Patterson, Gary Kent

Rodriguez, J. M.

Zakin, J. L.

English

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

Missouri University of Science and Technology Scholars' Mine Chemical and Biochemical Engineering Faculty Research & Creative Works Linda and Bipin Doshi Department of Chemical and Biochemical Engineering 01 Jan 1970 Drag Reduction Correlations Gary Kent Patterson Missouri University of Science and Technology J. L. Zakin Missouri University of Science and Technology J. M. Rodriguez Follow this and additional works at: https://scholarsmine.mst.edu/che_bioeng_facwork  Part of the Biochemical and Biomolecular Engineering Commons Recommended Citation G. K. Patterson et al., "Drag Reduction Correlations," AIChE Journal, vol. 16, no. 3, pp. 505 - 506, Wiley; American Institute of Chemical Engineers (AIChE), Jan 1970. The definitive version is available at https://doi.org/10.1002/aic.690160337 This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Chemical and Biochemical Engineering Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu. values. The mass transfer between fluid elements is considered to be caused mainly by coalescence and dispersion, and thus the overall mass transfer coefficient may be depen- dent on frequency of coalescence and dispersion, contact time, and contact area of fluid elements. Although U might be expected to vary with agitation speed, we as- sumed U to be a constant. The values of p were computed for parametric values of U, with k = 1.10, B = 0.10, K = 0.10, Z1 =297, and I 2  = 1.0. In  Figure 4, the calculated values of P are plotted vs. the period T together with the experimental results by using f = 0.5 for bafflled and 0.25 for unbaffled vessels. A comprehensive geometrical model has not been made. For periods greater than 10 sec., the calculated reaction rates are relatively insensi- tive to period but are dependent upon the mass transfer rate. At small periods, the reaction rates is insensitive to the mass transfer rates. The analytical and experimental methods for the chloral hydrate system indicate sufficient concurrence to merit reporting the simple approaches used for the model. ACKNOWLEDGMENT The authors appreciate the exchange arrangements made by the Takasaki Radiation Chemistry Research Establishment, Japan Atomic Energy Research Institute, and the University of Minnesota, and one of the authors wishes to thank Tomomichi Kasamatsu and Masaaki Takehisa for their en- couragement. NOTATION A = defined by Equation (3) ,  mole/(liter (mill.) or mho./(cm.) (min.) a = defined by Equation (9) B = defined by Equation (3) ,  min./rad. b, c = defined by Equation (9) C = product D = diameter of agitator, cm. d = defined by Equation (9) E = reactant f = coefficient of impeller discharge. G = G value of initiation reaction, mole/rad. Z = dose rate, rad./min. I1, la = dose rate of high and low dose rate portion, = average dose rate in the reactor, rad./min. i = number of cycle times per fluid element in the K = apparent rate constant of the propagation reac- k = apparent rate constant of the termination reac- kl, kz = propagation and termination rate constant, liter/ 1 = termination rate constant, liter/ (mole) (sec.) M = reactant N = agitation speed, rev./sec. n = concentration of chain carrier, mole/liter; [ R . ]  rad./min. reactor tion, sec.-l tion, sec.-l (mole) (sec.) = concentration of R *  designated by n - n p Q R .  = chain carrier radical r, s S = scavenger T t = time, sec.  U V = reactant volume, cc. X Y P Greek Letters K 0 Su bscriptr n = nonmixing p = perfect mixing = defined by Equation (6), mole/liter = defined by Equation (9) = discharge rate of agitator, cc./sec. = solutions of Equations ( 1 1 )  and (12)  = period of motion of fluid element in the reactor, sec. = overall mass transfer coefficient based on inter- action between fluid elements, sec.-l = rate of production of the product, mole/(liter) (min.) or mho./(cm.) (min.) = X/A,  dimensionless reaction rate = defined by Equation (9). = fraction of high dose rate portion in the reactor = mean circulation time of fluid in the reactor, sec. LITERATURE CITED 1. Hill, F. B., and R. hl. Felder, AZChE J., 11, 873 ( 1965). 2. Kawakami, W., K. Yoshida, and T. Sugo, preprint of the annual meeting of Soc. Cheni. Engrs., Japan, p. 297 (1965). 3. Fredrickson, A. G., A. H. Brown, R. C. Miller, and H. M. Tsuchiya, Am. Rocket SOC. J.,  31, 1429 (1961). 4. Hill, F. B., and N. Reiss, Can. J. Chem. Eng. ,  46, 124 (1968). 5. - and L. H. Shendalnian, AZChE J., 14, 798 (1968): 6. Yemin, L., and F. B. Hill, Znd. Eng. Chein. Fun&mentds, 8,210 (1969). 7. Andrews, H. C., and P. A. Shore, J. Chem. Phys., 18, 1165 ( 1950 1. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Freeman, G. R., A. B. Van Cleave, and J. W. T. Spinks, Can. J. Chem., 31,1164 (1953). Lotz, A. P., and K. Schmidt, Nature, 201,493 (1964). McIntosh, R. G., R.  C .  Eager, and J. W. T. Spinks, Can. J. Chem., 42,2033 ( 1964 ) . Kawakami, W., and H. S. Isbin, ibid., 47,4531 (1969). Hill, F. B., P. T. Fallon, A. C. Muller, and M. A. Rothbart, IAEA/SM-I23/44 ( 1969). Kamiwano, M., K. Yamamoto, and S .  Nagatn, Chem. Eng. Japan, 31, 365 (1967). Corrsin, Stanley, AIChE I., 10,870 ( 1964 ) . Mohr, W. D., “Proceeding of Thermoplastic Materials,” E. C .  Bernhardt, ed., p. 117, Reinhold, New York (1960). Rushton, J. H., D. E. Mack, and H. J. Everett, Trans. Am. Inst. Chem. Engrs., 42,441 (1946). Sacks, J. P., and -J. H. Rushton, Chem. Engr. Progr. 50,597 ( 1954). Nagata, .S., K. Yamamoto, and M. Ujihara, Mem. Fac. En Kyoto Uniu., 20,336 (1958). Ibis,  21,260 ( 1959). Marr, G. R., and E. F. Johnson, AZChE J., 9,383 ( 1963). Drag Reduction Correlations G. K. PATTERSON, J. L. ZAKIN, and J. M. RODRIGUEZ University of Missouri, Rollo, Missouri relation, which was proposed and tested for many poly- mers and solvents by Rodriguez, Zakin, and Patterson (2, 3 )  was of the form f / f P u  vs. U T ~ / D O , ~ ,  where 71 is the We would like to point out a more general drag reduc- tion correlation to which the one presented by Astarita, Greco, and Nicodemo ( I )  is very similar. The older cor- Vol. 16, No. 3 AlChE Journal Page 505 first-mode Zimm relaxation time of the polymer molecule in solution. The A, G, N correlation was f / f p v  vs. V / V ~ . ~ ,  where v = U N R ~ O . ~ ~ / V ,  and ~0.5 is the value of v at f / f P 1 ,  = 0.5. Thus, all drag reduction data were forced to fit the correlation at f / f p v  = 0.5, requiring a set of drag reduction data for each polymer solution to establish the value of V0.5. The R, 2, P correlation established that for polymers dissolved in polar or nonpolar solvents, the U T ~ / V O . ~  abscissa accounts for concentration and diam- eter effects, even though the levels of U T ~ / D O . ~  vary among polymer-solvent systems. Astarita, Greco, and Nicodemo obtained a linear log- log relationship between l /vo.S, relative measure of solu- tion relaxation time, and concentration up to a limiting value for their ET 597 solutions. They found that for dilute solutions, l /vO.j a It is not known whether this relation is applicable to other polymer solutions. The concentration effect is accounted for in the R,  2, P cor- relation by its effect on the calculated Zimm relaxation time 7'. All dilute polymer solutions tested so far are correlated by that relaxation time relationship, which requires viscosity-concentration data for calculation. The diameter effect on drag reduction is accounted for in each of the correlations by a velocity-to-diameter (to some power) ratio. In the A, G, N correlation, the abscissa contains the ratio U1.75/V0.25; the R, 2, P correlation contains U/VQ.2. Since the relative powers of U and D are the important factors, the A, G, N and R, 2, P cor- relations may be compared as U7/V and U5//D,  respec- tively. It is doubtful that this difference can be seen in drag reduction data over the velocity ranges normally encountered. The form of the A, 2, P correlation is supported by considerations of the turbulent energy dissipation spec- trum (4). For a Maxwell fluid and isotropic turbulence with turbulence intensity equal to that of a purely viscous fluid, the ratio of energy dissipation rates was found to be p JOw [ n2F ( n )  / ( 1 + &znz,k?2u'2T2/u2) ]d n  W WS pLS r n z F ( n ) d n  (1) -= Some phenomenological considerations led to the follow- ing relationship between wall shear stress and turbulent energy dissipation rate: (2)  Tw/ (721)) s = (w/ws) o ' 8 ( P s / P )  The friction factor ratio, which is the friction factor of the viscoelastic fluid divided by the friction factor of a purely viscous fluid of the same viscosity, may be ex- pressed as co n2F(n)dn 1 + n2C2T2 so" n2F(n)dn A simple energy spectrum function is F ( n )  a n-4 for n = kU (independent of diameter). If n, the relative frequency, is scaled such that at n < 1 the value of F ( n )  + 0, the above integrals may be evaluated from 1 to 00 to obtain f / f p u  = [ 1 - ( C U T )  ( n / 2  - tan-' ( C U T )  ) lo.* (4 )  The shape of this function is shown in Figure 1 and is compared with the shape of the correlation (which has been forced to have f / f P u  = 0.5 at CUT = 1.0) by choosing the proper value of C .  The A, G, N correlation is plotted with CUT = v/vo.,. The most obvious deficiency of the predicted f / f p v  function is that it is asymptotic to zero, whereas the cor- relation seems asymptotic to f / f y L '  = 0.2 to 0.4, depend- ing on the polymer-solvent system (1, 3 ) .  NOTATION c = concentration c = constant replacing 4?r2ur262/2 V = pipe diameter f = friction factor fl," = friction for purely viscous fluid F ( n )  = Eulerian energy spectrum function hl, = viscosity average molecular weight n = velocity fluctuation in Eulerian energy spectrum NRe = generalized Reynolds number R = thermodynamic constant T = Maxwell fluid relaxation time t = temperature u = time average velocity u' = root-mean-square fluctuating velocity U = bulk mean velocity W Ws - = turbulent energy dissipation rate for solution = turbulent energy dissipation rate for solvent Greek Letters /3 rlvp p = solution viscosity ps = solvent viscosity v ~ 0 . 5  = constant in Equation (1) = specific viscosity, ( p / p s )  - 1 1.0 Ak = eigenvahe in Zimm equation = eddy frequency used in A, G, N correlation = eddy frequency at f / f P v  = 0.5 = Zimm relaxation time, T,, = pipe wall shear stress for solution ( r W ) ,  = pipe wall shear stress for solvent M U ( ? s p / C ) P S  0.586 R t A k  0.2 LITERATURE CITED - 1. Astarita, Gianni, Guido Greco, Jr., and Luigi Nicodemo, 2. Rodriguez, J. M., M. S. thesis, Univ. Mo., Rolla (1966). 3. -, J. L. Zakin, and G. K. Patterson, SOC. Petrol. Engrs. Fig. 1. Comparison of R, Z, P and A, G, N correlations with theoreti- 4. Patterson, G. K., and J. L. Zakin, AlChE J . ,  14, 434 AlChE I . ,  15,564 ( 1969). 5.0 O O  1.0 2.0 3.0 4.0 J.,  7,325 ( 1967). CUT cal result. (1968). Page 506 AlChE Journal May, 1970  15475905, 1970, 3, Downloaded from https://aiche.onlinelibrary.wiley.com/doi/10.1002/aic.690160337 by Missouri University Of Science, Wiley Online Library on [12/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

Drag Reduction Correlations

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Drag Reduction Correlations

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