MIE334H1S_2018_NUMERICALMETHODSI_E

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Jan 9, 2024

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UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, Wednesday, April 18, 2018 DURATION: 2 and % hrs Third Year - Mechanical Engineering M1E334H1 - Numerical Methods I Calculator Type: 2 Exam Type: C Examiner — I. Filleter Rules and Guidelines: The examination is 150 minutes, and is comprised of six (6) questions, worth a total of 100 points. You should attempt all questions. The exam contains 6 pages, including this one. This exam is closed-book. A one-page (double-sided) aid sheet is allowed. No other external material is allowed, such as lecture or tutorial notes, quizzes, midterms, final exams, or books. You must answer all questions in the separate booklets provided. Write your name and student number on each booklet used. Clearly highlight all final answers, and show all intermediate workings. We cannot give partial marks if you do not. Question 1 112 Question 4 /16 Question 2 /18 Question 5 /16 Question 3 /18 Question 6 /20 1 of

1. (Short answers ONLY) (3 points) List any 3 numerical methods that are derived using the Taylor series. (3 points) For Romberg numerical integration: What other numerical integration method is it based on? What is the order of the error for the 1st iteration of the Romberg method? What type of numerical integration problem is it best suited for; Integrating tabulated data or a known equation? (C) (3 points) For temperature measurements taken along a heated steel beam: X (m) 1 0 10.5 11.0 11.5 12.0 T( 0 C) 190.00 164.49 148.90 138.78 135.00 (I) If a Newton quadratic interpolating polynomial was used to estimate the temperature of the beam at a position of 0.7 m. What data points would you select to use in calculating the estimate? (ii) If a Lagrange quadratic interpolating polynomial was used instead would the estimate be the same or not? Briefly explain. (d) (3 points) Name two differences between the bisection and false-position methods. Also, if curves (1) & (2) - demonstrate the true percent relative errors e associated with root estimates using the bisection and false-position methods as a function of the number of iterations for an arbitrary equation, which one would most likely belong to the bisection method and which one to the false-position method? 2. (18 points) For the following system of three linear equations: a 1 + 2a 2 + 9a 3 = 99 9a 2 + 2a 1 - 3a 3 = 36 a 3 - 3a 2 + 9a 1 = 18 Iterations Rearrange the equations in a form that would guarantee convergence if the Gauss- Seidel method is to be used to solve the system. What is a system in this form called? Solve for a 1 , a 2 , & a 3 using the Gauss-Seidel method. Consider an error tolerance of 10% in determining your solution. Use an initial guess of a 1 = a 2 = a 3 = 0. (Note: Retain 3 significant figures in your calculations) 2 of 6

3. (18 points) Reaction rate constants k can be represented as exponential functions of temperature: fE k = k 0 exp1 a -- where k 0 is a pre-exponential factor, E a is the activation energy, R = 8.314J/mol K is the gas constant, and T is the reaction temperature. Consider the rate constant versus temperature data as follows: k (s 1 ) 1 5.181 8.901 14.78 T(K) 1 300 310 320 Write the linearized form of the reaction rate equation given above. Use linear regression to estimate the rate constant at T = 316 K. Estimate the activation energy E a (in units of kJ/mol). (Note: Retain 5 significant figures in your calculations) 4. (16 points) If the work required to pull a block across a flat surface (in the x direction) is given by: w =X cos (9(x)) dx Where F(x) is the applied force as a function of distance and 9(x) is the angle of the applied force relative to the surface as a function of distance. For: F(x) = 1.6x - 0.045x 2 (in units of N when x input is in m) 9(x) = —0.00055x 3 + 0.0123 X2 + 0.13x (in units of rad when x input is in m) Use the Simpson's 1/3 rule (with 4 segments) to estimate the work done if the block is pulled from x = 0 to 3 m. (Note: Retain 4 significant figures in your calculations) If the force was instead applied parallel to the surface (i.e. 9(x) = 0) over the entire distance and the work was again estimated using the method in (a) would you expect your answer to be equal to the true value of the integral? Briefly explain your answer. (Note: you do not need to calculate the new estimate in (b)) 3 of 6

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5. (16 points)-The following relation can be used to obtain the pressure gradient dP/dx across a flat plate when a fully-developed laminar flow passes over it in a steady-state condition: dP d 2 dx = IL -- ; it = 1.79 x 10 -5 Pa s If the velocity is measured at various distances y from the surface, as summarized in the following table: Y (mm) 0 1 2 3 4 5 u(m/s) 0 0.05260 10.1793 0.3802 0.6552 1.0043 Use a 2 Id order accurate approximation to estimate the pressure gradient at the surface of the plate. Use a 2 nd order accurate approximation to estimate the shear stress r that the fluid exerts on the surface at y = 0 (Hint: t = i du/dy). (Note: Retain 4 significant figures in your calculations) -; U 4 of 6

6. (20 points) The following figure shows a thermal fin of length L = 0.1 m, with circular cross section of radius r = 2.5 mm, and thermal conductivity equal to k = 237 W/m K. The purpose of the fin is to enhance the heat transfer from the base (on the left) to the environment which is at T = 273 K with a convective heat transfer coefficient of h = 30 W/m 2 K. The other side of the fin is insulated. T , qb , To determine the performance of the fin, a researcher has measured the temperature on the insulated side of the fin to be T o = 337.49 K. The following 2 Id order differential equation (wl initial conditions) has been developed to find the temperature variation along the fin: (d 2 T 2h =m 2 (T—T) ; m 2 --- rk dT I 1 dx =0 t x =O Decompose the differential equation into two first-order differential equations Solve the system of ODEs obtained in the previous step from x = 0 to 0.1 m using Euler's method with a step size of 0.025 m. Summarize your results in a table. (C) If the exact value of the temperature Tb at the base of the fin can be calculated using the following relation, what would be the true percent relative error associated with the value of Tb you obtained in the previous step? (Tb - T) = (T 0 - T) x cosh(mL) (Hint: cosh: hyperbolic cosine) (d) Using temperatures estimated in step (b), estimate the heat flux q,' k dT I at the base of the fin using a 2nd order accurate backward approximation. (Note: Retain 5 significant figures in your calculations) 5 of 6

Finite-divided difference formulas: f(x11)-f(x1) f' (x1) f(x L + Z ) +4f(x1+1)-3f(x1) f" (xi) f(xj +2 )-2f(xt+ i ) +f(x) f' (x 1 ) - h ' - 2h j2 f"(x1) -f (x 13 ) + 4f (x 12 ) - 5f (x 11 ) + 2f (x i ) h 2 f'(x1) f(x1)-f(x_1) f'(x1) 3f(xj)-4f(xj1)+f(Xj_2) f"(x1) f(xi)2f(xi_i)+!(xi_2) h - h 2 - P (x1) f(xt1)-f(x11) f'(x1) -f(xj+2)+8f(x+l)-8f(xL_l) +f(x1-2) 2h , 12h f"(x1) f(xj1)-2f(x)+f(xj_1) f"(x1) -f(x2)+16f(x1)-3Of(x1)+16f(x1..1)-f(x1_2) - h 2 - 12112 6 of 6

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