Solving and Using Population Balances

Hounslow Discretization. Hounslow et al. (16) developed a relatively simple discretization method by employing an M-I approach (the mean value theorem on frequency). The population balance equations, such as Eq. 3.20, are normally developed using particle volume as the internal coordinate. Because of the identified advantages of length-based models, Hounslow et al. (16) performed the coordinate transformation to convert the volume-based model described by Eq. 3.20 to a length-based model as follows  

The continuous binder size distribution model described by Eq. 3.21 can also be discretized using a similar numerical scheme as follows (7)  

A particle of size v in the size range Xj and X/+1 can be represented by two fractions a(v, X/) and b(v, X/+i) associated with the two grid points Xj and X/+i, respectively. For the conservation of two general properties/i(v) and/2(v), these fractions satisfy the following equations  

By using this composition technique for particle properties, discrete equations for coalescence-only population balance model given by Eq. 3.20 have been formulated as follows  

tmax defined previously, djk is the Dirac-delta function, and rj is defined as follows  

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