Hw 5 solutions - Problem 2{55 pta Cunnider the dynamic model for sustained eeeillattene in yeast glyeehreie = en—eee=0te.e an 1 A T I =

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Unformatted text preview: Problem 2 {55 pta}. Cunnider the dynamic model for sustained eeeillattene in yeast glyeehreie: % = en—eee=0te.e an 1 A _ T I = 2k1fin‘1 — tip-Km +44 — fg{G,fi} where G and A are the intracellular concentrations of glumue and ATP1 respectively, Vin = llEfi is the constant flux of glucose into the yeaet cell, #1 = [3.02 is an enzyme activity, and the parameters hr, = ELI} and Km = 13.0I determine the kinetics ef ATP degradatien. 1. [LI] pts} Show that the parameter values produce a single steadywstate aulution: a 2 H115, .91 $1.”. 2. (20 pte) Show that the linearized model at this steadynstate point in: :1:=Au: §_ —0.0354 —0.203 0ft ‘ 0.0700 0.0435 where x = [G’ 0’19". 3. [15 pte} Determine the stability of the steady state by computing the eigenvalues of the A matrix. a. «I. w; - .5 f — J 1 _. 11 .. a U)(aj_5.d.3_)_ H... .. J . dé‘ .5” x x 21:; 1956934 gird} 6+ eff/M .4! :rmsimu‘ = r (aofl (D?) 6‘- (omJOmIJA‘E “0.04” gt 0,20”? .. da‘. 3 1 3 --:»-f;p;,= Mfl'f as“ 556+ cM' 6J’+ (mEMDJCIJ )A‘ (Knkfijl 2 15:,M“+ (24921 k.- ._.i I , ~. = mam (2m: fifgfg )4 : 1(0. 02) (l.77)_6\+ [056191) (“3'”) F 1 | I (CH-:- 13) x -- 0211+ L71)“ ’4 3-4: 2 6.0708 6+ 0.0513514“ (5] Ax [—DfiEI‘i Axial) ' ' . 4.! .._._ . a . _;___; . . 3 _ x x A a oH' c.0709 0.0mm X A my : Lima—m: 56-16? " " 5}! O-W ..¢-df$f~?‘. . i” 1*mwmwwmwwwfiru 1 ' ': pa} (aggfi'wflwfin 40-:mfiflmfiyu . . "= )1; '03:::3'1--)4-+"-s:n'!n $1?"){r-fOfiGfié'H—DI'HIJL . i _| .‘1 i { "#44:;51? Sgt-L"? f3” H'u‘ltll. L. :1 - tea-{rig J'fi."3;{'i.+ {*1 e ‘ 1 Problem 1 {W pts}. Consider the hatch operation of a flash drum for separating a. lunar},r liquid mixture. The drum has liquid molar holdup M and liquid mole fraction of the more volatile component re. A constant rate of heat Q is applied to the drum, producing a constant vapor molar flow rate V. The vapor mole fraction of the more volatile component is denoted y. A linear vapor—liquid equilibrium relation 3; = kit, where k 2} l, is assumed. Mass balances yield the following ordinaryr difierential equation system: dM a! Ma: '1. (25 .pts] Solve the dose analytically to obtain: MD—Vt Milt) = Iii} .2 ¢D(Mofl;DVt]t—L 2. [25 pts} Linearize the 013133 about the initial conditions to generate linear UDEs for M’H’J =' M[t) — Mn and :r’rjt) = fit} '— ou. atrialytioall}r solve the first linear ODE and formulate the second linear ODE to obtain: - , Milt}. =' no as“ V V v ' ; __ IE— " = ._..__ _ ( _ ) dt + u( 1):: Maw Use 1+ Mar How would you solve the second ODE? How would you expect pour-solution to compare to the analytical solution in part 1? ' ' . ' 3. {20 pts) Generate numerical solutions to the original CIDEs givenin part 1 by applying the forward Euler method with step size h. Show that: - Mr” 2 filo — th I; $n+l — In _ n-Lrh) 1);]:‘71 How would you expect your solutions to compare to the analytical solutions in part 1? |. ! L—I. I 1/ Ya = m y: fix k7! HJX e-fifil cl“ Hm} .____.._, Q]: n G - 2 ‘3 q JI_U NC I"? d+_-.Uy “UOJ K m _ rm T. u (l~k)("'lfi) [In (nu—wk fa mi) IE, (ii/go}: {#4) It» (mg?) ho _ U4. (Th-11 Hg Ho — f'ls ~15} Hui-1'): Kn - é) d C H H LU 24-”, 01de mqhm Gina—H: # ifiuc-U x :44 (fwd XMTKD h‘msmyn.) xic+J=gc+Hrn — 1 I Lima- ffi: 915-1: 4— [GIG-)1 : I“ G) “*4 M1 [3: 31h“ "P1 “qu 541':le3 [mg-rm “Inn JfJ'Til-‘fi fidufi. "£0" ‘i‘wty @ dn 1% «ELK. “4/1? hi1“: {Lin—Wm ® nFm—ua “wfirw: mm Ha: na-nua 1;; M [th = nnfiw I h. -1 d "F FM": {Earl—q; Mcfl: m. H; @6331 43 3'4; 1 q 71W“ f‘bI-AJFI 51A if Mt: xh _ A Eh“ (mm Gnarly £-+ 1H“ 51m 5.: {EL Huh-Ll] J-A‘fllw, ...
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This note was uploaded on 04/24/2010 for the course CHEM-ENG 361 taught by Professor Henson during the Spring '10 term at UMass (Amherst).

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Hw 5 solutions - Problem 2{55 pta Cunnider the dynamic model for sustained eeeillattene in yeast glyeehreie = en—eee=0te.e an 1 A T I =

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