We should like to thank W. R. Wilcox for his query concerning the turbulent region of axial non-Newtonian flow in annuli.
It seems interesting now to compare the predictions resulting from these two sets of definitions. For the case of laminar flow, comparison of the definitions of this author with ARNOLDG. FREDRICKSONthe analytical solution found by FredDEPARTMENT O F CHEMICAL ENGINEERING rickson and Bird ( 4 ) yields the following UNIVERSITY O F WISCONSIN relationship between f and -frRe(n,K) : MADISON, WIS.
R. BYRONBIRD
[fand
PHYSICHE TECHNOLOGIE TECHNISCHE HOGESCHOOL DELFT,HOLLAND
only enters to about the l i p power ( 3 ) ] it , would be extremely important in any prediction of the transition point between laminar and turbulent flow. There is also a more fundamental difference between these two sets of definitions in the case of turbulent flow. Those of this correspondence are fixed by the geometry and flow conditions of the system. However, Equations 1 and 2 require, in addition, some knowledge of the radius of maximum velocity, which may or may not be the same as for laminar flow, for which it can be analytically predicted ( 4 ) . For turbulent Kewtonian flow in annuli ,(8): it was found that X was about the same for turbulent and laminar flow, but was sometimes appreciably different fo? transition flow, which spread over a considerable range of Reynolds numbers. Considering this, those authors applied the hydraulic radius concept, which is exact for surfaces experiencing equal shear stresses, to the portion of the annulus outside the radius of maximum velocity. In this manner definitions and correlation of condition 2 above were obtained. Equations 1 and 2 appear to have the same basis, especially because they reduce to the same definitions for n = 1. The difficulty, of course, is that X may be a different function of n,K, and N R ~ ( n , ~ )than that found analytically for laminar flow. Once this function was found, however, the result could be a useful correlation for pressure drop prediction. Although rhese definitions are interesting material for speculation, an extensive experimental investigation of velocity profiles, pressure drops, and transition points is necessary to gain a clear and accurate picture of turbulent flow of power-law non-Yewtonian fluids in annuli.
NR,(n,K) defined by
Fredrickson and Bird]
LABORATORIUM VOOR
This, of course, is of the same form as that found for the usual case of laminar Newtonian flow in smooth tubes ( 3 ) . Likewise, comparison of Equations 1 and 2 with the analytical solution yields :
literature Cited (1) Bird, R. B., A.1.Ch.E. Journal 2, 428
(1956)
f
( 2 ) Bird, R. B., Stewart, W. E., Lightfoot, E. N.,“Notes on Transport Phenomena,”
= C(~,K)/-VRA w)
(5)
where C ( ~ , K=)
Chap. 6, Wiley, New York, 1958. (3) Chemical Engineers’ Handbook (J. H ,
Perry, editor), p. 382, Fig. 23, McGrawHill, New York, 1950. (4) Fredrickson, A. G., Bird, R. B., IKD.END. CHEM.50, 347-52 (1958). (5) Lamb, Sir Horace, “Hydrodynamics,:’ 6th ed., p. 586, Dover Publications, New York, 1945. (6) McCabe, W. L., Smith, J. C., “Unit Operations of Chemical Engineering,” p. 72, McGraw-Hill, New York, 1956. (7) Metzner, A. B., Reed, J. C., A.Z.Ch.E. Journal 1, 434-41 (1955). (8) Rothfus, R. R., LUonrad, C. C., Senecal, V. E., IND. ENG. CHEM.42, 2511-20 (1950).
[fand N R , ( ~ , defined K) as in Equations 1 and 21 Although this is not of the same form as Equation 4 and (3) it could equally well be used to predict P. For turbulent flow both sets of definitions yield similar results in the two known limiting cases : 1. I n the limit that K = 0 (powerlaw flow in circular tubes), both definitions reduce to the correlation of Metzner and Reed (7), found to work moderately well for power-law flow in tubes. 2. I n the limit that n = 1 (Newtonian flow in annuli) both definitions predict nearly the same P (Table I, lines 1 and 2, and correspondence of Fredrickson and Bird). The definitions of Equations l and 2 reduce to those proposed by Rothfus, Monrad, and Senecal ( 8 ) to correlate their data for Newtonian floiv in annuli.
SIR: Fredrickson and Bird recently suggested definitions for the friction factor and Reynolds number which may possibly enable prediction of pressure drops for turbulent flow of power law non-Newtonian fluids in annuli, using the usual friction factor-Reynolds number correlation (3) for smooth pipes. This author has also suggested possible definitions, which are:
For turbulent flow of power law fluids in annuli, however, the two defiWILLIAM R. WILCOX nitions, used in conjunction with the CHEMICAL ENGIXEERING DEPARTMENT correlation ( 3 ) yield very different values AND RADIATION LABORATORY for P (Table I). Table I also serves UNIVERSITY O F CALIFORNIA to point out the tremendous difference BERKELEY, CALIF. in Reynolds number that can occur when calculated by the two definitions. Although this is not too critical in calculating pressure drops [as X R t ( ~ , ~ )
where
CORRECTION Table 1.
Comparison of Calculations by Fredrickson and Bird (F&B) Method with Wilcox (W) Method
Purifying Thorium Nitrate by Solvent Extraction
P“/Lb. Force/Sq. rb
1 1 0.5 2
R 0.5 0.25 0.25 1.00
K
0.6 0.8 0.6 0.6
NR~(wY F&B W 23,859 33,500 5,970 8,760 80,500 38,900 4,560 48,000
FeetiFeet
fb
F&B 0.0063 0.0090 0.0047 0.0096
W 0.0058 0.0081 0.0057 0.0051
F&B 0.0587 0.335 0.0877 0.0447
W 0.0577 0.308 0.112 0.0259
~
~
f
in P 2
5 28 42
(appropriate lb. force-foot-second units); Ea = 1 foot/mcond; y = 60 1b.l Calculated from corresponding NE, and from relation for smooth tubes (1). Per cent difference in P based on P calculated by Fredrickson and Bird’s method. a
m = 1/1488
cubic foot.
~~
1 600
~~~
~~
INDUSTRIAL AND ENGINEERING CHEMISTRY
f
.
~ In the article “Purifying Thorium Nitrate by Solvent Extraction” [D. D. Foley and R. B. Filbert, Jr., IND. ENG. CHEM.,50, 144 (ISSS)], in the first table of column 3, and under the column heading “ 0 8 , data for the last three items 3.1, 97.7, and 106.4: respectively, should read: 0.82, 222.5, and 41.0, respectively.
’
