F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

(2.111)

so ordered subtraction of Eq. (2.111)from Eq. (2.109)is in order to yield

(2.112)

– where parentheses were taken out for convenience; after factoring out in the left‐hand side, and condensing similar terms in the right‐hand side, Eq. (2.112)becomes

(2.113)

One may now factor out k 1 k 2or (as appropriate) in Eq. (2.113)to obtain

(2.114)

where the content of the first parenthesis in the right‐hand side is but a geometric series containing n terms; insertion of Eq. (2.93)with u 0 = 1 unfolds

(2.115)

while division of both sides by 1 − k 2gives then rise to

(2.116)

A graphical representation of Eq. (2.116)is made available in Fig. 2.6– after recalling Eq. (2.72)and dividing both sides by u 0, i.e.

(2.117)

n , k 1/ u 0and k 2were consequently utilized as independent parameters. An increasing n systematically produces a higher‐value series – and a similar effect is triggered by a larger k 1or a larger k 2; there is a horizontal asymptote when k 2 = 0.5 (besides the trivial case of k 2 = 0), in much the same way horizontal asymptotes arose in Fig. 2.5.

Figure 2.6 Variation of value of n ‐term arithmetic–geometric series, S n, normalized by first term, u 0, as a function of n – for selected values of increment, k 1, normalized also by u 0, and ratio, k 2, for (a) k 1/ u 0 = 0, (b) k 1/ u 0 = 0.5, (c) k 1/ u 0 = 1, and (d) k 1/ u 0 = 2.

In general, one realizes that

(2.118)

with the aid of Eqs. (2.72), (2.73), and (2.117), or else

(2.119)

after direct application of the theorems on limits. If ∣ k 2∣ < 1, then Eq. (2.119)degenerates to

(2.120)

– where the first term entails an unknown quantity, while as n → ∞ was used to simplify the second term; after rewriting the second term in numerator of the first term of Eq. (2.119)as

(2.121)

one gets an unknown quantity of the type ∞ / ∞ – so one may apply l’Hôpital’s rule to get

(2.122)

Equation (2.122)degenerates to

(2.123)

since ∣ k 2∣ < 1; insertion of Eq. (2.123)allows final transformation of Eq. (2.119)to

(2.124)

which describes the vertical intercept of the horizontal asymptotes in Fig. 2.6for curves with . Therefore, the series is convergent when −1 < k 2< 1 – irrespective of the actual values of u 0and k 1, and divergent otherwise; as expected, Eq. (2.124)degenerates to Eq. (2.98)when k 1 = 0.

The trivial case, associated with k 1 = 0, transforms indeed Eq. (2.116)to

(2.125)

after having u 0factored out – and serves as descriptor of the curves plotted in Fig. 2.6a; Eq. (2.125)coincides with Eq. (2.93), corresponding to a plain geometric series, as expected from

(2.126)

that in turn stems from Eqs. (2.87)and (2.110)– thus fully justifying coincidence between Figs. 2.5and 2.6a. Conversely, k 2 = 0 converts Eq. (2.116)to

(2.127)

which serves as descriptor of the bottom curves in Fig. 2.6, labeled as k 2= 0. When k 2→ 1, Eq. (2.116)becomes

(2.128)