ExamFinalCHEE3334Fall2003Solutions2

ExamFinalCHEE3334Fall2003Solutions2 - CHEE 3334 Fall 2003...

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CHEE 3334 Fall 2003 Michael Nikolaou Student Name Final Examination Open-everything. 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 (but you may use anything you want to check numerical results)! PLEASE SHOW ALL RELEVANT WORK IN THE INDICATED SPACES. DO NOT RETURN ADDITIONAL PAGES. (You may use scratch paper only for scratch work.) 1. (40 pts.) Continuous biochemical reactors, such as the one shown on the right, are used in a variety of processes, from waste treatment, to alcohol fermentation, to production of antibiotics. In a biochemical reactor, cells (biomass) consume nutrients (usually called substrate ) such as glucose, to grow, multiply, and produce useful chemicals after consuming the substrate. A component mass balance on a continuous-flow biochemical reactor (frequently called a chemostat ) yields the ordinary differential equations ( 29 max 2 1 1 2 max 2 1 2 2 2 2 ( ) m f m x dx D x dt k x x x dx D x x dt Y k x μ = - + = - - + where 1 x = mass of cells per unit volume, 2 x = mass of substrate per unit volume. The parameters appearing in the model are 1 max 0.53 hr - = , 0.12 / m k g liter = , 0.4 Y = (yield), 1 0.3 D hr - = (dilution rate = volumetric flow rate / volume = F / V ), and 2 4 / f x g liter = (concentration of substrate in the feed stream). The initial conditions for this reactor are 1 (0) 1.5 / x g liter = and 2 (0) 0.5 / x g liter = a. Are the above differential equations linear or nonlinear? (Explain.) The equations can be written as ( 29 - - = + - - + = + max 2 1 1 2 max 2 1 2 2 2 2 0 0 ( ) m f m x dx D x dt k x x x dx D x x dt Y k x or ( ) 0 L x = where 1 2 x x x = Clearly, ( ) ( ) ( ) L x y L x L y + + , i.e. + 1 + + 1 1 1 1 2 2 2 2 ( ) ( ) ( ) x y x y L L L x y x y . Therefore, the equations are nonlinear. b. Perform one iteration of Euler’s method with time step 0.1 t ∆ = , to compute ( 29 1 1 x t and ( 29 2 1 x t . Euler’s method: - page 1 of 7 - F, x 2 f F, x 2 , x 1 V k tk x1(tk) x2(tk) mumax = 0.53 0 0 1.5 0.5 km = 0.12 1 0.1 1.519113 0.444718 Y = 0.4 2 0.2 1.536944 0.392865 D = 0.3 3 0.3 1.553234 0.345083 x2f = 4 4 0.4 1.567718 0.302028 dt = 0.1 0 0.5 1 2 0 1 2 3 4 5 6 x1 x2
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( 29 μ + + = + ∆ - + = + ∆ - - + max 2 1 1 1 1 2 max 2 1 2 2 2 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) k k k k m k k k k k k f m k x t x t x t t D x t k x t x t x t x t x t t D x x t Y k x t c. Substitute the derivatives 1 dx dt and 2 dx dt at time t k by backward finite divided differences. What numerical method would you use to calculate ( 29 1 k x t and ( 29 2 k x t in terms of known past values ( 29 1 1 k x t - and ( 29 2 1 k x t - ?
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This note was uploaded on 04/15/2008 for the course CHEE 3334 taught by Professor Nikolau during the Fall '06 term at University of Houston.

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ExamFinalCHEE3334Fall2003Solutions2 - CHEE 3334 Fall 2003...

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