can therefore be given:
Rough Estimation of Minimum Pressure Drop
Pressure drop per unit length may be found, using a previously published theory ( 1 6 ) . For a length A1 of the tube. pressure difference is 1p =
PmgU
+ Apt
It is certain that the mean value of the pressure drop will be larger than, or at least the same as the minimum found before, and therefore
(5)
or in certain cases minimal pressure difference can be fixed.
I 5ec m
0.6
O,L
G A S QUANTITY
t
Figure 1. After maximum liquid delivery i s reached, increasing the amount of gas introduced decreases liquid flow
152
INDUSTRIAL AND ENGINEERINGCHEMISTRY
cst
Figure 2.
Maximum oil transport by gas lifts
LIQUID T R A N S P O R T 4 Figure 3. Relation between fi and Reynolds number for long pipes Assumed average viscosity, 1 2 centistokes
10
01
b Figure 4. Relation between pressure drop and maximum liquid flow for short pipes 001
10
10
Calculation o f M a x i m u m Fluid Transfer
'The former estimate gives the possibility of further simplifications in Equation 4. Because the variables in this equation represent a complete set of dimensionless products: Equation 9 can be expressed as
data it may be assumed that viscosity is approximately the same, hoivever the exact values \\'ere not available for all of them. In Figure 2
*
[D
-
d ) D,'
is plotted. This is a quantity proportional t o p as a function of
I t can also be Jvritten as fs
=
0.3084
(:)*
(;)*
.
(12)
Ho\\.ever in this case, p does not represent a constant but rather a function:
f"
= f"(Tl:
7r3: T4)
(13)
fixed in this way can be found The empiricall>-. Feiv reliable measxements are available. and any conclusions drawn should be used \vir21 caution. For viscous flow in long tubes, it is reasonable to express as a function of the Reynolds number because in this case the frictional losses must have decisive influence. Therefore,
I' = I'
(14)
(Tl)
The data of various oil wells ivhich are raken mainly from a previous work 171) are given in Table I. For these
Table I.
2 2 1,':
86.'
9 95
'6
74P P
I'or comparison the known function #:Re) is also shoivn (Figure 3). The values in Table I \\-ere calculated Tvith an average viscosity of 12 cs. For other values of 1 ' p . the curve is displaced parallel to the curve shoivn. T h e Law of Blasius is also indicatrd in Figure 4. T h e diagram shoivs clearly the similar behavior of and f, In summary it can be said that maximum liquid transfer with viscous flow in long vertical tubes can be characterized by the correct choice of p . This value as defined here is a function of Reynolds numbers, as tvas the case \\it11 homogenous flow.
.4s previously mentioned for turhulrt:! flow in short tubes, important l o ~ e s other than those caused by friction occur, which cannot be expressed as a function of Reynolds number. Tests carried out with tubes 4 to 18 meters long indicate a general relation \vhirh is also useful in practice. T h e flow can be expressed heir with the dimensionless function 4. For density of the mixture! its fictitious average value is introduced: SP Pm = t Id '1-he density ratio can alro be exprcssed '
as
which is a function of kno\vn variables. For simplifying further investigations i t is useful to introduce ne\v variables I t is evident that the products
Data for Long Pipes Used in Oil Wells ( 2 1 )
Diameter, I n . Outqide Inside
3 4 6','4 7
Ivhich in turn is proportional to
M a x i m u m liquid Transport with Turbulent Flow in Short Tubes
2 21 ' 2 4 3 3
Length. 11. 500-700 490-1740 540-1290 560-~1920 1220-1920 940-1930 1280 1870-1970 1800-2000
AP
QinlnAl
P1.Q
C'u. 31. 'Be?.
36-190 35-330 46-126 32-480 88-450 32-480 35-103 97-370 39-285
0.0001-0.0007 0.0001-0.005 0.0002-0.0025 0.0001-0.0072 0.0006-0.0065 0.0002-0.011 0.0003-0.003 0.0015-0.0125 0.0005-0.0164
represent also a complete set of dimensionless products-i.e., rotvs of the matrix of exponent are linearly independent. The neiv characteristic equation is therefore
01
VOL. 53, NO. 2
FEBRUARY 1961
153
Acknowledgment I acknowledge the support and stimulation given by the late Ai.G . Pattantyus and express my gratitude to -%kosSzabb and Jozsef Gaal for their valuable assistance in measurements.
am
Geometrical dimensions occur in all products; however only three sets of data are available to find the powers m , n, P, and q. According to the experimental results for short tubes and for Reynolds number lo4, it can be assumed that
D mi
on005
>
#--0
Figure 5. Influence of pipe length on maximum liquid transport
This yields three equations, with the solution
that is QlmPr =
L k
a)
D6’2.g1’2,@ r1’,7r3’, rq’,B,
(
(16)
m =
3/2
n =
-11s
q =
-716
Equation 20 can also be written as
T h e experiments show that Q t m a x is proportional to 312 power of the total pressure drop :
for constant values of p i , p l , and c and for identical length and diameter of the tube (Figure 4). If the diameter is fixed, Qlmax is proportional to the -7,!6 power of tube length (Figure 5).
Using Equations 17 and 18, influence of tube diameter can be described:
Figure 6.
1 54
or with the introduction of the Laplace number
T h e measurements, shown in Figure 6, show a relatively small scatter. I t must be taken into consideration that in most measurements, gas has been introduced by means of circular openings in the wall of the tubes. This is of utmost importance because the results presented here should be used only for this method of introducing gas.
Maximum liquid transport for short pipes
INDUSTRIAL AND ENGINEERING CHEMISTRY
Nomenclature = length of the vertical tube = diameter of the tube d = diameter of the interior tube D, = - d2 reduced diameter k = absolute roughness Q = rate of flow p = density g = acceleration due to gravity V = absolute velocity of flow A# = pressure drop AP = pressure drop between gas inlet and outlet p = total pressure p = dynamic viscosity u = surface tension coefficient /, = friction coefficient La = Laplace’s Number
L D
R e = _ _ Reynolds number of Pt liquid flow Subscripts 1 = liquid s = gap m = mixture D = related to the cross section of the tube References (1) Anderson, L. E., Trans. A m . Soc. Mech. Engrs., No. 49-Pet-13, 1949. (2) Benfratello, G., Energia Elletrica 28, 80, 486 (1953). (3) Bergelin, 0. P., Chem. Eng. 55, 104 (1949). (4) Brown, R . .4, S., Sullivan, G. 4.. Govier, G. W., Can. J . Chem. Eng. 38, 62 (1960). (5) Chisholm, F., Laird, A. D. K., Trans. A m . SOC. Mech. Engrs., 80, 276 (1958). (6) Davis, I., Weidner, R., Bull. 667, Univ. of Wisconsin. 1914. (7) Dumitrescu, Z . angew. M a t h . u. .Mech. 23, 139 (1930). (8) Gibson, A. H., Phil. M a g . 26, 932 (1913). (9) Govier, G. W., Radford, B. A , , Dunn, J. S. C., Can. J . Chem. Eng. 35, 58 (1957). (10) Govier. G. W., Short, M’.Leigh, Ibid., 36, 195 (1958). (11) Grassmann, P., Lemaire, L. H., Chem. Ing. Tech. 30, 450 (1958). (12) Habermann, W. L., Morton, R. K., Proc. A m . Sac. Civil Engrs 80, 25, 387 (September 1954). (13). Langhaar, H. L., “Dimensional Analvsis and Theory of Models,” LViley, N e w York, 1951. ’ (14) Lockhard, R. W., Martinelli, R. C., Chem. Eng. 9 g r . 45, 39 (1949). (15) Ntmet, .4.:,Szab6, A., “L6gnyomC:os vizemelo . . . Xcad. Sci. Hung., 1952. (16) Pattantyus, .4. G., “VizszolgCltatAs mtlykutakb61,” -4cad. Sci. Hung., 1942. (17) Peebles, F. N., Garber, H. J., Chem. Eng. Progr. 49, 88 (1953). Tech. (18) Pickert, F., Dissertation Hochsch. Berlin, Essen, 1932. (19) Randall, J., Eng. News30,341 (1893). (20 Shaw, S. F., Eng. and Oferating, 1943, p. 47. (21) Shaw, S. F., “Gas-Lift Principles and Practices,” Gulf. Publ. Co., 1939. (22) Thein, K., Gliickauf 86, 313 (1950). RECEIVED for review January 4,1960 ACCEPTED September 16, 1969
