process control chp 6
DESCRIPTION
Process controlTRANSCRIPT
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
,