ExamFinalFall1998Solutions1

ExamFinalFall1998Solutions1 - CHEE 3334 Fall 98 Michael...

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Unformatted text preview: CHEE 3334 Fall 98 Michael Nikolaou FINAL EXAM Open-book, open-notes. Total points 100. Please, budget your time ! Do NOT try to read the book during the exam! Write neatly! Do NOT use your programmable calculator to find results you are asked to compute in a certain way--after all, the calculator may find the wrong answer or not all answers! Show all work. Return this sheet with your exam . GOOD LUCK! (30 pts.) 1. The reactions products side C B A _ + occur in a batch reactor. If y 1 ( t ) and y 2 ( t ) denote the concentrations of A and C inside the reactor, then the following ordinary differential equations (ODE) describe how y 1 and y 2 evolve with time: 2 1 2 2 2 2 1 1 ) ( 10 ) ( 2 ) ( ) ( 10 t y t y dt dy t y t y dt dy +- = +- = Let the initial conditions be ( 29 1 1 = y and ( 29 2 = y . Consider a grid of points t k t k = * , with 01 . = t and ,... 2 , 1 , = k a. For the above equations, do one step of Eulers method, to compute ) ( 1 1 t y and ) ( 1 2 t y . 2 1 1 1 2 1 2 2 2 2 1 1 1 1 2 1 ( ) ( ) 10 ( ) ( ) ( ) ( ) 2 ( ) 10 ( ) ( ) 0.9 ( ) 0.1 k k k k k k k k y t y t t y t y t y t y t t y t y t y t y t + + = + - + = + - + & = = b. Approximate the derivatives dt dy / 1 and dt dy / 2 at t 1 by centered finite divided differences, and develop a recursive formula for the solution of the above two ODE. Apply that formula to compute ) ( 2 1 t y and ) ( 2 2 t y . Use the values of ) ( 1 1 t y and ) ( 1 2 t y computed in part (a.) where necessary. 2 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 1 1 2 2 2 1 1 1 1 2 2 2 ( ) ( ) ( ) ( ) 2 10 ( ) ( ) 2 ( ) ( ) ( ) ( ) 2 2 ( ) 10 ( ) 2 ( ) 0.84 ( ) 0.158 k k k k k k k k k k k k dy y t y t y t y t t y t y t dt t dy y t y t y t y t t y t y t dt t y t y t +- +- +- +-- = + - + - = + - + = = Note dependence of current y on two past values of it....
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ExamFinalFall1998Solutions1 - CHEE 3334 Fall 98 Michael...

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