An-Najah National University Engineering Numerical Analysis (10626251) Final Exam – Spring 2020 Instructor: Eng. Mays Shadeed May, 10th 2020 Q1): (25 points) The enthalpy of a real gas is a function of pressure as described below. These data were taken for a real fluid. P (atm) 0.1 5 15 25 30 35 40 50 60 T= 360 K 220 4.1 2.2 1.35 1.1 0.90 0.68 0.61 0.54 V, L T = 400 K 250 4.7 2.5 1.49 1.2 0.99 0.75 0.675 0.6 T = 440 K 282.5 5.23 2.7 1.55 1.24 1.03 0.78 0.70 0.62 a) Use the best numerical differentiation technique available to estimate the value of at T = 400 K for each value of pressure. b) Use the best combination of Newton cotes integration to estimate the enthalpy of the fluid at 400 K. c) Use Newton’s third-order polynomial interpolation to find the volume at P = 20 atm and T = 360 K to the best possible accuracy. Q2): (15 points) A company manufactures two drugs: A and B. The drugs are produced by blending two chemicals only: 1 and 2. Drug A must contain 4 kg of chemical 1 and 1.5 kg of chemical 2, and drug B must contain 2 kg of chemical 1 and 5 kg of chemical 2. To make one kilogram, the company must invest 3 hours of skilled labor for drug A and 2 hours for drug B. The company has at most 150 hours of skilled labor and at least 80 kg of chemical 1 and 120 kg of chemical 2. The demand for drug B is at least half of that of Drug A. each type of drugs must be stored for 2 days before selling it. However, the storage tanks capacity is less than 120 kg. Drug A costs $4/kg and sells for $6/kg, while drug B costs $2.5/kg and sells for $3.5/kg. Determine how much of each drug should be produced to maximize the profit. What is the maximum profit? Set the linear programming problem to maximize the profit and solve the linear programming problem graphically using the manual and excel (all calculations and graphical works must be done using excel). Page 1 of 3 Q3): (20 points) Fit the data in the table to the following equations to get values for a and b for each equation using Excel and least-squares. Which equation is a better fit? Why? Provide evidence to support your answers. x y 0 0.381 20 0.264 50 0.18 65 0.151 150 0.086 1. 2. 3. Q4): (10 points) The following model can be used to compute the dissolved oxygen concentration in a river below a point discharge of sewage where C = dissolved oxygen concentration (mg/L), oxygen saturation concentration (mg/L), t = travel time (d), Lo = biochemical oxygen demand (BOD) concentration at the mixing point (mg/L), kd = rate of decomposition of BOD (d-1), ks= rate of settling of BOD ( d-1), ka = reaeration rate (d-1), and Sb = sediment oxygen demand (mg/L/d). Given the following values: Cs =10 mg/L Lo = 50 mg/L kd = 0.1 d-1 ka = 0.6 d-1 ks = 0.05 d-1 and Sb=1.0 (mg/L/d). If the critical minimum level oxygen concentration is (4±0.01) mg/L, use your numerical knowledge to check if the oxygen concentration drops below the critical limits along the time period ti=1.5 day and tf =5 days (Hint find the value of critical travel time (t) using Golden section search method). Page 2 of 3 Q5): (10 points) For the following function a) Fill in the following table by calculating y x 0 0.5 1 1.5 1 2 3 4 5 b) Use the most appropriate higher accuracy Newton cotes methods to find the integration for the tabulated data above. Q6): (20 points) One of the forms of Van der Pol equations is giving by the following second-order ODE: 𝜇 Subject to t=0 min, y=1, y'=0 and µ=10 Use the first-order Runge-Kutta method (Euler's method) to integrate the ODE from t=0 to t=10 mins with step size =2 and 5. Discuss your results. Page 3 of 3