Process Control

Report
Process Control
CHAPTER VI
BLOCK DIAGRAMS
AND
LINEARIZATION
Example:

Consider the stirred tank blending process.
X1, w1
X2, w2
I/P
AT
X, w1
AC
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

dV
1

( w1  w2  w)
dt

dx
w
w
 1 ( x1  x )  2 ( x2  x )
dt
V
V
dx
V
 w1 x1  w1 x  w2 x2  w2 x


 
dt
w
w
1.0
0
V
dx
 wx1  wx  w2
dt
0  w x1  w x  w2
dx
V
 w x1  w x  w2
dt
V dx
1
 x1  x 
w2
w dt
w




K
dx
 x1  x  Kw2
dt
 ( sX ( s)  
X (0))  X 1( s)  X ( s)  KW2( s)

0
X ( s)(s  1)  X 1( s)  KW2( s)
X ( s)
1
 G1 ( s) 
X 1( s)
s  1
X ( s)
K
 G2 ( s) 
W2( s)
s  1
X1(s)
1
s  1
W2(s)
X (s )
K2
s  1
Measuring Element
Assume that the dynamic behavior of the
composition sensor-transmitter can be
approximated by a first-order transfer function;

X m ( s )
Km

X ( s )  m s  1
when,    m , m can be assumed to be equal to
zero.
X (s)
Km
X m

Controller
P( s )
 KC
E ( s)
proportional

P( s )
1 

 KC 
1


E ( s)

s
I


P( s )
 K C 1   D s 
E ( s)
proportional-integral
proportional-derivative


P( s )
1
 KC 
1
 Ds 


E ( s)

s
I


proportional-derivative-integral

Current to pressure (I/P) transducer
Assuming a linear transducer with a constant steady
state gain KIP.
Pt( s )
 K IP
P( s )
P(s )
K IP
Pt(s)

Control Valve
Assuming a first-order behavior for the valve
gives;
Kv
W2( s )

Pt( s )
 vs 1
X d (s)
Change in exit composition due to change in inlet
composition X´1(s)
X u (s)
Change in exit composition due to a change in inlet
composition W´2(s)
X sp (s)
~
X sp ( 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;
qT (t )   AT (t )
4
k T (t )  k0 e
 E / RT ( t )
f p(t )  k p(t )

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,
2
dF
1d F
F ( x )  F ( xs ) 
x s ( x  xs ) 
2
dx
2! dx



( x  xs )  R
2
xs
xs is the steady-state value.
x-xs=x’ is the deviation variable.
The linearization of function consists of only the
first two terms;
dF
F ( x )  F ( xs ) 
dx
xs
( x  xs )
Examples:
F ( x)  x
1
F ( x )  xs
2
1
2
1  12
 xs
( x  xs )
2
k T (t )  k 0 e
E
RT ( t )
dk
k T (t )  k (Ts ) 
Ts T (t )  Ts 
dT
E
d 
RT ( t ) 
k T (t )  k (Ts ) 
k
e
0

dt 

 T Ts
k T (t )  k (Ts )  k 0 e
E
RTs
E
2
RTs
E
k T (t )  k (Ts )  k (Ts )
2
RTs
Example:

Consider CSTR example with a second order
reaction.
r A   kC A
2
Mathematical modelling for the tank gives;
dCA
2
V
 F (C A 0  C A )  VkC A
dt
The non linear term can be linearized as;
C A  C As  2C As (C A  C As )
2
2
The linearized model equation is obtained as;

dCA
2
V
 F (C A 0  C A )  VkC A  2VkC As (C A  C As )
dt

Example:

Considering a liquid storage tank with non linear
relation for valve in output flow rate from the
system;
dh
 qi  Cv
dt
h  h  hs

qi  qi  qi , s
A
h
Cv
d h

A
 qi 
h
dt
2 hs
Cv
1

R
2 hs
d h
1

A
 qi 
h
dt
R

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