UNIVERSIT DEGLI STUDI DI SALERNO Bachelor Degree in
UNIVERSITÀ DEGLI STUDI DI SALERNO Bachelor Degree in Chemical Engineering Course: Process Instrumentation and Control (Strumentazione e Controllo dei Processi Chimici) REFERENCE LINEAR DYNAMIC SYSTEMS First-Order Systems Rev. 2. 5 – May 17, 2019
FIRST-ORDER LAG I. C. : t=0 y(0)=0 see: Ch. 10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” First-order ODE, linear, non-homogeneous, with constant coefficients Forcing function: f(t) If a 0≠ 0 CANONICAL FORM in the time domain 30/11/2020 in the Laplace domain Process Instrumentation and Control - Prof. M. Miccio 2
FIRST-ORDER LAG position of the pole in the complex plane Im Characteristic polynomial: ps + 1 Characteristic eq. : ps + 1 = 0 Only 1 pole: p = -1/ p < 0 -1/ p Re NOTE: Self-regulating dynamic behavior 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 3
DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO STEP INPUT CHANGE Forcing funtion: f(t)=Au(t) A=const. > 0 F(s)=A/s t NOTE: Self-regulating dynamic behavior 30/11/2020 tp Process Instrumentation and Control - Prof. M. Miccio see: Ch. 10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 4
FIRST-ORDER PROCESSES Characteristic Parameters Static Gain ( Kp ): This is the ultimate value of the response (new steady-state) for a unit-step change in the input. ( Theorem of the final value) Time Constant (t p): The time constant of a process is a measure of the time necessary for the process to adjust to a change in its input. from: Romagnoli & Palazoglu (2005), “Introduction to Process Control” 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 5
FIRST-ORDER PROCESSES Effect of tp R F 0 h(t), m 1 h(t), m 0. 9 1 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 0 0 5 10 15 time 20 0 10 20 time Response of the level of a tank (first-order system) to a unit-step change in the input flow rate from: Romagnoli & Palazoglu (2005), “Introduction to Process Control” 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 6
FIRST-ORDER PROCESSES Effect of tp R F 0 The system eventually reaches a new equilibrium point (new steady-state). Two tanks with different cross sectional areas respond with different speeds (different time constants). Ø The tank with a small area responds faster (smaller time constant). Ø The tank with a larger area responds slower (larger time constant). from: Romagnoli & Palazoglu (2005), “Introduction to Process Control” 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 7
DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO STEP INPUT CHANGE Dimensionless diagram of the dynamic response NOTE: Self-regulating dynamic behavior 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio see: Ch. 10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 8
DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO UNIT IMPULSE Forcing function : f(t)= (t) F(s) = 1 Dynamic response: NOTE: Self-regulating dynamic behavior 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 9
DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO UNIT IMPULSE Dimensionless diagram of the dynamic response NOTE: Self-regulating dynamic behavior 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio see: Ch. 10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 10
DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO SINUSOIDAL INPUT Forcing function: Dynamic response for long time: - -f(t) Frequency Amplitude ratio AR: Phase lag 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio unchanged B/A < 1 <0 11
RESPONSE TO A SINUSOIDAL INPUT Homework: Diagram the DYNAMIC RESPONSE with the mod. Custom Process of LOOP-PRO Control Station 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 12
PURE CAPACITIVE SYSTEM or FIRST-ORDER INTEGRATOR If ao=0 CANONICAL FORM in the time domain 30/11/2020 in the Laplace domain Process Instrumentation and Control - Prof. M. Miccio 13
PURE CAPACITIVE FIRST-ORDER SYSTEM position of the pole on the complex plane Im Only 1 pole at the origin of the axes: Re s=0 DYNAMIC RESPONSE TO STEP INPUT CHANGE NOTE: marginally stable dynamic system 30/11/2020 NON-self-regulating for some inputs Process Instrumentation and Control - Prof. M. Miccio 14
THE OPEN TANK WITH A VARIABLE LEVEL AS A FIRST-ORDER SYSTEM R F 0 a) first-order OR 1 st-order after linearization b) pure capacitive see: Ch. 10 - Stephanopoulos, "Chemical process control: an Introduction to theory and practice" 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 15
NONLINEAR FIRST-ORDER SYSTEMS 1 st case: nonlinearity of the function of y(t) CANONICAL FORM where F(y(t)) is a nonlinear function of y(t) LINEARIZATION METHOD Taylor expansion at the first term: Where : ε=o(y-y 0)2 o indicates the “order of magnitude” y 0= ys= steady-state value of y(t) or y 0=0 or y 0= another point of interest of y(t) 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 16
LINEARIZATION METHOD APPLICATION TO THE TANK WITH A VARIABLE LEVEL Mass balance equations in dynamic condition: (1) at the steady-state: I. C. : t=0 ; h(0) = hs (1 bis) Linearization by Taylor: Replacing in (1) : R (2) F 0 At the steady state (h=hs; Accumulation=0): Thus, we have: 30/11/2020 (3) Process Instrumentation and Control - Prof. M. Miccio 17
LINEARIZATION METHOD APPLICATION TO THE TANK WITH A VARIABLE LEVEL Substracting eq. (3) from eq. (2) the deviation variable appears: I. C. : t=0 ; h’(0) = 0 Applying Laplace transform: L(I) = L(II) where: NOTE: The evaluated TF is valid only around the steady state. 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 18
LINEAR APPROXIMATION OF A NONLINEAR FUNCTION Due to the tangent theorem, the simulation of the linear approximation is satisfactory only around the point x 0 (hs) of linearization (it is not possible to extend the linearized function to the whole curve). Another linearization will be necessary to analyze the dynamic response of the system in another point. t 0 30/11/2020 t 1 Process Instrumentation and Control - Prof. M. Miccio t see: Ch. 6 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 19
ORIGINAL MODEL and LINEARIZED MODEL: COMPARISON BETWEEN THE DYNAMIC RESPONSES APPLICATION TO THE TANK WITH A VARIABLE LEVEL Initial Condition Theoretical response see: Ch. 6 - Stephanopoulos, "Chemical process control: an Introduction to theory and practice" 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 20
NONLINEAR FIRST-ORDER SYSTEM 2 nd case: nonlinearity in the derivative term d [y(t)]/d t ORIGINAL FORM: where is a nonlinear function of y(t) e. g. , LINEARIZATION METHOD The linearization method is still applicable, with the Taylor expansion at the first term, but consequent elaboration and calculations necessary to obtain the mathematical model in terms of deviation variables and the TF are more difficult and not treated here! 30/11/2020 Process Instrumentation and Control - Prof. M. Miccio 21
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