CBE 450 Chemical Reactor Fundamentals Fall, 2009 Homework Assignment #3

Transcription

1 BE 450 hemicl Rector Fundmentls Fll, 009 Homework ssignment #3 1. Numericl Solution of Sgle Nonler lgebric Eqution onsider contuous stirred-tnk rector (STR) which the followg dimeriztion rection goes from B with the followg mss blnce on t stedy stte 0 F F Vkdim er The let flow rte is 1 liter/sec, s is the let flow rte. The let concentrtion is 10 mole/liter. The rte constnt is 0.5 liter/mol/sec. The volume of the rector is 1 liter. () Solve this nonler eqution nlyticlly. (b) How mny roots re there? (c) Which root is the physiclly mengful root? Why? (d) Solve this nonler eqution usg Golseek Excel. Provide prt- of the spredsheet. (e) Solve this nonler eqution usg the Newton-Rphson method with numericl derivtives Mtlb. Note the itil guess you used to converge ot the physiclly mengful root. Solution: () Solve this nonler eqution nlyticlly. 0 F F Vkdim er This is qudrtic eqution. x with solution bx c 0 x b b 4c In our cse x F F 4Vk Vk dim er dim er F or

2 (b) How mny roots re there? There re two roots to qudrtic eqution. (c) Which root is the physiclly mengful root? Why? 3.58 is the physiclly mengful root. The root must be positive sce it is concentrtion. The root must lso be less thn 10.0, sce tht is the let concentrtion stisfies both these roots. (d) Solve this nonler eqution usg Golseek Excel. Provide prt- of the spredsheet. 0.5 V 1 kdimer 0.5 b 1 F 1 c 10 F 1 10 x f (e) Solve this nonler eqution usg the Newton-Rphson method with numericl derivtives Mtlb. Note the itil guess you used to converge ot the physiclly mengful root. I used the newrph_nd.m code. The put file ws: function f = funkevl(x) = x; F = 1; F = 1; V = 1; kdimer = 0.50; = 10; f = F*-F*-V*kdimer*^; The Mtlb put ws:» [x0,err] = newrph_nd(5) icount = 1 xold = e+000 f = e+000 df = e+000 xnew = e+000 err = e+00 icount = xold = e+000 f = e-001 df = e+000 xnew = e+000 err = e-00 icount = 3 xold = e+000 f = e-00 df = e+000 xnew = e+000 err = e-004 icount = 4 xold = e+000 f = e-006 df = e+000 xnew = e+000 err = e-007 x0 = err = e-007. Numericl Solution of System of Nonler lgebric Equtions

3 onsider contuous stirred-tnk rector (STR) which the followg meriztion rection goes from B with the followg mss blnce on t stedy stte 0 F F Vk The let flow rte is 1 liter/sec, s is the let flow rte. The let concentrtion is 10 mole/liter. The rte constnt is k k o exp E RT where k o = 1.0 sec -1 nd The stedy stte energy blnce is E = 500 J/mol/K. The volume of the rector is 1 liter. 0 F H F H H R Vk where the let enthlpy, H pt, where the het cpcity is 4.0 kj/liter/k nd the let temperture is 300 K, where the enthlpy of the rector is H pt, nd where the het of rection, H R, is -30 kj/mol. Fd the stedy stte concentrtion of nd temperture. Provide Mtlb eqution put file nd screen put. (You do not hve to provide the entire Newton Rphson code provided clss.) Solution: I used the NRNDN.m code with the syseqnput.m put file. The put file ws: function f = syseqnput(x); = x(1); % mol/liter T = x(); % K ko = 1.0; % 1/sec E = 500; % J/mol/K R = 8.314; % J/mol/K T = 300; % K k = ko*exp(-e/(r*t)); F = 1.0; % liter/sec F = 1.0; % liter/sec V = 1.0; % liter 3

4 = 10.0; % mol/liter p = 4000; % J/liter/K H = p*t; % J/liter H = p*t; % J/liter delthr = ; % J/mol f(1) = F* - F* - V*k*; f() = F*H - F*H - delthr*v*k*; The Mtlb put ws» [f,x] = NRNDN([5,300],1.0e-6,1) iter = 1, err = 1.48e+001 iter =, err =.03e-001 iter = 3, err = 5.43e-005 iter = 4, err = 3.85e-01 f = e-008 x = 1.0e+00 * The let concentrtion is 7.18 mol/liter nd the rector temperture is 31 K. 3. Numericl Solution of System of Nonler Ordry Differentil Equtions onsider contuous stirred-tnk rector (STR) which the followg meriztion rection goes from B with the followg mss blnce on V d dt F F Vk The let flow rte is 1 liter/sec, s is the let flow rte. The let concentrtion is 10 mole/liter. The rte constnt is k k o E exp RT where k o = 1.0 sec -1 nd E = 500 J/mol/K. The volume of the rector is 1 liter. The stedy stte energy blnce is V p dt F dt H F H H R Vk 4

5 H pt where the let enthlpy,, where the het cpcity is 4.0 kj/liter/k nd the let H pt temperture is 300 K, where the enthlpy of the rector is, nd where the het of rection, H R, is -30 kj/mol. The itil conditions with the rector re T(t=0) = 300 K nd (t=0) = 0.0 mol/liter. () Fd the trnsient behvior of the concentrtion nd temperture. Provide Mtlb eqution put file nd grphicl put. (You do not hve to provide the entire Runge-Kutt code provided clss.) (b) Wht re the long-time (stedy stte vlues) of the concentrtion of nd the temperture? (c) How do the vlues (b) compre with the stedy stte solutions obted problem? Solution: () Fd the trnsient behvior of the concentrtion nd temperture. Provide Mtlb eqution put file nd grphicl put. (You do not hve to provide the entire Runge-Kutt code provided clss.) I used the sysode.m code with the sysodeput.m put file. The put file ws: function dydt = sysodeput(x,y,nvec); = y(1); % mol/liter T = y(); % K ko = 1.0; % 1/sec E = 500; % J/mol/K R = 8.314; % J/mol/K T = 300; % K k = ko*exp(-e/(r*t)); F = 1.0; % liter/sec F = 1.0; % liter/sec V = 1.0; % liter = 10.0; % mol/liter p = 4000; % J/liter/K H = p*t; % J/liter H = p*t; % J/liter delthr = ; % J/mol dydt(1) = (F* - F* - V*k*)/V; dydt() = (F*H - F*H - delthr*v*k*)/(v*p); The commnd t the commnd-le prompt ws: [y,x] = sysode(,1000,0,0,[0,300]); The plot looked s follows: 5

6 y x The blck le is the concentrtion of. The red le is the temperture. You cn see tht the system is lredy t stedy stte t t = 0 sec. (b) Wht re the long-time (stedy stte vlues) of the concentrtion of nd the temperture? n exmtion of the put file gives the concentrtion nd temperture t t = 0 sec e e e+00 The fl concentrtion is 7.18 mol/liter. The temperture is 31 K. (c) How do the vlues (b) compre with the stedy stte solutions obted problem? These vlues gree to b 8 significnt figures with the stedy stte vlues obted problem. 6