Question

b) A first order reactionA→B, with the rate constant k is taking place in CSTR fed with A at concentration Car which remains unchanged. There are likely to be some deviations in the feed rate (F) of A. Derive linearized transfer function between. concentration of A in the outlet and feed rate of A assuming that volume V of reacting mixture remains unchanged.

Answer 1

 Is the model linear? If we decide to solve the model numerically, we do not have to linearize; in fact, the non-linear model would be more accurate. However, in this problem we seek the insight obtained from the approximate, linear model.

All terms involve a constant times a variable (linear) except for the following term, which is linearized using the Taylor series..

$(1) C0.5 ≈ (C0.5 ) + 0.5(C−0.5 ) (C − C )$

higher order terms AAs AsAAs

This approximation can be substituted into equation 2, and the initial steady-state model subtracted to obtain the following, with C’A = CA - CAS.

$(2) V dC'A = F(C' −FC' ) − Vk(0.5C−0.5 )C' dt$ A0

This linear, first order ordinary differential equation model can be arranged into the standard form, given in the following.

$τ(sC'A (s) − C'A (t) |t=0 )+ C'A (s) = KC'A0 (s)$