Balancing with Generating Functions

The following should be helpful in this regard for clarifying, how equations are to balance properly. It is shown that a deeper penetration of stoichiometry can literally point out also the impossibility of certain reactions for reasons of the balance. 

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As for many immobilised enz3nnes, the hydraulic behaviour Is not adequately described by classical fluid mechanics. It was, therefore, necessary to develop a detailed mathematical model of the column hydraulics which together with a laboratory test procedure, would provide data on the basic mechanical properties of the enzyme pellet. The model Is based on a force balance across a differential element of the enzyme bed. The primary forces involved are fluid friction, wall friction, solids cohesion, static weight and buoyancy. The force balance Is integrated to provide generating functions for fluid pressure drop and solid stress pressure down the length of the column under given conditions. 

The plausible way of solution of this problem is the combination of balancing with other chemical engineering calculations (simulation). So-called automatically generated streams are calculated as certain functions (e.g., fractions) of so-called reference streams. IBS generates such standard streams (flow rates) automatically when the reference stream is entered. This complementing of direct measurements is the only way of making balancing viable in older plants with insufficient instrumentation. 

Biagini and Parry [73] showed that the block copolymerization of di- and tripeptide-based NBE dicarboximide with PEG-functionalized NBE derivative using a first-generation Grubbs ruthenium initiator yielded water-compatible copolymers. The copolymers formed aggregates upon dispersion in water, and the type of aggregates (folded worms, interpenetrating networks) was directed by the specific peptide sequence rather than by the hydrophobic/hydrophilic balance [74]. 

One of the ways to solve such problems is to utilize the technique of the generating function, which is described in Appendix 3.1. With the help of the generating function G s, t), the mole balance relations in Eq. (3.3.5) are combined into one partial differential equation that has a numerical solution only. However, if the feed to the batch reactor is a pure monomer or has a distribution given by Eq. (3.5.7), it is possible to obtain an analytical solution. 

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