chbe2120fall2009finalExamSolutions

chbe2120fall2009finalExamSolutions - CHBE 2120 Final Exam...

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Unformatted text preview: CHBE 2120 Final Exam, Fall 2009 2:50 PM — 5:40 PM, December 8, 2009 By signing below, I am agreeing to abide by the rules of this exam and by the Georgia Tech Honor Code. In particular, I agree to use no printed materials other than a single 8.5x11” sheet of paper with notes on both sides, and I will only use standard scientific functions on my calculator (I will not use any programmable or graphir? functions on my calculator). Name: \ Signature: Write your answers in the spaces provided. Blank paper is available for scratch work. If you need additional room for your answers, staple extra sheets to the exam and clearly indicate the problem numbers for your solutions. In problems that ask for a specific calculated value, highlight your solution with a box. Note the credit will not be given for calculated results that are correct if incorrect reasoning for that result is used. You must turn oflall cell phones and other wireless devices for the duration of the exam. (for instructor use) Part A 20 points Part B 5 points Part C #1 Part C #2 20 points Part C #3 Part C #4 Part C #5 Part C #6 Part C #7 TOTAL (MAX. 150) Supplementary Information You may find some of the information on this page helpful in solving problems on the exam. Chemical En_ ineerin; Plant Cost Index Marshal! & Swift E I uiment Cost Index 2004 008 2 1179 i O O l The inverse of a matrix [:1 3] is: ‘1 1 _ i: (I; =ad—bcid b] Normal distribution tables and t distribution tables have also been included in this exam packet. Part A. Select the best answer for each question. (2 points each; 20 points total) 1) Between calculating a matrix inverse and performing Gaussian Elimination, which approach is generally faster for the same problem? a Matrix inverse ‘sian elimination 0) About the same (1) Unknown; the computational complexity of each is indeterminate 2) Calculate the computational complexity of the number of multiplications that would be performed by the following snippet of Matlab code. for i=l:n :7 ’4 15m9§ for j=3:m -~J.9 ,M’Z 91ng if (i <= n) qgwj 4mg 5;; )j/Ldf0 num = num * 2; ,- _ ,W. end ____::{; /JW///'r 7 end and £77 (/1 )(m «2) a 00; m) a) 0(n) b) 0(n2) c) nrn :di gone of the above/not enough information provided 3) Stiffness in a system of differential equations is generally associated with: a) Terms with coefficients that are different orders of magnitudes Wthat occur on different time scales /_____ mud—i2 d) None of the above 4) You use the implicit Euler approach to solve the following ODE: y”(x) — y'(x)y(x) + exp(t) = 0 : . Z ‘3. i: 23» At each step, this would require you to find: 1 j ’ a) The solution of a linear system of equations Cr b)" The sermon o a non inear system 0 equations” 2 7 a - ' - c) The el'genva ues ofa matrix —— ' Z 3' 2 ‘ 7* ll VL/EL'HJ i1”) - c . v ‘ rf (- d) None of the above/not enough mformation provrded b _ .4 I 2’2, cf'( gut? “LT—A[_ i ____ ’0er Zing” 4;” C “,7 Z, : :3, L r A «4-2,er 5) You are designing a plant that will use a s1ngle-train process to produce a chemical product. If you want to double the throughput of your plant design, it would be most cost-efficient to: a) Change all reactors from stainless steel to titanium-clad b) Use equipment-cost estimation equations designed with a different CE ind MEWEKMQJQEEfiC‘fl plants side-by-side‘“r.___._-WE§%‘\<£TE5 6’, / I (1 Size each piece of equipmentinuthepl—ant to handle twice as much throug pu ' 6) Cost indices like the CE index reflect: a) The benefits of straight-line depreciation elm acto in a c) The impacto c anges in thetaxcode d) The “economy of scale” 7) Your plant’s net after-tax earnings are $10 million/year, and the total depreciable capital cost cm was $20 million. The return on investment of this plant is: a) 50% ROI—’- ( / 't)(§~£) b) 25% [7-4: 0) 2rssr§7_uW~+——m Q 0 <14 XM C132 'None of the above/not enough informamx We‘d? ’7 ' (- C ., — " %f<5_ (reef 7705 8) You have just won the lottery and elected to take a $500,000 payout every year for the next 30 years. Assuming a standard annual interest rate of 5%, the present value of your lottery jackpot is: , F q f It as. 7a a) Approx1mately $500,000 / V. n L I S] b) Approximately $15,000,000 50%; ) Av; “$290 Elflgpproximately $33,219,000 ' - . w 37m “mm-I — 053% ‘ “ . * .0570?" 9) You perform a t-test with a significance levemhat is the likelihood that you reject your null hypothesis even though your nu ypothesis is actually true (a false positive)? (l/t ‘ . Z) ’05:: 5% {éée fla/éw: fig? ~ 0) 5% d) 95% 10) Which of the following is the LaGrange interpolating polynomial for the three (x,y) points 1,2; 2,6; 3,14 ? " _ {( ) ( ) ( )} #9 Ha.) __ ‘2)(36—3) (x—1)(x—3) (x—1)(x—2) a) 1300 _ (1—2)(1—3) (2—1)(2—3) (3—1)(3—2) __ (x—2)(x—3) (x——1)(x—3) (ac—1) x— b) f3CX) — 2 (1—2)(1:‘3) + 6 (2—1)(2—3) + 14 (3—1)(3—2) m _ am : gafz’iiilfi “452:; w «9/ >4 C)f3(x)_ (1-2>(1-3)+ (2—1)(2—3) @Trfi'g aarw {/9 / tax" " 1‘1: ,,.._ f7 \ ’ d) f3 (x) = W + 6% + Mgr—33mm 51—2)(1—3) (2—1)(2—3) (3—1)(3—2) Cy/wéz'ré of MMMZ War. Part B. All of these questions are taken straight from this semester’s quizzes. (1 point each; 5 points total) 1. The error of a fomard—difference finite difference approximation is 0(h). What is the error of a backward-difference approximation? on # 2. If you have an ODE integration method whose local error is of the order 0(h3), then what is the order of its global (propagated) error? \ '/ '2 7 3. You are using bisection to solve a one-dimensional root-finding problem. Your current bounds on the root are [2, 4]. You know that f(2) = -25, while f(4) = 33. According to your calculations, f(3) = 10. What are you the new bounds on your root, in the format [lowerBound, upperBound]? 0 ’25,“ 4. To determine unambiguously that a root lies in an interval, you need to evaluate the function at a minimum of two points. What is the minimum number of points/function evaluations necessary to determine unambiguously that a minimum or maximum lies in an interval? d .> 5. What is the term we used for a uniform amount of payment made at the end of every period? (Hint: we represented the annual maintenance costs of a pump as this.) [ta/(fit Part C. l) (12 points total) You are working with a continuously stirred tank reactor (CSTR) with volume V to convert compound A into compound B according to the following equation: A 9 B The volumetric flowrate into and out of the reactor is F (in L/rnin), the concentration of A in the inlet is CM“, the concentration of B in the inlet is 0, and the concentrations of A and B inside the reactor and in the outlet at any time are C A(t) and C30). The reaction rate of A to B is defined by the following rate equation: 7" = kCA The ambient temperature is T, and atmospheric pressure is P. a) Write transient balances on the concentrations of A and B 111 the reactor, in terms of k, F, CA,CB,CA,m,t, T, P, andV. (8 points) 7 E at “ icitt', ‘ 17;? {id—f '.LL(/WIi-l "J + m .50 1,27, cw}. b) Your boss tells you that in addition to the concentration of B in the inlet being 0, the initial concentration of B in the reactor should also be 0; that is, CB(t = 0) = 0. She also tells you that the concentration of B in the reactor at t = 30 should be 1. Is this an initial /puemf problem or a b7dary value problem? Why? (3 points) 3V MM/Téé/Cé‘ffiérémj we. ’ [50% (”35/8 [Qt M Maj” é?) 3:0 0)) fflMj/j gt, szyiméO/I c) Is your system ofequatlons linear or nonlmear? (1 point) Ml, 1V 6/ W6 {D 3/47 {4% Line/M 2) (20 points total) This problem deals with the following differential equation: a) Translate this equation( into a system of 1St order differential equations. (3 points) "Ejflfiz ‘ \j; ][9] ZELUH] 3/ of 1 and the initial conditions (7 points) 9(0)=3:’g(0)=6 fl [Jr ‘ 330: a &+,;Zi t}\ JGAZLM} [Mgr/)5- E:[/: 2. (+1 : $192? [Hgtdi 7%,?“ ’ my MC 71 :1+1.1a1}m1;: “H PM?! @444“ tit-1&1 «5—! (PM film 47.. 1'11‘71'1'4 é'z/ [Eu/92, +131mfl’ or: ((3:17: b) Integrate your system of differential equations to I using the Heun method, a step size h [3] c) What IS the local error of this single integration step in terms of the step size h? (1 point) OW) d) Integrate your system of differential equations to 1 using the Euler method, a step size 11 of 1, and the initial conditions (3 points) 9(0) = 3:9'(0) = 5 “\ e) What IS the local error of this single integratio step in terms of the step size h? (1 point) i l ”rift/L313? CZ/{Z “with f) Use the shooting method equation to find the value you would use for your next guess at 0 based th t t t f d b 3 t g’( ) on e we in egra ions you per orme a ove. ( poin s) B; 20 é), a 32/ .13“? L g: 9/ (g B) 26 =/0 5L AM; _, ‘§tfl \ g) Could you use a finite d1fference approximation for g”(x) and g’(x) to cast the o ginal 2"d-order differential equation into a set of linear equations that can be represented in A l ill 1711 format? Why or why_ not? (2 points) ' fleet/2 ,- [it fié/{Z/«éz/ czwza/ il (36.59:. 2111:“;- "(5 WME/léfiP/I/ //7 5C“ @flé} 3) (25 points total) You are doing undergraduate research in a new lab, but unfortunately there are no graduate students who are willing to train you. You are tasked with growing yeast cultures to a specific cell density characterized by its OD, or optical density, but you have no idea what affects the final density of a culture. You decide to test some possible factors, and recalling your ChBE 2120 training you design a two~factor factorial experiment to test the effect of shaker speed (factor A) and initial glucose concentration (factor B). You perform three replicate experiments for two different values of each factor. Your results are summarized below: (0 _ a) Calc late hegffect size forAand B. (6 points) £66 Z )c c _ . ’ _ L _ a 0 Q3 7‘ , ZW—w=aw Z. ‘L 575? an”? ‘EC 673 3/ 3W”?— ,O/ Elaflmd <42 {OJ Jri xii-”ink / -—*~L . l gill/H 3 [/15] 1 1/3 6 1 fl b ‘1' .' ' I JIWfiIi ) I 5 b) Calculate the interaction effect size, AB. (3 points) Ag :7 @174 [at +(() "4— ‘(J mm $1 1585 ,,__ ,04t/OZZ§+,0“‘/‘rfl/ : 034% 25— Wwi 1/? :. (9H6 '5 9:2??wa wt .4; glfla" a wen '33 “5:“? @1015 E: “7757751“155'3 so , LL / 1%“ $239 if -' “7%: =53? iiasfws / d) A grad student comes and looks at what you’ve done. She compliments yo work, but says your “effect size” calculations don’t make sense. She suggests that instead, you define two variables Ka and X}, which are each equal to +1 or -1 for an experiment depending on whether effect A or B is high or low, and then perform a multiple linear regression on your data to fit the equation y = axa + bxb + cxaxb + d. If you perform this regression, will the results of the regression have any relationship to the effect size calculations you just did? If so, what relationship will they have? If not, _why not? (3 points) Déwrlf 1 t (‘7' {'E/ (chi/l (ILL ') 'r;-f Q- MW 6) Test the hypothesis that the mean of the Ahigh, Blow distribution is 2.1. Use a significance K: 0301;;(35. (Sgint521022§_ # A352/ £5 f/t/i 2/ J 146‘ ~2/ lid" * W = #173 1) Two of your classmates of been working on the same problem, and one tested the hypothesis that the mean of the Ahigh, Blow distribution is 2.0, while the other tested the hypothesis that it was 1.8. Both friends claim that they accepted those null hypotheses. Based on your knowledge of statistics, choose whether: a) it is possible that both of their statistical tests were performed accurately, or b) one of them must have made a mistake in their calculations. Explain your choice (2 points) A711 MW / 1 1111 . 6F @llwiw/ M7 lfle Wig/«m Wtfl/p *6 (Kauai—Mean .. 4) (18 points total) You are trying to determine the dynamic flow characteristics in a new, state- of- the- art heat exchanger that has been purchased for your plant. Along with the heat exchanger comes some small, waterproof RFID chips that act as tracer particles within the exchanger fluid. You are able to take readings of the chips velocities at any given point in time. You make the following preliminary measurements of velocity as a function of time: mum In the following questions, use the centered-difference rule for finite difference approximations and the trapezoidal rule for integrations. a) Determine the total distance traveled by the microchip over the first 8 seconds using a step size, h, of 4. (3 points) I#A[%%)+2§Hflx )1» HM) 08¢ F0; {5‘}, (:92 it. ,n ll 17:41:44.; :2 ,1: . . . . . b) Setermme the total dlstance traveled by the mICI‘OChlp over the first 8 seconds usrng a step size, h, of 2. (3 points) U 6:0. {/2 f, 1‘, (7:6 Cl; g 1e .1: eta)? zgflwflm] ,1 M; ”I I flame '1 v 46 f ’0 c) What IS the error of your calculations expressed 1n terms of h? (1 point) 6) Calculate the acceleration of the chip at 4 seconds using a step size Ah of 4. (3 points) [Rodi/radar: 4% ti? 076 (M X 1%”56 ; H15 {XXL/425M: L [W89 fl) 7763- to —« 2(9) ‘ fid/i’T—fi t: I , _ x , . . . , / vKZ) -—- 3 5‘ ‘ 4,6717%!) ”fix/A; :2 [V [5) (’2 j 1: 7" , L: .r Zél Z ) 5" m \ l g) What 13 the error of your calculations expressed 1n terms of h? (l po1nt) . . . - . (N .2, g 06/; J h) Use only your answers from (c) and (f) to calculate a more accurate estimate 0 t 6 distance traveled by the chip. (2 points) 0 MA) lei/WA) flat/MM D; gfififlcéflgfi , ' S) (15 points total) The following function has only one root: 255? 1 5 " ’ f «- SJ- 3700 = xe-x 5599/1 M f WK K i a) Use 2 iterations of the Newton- Raphson algorithm to try to find the value of that root, starting from the initial guess Xo_ w 2. (6 points) / W2, 1 2 «25 ’ ’5 ‘31 jé") “216 t e e(_>c b) The obvious answer for the root of the initial equatio IS 0 A your guesses moving 41‘ 7‘66) < 3/é’m'c ds it? Why or why I? (2 points) /NJ§ $545349; 7; {exit-XV} m MflZ’a/ My 31> :r/ (M13265 63¢th Ill/W} flacfé a é‘gaé W f7; jwc‘Jf c) Using this information, what can you say about the convergence of Newton-Raphson ngeneral? (2 p ints) / (9/7 £6) 41% $06435 $11116 if 52%; jabs 5:35 @195; jib fj5% 4w) W1, 8/5 )T [”6” d) If you had instead used bisection to solve this problem with an initial interval of f-[ -,2 2], would you have been able to solve the problem? Why or why not? (2p ints) / f; if titty“ gag/5551404 Mir M5 42/ a; fdef //L a?!” 5/ «if/é l3 92/! j; 4/19” My (f wwéj% C57 Me yak/1 I. e) Cast this root- -finding problem as an optimization problem by defining the function 15'- 1% 2(x) that you would optimize. Then state an optimization method that could be used for N . this type of problem 2(3 ?:35) (366)): ‘L X Zevz’y 6) (31 points total) This problem deals with the following measurements of the volumetric flowrate of water, w, through your new heat exchanger when it is started up, measured every hour: You would like to fit this data to a curve. A coworker suggests that the data may follow the equation a) Transform the above equation into a form suitable for linear regression, such that it is analogous to the equation y = ax + b. Explicitly state what y, x, a, and b are in terms of w, c1, 02, and t. (6 points) r, M f -— Ltd/95. Mt: fl { ‘fiflz 6 Ca - “Lt {3L 2:: C‘f;5’f{:‘gjg"”, 4%; MTV/72, ACT/H {m1 b) You have second thoughts about the previous equation, 5&10u decide to instead fit the data to a simple curve: EKL' w= c1152 + c3 A L3 Use the normal equation to fit your data to the above equationh and detenning/Thekles ofclandc3 (lOpoints) Aflfflw/ %%&;M Elihu/EMA _%3C: 4 7:3 AB [ T‘TZTT TZZMTTT) J, 24:] + it?“ .‘ T W: 2; kWh/Er "f 5/ WW ’7‘? .L if 44/] :754’W Til—(7;; (:f]: Hé “(Cf WK] 5 (5” [ZMT’t‘fl/W .___[ l rT /] /+*Zr6+ltf fl]; ‘ T 39:37? 61%“ T T {241w omz a) {zé a t \i ’@ V f’ N A" Mi" M :x gin fi . (3) [won decide it may be more appropriate to fit your data to a more complete curve: at?“ w = c3 + czx + 61x2 7/ You use the normal equations and are left with the following system of linear equations to solve: 4 6 14 C3 23 [6 14 36Hc2]=[56] 14 36 98 Cl 152 4 d flf [7,85% the above equat‘ifgp using Gaussian elimination with fril pivoting. (10 points) " 1 r __ 4737": a ‘ . ‘ _, ' ' - jbf 5% i H '3é 46/} 5—2 ‘ . flatly/79) 74w 6 JfiZfi’é 41f ‘. flair??? p 0 £566” 5 --—-Z532 #332 _. “2'0- ?74 — (WY/47"”) [Z ’ ~Af,2f5é (1) Look at your results from (b) and (c); are Cl and 03 the same or different in your two answers? Why? (2 points) hi? WWI / V01 We ;4%31 75” @7771 we .0: a»; a (Wm M a; max I e) What if you need to calculate the parameters C], Cg, and 03 for hundreds of heat exchangers throughout your plant, where all of the exchangers have measurements of w at the same 4 time points t. What approach that we have learned would you use so that you could solve those hundreds of regression problems faster? (3 points) /4r/l m: 7%{4/ [j f/jéflfié’j d3 > MW/ Z) {me/w 7VW263. f0 dirt): e/lg} 7) (4 points total) This problem deals with the contour plot of a complex optimization function 2(x1,xz). a) On the following contour plot, draw three steps of how the gradient steepes ascent method would attempt to find the local maximum, starting m1 (2 points) 7% 79 b) On the following contour plot, draw three steps of how the univariate method would attempt to find the local maximum, starting from the “X” and keeping the x1 variable constant first. (2 pOintS) 42y X b ‘ 11 Q; / first ...
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