1. بسم الله 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) 1 Lecturer: M. A. Fanaei Ferdowsi University of Mashhad

2. 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) black box (empirical) Linear non-linear Static dynamic Distributed lumped Time domain frequency domain Continuous discrete For further reading refer to : R offel & Beltlem, “Process dynamics and control”, Wiley, 2006 2

3. A modeling procedure 1. Define goals Specific design decisions Numerical values Functional relationships Required accuracy 2. Prepare information Sketch process and identify system Identify variables of interest State assumptions and data 3. Formulate model Conservation balances Constitutive equations Rationalize Check degrees of freedom Dimensionless form 4. Determine Solution Analytical Numerical 5. Analyze results Check results for correctness Limiting and approximate answers Accuracy of numerical method Interpret results Plot solution Characteristic behavior Relate results to data and assumptions Evaluate sensitivity 6. Validate model Select key values for validation Compare with experimental results Compare with results from more complex model 3

4. Example 1. Isothermal CSTR F CAoF CAo F VCACA Define Goals 1.Dynamic response of a CSTR to a step in the inlet concentration. 2.The reactant concentration should never go above 0.85 mole/m 3 3.When the concentration reaches 0.83 mole/m 3, would a person have enough time to respond? What would a correct response be? 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

5. Example 1. Isothermal CSTR Prepare Information … 3.Assumptions Well-mixed vessel Constant density Constant flow in Constant temperature 4.Data F = 0.085 m 3 /min, V = 2.1 m 3 (C Ao ) initial = 0.925 mole/m 3,  C Ao = 0.925 mole/m 3 The reaction rate is r A = -kC A, with k = 0.04 min -1 F CAoF CAo F VCACA 5

6. Example 1. Isothermal CSTR F CAoF CAo F VCACA Formulate Model 1.Material balance: 2.Rationalize : 3.Degrees-of-freedom: One equation, one variable(C A ), two external variables (F and C Ao ) and two parameters (V and k). Therefore the DOF is zero, and the model is exactly specified. 6

7. Example 1. Isothermal CSTR F CAoF CAo F VCACA Analytical Solution 7

8. Example 1. Isothermal CSTR 8

9. Empirical Modeling (Step Testing) Final Control Element Process Sensor/ Transmitter Step Change Record m(t), %c(t), % Process Gain: 9

10. FOPDT Model Fit 1 : 10

11. FOPDT Model Fit 2 : 11

12. FOPDT Model Fit 3 : 12

13. Control Valve Control Valve Action Control Valve Characteristics Control valve Capacity m(t)m(t)vp(t)vp(t)Cv(t)Cv(t)f(t)f(t) 13

14. Control Valve 1.Control Valve Action is selected based on safety consideration Fail-Closed (FC) or Air-to-Open (AO) : Fail-Open (FO) or Air-to-Close (AC) : τ v : 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

15. Control Valve 2.Control Valve Characteristics Linear Quick-opening Equal percentage Rangeability parameter (50 or 100) 15

16. 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 (Δp v > 0.5 Δp o ). Choose an equal percentage valve if at design conditions the valve is taking less than half of the total pressure drop (Δp v < 0.5 Δp o ). Equal percentage valves are probably the most common ones. 16

17. 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: Where: f(t) = volume flow rate (gpm) Δp v = presuure drop across the valve (psi) G f = specicific gravity 17

18. Control Valve 3.Control Valve Capacity Gas Flow Subcritical flow: Critical flow: Where: f s (t) = Gas volume flow at standard conditions,14.7 psia & 60 o F (scfh) C f = Critical flow factor (0.6 – 0.95, typically 0.9) p 1 = Pressure at valve inlet (psia), T = Tempreture at valve inlet ( o R) G = Gas specific gravity 18