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solution hw3

# solution hw3 - Problem 1 From table 7.1 in the textbook the...

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Problem 1 From table 7.1 in the textbook, the Laplace transform of y(t)=te -t is: t 2 1 L te y(s) (s 1) ⎤ = = + (S1.1) Moreover, the Laplace transform of the unit-impulse δ (t) is: [ ] L (t) u(s) 1 δ = = (S1.2) Since u(t)=0 all the time but at t=t o , assuming that y(t) is in deviation form (i.e. y s =0), the transfer function of the system in exam can be written as follows: 2 y(s) 1 G(s) u(s) (s 1) = = + (S1.3) Problem 2 Assuming constant density and reactor volume, the overall material balance can be written as follows: 1 2 i i dh(t) A F (t) F F (t) 8h(t) dt = = (S2.1) As initial condition for equation (S2.1), we chose the steady state value of the hydrostatic pressure h s given by: 2 s is h(0) h F 64 = = (S2.2) Linearizing the right hand side of equation (S2.1) about the steady state value (S2.2), one obtains: [ ] [ ] 1 2 1 2 i i s s s F(t) 8h(t) F (t) F 4h h(t) h (S2.3) Defining the deviation variables H(t)=h(t)-h s and Q(t)=F i (t)-F s, and considering (S2.3), equation (S2.2) becomes 1 2 1 2 s s Ah h dH(t) H(t) Q(t) 4 dt 4 + = (S2.4)

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