hi 7i,, Re Rec
2U U
U’ V
e
P P V
Av x o
= unit vector in direction of incident beam
= unit vector in direction of scattered beam = Reynolds number based on bulk velocity, U D p / p = Reynolds number based on velocity at center of pipe = time-averaged longitudinal velocity = center line time averaged longitudinal velocity = bulk longitudinal velocity, volume flow rate per unit area = instantaneous longitudinal turbulent velocity component = root mean square of longitudinal turbulent velocity = instantaneous velocity vector in arbitrary direction = one half of angle between incident and scattered light beams = dynamic viscosity = density = mean frequency of laser-Doppler signal = full band width at half height of laser-Doppler signal = vacuum wave length of incident beam
literature Cited
Fabula, A. G., U. S. Naval Ordnance Test Station, Tech. Pub. 4226 (1966). \ - - - - I -
Fa&:A., Townend, H. C. H., Proc. Roy. SOC.AlS6, 656 (1932). Gadd, G. E., Nature 212, 874 (1966).
Goldstein, R. J., Hagen, W. F., Phys. Fluids 10, 1349 (1967). Goldstejn, R. J., Kreid, D. K., J . Appl. Mech. W E , 813 (1967). Goldstan, R. J., Kreid, D. K., in ‘Measurement Techniquae in Heat Trapsfer,” AGARD, 1969. University of Minnesota, HTL TR No. 85 (1968). Goren, Y., Norbury, J. F., J . Basic Eng. 8%D, 814 (1967). Jones, Ph.D. thesis, Universit of Illinois 1966. Laufer, J., Natl. Advisory dmmittee {or Aeronautics NACA Re t. 1174 (1954). Lumgy, J. L., A pl. Mech. Rev. 20, 1139 (1967). Mickelson, W. Natl. Advisory Committee for Aeronautics NACA Tech. Note 5670 (1955). Ri ken, J. F., Pilch, M., St. Anthony Falls H draulic Laboratory, Eniversit of Minnesota Tech. Re t. 42, 8er. B (1963). Ri ken, J. $, Pilch M., St. Anthony balls Hydraulic Laboratory, bniversity of Minnesota Project Rept. 71 (1964). Robertson, J. M., Martin, J. D., AZAA J . 4, 2242 (1966). Sandborn, V. A,, Natl. Advisory Committee for Aeronautics NACA Tech. Note 9286 (1955). Smith, K. A., Merrill, E. W., Mickley, H. S., Virk, P. S., Chem.
€f,
Eng. Sci. 22, 619 (1967).
Squire, W., Castro, W., Costrell, J., Nature 218, 1008 (1967). Virk, P. S., Merrill, E. W., Mickley, H. S., Smith, K. A., MolloChristensen, E. L., J . Fluid Mech. SO, 305 (1967). RECEIVED for review June 17, 1968 ACCEPTED January 27, 1969 Work sup orted by the University of Minnesota Graduate School unier Grant 429-0806-8520 and by the National Heart Institute, National Institutes of Health, under Contract PH-43-67-1122.
PRESSURE GRADIENT AND L I Q U I D HOLDUP IN IRRIGATED PACKED TOWERS J. E. BUCHANAN’ Departmenl of Chemical Engineering, University of New South Wales, Kensington, N.S.W . , Australia An examination of the influence of liquid holdup on effective pore size suggests that, for a fixed gas flow, kHJm5where Ht is the total holdup pressure gradient below the loading point should be proportional to ( 1 and k an empirical coefficient approximately equal to 2 / ( a X effective pore size), a being the packing external area per unit packed volume. Experimental results with air flow through $-inch Raschig rings having dry voidage 0.70 are satisfactorily correlated with k = 2.1. When total holdup is calculated as the sum of experimental static holdup and operating holdup calculated by the author’s recently proposed method, the value k = 2.0 is found. An initial part of the holdup has no effect on the pressure gradient. The final equation for these rings is:
-
(dg&/P&G2)
=
7.8(1
+ 52 pQ/uGdPG)[l - 2.O(Ht - 0 . 0 1 ) ~ ’
where A p is the pressure gradient, U G the superficial gasvelocity, d the nominal ring size, The influence of varying dry voidage is not considered.
PQ
gas density, and
P Q gas viscosity.
IN T H E design and operation of packed towers for gas-liquid mass transfer operations, one of the most important variables is the pressure loss in the vapor phase passing through the tower. In vacuum distillation low pressure loss is necessary for the success of the operation and an accurate knowledge of the pressure loss is essential for optimum design; in gas absorption-desorption operations when the gas must be supplied under pressure artificially, the economically best design is established by setting off the capital cost of the tower and packing against the capital and operating costs Present address, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 02139 502
I&EC
FUNDAMENTALS
of the blower required. Only in normal or high pressure distillation or when the gas phase is supplied under pressure at no cost is the pressure loss of little moment. It is surprising, therefore, to find that no formally sound and extensively tested equation is available for the prediction of the frictional pressure gradient in irrigated packet columna: no equation generalized in respect to liquid flow rate and properties and tested over wide ranges of those variables. The inadequacy of some of the available empirical relations is indicated in a recent report of plant tests by Clay et al. (1966).
The available theoretical treatments of the subject agree
that the pressure gradient equation for a given shape of packing should take the form:
where E is the actual voidage in the irrigated packing and the Reynolds number function can be evaluated from experiments on single-phase flow. In irrigated packed towers the actual voidage is the voidage of the dry packing less the total liquid holdup. e = EO - Ht Thus the problem may provisionally be divided into two parts: effect of variable holdup and prediction of holdup variations. Considering first the influence of voidage on pressure loss, we find that several approaches have been used. Cchida and Fujita (1938), in their pioneering work, derived t'he empirical rule that for a given packing and a fixed gas flow, pressui'e gradient should be proportional to e-15t. Later workers adopted equat,ions originally proposed, with some theoretical basis, to describe single-phase flow in porous media. Thus Morton et al. (1964) favor a form of equation attributed to Carman (1937), which requires pressure gradient to be proportional t,o e-3; Brauer (1960) and hlersniann (1965) use the rule proposed by Leva (1947), which implies a factor of e+ ( 1 - E ) . Each of these latter equations involves also a further correctioii term which is important only a t low gas Reynolds numbers and is, i i i fact, negligible in practical calculations. For beds of the common commercial packings, these rules give significantly different predictions. The primary aim of the prebent work, therefore, has been to t'est, these proposed equations against experimental measurenients over a wide range of operating conditions. None gives a satisfactory description of the experimental results and so a novel semitheoretical approach is developed. This analysis arrives at, essentially similar result,s, but by a radically different route. The proposed form of equation has greater flexibility than those mentioned above and admits more accurate correlation of the experimental data. Experimental data required for the testing of the correlations are values of pressure gradient and holdup measured simultaneously over a wide range of fluid flow rates and liquid properties, in part,icular liquid viscosity. S o such data are available in the published literature. The experimental work described below was designed to supply it. To avoid dealing, at this time, with the effects of variable packing shape arrangement and initial voidage, only a single packing was studied. Because of its simple, regular, and well defined shape and because there exists a large body of pressure gradient data for this form of packing, the Raschig ring type was chosen. But the general conclusions should be applicable to any packing of the film type. Only the coefficients in the describing equations-which are indeed shape factors-should need to be evaluated anew to extend the equations to packings of different shapes. Theoretical
The independent variables influencing the gas phase pressure gradient are: column diameter, D ; packing size, d ; gas flow rate, UG, density p ~ viscosity , p ~ liquid ; flow rate, U L , density p L , viscosity p ~ surface , tension a; and local gravitational field, g, and a simple application of the methods of dimensional analysis suggests that besides the shape a t least seven independent dimensionless variables are required to
describe the state of the system. One of the variables may be eliminated immediately. So long as the ratio D/d is sufficiently large-a minimum value of about 8 is often quoted-variation in column diameter has little further effect. The present discussion deals only with such cases. The six remaining dimensionless variables still stand in marked contrast to the two independent quantities, Re and e in Equation 1. It is desirable therefore, before dealing specifically with the theoretical model, to anticipate some difficulties which will appear later by showing, more clearly than has been done previously, how pressure drop can be predicted by the comparatively simple Equation 1, what assumptions are involved, and, more particularly, what are the limits of validity of the equation. In discussing the influence of irrigation on gas phase pressure gradient it is necessary to consider the interactions between the gas stream and the liquid film on the packing. These may be studied under three main headings, the different, though not necessarily independent, ways in which the streams affect each other. Geometric Interaction. The first and most obvious connection between the two streams is that they compete for flow space in their conduit; together they fill the packing void space. As liquid flow and hence liquid holdup increase, there is less room for flow of gas and a higher pressure gradient necessarily results. Buoyant Interaction. The presence of a flowing gas phase in the same conduit produces a change-usually a decrease-in the available head loss for the liquid stream. At any level in the packing the static pressures in the two streams must be the same. But because of its density and frictional losses, the pressure in the gas stream increases downward through the column. In flowing 1 foot down through the packing, the liquid loses not 1 foot head but a head reduced by (gpG/gc Ap)-the pressure rise in the gas phase. I n effect, this pressure gradient acts on the liquid against the gravitational force: I t is most easily taken into account by applying to the term g, wherever it appears in p ~ the liquid flow equations, a correction factor: (1 - p ~ / gcAP/gPL). If gas flows cocurrent, down through the packing Ap is p usually ~ very small, the available negative and, since p ~ / is head increases with gas flow. Interfacial Traction. Surface Drag. With two fluids moving in opposite directions through the same space, there must be some drag of each upon the other a t the separating surface. Whether this drag is significant in a given situation depends upon the other flow resistances present. In this regard the effects on the two streams may be considered separately.
+
LIQUID. When no gas is flowing, the liquid film is supported entirely by the solid packing. The only flow resistances acting upon it are due to liquid viscosity and the tortuous flow path imposed by the shape of the packing. Apart from the small buoyant interaction mentioned above, this is still true when a small countercurrent gas flow is established, as is shown by the fact, amply demonstrated by Elgin and Weiss (1939), Shulman et al. (1955), and many others as well as in the present experiments, that with moderate gas flows the liquid holdup is substantially independent of gas rate. If the natural effect of drag is absent, it may safely be assumed that no significant drag is operating. Evidently, within this range of gas flows, the drag imposed by the gas on the liquid surface is small compared with the flow losses within the liquid film itself. But as gas flow is further increased, the drag increases and VOL.
0
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AUGUST
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503
there comes a point when the holdup begins to increase. This is known as the loading point. It appears to represent the boundary of the region in which the drag of the gas on the liquid surface is negligible, the point at which the interaction becomes important. GAS. The flow situation in a packed tower is such that, as a result of the continuous changes in flow direction, crosssectional area, and shape of the gas flow path, form drag is greatly predominant over skin drag as a mode of pressure loss. Large rates of pressure loss do not necessarily entail correspondingly large values of shear stress a t the boundary. As a corollary, experiment shows that variations in skin drag have little influence on the total pressure loss. By varying liquid viscosity it is easy to produce a wide range of liquid surface velocities for a given liquid holdup. But for such cases the gas phase pressure loss a t a given gas flow is found to be constant. Again, a t least in the accuracy of pressure measurement used in the experiments described, there is usually zero pressure gradient at zero gas flow, confirming that there is negligible traction by the liquid film on the gas. With very high flows of low viscosity liquid, this is no longer quite true. Under these conditions a finite pressure gradient develops a t zero flow or, what is equivalent, a measurable gas flow with no pressure gradient. This is the phenomenon described by Uchida and Fujita (1938), which they called “sucking.” But even in this case the present experiments suggest that the behavior is not caused by simple traction, but rather by the formation of bubbles or cells of gas which are then carried down with the liquid. The mode of action corresponds closely to the pore closure model described by Lerner and Grove (1951), more closely indeed than does the performance near the flooding point which their account was meant to portray. It represents the onset of “slug” flow familiar in studies of two-phase flow in pipes.
A very wide range of operating conditions is available between the loading and sucking limits. Most commercial towers are designed to work within this range as a matter of practical convenience and economy. If the limits are exceeded, pressure loss begins to increase very rapidly, with little corresponding improvement in mass transfer efficiency; it is only a short step further to complete failure by tower flooding. Within the practical range there is negligible dynamic interaction between the streams. Since high liquid rate necessarily entails low gas rate a t loading, it is possible for the operating limits of loading on the one hand and sucking on the other to merge. At very high liquid rates no gas rate may exist for which the interaction between the streams is negligible. For these reasons the present study deals only with the flow regime between the stated limits. I t is the absence of dynamic interactions which allows the equation to be drastically simplified. Within the specified operating limits the liquid flow variables act independently of the gas flow variables. The two may be considered separately and the pressure gradient equation may be written in a general way as: AP = $Ed, L r ~PG, , PG, Shape,
(4 g, UL,PL, P L , r,Shape)]
or, using the conventional diniensionless forms :
f
=
$[Re@, Shape, (ReL, Fr, a/pLg#, Shape)]
(2)
including now only four independent dimensionless variables. The only significant interaction between the streams in this flow regime is a purely geometrical one. The irrigating liquid occupies some of the void space in which gas would otherwise flow. In conjunction with the solid packing, the liquid merely establishes the boundary of the gas flow conduit, 504
l&EC
FUNDAMENTALS
the initial packed assrmbl~7modified by the presence of a liquid film on the packing surfaces. Now all the details of this film are defined by the liquid flow variables given, and within the preload range Equation 2 is exactly true. That the required function of the liquid flow variables is the bed voidage is the natural and simple next assumption. But it is only an assumption, and although it has proved useful in this and in associated fields it must be tested experimentally. It would seem that the distribution of the liquid film should be as important as its total quantity and that this distribution might show wide variation in different flow conditions. Rut accepting the assumption, the voidage is a function of the fluid flow variables listed. E
=
to -
H t = J (Rer, Fr, u/pLgd2, Shape)
(3)
and Equation 2 becomes:
(4 or, what is really the same: f = $ (Reo, Ht, Shape)
(4a )
The final step, the complete separation of these two remainiiig variables, requires several further minor assumptions. In the range of gas Reynolds number ( U G ~ P G / P involved, G) the friction factor for a dry tower is only a weak function of Reynolds number. With fluid int,eractions ignored, the situation is just that of a dry tower with slightly changed dimensions and shape. The assumption that these slight changes cause no significaut change in the Reynolds number function seems reasonable and allows further useful simplification. Equation 4 is now brought to the required form:
I t is appropriate now to examine how holdup influences the pressure gradient or, more exactly, how it affects the various terms in the pressure drop equation. The quantities primarily affected are the effective pore size, 6, and the mean gas velocity in the pores, v. Neither is measurable, nor, indeed, exact15 definable, so they are represented in the flow equations by the packing size d, and the superficial gas velocity, UQ,both quantities being clearly defined and accurately measurable. In a packed assembly of specified shape any definable linear dimension or velocity will be simply related to these quantities:
6 = ad v=
When liquid is applied to the packing, the effective pore size is reduced. If a film of mean thickness A is spread over the packing surface, 6 will be reduced to a value about 6 - 26. The term d representing it in the equation is replaced by d’, where d‘
Y
(6 - 2 A ) / a = d - 2A/a
(5 )
With the flow passages thus restricted, the pore velocity is naturally increased. The flow area is reduced by a factor about (d’/d)2 and the pore velocity increased in the same ratio. The term Vc representing it in the equation becomes UG(did')'. Now the pressure gradient in a dry tower is given by: g b p o = PGCG’/d*$
(Reo, Shape)
(6
and if it can be assumed that the Reynolds number function is the same in both cases-that is, that the shape is virtually
unchanged-the be given by:
pressure gradient in the irrigated tower will
A p = Ap0( d / ~ ? ' ) ~ g o b = ( d / d ' ) 5 ( P G L ~ G * / ~(ReG, ) * X Shape)
Noting that d' in the form:
=
d when A
=
(7)
0, Equation 13 may be put
d'/d = 1 - y A / d
(8 1
where y = 2/a
Also
Amem= H t / a where H t is the total holdup and a the packing interfacial area. For a given shape of packing, the product ad is a 5.) dimensionless constant. (For Raschig rings ad Substituting in Equation 8
-
d'jd
=
1 - kHt
where k is a constant given by: k - y/ad
or k -
(9 1
2/aad
Substituting now in Equation 7, gcAp = (1 - k H t ) - 5 * (PGUG2/d)*X((Rec,Shape)
(10)
or, if Ap0 is the pressure gradient for the same gas flow in a dry tower , Apo/Ap = (1 - k H t ) 5 (11) that is, for this model: +'(e)
= (1
- IcHt)-5
(12)
Experimental
Apparatus. The general arrangement of the apparatus is shown in Figure 1. The column was a tube of acrylic plastic 5%inches in i.d., packed t o a depth of 55 inches with 5/8-inch (nominal)
ME TER
PUMP
Figure 1.
General arrangement of apparatus
porcelain Raschig rings supplied by Hydronyl, Ltd. The properties of the packing were: Mean ring height Mean ring diameter Mean wall thickness (calcd.) Number per cubic foot Yoidage
0.639 inch 0.629 inch 0.096 inch 5270 0.701
The packing size, d, was taken to be 0.634 inch, the mean of the ring height and diameter. Below the column were an air inlet tee and the liquid tank, a 30-inch length of 6-inch glass pipe. Inside the tank were fitted a thermometer and an internal gage glass, a length of tube extending from the bottom of the tank to a return bend well above the liquid level and fixed to the tank wall with epoxy cement. The liquid content in the tank could be observed in this tube without disturbance from surface waves and taking no account of suspended air bubbles. Liquid flow followed a closed circuit. The liquid was drawn from the liquid tank and pumped through the flowmeter up to the distributor, from which it flowed over the packing and back into the tank. Liquid flow was metered using the movement of a Parkinson-Cowan S3G 1-inch rotary piston positive displacement meter. The counter gears and dial were removed and the meter was fitted with a pulse-generating device from which electrical pulses were sent to a timer-counter a t rates of either 2 or 20 pulses per revolution of the piston. This arrangemeat gave a meter which was easily calibrated; the calibration was almost independent of viscosity and of flow rate over a range of about 50 to 1 for a given liquid. The minimum usable rate fell as liquid viscosity increased, which also fitted in well with the packed column characteristics. Liquid was distributed over the packing from 29 copper tubes 1/4 inch in 0.d. and 0.049-inch wall thickness set at 25/32-inch square pitch and projecting 4 inches from a brass tube plate. The lower ends of the tubes were about 3 inch above the top of the packing and were all cut off a t the same level. This arrangement had the useful result that, with aqueous solutions, when the liquid flow was stopped by closing the control valve, surface tension forces at the tube openings prevented the entrance of air; the distributor and associated piping remained full of liquid and could be held in that state for as long as was necessary for the packing to be fully drained and the liquid level at zero operating holdup to be established. Because of its low surface tension, the hydrocarbon oil could not be held back for long times and, although it was quick-draining, its operating holdup could not be measured so accurately. Air was supplied to the column from a blower and metered by a bank of calibrated rotameters. Pressure drop was measured over a 36-inch gaging section from 13 inches below the top of the packing to a level 6 inches above the packing support screen. The pressure sampling points were located within the bed and well away from the wall, using the fitting shown in Figure 2. Except that the piece nearer the wall was not hollow, the parts inside the column corresponded in shape to two Raschig rings. When air was passing through the tower, the pressure gradient was measured with one of two slack diaphragm draft gages: a Dwyer Magnehelic 0- to 0.5-inch water gage meter for the lower pressures and a Negretti & Zambra 0- to 9-inch water gage meter for the higher. For the single-phase runs an inverted U-tube was used with air over the flowing liquid. For the measurements of static holdup, a 12-inch length of the same tube used for the column was fitted with a screen bottom and hung from an Ohaus Model 1122 balance. This container hung inside a tank which was covered to minimize drafts and liquid evaporation. Materials. The irrigating liquids used were water and several concentrated solutions of sucrose in water and light diesel fuel oil (Dieseline). Approximate properties a t the VOL.
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NO.
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AUGUST
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505
Figure 2.
Table 1.
Approximate Properties of Test liquids Density,
Solution Water 455; sucrose
SOc& sucrose 67.5y0 sucrose Dieseline
Pressure tapping
G./Nl.
1.OOO 1.204 1.263 1.331 0.810
Viscosity, Centipoises
1.0 5.5 45
200 2
Surface Tension, Dynes/Cm.
Static Holdup (Calcd.)
71 65 69 69 27
0.026 0.022 0.022 0.021 0.017
operating bemperature of the solutions used in gas pressure drop runs are given in Table I. Densities were measured by hydrometers. Viscosity was measured with Ostwald-type viscometers and for the sucrose solutions agreed well with the standard values reported by Bates (1942). Surface tensions were measured with a Cambridge-Du Nouy platinum ring tensiometer. For single-phase pressure drop measurements water and a sucrose solution of viscosity 10.9 centipoises were used. For the measurements of static holdup, a set of solutions was used, similar to those in the pressure drop runs. In one case, as a result, apparently, of biological action, a sucrose solution exhibited an unexpectedly low surface tension. Otherwise the properties were much the same. Procedure. STmw HOLDUP.The empty bucket was first wett,ed with the liquid under test and drained. It was then filled with a known weight of air-dried packing and the whole was onze more thoroughly wetted by pouring the liquid over it. The packing was then drained to constant weight, but in no case for less than 24 hours, and the holdup (pounds of liquid per pound of packing) was measured as the difference between the weights of dry packing and drained packing in the drained bucket in each case. In calculating the holdup in the usual volumetric terms (cubic feet of liquid per cubic foot of packed volume) it was assumed that holdup was proportional to the weight of packing and the final results were brought to a common basis of the voidage 0.701 observed in the tower. SINGLE-PHASE PRESSURE DROP. The tower was filled with the test liquid to a level well above the upper pressure tapping. The liquid was circulated through the regular liquid pump and flowmeter while pressure drop readings were taken. TWO-PHASE PRESSURE DROPAND LIQUIDHOLDUP. E X perimental work with a new liquid phase was begun by pumping the liquid over the packing at a high rate and for a long time, so that the packing was saturated with the liquid. The liquid flow was then stopped by closing the control valve and 506
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FUNDAMENTALS
the liquid was allowed to drain from the packing until drainage ceased while the pipes and distributor remained full, as described above. Thus the liquid level for zero operating holdup was established. The liquid flow was then resumed and the packing once again saturated and allowed to drain in the same way, but this time for only a short time, from 15 t o 30 minutes. Thus was established a subsidiary standard holdup which could be used later to check quickly for liquid evaporation or leakage. In the case of the Dieseline it was found more practical to establish this standard level with a small known liquid flow. After these preliminaries the two-phase pressure drop measurements were begun. The liquid flow was set and the air flow was increased in steps up to the maximum flow while readings were taken of air flow, liquid flow, pressure drop, liquid level in the tank, liquid temperature, and air pressure a t the rotameters and in the column. After each run air supply temperature and humidity were measured. With the air pressures these were used for calculating air density in the rotameters and in the column. The air flow rate increments were chosen to give about equal increments in (AP)"~. After each run the standard holdup was established and the liquid level checked. Any losses-and they were always small-were made up before the next run. Measured holdups were adjusted by a small correction to take account of the liquid in free fall between the packing and the liquid standing in the tank. Results
Static Holdup. Static holdup was measured over a rather wider range of liquid properties than in the pressure drop experiments. The results were required for, and apply only to, the Raschig rings used in these experiments. For a fixed shape of the packing pieces the static holdup is described by H8
=
x
b/PLg#)
(13)
Figure 3, a graph of the experimental points plotted according to Equation 13, shows also the best fitted straight line. The values of H , for the pressure drop runs (Table I ) were interpolated from this graph. As suggested by Figure 3, the claimed accuracy of static holdups is not high, mainly because of doubts about the amount of evaporation which may have occurred during draining. The estimated accuracy of f0.002 is adequate for the present purpose. Pressure Gradient, Holdup. SINGLE-PHASEFLOW. The experimental results are summed up in the simplest way 011 Figure 4, where experimental values of the friction factor (dApg,/pU2) are plotted against the Reynolds number (d UP/P1. These results were correlated by an empirical expression
1
Figure 3.
2
3 ( u / e g d r )x 1 0 0 Static holdup results
100 50
f
20 10
5
Re Figure 4.
Friction factor in single-phase flow
first proposed by Reynolds (1900) for flow in pipes and for flow in porous media by Forchheimer (1901); it takes the form : f = F(1-l- C/Re) (14) and has been found very successful in dealing with experimental data. The best values of the constants were calculated by a linear regression of the experimental values of f and l/Re. The calculated values led to the equation:
f = 8.6(1 4-52/Re)
(15)
The curve of this equation is shown on Figure 4. The friction factor, f , is the same quantity as the function 4 (Re) of Equation 1 . In regard to the consistency of the experimental results, Figure 4 speaks for itself. The accuracy of the final equation depends upon the absolute accuracy of the flow measurements and the goodness of fit of the chosen form of equation to the experimental results. Considering these matters, it is estimated that within the range of investigation Equation 6 should describe the function with an accuracy of f50j,. TWO-PHASEFLOW.The results of the two-phase flow experiments are analyzed using the assumptions stated above that the pressure gradient equation takes the general form: dApgc/pcUG2 = +(Re).4’(e)
(1)
and that the Reynolds number function can be evaluated from experiments with single-phase flow such as those described. Substituting from Equations 3 and 4, the relation can be expreqsed in the more explicit form:
P = [dApgc/pG(l where
+ 52/Re~)]”*=
m = [F+’ (e)]1’2
(16) (17 1
(It is undesirable to introduce a t this point the value of F from the single-phase experiments.) Below the loading point the liquid holdup and hence the voidage are substantially independent of gas rate and 4’ (e) is a constant. Therefore, if the experimental values of P are plotted against corresponding values of the gas velocity, UG,below the loading region, the points should lie on straight lines of slope m passing through the origin. Figure 5 is a typical plot of experimental data treated in this way. For the experimental run with air flow through dry packing and for runs in which the packing was wet but there was no liquid flow, a straight line through the origin gave an excellent fit to the experimental points. In the other runs, with some exceptions noted below, after several points were set aside as being in a transition region, the remainder clearly defined two straight lines, one passing through the origin and covering the range of flows below the
Figure 5. results
Pressure gradient in irrigated tower, typical PL
-
45 cp.
p~
-
1.283 g./ml.
UL, Ft./Sec. X
Curve
A
loa
0 0.61 4.6 15.9 24
B
C D E F
37
loading point, the second describing performance in the loaded region. The gas velocity a t the intersection of these lines is the loading velocity. This method of data selection is partly subjective and may be criticized on that score; but the implied loading points show reasonably consistent trends on the graphs of P and of holdup against gas velocity. In any case, the procedure entails the rejection of high velocity points only and has little effect on the value finally calculated for the preload slope. In a few cases the gas velocity was not taken to a level sufficiently high for the loaded range to be clearly defined. For these runs it was assumed that points for which A p was less than 0.7 inch of water per foot were definitely in the preload range and only these points were used for correlation. The experimental results for the two highest water flows were exceptional, in that no straight line through the origin was clearly defined. They were anomalous also in other ways, discussed a t greater length below. A typical plot of holdups measured a t the same time as the pressure drops is shown in Figure 6 complementary to the P graph of Figure 5. The plotted values are total holdups, each being the sum of a measured operating holdup and a static holdup estimated from the correlating line of Figure 3. No correction for liquid evaporation has been applied; the amount of evaporation was small in every case and negligible in the preloading range. Holdup was found to follow the course described by many workers. It was substantially constant over the preloading range and began to rise sharply just before the estimated loading point. Again the runs for the two highest water VOL.
8
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507
'JI
.15
I
p,
.......
\
Wottr
.........
'.
0.128
The value 0.128 for m-2 in single-phase flow correspond to a value of 7.8-i.e., 1/0.128 for F . It leads to:
4 (Re) = 7.8 (1
+ 52/Re)
Also shown in Figure 10 is the extrapolation of the curve of Equation 26 into the high holdup range. It gives a good fit for the data for high viscosity fluids but low prediction of pressure gradient for the low viscosity fluids a t high flow rates. Using Equation 24 as the final predicting expression, the equations may now be brought together into an almost complete set which should give useful predictions for values of total holdup up to about 0.1.
HQ H, Ht
4 ’ ( ~= ) [l - 2.O(Ht - 0.01)-J5
m
= d P / d U G , dimensionless
P
= [dApg,/pQ (1 52/Re~)1’/’LT-~, ft./sec. = pressure gradient, ML-2T-2, lb/sq. ft., ft.
+
AP U
= voidage, fractional free space, dimensionless
B
= voidage of dry packing = dynamic viscosity, ML-lT-l,
(24)
The only doubtful quantity is the static holdup. No general equation is available ; it should be evaluated experimentally for each packing and liquid. In the absence of such data for large rings, the static holdup may be ignored, with usually only minor error in the prediction of pressure gradient. Because the measurements were made in a small-scale tower and the influence of variable initial voidage has not been investigated, it would be premature to offer the equations which have been developed as the final correlation for pressure gradient in ring-packed towers. But the proposed theoretical approach does supply a possible methodology for the final solution. The equations can give reasonably accurate predictions, if their use is confined to the moderate holdup region. Nomenclature
Fr Re
a C d
D
f
B F
Q Bc
Froude number, C:L2/gd, dimensionless Reynolds number, U d p / p , dimensionless interfacial area of packing, L-l sq. ft
