Unstructured Models of Pellet Growth

The growth of pellets is not always exponential because of mass transfer limitations (Metz and Kossen, 1977 Pirt, 1975 Whitaker and Long, 1973). Increasing the size of pellets results in a significant decrease in reaction rate, and the apparent Kg value increases (e.g., Kobayashi et al., 1973). To describe the growth of pellets, one can use either the exponential law or the Monod relationship. However, the logistic equation (Kendall, 1949) in the form of Equ. 5.74 or the Gompertz equation (Chiu and Zajic, 1976) 

Most often, the model used to describe the growth of pellets is the cube root equation (cf. Metz and Kossen, 1977 Pirt, 1975 Trinci, 1970). Considering the cell mass to be related to the pellet radius, R, and the number of peUets, N, according to Equ. 5.256 

The cube root law results from a combination of exponential growth with mass transport limitations in the solid phase, which is expressed in the concept of dp as shown by Pirt (1975). 

The predictions of different kinetic model equations for pellet growth are compared in Fig. 5.76 (van Suijdam et al, 1982). The main objection against the models presented is that they are autonomous, that is, the biomass rate equation is not related to the concentration of the limiting substrate, and that mass transfer limitations are not explicitly considered in these macrokinetics. To introduce these phenomena, a combination of mass transfer limitation effects and biokinetics must be used. Such an integrated model for pellet growth was developed (Metz, 1975) and extended (van Suijdam et al., 1982), based on balance equations for X, S, and O2 in liquid and solid phase. 

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