Mathematical Modelling for Process Control

Dr. Haresh Manyar

To develop a control system for a chemical process which will

guarantee that the operational objectives of our process are

satisfied in the presence of ever changing disturbances.

Experimental approach:

- physical equipment in place

- change input variables to get data for output variables

- costly and time consuming

- can not be done before chemical process construction

Theoretical approach:

- we need a set of mathematical equations (differential,

algebraic) whose solution yields the dynamic or static

behaviour of chemical process.

State variables and state equations for a chemical process

To characterize a system and understand its behaviour, we need:

A set of fundamental dependent quantities to define the

natural state of a given system

a set of equations in the above variables-which will describe

how the state of the system changes with time

- Mass

- Energy

- Momentum

Convenient measurements:

Density

Concentration

Temperature

Flowrate

State variables

State equations are derived from the Principle of Conservation of

Fundamental Quantities.

The quantity S can be one of the following fundamental quantities:

- total mass,

- mass of individual components

- total energy, and

- momentum.

period time

system the

withinconsumed

S ofAmount

period time

system the

withingenerated

S ofAmount

period time

system theofout

S of Flow

period time

system in the

S of Flow

period time

system awithin

S ofon Accumulati

outletj

jjinleti

iiFF

dt

V)d(

Balance Mass Total

rVFcFccn

outletjjA

inletiiA ji

dt

V)d(

dt

)d(

Acomponent aon Balance Mass

AA

gjoutletj

jiinleti

iWQhFhF

Eji

dt

P)Kd(U

dt

d

BalanceEnergy Total

Commonly used forms of balance equations

(a)Total mass

Assuming constant density (independent of temperature),

FFdt

dhA

i

State Equations for the Stirred Tank Heater

Lets apply the conservation principle

on the total mass and total energy

FFdt

Ahii

time

system theofout

mass of Flow

time

system in the

mass of Flow

time

mass of

on Accumulati

piiC

QTTF

dt

dTAh

)(

(b) Total Energy

Where Cp is the specific heat capacity of liquid in tank and Tref is the reference

temperature where specific enthalpy of liquid is assumed to be zero.

Rearranging the above equation gives following state equation:

QTTFCTTCFdt

TTAhCdrefprefipi

refp

)()()]([

time

steamby

suppliedenergy

time

energy total

ofoutput

time

energy total

ofInput

time

energy totalof

on Accumulati

State Equations for the Stirred Tank Heater

State Equations for the Stirred Tank Heater

p

iiC

QTTF

dt

dTAh

)(

FFdt

dhA

i

State equations

State variables: h,T

output variables: h,T (both measured)

Input variables:

- disturbances: Ti, Fi

- manipulated variables Q, F (for feedback control)

Fi (for feedforward control)

State equations along with state variables constitute mathematical model

(left hand side indicates rate of accumulation of mass or energy over time)

Analysis of the

dynamic and static behaviour

of the stirred tank heater

Assuming that the tank heater is at steady-state,

the situation is described as follows:

Rate of accumulation is set zero.

0)(,,

p

s

ssisiC

QTTF

0

,

ssiFF and

Case 1: disturbance: the inlet temperature decreases by 10%.

liquid level in the tank remains same, h=hs. (check state equation of mass)

temperature of liquid in the tank, T will start decreasing with time.

We notice that after a certain time, the tank

heater again reaches steady state conditions.

p

iiC

QTTF

dt

dTAh

)(

FFdt

dhA

i

steady-state situation:

0)(,,

p

s

ssisiC

QTTF

0

,

ssiFF and

Case 2: disturbance: the inlet Flowrate decreases by 10%.

liquid level in the tank decreases with time

temperature of liquid in the tank, T will start increasing with time.

p

iiC

QTTF

dt

dTAh

)(

FFdt

dhA

i

Again, we notice that after a certain time, the tank heater again reaches

steady state conditions.

Dead-Time ( transportation lag or pure delay or distance-velocity lag):

So far, it has been assumed that whenever a change takes place in

one of the input variables (disturbances, manipulated variables), its effect

is instantaneously observed in the state variables and the outputs.

This is contrary to the physical experience, which indictates that

whenever an input variable of a system changes, there is a time interval

(short or long) during which no effect is observed on the system itself.

This time interval is called Dead time.

)t-(tT(t)T

seconds.

.

dinout

avav

dU

L

UA

LA

rateflowvolumetric

pipetheofvolumet

Degrees of Freedom Analysis

To simulate a process, we must first make sure that the

out-put variables of model equations (variables on left

hand side) can be solved in terms of the in-put variables

(variables on the right hand side).

In order for the model to have a unique solution, the

number of unknown variables must equal the number of

independent model equations.

f = (number of variables)- (number of equations)

p

iiC

QTTF

dt

dTAh

)(

FFdt

dhA

i

Degrees of Freedom Analysis

for Process Control

The degrees of freedom NF is the number or process variables that must be specified in order to be able to determine the

remaining process variables.

If a dynamic model of the process is available, NF can be determined from a relation that was introduced earlier,

where NV is the total number of process variables, and NE is

the number of independent equations.

EVF NNN

For process control applications, it is very important to determine the

maximum number of process variables that can be independently

controlled, that is, to determine the control degrees of freedom, NFC:

Definition. The control degrees of freedom, NFC, is the number of

process variables (e.g., temperatures, levels, flow rates, compositions)

that can be independently controlled.

In order to make a clear distinction between NF and NFC, we will refer to NF as the model degrees of freedom and NFC as the control

degrees of freedom.

Note that NF and NFC are related by the following equation,

where ND is the number of disturbance variables (i.e., input variables that

cannot be manipulated.)

DFCF NNN

General Rule. For many practical control problems, the control

degrees of freedom NFC is equal to the number of independent material

and energy streams that can be manipulated.

Figure : Two examples where all three process streams

cannot be manipulated independently.

Example: Determine NF and NFC for the steam-heated, stirred-tank

system modeled by Eqs. 1-2. Assume that only the steam pressure Ps

can be manipulated.

p

iiC

QTTF

dt

dTAh

)(

FFdt

dhA

i

Solution

In order to calculate NF from Eq. 1, we need to determine NV and

NE. The dynamic model contains two equations (NE = 2) and six

process variables (NV = 5): Fs, Ti, Fi, F and T.

Thus, NF = 5 2 = 3.

If the feed temperature Ti and flow rate Fi are considered to be disturbance variables, ND = 2 and thus NFC = 1 from Eq. (10-2).

It would be reasonable to use this two degrees of freedom to control temperature T by manipulating steam pressure, Fs and .