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1. The level in a tank with a free-draining gravity discharge can be represented by the model: 10 +2√/h(t) = Fin(t) dh(t) dt The symbols have their usual meaning, and the system starts at steady state with Fin = 4 (don't worry about units for this example) for t < 0. a) What is the steady state value of h? b) Derive the linearised version of this equation using deviation variables for h(t) and Fin(t). Identify the time constant and the steady-state gain. c) What is the transfer function relating changes in h(t) to changes in F(t)? d) From this, calculate the time domain solution of the response of the level h(t) to a step change at time t = 0 in the inlet flowrate to 4.2 using the Laplace method.

1. The level in a tank with a free-draining gravity discharge can be represented by the model: 10 +2√/h(t) = Fin(t) dh(t) dt The symbols have their usual meaning, and the system starts at steady state with Fin = 4 (don't worry about units for this example) for t < 0. a) What is the steady state value of h? b) Derive the linearised version of this equation using deviation variables for h(t) and Fin(t). Identify the time constant and the steady-state gain. c) What is the transfer function relating changes in h(t) to changes in F(t)? d) From this, calculate the time domain solution of the response of the level h(t) to a step change at time t = 0 in the inlet flowrate to 4.2 using the Laplace method.

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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1. The level in a tank with a free-draining model: gravity discharge can be represented by the 10 dh(t) dt 2√h(t) = Fin(t) The symbols have their usual meaning, and the system starts at steady state with Fin = 4 (don't worry about units for this example) for t < 0. a) What is the steady state value of h? b) Derive the linearised version of this equation using deviation variables for h(t) and Fin (t). Identify the time constant and the steady-state gain. c) What is the transfer function relating changes in h(t) to changes in F(t)? d) From this, calculate the time domain solution of the response of the level h(t) to a step change at time t = 0 in the inlet flowrate to 4.2 using the Laplace method.
Transcribed Image Text:1. The level in a tank with a free-draining model: gravity discharge can be represented by the 10 dh(t) dt 2√h(t) = Fin(t) The symbols have their usual meaning, and the system starts at steady state with Fin = 4 (don't worry about units for this example) for t < 0. a) What is the steady state value of h? b) Derive the linearised version of this equation using deviation variables for h(t) and Fin (t). Identify the time constant and the steady-state gain. c) What is the transfer function relating changes in h(t) to changes in F(t)? d) From this, calculate the time domain solution of the response of the level h(t) to a step change at time t = 0 in the inlet flowrate to 4.2 using the Laplace method.
1. The level in a tank with a free-draining gravity discharge can be represented by the model: 10 +2√h(t) = Fin(t) dh(t) dt The symbols have their usual meaning, and the system starts at steady state with Fin = 4 (don't worry about units in this example) for t < 0. a) What is the steady state value of h? dh dt = 0 NB hss=, so the steady state gain is Let's linearise the non-linear term: √h = √hss + 2√ √hss=4, hss = 4 b) Derive the linearised version of this equation using deviation variables for h(t) and Fin (t). d√h] dh 20 dhss dFss (√h) = √hss h(t) = hss + h' (t) = Fss = 2. Jhss Take the deviation variable of the non-linear term xh* = √√hss + 1 2√hss 1 2√hss =h* - √hss = =h² dh' (t) dt -+h(t) = 2 Fin (t), NB See that steady state gain is 2, and time scale T = 20. Note also that we cannot heedlessly take the Laplace transform of a non-linear function. = hss L{√hss + h* (t)} # + h*(s) S h*(s) 2 F*(s) 1+20s t> 0 We must linearise before taking the Laplace transform! c) What is the transfer function relating changes in h(t) to changes in F(t)? :h* d) From this, calculate the time domain solution of the response of the level h(t) to a step change at time t = 0 in the inlet flowrate to 4.2 using the Laplace method. By the same method as previously, h(t) = 4 +0.4[1- exp(-/20)]
Transcribed Image Text:1. The level in a tank with a free-draining gravity discharge can be represented by the model: 10 +2√h(t) = Fin(t) dh(t) dt The symbols have their usual meaning, and the system starts at steady state with Fin = 4 (don't worry about units in this example) for t < 0. a) What is the steady state value of h? dh dt = 0 NB hss=, so the steady state gain is Let's linearise the non-linear term: √h = √hss + 2√ √hss=4, hss = 4 b) Derive the linearised version of this equation using deviation variables for h(t) and Fin (t). d√h] dh 20 dhss dFss (√h) = √hss h(t) = hss + h' (t) = Fss = 2. Jhss Take the deviation variable of the non-linear term xh* = √√hss + 1 2√hss 1 2√hss =h* - √hss = =h² dh' (t) dt -+h(t) = 2 Fin (t), NB See that steady state gain is 2, and time scale T = 20. Note also that we cannot heedlessly take the Laplace transform of a non-linear function. = hss L{√hss + h* (t)} # + h*(s) S h*(s) 2 F*(s) 1+20s t> 0 We must linearise before taking the Laplace transform! c) What is the transfer function relating changes in h(t) to changes in F(t)? :h* d) From this, calculate the time domain solution of the response of the level h(t) to a step change at time t = 0 in the inlet flowrate to 4.2 using the Laplace method. By the same method as previously, h(t) = 4 +0.4[1- exp(-/20)]
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