chbe2120Fall2010Exam1Solutions

chbe2120Fall2010Exam1Solutions - CHBE 2120 Exam 1, Fall...

Info iconThis preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CHBE 2120 Exam 1, Fall 2010 7:15 PM ~— 9200 PM, September 20, 2010 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 one side, and I will only use standard scientific functions on my calculator (I will not use any programmable or graphing functions on my calculator). Name: Signature: Write your answers in the spaces pro ' . ank 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 that credit will not be given for calculated results that are correct if incorrect reasoning for that result is used. You must turn of all cell phones and other wireless devices for the duration of the exam. (for instructor use) Part 1 - Part 2 Part 3 Part 4 Part 5 Part 6 (max. 26) TOTAL - (MAX. 100) Part 1. Choose the best answer for each problem. (2 points each, 12 points total) 1) You need to perform Gaussian elimination on the following system: 0 4 8 x1 1 3 6 4 x 3 7 How do you proceed? 64?8 2 0 7 x1 1 K 6’ VQC A) Pivot the system so that it looks like this: [0 4 8] [x2] = [4] 3 6 4 x3 7 3 6 4- x1 7 B) Pivo the system so that it looks like this: 2 0 7 [x2] = [4] 0 4 8 x3 1 3 6 4 xgp%%(yég C) Pivot the system so that it looks like this: 2 0 7 352] = 4] 0 4o 8 x1 1 D) No pNng is necessary; calculate the first coefficient for row subtraction E) Mum answers above are correct/valid 2) “Partial pivoting” is defined as which of the following? A. Only perform a row pivot if the diagonal element is equal to zero B. Only perform a row pivot if the diagonal element is approximately equal to zero n1 exchange matrix rows, not the b vector elements D. Pérfo a row pivot at every step 3) What is the order of complexity of the number of operations in the following MATLAB code? foo = 2, for 1 = 1 n §$;-*’—‘—“ /1 for j = l:m {=;*-_‘—“-/h for k = l:n. fifi:‘—~——-' /) foo = foo + i*j*k; end ‘Z end (:j) /5 /fi foo = foo + i*foo, end A) 0013) ) 0(n m Hm ) D) on?) 4) Consider the following MATLAB code; what is the value of bar after executing this code? bar 2; {‘I z 3 for 1f:r132~ 1'2 .72 affz; 5’- _ ‘ J bar = bar + i*j; 5'2}. end bar = bar + i*bar; 7‘3 ' 3 Z/I/Z. end 0 j;— A 36 . @339 w _ . C) 54 [6% 746 " 43/ D) 66 E) None of the above 5) Given: §=y5+cost; y(0) = 1. You are instructed to use the Multipoint Heun method to integrate this equation with h = 0.01. Which of the following is true? A) You can’t immediately use any Heun method because the ODE is nonlinear B) You can’t immediately use the Multipoint Heun method because the step size is too II a ma ’t immediately use the Multipoint Heun method because you don’t have - ough initial conditions/information/data points D) You can immediately proceed with a Multipoint Heun integration 6) An nth order ODE needs how many boundary conditions or fixed initial values in order to be solved? A) 1 B) n—2 Cg n—l E)n+l Pan 2. (12 points total) Suppose you have 3 matrices: A=[1 2 3].B=H,C=[§ i];0=[§ 5 6 Perform each of the following operations or state that they cannot be performed/are undefined. (2 points each) H- § 1) AB 4 _ 4) Give an example of an upper triangular matrix. (1 point) I23 0%? 004 5) Give an example of an identity matrix. (I point) [00" 010 [mil 6) You are integrating an ODE from t = 0 using the Euler method and a step size of h = 0.5. You find that y(0.5) = 1. Your friend is integrating the same ODE using the Heun method and a step size of h = 0.5. She finds that y(0.5) = 11. Is it reasonable for you to continue integrating to t = 10 using the Euler method and h = 0.5? / If, If so, justify why, and why the Euler method is a good choice here. //’ / If not, justify why not, and say what you would do if you were forced to continue integrating V5, with the Euler method. 3 points) W ‘ a7 2 ‘" gag/M dé g d 7 6 W p W0f f/l .WZQMM (‘5 fig} 50 jwyflm/ae veg/W gimm- //’/»//-=/O < , :— 3 % 070/; 9Wf@{’/a ' 245/6/ :DP w 7) You are given the following differential equation: You are told that y(3) = 10. You are asked to integrate from t = 3 to t = 10. r! ./ _+l 0 Elk/54 QM 114$: 5‘7? $226 aw _ 2_ dt—3t+4y 2 Is this an initial value problem? Why or why not? (1 point) %5. [Mafia Mg M M WW 742 55 H: M 530/ p 27C 993/ M1 W a 5 M9 3);, M 7 5‘86 x/ (M M Part 3. (24 points total) The following system of equations describes the operation of a new reactor setup that your coworkers have developed. ' 3’ —4-x = — 8 '3: I 63‘3 3— x1 = 22"“ 8172 ’K ‘*%L+67{? — l 5x1‘3x3+10=° 57, ~37t5 ‘=- —/(J t/ J , :21) Can you translate these problems into matrix-equation form that can be solved using r~ ‘ fl 6”“ { . . s . n . . - c) '{ Gaussum el1mmat10n’? If so, translate it into matrix form. If not, explaln why you can’t. WW 6’ (4 points) 0 20 All 7‘} a/ Q L 1" Z _ 5’ 0 "-3, 7‘3 '70 , ,l WAR WM r; 5L2) Now, use Gaussian elimination to solve the following system of equations. Pivot only 2 2 2 x1 8 when necessary. (10 points) 6 4 1 x2] = 22] 4 0 9 x3 —3 2%) + 62H) :— 13 / U "middle; Wj'fwg') ULing your previous results, perform LU decomposition on the system above. Clearly MU {119mg 21indicate what the L and U matrices are. Do not re-solve the system. (4 pomts) ,. {00 . 222 [5-3/0 Wow-26' 2 Z/ O 0/5“ _H':/l ' F‘ _f (21" {695/ [iota-‘7 [whiff gm r {5.5%} l“ LVJI_,-.‘H'ljg.t;f nir'fl' 4; Dem nstrazte that yOstjLU ecorrigcdiéitiorgilsaébrrect s’igp che 3 points) era—'5’ 0 0/“; 2 [@O 7’2 {00 22 H 3/0 0-4-623/0 éb’zw’éd/ .22) 00 5 21/ 4,1 ‘M‘fflaf Lfooz LUZ 1’ 5) If you had 100 other b vectors to solve for, would it be easiest to use Gaussian if 1L, / Elimination for each of those problems, would it be easiest to use your LU decomposition H to solve for each vector, or would both approaches be equivalently easy? Justify your [1‘ LU J35 y/w’ 75} mag /¢ M U _, Part 4. (ll points total) M g” for”, .' " / fl) Make a Taylor expansion of a function y(x) around x using a distance of Ax. Keep three L} m {x ,‘nst’ exact terms in your Taylor expansion, and include an error term for the fourth term that p 77’ * ,r' M 6') approximates the order of the error. (3 points) a J 2 9 J 6 ci(.4;[£:A jasj (1)1" ditjlw + m 2) Let’s say you know the value of y’(x) and of y’(x+Ax). Use the definition of a slope between two points to approximate the value of y’ ’(x). (2 points) a [2 , M «j [+4 J’Z'C‘c) L f“ f (.y , a : 2: 3r 5/0195 flm @WQK “A”; 30 AK I'n , I 9/ MfiWéw/IyJ) Elug thllS aggrommanon Into your Taylor expan81pn, and reduce the equatlon untll you '/ 6r M fen” ave on y ee exact terms (one in yet? 1n y (x), and oneln y (x+Ax)), plus one fifth) {72331343333249 (A: 17%“ ’5" + 0&ng mi—Ax a, +4; + ' wro/mj If 3 #1111 instead, you would likely use an approximation or guess of what that value is. Keeping iML {It (‘7’ 1" i this in mind, look carefully at the equation you have derived, and indicate which I»— q r , { 7 + I integration method that you have learned is identical to this approximation. What is the / (“If "‘9" _ . local error of that mtegration method? Is that co srstent W1th your V2.1?ko 655% ‘" 0 Part 5. (15 points total) Your roommate, a biology major, has created a strain of yeast that turns essentially all of the glucose it is given into acetic acid. For every mole of glucose that it is supplied, the yeast cells create 3 moles of acetic acid. The influx of glucose into a yeast cell is driven by the concentration differenc of glucose between the interior and exterior of the cell times a constant, K... (with units [5]). Since these cells grow in a huge vat of well-stirred liquid, the extracellular concentration is essentially constant with a value of Ge (in [mol/LD. This makes the equation for the rate of glucose influx: rateGlucose influx = m(Ge _ G) Where G is the concentration of glucose in [Incl/L]. When exposed to high concentrations of glucose, this strain of yeast turns on genes that export acetic acid into the liquid reactor solvent. As these genes are being turned on, the kinetics of this acetic acid transport are as follows: at TateAcettc Acid export = b + t * A I Where a is a constant with units Eds], b is a constant with units [3], t is the time (in [3]) since the reaction was started by adding glucose, and A is the concentration of acetic acid in [moi/L]. Glucose never leaves the cell, and acetic acid cannot reenter the cell. The set of reactions from glucose to acetic acid can be approximated reasonably as one pseudo~ reaction that is first order and has a rate constant of k with units Ms]. This can be represented by the following diagram: M restore, :5 (the éwc -~ v 1) Write two transient balancd‘on an individual cell in terms of the concentrations of | f ‘ glucose and acetic acid in the cell and all relevant parameters. One balance should be on 1 13 1 5M? tgflucose in he cell, the other should be on acetic acid in the cell. (8 points) _ J . I ,' " __ Clam .2: Ill/~007’f’ 6’54/~ 604/5 Mose: Va 49-Alen (Kg-é) - 0+0 are 2) Is your glucose balance a linear or non inear 1 erential equation? (1 point) Ltd/[W 3) Is your acetic acid balance a linear or nonlinear differential equation? (1 point) (“MW Now assume the following values of constants: Km = 10, Ge = 10, k = 100, a = 0.2, b = 2. 4) Is your glucose balance clearly a stiff equation? Explain why or why not (2 points) _ ([00406‘1000 5) Is your acetic acid balance clearly a stiff equation? Explain why or why not. (Hint: to decide whether it is stiff, consider what the equation looks like wherfi is extremely small! _Y and when t is extremely large.) . (2 points) 4'} C’ 77 a 2A? & th ‘,Zé 7/7 «1.x, .. '— -f— - +d ' ' /,%“‘(W€ 2; WZ/ git/(6066’ ‘Qkfifi Part 6. (26 points total) Given the following differential equation: d4y d2 y WWW” y(0) = 1; HO) =4; y"(0) = 2; y”’(0) =3 L 1 f' - 41) Circle all of the following terms that describe the above differential equatlon: (2 pomts) _ /’ .iX o rt‘ V ' . layfiwmnw 18‘ order it (w? PDE Q 2) Transform this equation into a system of coupled first—order ODEs that DOES NOT contain the original y variable in any form. Make sure to clearly define all variables that you use. Also include the initial condition for our 5 stem of first-order OD R I a ‘ m "a 3) Use the Euler method with a step size of h =. 0.1 to inte . (8 points) ' \ : .. A - gt ‘- 80” L If - y‘/ c) " Sillgof/l JCGOIéOJ .7 ’13 ,' ,rflr'rzl M75 39% y l : 3d 1'22: {3%} (*5) 5/ :"y° w, (.r 2t" 111" yr ’5’ - I ' A [2/7” A 711% / yd?! : 9‘; f Aligygttz) fat/L) l"? ‘M We Use the midpoint method with a step size of h = 0.2 to integrate this system to t = 0.2. “' _ ,bi (10 points) ' N“ W” “"0 ' " 1 yo T % féox'gd) J fwd" L, - “NZ 6 {a L ck FA F/éirz) #1 JW . / .’//@/wth : 0 1' 5) If you had decreased the time step for each of parts (3) and (4) by a factor of 1—10- how much more accurate would your results have been for each part? (Hint: Don’t redo the integrations, but describe what the impact on the error would be.) (2 points) 2551/, 50 error WM I? 3446/ “My; HH/ M1] errant, «J; ill-win 72? Mr . asea/jééw/wfln} ’/ ...
View Full Document

Page1 / 12

chbe2120Fall2010Exam1Solutions - CHBE 2120 Exam 1, Fall...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online