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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
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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
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gut? “LT—A[_ i ____ ’0er
Zing” 4;” C
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Z, : :3, L r A «42,er 5) You are designing a plant that will use a s1ngletrain process to produce a chemical product.
If you want to double the throughput of your plant design, it would be most costefﬁcient to: a) Change all reactors from stainless steel to titaniumclad
b) Use equipmentcost estimation equations designed with a different CE ind MEWEKMQJQEEﬁC‘ﬂ plants sidebyside‘“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 straightline depreciation elm acto in a
c) The impacto c anges in thetaxcode d) The “economy of scale” 7) Your plant’s net aftertax 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% [74:
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 '  .
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9) You perform a ttest 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)_ (12>(13)+ (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 backwarddifference 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 onedimensional rootﬁnding 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(/WIil "J
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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“ tit1&1
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+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"dorder 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)
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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
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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
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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/
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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 centereddifference 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
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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 ;
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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 = xex 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)
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(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 NewtonRaphson 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
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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
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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)
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#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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