ENGSCI 313 Page 1 of 13 THE UNIVERSITY OF AUCKLAND SEMESTER ONE 2018 Campus: City ENGINEERING SCIENCE Mathematical Modelling 3ECE (Time allowed: THREE hours) NOTE: TOTAL MARKS: 180 Answer ALL questions. ENGSCI 313 Page 2 of 13 SECTION A Complex Analysis Question 1 (20 marks) (a) Put the complex number = 4 + 53 + 2 in the form jba + . (3 marks) (b) Consider the complex number 2 + 2 (i) Evaluate the modulus of this complex number. (ii) Evaluate the (principal) argument of this complex number (in terms of π ). (iii) Express the complex number in polar form, (cos() + sin()) (iv) Express the complex number in exponential polar form (4 marks) (c) Consider the function () = + 2 (i) Re-write in the form ( ) ( , ) ( , )f z u x y jv x y= + . (ii) Use the Cauchy-Riemann equations ,u v u v x y y x ∂ ∂ ∂ ∂ = = − ∂ ∂ ∂ ∂ to test whether ( )f z is an analytic function. (6 marks) (d) Write 4+ in the form jba + . (2 marks) (e) Sketch in the complex plane the curve 1 + −, 0 ≤ < 2. Label clearly the starting point of the curve. Use an arrow to indicate the direction of the curve as t increases. (5 marks) ENGSCI 313 Page 3 of 13 Question 2 (20 marks) Evaluate the following four integrals (i)-(iv). In each case, sketch the contour of integration and any singular points of the functions to be integrated. Do NOT use the Residue Theorem. Simplify your resulting expressions as far as possible. State which of the following (A, B, C or D) you are using to solve each problem: A. Cauchy’s Integral Theorem B. Cauchy’s Integral Formula: )(2)( 0 0 zjfdz zz zf C π= −∫ C. The Formula for Derivatives: ( ) 0 ! 2)( 1 0 z n n C n dz fd n jdz zz zf π = −∫ + D. None of the above (i) ∫ +1 where is the circle | + 1| = 1 (5 marks) (ii) ∫ −1 2 where is the circle | − 1| = 2 (5 marks) (iii) ∫ sin () where is the circle | − 2| = 1 (5 marks) (iv) ∫ 2+1 where is the circle | − | = 1 (5 marks) ENGSCI 313 Page 4 of 13 Question 3 (20 marks) (a) Consider the function () = 2 − − 6 (i) Sketch the complex plane and on it plot any singular points of )(zf . (ii) How many power series of )(zf are there about the centre 0z = ? (iii)Sketch the centre and the ROC’s of these series. (iv) For each series, say whether it is a Taylor or Laurent series. (v) Express the ROC’s in the form Rzz <− 0 , 1 0 2R z z R< − < , or 0R z z< − . (5 marks) (b) Use the Residue Theorem to evaluate the integral � ( − ) where C is the contour 2=z . (5 marks) (c) A digital filter has transfer function () = 1 − 1/2 (i) Evaluate the response function of the filter, )()()( zHzXzY = , for the sequence nanx =)( . (Use the geometric series 11 0 k c k c∞− == ∑ .) (6 marks) (ii) By using partial fractions, determine the response of the filter, ( )y n , to the input ( ) nx n a= . (2 marks) (iii) What is the response to the input data () = (1)? [Note: the Z – transform of a sequence )(nx is defined as ∑∞= −= 0 )()( n nznxzX . The inverse Z – transform of )/(1)( bzzX −= is the sequence 1)( −= nbnx .] (2 marks) ENGSCI 313 Page 5 of 13 SECTION B Modelling with Differential Equations Question 4 (20 marks) (a) A thin rod of length is covered in insulation along its length. Imagine that this rod has spent a substantial amount of time in an industrial oven at 250°C. The rod is then removed from the oven and allowed to cool. (i) Calculate the diffusivity (2) for this material. Use the following properties: Specific heat = 154 /(°) Density = 20 /3 Thermal conductivity = 310 /( ∗ ∗ °) (4 marks) The temperature distribution along the rod (, ) for the 1D heat equation: = 2 2 2 is given as: (, ) = −λ22�(λ) + (λ)� where 2 is the diffusivity for this rod and λ is a constant. Now assume that the rod was removed from the oven. One end at = 10 is covered in a cap made from an insulating material (10,) = 0 and the other end ( = 0) is fixed at temperature (0, ) = 0° C. (ii) Derive an expression for the temperature along the rod (, ) in the format of an infinite sum. (12 marks) (b) Assume the following recurrence relation from the finite difference for the one dimensional heat equation: +1 = (1 − 2) + −1 + +1 ; where = 2Δ(Δ)2 If diffusivity 2 = 0.2 2 � ∆ = 0.4 What can be the minimum value for ∆ to avoid instability? (4 marks) Rod Insulation ENGSCI 313 Page 6 of 13 x=L Question 5 (20 marks) (a) Write down 2 assumptions that apply to tube model for the aquatic wave. (2 marks) (b) Write down expressions for the two boundary conditions associated with air movement in the tube that is open at x=0 and blocked at x=L depicted below. x=0 x=L (2 marks) (c) Write down boundary conditions for the temperature (, ) in a 0.5 thick wall that is insulated at = 0 and NOT insulated at = 0.5. (2 marks) (d) Assume that we need to determine the electric potential within a long box standing along z direction. Now a cross section through it on the x-y plane is shown below. The boundary conditions are given as: (0,) = 0 (,) = sin � � (, 0) = 0 (, ) = 0 If the solution is in the form of (, ) = () ∗ (), then what type of solutions would you initially guess for the () and ()? Why? Justify your answer. (4 marks) (e) A vibrating string that is 50 long and fixed at both ends and is modelled to be: (, ) = � � � � � � + � ��∞ =1 Initial conditions are given as: (, 0) = 0 ; 0 < < 50 (, 0) = 2 �5 � (i) Find the constants and showing the steps you took to get them. (Hint: Use = 2 ∫ (, 0) 0 ) (8 marks) (ii) Write down the solution (, ) using the values of your constants if = 1. (2 marks) y=L y x ENGSCI 313 Page 7 of 13 SECTION C Fourier Analysis For Section D you may use the Fourier Series & Fourier Transform Formulas below A Fourier Series approximation to function f(t) is given by :      +     +≈ ∑∑ == T ntB T ntA a tS N n N N n NN ππ 2sin.2cos. 2 )( 11 0 Where, , , dt T nttf T Bdt T nttf T Adttf T a T TN T TN T T ∫∫∫ −−−     =     == 2/ 2/ 2/ 2/ 2/ 2/0 2sin)(22cos)(2)(2 ππ A Complex Fourier Series approximation to function f(t) is given by : ∑ ∞ ≠ −∞= += 0 0 0)( n n tinw necctf Where, ∫ ∫ === − b a b a tinw n fwdtetfT cdttf T c , , π2)(1)(1 00 0 The Fourier Transform equation is given by : ∫ ∞ ∞− −= dtetfwF iwt)()( where, fw π2= The Inverse Fourier Transform equation is given by : ∫ ∞ ∞− += dwewFtf iwt)( 2 1)( π where, fw π2= The Forward Discrete Fourier Transform (DFT) is given by : )1(,...,3,2,1,0 , ].[][ 1 0 2 −== ∑ − =      − NKenxkX N n n N Ki π The Inverse Discrete Fourier Transform is given by : )1(,...,3,2,1,0 , ].[1][ 1 0 2 −== ∑ − =      + NKekX N nx N k n N Ki π ENGSCI 313 Page 8 of 13 Question 6 (25 marks) The function, (), is periodic and is defined such that: () = � ≤ < � � ≤ < (a) Sketch an odd extension of the function over the range [−2, 2]. (2 marks) (b) Determine the Fourier series approximation, (), of the odd periodic function and give the expression for the (). Determine the first 3 terms in your approximation. (8 marks) (c) What is the DC offset of ()? (1 mark) (d) What is the frequency, in Hz, of the third harmonic of (). (1 mark) (e) Which of these expansions will give you a better approximation to the function? Justify your answer. - Fourier series approximation using 32 terms - Fourier series approximation using 3 terms - Fourier series approximation using 23 terms (1 mark) Assume a new periodic signal, (), is defined such that: () = � ≤ < (f) Calculate the fundamental frequency, , and fundamental angular frequency, 0, of this signal; (1 mark) (g) Assume that the complex Fourier series approximation coefficients of this signal, over the interval [0, 2] are given by: 0 = 12 = 2 (i) Write down the complex Fourier series approximation for = [−3, 3] and determine the corresponding frequencies. (7 marks) (ii) Plot the two-sided scaled magnitude vs (), for = [−3, 3]. Scale factor: (4 marks) ENGSCI 313 Page 9 of 13 Question 7 (15 marks) A signal () is given by () = 30 + 4 (10) + 16(40) + (400) (a) What are the frequency components of the signal, (), in Hz? (2 marks) (b) What is the minimum sampling frequency, , required to reconstruct this signal? Explain your answer. (4 marks) (c) Sketch the two-sided magnitude spectrum of the signal () sampled at (call it plot I). (3 marks) (d) Sketch the two-sided magnitude spectrum of the signal () sampled at 2� (call it plot II). (3 marks) (e) Explain how the frequencies have been altered by this new sampling rate. (3 marks) ENGSCI 313 Page 10 of 13 SECTION D Models for Optimisation Questions 8 and 9 refer to the following problem description: Steam Power Inc (SPI) is a geothermal power generator. They are interested in reducing the cost of power generation in part of their network. Two phase liquid (steam and water) is pumped from 4 well groups (W1, W2, W3, W4) to separators (S1, S2) where the water and steam are separated – based on the current enthalpy estimate this results in 80% water and 20% steam. The steam is sent on to the turbines (T1, T2) where it is converted to electricity. The water is sent: either to the binary plant (B) where it is used to boil pentane, which in turn drives a turbine to produce further electricity; or to the reinjection wells (R) where it is returned to the geothermal field. A diagram of the setup is shown below, with the megalitres per day (ML/day) of two phase liquid available from each of the well groups shown on the left, and the required volumes for production at the turbines shown on the right. Note these volumes are either water, or quantity of two phase liquid converted to steam (so flow is conserved in the network). The cost of transhipment ($1000 per megalitres) is shown in the table below: S1 S2 T1 T2 B R W1 15 12 - - - - W2 18 12 - - - - W3 8 16 - - - - W4 20 14 - - - - S1 - - 5 4 12 10 S2 - - 8 16 14 8 B - - - - - 4 Question 8 (20 marks) (a) Define the decision variables for this optimization problem (including any units). (3 marks) (b) Formulate this problem as a linear programming problem. (12 marks) S1 S2 T1 T2 B 40 50 15 25 ≥120 W1 W2 W3 W4 50 80 R ENGSCI 313 Page 11 of 13 (c) The solution from Solver (in Excel) is shown below. Summarise the cheapest generation plan available to SPI for this part of their network. (5 marks) Objective Cell (Min) Cell Name Original Value Final Value $B$1 Cost 5080 5080 Variable Cells Cell Name Original Value Final Value Integer $A$5 WG1_S1 40 40 Contin $B$5 WG2_S1 0 0 Contin $C$5 WG3_S1 50 50 Contin $D$5 WG4_S1 0 0 Contin $E$5 WG1_S2 0 0 Contin $F$5 WG2_S2 50 50 Contin $G$5 WG3_S2 0 0 Contin $H$5 WG4_S2 60 60 Contin $I$5 S1_T1 0 0 Contin $J$5 S1_T2 18 18 Contin $K$5 S1_B 72 72 Contin $L$5 S1_R 0 0 Contin $M$5 S2_T1 15 15 Contin $N$5 S2_T2 7 7 Contin $O$5 S2_B 48 48 Contin $P$5 S2_R 40 40 Contin $Q$5 B_R 120 120 Contin Constraints Cell Name Cell Value Formula Status Slack $D$10 >= WG1 WG4_S1 40 $D$10<=$B$10 Binding 0 $D$11 >= WG2 WG4_S1 50 $D$11<=$B$11 Binding 0 $D$12 >= WG3 WG4_S1 50 $D$12<=$B$12 Binding 0 $D$13 >= WG4 WG4_S1 60 $D$13<=$B$13 Not Binding 20 $D$15 "= S1_Steam" WG4_S1 0 $D$15=$B$15 Binding 0 $D$16 "= S2_Steam" WG4_S1 0 $D$16=$B$16 Binding 0 $D$17 "= S1_Water" WG4_S1 0 $D$17=$B$17 Binding 0 $D$18 "= S2_Water" WG4_S1 0 $D$18=$B$18 Binding 0 $D$19 "=Binary" WG4_S1 0 $D$19=$B$19 Binding 0 $D$21 <= 1 WG4_S1 15 $D$21>=$B$21 Binding 0 $D$22 <= 2 WG4_S1 25 $D$22>=$B$22 Binding 0 $D$23 <= 3 WG4_S1 120 $D$23>=$B$23 Binding 0 ENGSCI 313 Page 12 of 13 Question 9 (20 marks) After using Solver (in Excel) to solve the linear programme from Question 8, the sensitivity analysis below is obtained. Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $A$5 WG1_S1 40 0 15 1 1E+30 $B$5 WG2_S1 0 2 18 1E+30 2 $C$5 WG3_S1 50 0 8 10 1E+30 $D$5 WG4_S1 0 2 20 1E+30 2 $E$5 WG1_S2 0 1 12 1E+30 1 $F$5 WG2_S2 50 0 12 2 1E+30 $G$5 WG3_S2 0 12 16 1E+30 12 $H$5 WG4_S2 60 0 14 2 2 $I$5 S1_T1 0 9 5 1E+30 9 $J$5 S1_T2 18 0 4 5 10 $K$5 S1_B 72 0 12 1.25 2.5 $L$5 S1_R 0 4 10 1E+30 4 $M$5 S2_T1 15 0 8 9 110 $N$5 S2_T2 7 0 16 10 5 $O$5 S2_B 48 0 14 2.5 1.25 $P$5 S2_R 40 0 8 4 27.5 $Q$5 B_R 120 0 4 1.32856E+17 10 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $D$10 <= WG1 40 -3 40 35 20 $D$11 <= WG2 50 -2 50 60 20 $D$12 <= WG3 50 -10 50 35 20 $D$13 <= WG4 60 0 80 1E+30 20 $D$15 "Conserve S1_S" 0 114 0 4 7 $D$16 "Conserve S2_S" 0 102 0 4 10 $D$17 "Conserve S1_W" 0 -6 0 40 48 $D$18 "Conserve S2_W" 0 -8 0 40 1E+30 $D$19 "Conserve Binary" 0 -4 0 120 1E+30 $D$21 >= T1 15 110 15 4 10 $D$22 >= T2 25 118 25 4 7 $D$23 >= B 120 10 120 40 48 ENGSCI 313 Page 13 of 13 (a) It is possible that a relaxation of government regulations on decibel level could reduce the cost of transshipment from Separator 2 to Turbine 2 to $12,000/ML. Is it possible to evaluate the impact of this change on the current solution from the sensitivity analysis report? If so, what would be the effect of an upgrade? (Be sure to justify your answer.) (3 marks) (b) Similarly, it is possible that a relaxation of the same government regulations could reduce the cost of transshipment from Well group 4 to Separator 1 to $16,000/ML. Is it possible to evaluate the impact of this change on the current solution from the sensitivity analysis report? If so, what would be the effect of an upgrade? (Be sure to justify your answer.) (3 marks) (c) What is a shadow price? Why is the shadow price 0 for transshipment from Well group 4? (3 marks) (d) If the daily demand at Turbine 1 were to increase to 25 ML is it possible to determine the effect on the optimal transmission plan using the sensitivity analysis report? If so, what is the effect? (2 marks) (e) If the daily demand at the Binary plant were to increase to 150 ML is it possible to determine the effect on the optimal transmission plan using the sensitivity analysis report? If so, what is the effect? (2 marks) (f) If Separator 2 has a limit of 100 ML on how much liquid it can process per day, how would you model this? (2 marks) (g) The volume of two phase liquid SPI can draw from Well group 1 is currently limited by resource consent. If the appropriate government official is made a director of SG1 (at a salary of $10,000 per day) the resource limit on WG1 could be increased to 50 ML/day. Each ML of liquid over the original 40 ML would cost an additional $2,000 over the original cost to ship from the Well group. How would your formulation (from question 8b) change to include this decision about relaxing WG1’s resource limit? (4 marks) (h) In reality the transportation cost is not linear, but rather the relationship between the volume shipped and cost can be more accurately modelled using a cubic function. After modelling this cost appropriately and solving again the optimal solution is actually found to have an objective function value of $6813. Why should we be careful about implementing this solution? 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