I
JOHN LOHRENZ and FRED KURATA The University of Kansas, Lawrence, Kan.
A Friction Factor Plot
. . . for Smooth Circular Conduits, Concentric Annuli, a n d Parallel
Plates
This new approach to the old annulus problem presents a friction factor correlation which can be used without difficulty and which yields a satisfactorily constant value of the critical Reynolds number
A
HYDRAULIC radius concept is used in all previous correlations of fluid flow in concentric annuli. Theoretical justification of the hydraulic radius concept is not clear. Davies and White (70) note that the hydraulic radius “takes into account to some extent the shape of the cross-section; moreover, it is the simplest quantity of this nature which has the necessary dimensions of length, and for noncircular pipes it is a convenient substitute for (the characteristic dimension) in the Reynolds number.” Based upon this argument of convenience, the use of the hydraulic radius has become customary for annuli and other geometries. As the hydraulic radius approach is empirical, a new study was made of the theory and experimental results of fluid flow through concentric annuli without introducing the hydraulic radius concept. An equivalent diameter for annuli has been defined which can be mathematically extended to the limiting cases of circular conduits and parallel plates. A friction factor plot has been developed for smooth circular conduits, concentric annuli, and parallel plates. Experimental data yield a good correlation. %
The Friction Factor Plot According to Equation 18 (p. 706), a plot o f f us. NEe with the parameter de/(& - &), should correlate all the pressure drop data for smooth circular conduits, concentric annuli, and parallel plates. I n the completely laminar region only, the parameter is unnecessary and Equation 12 applies. For all of the investigations listed in Tables I and I1 for which the original data were available, values of f, NRo and de/(& - d1) were calculated using a drum digital computer. A total of 1305 original data points for annuli and 77 original data points for parallel plates were computed. Some of the data were not considered further because the following criteria
were not satisfied: Any scatter of the data must be small, both walls of the annulus must be smooth, data must extend into the completely laminar region and agree with Equation 12. T h e original data which met these criteria (2, 4, 7, 8, 76, 26, 28, 33, 34) were used to develop the correlation. Plots o f f us. NRe were prepared for each of the annuli studied in these data. Figure 1 represents the results of this study. Not all of the original data considered is plotted o n Figure 1. Only sufficient d a t a to show the correlation, illustrate the reproducibility of the data, and allow use of the drawing for design calculations are represented. T h e upper curve on Figure 1 is the friction factor curve of Moody (24) for smooth circular conduits, excepting the range of NRe from 1000 to 4000, where the curve is from the data of Seneca1 and Rothfus (32)in the transition region. T h e curve for circular conduits, for which the geometric parameter, d./(dZ - d J , is unity, also correlates the concentric annuli data for values of the geometric parameter greater than about 0.85. T h e bottom curve of Figure 1 applies to parallel plates and small concentric annuli which approach the limit defined by Equation 13. Using a characteristic dimension of 2s or (dz - d l ) in the Reynolds number, Davies and White (77) found a critical Reynolds number, where flow deviates widely from laminar behavior, of 2880 for parallel plates. This is compared with their value of 2300 for circular conduits. They concluded that the critical Reynolds number “does hot occur a t a precisely constant value of (the Reynolds number) for pipes of different forms.” If Davies and White’s (77) value of 2880 is multiplied by 4 2 7 3 , a critical NRa of 2350 is obtained. This is good agreement with the expected value of 2300 obtained for cifcular conduits. This criteria for the Reynolds number is not satisfied for the function of Fredrickson and Bird (741,
T h e curve for d,/(dz - d l ) equal to 0.824 is based upon the data of Braun (4) and Walker, Whan, and Rothfus (33). T h e data from both sources lie on a single curve. This curve may be used as a basis for interpolation between the curves for circular conduits and parallel plates for design calculations. T h e original data yielding this curve are not presented to prevent crowding. Excellent agreement also exists between the data of Rothfus (28) and Walker, Whan, and Rothfus (33) for annuli where d,/(dZ - d l ) was 0.836. This agreement is especially interesting in view of the fact that the annuli of the two investigations had very different dimensions, and different test fluids, water, and air were used.
Previous Correlations According to the hydraulic radius concept, four times the cross-sectional area divided by the wetted perimeter can be used as the characteristic dimension in the Reynolds number. The Reynolds number obtained in this way should indicate the relative degree of turbulence for all geometries. For concentric annuli, the characteristic dimension defined by the hydraulic radius concept is (d2 - d l ) . Wiegand and Baker ( 3 5 ) recommended the use of this characteristic dimension after making a survey of the available data of pressure drop for fluid flow through concentric annuli. Using this characteristic dimension, Knudsen and Katz ( 7 7 ) find
0.07(j
I-(& -
lo.,,
d1)Vp
(1)
in the turbulent region. As expected from t h e hydraulic radius concept, Equation 1 is almost equivalent to that of Blasius ( 3 ) for circular conduits. Knudsen and Katz (77) note deviations as high as 3570. These deviations become particularly significant as (dz - d l ) becomes .small. Cald: well (6) notes that there “is no a priori reason why”
VOL. 52, NO. 8
AUGUST 1960
703
for annuli should give the same function as for circular conduits. Dimensional analysis shows that Equation 2 is an incomplete function. An additional parameter is necessary. The data studied by Wiegand and Baker (35)clearly show the effect of an additional geometric parameter. Davis (72)made a study of the available data of fluid flow through concentric annuli. He found by dimensional analysis:
a- J a N &< a" c.l
where for laminar flow u = +1;
(NKe
< 1000):
6 = +0.1
and for turbulent flow (NRe> 2100): u = +0.2;
Figure 1 may be used for design calculations in exactly the same way as other friction factor chartse.g. the Moody (24)chart. The agreement of the curves of Figure 1 with the experimental data i s about the same as for the Moody (24)chart for smooth circular conduits. Figure 1 i s considerably more accurate and generally more applicable than Equations 1 and 3. Although no increase in accuracy i s claimed for Figure l over the correlation of Walker, Whan, and Rothfus (33),several advantages may be cited. ,Figure 1 i s based on the data o f several investigations using different annuli and parallel plates with fluids o f widely different properties. Thus, experimental verification is more complete. ,Figure 1 i s not dependent on the experimental observation that the predicted and experimental radii of maximum velocity coincide for turbulent flow. Experimental observation of this coincidence, although established in limited ranges, becomes increasingly difficult for more turbulent flow and smaller annuli. ,Design calculations are simpler using Figure 1, especially when Figure 2 which gives the geometric parameter, de/(& - dl), as a function o f dl/dz i s used.
-
b = -0.1
Davis (72) plotted f" us. the Reynolds number using (dz - d ~as) the characteristic dimension, and found a correlation excepting a discontinuity in the transition region. Recent work (32-34) has, however, shown the reproducibility of pressure drop data in the transition region for fluid flow in circular conduits, concentric annuli, and parallel plates. Also the use of Davis' (72) equation does not seem advisable in the laminar region, where theoretical analysis yields a better representation of the pressure drop data. Rothfus, Monrad, Sikchi, and Heideger (30) were the first to discard the use of (dp - d l ) as a characteristic dimension. They noted experimmtal evidence ( 78, 29) that the radius of maximum velocity for highly turbulent flow appears to be the same as that which would be predicted for laminar flow by theoretical analysis. Applying the hydraulic radius concept to the fluid between the outer wall of the annulus and the radius of maximum point velocity, a characteristic dimension is:
in which, from laminar flow analysis, (5)
In the portion of the annulus considered, highly turbulent flow should be the same as that in a circular conduit with a diameter equal to the characteristic dimension defined by Equation 4. Walker, LYhan, and Rothfus (33) tested the characteristic dimension of Equation 4 using only their own data for six different annuli with water as the test fluid. Agreement was found between the friction factor us. Reynolds number curves for the annuli studied and smooth circular conduits in the highly turbulent region. As required from theory, separate lines on their friction factor plots are given for each annulus in the laminar region. Further, since the coincidence of the calculated and experimental radii of maximum point velocity does not occur in the transition and slightly turbulent regions, a separate curve was necessary for each annulus in these regions.
v
Figure 2. The shape factor d,/(dz-dJ as a function of dl/dz
-6.1
Theoretical Analysis
0.8
Lamb (20) derived the equation giving the point velocity profile for laminar flow in a concentric annulus:
.=--(- + g,APl 2pL
r2 2
I '
r:ax In
2)
(6)
where rmar is defined by Equation 5. By integrating Equation 6 between the
704
INDUSTRIAL AND ENGINEERING CHEMISTRY
FRICTION FACTOR CORRELATION Table 1.
Literature Survey of Studies of Pressure Drop through Concentric Annuli da
Investigators Becker ( 2 )
Lonsdale (23)
Winkel ( 3 6 ) Atherton ( I ) Caldwell ( 6 ) Kratz, Macintire, and Gould (19) Lea and Tadros ( 2 1 )
Schneckenberg (31)
- di)
dz, Feet
d1, Feet
0.07231 0.07231
0.06824 0.06955
0.816 0.816
0.38136 0.43428 0.48651 0.59114 0.64419
0.27992 0.43327 0.48546 0.53799 0.59029
0.816 0.816 0.816 0.817 0.817
Air
0.06227 0.06227 0.06227 0.08268 0.08268 0.08268 0.13031 0.13031 0.13031
0.04249 0.02605 0. 00062a 0.06253 0.042490.02605 0.10453 0.09383 0.08333
0.817 0.822 0.894 0.817 0.819 0.825 0.817 0.817 0.818
Water
(dz
0.0984
0.0820
0.817
0.1927 0.1927 0.1927
0.1542 0.0874 0.0700
0.817 0.821 0.823
Test Fluids Water
de = 2 / d i
Water Water, air, oils
0.0521
0.817
0.1383 0.1083 0.0833
0.817 0.818 0.819
0.02112 0.02112 0.02112 0.02112 0.02112
0.00092 0.00178 0.00392 0.00545 0.01033
0.864 0.850 0.834 0.828 0.820
Water
0.0266 0.0266 0.0266 0.0266 0.0266 0.0266 0.0266
0.0233 0.0246 0.0253 0.0258 0.0260 0.0262 0.0265
0.817 0.816 0.816 0.816 0.816 0.816 0.816
Water
Water Water Air
0.852
0.04226 0.04226 0.04226 0.04226 0.04226 0.04226 0.04226 0.04226 0.04226
0.000072 0.000246 0.000656 0.000853 0.001312 0.002989 0.005495 0.007743 0.01224 0.02083 0.02582
0.920 0.903 0.886 0.880 0.872 0.854 0.841 0.834 0.826 0.829 0.818
Zerban (37)
0.1195 0.1613 0.2033 0.2503 0.3908
0.1035 0.1244 0.1454 0.1689 0.2392
0.817 0.817 0.817 0.818 0.818
Air
Carpenter, and others (7) Govier (16) Rothfus (88)
0.0695
0.0521
0.817
0.3355
0.1250
0.823
0.2565 0.2565
0.1667 0.0417
0.818 0.836
Water Steam, air Air
Walker, Whan, and Rothfus (33)
a
0.0518
0.827 0.824
0.0312 0.0312 0.0312 0.0312
0.0283 0.0254 0.0225 0.0200
0.817 0.817 0.817 0.818
0.06250 0.06250 0.06250 0.06250 0.06250 0.06250
For laminar flow,
I NRe
0.816
0.0417
Substituting Equation 9 into Equation 7 and rearranging, a friction factor and a Reynolds number using d, as the characteristic dimension are defined as follows :
f = - 16
0.0039
0.1867
(dQ/dl)
Water Water, CaCl2 Brines
0.0617
0.1235
I
+ d f - ( d i - d:)/ln
(9)
0.328083
Knudsen ( 1 6 ) Braun (4) Nootbaar (36)
Instead of introducing the hydraulic radius concept, a more rigorous approach to the correlation of fluid flow in concentric annuli would be to define an equivalent diameter by direct analogy to the HagenPoiseuille equation for laminar flow through circular conduits,
Thus, one defines
0.1717 0.1723 0.1717
Fage and Townend (13) 0.0521 Cornish (9) 0.331115 Piercy, Hooper, and 0.04226 Winny (37) 0.04226
two walls of the annulus, a n equation in terms of the average bulk velocity, V , is obtained: APf
.
Limiting Cases. Two limiting cases exist. First, the annulus might be infinitely small, and second, the core of the annulus might be vanishingly small. I n the first case, as the annulus becomes smaller, curvature has less effect and fluid flow should approach that between parallel plates. Rothfus (28) states that “while the flat plate criterion is easily visualized, its mathematical justification is obscure.’’ Using de, such mathematical justification is clear. By the successive application of L’Hopital’s rule, one obtains lim [d,/(dz - d l ) ] = 4/2/3 = 0.816 (dz - d i ) + 0 Therefore, for parallel plates
(13)
1 I
de = 2 / 2 7 (dz - d l ) = 2//8/3s (14) For the second limiting case, it may be readily shown that,
Water Water Water, methylcellulose solutions
0.00163 0.875 Water 0.00417 0.855 0.00782 0.842 0.01033 0.836 0.02070 0.824 0.03117 0.820 The stated measurement appears to be in gross error, probably due to a misprint.
1 I
lim [d,/(d* - d l ) ] = 1 (15) 4 dz Thus, as the core vanishes, the annulus becomes a circular conduit and
’
+
I
de = dz (16) I n all three geometries considered, Equation 1 2 applies for laminar flow, when the correct equivalent diameter defined by either Equation 9, 14, or 16 is used.
I
Dimensional Analysis
~
The previous theoretical analysis cannot be extended into the regions of transition
and turbulent flow. Dimensional analysis has generally been used to correlate fluid flow in these regions. I t would be advantageous to use the equivalent diameter defined in the previous section in the dimensional analysis. Thus, + z [ ( A P t / L ) , V , P , P, gc, de, (dz - d i ) ] = 0 (17) Since there are seven variables and four primary quantities, three dimensionless VOL. 52, NO. 8
AUGUST 1960
705
1
,1
I
,1 I
groups will yield a complete function. Thus,
Other dimensions could have been used in Equation 17 instead of (dz - d l ) , for example dz or di. The dimension used, however, has the advantage of yielding a dimensionless shape factor, d,/(d* - di), which has two finite limits defined by Equations 13 and 15. The form of Equation 18 is similar to that of Fredrickson and Bird (74), who took the friction factor based on the hydraulic radius concept, f ’, as a function of NRe times the shape factor d,/(dz - d l ) . No particular advantage may be claimed at this point for either Equation 18 or the method of Fredrickson and Bird. Both require consideration of an additional geometric parameter in the turbulent region.
literature Survey A literature survey of the experimental studies of pressure drop for fluid flow through concentric annuli was completed. Table I gives various investigators arranged in chronological order, dimensions of the annuli used, and the test fluids. The literature survey shows an interesting controversy with regard to the experimental confirmation of Equation 12 for laminar flow in concentric annuli which approach either of the two limiting cases. Becker ( Z ) , Lonsdale (23), Winkel ( 3 6 ) , Caldwell (6), Schneckenberg (37), Cornish ( 9 ) , and Carpenter and others (7) worked with annuli which approached the limit of Equation 13-i.e., very small annuli. They found agreement with Equation 12 in the laminar flow region. The data of Nootbaar ( 2 5 ) for small annuli are in disagreement with Equation 12. Nootbaar’s results indicate his pressure drops were consistently lower than those predicted by Equation 12. Rothfus (28) noted four sources of error in the data of pressure drops through annuli. They are: Roughness of the inner and outer pipes, insufficient calming lengths, eccentricity of the core, physical impediments in the fluid stream. For small annuli, another source of error is small inaccuracies in the uniformity and measurement of the dimensions of the
Table 11. Literature Survey of Studies of Pressure Drop through Parallel Plates Aspect Test References s, Feet Ratioa Fluids (5) 0.0787 32 Water (11)
0.0375 0.0114
68 223
0.000495 0.000673 0.000682 0.000758 0.000794 0.00115 0.00120 0.00134 0.00156 0.00223
168 124 122 110 105 73 69 62 53 37
Water
Air Air 20 Water (54) 0.0583 The aspect ratio is the width of the experimental rectangular channel divided by height, s. (8)
(88)
706
0.0554-0.0580 0.0570-0.575
17 17
annuli. For falling cylinder viscometers, which are analogous to concentric annuli, Lohrenz, Swift, and Kurata (22) pointed out the extreme sensitivity to physical dimensions when the cylinder diameter approaches that of the tube diameter. Schneckenberg ( 3 7 ) recognized this sensitivity to measurement errors when he corrected the results for his four smallest annuli to account for some deviations from Equation 12. For the data of Nootbaar (25), if the outer pipe was slightly nonuniform so that the actual effective diameter was 0.002 inch larger than stated, fair agreement with Equation 12 would have been obtained. Lea and Tadros (27) studied annuli where the core was a wire or filament and approached the limiting case of Equation 15. Their results differed by as much as 28% from that predicted by Equation 12 for laminar flow. Lea and Tadros speculate that the reason for the disagreement was slip at the core of the annulus. Fage and Townend (73), Piercy, Hooper, and Winny (27), and Walker, Whan, and Rothfus (33) used annuli with small cores and found agreement with Equation 12. Some of the annuli of these investigators approached the limiting case of Equation 15, closer than any of those of Lea and Tadros (27). Piercv, Hooper, and Winny (27) do not state the calming length used. Incipient turbulence due to an insufficient calming length is another possible cause of the disagreement. The bulk of the experimental data for laminar flow in concentric annuli support the validity of Equation 12, even when the limiting cases are approached. A literature survey of the fluid flow pressure drop studies through parallel plates was also made. Table I1 lists the various references, the dimensions of the system, and the fluids used.
Nomenclature exponential constants in Equation 3, dimensionless dl = diameter of the core of the annulus, feet d z = inside diameter of the outer tube of the annulus, feet de = equivalent diameter of the annulus defined by Equation 9, feet f‘, f”,f = friction factors defined by Equations l , 3, and 10, respectively, dimensionless g c = universal gravitational constant, (1b.-mass) (feet)/(lb.-force) (secz) L = length of the annular passage, feet NRe = Reynolds number defined by Equation l l , dimensionless A P , = frictional pressure drop, 1b.-force/ square feet r = general radius in the annulus, feet rt = radius of the core of the annulus, feet r2 = inside radius of the outer tube of the annulus, feet rmax = radius of maximum point velocity for laminar flow defined by Equation 5 , feet s = perpendicular distance between parallel plates, feet u = point velocity in the annulus, feet/ sec. V = average velocity in the annulus, feet/sec. a,b
=
Greek letters: = proportionality constant in Equation 3, dimensionless y = viscosity of the fluid, 1b.-mass/feetsec. p = density of the fluid, 1b.-mass/cu. feet = a general function a
INDUSTRIAL AND ENGINEERING CHEMISTRY
literature Cited (1) Atherton, D. H., Trans. A m . SOC. Mech. Engrs. 48, 145 (1926). (2) Becker, E., Zeit. Ver. deut, Zngr. 51, 1133 (1907). (3) Blasius, H., M i t t . Forschungsarbeiten, No. 131, l(1913). (4) Braun, F. W., Jr., M. S. thesis, Oregon State College, Corvallis, Ore., 1951. (5) Buckingham, E., Engineering 115, 225 (1923). (6) Caldwell, J., Journal Roy. Tech. Coll. (Glasgow) 2, 203 (1930). ( 7 ) Carpenter, F. G., Colburn, A. P., Schoenborn, E. M., Wurster, A., Trans. A m . Znst. Chem. Engrs. 42, 165 (1946). (8) Corcoran, W. H., Page, F., Jr., Schlinger, W. G., Sage, B. H., IND. ENG.CHEW44, 410 (1952). (9) Cornish, R. J., Phil. M a g . 16, 897 (1933). (IO) Davies, S. J., White, C. M., Engineering 128, 71 (1929). (11) Davies, S. J., White, C. M . , Proc. Roy. Soc. (London) A l i 9 , 92 (1928). (12) Davis, E. S., Trans. Am. SOC.Mech. Eng. 6 5 , 755 (1943). (13) Fage, A.: Townend, H. C. H., Phil. Ma,q. 14, 500 (1932). (14) Fredrickson? A. G., Bird, R. B., IND. ENG.CHEM.50, 1599 (1958). (15) Govier, G. W., D.Sc. thesis, University of Michigan, Ann Arbor, 1948. (16) Knudsen, J. G.: Ph.D. thesis, University of Michigan, Ann Arbor, 1949. (17) Knudsen, J . G., Katz, D. L., “Fluid Dynamics and Heat Transfer,” McGrawHill, New York, 1958. (18) Knudsen, J. G., Katz, D. L., Proc. Midwestern 7st Conf. on FZuid Dynamics, University of Illinois, Urbana, p. 175, 1950. (19) Kratz, A. P., Macintire: H. J., Gould, R . E., Univ. Illinois Eng. Expt. Sta. Bull. 222 (1931). (20) Lamb, H.: “Hydrodynamics,” 6th ed.: Dover Publications, New York, 1932. (21) Lea, F. C., Tadros, A. G., Phil. M a g . 11, 1235 (1931). (22) Lohrenz, J., Swift, G. W., Kurata, F., to be published, A.1.Ch.E. Journal. (23) Lonsdale, T., Phil. M a g . 46, 163 (1923). (24) Moody, L. F., Trans. Am. Soc. Mech. Eng. 66, 671 (1944). (25) Nootbaar, R. F., M.S. thesis, Illinois Institute of Technology, Chicago, 1951. (26) Page, F., Jr., Schlinger, W. G., Breaux, D. K., Sage, B. H., IND. ENG. CHEM.44,424 (1952). (27) Piercy. N. A. V., Hooper, M. S., Winny, H. F., Phil. M a g . 15, 647 (1933). (28) Rothfus, R. R., D.Sc. thesis, Carnegie Institute of Technology, Pittsburgh, 1948. (29) Rothfus, R. R., Monrad, C. C., Senecal, V. E., IND.ENG. CHEM.42, 2511 (1950). (30) Rothfus, R. R., Monrad, C. C., W. J., Ibid., Sikchi, K. G., Heideger, 47, 913 (1955). (31) Schneckenberg, E., Zeit. angew. Math. u. Mech. 11, 27 (1931). (32) Senecal, V. E., Rothfus, R. R., Chem. Eng. Proqr. 49,533 (1953). (33) M‘alker, J. E., Whan, G. A,, Rothfus, R . R., A.Z.Ch.E. Journal 3, 484 (1957). (34) Whan, G. A., Ph.D. thesis, Carnegie Institute of Technology, Pittsburgh, 1956. (35) Wiegand, J. H., Baker, E. M., Trans. A m . Znst. Chem. Eng. 38, 569 (1942). (36) Winkel. R., Zeit. angew. Math. U. Mech. 3, 251 (1923). (37) Zerban, A. H., Ph.D. thesis, University of Michigan. Ann Arbor, 1940. RECEIVED for review December 17, 1959 ACCEPTED April 11, 1960
