بسم الله الرحمن الرحيم

Advanced ControlLecture 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)

1Lecturer: M. A. Fanaei Ferdowsi University of Mashhad

01_5Modeling (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 goalsSpecific design decisionsNumerical valuesFunctional relationshipsRequired accuracy

2. Prepare informationSketch process and identify systemIdentify variables of interestState assumptions and data

3. Formulate modelConservation balancesConstitutive equationsRationalizeCheck degrees of freedomDimensionless form

4. Determine SolutionAnalyticalNumerical

5. Analyze resultsCheck results for correctness Limiting and approximate answers Accuracy of numerical methodInterpret results Plot solution Characteristic behavior Relate results to data and assumptions Evaluate sensitivity

6. Validate modelSelect key values for validationCompare with experimental results

Compare with results from more complex model

3

Example 1. Isothermal CSTR

F

CAo

F

V CA

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/m3

3. When the concentration reaches 0.83 mole/m3, 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

Example 1. Isothermal CSTR

Prepare Information …

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

F

CAo

F

V CA

5

Example 1. Isothermal CSTR

AAAoA kVCFCFC

dtdCV

F

CAo

F

V CA

Formulate Model

1. Material balance:

2. Rationalize :

VKFVC

VFC

dtdC

AoAA

where1

3. 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

CAo

F

V CA

Analytical Solution

min4.12,503.0

)1]()([)( /

VkFFKwhere

eCCKCC

p

tinitAAopinitAA

7

Example 1. Isothermal CSTR

8

Empirical Modeling (Step Testing)

Final Control Element

Process Sensor/ Transmitter

Step Change Record

m(t), % c(t) , %

1)()(

:

0

seK

sMsC

timedeadplusorderfirst

st

Process Gain:

mcK s

DD

9

FOPDT Model

Fit 1:

10

FOPDT Model

Fit 2:

11

FOPDT Model

Fit 3: 2012 ,)(23 tttt

12

Control Valve

Control Valve Action

Control Valve Characteristics

Control valve Capacity

m(t) vp(t) Cv(t) f(t)

13

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) :

100)()(

)( tmtvdt

tdvp

pv

100)(1)(

)( tmtvdt

tdvp

pv

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

)()( max, tvCtC pvv

1)(max,)( tv

vvpCtC

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 :

Where: f(t) = volume flow rate (gpm)

Δpv = presuure drop across the valve (psi)

Gf = specicific gravity

f

vv G

ptCtf D )()(

17

Control Valve

3. Control Valve CapacityGas Flow

• Subcritical flow:

• Critical flow:

Where:

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

5.1)148.0()(836)( 31 yforyyGTpCtCtf fvs

1

63.1pp

Cy v

f

D

18

5.1)(836)( 1 yforGTpCtCtf fvs