Turbulence Properties in the Core of Pipe Flow
Equations are derived to relate the eddy diffusivity of momentum to energy dissipation and ?heshear velocity. Experimental data for pipe flow and for a turbine impeller stream show good agreement with these derivations.
The experimental data that have been reported for tracer dispersion in the turbulent core and for the velocity profile of the core provide a basis for evaluation of core turbulence properties for smooth pipe. Turbulence properties of interest are a turbulent dispersion coefficient or eddy diffusivity, ED, the Lagrangian. integral time scale, T L ,and the microscale, A, as defined by Taylor (1921) in relation to the mean fluctuating velocity in the direction transverse to mean flow. Turbulent pipe flow has the additional characteristic that core turbulence is nearly isotropic. The turbulence properties are related by the equation
E D = ( ; I ~ ) T=L(;YZ)l'*X
For flow in the core of a circular nine this derivation results in the equation
Eddy diffusivity for momentum is represented by
and for ED>> v
(1)
Eddy Diffusivity Derivation Relatively constant eddy diffusivities for momentum and heat are reported for the pipe core by Johnk and Hanratty (1962). Hughmark (1972) showed a heat transfer, mass transfer, and momentum relationship for the core region hEC
-= k E C = uf
(2)
Combination of eq 5,6, and 7 with integration to determine the average value of the energy dissipation for the core gives t~
= 1.67u*'u/D
(8)
Substitution in eq 5 provides the eddy diffusivity in terms of the energy dissipation
PCP
and that experimental data indicate that eddy diffusivities for heat and momentum are equal in the fully turbulent core region (Hughmark, 1972). Heat transfer for the conditions of constant eddy diffusivities of heat and momentum with circular cross section represents an analytic solution of the equation such as that presented by Longwell (1966) h
ua
PCp
24
-=-
2
4 =A -ua2
(3) Adr dr dr
with the velocity variation corresponding to constant eddy diffusivity (4)
Combination of eq 2,3, and 4 provides the eddy diffusivity for constant and equal eddy diffusivities of heat and momentum in a circular cross section
ED = 0.15Duf
=
cp
4u*'u
=-
D So substitution of eq 10 in eq 9 represents a constant eddy diffusivity with the energy dissipation corresponding to pipe flow
E D = 0.72Du*'/u
La
A = I Qo LrED (Lrdrdr)
Energy dissipation in pipe flow is
O.~DU*'/U
(5)
In turbulent pipe flow constant eddy diffusivity exists only in the core region; therefore, eq 5 must be modified for this condition.
Application to Pipe Flow Batchelor (1959) represented the decay of isotropic grid generated turbulence by the equation tLe = 1.65(ui)3/2 Assumption of a similar relationship for the microscale indithat the eddy diffusivity could be a function of the en,ergy dissipation. Venezian and Sage (1961) show the derivation for energy dissipation with a turbulent velocity profile.
(11)
Published eddy diffusivity data include those of Baldwin and Walsh (1961), Becker e t al. (1966), Boothroyd (1967), Groenhof (1970),Johnk and Hanratty (1962),Patterson and Zakin (1967), Quarmby and Anand (1969), and Towle and Sherwood (1939). These data represent pipe diameters from 2.5 to 30 cm, a Reynolds number range of 17 700 to 680 000 and water, air, and hydrocarbons as fluids. An average coefficient of 0.68 is obtained from these data with the model represented by eq 11. An average absolute deviation of about 12% is shown between values calculated from eq 11 with the 0.68 coefficient and the experimental values. Thus, eq 11 shows excellent agreement with the experimental data for turbulent pipe flow in which constant eddy diffusivity exists only in the core region.
Turbine Impeller Stream Turbulence in a turbine impeller stream has been shown by Farritor and Hughmark (1974) to give mass transfer response analogous to pipe flow. Levins and Glastonbury (1973) report fluid velocities, fluctuating velocities, and microscales for water with 7.5 and 10-cm diameter turbines in a 25-cm diameter vessel a t 350 rpm. The region in the stirred vessel represented by this would could be expected to have a local energy dissipation rate about three times the average rate as shown by Cutter (1966). Assumption of the impeller stream diameter equal to the impeller blade width and use of eq 9 results in calculated eddy diffusivities that are within 50% of Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
307
the experimental values. This result indicates that eq 9 is applicable to conditions of high energy dissipation in an impeller stream as well as pipe flow. Fluctuating Velocity Robertson et al. (1968) obtained data with air for 7.6 and 20.4-cm diameter pipe with rough surfaces and concluded that these data along with literature data for air and water are represented by (?)1/2
= 0.78~'
(12)
In addition to the data considered by Robertson et al., Becker et al. report data for air with 20.4-cm diameter pipe and Boothroyd for air with 5 and 7.6-cm diameter pipe. Data with water are provided by Goldstein et al. (1969) for 1.4-cm diameter and by Taylor and Middleman (1974) for 5-cm diameter pipe. Patterson and Zakin (1967) report fluctuating velocity data with several hydrocarbons. The data from these references show an average absolute deviation of about 7% from values calculated by eq 12. This is additional confirmation of the validity of eq 12. Taylor and Middleman also report data for 50 ppm of Polyox in water which shows drag reduction of about 35% a t a Reynolds number of 55 000. Seyer and Metzner (1969) obtained data with 0.01% ET-597 in water at a Reynolds number of 144 000 in a 2.54-cm diameter pipe which shows drag reduction of about 30%. These data show an average absolute deviation of about 9% in comparison to values calculated from eq 12. Equation 12 also appears to apply to the fluctuating velocity for the impeller streams reported by Levins and Glastonbury (1972). Estimated ratios of (?)1'2/u* of 0.64 and 0.66 are obtained with the assumptions for energy dissipation and stream diameter as described earlier in this paper. Schwartzberg and Treybal (1968) also report data for a region above a turbine impeller. The data represent an impeller diameter range of 10.2 to 22.9 cm and a tank diameter range of 22.9 to 43 cm. Average vertical plane fluid velocities were correlated by the equation
u = 1.39-
nD
and average fluctuating velocities by the equation nD ~
(T2H)ll3
These equations with the same assumptionsas applied to the / U * of Levins and Glastonbury data provide a ( U ~ ) ~ / ~ value 0.63. Thus, the values for impeller streams are within 20% of eq 12. Microscale Equations 1, 11, and 12 result in the equation for the microscale
X = 0.92Du*/u
(13)
Microscale data are reported for pipe by Becker et al., Boothroyd, Taylor and Middleman, and Patterson and Zakin. Microscale values calculated from eq 13 show about a 9% average absolute deviation with the experimental values. Conclusions Assumptions of constant and equal eddy diffusivity of heat and momentum in turbulent flow for the core region of a circular pipe are used to derive equations for the eddy diffusivity
308
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
Nomenclature a = pipe radius C, = fluid specific heat D = pipe diameter E D = eddy diffusivity f = friction factor H = liquid height in vessel h = heat transfer coefficient k = mass transfer coefficient L e = Eulerian length scale n = impeller speed r = radial distance from pipe centerline ro = piperadius T = vessel diameter T L = Lagrangian integral time scale u = average stream velocity (u2)1/2 = mean fluctuating velocity Li = time average axial velocity at any radial position u* = shear velocity Greek Letters 6 = energy dissipation X = microscale p = fluiddensity Y = kinematic viscosity T = shear Subscripts C = core EC = eddycore P = pipe
Literature Cited
( T2H)l l 3
( u ~ ) ~= '0.5 ~
of momentum related to energy dissipation (eq 9) and the shear velocity (eq 11).Experimental data for eddy diffusion in pipe flow show excellent agreement with eq 9. Experimental data for eddy diffusion in a turbine impeller stream show good agreement with eq 11. Combination of an equation relating fluctuating velocity to shear velocity with the derived eddy diffusion equation results in eq 13 for the microscale which shows excellent agreement with experimental data.
Baldwin, L. V., Walsh, T. J., A.I.Ch.E. J., 7, 53 (1961). Batchelor. G., "Theory of Homogeneous Turbulence," p 103, Cambridge University Press, 1959. Becker, H. A., Rosensweig, R. E., Gwozdz. J. 6.. A.1.Ch.E. J., 12, 964 (1966). Boothroyd, R. G., Trans. Inst. Chem. Eng., 45, T297 (1967). Cutter, L. A., A.1.Ch.E. J., 12, 35 (1966). Farritor, R. E., Hughmark, G. A., A.LCh.E. J., 20, 1027 (1974). Goldstein, R. J., Adrian, R. J., Kried, D. K., Id.Eng. Chem., Fundam., 8, 498 (1969). Groenhof, H. C.. Chem. Eng. Sci.. 25, 1005 (1970). Hughmark, G. A., A.I.Ch.E. J., 18, 1072 (1972). Hughmark, G. A., A.I.Ch.E. J., 20, 172(1974). Johnk, R. E., Hanratty, T. J.. Chem. Eng. Sci., 17, 867 (1962). Levins. D. M., Glastonbury, J. R., Trans. Inst. Chern. Eng., 50, 32 (1972). Longwell, P. A.. "Mechanics of Fluid Flow," p 356, Waw-tiill, New Y a k , N.Y., 1966. Patterson, G. K., Zakin, J. L.. A.I.Ch.E. J., 13, 513 (1967). Quarmby, A., Anand. R . K., J. FIuidMech., 38, 457 (1969). Robertson, J. M.. Martin, J. D., Burkhart, T. H., Ind. Eng. Chem., Fundam., 7,253 (1968). Taylor, A. R., Middleman, S..A./.Ch.E. J., 20, 454 (1974). Taylor, G. I:, Roc. London Math. Soc.,20, 196 (1921). Towle, W. L., Sherwood, T. K., Id.Eng. Chem., 31, 457 (1939). Schwartzberg, H. G., Treybal, R. E., Ind. Eng. Chem., Fundem., 7, 1 (1968). Seyer, T. A.. Mtzner. A. B., A.I.Ch.E. J., 4, 393 (1958). Venezian. E.. Sage, 8. H., A.I.Ch.E.J., 7, 688(1961).
Research and Development Department Ethyl Corporation Baton Rouge, Louisiana 70821
Gordon A. Hughmark
Received for reuiew September 21,1976 Accepted February 4,1977
