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بسم هللا الرحمن الرحيم Advanced Control Lecture two 1- A modeling procedure (Marlin, Chapter 3) 2- Empirical modeling (Smith & Corripio, Chapter 7) 3- Control valve: Action, characteristics and capacity (Smith & Corripio, Chapter 5) Lecturer: M. A. Fanaei Ferdowsi University of Mashhad 1 Modeling (relation between inputs and outputs of process) We can tune the controller only after the process steady-state and dynamic characteristics are known. Types of model • White box (first principles) n black box (empirical) • Linear n non-linear • Static n dynamic • Distributed n lumped • Time domain n frequency domain • Continuous n discrete For further reading refer to : Roffel & Beltlem, “Process dynamics and control”, Wiley, 2006 2 A modeling procedure 1. Define goals 5. Analyze results Specific design decisions Check results for correctness Numerical values Limiting and approximate answers Functional relationships Accuracy of numerical method Required accuracy Interpret results 2. Prepare information Plot solution Sketch process and identify system Characteristic behavior Identify variables of interest Relate results to data and assumptions State assumptions and data Evaluate sensitivity 3. Formulate model 6. Validate model Conservation balances Select key values for validation Constitutive equations Compare with experimental results Rationalize Compare with results from more Check degrees of freedom complex model Dimensionless form 4. Determine Solution Analytical Numerical 3 Example 1. Isothermal CSTR F Define Goals F V CAo 1. Dynamic response of a CSTR to a step in the inlet concentration. 2. The reactant concentration should never go above 0.85 mole/m3 3. When the concentration reaches 0.83 mole/m3, would a person have enough time to respond? What would a correct response be? CA 1. The system is the liquid in the tank (as shown in Fig.). 2. The important variable is the reactant concentration in the reactor. Prepare Information 4 Example 1. Isothermal CSTR Prepare Information … F F CAo V CA 3. Assumptions • Well-mixed vessel • Constant density • Constant flow in • Constant temperature 4. Data • F = 0.085 m3/min , V = 2.1 m3 • (CAo)initial = 0.925 mole/m3 , DCAo = 0.925 mole/m3 • The reaction rate is rA = -kCA , with k = 0.04 min-1 5 Example 1. Isothermal CSTR F Formulate Model 1. F V dC A dt 2. FC Ao FC A kVC CA A Rationalize : dC A dt 3. V CAo Material balance: 1 CA F V C Ao where V F VK Degrees-of-freedom: One equation, one variable(CA), two external variables (F and CAo) and two parameters (V and k). Therefore the DOF is zero, and the model is exactly specified. 6 Example 1. Isothermal CSTR F F CAo V CA Analytical Solution C A ( C A ) init K p [ C Ao ( C A ) init ](1 e where K p F F Vk 0 . 503 , t / ) 12 . 4 min 7 Example 1. Isothermal CSTR 8 Empirical Modeling (Step Testing) Step Change Record m(t), % Final Control Element Process Sensor/ Transmitter c(t) , % first order plus dead time : C (s) M (s) K e t0 s s 1 Process Gain: K Dcs Dm 9 FOPDT Model Fit 1 : 10 FOPDT Model Fit 2 : 11 FOPDT Model Fit 3 : 3 2 ( t 2 t1 ) , t0 t2 12 Control Valve m(t) vp(t) Control Valve Action Cv(t) Control Valve Characteristics f(t) Control valve Capacity 13 Control Valve 1. Control Valve Action is selected based on safety consideration • • Fail-Closed (FC) or Air-to-Open (AO) : v Fail-Open (FO) or Air-to-Close (AC) : v dv p ( t ) dt dv p ( t ) dt v p (t ) m (t ) 100 v p (t ) 1 m (t ) 100 τv : Time constant of valve actuator (3-6 sec for pneumatic actuator) The gain of FC (AO) valve is positive The gain of FO (AC) is negative 14 Control Valve 2. Control Valve Characteristics • Linear • Quick-opening • Equal percentage C v ( t ) C v , max v p ( t ) C v ( t ) C v , max v p ( t ) 1 Rangeability parameter (50 or 100) 15 Control Valve 2. Control Valve Characteristics : How we must select the correct valve characteristics (Linear or Equal percentage) The correct selection requires a detailed analysis of the installed characteristics As a rule of thump: Choose a linear valve if at design conditions the valve is taking more than half of the total pressure drop (Δpv > 0.5 Δpo ). Choose an equal percentage valve if at design conditions the valve is taking less than half of the total pressure drop (Δpv < 0.5 Δpo ). Equal percentage valves are probably the most common ones. 16 Control Valve 3. Control Valve Capacity The control valve capacity is : The flow in U.S. gallons per minute (gpm) of water that flows through a valve at a pressure drop of 1 psi across the valve Liquid Flow: f (t ) C v (t ) Dpv Gf Where: f(t) = volume flow rate (gpm) Δpv = presuure drop across the valve (psi) Gf = specicific gravity 17 Control Valve 3. Control Valve Capacity Gas Flow • Subcritical flow: f s ( t ) 836 C v ( t ) C f • Critical flow: f s ( t ) 836 C v ( t ) C f Where: y 1 . 63 Dpv Cf p1 p1 ( y 0 . 148 y ) 3 for y 1 . 5 GT p1 for y 1 . 5 GT fs (t) = Gas volume flow at standard conditions,14.7 psia & 60 oF (scfh) Cf = Critical flow factor (0.6 – 0.95 , typically 0.9) p1 = Pressure at valve inlet (psia), T = Tempreture at valve inlet (oR) G = Gas specific gravity 18