Lumped equation

An extension of this one-dimensional heterogeneous model is to consider intraparticle diffusion and temperature gradients, for which the lumped equations for the solid are replaced by second-order diffu-sion/conduction differential equations. Effectiveness factors can be used as applicable and discussed in previous parts of this section and in Sec. 7 of this Handbook (see also Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990). 

There are n equations for the n lumps of the membrane system, and two equations for the adjacent reservoirs (0 and n+1). All the lump equations have the following common form ... 

The corresponding lumped reaction can be obtained by multiplying each reaction by a coefficient and adding them and stating that the quasi-stationary species do not appear in the lumped equation for the reaction the following system is then obtained ... 

Computational method and estimation of parameters. The system of three differential equations which pre-sents the design model is nonlinear and subject to boundary conditions. For solving numerically the model equations the method of orthogonal collocation was used (90, 91). As collocation functions the so-called shifted Legendre polymonials were applied. As a rule the collocation was done for 5 inner points. The lumped equations were solved by means of the Newton-Raphson iteration method. 

Due to the complex nature of the emulsion polymerization process and the associated set of distributed and lumped equations, the control vector parameterization approach can be adopted along with the -constraint optimization technique in a multiobjective form to account for the various control objectives. Several constraints should be included to define the desired final particle size and to account for different process and recipe limitations. 

The solution of the gasoline equation requires the input from the six oil lump equations resulting in seven simultaneous differential equations which can be solved using a Runge-Kutta algorithm. The solution strategy for the eight lump model is presented by Kraemer (1991). 

The overall cracking constants for the pure hydrocarbon groups (P, Nj, and A,) are presented in Table 5. Values obtained using both the exponential decay function and the power law function are presented. These constants were obtained by solving the light lump equation (equation 3.13) in the 8-lump model with Yj set to zero. This is equivalent to using the three lump model for each separate light lump oil mixture with first order kinetics. 

The corresponding lumped equation system is then given by... 

However, in order to express the right-hand side in terms of the new lumped variables c, then the generalised inverse needs to be found. The inverse will not be unique, but its form does not affect the form of the lumped equations. We can arbitrarily choose... 

Here k and k are the lower and upper limits for the particular mixture. The product g(k,t)dk is the total concentration of a species with rate constants between k and k + dA, and should be interpreted as a concentration distribution function. As the number of species within the mixture grows and approaches infinity, then the separation between k and k becomes larger, and for convenience, it is assumed that k 0 and k - oo. This leads to the conventional form of the lumping equation for continuous mixtures ... 

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