Population Balance Modeling of Turbulent Mixing for Miscible Fluids

[+] Author and Article Information
Giridhar Madras

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, Indiagiridhar@chemeng.iisc.ernet.in

Benjamin J. McCoy

Department of Chemical Engineering,  Louisiana State University, Baton Rouge, LA 70803.

J. Fluids Eng 127(3), 564-571 (Mar 01, 2005) (8 pages) doi:10.1115/1.1899174 History: Received July 20, 2004; Revised December 17, 2004; Accepted March 01, 2005

Blending one fluid into another by turbulent mixing is a fundamental operation in fluids engineering. Here we propose that population balance modeling of fragmentation-coalescence simulates the size distribution of dispersed fluid elements in turbulent mixing. The interfacial area between dispersed and bulk fluids controls the transfer of a scalar molecular property, for example, mass or heat, from the dispersed fluid elements. This interfacial area/volume ratio is proportional to a negative moment of the time-dependent size distribution. The mass transfer coefficient, in the form of a Damkohler number, is the single geometry- and state-dependent parameter that allows comparison with experimental data. The model results, easily realized by simple computations, are evaluated for batch and flow vessels.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

Variation of Pavg(θ) with time θ for γ=10−4,10−3,10−2, 0.1, 0.5 and 1 on a log-log plot showing the asymptotes, γ1∕2, at large time θ and the asymptotic behavior, (1+θ)−1, for small and intermediate time

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Figure 2

Variation of the νth moment, P(ν)(θ), with scaled time θ for (a) ν=−1,−2∕3, and −1∕3 with γ=0.01; (b) γ=10−4,10−3,10−2, 0.1, 0.5 and 1 with ν=−2∕3

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Figure 3

Variation of the intensity of segregation, Is, with time, θ, [Eq. 313] for κM=1, 10, 50, 100, 500, 103 and 104 and α=0.2,ν=−2∕3,γ=0.01

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Figure 4

Variation of the extent of mixing, Em(θ), (a) with κMτM [Eq. 315] for α=0, 0.2, 0.4, 0.6, 0.8, 0.9 and 0.95; (b) with θm for κM=0.1, 0.5, 1, 5, 10 and 100 and α=0.2,ν=−2∕3,γ=0.01; (c) with θm for γ=10−4,10−3,10−2, 0.1, 1 and α=0.2,ν=−2∕3,κM=1; (d) with θm for γ=10−4,10−3,10−2, 0.1, 1 and α=0.2,ν=−2∕3,κM=0.01

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Figure 5

Variation of τ with time, θ, [Eq. 316] with γ (=10−4,10−3,10−2, 0.1, 0.5 and 1) as a parameter for ν=−1,−2∕3, and −1∕3

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Figure 7

Variation of the mixing time with κM for ν=−1∕2,−2∕3, and −0.9 and α=0.2,Em=0.95

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Figure 8

Comparison of the model [Eqs. 319,320] with γ=0.01 and ν=−2∕3 and the experimental data of Marchisio (26). The other parameters are listed in Table 1.

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Figure 6

Dispersed and bulk fluid concentrations in a CSTR: (a) Time variation of the dispersed phase concentration, C(θ), and bulk fluid concentration, Cb(θ), in a CSTR for γ=0.01,ν=−2∕3, and κM=100; (b) Time variation of the bulk fluid concentration, Cb(θ), in a CSTR for κM=0.1, 10, 1000 with γ=0.01,ν=−2∕3; (c) Time variation of the bulk fluid concentration, Cb(θ), in a CSTR for ν=−1∕3,−2∕3,−1 with γ=0.01, and κM=1; (d) Time variation of the bulk fluid concentration, Cb(θ), in a CSTR for γ=10−4,10−2, 1 with ν=−2∕3 and κM=1




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