The integrals cannot be solved in a closed form

The integrals given in the previous section can be solved in a closed form only in very few cases. Under such conditions the integrals in eq. (3.109) 

The inner integral J can be solved in a closed form using z = constant as explained in Section 3.4.1. It is given by 

In Fig. 3.16 the relationship between rj and x is given for some values of r at constant degree of advancement = O.I. T is calculated according to eq. (3.118) for some assumed values of (0,t). T] is determined by numerical integration using eq. (3.117). 

In Fig. 3.17 a different presentation is chosen. One assumes at turnover = 0.95 at= 0 and the corresponding time (t = 3.00) is kept constant. Assuming different values of r the turnover is given as a function of ij. These values can be obtained by numerical integration according to eq. (3.117) for a constant lower limit and a variable upper limit. The values for r— oo ( =0) cannot be obtained by numerical integration. Under these conditions the exceptional case happens where the integral can be solved in a closed form. 

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