mathematical modelling of chemical processes

8/9/2019 mathematical modelling of chemical processes

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CHE334 Instrumentation andProcess Control

Lecture 5Chapter 4 Mathematical Modeling of

behavior of Chemical Process

By Dr. Maria MustafaDepartment of Chemical Engineering


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Lecture Content

Mathematical Modeling of the behavior

of Chemical Process

Development of Mathematical Model

2


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Modeling the Dynamics and static

Behavior of Chemical Process

Development of Mathematical

representation of the chemical andphysical phenomena.


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Modeling the Dynamics and static

Behavior of Chemical Process

It requires concepts of Thermodynamics

Kinetics

Transport phenomena

Its is a prerequisite to the design of its controller

Generally two approaches are used to analyze

of chemical process behavior with time

Experimental Approach

Theoretical Approach


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Need of Mathematical Model of

Chemical Process In simplified way, to analyze how process

reacts to various input in the mathematical

form to provide it to the control designer.

Lets take Example of Feedforward controller

for system

How to change the manipulated variable to

cancel the effect of disturbances ?

Output = f ( disturbances )Output = f ( manipulated variable )

Equating two equations we have

f ( disturbances ) - f(manipulated variable ) 0

MathematicalModelofProcess


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State Variables and State Equations

State Variables : The characterizing variables that

define the fundamental quantities such as mass,

energy and momentum. Examples :Density, T, P,

Concentration and Flow rate

These state variables define the behavior of aprocessing systems or we can say state of system.

The equation that relates the state variables

(dependent variable) to various independentvariables by some conservation principle applied

on the fundamental quantities are called state

equation. It constitute the mathematical model.


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General Principle of Conservation The principle of conservation of a quantity S states

that:

=


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The quantity S can be on of the fundamental

quantitates

Total mass

Mass of individual components

Total energy

Momentum

1

2

3

N

1

2

3

N Ws

Q

System


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For total mass balance

() = : :

Mass Balance on component

() = () = : :

Total enegry balance

() = ( + + )=

:

:


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The above three equations are set of

differential equation that show how state

variables changes with time. This determinesthe dynamic of variables.

If the state variable do not change with time

then process is at steady state i.e S per unittime is zero and the equations results in a set

of algebraic equation.


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Mathematical Model of Heat Tank

Assumptions

The momentum of

the heater remain

constant. Tank does not move

so kinetic and

potential energy is

zero.

For liquid systen

dU/dt =dH/dtFs

Fi, Ti

h

F, T

T

Q


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Applying mass and energy balance around the

heater tank systemFor total mass balance

()

=

() =

Total energy balance

() =

( + + )=

+


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() = +

Rate of change of Enthalpy = = = Putting the H value in above equation we have

[ ] == + ()

= ( ) +


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Summarize

State Equation()

=

() = ( ) + State Variables : h,T

Output Variables : h,TInput Variables

Disturbances : Ti, Fi

Manipulated Variables = Q,F, Fi


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Static and Dynamic behavior of heated

stirred tank Suppose heated tank system is at steady state ( nothing is changing ) After to time, the Ti change by 10 %, T at output will also change and

come to a new steady state vale. The new value of steady state can

be calculated by using initial Condition that T(t=0)=Ts ( known )

to Time to Time

Ts Ts

Temperature

Temperat

ure

New steady state

New steady state


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Suppose that heated tank system is at steady state ( nothing is

changing )

After to time, the Fi change by 10 %, T and h both at output will

also change and come to a new steady state vale. The new valueof steady state can be calculated by using initial Condition that

T(t=0)=Ts ( known) and h(t=0)=hs ( known) .

to Time to Time

Ts hs

Temperature

Heighto

ftank

New steady state

New steady state


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Additional Elements of the

Mathematical Models

In addition to balanceequations, additionalthermodynamic equilibria,

reaction rates, transportrates relationships for heat,mass, momentum etc arerequired. These

relationships are needed tocomplete mathematicalmodeling and can beclassified as follows

1. Transport Rate

Equations

2. Kinetic rate

Equations

3. Reaction and

phase equilibria

relationships

4. Equation of state


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d


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Dead TimeWhenever an input variable of a system changes,

there is a time interval ( short or long) during which

no effect is observed on the output variable of the

system. This time interval is called dead time,

transportation lag, or pure delay, or distance-

velocity lag.

T

t

Curve A

Curve B

td

Fl f i ibl i li id


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Flow of incompressible non-reacting liquid

through a thermally isolated pipe

ATin Tout

L

Delay Response of exit temperature to inlet Temperature


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Mathematical model development of

Continuous stirred tank Reactor

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cA, Ti, Fi

Tci, Fc

Tco, Fc

cAT, F

A + B ---> C ( exothermic reaction)

Chemical Process Operation


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Next Lecture

Modeling Considerations for Chemical

Process

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mathematical modelling of chemical processes