Assignment 2 Solutions 2009

Outputscan betheoutlet composit ion and the vo

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Unformatted text preview: me in the tank. The model equat ions are given as: dC Fi n = ( n - C ) - kC 2 Ci dt V dV = Fi n - F = Fi n - bV dt Here C is the tank concentration, V is the tank vo lume, F is the flow rate, and the subscript in refers to inlet condit ions. k and b are constants. a. What are the state variable(s), input variable(s) and output variable(s)? b. Obtain a linear state­space model for this system. Solution: The state variables are co mposit ion C and vo lume V. Input variables would be inlet concentration and inlet flow rate. The output variables would depend on contro l object ives. We would t ypically be interested in maintaining a constant yield in the reactor (hence constant outlet composit ion) and constant level (or volume) to ensure constant residence t ime. Outputs can be the outlet composit ion and the vo lume (the state variables). The first equation (component balance) can be classified as nonlinear, hence requiring the applicat ion o f Taylor expansio n. The second equation (total mass balance) is already in linear form. dC Fi n = (C n - C ) - kC 2 = f ( Fi n , C in , , C ) i V 1 dt V dV = Fi n - F = Fi n - bV = f 2 ( Fi n , C in , , C ) V dt The Taylor expansio n of the first equation yie lds: ¶f dC @ f ( Fi n , s , C in , s , , C s ) + 1 Vs 1 dt ¶C in F ( in - C in, s ) + C Cin Vs C s in , s , , s , , ¶f 1 ¶Fi n F ( Fi n - Fi n , s ) Cin , Vs C s in , s , s , , ¶f ¶f 1 ( - C s ) + 1 C (V - V ) s ¶C Fin , s , Cin , s , s , C s ¶V Fin , s , Cin , s , s , C s V V The derivat ives can be calculated as fo llows: é F , s ù ¶ f1 in =ê ú = a ¶C F , C , , C ë V s û in Vs s in , s in s , ¶f 1 ¶F in...
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