HWK3514_HWK_7_solution_Fall_2007

HWK3514_HWK_7_solution_Fall_2007 - ME 3514 System Dynamics...

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ME3514 - HWK - Burdisso - 68 ME 3514 – System Dynamics HOMEWORK #7-Solution DUE: 10/31/07 1. (10pts) In the liquid level system shown in the following figure, the head is kept at 1m for 0 t < . The inflow valve opening is changed at 0 = t , and the inflow rate is sec 05 . 0 3 m for 0 t . Determine the time needed to fill the tank to 2.5-m level. Assume that the outflow rate Q s m 3 and head H m are related by H Q 02 . 0 = The capacitance of the tank is 2 2 m . Remark: To solve this problem, please linearize the equation of motion around 1 H m = . Figure: Liquid-level system Solution: The equation of motion in terms of the absolute level ( ) H t is (conservation of mass) ( ) ( ) i dH C Q t Q t dt = The valve equation is then ( ) 0.02 ( ) Q t H t = There are two options: a) Replace ( ) 0.02 ( ) Q t H t = into EOM: ( ) 0.02 ( ) ( ) i dH t C H t Q t dt + = (1a) b) Differentiate ( ) 0.02 ( ) Q t H t = w.r.t. and then replace: ( ) 0.02 ( ) 2 ( ) dQ t dH t dt dt H t = 2 ( ) ( ) ( ) ( ) 0.02 i H t dQ t C Q t Q t dt + = (1b) NAME: ( ) ( ) Q t Q q t = + ( ) ( ) i i Q t Q q t = + ( ) ( ) H t H h t = +
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