Homework Assignment Number Eight Solutions

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1 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle Homework Assgmet Number Eght Solutos Problem Fd the soluto to the ollowg system o olear algebrac equatos ear () Soluto: s Sce ths s a system o three olear algebrac equatos wth three ukows we ca use the multvarate Newto-Raphso wth umercal appromatos to the dervatves We wll set a tolerace o 0-6 The put ucto looks lke: ucto = ukeval() these two les orce a colum vector o legth = ma(sze()); = zeros(); eter the uctos here () = () + *() + *() - ; () = ()^ - *()^; () = () - s(()); The commad le prompt ad output yelds >> [err] = rd([]0e-6) ter = err = 8e-0 = 08e+00 ter = err = 97e-0 = 5e-0 ter = err = 50e-0 = 66e-0 ter = err = 8e-06 = 95e-0 ter = err = e-06 = 97e-0 ter = err = 80e-07 = e-0 = err = = e e-0 Thereore the soluto s ad 0 Problem Perorm teratos o multvarate Newto-Raphso o the ollowg system o olear algebrac equatos (k ) Show the value o the acoba resdual ad at each terato Use ad tal guess o ( ) = () Report the values o ( ) or the rst two teratos Soluto:

2 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle I order to perorm the multvarate Newto-Raphso method we must rst determe the uctoal orm o the partal dervatves The ollowg the algorthm outled above: Step Oe Make a tal guess I ths case the tal guess s gve the problem statemet: = 0 ad = 0 Step Two Usg that tal guess calculate the resdual ad the acoba ad 0 R Step Three Solve ) ( ) ( ) ( k k k R (Usg Lear Algebra) 0 0 Step Four Calculate ew values or () Step Fve Loop back to Step ad repeat utl coverged Here are what urther teratos yeld terato R e e e e - Problem A uderstadg o thermodyamcs allow us to descrbe chemcal equlbrum I we put chemcals A ad B a pot ad they react to orm chemcal C we ca determe how much o A B ad C are preset uder ether (a) sothermal codtos where the pot s mataed at a costat temperature or (b) adabatc codtos where the pot s sulated ad o heat s allowed to escape I we have three chemcals we ca cosder that we have three

3 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle ukows the mole racto o each compoet but t turs out those three ukows ca be related to a sgle varable the etet o reacto The relatoshp betwee the umber o moles ad the etet o reacto s al tal where we have to kow the tal umber o moles the pot ad the stochometrc cocet I we wat the al mole ractos we ca relate them to the al moles va: j j I the sothermal case the oly equato s a epresso or the chemcal equlbrum betwee reactats ad products Ths epresso s coded up a MatLab put le or use wth rdm ad provded below I the adabatc case we have a addtoal varable the ukow adabatc temperature whch s determed through a eergy balace whch s also provded the put le below Cosder the reacto: ½ N + / H NH Cosder a batch reactor (a closed pot) wth tally 05 moles o N 5 moles o H ad 00 moles o NH all tally at 5 C (a) Solve or the etet o reacto the reactor s operated sothermally at T so = 500 K ad the pressure s atm (Set the varable adabatc = 0 or ths case) Also use the equatos provded above to report al mole ractos o each compoet (b) Solve or the etet o reacto ad adabatc temperature the reactor s operated adabatcally K ad the pressure s atm (Set the varable adabatc = or ths case) Also use the equatos provded above to report al mole ractos o each compoet To solve ths problem you requre a buch o physcal costats These are already etered the put le but are provded here aga or the sake o completeess Heats o ormato (Kcal/mol) troge hydroge ammoa H = [0; 0; -0960]; Free eerges o ormato (Kcal/mol) troge hydroge ammoa G = [00; 00; -90]; Heat capacty costats (a rst row b secod row c thrd row d ourth row) troge hydroge ammoa Cpco = [ ]; Soluto: (a) Isothermal Problem For the sothermal problem we have oe ukow: the etet o reacto Usg the put ucto provded below we have

4 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle >> [err] = rd([05]0e-6) ter = err = 7e-0 = e+00 ter = err = 5e-0 = 70e-0 ter = err = 658e-0 = 86e-0 ter = err = 906e-0 = 0e-0 ter = 5 err = e-0 = 08e-0 ter = 6 err = e-08 = 7e-08 = err = = e e-08 The problem coverged ad the etet o reacto s 0085 moles The al moles o ay speces s determed by addg the tal moles ad the stochometrc coecet tmes the etet o reacto al tal N tal N N al H al H tal H al NH tal NH NH We ca covert these moles to mole ractos va 008 j j N 057 H NH (b) Adabatc Problem For the adabatc problem we have two ukows: the etet o reacto ad the al temperature We use the same put as problem (a) but chage two les: pressure = ; adabatc = ; >> [err] = rd([050]0e-6) ter = err = 09e+0 = 550e+0 ter = err = e+0 = 60e+00 ter = err = 07e+0 = 0e+00 ter = err = 8e+0 = 60e-0 ter = 5 err = 9e+00 = 0e-0 ter = 6 err = 0e-0 = 8e-0 ter = 7 err = 6e-0 = 5e-06 ter = 8 err = 75e-09 = 6e- = 0e+0 * err = = e e-

5 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle The problem coverged The etet o reacto s The adabatc temperature s 685 K The al moles o ay speces s determed by addg the tal moles ad the stochometrc coecet tmes the etet o reacto al tal al N tal N N al H tal H H al NH tal NH NH We ca covert these moles to mole ractos va j j N 090 H NH Iput le rom Problem (a): ucto = ukeval() these two les orce a colum vector o legth = ma(sze()); = zeros(); eter the uctos here adabatc chemcal reacto equlbra calculato troge plus hydroge to ammoa (Sadler Illustrato 9- page 509) Checklst () Make sure you have etered the correct umber o reactos (r) () Make sure you have etered the correct umber o compoets (c) () Set adabatc = or adabatc system set adabatc = 0 or sothermal system () I the system s sothermal set the sothermal temperature (Tso) (5) Eter tal mole amouts o each compoet (low) (6) Eter tal temperature o each compoet (T) (7) Eter reactor pressure atmospheres (pressure) (8) Eter the stochometrc matr (9) Eter the heats o ormato (Kcal/mol) Eter the ree eerges o ormato (Kcal/mol) () Eter the heat capacty costats STEP ONE ENTER PROBLEM SPECIFICATIONS () umber o depet chemcal reactos r = ; ()umber o compoets c = ; () determes whether the system s adabatc or sothermal 5

6 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle adabatc = 0; adabatc = ; () sothermal temperature Tso = 50; (5) let lowrates low=zeros(c); low() = 05; low() = 5; (6) let temperatures (K) T=zeros(c); TempC = 5; T() = TempC+7; T() = TempC+7; (7) reactor pressure atmospheres pressure = ; put ths atmospheres (8) stochometrc matr the order s N H NH u=[-/ -/ ]; (9) heats o ormato at the reerece temperature rom Sadler (Kcal/mol) N H NH H = [0; 0; -0960]; ree eerges o ormato at the reerece temperature rom Sadler (Kcal/mol) N H NH G = [00; 00; -90]; () heat capacty costats rom Sadler App Each colum cotas parameters or a gve compoet The our rows cota the our parameters a b c d or Cp = 8*(a + b*t/0^ + c*t^/0^5 +d*t^/0^9) [oules/mole/k] N H NH Cpco = [ ]; you wo't eed to touch aythg below here STEP TWO: LET THE PROGRAM DO ITS WORK STEP TWO A: Idety ukows r etets o reacto X=zeros(r); or = :r X() = (); Temperature (adabatc == ) T = (r+); else 6

7 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle T=Tso; STEP TWO B DEFINE KNOWNS H = H*8*000; (oules/mol) G = G*8*000; (oules/mol) reerece temperature Tre = 5 + 7; costats R = 8; [/mol/k] RT = R*T; heats o reacto at the reerece temperature DHrre = u*h; ree eerges o reacto at at the reerece temperature DGrre = u*g; scale heat capacty costats or k = :c Cpco(k) = Cpco(k)*0^-; Cpco(k) = Cpco(k)*0^-5; Cpco(k) = Cpco(k)*0^-9; Cpco = 8*Cpco; Cp = 8*(Cpco() + Cpco()*T/0^ + Cpco()*T^/0^5 +Cpco()*T^/0^9) ethalpes at arbtrary T (eed ths or low out o the reactor term eergy balace) or k = :c Cpt(k) = Cpt(TCpcok) - Cpt(TreCpcok); heats o reacto at arbtrary T (eed ths or eergy balace) or = :r DHr() = DHrre(); or k = :c DHr() = DHr() + u(k)*cpt(k); calculate equlbrum costats at the reerece temperature or = :r Kare() = ep(-dgrre()/(r*tre)); tegrate the heat o reacto over RT^ rom Tre to T (eed ths to calculate Ka as a ucto o Temperature) or = :r term = DHroRTtuk(TTreCpcouDHrreRc); term = DHroRTtuk(TreTreCpcouDHrreRc); DHroRTt() = term - term; calculate equlbrum costats at arbtrary temperature or = :r Ka() = Kare()*ep(DHroRTt()); 7

8 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle dee molar composto based o etet o reactos E = low; or = :r or k = :c E(k) = E(k) + u(k)*x(); E = sum(e); E = E/E; stream ethalpes heat = zeros(c); or k = :c heat(k) = low(k)*(cpt(t(k)cpcok) - Cpt(TreCpcok)); heattot = sum(heat); heatout = zeros(c); or k = :c heatout(k) = E*E(k)*(Cpt(TCpcok) - Cpt(TreCpcok)); heatouttot = sum(heatout); Step Three Wrte dow r+ equatos r equlbrum cotrats or = :r () = ; or k = :c () = ()*(E(k)^u(k)); () = () - (pressure^(-sum(u(:))))*ka(); c materal balaces mole = low + u'*x - E*E; or k = :c (k+r) = mole(k); mole racto costrat (r+c+) = sum(e) - ; eergy balace covad = 00; heatr = 00; or = :r heatr = heatr + DHr()*X(); (adabatc == ) (r+) = covad*(heattot - heatouttot - heatr); prt ('E = 5e 5e 5e \' E() E() E()) tegrated heat capacty o compoet k ucto y = Cpt(TCpcok) Cp = 8*(a + b*t/0^ + c*t^/0^5 +d*t^/0^9) [oules/mole/k] y = Cpco(k)*T + Cpco(k)*T^/ + Cpco(k)*T^/ + Cpco(k)*T^/; [oules/mole] 8

9 D Keer MSE 0 Dept o Materals Scece & Egeerg Uversty o Teessee Kovlle tegrate the heat o reacto over RT^ rom Tre to T ucto y = DHroRTtuk(TTreCpcouDHrreRc) terma = -DHrre()/(R*T); termb = 00; or k = :c tb = Cpco(k)*log(T) + Cpco(k)/*T + Cpco(k)/6*T^ + Cpco(k)/*T^; tb = Cpco(k)*Tre + Cpco(k)*Tre^/ + Cpco(k)*Tre^/ + Cpco(k)*Tre^/; termb = termb + u(k)/r*(tb + tb/t); y = terma + termb; 9