AXIAL MIXING OF FLUIDS I N TURBULENT FLOW THROUGH CONCENTRIC ANNULI JOSEPH V.
SMITH’ AND JOSEPH J. PERONA
University of Tennessee, Knozville, Tenn. 37916 Axial dispersion during turbulent flow in concentric annuli was studied experimentally and theoretically. Experimental results were obtained for annuli with diameter ratios of 0.182, 0.272, and 0.454, and for straight pipe. The experimental technique consisted of injecting a dilute solution of sodium nitrate into the annular conduit and measuring the concentration distribution along the axis with electrical conductivity cells installed at two axial locations. The axial dispersion coefficient, mean velocity, and hydraulic diameter were grouped to form an inverse Peclet number, called the axial dispersion number, which was correlated with the Reynolds number. The axial dispersion number for annuli was the same as that for pipes when the hydraulic diameter was used as the length parameter. Equations based on the Taylor diffusion model were developed for predicting the degree of dispersion, theoretically. Solution of the equations requires a knowledge of the mean radial velocity distribution, the friction factor, and the Schmidt number. Theoretical values of the dispersion number were too sensitive to the radial velocity distribution to permit the use of generalized velocity distributions. The predicted dispersion numbers were about 60 to 70% of the experimental values.
of the most useful models for describing nonideal flows het::‘ dispersion model, This has several forms, each of which involves a different level of simplification. These models are useful for systems in which the flow more nearly approaches plug flow than the opposite extreme, backmix flow. Highly turbulent flows approximate plug flow and are thus amenable to description by the dispersion model. In turbulent flow the velocity fluctuations are rapid, numerous, and small with respect to vessel size; therefore they may be considered random. This consideration suggests the use of the analogy between the actual fluid mixing and the mixing that is entirely due t o a diffusional process. I n this way a n apparent diffusivity may be defined, and a linear second-order differential equation used to describe the mixing. The equation may be solved by classical methods. Khile dispersion models provide phenomenological descriptions of turbulent mixing and offer no insight into the basic nature of the fluctuations, they are of value for design calculations. Taylor (1964) published the first analysis of turbulent mixing in pipe flow based on velocity profiles. By employing Reynolds’ analogy between mass and momentum transfer and assuming a universal velocity profile for pipe flow, based on the data of several investigators, he estimated the effective coefficient of diffusion, K , to be 3.5 d ~ f - l ’ ~ . Aris (1956) presented a new basis for Taylor’s analysis by describing the distribution of solute in terms of its moments in the direction of flow measured above the mean velocity. He showed that the rate of growth of the variance is proportional to the sum of the molecular coefficient of diffusion and the Taylor diffusion coefficient, K . Aris further showed that a finite distribution of solute tends to become normally distributed in highly turbulent flow. A slightly different version of Taylor’s theoretical analysis, presented by Tichacek, Barkelew, and Baron (1957), included the effect of molecular diffusion, Which Taylor had ignored, and the velocity profiles used were actual experi‘Present address, Tennessee Eastman Co., Kingsport, Tenn.
37662.
mental curves rather than a generalized profile. The solution was in the form of integrals that could be easily evaluated by graphical means. Taylor’s analysis, on the other hand, involved much more difficult numerical integrations. The effect of molecular diffusion on the total dispersion mas found to be negligible whenever the Schmidt number was larger than 100. Taylor showed that the calculated dispersion coefficient was extremely sensitive to the velocity profilesthat is, a variation of less than 3% in the velocity profile caused a difference of 60% or more in the diffusion coefficient. Levenspiel (1958) showed that mixing data, accumulated under widely varying conditions, could be correlated when plotted as a dimensionless dispersion number E/GDH (inverse Peclet number) us. Reynolds number. Levenspiel and Bischoff (1963) presented a detailed analysis of all dispersion mechanisms and models that had been developed a t that time. They also reviewed the methods available for measuring the degree of mixing in various flow systems. Experimental
The experimental data were treated according to the axial-dispersed plug-flow model, which is described mathematically by
When the two points where measurements are taken are infinitely distant from the ends of the vessel, Aris (1959) showed that the differences in the variances of the concentration distributions a t these two points are n z (a’)z = Npe
(2 1
Bischoff and Levenspiel (1962) presented charts shelving the magnitude of the error encountered in using Equation 2 to compute the mixing coefficient, when the second measureVOL.
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ment point is located at various distances from the end of the vessel. They plotted (Nip, - hrp,,)/ATpe (where hTpem refers to the double infinite pipe case) against the ratio of the distance between the second measurement and the end of the test section to the distance between the two measurement points. In this investigation, the length of the end section was 1.5 feet, and the distance between measurement points was 13.58 feet; thus, the ratio was 0.11. Parameters on the charts were n’p, and b, which is the ratio of the dispersion coefficient in the end section to that in the test section. From Cassell’s (1966) work, the length Peclet number for pipe flow with the present equipment is about 40. If it is assumed that the length Peclet number for flow in annuli is about the same as that for pipe (the experimental results indicate that this is a valid assumption), and that the mixing in the exit section is not greatly different from that in the test section, so that b lies approximately between 0.5 and 3, it can be concluded that end effects at the downstream probe will not lead to an error greater than 2%. The flow system was basically that used by Cassell (1966). It consisted of a water supply, a pump, a rotameter, and a tracer injection section ahead of the test section. The annular test sections were made with three different inner tube sizes in a la-inch pipe, in order to obtain a range of diameter ratios. Also, one set of data was taken for straight pipe with no core. The inner tube diameters were 1/4, 3/8, and 5/8 inch, yielding diameter ratios of 0.182, 0.272, and 0.454, respectively. The upstream end of each tube was tapered to about a 20’ angle relative to the axis of the tube. Stainless steel struts 1/16 inch in diameter were located a t intervals of 32 inches to support and center the cores. Three struts on a triangular pitch were used a t each interval. The first set of struts was located 3.5 inches from the upstream end of the core in all three cases. Each strut was either soldered or fastened to the tube with epoxy glue. It was filed until it made a sliding fit with the inner wall of the pipe. Struts for a given tube were filed to the same length, within 0.003 inch. The first set of struts on the cores helped to break up any large eddies that were introduced as the fluid entered the test section. I n the case of a run with no core, a steel wool pad was used to break up the eddies. Two 1/8-inch static pressure taps were drilled 100 inches apart in the 19-inch pipe in order to obtain pressure drop data (to determine friction factors). The first tap was located 76.5 inches from the injection section. -1 U-tube manometer was used t o measure pressure differences. The experimental test section length, defined as the length between the two conductivity probes, was 13.58 feet in all runs. An entrance length of 6 feet 11 inches preceded the test section. This length is equivalent to 50 hydraulic diameters of pipe with no core and to 110 hydraulic diameters for the annulus having the largest diameter ratio. Since Olson and Sparrow (1963) have reported that, for annuli, 20 to 25 hydraulic diameters are sufficient for obtaining a t least 957, of fully developed flow, the selected test section length (above) ensured that the flow would be well developed at the upstream probe. An exit length of 1.5 feet was provided. Further details are given by Smith (1967). Experimental Results
The experimental results, with the results compiled by Levenspiel (1958) for straight pipe, are shown in Figure 1. The axial dispersion number, E/D& is an inverse Peclet number and is plotted against the Reynolds number. Each data point on Figure 1 actually represents the average of three runs. The variances recorded a t the two measurement points were averaged over three identical runs before being 622
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FUNDAMENTALS
c
c
4
LEVENSPIELS COLLECTED DATA RANGE FOR P I P E
0 0.182 DIAMETER RATIO ANNULUS
0 0.272 DIAMETER RATIO ANNULUS A 0.454 DIAMETER RATIO ANNULUS
O.! 104
D O , NRe
105
= 77
Figure 1. Axial dispersion numbers for flow in annuli and pipes 1000
800 600
400 200 0
0
.I
-2
1000,
,
,
0
.I
.2
-3 -4 -5 -6 T I M E , SECONDS
.3
.4
.5
.6
.7
-8
-9
10 .
.7
.E
.9
10 .
1.1
I
T I M E , SECONDS
Figure 2. Concentration pulses at upstream and downstream measurement points at N R = ~ 39,800
used in Equation 2. This was necessary because small errors in the variances can lead to large errors in their differences. Typical tracer concentration curves are shown in Figure 2. The maximum difference between the average variance of three runs and the variance of F, single run was, in each case, less than 11%. The average of these maximum differences over all the data was less than 4%. Within the precision of the data, the axial dispersion number for annuli is independent of the annulus diameter ratio and is the same as that for straight pipes when the hydraulic diameter is used as the length parameter. Prediction of Dispersion Coefficient from Velocity Profiles
The general dispersion model provides the starting point for the present analysis. It is convenient to assume an origin moving with the mean velocity of flow, ii. Thus, a new
variable, y, is defined such t,hat y=z--.iit
I n terms of this new coordinate system, and with no source in the region of consideration, the dispersion model can be represented by
-
(u-6,)-
ac aY
(3)
An expression for the effective axial dispersion coefficient, E t , was obtained from Equation 3 by the procedure of Tichacek et al. The equation was simplified and integrated twice with respect to r, leading to Equation 4, which corresponds to Tichacek’s Equation 6 with their (radius)2replaced by R 2 ( 1- K ~ ) .
By means of Reynolds analogy, E , was expressed as a function of the shear stress, 7. The distribution of shear stress in a concentric annulus is given by
[
r2 -rfPA2]
pf2 7-=
2R(1 -
Substituting for E, and variable, 11, such that
7,
K)
(5 1
and introducing the dimensionless
r 11=
R(1 -
K )
Equation 4 can be writ,t,en in dimensionless form
Et Et --=-2R(1 - K ) U Dgz7
2(1 (1
+
K) K )
The group E1/DIf6, is an inverse Peclet number based on diameter; it is called the axial dispersion number. Solutions to Equation 7 may be obtained numerically by using experimental values for velocity profiles and friction factors. I n the foregoing development, the effect of axial eddy diffusion on the total longitudinal dispersion was neglected in order to simplify the mathematics. However, it is possible to estimate the order of the effect by assuming that the axial eddy diffusion coefficient is equal to the radial eddy diffusion coefficient. Again following the procedure of Tichacek et al., the mean coefficient of diffusion due to the longitudinal fluctuations of turbulent velocity may be written as:
/l’l-K
K/~-K
[j[”?
- k2/(1 - K ) 2 11
The value of Et’ is small compared with E,, but it is additive; thus the corrected value of E,, allowing for longitudinal turbulent diffusion, is the sum of the expressions given in Equations 7 and 8. The reduced velocity, u/6,is primarily dependent on the Reynolds number and the roughness of the wall, although there is some slight dependence on the annulus diameter
ratio for certain values of K . Thus, the dimensionless dispersion number depends only on the Reynolds number, the friction factor (or roughness), the Schmidt number, and the annulus diameter ratio, K . For liquids, the Schmidt number is of the order of 100, and affects dispersion number only slightly, as can be seen by inspection of Equation 7. Theoretical Results
Equation 7 was evaluated numerically for three cases in order to obtain theoretical values of the axial dispersion number. Evaluation of the equation requires knowledge of the friction factor of the system, the mean radial velocity distribution, and, in general, the Schmidt number. For liquids, the term containing the Schmidt number can be neglected, since it is of the order of 100. Since there are some small differences in the friction factor for annuli, as reported in the literature, we decided to determine experimentally the friction factor for at least one of the annuli studied. Friction factor data that were obtained for the 0.272 diameter ratio annulus were almost identical to those of Brighton (1963). Since Brighton’s data were taken for several annulus diameter ratios, it was concluded that the data obtained for the 0.272 diameter ratio annulus could be used for all annuli. Several investigators (Brighton, 1963; Knudsen, 1951; Okiishi and Serovy, 1964; Rothfus, 1948) have presented velocity profile data for annuli in generalized form. These data, unfortunately, exhibit a wide degree of scatter. Their application to Equations 7 and 8 presents serious problems because of the extreme sensitivity of the integrals to the velocity profile. It was, therefore, concluded that the generalized velocity profiles could not be used to obtain any meaningful results. In lieu of sufficiently precise generalized profiles, the theoretical equations were evaluated from actual experimental velocity data a t three conditions. Of all the data in the literature on flow in annuli, those of Brighton (1963) seem the most precise. Also, Brighton’s data include the velocity gradients near the point of maximum velocity, which are important to an accurate solution. For these reasons, the velocity profiles used were taken from his work. The velocity data used were for flow in annuli having diameter ratios of 0.125, 0.375, and 0.562 a t Reynolds number of 89,000, 65,000, and 46,000, respectively. The mean velocity, U,must be known very accurately. It was necessary, therefore, to compute its value by integrating the point velocities over the cross section rather than by simply using measured values. Also, the point of maximum velocity must be known accurately for satisfactory results. Table I shows the results of the numerical solution of Equations 7 and 8, along with some of the results for liquids presented by Tichacek et al. for mixing in pipe flow. The effect of axial eddy diffusion on the total axial dispersion number, as determined from the solution of Equation 8, was computed for only one case, the 0.125 diameter ratio annulus. The axial eddy diffusion contributed less than 3y0 to the axial dispersion number. Taylor and Tichacek et al. have shown that the effect of axial eddy diffusion in pipe flow is also negligible. At first examination of Table I, there appears to be a n effect of annulus diameter ratio and/or Reynolds number on the theoretical axial dispersion number; however, the apparent effects could be accounted for by errors in the velocity profiles used in the computations. The actual mean point velocities were not accurate to more than 1 or 2%, and the velocity gradients were not known, in many cases, with an accuracy VOL.
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Table 1.
Theoretical Values of Axial Dispersion Number for Turbulent Flow in Annuli and Pipes Annulus Diameter Ratio
0.125 0.375 0.562
GQ 0 0 0 a
Reynolds Sumber
El /CDH
89,Ooo 65,000 46,000 40,000 43,400 91,Ooo 205,000
0.22 0.20 0.17 0.28 0.18 0.19 0.17
All results for diameter ratio of zero taken from Tichacek et al.
(1957).
Nomenclature
c CM
C
DH
EL
of more than 10%. These errors could cause a 25% error in the dispersion number, which is the approximate difference between the maximum and the minimum values in the table. Table I shows a sharp change in the axial dispersion number for pipe f l o between ~ Reynolds numbers of 40,000 and 43,000. Such a sharp change in this region is inconsistent with experimental data and is probably due to errors in the velocity profiles. By taking into account the degree of error that is possible in the numerical calculations, it is concluded that no statistically significant difference exists in the theoretical values of the axial dispersion number for annular flow and those for pipe flow. The theoretical values of the axial dispersion number for both annular and pipe flows are somewhat lower than those obtained experimentally. Taylor’s theoretical work for pipe flow shows slightly larger values than those obtained by Tichacek et al.; however, the form of Taylor’s equation necessitated the numerical integration of a function that increased very rapidly near the pipe wall. It is difficult to integrate such a function numerically with accuracy; consequently, the values of Tichacek el al. are probably more accurate. The theoretical values are typically about 60 to 70% of the experimental values. This difference is explained partly by the computational difficulties discussed above and partly by the failure of the assumptions made in developing the model to describe the real system. The assumption that mass and momentum transport are analogous may not be entirely valid, particularly at the lower Reynolds numbers. Also, the assumption that the axial concentration gradient is independent of radial position is not strictly true. The error introduced by this assumption can be estimated by a second-order approximation. The analogous equation for pipe flow mas evaluated numerically for two Reynolds numbers by Tichacek et al. Since the pipe flow equation and the annular flow equation differ only in boundary conditions, and the axial dispersion numbers for the two geometries are the same, the same degree of error probably exists for each case. The accuracy of the first-order approximation depends on the shape and the scale of the concentration curve along the axis of the vessel. Tichacek et al. estimated the error in the first-order approximation for three types of axial concentration profiles. The error for the pulse-type injection was not greater than 25% when the characteristic concentration length was greater than 50 to 100 pipe diameters. For smaller characteristic lengths, the dispersion number could be much larger than the model predicts. The characteristic length for a pulse injection was defined as the length of the region in which the concentration of tracer is between 1 and 99% of its maximum value. Thus, in view of the above analysis, it is 624
not at all surprising that the theoretical dispersion numbers for flow in annuli and pipes differ somewhat from the experimental values.
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FUNDAMENTALS
E, Et Et‘
f Q’ R T
t U
ii 2
Y z
= point concentration of tracer, moles/length3
= mean concentration of pulse if uniformly distri-
buted between measurement points, moles/ length3 = dimensionless concentration, C/C,), = hydraulic diameter of annulus, 2 ( R - K R ) , length = axial component of total dispersion coefficient (axial dispersion coefficient), general dispersion model, length2/time = radial dispersion coefficient, general dispersion model, length2/time = first-order approximation to effective axial dispersion coefficient in theoretical analysis determined from radial variations of velocity, length2/time = mean diffusion coefficient in theoretical analysis accounting for mixing due to longitudinal velocity fluctuations, 1ength2/time = Fanning friction factor, dimensionless. = rate of transport of solute across plane moving with mean speed of flow due to axial diffusion, moles/ time = pipe radius, length = radial distance, length = time = axial point velocity, length/time = mean velocity of flow in axial direction, length/time = axial position variable relative to fixed reference point, length = axial position variable relative to plane moving with mean speed of flow, length = dimensionless axial position, z / L
GREEKLETTERS
0 K
= dimensioiiless time, Qt/L = ratio of radius of inner wall of annulus to radius
of outer wall, dimensionless A
= shear stress, mass/length-time2 = ratio of radial position of maximum velocity t o
p
= density of fluid, lb.,,s/cu.
T
a’s
radius of outer wall, dimensionless ft. = difference in variances (second moment about mean) of dimensionless concentration-dimensionless time curves measured a t two points, dimensionless
DIMENSIOKLESS GROUPS = length Peclet number, f i L / E N R ~= Reynolds number, Dfrii/v Ng, = Schmidt number, v / D E l i i D ~=dispersion number for annulus (inverse Peclet number based on diameter) Np,
literature Cited
Aris, R., Chem. Eng. Sci. 9, 266-7 (1959). Aris, R., Proc. Roy. SOC.(London), Ser. A, 236, 67-77 (1956). Bischoff. K. B.. Levensuiel, O . , Chem. Eng. Sei. 17, 245-55 ~. (1962j. Brighton, J. A., “Structure of Fully Developed Turbulent Flow in Annuli,” unpublished Ph.D. dissertation, Purdue University, Lafayette, Ind., August 1963.
Cassell, R. E , , Jr., “Axial Dispersion in Turbulent Flow through a Piping System Containing 90” Elbows,” master’s thesis, TTniver4tI. o.,.f Tennessee Marrh 1966. - ...~~, .~~ _. . - .._. .___ - _...._____ , Knnxville. ~ - -,- Tenn.. Knudsen, J. G., Katx, I). L., “Velocity Profiles in Annuli,” First Midwestern Conference on Fluid Dynamics, J. W., Edwards., ed.., .. D D . 175-203. Edwards Bros., Ann Arbor, Mich., ~
~~
~
~
1951.
Levenspiel, O., Ind. Eng. Chem. 60, 343-6 (1958). Levenspiel, O., Bischoff, K. B., Advan. Chem. Eng. 4, 95-199 (1963). Okiishi, T. H., Serovy, G. K., “Experimental Velocity Profiles for Fully Developed Turbulent Flow of Air in Concentric Annuli,’ A.S.M.E. Paper 64 W.4/FE-32, New York, 1964.
Olson, R. M., Sparrow, E. M., A.I.Ch.E. J. 9, 766-70 (1963). Rothfus, R. R., “Velocity Gradients and Friction in Concentric Annuli.” unoublished Ph.D. dissertation.’ Carneeie Institute of Technolo&, Pittsburgh, Pa., 1948. Smith, J. V., “Axial Mixing of Fluids in Turbulent Flow t,hrough Concentric Annuli,” master’s thesis, University of Tennessee, Knoxville, Tenn., June 1967. Taylor, G. I., Proc. Roy. Soc. (London), Ser. -4, 223, 446-48 ( 1954). Tichacek, L. J., Barkelew, C. H., Baron, T., -4. I.Ch.E. J . 3, 439-42 (1957). RECEIVED for review August 28, 1968 ACCEPTEDMay 27, 1969 0
FIBROUS B E D C O A L E S C E N C E OF W A T E R Steps in the Coalescence Process R. N . H A Z L E T T Naval Research Laboratory, Washington, D. C. 20390
The coalescence phenomena in a fibrous filter bed have been analyzed on the basis of three major steps: approach of a dispersed water droplet to a fiber, attachment of the droplet, and release of enlarged droplets from the downstream side of the filter. For approach the interception process is important, while diffusion and inertial impaction processes ore insignificant in the total efficiency of this step. Surfactants that increase the water-hydrocarbon interfacial viscosity may interfere with the attachment of the small water droplets in the initial part of the filter or the attachment of a coalesced water thread as it snakes through the fibrous bed. They also interfere with the release process by altering the interfacial tension. The latter property influences the size of a water thread at rupture and thus controls released water droplet size. Rayleigh instability can occur when the interfacial tension is decreased.
HE removal of emulsified water from an organic liquid is a Trequirernent in many phases of today’s technology. The requirement of removal of free water is particularly stringent with aircraft fuels, since free water in a fuel system results in corrosion and microbiological growth as well as icing problems a t low operating temperatures (Krynitsky and Garrett, 1961). Particulate, or free, water is readily emulsified in fueling systems which use high-speed centrifugal pumps and are characterized by turbulent flow in fuel lines. A centrifugal pump readily produces a water-in-fuel emulsion in which 90% of the droplets are less than 4.8 microns and 100% are less than 8.6 microns in diameter (Bitten, 1967). Water droplets of this size do not settle out of fuel in a reasonable length of time, particularly if there is any agitation or sloshing in the storage tank or if thermal gradients are present. The drop size may be increased by passage of the emulsions through a fibrous bed. I n this situation, small droplets are retained until growth occurs by coalescence. The larger drops are then swept from the fiber bed by dynamic forces and ideally are of such size that they can separate by gravitational forces. Fueling equipment employing this principle is known as a filter-separator. I n addition to coalescing and separating emulsified water, this equipment serves as a filter for solid matter which is retained in the fibrous bed. A filter-separator typically consists of two
stages: a resin-bonded glass-fiber bed which retains solid matter and coalesces water drops and a hydrophobic screen or paper, included to strip water drops of an intermediate size which are too small to settle by gravity forces but too large to pass through the pores of this separator stage along with the continuous fuel phase (Redmon, 1963). Filter-separators with fueling rates of 10 to 600 gallons per minute are typical. The physical size of a particular installation is controlled by the fiber-bed flow velocity, which has an upper limit of 0.5 to 1.0 cm. per second. The performance of a filter-separator may be inadequate if the jet fuel contains surface active materials (Krynitsky and Garrett, 1961) This requires care on the part of the refiner, the transporter, and the user to ensure absence of undesirable surfactants from the fuel. Another consequence of this behavior is that additives, such as corrosion inhibitors and antioxidants, which might otherwise be useful in controlling certain fuel properties, must be selected carefully. The complex events occurring in a filter-separator are not adequately understood. The work described here examines the steps in the coalescence process in an effort to understand the physical and chemical phenomena involved and to determine the critical factors for coalescence. The process of coalescence in a fibrous bed may be divided into three main steps: approach of a droplet to a fiber or to a droplet attached to a fiber, attachment of a droplet to a fiber I
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