che446_10_hw2

# che446_10_hw2 - Homework#2 ChE 446 Fall 2010 Problem 1...

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Homework #2 ChE 446 Fall 2010 Problem 1. Consider the following set of diﬀerential equations, dy 1 dt = - 2 y 1 - y 2 + u ( t - 2) dy 2 dt = 3 y 1 + y 2 - u ( t - 2) where: y 1 (0) = y 2 (0) = 0. 1. Derive the following transfer function: Y 2 ( s ) U ( s ) = G ( s ) = - s + 1 s 2 + s + 1 e - 2 s Find the pole(s) and zero(s) of the transfer function and plot them in the complex plane. 2. Considering the following input change: u ( t ) = e + t . Derive the time domain solution y 2 ( t ) and determine: lim t →∞ y 2 ( t ). Explain why the output converges to a constant even though the input is an exponentially increasing function of time. Problem 2. Consider a process described by a ﬁrst-order transfer function with a steady- state gain of 1 3 and a time constant of 1 3 . 1. Use partial fractions to derive the open-loop response y 0 ( t ) for the following input change: u 0 ( t ) = 4 te - 2 t . 2. Use the y 0 ( t ) relation to determine lim t →∞ y 0 ( t ). Verify the result by applying the ﬁnal value theorem. Problem 3. Consider a catalyst delivery system for a polymerization reactor. The catalyst

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che446_10_hw2 - Homework#2 ChE 446 Fall 2010 Problem 1...

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