mathematical modeling of chemical processes · performance specifications of a second-order system...

Mathematical Modeling of

Chemical Processes

Part II

Continuous Stirred-Tank Reactor

(CSTR)

Continuous Stirred-Tank Reactor

(CSTR)

Assumptions

– neglected heat capacity of inner walls of

the reactor, constant density and specific

heat capacity of liquid,

– constant reactor volume, constant overall

heat transfer coefficient, and

– constant and equal input and output

volumetric flow rates.

– the reactor is well-mixed.

Control loop for the Stirred

Heating Tank

Mathematical model of a

thermocouple

q0

qj

qiqi

q0

a) bare thermocouple b) thermocouple with protect jacket

Blending system Control Method

Modeling the pneumatic control

valve

p1

p2

q

Element Time Responses

• first order element

– Transfer function

)()()(

tKxtydt

tdy

1)(

)()(

s

K

sX

sYsG


• Step input x(t) = MU(t)

– When M =1 (unit step input),

s

M

s

KsXsGsY

1)()()(

t

eKMs

M

s

Kty 1

1)(

1L

632.01)(

t

t

eKty


• Second order element

)()()(

2)(

2

22 txty

dt

tdy

dt

tydmm

12

1

)(

)()(

22

sssX

sYsG

mm

2

00

2

2

0

2

ss

Second order element

• Given a step input x(t) = MU(t),

• Use notations in Chapter 4, and let M=1

(unit step)

s

M

sssXsGsY

2

00

2

2

0

2)()()(

22

2

2)(

nn

n

ssssC


• Factoring the denominator

• Case A damping ratio equals unity = 1

11)(

22

2

nnnn

n

ssssC

2

2

)(n

n

sssC


• Partial fraction expansion

• The time domain responses of output

nns

K

s

K

s

KsC

3

2

21)(

nn

n

sss

112

tt

nnn etetc

1)(


• Case B damping ratio greater than

unity > 1

11)(

2

3

2

21

nnnn s

K

s

K

s

KsC

1

112

1

1121)(

2

122

2

122

nnnn ssssC



• Case C damping ratio less than unity < 1

Let

t

t

n

n

e

etc

1122

1122

2

2

112

1121)(

sin1 thereforeand ,cos2


• Partial fraction expansion


12

12

1sin2

1sin2

1)(

nn

j

nn

j

jsj

e

jsj

e

ssC

te

tc n

tn

2

21sin

11)(

Step responses of a second

order element over damped ≥1

Step responses of a second order

element under damped <1

Second order underdamped

response specifications

Performance specifications of a

second-order system

• Let

• We have

• therefore

0)(

dt

tdc

2 , ,012

tn

2

2max

1exp1

sin1

1)(

tnetc


second-order system

• Peak time Tp, the time required to reach

the first peak

• Percent overshoot, %OS is the amount

that the waveform overshoots the final

steady-state

21

n

pT

21

exp%

OS


second-order system

• Settling time Ts , the time required for damped oscillations to reach and stay within±2% of the steady-state (final) value

• Rise time Tr is the time required for the waveform to go from 0.1 to 0.9 of the final value

n

sT

4

Location of the Roots in the

s-plane and the Transient

Response

s-plane

= 0

0 < < 1

≥ 1

s

s < 0s > 0

Element Time Response

• Proportional element

It is a step with KM as its magnitude

• Integral element

It is a ramp at the slop of KM/i

• Differential element

It is a impulse

• Delay element

It is a step after a time delay of

Development of Empirical

Dynamic Models from Step

Response Data

Higher order system and dead time

Higher order system and dead

time

Approximate using first-order-

plus-time-delay model

• The response attains 63.2% of its final response at one time constant (t = q )

• The line drawn tangent to the response at maximum slope (t = q) intersects the 100% line at (t = q ).

• K is found from the steady state response for an input change magnitude M. The step response is essentially complete at t = 5.

Approximate using first-order-

plus-time-delay model

q

Inflection point

Time Constant

Inflection point

Time Constant

Sundaresan and Krishnaswamy’s

Inflection point of the process reaction

curve is too arbitrary and difficult to

determine when data is noisy

• Step 1 take 35.3% response time t1

• Step 2 take 85.3% response time t2

• Substitute into the equations

12

21

67.0

29.03.1

tt

tt

q

Time Constant

35.3%

85.3%

t1 t2

63.2%

Second-order Model

In general, a better approximation to an

experimental step response can be

obtained by fitting a second-order

model to the data

The larger of the two time constants, 1, is called the dominant time constant

11)(

21

ss

KsG

Second-order Model

• Two limiting cases:

– 1/ 2 = 0, where the system becomes first order, and,

– 1/ 2 = 1, the critically damped case ( =1)

• Determine t20 and t60 from the step response.

• Findζand t60 / from Figure 14.

• Find t60 / from Figure 14 and then calculate (since t60 is known).

Second-order Model

t20 t60

Second-order Model

Modeling Second-order Through

Least Square Fit

• For unit step input

• Assume c(t) takes the form,

where css is the final-value of c(t)

)()(

)(

)()(

1ssC

sC

sR

sCsG

s

btat

ss eKeKctc 21)(


Least Square Fit

• Step 1: Least square fit first term

The intercept is log K1

The slope is 0.4343a

at

ss eKctc 1)(

atK

eatKctc ss

4343.0log

loglog)(log

1

1


Least Square Fit

• Step 2: Subtract the line fitted from the

experimental data

The intercept is log K2

The slope is 0.4343b

btKeKctc at

ss 4343.0loglog)(log 21


Least Square Fit

• Step 3: Adjustment to have c(0) = 0,

– Let

– Now we have,

adKK

adjustment

2

1 11

adKK

adKK

2

'

2

1

'

1

btat

ss eKeKctc '

2

'

1)(

Time Responses Using State

Variable Method

• For non zero second order system,

• Divide s2 on both numerator and

denominator,

22

2

2)(

)(

nn

n

sssR

sC

221

22

222

22

21

)2()(

)(

ss

s

sss

s

sR

sC

nn

n

nn

n


Variable Method

• Define,

• Therefore,

22121

)()(

ss

sRsE

nn

)()( 22 sEssC n

)()(2)()( 221 sEssEssRsE nn


Variable Method

The state-variable signal-flow graph,

n2ζω

2

nω

R(s)

X2(s) X1(s)

C(s)2

nωS-1S-11

S-1 S-1

x2(t0) x1(t0)

E(s)

• Example

• Laplace transform,

• then

)()(3)(4)( trtctctc

21

2

2

341

34

1

)(

)(

ss

s

sssR

sC

21 341

)()(

ss

sRsE

)(3)(4)()(

)()(21

2

sEssEssRsE

sEssC

4

3

R(s)

X2(s) X1(s)

C(s)2

nωs-1s-11

s-1 s-1

x2(0) x1(0)

E(s)

• Quiz 15 min

• Use Mason’s theory to find X1(s) and X2(s)

mathematical modeling of chemical processes · performance specifications of a second-order system...

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