PCT 301T Modeling(1)

8/22/2019 PCT 301T Modeling(1)

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PCT 301T

MODELING

Modelling is used to study the dynamic behaviour, process design,

model-based control, optimization and predictions of the processes.

Modelling principles

Dynamics models of chemical processes consist of ODE (ordinary

differential equations) and / partial differential equations (PDE), plus

related algebraic equations.

For process control problems, dynamic models are derived using

unsteady state conservation laws.


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A systematic approach for developing dynamic models

1. State the modelling objectives and the end use of the model.2. Draw the schematic diagram of the process and label all

process variables.

3. List all the assumptions involved in developing the model.

4. Write appropriate conservation equation (mass, component,

energy, and so forth).5. Introduce equilibrium relation and other algebraic equation

(from thermodynamics, transport phenomena, chemical kinetics

etc).

6. Perform degree of freedom.

7. Simplify the model. (Grouping of like terms).8. Classify the inputs as the disturbance or manipulated

variables.


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Conservation laws

Theoretical models of chemical processes are based onconservation laws such as the conservation ofmass and energy .

Conservation of Mass

Rate of mass accumulation = rate of mass in

rate of mass out

Conservation of Component i

Rate of = rate of rate of + rate of

mass mass in mass out mass

accumulation component i produced


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Conservation of Energy

Rate of = rate of rate of + net rate of

energy energy in by energy out by heat addition

accumulation convection convection to the system

+ net rate of work performed on the system

by the surrounding


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Uses of Mathematical Modeling

To improve understanding of the process

To optimize process design/operating conditions

To design a control strategy for the process

To train operating personnel


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Developm ent of Dynam ic Models

Il lustrat ive Example: A Blending Process

FIRST ORDER SYSTEM RESPONSE FOR UNSTEADY STATE

PROCESS

Example 1

Liquid - level system (Mass storage)

Object ive is to measure the height level (h) of the system and

determine the transfer function.


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Assumpt ion made: out flow (Q0) is linearly related to the hydrostatic

head of the liquid level through the resistant (R)

Conservat ion equat ion :Mass

Accumulation = Input - output

Degree of freedom analys is: 4 -3 = 1

2 input and 2 output

Q0

h(t)

Rh

Qi

x(t)


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Accumulation = Input output

outin mmdt

dM

...Eq(1)

0

)( QQ

dt

Ahdi ..Eg(2)

R

hQ

dt

Adhi

. ...Eq(3)

RQhdtdhRA i

..Eq(4)


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Apply Laplace transformation in Eq (4)

And find the transfer function.

11)(

)(

s

K

RAs

R

sX

sH p

...Eq(5)

Apply the inverse to determine the process response

DO EXAMPLE (B) FROM YOUR NOTES

Process Gain (Kp)

Process time

(tau)


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Example 2

Consider a perfectly mixed stirred-tank heater; with a single feed

streams and a single product stream, as shown below. Assuming that

the flow rate and temperature of the inlet streams can vary, that the

tank is perfectly insulated, and the rate of heat, Q added per unit timecan vary, heat losses are negligible, density and heat capacity of the

liquid are assumed to be constant. Develop a mathematical model to

find the tank liquid temperature as a function of time and its time

response when the inputs are subjected to the step change.


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Fig: 1 Stirred tank heating process with constant volume.

Object ive is to measure the temperature (T) of the system anddetermine the transfer function.


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Assumpt ion made: the outlet and the inlet flow are equal, volume,

density and the heat capacity of the liquid are constant and heat losses

are negligible.Conservat ion equat ion :Energy

Rate of = rate of rate of + net rate of

energy energy in by energy out by heat addition

accumulation convection convection to the system

+ net rate of work performed on the system

by the surrounding


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WQTfCTfCdt

dHpip .Eq(1)

QTTfCdt

dTCV ipp ...Eq(2)

QTfCfCpTdt

dTCV

ipp

.Eq(3)

QfC

TTdt

dT

fC

CV

p

i

p

p 1

...Eq(4)


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Q

fC

TT

dt

dT

fC

CV

p

i

p

p

11

Apply the Laplace transformation and find the transfer function

So,Q

sf

V

fCsT

sf

VsT

p

i

1

1

)(

1

1)(

...Eq(5)

Process time

(tau)

Process Gain (Kp2)

Process Gain (Kp1)


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)(1

)(1

)(21

sQs

KsT

s

KsT

p

i

p

...Eq(6)

Two transfer functions

T.F 1 1)()( 1

sK

sTsT p

i

T.F 2 1)(

)( 2

s

K

sQ

sT p


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An unsteady-state mass balance for the blending system:

rate of accumulation rate of rate of(2-1)

of mass in the tank mass in mass out

1 2

(2-2)

d Vw w w

dt


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The corresponding steady-state model

The Blending Process Revisited

For constant density:

1 2

1 1 2 2

0 (2-4)

0 (2-5)

w w w

w x w x wx


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Equation 2-7 can be simplified by expanding the accumulation term using the

chain rule for differentiation of a product:

Substitution of (2-7) into (2-8) gives:

Substitution of the mass balance in (2-6) for in (2-9) gives:


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After canceling common terms and rearranging (2-6) and (2-10), a more

convenient model form is obtained:


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Typical question

A stirred-tank heating process described by)(

1

1)('

1

/1)(' ' sT

s

w

msQ

s

w

m

wCsT i

, is used to

preheat a reactant containing a suspended solid catalyst at a constant flow rate of 1000

kg/h. the volume in the tank is 2 m3, and the density and specific heat of the suspended

mixture are, respectively, 900 kg/m3

and 1 cal/g0C. The process initially is operating

with inlet and outlet temperature of 100 and 1300C. The following questions concerning

process operations are posed:


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(a) What is the heater input at the initial steady state and the value of K and

TAU?

(b) If the heater input is increased by 30%, how long will it take for the tank

temperature to achieve 99% of the final temperature change?

Assume the tank is at its initial steady state. If the inlet temperature in increased

suddenly from 100 to 1200C, how long will it take before the outlet temperature

changes from 130 to 1350C.

PCT 301T Modeling(1)

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