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The Population Balance Equation

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Chemical Reactor Modeling

Abstract

The chemical engineering community began the first efforts that can be associated with the concepts of the population balance in the early 1960s.

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Notes

  1. 1.

    Smoluchowski [121], Williams and Loyalka [136], Friedlander [27] and Kolev [48] (p. 170), among others, considered very small particles and interpreted the cross sectional area in a slightly different way representing an extrapolation of the concepts in kinetic theory of gases. In these analyzes the traveling particle is treated as a point in space without cross sectional area, thus the effective area becomes equal to the cross sectional area of the stagnant particle. This alternative cross sectional area approximation becomes: \(\sigma '_{A_T} = \frac{1}{4} \pi d^2\).

  2. 2.

    The collision tube concept is familiar from the kinetic theory of gases. Consider a particle in such a cube, moving with a relative speed with respect to the other particles which are fixed. The particle in the tube sweeps a volume per unit time (\(\mathrm{m}^3 \mathrm{s}^{-1}\)). Venneker [132] named the rate of volume swept by the particle for the effective swept volume rate. Henceforth this name is used referring to this quantity.

  3. 3.

    Note that the parameter value used by Luo [79] is a factor of 4 larger than the one in (9.17). The reason for this deviation is that Luo used a different definition of the effective total cross-sectional area \(\sigma _{A_T}\).

  4. 4.

    Luo and Svendsen [80] did not distinguish between the experimentally determined relation (9.32) and the Kolmogorov structure function (9.15). In their work the theoretical parameter value \(C\) was calculated as \(C=(3/5)\varGamma (1/3)C_k\approx (3/5)\times 2.6789 \times 1.5 = 2.41\). It follows that their mean droplet velocity estimate is \(\bar{v}_\text {drops} \approx (\frac{8 \overline{\delta v^2 (d)}}{3\pi })^{1/2} = \sqrt{8 \times C/(3\pi )} (\varepsilon d)^{1/3} =\sqrt{2.046}(\varepsilon d)^{1/3}\).

  5. 5.

    Politano et al. [98] adopted the kernel functionality of Luo and Svendsen but modified the definition of the effective collision cross-sectional area, \(\sigma _{A_T} = \frac{\pi }{4} \left( \frac{d_i + \lambda }{2}\right) ^2\), in accordance with other work in nuclear engineering (e.g., Kolev [48], p. 168).

  6. 6.

    Luo and Svendsen [80] did not distinguish between the value of the theoretical parameter \(C\) in the structure function relation (9.15) and the empirical parameter \(\beta = 2.0\) in the droplet rms velocity relation (9.32) when calculating the mean eddy velocity from (9.39). To reproduce the parameter value in the eddy number density relation used by Luo and Svendsen, the value of the \(\beta \) parameter is computed using the structure function relation (9.15) instead of the empirical relation (9.32). That is, they let \(\beta = \frac{8 C}{3\pi }\), and \(C=\frac{3}{5}\varGamma (1/3)C_k \approx \frac{3}{5}\times 2.6789\times 1.5=2.41\). The parameter in the eddy number density of eddies was thus approximated as: \(\frac{9 C_k}{C 2^{5/3} \pi ^{2/3}} = 0.822\).

  7. 7.

    Luo and Svendsen used the parameter values \(C=\frac{3}{5}\varGamma (1/3)C_k = 2.41\), and \((8C/3\pi )\approx 2.045\), hence their parameter value in (9.45) becomes \(\frac{\pi }{4} (0.822) \sqrt{2.045} \approx 0.923\).

  8. 8.

    Notice that this efficiency function is not necessary volume or mass conservative as Luo and Svendsen [80] did consider the breakage efficiency function being equal to the kinetic energy distribution function. It would probably be better to consider the breakage distribution function purely proportional to the empirical kinetic energy distribution function, and determine the probability constant by requiring bubble volume or mass conservation within the breakage process.

  9. 9.

    Luo [79] employed (9.16) with \(C\approx 2.41\) and obtained \(\bar{e}(d_i,\lambda ) = \rho _c \frac{\pi }{6} \lambda ^3 \frac{\bar{v}^2_\lambda }{2} = \rho _c \frac{C \pi }{12} \lambda ^{11/3} \varepsilon ^{2/3} = \rho _c \frac{C\pi }{12} \xi ^{11/3} d_i^{11/3} \varepsilon ^{2/3}\). Moreover, with this parameter value and bubble velocity estimate the critical energy ratio becomes \(\chi _{cr} \approx e_s (d_i,d_j)/\bar{e}(d_i,\lambda ) = 12C_f\sigma _I/(C \rho _c \varepsilon ^{2/3} d_i^{5/3} \xi ^{11/3})\).

  10. 10.

    Luo and Svendsen [80] employed the parameter value \((8C/3\pi )\approx 2.045\) instead, hence the parameter in (9.52) becomes \(0.923\).

  11. 11.

    A local space dependency of the \(P_B\)-function is retained as the integration is over a differential microscopic volume \(dV_{r'}\), thus the integrated \(P_B\)-function varies locally in space \(\mathbf {r}\) but the \(P_B\)-function does not differ over the microscopic differential volume \(dV_{r'}\) of the different particles.

  12. 12.

    A local space dependency of the \(\kappa \)-function is retained as the integration is over a differential microscopic volume \(dV_{r'}\), thus the \(\kappa \)-function varies locally in space \(\mathbf {r}\) but the \(\kappa \)-function does not differ over the microscopic differential volume \(dV_{r'}\) of the different particles.

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  1. Department of Chemical Engineering, Norwegian University of Science and Technology, Trondheim, Norway

    Hugo A. Jakobsen

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Jakobsen, H.A. (2014). The Population Balance Equation. In: Chemical Reactor Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-05092-8_9

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