(V 1. (T i. )- FrC p. ))= 0 = FrC p (T 1. (T 1s. )+ UA(T os. (T is
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1 . Yu are repnible fr a reacr in which an exhermic liqui-phae reacin ccur. The fee mu be preheae he hrehl acivain emperaure f he caaly, bu he pruc ream mu be cle. T reuce uiliy c, yu are cniering inalling a hea recvery yem in which he pruc prehea he fee. A Of cure yu will eign an elec he hea exchanger hanle he eire eay-ae hea uy, bu yu wul al like cnier he cnrl yem a par f he eign. Tha mean yu wan uneran he yem ynamic - hw variain in he inle emperaure an reacr hea la affec he ule emperaure. Yu herefre repreen he yem a a cllecin f fir-rer lag. Here he hea exchanger i repreene a w irre ank cnnece by hea ranfer area A. Each ank will have i wn vlume. The hea f reacin i repreene by a heaing cil ha eliver pwer Q. Q fee pr T a. Mel he yem: begin wih M&E balance an bain anar-frm ifferenial equain. Claify yur variable a inpu, inermeiae, an upu. Yu have ank - hw many equain yu expec? Three equain, f cure - we mu wrie an energy balance n each ank. Fr example, n he ank repreening he fee ie f he hea exchanger, V - T r = F - F UA - 0 = T r i a reference emperaure fr efining enhalpy f he flwing liqui. In ur implifie mel, hea ranfer ake place beween w well- mixe ank, here i n nee fr lgmean emperaure in efining he hea ranfer cefficien. Wriing he balance a eay ae give V = 0 = F - F - T r UA - Frm he eay ae balance we can calculae he upu an inermeiae variable frm he inpu variable an peraing cniin. Subracing he eay balance frm he riginal ODE give eviain variable, remving he reference emperaure T r in he
2 prce. We have al preume ha eniy an hea capaciy n vary wih emperaure, which i accepable fr ur implifie analyi. V = F - F UA - 0 = 0 Nw we place hi in anar frm fin T T = T i T T 0 = 0 T T = CQ T 0 = 0 T = T 0 = 0 V j j = j =, = V UA = F F F C = Fr C p b. Cmbine he equain in a ingle equain ha relae upu inpu variable hrugh ranfer funcin. D hi by aking Laplace ranfrm an algebraically cmbining eliminae he inermeiae variable. Wha rer yu preic fr he final ranfer funcin? The ranfrm i raighfrwar. Fr example = Ti T Thi hw u ha epen n w inpu, prcee hrugh ranfer funcin wih ienical ynamic funcin f bu iffering gain. We ue w f hee equain eliminae he inermeiae variable an T, leaving u wih he ranfer funcin ha relae inpu an Q upu. = G i G Q Q G i = D G Q = C C D D = - Cmbining hree fir-rer equain ha given u a hir-rer equain.
3 c. Reurn he Laplace ranfrm f he cmpnen equain. Draw a blck iagram repreen he ignal flw hrugh he cmpnen ranfer funcin, hwing hw inpu are prcee becme upu. Then ue hi iagram erive a ingle equain relaing upu inpu. Shul he reul be he ame a in par b? Wa i eaier han b? T i T T Q C T Drawing hi iagram in eence guie yu hrugh he algebraic ubiuin, ha I fun he algebra eaier han in par b. egin wih he upu an wrk backwar. le G j = j
4 = G G T = G T = G G G CG G Q G [G T CG Q ] = G G G [G T i G T ] CG G Q T - G G - G G G = G G G G G T i CG G Q Ł ł Thi la equain hul be equivalen he reul f par b.. T really rive i in he grun, reurn yur riginal cmpnen ODE. Cmbine hem in he ime main ha he epenen variable i relae inpu frcing funcin. Yu will have iffereniae me f he equain pruce furher inepenen equain ha yu can eliminae inermeiae variable. Wha rer yu preic fr he reuling ODE? Can yu ee he relainhip beween hi equain an he ranfer funcin exprein erive earlier? Tha hir-rer Laplace ranfrm equain i cmpleely equivalen he hir-rer ODE ha we ge by cmbining hree fir-rer ODE. T T = T i T T 0 = 0 T T = CQ T 0 = 0 T = T 0 = 0 Fir, iffereniae. = T a In a, ue an eliminae erivaive f an T, an hen eliminae T. Nw i relae inpu an Q, an inermeiae variable. 4
5 T T Ø Œ - ø œ º ß 4 = C Q - Then iffereniae 4, eliminae he erivaive f wih, an eliminae wih 4. The reul i T T Ø ø Œ - œ T ß º = Ti C Q CQ 5
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