Pseudo distribution

At some point in time, a pseudo-distribution equilibrium is attained between the tissues and fluid of the central compartment and the poorly perfused or less-readily accessible tissues (peripheral compartment). Once such a pseudodistribution equilibrium has been established, loss of drug from plasma can be described by a... 

CLD/DBD for Mixed-metallocene Polymerization of Ethylene 9.6.2.1 Formulation of Pseudo-distribution Problem... 

Propagation step In this case the degree of branching is not altered, so this is a simple step from this point of view. The contribution to the balance of oth branching moment distributions of living chains in reduced or pseudo-distribution notation follows, by multiplication with t and taking the summation over the branching index t ... 

Transfer to monomer, /I-hydride elimination Again, the degree of branching is not affected, hence the contributions to the pseudo-distribution balances of the living and dead chains are given by ... 

The principles of the pseudo-distribution model are applicable to any reactor. Results are presented for a semi-batch reactor using kinetic data from a realistic mixed-metallocene system in Figure 9.8. 

We conclude that the pseudo-distribution approach can be appUed successfiilly, provided a good approximation can be made for the branching distribution at given chain length from the branching moments. The method is valid for batch reactors as well, in contrast to, for example, the pgf-cascade method (Section 9.7), which is restricted to steady state reactors. It is to be preferred over classes methods in cases, like the metallocene one, where the second distribution dimension may assume high values as well. 

This problem has been introduced in the discussion of the classes approach. For reaction equations and a full set of population balances, see Tables 9.5 and 9.6. Here, we address the more general problem of more than one TDB per chain [9]. This occurs as a consequence of insertion of TDB chains created by disproportionation or of recombination termination. We start with the full 3D set of Table PVAc2 and then reduce it to a ID formulation by developing the TDB and branching moment expressions. The (N, M)th branching-TDB moments or pseudo distributions for living and dead chains are defined by ... 

Performing the corresponding summations on the equations in Table 9.6, one obtains the (N, JVf)th moment formulation of Table 9.11. Some of the summation terms in these equations will not be evaluated for the general (N, M) case, but we will determine them by assigning values to N and M. Since we will not address branching, we take M = 0 here, but in principle this can be treated in a similar way. We will focus now on the TDB moment distributions and successively derive the model equations for the zeroth, first, and second moments, or N = 0,1, and 2. Solving the model thus essentially means solving the population balances of the real concentration distributions and P and the pseudo-distributions and... 

Next, we vill derive the higher moment equations. For N = 1 and M = Owe obtain the set of population balance equations for the pseudo-distributions of order (1,0) as listed in Table 9.13. 

This implies that under the condition of a maximum of one TDB per chain, the set of population balance equations of the TDB branching moment variant of the model is solvable without requiring any additional closure assumption. The results obtained with the pseudo-distribution model are identical to those obtained with the classes model shown before (see Figure 9.6). 

TDB Pseudo-distribution Approach for More than one TDB per Chain... 

The 1D population balance for the pseudo-distribution or first branching moment distribution follows by taking the first branching moment of Eq. (78) ... 

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