eter estimates which lead to relatively large phase angle errors could be accepted by amplitude ratio fitting as the best estimates. As the previous example illustrates, such estimates can differ considerably from the true parameter values. Although the quantitative results obtained in this study are based on two relatively simple models, the shapes of the simulated responses are characteristic of a broad class of physical systems, and the qualitative results should also be applicable to models of greater complexity. In summary, the computational simplicity of amplitude ratio fitting does not compensate for the relatively high sensitivity of this technique to response measurement errors, and transfer function fitting should therefore be used in preference to amplitude ratio fitting for frequencydomain parameter estimation.
P = Pecletnumber Re = real part of a complex number t = time At = time-domain sampling increment x ( t ) = process input y ( t ) = experimental process output y c ( t ) = modeloutput ym = maximum value of y ( t ) Greek Letters y = objective function for transfer function fitting T = mean residence time 4 0 ' ~ )=. G c 9 w ) - G G w ) x = objective function for amplitude ratio fitting w = frequency wmax = maximum value for which GCjw) is calculated
Nomenclature A R ( w ) = experimental amplitude ratio AR,(w) = model amplitude ratio C( t ) = unit impulse response C, = maximum value of C( t ) GCjw) = experimental frequency-domain transfer function G,Cjw) = model frequency-domain transfer function Im = imaginary part of a complex number
Carnahan, B., Luther, H. A,, Wilkes. J. 0.. "Applied Numerical Methods. pp 548-549. Wiley, New York, N. Y . . 19.69. Clements, W. C., Jr., Chem. Eng. Sci.. 24, 957 (1969). Clements, W. C. Jr., Schnelle, K. B., Ind. Eng. Chem.. Process Des. Develop. 2, 94 (1963). Felder, R. M., Harrison, R. E.. Rousseau, R. W., Chem. Eng. Commun., in press, 1974. Harrison, R. E., Ph.D. Thesis, North Carolina State University, Raleigh, N. C., 1974. Hays, J. R., Clements, W. C., Jr., Harris, T. W.. AlChE J.. 13, 375 (1967). Lees, S.,Dougherty. R. C.. J. Basic Eng., 89, 445 (1967).
J = v=I
Literature Cited
N = number of tanks in series n ( t ) = noise function
Received for reuieu: August 29,
1973 . Accepted June 13, 1974
Axial, Laminar, Non-Newtonian Flow in Annuli Christopher P. Russell and E. B. Christiansen* Department of Chemical Engineering, The University of Utah, Salt Lake City, Utah 847 72
A more general and convenient method for the prediction of the pressure loss-flow rate relationship for the axial flow of time-independent non-Newtonian fluids in annuli, based on tube-flow correlations, is described. The equations of motion were solved numerically for Powell-Eyring flow in annuli and it is demonstrated that plots of a reduced shear stress at the outer wall of the annulus vs. a reduced pseudo-shear rate superimpose almost exactly on analogous plots for tube flow. Values of the ratio of the radial distance at which the shear rate is zero to the outer radius of the annulus relative to the same ratio for Newtonian flow are given for ratios of the inside to outside radius from 0.05 to 0.8 and for mildly to highly non-Newtonian flow. The results compare reasonably well with previously published experimental data.
Axial non-Newtonian flow in annuli is important in such applications as flow in double-pipe heat exchangers and the flow of polymers and polymer solutions in extruders. Analytical solutions for axial, laminar flow in annuli have been reported by Fredrickson and Bird (1958), Laird (1957), and Slibar and Paslay (1957) for Bingham flow; by Fredrickson and Bird (1958) for power-law flow; and by McEachern (1966) for Ellis-model flow. More recently, Nebienskjr, et al. (1970), reported data for a limited range of Powell-Eyring flow, which were obtained by a numerical solution of the equation of motion. Also, Fredrickson (1959) and Coleman and No11 (1961) have proposed a general solution that requires graphical or numerical evaluation of integrals by use of experimental data. The relationship between the shear rate, i., and the
shear stress, r , given by the power law = wl(;)n-*~ (1) predicts infinite viscosities and zero viscosities in the limits of very low and very high shear rates, respectively. The low-shear-rate prediction is relatively unimportant in application to tube flow, where low shear rates occur at small radii. However, large errors may occur in application of the power law to flow in annuli, since the lowshear-rate flow occurs at large radii. Thus, large differences have been noted (McEachern, 1966; Vaughn and Bergman, 1966) between analytical results based on the power law and experimental results. McEachern (1966) has shown that analytical results based on the Ellis model and results from the generalized method agree well with 7
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3% AOUEOUS CARBOPOL FLOW DATA (BERGMAN, 1962)
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TUBE A DIAMETER 0 I15 cm TUBE 8 , DIAMETER 0 0 8 7 a 0 TUBE C DIAMETER 0 0 3 5 c m ANNULUS , K = 0 5 0 7 , LARGER DIAMETER 0 0
,
I
-x
