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Unformatted text preview: ChBE 2120
Professor Gallivan, Summer 2007 i I I Exam 1 1 v /10 ( l) (10 pts) Consider a mass balance on the system shown below. Assume that the mass ﬂow M l is 1 kg/s. Set up mass balances for the system. Clearly state your unknowns.
Write this system of equations as a matrix equation.
Does this matrix equation have a unique solution, no solution, or inﬁnite solutions? Explain. A, mm)
B. ‘l o I 0 M1 ’
W13 0
l —] 0 0 m ,  .. 9 ’
o O l l 0
O .1 o l "45 I C. By :‘nspwtﬂn Jr Gut/ifs bﬂ‘m‘naﬁmn) 4.69:3 am? r14,
out)? M "Ma M of msi}. 6.15.
ML W 19—9. RHWVB,+W I‘M}; iNFW/Tﬁ Nvmﬁék 0F SOLUﬂan/S/ I, ‘i l (2) (30 pts) Solve the matrix equation A x = b for x using Gauss elimination. Show all steps.
—4 2 l 5
A: 2 1 0,1): 3
1 l 3 —7 o I It “123
x: '27.7
gamma Mmﬁﬂm \‘ 4, g u
6X23*13”“233 "/GG ail/A'ng W 't (3) (40 pts) Consider a house in Atlanta on a hot summer day. The interior of the house
is T = 65 C and outside the temperature is Tout = 90 C. At time t = 0 hours, the air
conditioning fails. Over the next 6 hours the outside temperature decreases linearly
from 90 C to 75 C. The energy transferred from the inside to the outside is Q = 1000
(T — Tout) kW. The thermal mass of the house is (m cv) = 1000 kJ/K.  A. Write an energy balance for the temperature in the house at time t (where t is
somewhere between 0 hr and 6 hr). Set it up in a form for use with the Euler
method or any other Runge Kutta method. Do NOT solve. B. Now assume that a more detailed heat transfer analysis was performed, resulting in the equation: ‘57: = 0.047(90e'"21 — T) ‘75 and T(0) = 65C . T is in units of degrees C and t is in hours. Solve for three iterations using a time
step of 0.1 hr using the Euler method. i if. 7? f (e, '71) A Hajj/q
o o 65 _ 5.375 ' ~ M375 '
n 0.1 ' 99,59 5,520 0,552.0
7' a7— GGl‘r’ _ 5.187 _ 05137.
3 lo.3 Geog :  .  Wff Em “swims: Hem): act/700$ hi“ 7:}; I I
6"“: '1'; I I *Wiﬂiii C. Solve for three iterations using a time step of 0.1 hr, but this time use the
midpoint method. (4) (10 pts) Describe the differences between the Euler method and the midpoint method.
Be sure to address what are the differences in the logic, the computationat
requirements, and the accuracy. (A sketch might be helpful in describing the logic.) Y NW
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+3 Acouer Moot“ T070? uya f6 W Mid/aﬁﬂf WW Tag/W oy/ﬂ So w’dpM‘WF 1'! mm arc/Wade) £5735“. [Al/Um, ' (5) (10 pts) Consider the differential equation d3y dy . dy dzy
—+t ——=0W1th' 0=2,————0=O,~—————0=—l
of? Val: ﬂ ) dt( ) dtz ( ) Explain how you would solve for yﬂ). You should specify a numerical method we
learned in class, and set up the equations in the form needed to use it. However, you
do NOT need to actually calculate any numerical answers. NM 2 vat/gm: out
051:2}, = {:(Uui‘=£9 ® = iguana; a) 3:3: —61/;3:'=1f3(ﬁé,,21,a3 ...
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 Summer '07
 Gallivan
 Numerical Analysis, Mass Balance, Runge–Kutta methods, Numerical ordinary differential equations

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