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Q2. The response of the level of a tank to changes in two input flowrates can be represented by the equation: 5 dh - + 2√h = F₁+F₂ dt a) It is proposed to set up a level controller on the tank by using the flowrate F₁ to regulate against changes in F2. Convert the equation to the s-domain and show the controlled system on a block diagram. b) What is the minimum gain required to limit changes in the level to ±0.1 m when the flowrate, F2, changes by ±0.5 m³ min-¹. The nominal operating level of the tank is 3 m. c) If the gain suggested above is used and the flowrate changes in a step by 0.2 m³ min-¹, what is the final change in the level and how long will it take for the system to come to steady-state?

Q2. The response of the level of a tank to changes in two input flowrates can be represented by the equation: 5 dh - + 2√h = F₁+F₂ dt a) It is proposed to set up a level controller on the tank by using the flowrate F₁ to regulate against changes in F2. Convert the equation to the s-domain and show the controlled system on a block diagram. b) What is the minimum gain required to limit changes in the level to ±0.1 m when the flowrate, F2, changes by ±0.5 m³ min-¹. The nominal operating level of the tank is 3 m. c) If the gain suggested above is used and the flowrate changes in a step by 0.2 m³ min-¹, what is the final change in the level and how long will it take for the system to come to steady-state?

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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2. a) The given model is non-linear so we must linearise the square root term, as you've done earlier in the module. This leads to: dh* 5. 5 d² + (1) ₂ h* = F₁* + F₂* dt We're told that the steady-state level in the tank is 3 m. Taking the Laplace Transform of this gives: 1.73 1.73 8.66s + 1 8.66s + 1 b) Using a pure proportional controller (reverse acting), F₁ = -Kcε* Thus, h*(s) = 1.73 h* (s) 8.66s +1+1.73K -F₁*(s) + -Kchsp (s) +; 1.73 8.66s +1+1.73K -F₂* (s) h* (t→∞o) = Since this is a first order process (with pure-P control), it won't overshoot its final steady-state value, so we can just look at the steady-state to get the maximum deviation in output after step disturbance: -F₂* (s) 1.73 1 + 1.73Kc If F₂ changes by ±0.5 we want to limit changes in the level to ±0.1. Thus, Kc = 4.42. c) Closed loop disturbance gain = 0.2, So, the tank level will increase by 0.04m Rearrange the equation into standard form for a first order transfer function to find a closed loop time constant of 1 min. -F₂ So the response will be 95% complete in approximately 3minutes and 99% complete after approximately 5 minutes (because (1-exp(-3)) = 0.95 etc.)
Transcribed Image Text:2. a) The given model is non-linear so we must linearise the square root term, as you've done earlier in the module. This leads to: dh* 5. 5 d² + (1) ₂ h* = F₁* + F₂* dt We're told that the steady-state level in the tank is 3 m. Taking the Laplace Transform of this gives: 1.73 1.73 8.66s + 1 8.66s + 1 b) Using a pure proportional controller (reverse acting), F₁ = -Kcε* Thus, h*(s) = 1.73 h* (s) 8.66s +1+1.73K -F₁*(s) + -Kchsp (s) +; 1.73 8.66s +1+1.73K -F₂* (s) h* (t→∞o) = Since this is a first order process (with pure-P control), it won't overshoot its final steady-state value, so we can just look at the steady-state to get the maximum deviation in output after step disturbance: -F₂* (s) 1.73 1 + 1.73Kc If F₂ changes by ±0.5 we want to limit changes in the level to ±0.1. Thus, Kc = 4.42. c) Closed loop disturbance gain = 0.2, So, the tank level will increase by 0.04m Rearrange the equation into standard form for a first order transfer function to find a closed loop time constant of 1 min. -F₂ So the response will be 95% complete in approximately 3minutes and 99% complete after approximately 5 minutes (because (1-exp(-3)) = 0.95 etc.)
Q2. The response of the level of a tank to changes in two input flowrates can be represented by the equation: 5 dh dt +2√h = F₁+F₂ a) It is proposed to set up a level controller on the tank by using the flowrate F₁ to regulate against changes in F2. Convert the equation to the s-domain and show the controlled system on a block diagram. b) What is the minimum gain required to limit changes in the level to ±0.1 m when the flowrate, F2, changes by ±0.5 m³ min-¹. The nominal operating level of the tank is 3 m. c) If the gain suggested above is used and the flowrate changes in a step by 0.2 m³ min-¹, what is the final change in the level and how long will it take for the system to come to steady-state?
Transcribed Image Text:Q2. The response of the level of a tank to changes in two input flowrates can be represented by the equation: 5 dh dt +2√h = F₁+F₂ a) It is proposed to set up a level controller on the tank by using the flowrate F₁ to regulate against changes in F2. Convert the equation to the s-domain and show the controlled system on a block diagram. b) What is the minimum gain required to limit changes in the level to ±0.1 m when the flowrate, F2, changes by ±0.5 m³ min-¹. The nominal operating level of the tank is 3 m. c) If the gain suggested above is used and the flowrate changes in a step by 0.2 m³ min-¹, what is the final change in the level and how long will it take for the system to come to steady-state?
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