Question

Problem: A reaction A+ R takes place in a constant-volume isothermal CSTR. The component mass balances are: 1401 =CH-C, -kC,C, de=C-Cp+kC/CR dt where: Cai feed concentration of A (input); CRI: feed concentration of R (assumed constant); CA, CR: concentrations of A,R, respectively, in the reactor and outlet; t, k:, positive constants. a) Linearize the equations around the design steady-state (0.0...) (and put them in deviation variable form. b) Calculate the transfer function between CR and CA c) Under what conditions on the process parameters and steady state values is the above transfer flmction stable? d) what is the order of this transfer function ?

Answer 1

at ... ☺ Co-Ğ = cé) , e da Cai - Ca klas.... 0 z dCA = Cei - GQ + k CA CR... @ At Steady state Jeda - Chi - A - K CA CO ... ③ 7 d६.८-६+६ 7 Subtracting from 0 & © from @ zda = Cai -Á - (k (ALB - k A Go) (CA - CA - CA z dle cei - C6 + (k ale-k (ACR) a i f109) Hay.) +() Ca=7o) (24), (y-yo) Neglecting other torms - =6, Y= 68 LAG = 5 + Ğ CCA- A) + Å (Co-CG). AG - A G = CG A + A Co. Similarly CALE=GR Å + G (AM) + CA (Co-Q) no, yo 72. Yo CA CR- Te A = CR CA + Cale Putting these in ☺ & © Eda & Car - CA - kg Au - kA ... ③ at edco' - Cicé + kłR CA + k Alk... ®

© Taking Laplace Transform of ®*® ESCACS) - CAO ) - CA (3) - k C 5 (s) - K C6 (3) (cs +1+) CC - āí (s) – kĞ Coc.). A '(s) = Chi(s) k a técs) est (1tk Co ist (1tk J. ESTÁ (s) = CR; (3) - CR (3) + k CR CA (s) + k CA C6 (o) (: Cé(0)=0) (estl-ki) G (s) = Co (s) + k EQ CÁCs) Gel - Chi (s) — + ka WCB est 1- KIA, est- kA) Now putting CA (s) here, (s) = Ri (3) + k u I Ci (s) - kLA Có (3) estl- k a ēstl-kto lest (1+kGB) Zst (1tk) : In between C (s) & Chi(s). . Putting Cri (3) = 0 &, C6 (3) = 0 Cé(s) = k Q Chi (3) es + (109 K E) (est 1tk) s). k C D com [cs+(1+k%)][ <3 +(1+k)]

acs) - Es+l-kĞ (ts +1+k8 k C 7s2+2 (1-67) s +T (1+)3 +(1-kā)(1+) - kr Zs+ t2-k (6-6)]s + [1-kā teče -kace Taking Routh Array of the characteristic Equation. 22 (1-k (EA-(6) - k ² (ACE) { [2-k(6-6)] z[2-k (50-56) (1 - 21G-56)-15A (B) - 7x0 [2-k (A - (b)] . for the system to be stable all elements in the Routh's array of first columm should be yo. fa-keth (6) ][ - k (62-66) -k" AC6]>0. [2-ketta - to 1-k (CA-C6) - k? TA CB YO Tk (A - (G) + k² GB < 1 ) and for any values of the system is stable d) The order of the system is 2.
the order of system is 2 since the highest exponent of s is 2