kL
= over-all mass transfer coefficient for gas from bubble to
P
=
QA
=
R
=
r
=
s
=
t,
=
’
=
V
=
liquid, ft./hr. froth density, Ib./cu. ft. average quantity of gas absorbed per unit time, Ib.moles/hr. ratio of quantity of carbon dioxide absorbed into 0.085MN a O H to quantity absorbed into pure water under same flow conditions mean radius of bubbles in froth on sieve tray, ft. fractional rate of surface renewal, set.-' time of penetration of gas into liquid, sec. total volume of froth on the sieve tray, cu. ft. system of chemical composition different from pure water
(2) Coppock, P. D., Meiklejohn, G. T., Trans. Inst. Chem. Engrs. 29. 75 (1951). (3) Danckwer&, P. V.. 4.Z.Ch.E. J. 1 , 456 (1955). (4) Danckwerts. P. V., Kennedy, A. M., Chem. Eng. Scz. 4, 202 (1958). (5) Harned. H. S., Davis, R., J . A m . Chem. Soc. 6 5 , 2030 (1943). (6) Higbie, R., Trans. .4m. Inst. Chem. Engrs. 31, 365 (1935). (7) Leibson, I., Holconib, E. G., Cacoso, A. G., Jacmic, J. J., A.I.Ch.E. J . 2, 296 (1956). (8) Rennie, J., Evans. F.. Rrzt. Chem. Eng. 7 , 498 (1962). (9) Robinson. D. G.. Gerster. J. A , “Tray Efficiencies in Distillation Columns.” Final Report from Unixersity of Delaware, Am. Inst. Chem. Eng., New York. 1958. (10) Smith, R. K., Ph.D. thesis, Virginia Polytechnic Institute, 1965.
literature Cited
(1) Bowman, C. W., Johnson, A . I., Can. J . Chem. Eng. 40, 139 (1962).
RECEIVED for review July 27: 1965 ACCEPTED October 18, 1965
LAMINAR FLOW O F NON-NEWTONIAN FLUIDS IN CONCENTRIC ANNULI ROBERT D . V A U G H N ’ A N D P E R R Y D. BERGMAN Department of Chemtcal Engineering, Purdue Unicersity, Lafayette, Ind.
Experimental data on two non-Newtonian fluids over a 5000-fold range of flow rate confirm the failure of the power law model to predict pressure loss and flow rate in concentric annuli. A design method which uses circular capillary rheometer data on non-time-dependent fluids is developed and illustrated with experimental data.
flow of non-Newtonian fluids in concentric annuli of importance in plastic extrusion operations (751, in mixing of very viscous liquids ( 7 4 , and in well bore fluid circulation (9), and has received much attention in recent years. Theoretical solutions for various models (power law, Bingham plastic, Reiner-Rivlin, and Rabinowitsch) have been published (7, 72); however, no data have been published to substantiate the results. T h e general approach of No11 (simple fluid) has been extended to the case of helical flom (5), which includes concentric annular flow as a special case. More practical applications of these results have been undertaken (4, 73), again without experimental verification. The only experimental data which are known to the writers are for the flow of non-Newtonian fluids in the parallel-plates limiting case of concentric annulus (78, 79). I n both cases the authors used very specific models and, as a result, their work lacks generality from both fluid model and geometrical considerations. Moreover, the theoretical work on models other than the power la\\ and the No11 simple fluid are not considered in this paper. since the utility of these models has not been satisfactorily demonstrated. T h e utility of the power law model has been demonstrated for flow of non-Kewtonian fluids in circular tubes (3, 7 7 ) . I n view of the success of the power law model (Equation 3) in predicting pressure loss and flow in circular pipes, it was surprising that critical comments on the solution of Fredrickson and Bird (7) began to appear almost immediately. Metzner (70) noted that the power law solution requires that the constant, K , and the exponent, n. “be constant over the entire AMINAR
L is
1
44
Present address, LV. R . Grace & Co., Clarksville, Md. I&EC PROCESS D E S I G N A N D DEVELOPMENT
range of shear stresses from r R (which is positive) through zero, to T K R (which is negative).” Metzner shows that this cannot occur for non-Newtonian fluids and that the power law solution will, a t best, be an approximation. Metzner (70) also denies the recommendation of Fredrickson and Bird to consider the use of annular-flow data for viscometric calculations. The power law model predicts infinite apparent viscosity at zero shear stress; however, real non-Newtonian fluids exhibit a finite and constant viscosity (Newtonian) at zero shear stress. For a circular pipe this point corresponds to the center line. a t which point the fluid elements contribute little to the volumetric flow, since the product of the local velocity and radius appears in the definition of the volumetric flow, Q.
Q
=
2
JR
(UT)
K
dr
(1)
However, in an annulus the region of zero shear occurs at a finite value of the radius. Y = XR (see Figure l ) , and. as a result, the erroneous velocity prediction contributes substantially to the volumetric flow.
Q
=
2
?r
L
(UT)
dr
(2)
Since the contributions of the region near zero shear increase as the value of n decreases toward the “infinite pseudoplastic” limit. n = zero. the error becomes greater as the fluid becomes more non-Newtonian. Only in the limiting cases of flow through a circular pipe ( K + 0) is the rrror eliminated Savins ( 7 3 ) in an analysis of the parallel planes case found that for \ride ranges of n and K
the predictions of flow in annuli by the Fredrickson and Bird solution was astonishingly close to that of the limiting case of parallel planes for a power law fluid. Specifically, for values of K between 0.3 and 1 and for values of n bet\veen 0.33 and 1, the differences in the predicted floiv rate are less than 57c. I t appears that the Fredrickson-Bird solution of the po\ver law model describes an annulus as two parallel planes even for values of K as low as 0.3. The picture is at variance with physical reality. Since real fluids exhibit a finite viscosity as the shear stress and shear rate approach zero, Lvhereas the po\ver la\v model predicts an infinite viscosity, the poxver law model tends to predict lo\ver volumetric flolv rates than ~vouldbe observed with a real fluid-both fluids a t constant pressure loss. This conservative error is the source of the observation of Savins (73). If the flo\v rates predicted by the powrer laiv solution of Fredrickson and Bird were arbitrarily raised to conform with the behavior of real fluids, the similarity of the annulus solution to the parallel planes case, Ivhich was studied by Savins! \vould cease a t higher values of the diameter ratio, K : than the value of 0.3. Aside from the power law model approach, only the No11 simple fluid merits discussion. The approach of Coleman and No11 (5) and any other method of integration of experimental shear stress-shear rate data fail in the case of the annulus for a similar reason. The po\ver la\v model fails to predict the correct velocity a t the important shear stress (near zero), while the integration methods fail because it is difficult to measure shear rate-shear stress data experimentally over the entire range of shear stress, 7 = 0 to 7 = r R . The method developed in this paper requires data only in the region of 7 =
re.
Experimental
Flow of non-Newtonian fluids in circular and concentric annular tubes was studied to confirm the previous conclusions about the power law model and to provide a basis for developing a method of predicting pressure loss and flow rates for non-New tonian fluids in concentric annuli.
Equipment, T h e flow data were obtained with a n extrusion rheometer with changeable capillary tubes. Xitrogen gas from cylinders was used as the motive fluid. Circular and annular stainless steel capillary tubes could be inserted in the
+
VELOCITY PROFILES
Limit of Dilatant Fluid Newtonian Fluid Limit of Pseudoplastic Fluid Shear Stress Distribution
Figure 1. Shear-stress, typical velocity profiles, and geometrical parameters for axial annular flow of fluids
Table 1.
Dimensions of Tubes
T u b e Ah. A B C
D Annulus Inner tube Outer tube
Diameter, Ft. 0 00378 0 00285 0 00115 0 000685
Calculated Length, Ft 0 917 2 13 2 21 0 383
0.00521 0.01025
1.05 1 .05
bottom of the rheometer. Pressure drop was measured by calibrated pressure and a manometer, which measured the pressure in the gas space above the fluid. The pressure a t the exit of the tubes was atmospheric. Corrections were made for the static head of fluid above the tube inlet and for the kinetic energy losses. All flow measurements were made a t room temperature, 70' to 75' F. The circular capillary tubes were made from hypodermic tubing. The dimensions of the tubes used are given in Table I. All tubes were calibrated with a National Bureau of Standards oil of known viscosity. Any corrections because of imperfections or inlet and exit losses were allowed by using a corrected length (Table I ) . Experimental data \vere taken on three capillary tubes of different lengths to check for tube-end and time-dependent effects. None Lvere observed for either fluid studied. The concentric annulus was constructed by insertion of a i/l,-inch brass rod into a '/(-inch tube of approximately I/*inch wall thickness. Sleeves a t each end of the annulus were used to center the core of the annulus. The core was sealed to the sleeves while under tension to assure concentricity. Slots were drilled into each end of the outer tube to permit influx and efflux of the fluid under test. The holes were nearly ellipses \cith major axis of '/2 inch and minor axis of inch. The annulus was also calibrated with the NBS oil. Small end effects were incorporated into a revised tube length. From the diameters given in Table I , the value of K is seen to be 0.507. According to Vaughn (76) experimental studies with annuli of this radius ratio are of little value for establishing the position of maximum velocity (XR) in a concentric annulus; however, such studies are suitable for developing a design method for extrusion operations, etc. The fluids studied in this work were dilute aqueous solutions of sodium carboxymethylcellulose (CMC) (Du Pont) and Carbopol (CP) 934 (Goodrich). Details on the experimental work are available (2).
Results. Experimental data were taken on two fluids, 1.64% C M C and 3.0% CP, in both circular and annular tubes. The circular tube data were used to evaluate the power law constants, K and n, in Equation 3. 7
=
-K
($)"
(3)
From these constants, the value of K = 0.507, and the theoretical results of Fredrickson and Bird (7) the predicted pressure drop and flow rate Lvere calculated for the po\ver law model. These results are compared with the annular experimental results for the t\vo fluids in Figures 2 and 3. The po\ver la\v model is seen to be in error by about 25% at the higher flow rate and by about 1007, at the lolver flow rate. Unpublished experimental data of Fredrickson ( 6 ) show substantially the same conclusions. Proposed Design Method
The proposed design method combines the separate observations of previous studies on non-Nerctonian fluids in circular tubes and on Ne\\ tonian fluids in annuli Alves Boucher, and Pigford ( 7 ) obseived that for both Ne\\tonian and non-Newtonian fluids a logarithmic plot of APD 4L LS 324 TD?\ ields a VOL. 5
NO. 1
JANUARY
1966
45
'7
II
1
I
4 II
Figure 2. Comparison of experimental annular flow data on 1.64% carboxymethylcellulose solution with power law model
-1
I
I
Figure 4. Design chart for annular flow of 1.64yo carboxymethylcellulose solution
6 POWER L A W
n
2
..
I
-
- 1
:0.55
K .O
18
I
I
/
-
x
7
1
I
I
/
/ -Y , .,
IO
I \EXPERIMENT. lX:O 507
,t
I A-
I 1
I
Figure 3. Comparison of experimental annular flow data on 3.0% Carbopol solution with power law model
Figure 5. Design chart for annular flow of 3.0y0Carbopol solution
single line for a given fluid. For a Nexvtonian fluid these quantities are the ivall shear stress and wall shear rate, respectively, ~vhilethe slope is the reciprocal of the viscosity. For a non-Ne\vtonian fluid the definition of the \Val1 shear stress is unchanged, while the slope has been defined as n' (77). T h e slope, n ' , has been used to determine the wall shear rate ( 7 ) and to correlate fluid flo~vdata in circular pipes ( 7 7 ) . Rothfus et ai. (77) have observed that data on Newtonian flo~vin concentric annuli can be correlated using a friction factor based only on the outer \vall shear stress. The correlation achieved is identical Ivith that for circular tubes. These facts suggest that flow data for non-Newtonian fluids in circular tubes and concentric annuli can be made indistinguishable by plotting the outer wall shear stress, r R , and a modification of the term 3 2 Q / r D 3 , which is suggested by the Equation 4. form of the Newtonian result (8),
in both round tubes and concentric annuli are plotted in Figures 4 and 5. The only term in these parameters which is not \vel1 defined is the radius ratio of niaximum velocity, A. I n the absence of better results, the values predicted by the polver la\v (7) are used. I n spite of the deficiency of the power law in predicting flow rate and pressure loss for concentric annuli, the values of h \vhich are predicted reduce to the appropriate limits (76). Also, the value of X at K = 0.507 differs about +2.3% from the Neivtonian value. Only for values of ti less than 0.3 does the value of X for a non-Newtonian fluid differ significantly from that of a Newtonian fluid (76). T h e term: A, depends only upon K and n and is estimated by using the value of n' from circular tube data a t the corresponding value of the shear stress, T ~ . T h e results for both flow devices are seen to be well defined by a single line over a 5000-fold range of flow rate. T h e results of Wiley and Pierce (78) and Fredrickson ( 6 ) for circular tubes and concentric annuli show close agreement when plotted on the coordinates proposed in this paper. \L'hile it is as yet not possible to develop a general correlation, such as that which exists for circular tubes ( 7 7 ) , by means of graphs such as Figures 4 and 5, pressure loss and flow rate for concentric annuli can be predicted from pipe-flow data. Thus data from a capillary tube rheometer may be used to predict results for concentric annular tubes. The method, which is exemplified by Figures 4 and 5, involves a minimum number of assumptions and no model for describing the entire flow situation. T h e method relies primarily on scale-up of experimental data. Only in the estimation of the parameter,
These tw'o terms are:
(5) and
Data on the two non-Neivtonian fluids which were studied 46
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
A, is an unproved theoretical result from a model for nonNeutonian behavior used. T h e reliability of the po\ver law result in this particular instance is reinforced by the independent analysis of Vaughn (76). T h e following calculation illustrates the method.
X = 0.727
T h e abscissa coordinate may be calculated : 32 X 0.1
K
K ~ )
D n
Problem. Determine the pressure loss for 0.1 cu. foot per second flow of a 1.647, CMC solution through 10 feet of a concentric annulus with a n inside diameter of the outer tube of 0.1 foot and a K of 0.507. Data. Circular capillary tube data for the fluid are plotted in Figure 2. For a circular tube K = 0 and h = 0 and 1 - X2 = 1 - K~ = 1. As a result, the terms may arbitrarily be added to the usual coordinates, A P D / 4 L and 3 2 4 ’*D3, to yield those of Figure 4. Thereby the capillary tube data are converted into the coordinates for concentric annuli. Calculation. Under these conditions n = 0.33. For K = 0.507 and n = 0.33 the data of Fredrickson and Bird (7) yield :
324 (1 -
Nomenclature
K L
Example Calculation
iD3
acknowledge the support of E. I. d u Pont de Nemours 8r Co. and the B. F. Goodrich Co., which supplied the polymers.
X (0.1)3 X [ ( I - (0.507)*]
=
1370
Q r
R u
Z K
X p T
tube diameter, ft. constant defined by Equation 3 tube length, ft. power law exponent, dimensionless volumetric flo~vrate, cu. ft.//sec. variable radius, ft. radius a t outer \vall> ft. velocity, ft. ’sec. variable axial distance! ft. ratio of radii in annulus, K R.R, dimensionless position of plane of maximum velocity in annulus, dimensionless = Neivtonian viscosity, Ib.,-sec./sq. f t . = shear stress! Ib.,/’sq. f t . = = = = = = = = = = =
literature Cited (1) Alves, G. E., Boucher, D. F., Pigford, R . L., Chem. En?. Progr. 48, 385 (1952).
12) Bereman, P. D., M.S.Ch.E. thesis. Purdue Lniversitv, 1962. (35 Birdu. R. B.. A.Z:Ch.E. J . 2. 428-9’11956). (4j Brodkey. R’. S., 2nd. Eng. Chern. 54,44 (1962). (5) Coleman, E. D., Noll, LV., J . Appl. Phys. 30, 1508 (1959); 35, 2276 (1964). (6) Fredrickson, A. G., Ph.D. thesis, University of ivisconsin. 1958. 17\ Fredrickson. A. G.. Bird. R . B.. Znd. Ene. Chem. 50. 347 119581. (8j Lamb, H., “Hydrodynamics,” 6th ed:. p. 586. Doxer‘. N e h York 1945. ~
~
(9) Melrose, J. C., Savins, J. G., Foster, I$’. K., Parish. E. R.. Trans. A.Z.M.E. 213, 316 (1958). (10) Metzner, A. B., “Processing of Thermoplastics,” E. C.
From Figure 4 the corresponding ordinate is
Bernhardt, ed., Reinhold. New York, 1960. i l l ) Metzner. A. B.. Reed. J. C.. A.Z.Ch.E. J . 1. 434 (1955). (l2i Rotem, Z., J . Appl. Mech. 29, 421 (1962). ’ (13) SaLins, J. G., Trans. A.Z.,M.E. 213, 325 (1958). (14) Schrenk, it‘. J., Cleereman. K. J.. Alfrev. T.. Jr.. T r a n s . doc. Plastzcs Engrs. 3, 192 (1963). 115) Thomas. D. G.. Chern. Enu. Proer. 60. 35-6 11964) (16j Vaughn, R. D., SOG. P Z r z . Enirs. J.’3, 2?4(1965). (17) Walker, J. E., Whan, G. A., Rothfus, R. K., A.Z.Ch.E. J . 3, 484 (1957). (18) Wiley, R . M., Pierce, J. F., Chern. Eng. Progr. 47, 432 (1951). (19) Williams, M. C., Bird, R. B., A.Z.Ch.E. J . 8, 378 (1964). \
The pressure loss may be calculated:
I
~
6 X 4 X 1 0
Ap =
~
0.1 X [ l
-
(0.727)2]
X
1 -
144
= 35 Ib., per sq. inch
Acknowledgment
T-he authors are indebted to the Lubrizol Co. and the Purdue Research Foundation for financial support. They also
RECEIVED for review March 1, 1965 ACCEPTED September 7. 1965
VOID FRACTION VARIATION IN T H E SPOUTED BED ANNULUS An Appl-oximation L0 U IS A
.
MA D0 NNA
,
Pennsylvania Military College, Chester, Pa.
The void fraction variation in the annulus of the spouted bed is difficult to ascertain experimentally. However, a differential expression may b e obtained from the equation of continuity and, with the use of the solids velocity curves of Gishler et a/., the expression can b e integrated to obtain an equation that relates bed porosity or void fraction to bed depth as an approximation.
spouted bed was initially investigated by Xlathur and ( 2 ) . However: some problems \vere not completely covered by any of the workers in this field. Among the most important concepts introduced was that there existed in thr spouting bed a void fraction variation or profile in the annulus. but the theoretical significance of this variation was nrver pursued. This publication is written to develop a n approximate expression relating a solid void fraction gradient HE
Tc-.,ishler .
or profile to bed depth. T h e void fraction is difficult to obtain experimentally while the bed is operating; howevrr, if solids in a containing vessel can be thought of as in approximate slug flow, a calculation can be made. This assumption should be valid because solids flowing through a containing vessel indicate a pseudolaminar flow-that is, each particle has only one component velocity vector. This is experimentally observed in the annulus of the spouted bed, where the motion of the VOL. 5
NO. 1
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47
