KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec15

# KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec15

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1 Chapter 5 The standard form for a first-order TF is: where: Consider the response of this system to a step of magnitude, M : Substitute into (5-16) and rearrange, First-Order System ( 29 ( 29 (5-16) τ 1 Y s K U s s = + steady-state gain τ time constant K @ @ ( 29 ( 29 for 0 M U t M t U s s = = ( 29 ( 29 (5-17) τ 1 KM Y s s s = +

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2 Chapter 5 Take L -1 (cf. Table 3.1), ( 29 ( 29 1 (5-18) t y t KM e - = - Let steady-state value of y ( t ). From (5-18), y Y @ . y KM Y = t ___ 0 0 0.632 0.865 0.950 0.982 0.993 y y Y τ Note: Large means a slow response. τ y y Y τ t
3 Chapter 5

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4 Chapter 5
5 Chapter 5

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6 Chapter 5
7 Chapter 5

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8 2 2 ) ( ϖ + = s s U 2 2 2 2 2 1 0 2 2 p s s s 1 s s 1 s K ) s ( Y ϖ + α + ϖ + α + + τ α = ϖ + ϖ + τ = 1 K 1 K 1 K 2 2 p 2 2 2 p 1 2 2 2 p 0 + τ ϖ ϖ = α + τ ϖ τ ϖ - = α + τ ϖ τ ϖ = α For a sine input (1st order process) output is. .. By partial fraction decomposition, Chapter 5
9 ) t sin( 1 K e 1 K ) t ( y 2 2 p t 2 2 p φ + ϖ + τ ϖ + + τ ϖ ϖ τ = τ - ) arctan( ϖτ - = φ note: φ is not a function of t but of τ and ϖ . For large t, y(t) is also sinusoidal, output sine is attenuated by… (fast vs. slow ϖ ) Inverting, this term dies out for large t 1 1 2 2 + τ ϖ Chapter 5

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10 Chapter 5
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Unformatted text preview: Consider a step change of magnitude M . Then U ( s ) = M / s and, Integrating Process Not all processes have a steady-state gain. For example, an “integrating process” or “integrator” has the transfer function: ( 29 ( 29 ( 29 constant Y s K K U s s = = ( 29 ( 29 2 KM Y s y t KMt s = = Thus, y ( t ) is unbounded and a new steady-state value does not exist. L-1 11 Chapter 5 Consider a liquid storage tank with a pump on the exit line: Common Physical Example:-Assume: 1. Constant cross-sectional area, A . 2.-Mass balance:-Eq. (1) – Eq. (2), take L , assume steady state initially,-For (constant q ), ( 29 q f h Y (1) (2) i i dh A q q q q dt =-=-( 29 ( 29 ( 29 1 i H s Q s Q s As Y =-( 29 Q s Y = ( 29 ( 29 1 i H s Q s As Y =...
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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec15

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