MIE334H1S_2016_NUMERICAL METHODS 1

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Mechanical Engineering

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Jan 9, 2024

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UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, Friday, April 22, 2016 DURATION: 2 and Y2 hrs Third Year — Mechanical Engineering MIE334H1 — Numerical Methods | Calculator Type: 2 Exam Type: C _ Examiner- - T. Filleter Rules and Guidelines: e The examination is 150 minutes, and is comprised of six (6) questions, worth a total of 100 points. You should attempt all questions. e The exam contains 4 pages, including this one. e This exam is closed-book. A one-page aid sheet is allowed. No other external material is allowed, such as lecture or tutorial notes, quizzes, midterms, final exams, or books. e You must answer all questions in the separate booklets provided. e Write your name and student number on each booklet used. e Clearly highlight all final answers, and show all intermediate workings. We cannot give partial marks if you do not. Question 1 /12 | Question 4 /20 Question 2 /16 | Question 5 /16 Question 3 /12 | Question 6 /24 1of4

1. (12 points) Derive a 1st order backward approximation of f "(x) = Z—;—’; based on f(x;), f(x;—1), and f(x;_;) using the Taylor series expansion. Clearly show why the relation you obtained is 1st order accurate. (note: x;_1 = x; — h,x;_ = x; — 2h). 2. (16 points) For the following system of three linear equations: x1+4-x2 —X3.= 6 X1 — 2%, +4x3=9 6x1—x2+x3=7 Use the Gauss-Seidel method to solve for x4, x,,%3 to an error tolerance of & = = 6%. If necessary, first re-arrange the equations to guarantee convergence. In all of your calculations retain 3 significant figures. 3. (4 points each) (a) Suppose the solution to an ordinary differential equatlon (ODE) is a 2nd-order polynomial. If we want to obtain perfect results, which method — Euler's or Heun's — should we use? Brleﬂy explain your answer (1-2 sentences ONLY). (b) Which of the following statements about root finding methods are true? | Open methods always converge on the solution il The Newton-Raphson method exhibits a quadratlc convergence rate ! The false-position method requires two initial guesses v Brent's method uses a combination of the Newton-Raphson method and the bisection method (c) What is the main advantage of the Secant Method over the Newton-Raphson method? (Briefly answer in 1 sentence ONLY) 20of4

. (20 points) The following data defines the sea-level concentration of dissolved oxygen for fresh water as a function of temperature. ' (a) Determine the best estimate of the concentration of oxygen at 27°C using Newton’s polynomial interpolation. Determine the true error if the exact value is 7.968 mg/L. In all of your calculations retain 4 significant figures. (b) Write out the system of equations (in matrix form) that you would need to solve to estimate the concentration of oxygen at 27°C using a quadratic spline. (You DO NOT need to solve the equations or the estimate). Would you expect this estimate to be better or worse than that found in (a)? Briefly explain why? T (°C) 16 24 32 40 O,(mgll) 9.870 8418 7.305 6.413 . (16 Points) Consider steam condensing on the outside of a heat exchanger tube in which water is flowing. If the following equation describes the variation of the temperature of the water as it flows through the heat exchanger tube at steady-state conditions: : with the following condition at the inlet of the tube: T(0) =24 °C where v is the average velocity of the water stream (1.5 m/s), T is the temperature of the water stream (°C), x is the distance down the heat exchanger tube from the entrance (m), h is the heat transfer coefficient (1,000 W /m?-°C), p is the density of water (1,000 kg/m?), C, is the heat capacity of water (4181 J/kg-°C), D is the inside diameter of the tube (22 mm) and T, is the temperature of the condensing steam (120 °C). If the length of the heat exchanger tube is 20 m, use Eulers Method (with a step size of 4m) to determine the outlet water temperature. In all of your calculations retain 4 significant figures. 3of4

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6. (24 Points) The outflow concentration from a reactor is measured at a number of times over a day: t (hr) 0 1 5.5 10 13 16 19 24 c (mg/L) 1 1.5 2.3 2.1 4 5 55 1.2 The flow rate for the outflow in m?®/s can be computed with the following equation: Q(t) = 20 + 10sin (2—’; (t— 10)) (Hint: t is in hours) The flow-weighted average concentration ieaving the reactor over the 24-hr period can be obtained using the following relation: _ Jo Qe J,Q@®)dt (a) Use a combination of the Trapezoidal, Simpson’s 1/3, and Simpson’s 3/8 rules to obtain a best estimate of the flow-weighted average concentration over the 24-hr period. (Hint: you should use the Trapezoidal rule more than once during this procedure). In all of your calculations retain 3 significant figures. C - (b) Estimate the rate of change of outflow concentration at t = 16 hr? Obtain the rate with a 2™ order accurate method. Finite-divided difference formulas: f,(xi) ~ f(xi+131_f(xi)’ f,(xl) ~ “f(xi+_.2)+4f2(;ci+1)—3f(xi)’ f,,(xi) ~ f(xi+2)—2f’f:i+1)+f(xi) . i-_l. - 4 ~ 3 i) —4 i— i— 1174 ~ i)—2f(xXj—1)+ i— F(x;) Ef(-’C) ’]:(x 1), Fx) = F(xo) f(’zc;l1)+f(x 2), F7(x) =f(x1) f(x}:_zl) F(xi-2) ' o FO)=FXic) pren y o = Xir2)+8F (Kiga) -8 (i1 )+f (Xi2) f (xi) = 2h > f (xl) = 12h ~ i) =2 (xi)+f (x4 ~ —f(xig2)+16f( )—30f(x-)+16f(xi- )—f(xi-2) f"(xi) o~ i+1 hzl f(xi 1)’ fll(xi) a~ f(Xit2 Xi+1 12h21 1 i—2 4 of 4