Process Control

CHAPTER VI

BLOCK DIAGRAMS

ANDLINEARIZATION

Example:

Consider the stirred tank blending process. X2, w2

X, w1

X1, w1

AT AC

I/P

xsp

Control objective: regulate the tank composition x, by adjusting w2.

Disturbance variable: inlet composition x1

Assumptions: w1 is constant System is initially at steady-state Both feed and output compositions are dilute Feed flow rate is constant Stream 2 is pure material

Process

21

21

21

21

21

0

2

0.1

22111

22

11

21

1

0

)()(

)(1

wKxxdt

xd

ww

xxdt

xd

w

V

wxwxwdt

xdV

wxwxw

wwxwxdt

dxV

xwxwxwxwdt

dxV

xxV

wxx

V

w

dt

dx

wwwdt

dV

K

ww

1)(

)(

)(

1

1)(

)(

)(

)()()1)((

)()()())0()((

22

11

21

21

0

s

KsG

sW

sX

ssG

sX

sX

sWKsXssX

sWKsXsXXsXs

)(1 sX

)(2 sW )(sX

1

1

s

12

sK

Measuring Element

Assume that the dynamic behavior of the composition sensor-transmitter can be approximated by a first-order transfer function;

when, can be assumed to be equal to zero.

1)(

)(

s

K

sX

sX

m

mm

mm ,

)(sX mX

mK

Controller

ss

KsE

sP

sKsE

sP

sK

sE

sP

KsE

sP

DI

C

DC

IC

C

11

)(

)(

1)(

)(

11

)(

)(

)(

)(proportional

proportional-integral

proportional-derivative

proportional-derivative-integral

Current to pressure (I/P) transducer

Assuming a linear transducer with a constant steady state gain KIP.

IPt KsP

sP

)(

)(

)(sP )(sPtIPK

Control Valve

Assuming a first-order behavior for the valve gives;

1)(

)(2

s

K

sP

sW

v

v

t

)(sX d

)(sX u

)(sX sp

)(~

sX sp

Change in exit composition due to change in inlet composition X´

1(s)

Change in exit composition due to a change in inlet composition W´2(s)

Set-point composition (mass fraction)

Set-point composition as an equivalent electrical current signal

Linearization A major difficulty in analyzing the dynamic

response of many processes is that they are nonlinear, that is, they can not be represented by linear differential equations.

The method of Laplace transforms allows us to relate the response characteristics of a wide variety of physical systems to the parameters of their transfer functions. Unfortunately, only linear systems can be analyzed by Laplace Transforms.

Linearization is a technique used to approximate the response of non linear systems with linear differential equations that can than be analyzed by Laplace transforms.

The linear approximation to the non linear equations is valid for a region near some base point around which the linearization is made.

Some non linear equations are as follows;

)()(

)(

)()()(/

0

4

tpktpf

ektTk

tATtTqtRTE

A linearized model can be developed by approximating each non linear term with its linear approximation. A non linear term can be approximated by a Taylor series expansion to the nth order about a point if derivatives up to nth order exist at the point.

The Taylor series for a function of one variable about xs is given as,

xs is the steady-state value.

x-xs=x’ is the deviation variable. The linearization of function consists of only the

first two terms;

Rxxdx

Fdxx

dx

dFxFxF sxsxs ss

22

2

)(!2

1)()()(

)()()( sxs xxdx

dFxFxF

s

Examples:

2

20

)(0

)(0

21

21

21

)()()(

)()(

)()(

)()()(

)(

)(2

1)(

)(

s

ss

s

RTE

s

TT

tRTE

s

sTs

tRTE

sss

RT

ETkTktTk

RT

EekTktTk

ekdt

dTktTk

TtTdT

dkTktTk

ektTk

xxxxxF

xxF

s

s

s

Example: Consider CSTR example with a second order

reaction.

)(2)(

)(2

)(

2

0

22

2

0

2

AsAAsAAAA

AsAAsAsA

AAAA

AA

CCVkCVkCCCFdt

dCV

CCCCC

VkCCCFdt

dCV

kCr

Mathematical modelling for the tank gives;

The non linear term can be linearized as;

The linearized model equation is obtained as;

Example:

Considering a liquid storage tank with non linear relation for valve in output flow rate from the system;

hR

qdt

hdA

h

C

R

hh

Cq

dt

hdA

qqq

hhh

hCqdt

dhA

i

s

v

s

vi

siii

s

vi

1

2

1

2

,