I
OCTAVE LEVENSPIEL Bucknell University, Lewisburg, Pa.
Longitudinal Mixing of Fluids Flowing in Circular Pipes Design charts incorporating data from the literature can be applied to pipeline studies and design of chemical reactors
THE
problem of fluid self-mixing in vessels has been studied by Danckwerts ( 3 ) ,who considered it in the light of the distribution of, or spread in, the residence time of the fluid in the equipment. Danckwerts ( 4 ) also summarized experimental techniques for the measurement of the distribution of residence times. These methods all involve introducing tracer material into the incoming fluid stream and measuring its concentration in the outgoing fluid . stream, but they differ in the way the tracer is introduced. It may be introduced continuously but in amounts varying sinusoidally with time (6, 70) or in any cyclic manner. It may also be introduced continuously in concentration Co after an initial time, before which no tracer is introduced a t all. Finally it may be injected in a slug, essentially instantaneously. Typical tracer input and output concentrationtime diagrams illustrating these methods of tracer introduction are shown (Figure 1). Danckwerts called the dimensionless form of the output signal (Figure I$), in which the ordinate is measured as C/Co and the abscissa as vO/V, the “F-curve.” He also called the dimensionless form of the output signal (Figure l,c), in which the ordinate is measured as CV/Q and the abscissa as &/V, the “C-curve.” The variable nO/V, which may be called reduced time, is unity for the period required to pass one void volume of fluid through the vessel. With the above choice of variables the F-curve rises from 0 to 1, the area under the C-curve is always 1, and when integrated the C-curve yields the corresponding F-curve. Though originally developed for vessels, these tracer diagrams are also applicable to flow in pipes. Numerous investigators (3, 6, 9-12, 78-22) have pointed out that longitudinal fluid mixing may be characterized
by a diffusion-type model governed by Fick’s law of diffusion
ac a2c 8 = D -dX2
(1)
where D , which may be called the longitudinal dispersion coefficient, uniquely characterizes the mixing process., Assuming Fick’s law to be applicable, equations have been derived (73, 79) relating D to the expected C-curve. Thus
(2)
which is a family of C-curves with D/uL as parameter. Similarly the Fcurve, which is the integral of the C-curve, is an unique function of DluL. I t has been shown (73) that the spread or width of the C-curve as measured by its variance d is related to D/uL by
D
UL
=;
( d * m -1)
(3)
For the special case that D/uL < < 1, the C-curve as given by Equation 2 approaches the normal or Gaussian error curve (1
4
-;>t
($)
(4)
in which case Equation 3 reduces to
D _ UL
-1 -
zu2
(5)
Equations 3 and 5 afford a convenient method for finding D from experimental F- and C-curve data; D can also be
found by methods suggested by cyclic tracer injection experiments (Figure 1,a).
Fluid Mixing in Pipes The diffusion-type model rests on the assumption of homogeneous fluid mixing, and under certain conditions it may be applicable to pipe flow. When applicable unique functional relationships would be expected between the system and fluid variables and the longitudinal dispersion coefficient, D. If mixing is not homogeneous, as in the case of a short pipe (end effects) or in a pipe with bends, average D values result. These would not be expected to correlate directly with the system variables unless the actual flow patterns were accounted for. Because of the complexity of such situations only flow through straight pipe is considered. Dimensional Analysis. Consider the variables affecting the mixing process during steady state flow in a straight pipe from the point of view of their dimensions. These would include the variables usually pertinent to flow problems: the fluid density, p ; the viscosity, y; the velocity of flow, U; the pipe diameter, d, and the roughness, E , which is a measure of the depth of its surface irregularities. The roughness has been shown to be important only in turbulent flow (76). In streamline flow the radial interchange of material between adjacent fluid layers moving at different velocities may also be expected to affect D. This effect involves the molecular self diffusion of the fluid as characterized by the molecular diffusion coefficient D,. Thus, in general, for both streamline and turbulent flow = *~[P,P,U,d,e,Dvl
According to the Buckingham TI-theorem, this functional relationship may be VOL. 50, NO. 3
MARCH 1958
343
expressed groups:
in terms of dimensionless
.-.s
.-a
Equation 6 shows that a dimensionless group involving the longitudinal dispersion coefficient should be a function of the Reynolds number, the Schmidt number, and a relative roughness number which are measures of the viscous, molecular diffusion, and roughness effects, respectively. Streamline Flow. As pipe roughness is not a significant variable in streamline flow Equation 6 in this case should reduce to
.d
a
3 2
'B
Q,
P
5.-
*
E
tj
D Ud =
+,C(?)> (A)]
(7)
and consequently a plot of D/ud us. dup/p (Figure 2) should result in a family of curves with p / D , p as a parameter. In a series of articles (78-20) Taylor considered the self-mixing of fluids flowing in pipes from a theoretical point of view. For streamline flow he showed (78, 20) that when L
d2
u %mE radial mixing is great enough compared to longitudinal mixing to assure a uniform cross-sectional concentration of material everywhere. For this situation the diffusion-type model is applicable, and the longitudinal dispersion coefficient, D , can be found by relating the radial to the longitudinal mixing of material. The radial mixing of material is assumed to be caused solely by molecular diffusion, while the longitudinal mixing of material is assumed to be caused
L
Q)
P Ly X
Lo
bn
+ 0
x
to
CQ N
10 .m t
$
4
0
0
a
0
9 ;?;;?; 3 o o
. .
?
?
z
-U ,
l-
a
DI b-
m
z
10
W
0 m
o
2 %
sa
'9 0 0
m
Z O 0
Y I m n
0
9 2.0 C N . N
N
S
" 8 9 2
00'
G 0
m
m
0:
2
k INPUT
I
V
- TIME
Figure 1. Typical recordings for various modes of tracer injections a.
b. C.
344
INDUSTRIAL AND ENGINEERING CHEMISTRY
Cyclic input Step input Jump of instantaneous input
in turbulent flow, Taylor (79) considered the diffusion model to be universally applicable, in which case the longitudinal dispersion coefficient is given by
solely by the parabolic velocity distribution of flow in a pipe with no end effects. The final result of Taylor's analysis is
D=-- d2u2 192 D,
e = 3.57 dj
(9)
ud
which, in the dimensionless form of Equation 7, yields
The condition of applicability Equation 10 is that
where f is the Fanning friction factor. The analysis is based on the assumption that Reynolds' analogy holds. According to this assumption the intraphase transfer of heat, momentum, and matter by a fluid in turbulent flow are analogous. Hence the extent of fluid mixing (transfer of matter) can be calculated by shear stress and pressure drop information (transfer of momentum). To find D from Equation 13 the value of f must be known. Experimentally, f has been found to be a function of the flow and system variables, or
of
This is Equation 8 in dimensionless form. When plotted (Figure 3), Equation 10 predicts that all experimental data should fall on a straight line of slope 1 and intercept 1/192. When Equation 10 is plotted as in Figure 2 a family of curves of slope 1 and intercept (1/192) (p/pD,) should result. Turbulent Flow. In turbulent flow molecular diffusion should not be expected to contribute significantly to the transport of material, this process being greatly overshadowed by turbulent eddy mixing. Thus, Equation 6 should reduce to
2 ud = a4 [(?),
cient information to allow them to be reanalyzed are considered. Experiments in the streamline flow region, done by Taylor (78) and Fowler and Brown (7), are shown in Figure 3. Taylor's three runs were made by measuring the dispersion of potassium permanganate in water flowing in a tube 0.0504 cm. in diameter. The conditions of Equation 11 were satisfied, and Figure 3 shows that the points are close to the theoretical curve. In plotting the points a problem arose in choosing a value from the molecular diffusion coefficient, Do, for the potassium permanganate-water system as this varies considerably with concentration (8). The value chosen was that reported at the average potassium permanganate concentration. By contrast to the good agreement found by Taylor, the data of Fowler and Brown are much below the theoretical curve. However, this is not surprising as the condition for applicability of the diffusion model, Equation 11, was far from being satisfied. For the points further away from the curve
(;)I
thus checking Equation 12. A number of analytical expressions of the above relationship are available (75, 76). However, as these are relatively complex compared to a graphical representation, the latter is used. The result of Taylor's analysis is shown in Figure 2 by a curve based on Equation 13 and a value of f for commercial pipe given in Perry (75). A family of curves for different values of the parameter e/d are obtained if the charts for f presented by Moody are used (74); however, for the present purposes the one curve found in Perry (75) is sufficient.
(12)
and consequently a plot of D/ud us. dup/fi should result in a family of curves with E / D as a parameter. In his theoretical analysis of mixing Figure 2. Correlation for both streamline and turbulent flow ranges of D/ud, a longitudinal dispersion number, as a function of Reynolds number
L 1 28.8 d- w 220
while for the points closest to the curve
As the requirements of Equation 11 are approached the experimental points approach the curve; however, in no case are the conditions of Equation 11 satisfied. The transition from streamline to turbulent flow is shown by Fowler and Brown to occur a t a Reynolds number of about 2300. A marked change in the magnitude of mixing as measured by D/uL accompanies this transition.
Discussion To test the reliability of the predictive equations they should be checked against experiment. For this purpose all published investigations which present suffi-
@,0,0.ideal conditions
&&A.
Data obtained in pipe with few bendse.g., commercial pipeline V,V. Mixing between fluids of different physical properties 0. Flow in artificially roughened pipe 0. Curved pipe
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.
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VOL. 50, NO. 3
MARCH 1958
345
For turbulent flow conditions considerable data are available under widely varying flow conditions. Figure 2 and Table I summarize this information. However, not all the experiments were done under the ideal conditions of straight pipe and uniform fluid properties, for much of the information was obtained in commercial pipelines. When fluids of different physical properties were used the Reynolds number chosen was that for the 50 to 50 mixture. Figure 2 shows that as the Reynolds number decreases from 10,000 to 2300, experimental findings deviate increasingly from the theoretical curve. This discrepancy may be explained by considering the assumption on which the analysis of dispersion in the turbulent flow region is based. The assumption is that molecular diffusion plays an insignificant role in the mixing process. Now in the laminar sublayer, molecular diffusion does play a role in the mixing process so that when this layer has an appreciable thickness, as it does for Reynolds number below 10,000, deviations from theory might be expected. Thus one might expect the Schmidt number to appear as another parameter with increasing importance as the Reynolds number decreases to 2300. The only data available in this region are for a narrow range of Schmidt numbers, so the significance of this parameter cannot be determined until further experiments are carried out, Beyond a Reynolds number of 20,000 there is but one point obtained under ideal conditions. Nevertheless. such data that are available allow the inference to be drawn that the theoretical curve is the best estimate for mixing in this region. S o t e that most data scattering occurs in experiments with mixing of two different fluids. Additional data (2) not included in Table I on fluid mixing as a function of pipe length show that the length of gasoline-gasoline and kerosine-gasoline contamination in successive flow varies, respectively, as the 0.482 and 0.529 power of the pipe length for pipes from 4 to 12 inches in diameter. This checks closely with the diffusion mode1 which predicts (see Equation 5) that the length of contaminated section varies as the 0.5 power of the pipe length. Example. A pipe 1 foot in diameter and 100,000 feet long is to pump two Auids successively. The Reynolds number of a 50 to 50 mixture of the two fluids is 20,000. Determine the amount of fluid at the pipeline outlet which is contaminated due to intermixing of components. Fluid is considered to be contaminated if the composition of the components is between 5 and 95%. From Figure 2 for the given flow conditions _ -- 0.3 ud
346
co
But d
L = Therefore
$ = ($)
(e)
=
3
x
10-6
For this low value of D/uL, Equation 5 may be used to find the length of the mixing zone. Thus =
2.45
x 10-3
where u is measured in pipe lengths of fluid. But the central 90y0 of the area under the normal curve will lie within 3.29 u. Therefore, the section contaminated because of intermixing is = = =
(3.29)(2.45 X 10-3) 8.06 X 10-3 i e lengths of fluid (8.06 X IO+? ?105)(~/4) 634 cubic feet of fluid
Conclusions
If the longitudinal dispersion coefficient is a variable which actually characterizes the mixing process, unique functional relationships with usual flow variables should yield definite correlations among the variables. Thus charts based on the assumption that longitudinal fluid mixing is analogous to the process of diffusion should yield definite correlations among the variables. If the longitudinal dispersion coefficient does not characterize the mixing process the charts should show a scattering of data. Results of numerous investigations under widely varying conditionse.g., pipe diameter, a factor of 300; Schmidt number, a factor of 1000-are brought together into a narrow band thus validating the use of the diffusion model. O n the basis of theory and experiment these charts are universally applicable in the turbulent region, but they can only be used under certain conditions in the streamline region. In the streamline region the few data available indicate that the theoretical predictions are reliable. However, much further experimental work needs to be done, The scatter in the turbulent region data can be explained by the fact that the conditions of applicability of the diffusion model, that of using straight pipe, were not satisfied. Further experiments should be carried out in this region to narrow the range of uncertainty in the recommended curves, to determine the possible roIe of the Schmidt number at low Reynolds numbers, and to investigate the effects of pipe roughness and curvature; these variables have been shown to increase the extent of mixing above that found in straight smooth pipe. Nomenclature
C = concentration of tracer, fluid volume of tracer per volume of fluid, dimensionles:
INDUSTRIAL AND ENGINEERING CHEMISTRY
concentration of tracer a t point of introduction into fluid, dimensionless d = diameter of pipe, L D = longitudinal dispersion coefficient defined in Equation 1, L2/ T D, = molecular diffusion coefficient, =
f =
L2/T Fanning friction factor, dimensionless, defined by T~ = fPU2/2
L = a length of pipe
-
dimension of length M = dimension of mass Q = volume of tracer of unit concentration which would correspond to actual amount of tracer introduced into fluid, L3 T = dimension of time u = vL/V, average flow velocity, L / T v = volumetric fluid flow rate, L2/T volume of pipe, L3 x = space coordinate surface roughness, L E = B = time, measured from time of injection of tracer into the flowing fluid, T P = viscosity, M / L T P = density, MIL3 uz = variance of C-curve. in terms of pipe lengths, dimensionless T O = shear at pipe wall, M / L T i
v =
literature Cited
(1) Allen, C. M., Taylor, E. A., Trans. Am. SOC. Mech. Engrs. 45, 285 (1923). ( 2 ) Birge, E. A,, Oil Gas J . 176 (Sept. 20, 1947). (3) Danckwerts, P. V., Chem. Ens. Sci. 2, l(1953). (4) Danckwerts, P. V., 2nd. Chemist 3, 102 (1954). ( 5 ) Davidson, J. F., Farquharson, D. C., Picken, J. Q., Taylor, D. C., Chem. Eng. Scz. 4,201 (1953). ( 6 ) Deisler, P. F., Jr., Wilhelm, R. H., IND. ENG.CHEY.45,1219 (1953). (7) Fowler, F. C., Brown, G. G., Trans. Am. Inst. Chem. Engrs. 39, 491 (1943). (8) Furth, R., Ullmann, E., Kolloid-Z. 41,307 (1927). ( 9 ) Hull, D. E., Kent, I. Mi., IND.ENG. CHEM.44, 2745 (1952). (10) Klinkenberg, A,, Sjenitzer, F., Chem. Eng. Sci. 5 , 258 (1956). (11) Kramers, H., Alberda, G., Zbid., 2, 173 (1953). (12) Lapidus, L., Amundson, N. R., J . Phys. Chrm. 56, 984 (1952). (13) Levenspiel, O., Smith, W. K., Chem. Eng. Sci. 6 , 227 (1 957). (14) Moody, L. F., Trans. Am. Sac. Mech. Engrs. 66, 671 (1944). (15) Perry, J. H., “Chemical Engineers’ Handbook,” 3rd ed., Sect. 5, McGraw-Hill, New ’T’ork, 1950. (16) Rouse, H., “Elementary Mechanics of Fluids,:’ chap. 7, Wilep, New York, 1946. (17) Smith, S. S., Schulze, R. K., Petroleum Engr. 19, 94 (1948); 20, 330 (1948). (18) Taylor, G. I., Proc. Roy’.Soc. 219A, 186 (1953). (19) Zbid., 223A, 446 (1954). (20) Ibid., 225A,473 (1954). (21) Wehner, J. F., Wilhelm, R. H., Chem. Eng. Sci. 6,89 (1956). (22) Wilhelm, R. H., Chern. Erg. Prarogrr. 49,150 (1953). RECEIVED for review March 25, 1957 ACCEPTED June 28, 1957
