Process Control and Instrumentation by Prof. a. K. Jana Department of Chemical Engineering Indian...

8/18/2019 Process Control and Instrumentation by Prof. a. K. Jana Department of Chemical Engineering Indian Institute of Te…

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Process Control and Instrumentation

Prof. A. K. Jana

Department of Chemical Engineering

Indian Institute of Technology, Kharagpur

Lecture - 1Introduction to Process Control

The name of this course is Process Control and Instrumentation. Initially, we will cover

the process control and then the instrumentation course.

(Refer Slide Time: 01:02)

So, the first topic is introduction to process control, this topic we will cover first. Now, in

chemical engineering, we have a number of chemical units; for example reactors, for

example distillation column, like pump, compressor, etcetera, these are the different units

which are extensively used in chemical engineering. Now, to constitute a chemical plant,

we need to assemble few of these units if we assemble few of these units then we can

constitute a chemical plant.

Now, what is the objective of a chemical plant suppose, we have a plant now, it receives

raw material so input to this plant is, raw material and output is product. So, basically,

the plant receives raw material using different available sources of energy, the plant

produces products in the most economical way, this is a objective of a plant. It receives

raw material, it uses available sources of energy then it produces product in the most

economical way, this is the objective.


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Now, to meet this objectives, we need to meet some requirements porter these

requirements, the primary requirement is safety, say for example, a reactor which is

design to operate within 100 psig pressure. We have one reactor, which is designed to

operate within 100 psig pressure, now to maintain this pressure, we need some external

intervention so that, the reactor operates below this limit, this is the first requirement that

is, safety.

Second one is the production specifications, a process must produce desired amounts of

product and desired quality of product. So, first one is quality and second one is quantity,

this the product specifications, we need to maintain the quantity as well as the quality,

that is a product specification. Third important point is environmental requirements so

there are a number of state and federal laws, which enforce to maintain say for example,

the concentration of chemicals.


Say for example, it is require to maintain the concentration of chemicals in the effluent

stream now, another example is sulphur dioxide concentration in the stream, which is

rejected to the atmosphere. Second environmental requirement is, to maintain the sulphur

dioxide concentration, this is example which is rejected to the atmosphere. And third

example is the waste water which is retained to the river or lake, these are three examples

under the environmental regulations.


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Next important requirement is operational constraints now, the plant have certain

constraints inherent to the operation say for example, the distillation column, this is the

first example. The distillation column should not be flooded, this is one operational

constraints second example is, tanks should not overflow or go dry. Third example is, the

temperature in a catalytic reactor should not exceed the upper limit, if we consider a

catalytic reactor.

So, the temperature should not exceed the upper limit because, if the temperature is

higher than the upper limit, the catalyst may be destroyed so this is all about the

operational constraints. Fifth one is the economics so it is require to control the operating

conditions at given optimum level of minimum operating cost, maximum profit, etcetera.

Say for example, we need to judiciously use the raw materials, the energy used in the

process the human level so that, we can minimize the operating cost and maximize the

profit so these are all about the requirements to meet the plant of ((Refer Time: 09:08)).

Now, it is very obvious from this discussion, that to maintain all these requirements,

there is some external intervention is required and that external intervention is nothing

but, the control system so to maintain, to meet all these requirements, we need to devise

a control system.


Next, we will discuss three important issues which can dealt by the control system, what

are these issues. First issue is, the influence of the external disturbances, this is the first


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issue second one is, the stability of a chemical process, this is the second issue and third

one is, the performance of a chemical process. So, these are the three important issues

which should be dealt by the control system.

Now, the basic aim of a controller is, to separates the influence of external disturbances,

second one is to ensure the stability of a chemical process and third one is, to optimize

the performance of a chemical process. So, we need to device a control system, which

separates the influence of external disturbances, it ensure the stability of a chemical

process and it can optimize the performance of a chemical process. We will discuss all

these three issues in brief with the help of control system.

So, first we will discuss the external disturbance so to discuss this external disturbance,

we will consider a simple example that is, a tank heating system. The schematic of the

tank heating system is something like this, this is a tank, a liquid steam which is entering

this tank with a flow rate of F suffix i, temperature of this inlet steam is T suffix i. Now,

this liquid is while stared, the outlet steam flow rate is F and temperature is T now, the

liquid in this tank has the height of h definitely, the temperature is also T because, we

were considering starrer.

Now, to heat up this process, we need to introduce one coil, through which steam is

going with a flow rate of F suffix s t. So, this is a heating tank system, a liquid is entering

the process with flow rate F i and temperature T i, the outlet flow rate has the rate of F

and temperature T, one coil is introduce for heating the liquid and through the coil, steam

is passing with a flow rate of F s t.

Now, what is the objective of the this process, that we need to mention first, first

objective is, to maintain the liquid temperature T at it is desired value, that desired value

is suppose, T suffix d this is the first objective. The temperature of the liquid should be

maintained at it is desired temperature that is, T d. Similarly, the second objective of this

process is, we need to maintain the height of liquid in the tank at it is desired value that

is, h suffix d so these are the two objectives.

Now initially, what are the steps we need to follow for any chemical process, first is, we

have to follow the startup procedure of a process. After starting of the process, it reaches

at steady state, we are presenting that by ss so after starting up the process, it reaches atsteady state. Now, suppose, the process is at steady state and if there is no external


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intervention I mean, if there is no change of F I, there is no change of T i, there is no

change of F s t, the process remains at steady state all the times.

So, if there is no change of any input variables then the process remains at steady state so

there is no need of any control system but, this is not the case in practice. Usually, the

input variables may change with time, not regularly but, maybe in some interval say, for

example this heat, inlet flow rate which is coming from one upstream unit. So, we do not

have any control on F i and T i exactly so this is the practical case where, the input

variables may change with time and that is why, we need to devise a control system.

Now question is, how we can devise that control system and how, we can keep this

objectives say, for example, we are considering the first objective I mean, we need to

maintain the temperature at it is desired value, how we can maintain that. First of all, we

have to measure this temperature of this liquid steam, we can measure the temperature

using one thermocouple say, this is a thermocouple employing, which we can measure

the temperature of this liquid.

And suppose, the thermocouple outlet I mean, the temperature which is measured by the

thermocouple, that is exactly the liquid temperature T. Now, this temperature is then

compared with the desired value that is T d, this input T d is given by the person who is

in charge of operation or you can say, value of this T d is specified by the control

engineer. Now, we compare this T d and T, and the outlet of this comparator is e, that is

nothing but, the error and that error is T d minus T.

So, we can do one thing, we can put here positive sign and here negative sign, this error

signal goes to a controller, this error signal is the input to the controller then this

controller produce or calculates control actions and that control action is implemented

through this control valve. So, this is a heating tank system, we are considering only the

first objective that is, we need to maintain the temperature at it is desired value. We are

not considering the second case so initially, we need to measure the liquid temperature

using one temperature sensor that is, thermocouple.

We have assume, that the thermocouple outlet temperature is exactly equal to the liquid

tank temperature then this temperature is compared with the desired temperature that is,

Td. So, we have used one variable that is, error which is represented by e, e equals to T dminus T then this error signal goes to the controller, controller calculates and that


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calculated action is implemented through this valve. Now suppose, error is greater than 0

this is the first case, error is greater than 0 it means, T d is greater than T that means, the

desired temperature is higher than the temperature exist in the liquid system. So, what is

require to do for the controller?

Student: ((Refer Time: 20:41))

Yes, the controller will increase the F st so it is require to increase the steam flow rate

that means, more steam is required to flow through the control valve, this is one case.

Similarly, if error is less than 0 in that case, the situation is just opposite I mean, T d is

less than T that means, the control should reduce the stem flow rate. Now, we have to

introduce this fact in terms of the disturbance, here the disturbance is basically, the feed

flow rate and temperature.

Suppose, feed flow rate has been increased, this F i feed flow rate increases now, there is

no change of the inlet steam flow rate, only change is in F i. If F i increases and F st

remains same then what happens, temperature increases or decreases, temperature

decreases. So, in this situation, what the controller will do, the controller will increase

the steam flow rate that means, the controller will increase this F s t. So, this is

description in terms of external disturbance so it is very clear from this discussion that,

the controller which separates the effect of external disturbance.



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Next, we will discuss the stability of a chemical process so we will first draw a plot, we

will make a plot, this is x and this is time t. This x may be temperature, x may be

concentration now, s i mention that, initially the process is at steady state. Now, at time t

equals to t naught, the x is disturbed, this is the steady state value of x. Now, first x is

disturbed at t equals t naught now as a result, the x show this type of response that

means, x returns automatically to the steady state.

If this is the case, this type of process is called stable or self regulating process, x returns

automatically to the steady state value and this type of process is called stable process or

self regulating process. So, for this case, there is no need of any external intervention I

mean, no need of any controller for this stable system. We will consider another case

which is unstable so this is x versus time that is t, initially the process is at steady state

and x is disturbed at time t equals to t naught.

Now, in this case, the response is like this so this figure indicates that, x does not return

to the steady state, this figure clearly indicates that x does not return to the steady state

and this type of processes are called unstable process. And for this process, there is a

need of external intervention I mean, there is a need of controller. Now, from this

discussion, we cannot control that for this stable process, there is no need of controller,

we cannot conclude that.

Because, we have mention three important issues so this is only one issue, for other two

issues like the separation of external disturbance and third one is the optimum

performance of the process, for those two issues, we may need the controller. Not only

that, even for this stable process to reach at steady state with a short period of time, we

may need a controller. So, we can say that, here also the need of controller we realize,

next issue is the performance of a chemical process.


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Or optimize the performance of a process this is a third issue now, the main operational

objectives, main objectives are, first one is the safety and second objective is production

specifications, this is the second objective. Once these are achieved, once these two

objectives are achieved, the next goal is to make the operation more profitable. Now, in

that direction, we will consider one example that is, continuous stared tank reactor.

Suppose, this is a jacketed reactor, this is a schematic of a reactor reactants enter the

process, this is the reactant and this is the product. Now, in this process, two consecutive

reactions A to B and then B to C occur, this is two consecutive exothermic reaction

occur. A is the reactant, B is the desired product and C is the undesired product so this is

basically products, A is the reactant, B is the desired product and C is the undesired

product.

Now, what is the economic objective, the economic objective for this process is to

maximize the profit. Now, the profit function is phi and this is integration of 0 to t, t is

nothing but, the operational time and here, one function will be there, this function is in

terms of revenue from the sales of the product B then it includes the cost of reactant A

and then the cost of coolant. Basically, if this is the exothermic reaction, we need to

introduce here coolant to take out the exothermic heat and this is coolant in, this is

coolant out.


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So, the profit function which includes the revenue from the sales of product B, cost of

reactant A and the cost of coolant then we need to maximize the profit. And then we can

maximize the performance we can say so these are all about the three issues, which can

be dealt by the control system. Next, we will discuss the classification of variables, the

variables which are extensively used in process controlled course, that we will discuss in

the next.


So, next topic is classification of variables, the variables is usually 2 types, one is input

variable and second one is output variable. Now, input variable is again 2 types, first one

is disturbance or load variable and it is conventionally represented by LV, second input

variable is manipulated or adjustable variable or some time it is called control variable.

So, input variable is 2 types, one is disturbance or load variable and second one is

manipulated variable or control variable, this we can represent by MV.

Similarly, the output variable again 2 types, one is measured output and second one is

unmeasured output, another variable is also used that is, controlled variable. So, do not

confuse with this controlled variable, controlled variable is manipulated variable and

controlled variable we will discuss that. But at this point, I can say, that the controlled

variables are usually the measured output. Sometimes this is also unmeasured output so

this the controlled variable so these are all about the variables. Now, we will take


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different examples and we will select the different variables for that specific example so

we will start with a simple liquid tank system.


We will first start with a liquid tank so this is a liquid tank system, the input to this

process is the input steam, has a flow rate of F i, outlet flow rate is suppose, F naught.

The liquid in this tank has the height of h and the cross sectional area of tank is A now,this is a simple liquid tank system. So, what is the objective of this process? The

objective of this process is, to maintain the liquid height in the tank at it is desired value

h d ,this is the objective of this tank system now here, the control variable is height.

The controlled variable we will represent by CV, the controlled variable CV here is

liquid height. Now, if we considered this process so can you classify the variables so

which one is the input variable for this process? F i. Input variable is F i, what are the

output variables, one is F and another one is liquid height, the output variables are one is

outlet flow rate F and second one is liquid height. Now, for this example system, can you

make a pair in between controlled variable and manipulated variable, we have decided

that, for this particular system, liquid height in the tank h is the controlled variable.

So, what will be the corresponding manipulated variable I mean, to maintain this liquid

height, which variable we can adjust F i or F o? ((Refer Time: 40:20))


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I think both we can do, F i and F naught. I am just making one controlled configuration

suppose, this F naught is the manipulated variable corresponding to liquid height h so

what will be the control configuration, as we have drawn for the heating tank system. So,

first of all, we need to measure this height by level sensor so here a level sensor we can

place, which can measure the liquid height.

Suppose, this is height then it is compared with the desired value h d, this carries positive

sign and this is negative sign. Now, the output of this comparator is error, error is

basically desired height minus height then this error signal goes to controller and this

control action is implemented here, this is a control valve. So, F naught is the

manipulated variable in this control configuration now, can you tell me, F naught is input

variable or output variable? ((Refer Time: 42:13))

You see the classification, manipulated variable is under input variable or output

variable? Input variable. So, if we consider F naught as the manipulate variable then F

naught is input variable. In this example, it is clear that, if F naught is manipulated

variable then that is input variable, not output variable. Because, manipulated variable is

one type of input variable that is why, you write within bracket F naught but, if we

consider F i is the manipulated variable in that case, F naught is output variable. Next we

will take another example, to know about all these variables.



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That example we have consider heating tank system so this is if heating tank system, the

inlet steam has flow rate F i, temperature T i and it has the outlet flow rate of F and T,

steam is introduced through this coil, it has the flow rate of F st and liquid height here h,

temperature T so this is the heating tank system. Now, what are the inputs in this case, F

i, F st, T i these are the inputs and what about F, may be input and what about the outputs

F, T and h.

We have mentioned the objectives of this process, first objective is to maintain T at it is

desired value and second is we need to maintain height at it is desired temperature. Can

you classify the, can you make the control pairs I mean, manipulated variable and

corresponding controlled variable pairs. So, one controlled variable is temperature

because, that is our objective, second control variable is height, how you can maintain

this temperature, by adjusting F st and for the case of height, how we can maintain the

height, by adjusting F i and F naught, F i and F.

If F is the manipulated variable, as we discuss for the previous example then F will be

one input variable, if F i is the manipulated variable in that case, F will be output

variable. So, F i and F st, these two are suppose for this case, manipulated variable and

this T and height, these two are controlled variable. So, which one is disturbance variable

or load variable, inputs variables are 2 types, one is manipulated variable and another

one is loaded variable, F i and F st they are manipulated variable so rest is T i so this is

load variable. So, these are all about to the variables, different variables next, we will

discuss the controlled configurations. Although we did not discuss the control

configurations in details but, before that, we want to know the different configurations.


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Control configurations, we will consider the distillation example, which is quite complex

compare to this liquid tank and heating tank system. Now, first we will consider one

control scheme that is, the feedback control scheme. So, distillation column, this is the

tower of a distillation column, feed is introduced here with flow rate F and composition

Z, the overhead vapor which leaves the top tram is condensed in this condenser, this is a

condenser. Then, the condensed liquid is accumulated in a drum, which is the reflux

drum, this is also called reflux accumulator.

A part of this liquid is recycled back to the top section of the column and a part is

withdrawn as distillate with flow rate D and composition x D, composition means here

mold fraction. Similarly at the bottom, the liquid is withdrawn and it is subjected to a

reboiler, the produced vapor is recycle back to the bottom tray and some amount of

liquid is taken out as bottoms with flow rate B and composition x B. So, feed is

introduced with flow rate F, composition Z, this is a feed tray, this is the top tray, this is

the bottom tray.

Now, the overhead vapor goes to a condenser, condensation occurs then the condensed

liquid is accumulated in this reflux drum and part of this liquid is recycle to the top tray

as reflux, this is called reflux rate, reflux flow, reflux steam. And another part, a part of

this accumulated liquid is withdrawn as distillate with flow rate D and composition or

you can say mold fraction x D.


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Similarly at the bottom, this liquid which is coming from the bottom tray, it goes to a

reboiler, the produced vapor is recycle back just below the bottom tray, this is bottom

tray. And some amount of liquid is withdrawn as bottom product or bottoms, with flow

rate B and composition x B. First we have to know, what is the control objective of this

process, there are basically 2 product, one is distillate another one is bottoms, we will

consider presently the top product.

So, what is the objective, objective is to maintain the top product composition at it is

desired value, this is the objective if you consider the top section only, we need to

maintain the top product composition x D at it is desired value. So, what will be the

control configuration for this case, if this is our control objective that means, this is the

controlled variable so controlled variable manipulated variable pair we have to make. If

x D is the controlled variable then corresponding manipulated variable is reflux flow rate

R.

Can you make the control configuration now, yes we can make it suppose here, the liquid

has composition of x D so we need one composition analyzer to measure x D. Then, this

analyzer gives the value of x D then that x D is compared in this comparator, this is

negative and this is x D, this is capital D desired this is positive then we get the error

signal. This error signal goes to the composition controller then the control action is

implemented through this valve so this is the control configuration.

So, this is basically the feedback control scheme, in the feedback control scheme, the

controlled variable is measured. Anyway, today we do not have time so in the next class,

we will discuss other two control schemes, they are feed forward control scheme and

inference sale control scheme, along with other topics.


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Prof. A. K. Jana



Lecture - 2Introduction to Process Control (contd.)

Today, we will continue our discussion with the example of distillation column.


Again I am drawing the schematic of the distillation column, this is the tower now, feed

is introduced in the feed tray. Feed flow rate is F, composition is z then, the operate

vapor lifts the top tray and then, that is condensed in the operate condenser. The

condense liquid is then accumulated in this reflux drum, this is the reflux drum, this is

condenser then, a part of this liquid is taken out as distillate, this is the top product with

flow rate D and composition x D.

Now, another part of this accumulated liquid is recycle back to the top tray, this is the top

tray so, this is the connection for the reflux rate and the bottom liquid which left the

bottom tray, this is bottom tray is reboiled in the reboiler, this is the reboiler. Then, the

produced vapor is recycled back just below the bottom tray and a part of the liquid is

withdrawn as bottoms, this is the bottom product which has the flow rate of B and

composition of x B.


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We have discussed with this example and one control configuration we have drawn in

the last class, that control configuration was the feedback control configuration. That was

the feedback control configuration, in that feedback control configuration, we have

majored x D basically for this particular process, the controlled variable is x D and

corresponding manipulated variable is reflux rate, this is reflux rate. We are considering

only the single loop at the top, we are not presently the considering the bottom loop.

Now, in the feedback control configuration we have seen that, the controlled variable

which is x D was directly measured to manipulate the reflux rate. Today, we will

consider another control scheme that is, feed forward control scheme. First, we will

configure this feed forward control scheme for this distillation column, in this feed

forward control scheme, the disturbance is measured. Here, disturbance is the feed

composition as well as feed flow rate but, we are considering this is the majored variable.

Now, for measuring this feed composition, we need one composition analyzer, this is

composition analyzer, which is measuring basically the feed composition z. We are

assuming that, this composition analyzer output is exactly identical with the feed

composition. Then, this information z goes to the feed forward controller, this is the

block for feed forward control scheme then, these feed forward controller calculates the

control action based on this feed composition and this controller action is implemented

through this control valve.

For the feed forward control scheme, disturbance variable or load variable is measured

using one composition analyzer then, that measured feed composition information goes

to the feed forward controller. Then, the controller calculates the control action and that

control action is implemented through this control valve. So, the major difference

between the feedback control and feed forward control is, for the feedback controlscheme, we measured directly the controlled variable and for feed forward control

scheme, we measured the load variable or disturbance variable. There are many other

differences, which we will discuss in the subsequent classes, another control scheme we

will discuss next with this distillation example that is, inferential control scheme.


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So, we will considers the same distillation column to discuss inferential control scheme,

this is a schematic of the same distillation column. Feed is introduced here then, the

operate vapor is condensed and condense liquid is accumulated in the reflux drum, a part

of that liquid is the recycled back to the top tray as reflux stream and a part of the liquid

is withdrawn as distillate. At the bottom section, we incorporate one reboiler, the

vapourised stream is introduced just below the bottom tray and at the bottom, the bottom

product is withdrawn.

Now, in this inferential control scheme, the x D which is controlled variable is not

directly measured, most composition analysis provide large delays in the response.

Secondly, it provides high investment and maintenance cost that is why, sometimes this

composition I mean, the product composition is not directly measured. In that case, we

search for the secondary measurements and then, using those secondary measurements,

we can infer that product composition, that is the purpose here.

So, to estimate x D ,we need to select the secondary variables for this particular example

suppose, this is first tray, this is second tray, this is third tray. So, the temperature of this

first tray is suppose T 1, this is T 2, this is T 3 it is quite easy to measure the temperature

compare to the composition measurement. So, we can say that, these three tray

temperatures we can consider as secondary measurements so, we will measure these

three temperatures using thermocouples.


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Then here, we put one block for estimator, these three temperature after measurement,

the temperature information goes to this estimator, this estimator basically nothing but,

one algorithm which consists of some equations and correlations. These correlations

basically, calculate the x D based on the information of the major temperature. So, here,

some mathematical equations are included within this estimator and using those

equations, we can directly calculate based on this measured temperature information x D

value.

Basically, this composition is inferred based on the measure temperatures then as usual,

this estimated composition x D goes to the controller and the controller calculates the

action and those control actions are implemented through this control valve. Since the

product composition, which is the controlled variable is inferred that is why, the

controller is called inferential controller. So, we can write that, the unmeasured output

basically is the function of secondary measurements.

Here, the unmeasured output is x D and the secondary measurements are three

temperatures T 1, T 2 and T 3. So, I have mentioned earlier that, why we prefer this

control scheme, inferential control scheme because, if we directly one to measure this

product composition in that case, we need composition analyser. And most composition

analysis provide large delays in the response, this is the first drawback, most product

composition analysers provide large delays in the response.

Second drawback is high investment and maintenance cost, this is the second drawback

for the use of composition analyser. So, this is all about the different controlled

configurations with the example of distillation column. In the next, we will discuss the

different hardware elements of a control system.


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We will discuss different hardware of a control system, to discuss this topic it is better to

take one example. We will continue the heating tank system, to discuss this topic it is

better to consider one example and we will continue the heating tank system, which will

discussed in the first class. This is the heating tank system, feed is introduced with a flow

rate F i, temperature T i and product is withdrawn with a flow rate of F and temperature

T.

Now, to heat the liquid in this tank, one steam stream is introduced with a flow rate

suppose, F suffix st and for this particular system, we have this control pair I mean,

controlled variable and manipulated variable pair, we want to maintain the temperature

through the manipulation of steam flow rate. Our control objective is to maintain the

temperature of this tank and that is why, this is our controlled variable, corresponding

manipulated variable is steam flow rate.

So, this is basically a close loop system I mean, this is the open loop system if we want

to make it close, we need to include the controlled scheme, this is open loop now, we

want to close it by introducing controlled scheme. So here, temperature is a controlled

variable so, we need to first measure the temperature by using one thermocouple. This is

a thermocouple then, this measured temperature T goes to the comparator, this is the

comparator.


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The measured temperature is compared with the desired temperature, this is positive, this

is negative, output of this comparator is the error signal. Then, this error signal goes to

the controller then, the controller calculates the control action and that control action is

implemented through this control valve. So, this is basically the close loop system now,

we will discuss what hardwares are involved in this close loop system so, first one is the

process.

In this process, physical and chemical operations occur, this heating tank system is a

process, this is the first hardware involved in this close loop system. Second hardware is,

measuring instruments or sensors, this measuring instruments or sensors are used to

measure, these are used to measure first one is load variable. These sensors are used to

measure controlled variable and secondary output, third one is secondary output.

Although we have shown here, only the measurement of controlled variable but, we can

measure also the disturbance variable and secondary output by the use of the measuring

device or sensor. So, we will just discuss in brief what are the different measuring

devices used for different variables. So, we can write them in a table form, variable and

sensor used for measuring that variable like first variable is temperature say.

Temperature we can measure by the use of thermocouple, we can use thermocouple, we

can use resistance thermometer.

We can use thermocouple for measuring temperature, we can use resistance thermometer

but, mercury thermometer is not good because, the measurement cannot be transmitted

readily. Mercury thermometer is not good option because, the measurement signal

cannot be readily transmitted. Next we will consider another variable that is pressure,

pressure we can measure by the use of manometer, by the use of diaphragm element. For

measuring the pressure, we can use manometer, we can use diaphragm element.


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Next, we can consider another variable that is a flow rate, flow rate we can measure by

the use of orifice meter, flow rate we can measure by the use of venturi so, these are two

measuring devices, which are extensively use for measuring the flow rate. Like, another

variable is liquid level, liquid level is measured by the use of differential pressure cell,

DP cell. Liquid level in the tank is measured by the use of differential pressure cell,

another one is composition, composition is measured by using chromatographic

analyzer.

Although we have started all these things in the instrumentation part in details but, just to

know, which devices can be use for different variables, we have just discussed in brief

so, our second hardware was the measuring instruments or sensors. Now, third hardware

is transducer now, measurement signal, measurements cannot be used for control until

they are converted to physical quantities. Physical quantities say for example voltage,

say for example current, say for example pneumatic signal.

So, measurements cannot be used for control until they are converted to physical

quantities, which are readily transmitted, these signals we can readily transmit. And this

is the purpose of the use of transducer basically, the transducer physically convert the

measurement signal to the physical quantities. Next one is transmission lines, this is use

to carry the measurement signal from sensor to the controller and from the controller to


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the control valve. The transmission lines are used to carry the measurement signal from

sensor to controller and then, from controller to control valve.

If you see the schematic representation of the close loop system, I think you can clearly

understand this so, this is the usage of the transmission line. Now, measurements signal

is sometimes very weak, sometimes the measurement signal is very weak say for

example, in few milli volts. In that case, this transmission line is equipped with

amplifier, when the measurement signal is very weak in that case, the transmission line is

equipped with amplifier, this is the fourth hardware which is involved in the close loop

system.


Now, fifth hardware is the controller, this is another hardware which receives

measurement signal from the sensor. The controller basically receives measurement

signal from sensor and then, it decides, what action should be taken based on the major

values. So, the controller basically receives the measurement signal from the measuring

device in the next step, the controller calculates the control action based on the measured

signal comparing with the desired value of that control variable.

Sixth hardware is the final control element now, for the case of controller I told, that the

controller basically calculates the control actions. Now, that control action is

implemented through the final control element so, the control action, which is calculated

and produced by the controller that is, physically implemented through this final control


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element. Can you give any example of the final control element, yes first example is say,

control valve, another example is variable speed pump, variable speed compressor.

Why variable speed, if the speed is fixed then, that cannot be changed that is why,

variable speed pump and variable speed compressor. Now, next hardware element is a

recording device, this is used to visualize the plant behavior through the measurement

signals. If we used to visualize the plant behavior at different situations, we need one

recording device and that realization we can achieve through the measured values. And

for this purpose, we can use video display unit, which is usually accommodated in the

control room.

Video display unit can be used for this recording purpose, which is usually

accommodated in the control room. So, these are about the hardware elements for a

particular process so far we have discussed, the first topic that is, introduction to process

control. In the next, we will discuss the second topic that is, the mathematical modeling

and the use of mathematical modeling in process control.


So, we will start to discuss the second topic that is, mathematical modeling first we will

know in brief, what is the mathematical model. Suppose, we have one experimental

setup so, we will represent the experimental setup, we will use one block to represent the

experimental set up. Now, to this experimental setup, the input is introduced I mean,


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input is introduced to the setup and if we run the setup, we get output this is a common

thing in our laboratory which happened.

So, what is basically the model, can we represent this experimental setup by some

mathematical equations or correlations we can? We can represent this mathematical

setup by some mathematical correlations or equation. So, suppose, some mathematics are

involved then, we are giving the same input to this block, we will get some output. Now,

this block is the representation of the model, we are just representing this experimental

setup by mathematical correlation.

Now, if these two outputs are close enough then, we can say that, this model is a good

model so, model is the mathematical representation of a process, intended to promote

understanding of the real system. A real system, we can represent by using some

mathematical equations and correlations so, this is the definition of model. And next, we

will discuss about, what is the use of this model, why we will develop the model, what is

the use of mathematical modeling in process control, that we will cover in the next.


So, use of the process model, first we can write to understand the process behavior

suppose, we have the model and we have the solution of this model, also we are giving

some input to this model, we can get the output only if, we have the solution otherwise,

we cannot get the output. We have develop the model structure for a particular process,


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we are giving some input information to the model basically, the input information is

specified and then, if we solve the model then, we can get some output.

Now, suppose, the model initially is at steady state I mean, after start of we can reach at

steady state. Now, at steady state, we are giving some change in this disturbance variable

then, the process will shift from steady state to another state. Basically, if we change in

load variable, we will get some transient response and that transient response we can get

by performing this simulation of this model without performing experiment, is not it. So,

we can understand the process behavior by some change in load variable, to understand

the transient behavior without performing the experimental setup.

So, this is the first purpose of the model secondly, to train the operating personnel, the

model we can use for training purpose, without running the experimental setup or

without running the plant. Suppose, we have the model structure same thing which we

have drawn earlier, we have some input, we have output. Now, some situations can be

irritated using this model like, the feed is introduced to a particular process here, we are

directly using some value for the feed flow rate.

Now, suppose, the pump is not delivering the feed to the process what will happen,

suppose in the distillation column, we have some minimum reflux rate. If we consider

lower than that minimum reflux rate, what will happen so, these types of emergency

situations, we can irritate by the use of this model, without disturbing the process. Or

even some situations cannot be permitted to irritate in the real process so, to train the

operating personnel, we can also use the simulated model.


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The next purpose is selection of control pairs, we have taken few examples and we have

discussed the controlled variable and manipulated variable pair. But, this controlled

variable and manipulated variable pairs, we can select by knowing the model of that

particular process. So, if we know the model of a particular process, we can determine

the pair between controlled variable and manipulated variable so, for this purpose also,

we need the process model.

Fourth purpose is, to develop the model based controller, you know most of the

advanced controllers are model based controller and the name clearly suggests that, the

controller which consists of or which includes the process model, those controllers are

basically model based controller. So, most of these advanced controllers, use the process

model so, in that sense we can say that, we need the process model for the development

of advanced controllers.

Fifth one is, optimize the process operating conditions, to determine the most profitable

operating condition, we need the process model and economic information. So, we need

process model and some economic information to determine the most profitable

operating condition. So, we need the process model, we need some economic

information to determine the most profitable operating condition. So, these are the issues,

for which we need the process model next, we will discuss the classification of process

model.


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There are different ways to classify the process model but here, we will discuss the

classification of process model based on, how they are obtained. There are three different

models I mean, we can classify in three different ways, three different models are there,

first one is theoretical model. The theoretical model is basically developed based on

principles of conservation, this is developed based on the principle of conservation.

So, first type of model that is, theoretical model and this model is develop principle of

conservation, I think you know the principle of conservation like, mass conservation,

energy conservation, momentum conservation. And another type of model that is

empirical model, this model is obtained by fitting experimental data, the second type of

model that is empirical model, this empirical model we can obtain by fitting

experimental data.

Basically, if we have the experimental setup, we have different sets of input output data

now, if we have the input output data by fitting those experimental data, we can

determine the coefficients. By that way, we can construct a model and that is the

empirical model. Third one is just the combination of these two, theoretical and

empirical model, that is called semi empirical model, sometimes this is also called hybrid

model.


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Semi empirical model is the combination of theoretical model and empirical model, semi

empirical model is the combination of these two. Next, we will discuss in brief, what are

the advantages and disadvantages of this models.


So, first is the theoretical model, which is developed based on the conservation principle

so, what is the advantage of this model. First advantage is, it provides physical insightinto process behavior, this is the first advantage. Second advantage is, it is applicable for

a wide range of conditions, this is the second advantage. Similarly, what are the

disadvantages, it is time consuming to develop so, it leads to be time consuming to

develop because, particularly for the complex systems, the theoretical model is too large.

So, in that sense we can say that, this is time consuming and another drawback is, some

model parameters are not readily available, some model parameters for example, reaction

rate coefficient, overall heat transfer coefficient, these are not readily available. In that

case, we have to follow the empirical technique so, these are basically, the two

drawbacks and two advantages of the theoretical model. Now, what about the empirical

model, I think we can say something for this empirical model, based on the discussion of

this theoretical model.

So, what is the advantage, it is easier to develop, this is the advantage because, if you

have the input output data set then, we can feed some correlations or equations using

those input output data set. So, for complex process, we do not have so many equations


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like theoretical model so, this is only the advantage for this case. And what are the

disadvantages, this is applicable for a narrow range of conditions, this is applicable for a

limited range of conditions.

So, this is the advantage and this is the disadvantage for the empirical model similarly, I

think you can find, what are the advantages and disadvantages of the semi empirical

model. Now, for the process control, we have discussed earlier different variables I

mean, input variables, output variables then, input variables are again 2 types

manipulated variable and load variable, and output variables are 2 types majored output

and unmajored output. At this time, we will discuss another variable which is extensively

used in process control that is, state variable that we will discuss now.


State variable basically describes the natural state of a process now, there are basically 3

fundamental quantities. What are these quantities mass, energy and momentum, these are

three fundamental quantities. Now, these fundamental quantities are usually not directly

measured, these three fundamental quantities are not directly and conveniently measured.

And these three fundamental quantities are usually characterized by say, temperature, by

pressure, by composition, by flow rates, etcetera.

There are 3 fundamental quantities mass, energy and momentum, these three

fundamental quantities are not directly and conveniently measured and they are

characterized by these variables temperature, pressure, composition, flow rate and these


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variables are called state variable because, they describe the natural state of the system.

So, these are called state variable and it arises naturally in the accumulation term, we will

discuss in the next, the conservation principle and there we will see that, the state

variables usually present within the accumulation term so, it arises in the accumulation

term.

Within the accumulation term the state variable is present and since we have discussed

the state variable, in the next we will discuss state equations. These equations basically

derived by the application of conservation principle on the fundamental quantities to

relate the state variables with other variables are called state equations. The state

equations are derived by the application of conservation principle on the fundamental

quantities to related the state variables with other variables including other state variables

are called state equations.

These are about the state variable and state equations and how, we can see the state

variable within the state equation. I mean, first we will go for the development of state

equation then, we will see which one is the state variable within that state equation. Now,

for that purpose, we have to know the conservation principle.


The conservation principle, we can represent in general form like rate of accumulation

equals to rate of input minus rate of output plus rate of generation, this is the general

form of the conservation principle. Now, I told about the three fundamental quantities


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mass, energy and momentum so, if we include here rate of mass accumulation equals to

rate of mass input minus rate of mass output, that is the conservation of mass.

For the case of conservation of mass, there is no existence of this rate of generation term

and for the energy conservation similarly, we can write rate of energy accumulation

equals to rate of energy input minus rate of energy output plus rate of energy generation.

So, this is about the conservation principle and the state equations, state variable, the

description on that I mean, by the derivation of the state equation, we will know which

one is the state variable, this thing we discuss in the next class.

Thank you.


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Prof. A. K. Jana



Lecture - 3Mathematical Modeling (Contd.)


We will continue the topic Mathematical Modeling. In the last class, we have discussed

about the state variables and state impressions and in this class, we will develop the

mathematical model, first for a CSTR, Continuously Stirred Tank Reactor. So, you will

take CSTR example and will derive the mathematical model, this is if jacketed CSTR,

this is a tank, this is a jacket. Now, one medium is introduced here and that medium is

coming out here, first of all the feed which is entering to the reactor has the flow rate of

F i, concentration of the feed is C A i and temperature is T i.

The suffix i indicates the input, A is a component, I mean C A i is the input

concentration of component i and T i is a input temperature. The product steam which is

coming out from this CSTR has the flow rate of F, concentration in terms of component

A is C A and temperature is T. Now, before deriving the mathematical model for this

CSTR, first we will know the units of different flows like F i and F along with the

coolant medium flow rate kept F c.


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So, all these flow rates are basically volumetric flow rate, here temperature is T c i, outlet

flow rate is F c and outlet coolant temperature is T c naught. Concentration C A and inlet

concentration C A i, both are molal concentration I mean, unit is say mole per unit

volume. C A and C A i both are molal concentration for example, they are in mole per

unit volume. Next, you will consider the assumptions, so first assumption is perfect

mixing that means, the temperature of this outlet steam T and compositions C A, they are

same with that of the reaction mixture.

The perfect mixing indicates, everywhere in this tank temperature and concentration,

they are identical and the outlet temperature composition also identical with the

temperature and concentration of reacting mixture. Second assumption is liquid density

rho and the heat capacity C p they are constant, third assumption is we are considering a

simple exothermic first order reaction. And to remove this exothermic heat, the coolant

steam is introduced in this jacket, this is the coolant steam.

Fourth assumption is, the reactor is perfectly insulated that means, there is no heat loss

from the reactor to the surroundings it means, no heat loss from the system to the

surroundings. Fifth one is, coolant is perfectly mixed in the jacket and last assumption is,

we will not consider any energy balance for the jacket, there is no energy balance for the

jacket. These are the assumptions adopted for this CSTR system and based on these

assumptions, we will derive the modeling equations, now first we will go for the overall

mass balance.


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First, we will go for overall mass balance so, to develop the overall mass balance of the

CSTR system, we need the conservation of mass. What is that? Conservation of mass is

rate of mass accumulation equals to rate of mass input minus rate of mass output, this is

the conservation of mass. Now, for this CSTR system, what is this term I mean, how we

can represent the mass of accumulation see in this reactor system, you consider the

volume of the liquid is v.

The volume of the liquid in the reactor is v now, if v is the volume, if we multiply with

density so, this whole terms becomes mass. Now, differentiation of this is the mass flow

rate d d t v rho, that is the rate of mass accumulation. Now, rate of mass input, input to

the system is basically F i but, we know that, F i is the volume at the flow rate so, we

have to multiply with rho. So, F i multiplied by rho that is the input mass rate similarly,

what will be the output, output flow rate I mean, the volumetric flow rate that is, F.

So, if we multiply with rho then, this is the output rate now, we have assume that, rho

and C p both are constant. If rho is constant then, we can write this rho d v d t equals to F

i rho minus F rho that means, d v d t equals to F i minus F. Suppose, this is equation

number 1 so, this is a mass balance equation similarly, we can go for mass balance of

component A or you can say, that component mass balance. So, we will consider in the

next, mass balance or component mass balance and the component is here component A.


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So, what is the conservation principle for this rate of accumulation of component A, this

is a accumulation term. Then, rate of input of component A then, rate of generation of

component A minus rate of output of component A, this is a conservation principle for

component mass balance. Now, we have to write all these individual term accumulation,

input, output and generation now, for this CSTR system, what is the accumulation term.


Volume multiplied by C A now, if we one to represent in terms of rate then, d d t of v C

A this is mole per unit volume C A and v is volume so, mole per unit time overall. So,

what is the input, F i is a flow rate of input stream and concentration is C A i so, F i

multiplied by C A i, that is the rate of input of component A. Now, what is a generation,

minus r A into v see, minus r A is the rate of disappearance of A. If we multiply with

minus sign then, we get the generation, minus r A is the rate of disappearance of A so, if

we multiply with minus then, that becomes generation.

And what is the output, output is F multiplied by concentration of output steam that is, C

A so, from the conservation principle, we got this equation. Now, we can write this

equation in this form C A d v d t plus v d C A d t equals to F i C A i minus F C A minus,

minus of r A into v. Now, we know the d v d t term I mean, if we substitute equation 1

then, this equations becomes C A F i minus F plus v d C A d t equals to F i C A i minus

F C A minus, minus of r A into v.


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Now, this F multiplied by C A and this F C A will be canceled out then, we can

rearrange this equation to v d C A d t equals to F i C A i minus C A minus, minus of r A

into v. If we divide again both sides by v, we will finally get d C A d t equals to F i

divided by v, C A i minus C A minus, minus of r A, minus of r A is the rate of

disappearance of A. Now, if we consider Arrhenius equation then accordingly, we can

write minus of r A equals to k naught exponential of minus of E divided by R T into C A.

According to the Arrhenius principle, the reactions rate equals to pre exponential factor

then, exponential of minus E divided by R T, C A is the activation energy and R is the

universal gas constant. If we substitute this reaction rate expression here then finally, we

get d C A d t equals to F i divided by v, C A i minus C A minus k naught C A

exponential of minus of E divided by R T. So, this is the final form of component mass

balance equation so, these two equations we have derived based on the conservation of

mass in the next, we will consider the energy balance equation.


Energy balance equation for the example CSTR system, what is the conservation

principle of energy, conservation principle we can write in this form, rate of energy

accumulation equals to rate of energy input minus rate of energy output minus rate of

energy removed by the coolant plus rate of energy added by exothermic reaction. So,

here 4 terms are involved accumulation, input, output, energy removable and energy

added by the exothermic reaction.


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So, next we have to representation all these terms using term mathematical terms I mean,

variables. So, what is the energy accumulation say, volume in the tank is represented by

v and if we multiply with rho, this becomes m then C p, then d t, energy term we can

write in terms of m C p d t. Now, here we are considering reference temperature equals 0

now, if we write here d d t of v rho C p T then, this becomes rate of accumulation of

energy.

Now, next we will consider the rate of energy input, volumetric flow rate of input steam

that is, F i if we multiply with rho then, that becomes mass flow rate. Similarly, C p then,

d t I mean, T i minus T reference here, reference temperature we are assuming 0. So, T i

and you recall, we have consider the C p that is constant, what will be the output rate of

energy output. The flow rate of outlet steam I mean, the volumetric flow rate that is, F

similarly, if we multiply rho, this becomes mass C p and outlet temperature is T.

Now, rate of energy removed by the coolant this one, this will represent by Q, energy

removed by the coolant is represented by here Q. What is Q, how we can calculate Q, we

know flow rate F c, rho c, C p c and then, temperature difference is outlet temperature of

coolant minus inlet temperature of coolant. What is the last term I mean, how we can

represent the last term, last term we have to consider in this way, minus del H that is,

heat of reaction then, minus r A multiplied by v.

Minus del H here is heat of reaction, it is well known to ask that, heat of reaction is

negative I mean, this negative term is used for the case of exothermic reaction and for

endothermic reaction, we use here positive sign. So, this term represents the energy

added by exothermic reaction next, we need to simplify this equation. Now, dividing

both sides by rho C p, v d capital T d small t plus temperature d v d t equals to F i T I,

since we have we are dividing both sides by rho C p.

Next term is F multiplied by T then, Q divided by rho C p and finally, minus del H minus

r A v divided by rho C p. Now, you will substitute this term d v d t, we have the equation

of d v d t obtained from the total mass balance, if we substitute d v d t then, we can write

it like this way, T multiplied by F i minus F equals to F i T i minus F t minus Q divided

by rho C p plus minus of del H minus of r A into v divided by rho C p now, this F T and

this F T we can cancel.


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And then, we get v d T d t equals to F i T i minus T minus Q divided by rho C p plus

minus of del H minus of r A into v divided by rho C p. And finally, we will divide both

sides by this volume term then, we get d T d t equals to F i divided by v T i minus T

minus Q divided by v rho C p plus minus of del H minus of r A divided by rho C p.

Now, again we will substitute here the Arrhenius law then, d T d t equals to F i divided

by v T i minus T minus Q divided by v rho C p plus minus of del H k naught C A

exponential of minus E divided by R T whole divided by rho C p so, this is a energy

balance equation.

So, for the example CSTR system we got three equations, one is based on total mass

balance then, component mass balance and last one is based on energy balance.


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Now, the modeling equations I am writing here, one we got d v d t that is equals to F i

minus F, second equation d C A d t that we got F i divided by v, C A i minus C A minus

k naught C A exponential of minus E divided by R T. And energy balance equation we

got that is, d T d t equals to F i minus v into T i minus T minus Q divided by v rho C p

plus minus of del H C A k naught exponential of minus E divided by R T whole divided

by rho C p.

Here, I have mentioned that Q, Q equals to our coolant flow rate is F c now, density is

suppose rho c, heat capacity if we will consider C p c multiplied by the temperature

difference. What is the outlet temperature of this, T c o minus T c i so, this is the

expression for Q. Now, you will just classify, we will just see what are the different

variables involved in the modeling equations here, what are the input variables. Input

variables are C A i then, F i then, T i then, Q.

We are not considering F c, T c i we are considering Q and F definitely, F will be input

variable. If this is considered as the manipulated variable for controlling the liquid height

or liquid volume so, these are the input variables. What are the output variables, output

variables are here v, C A and T see, in this three modeling equations v, C A and T, they

are present within the accumulation term so, these three variables are also state variables.

So, we can write here these are also state variables because, they are present within the

accumulation term.


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Now, among this input variables, which are the manipulated variables, for the example

CSTR system, if we consider liquid volume is a first I mean, one control variable and

another one is say, temperature. Now, the corresponding manipulated variables are, if we

consider F as the manipulated variable for v and Q as the manipulated variable for

temperature then so, Q and F, these two are basically manipulated variables.

So, these three I mean, the rest input variables are load variables or disturbance variables,

among these five input variables, two are the manipulated variables and other three are

the load variables. So, this is the development of model structure for the sample CSTR

and we have seen the different variables, which are involved in this example CSTR.

Before going to discuss another system, we will study about the degrees of freedom

analysis.


So, we will next study the degrees of freedom so, so far we have discussed about the

modeling of chemical processes, after deriving the mathematical model of a process, we

need to solve those modeling equations. The solution of a model structure is basically

called simulation, we need to stimulate the modeling equations. Now, for the stimulation

of a modeling equation, we need to describe this degrees of freedom. Degrees of

freedom, which suppose is represented by F then, we can write F equals to v minus E,

degrees of freedom we are representing here by F then, F equals to v minus E.


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V is the total number of independent process variables and E is the total number of

independent equations. So, degrees of freedom basically, total number of independent

variables minus total number of equations. Now, in the analysis of degrees of freedom,

we will consider three different cases, in the first case we will consider say, F equals to

0. It means, number of independent variables equals to number of independent equations

that means, the system is exactly specified.

When degrees of freedom equals 0 then, we can write v equals to E that means, number

of independent process variables equals the number of independent equations. In this

situation, we can say the system is exactly specified I mean, there is no problem to find

the solution of the modeling equations. In the second case, we will consider F is greater

than 0 that means, v is greater than E so, in this case, the system is called under

specified. How we can make it exactly specified system by the inclusion of F number of

additional equations.

So, to make it exactly specified, we need F additional equations, to make the under

specified system exactly specified, we need F number of additional equations then, we

can only get the solutions of the modeling equations. Last case is, F is less than 0 that

means, total number of independent process variables is less than total number of

independent equations and in this case, the system is called over specified. So, to make

this over specified system exactly specified, we need to remove F number of equations.

Usually in practice, the case 2 is the common I mean, F greater than 0 is the common

case in practice, F greater than 0 that means, v greater than 0. Now, thing is that, if this is

the situation F greater than 0, how we can make it exactly specified basically, there are 2

options, first option is we can specify more number of disturbance variables. So, by

specifying more number of disturbance variable, if we can specify more number ofdisturbance variables then, number of unknown variables is reduced.

So, this is one option and in the second option, by incorporating more number of

controller equations. This is a second option, either we have to reduce the number of

unknown variables or we have to increase the number of equations to make the F equals

0. Anyway, we will discuss this degrees of freedom with taking one simple example, we

will consider the stirred tank heater.


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To describe this degrees of freedom analysis, stirred tank heater example for degrees of

freedom analysis. So, before going to analyze the degrees of freedom, we need the model

so, first we will develop the model for the system and then, we will go for the degrees of

freedom analysis. The schematic of this system, we have to draw we have to develop

first so, this is F i and temperature is T i. Now, steam is introduced here through the coil

for heating purpose, this is steam suppose, flow rate is Q, outlet flow rate is F and

temperature is T.

Now, liquid in the tank has the height of h, temperature here also T, cross sectional area

of this tank is A. Now, before going to develop the model, we need to consider some

assumptions so, what are these assumptions, first assumption is, the tank is perfectly

mixed. Second assumption is rho and C p both are constant, third assumption is the tank

is perfectly insulated that means, there is no heat loss from the tank to the surroundings.

So, first we will develop the total mass balance equation so, what is the accumulation of

mass d d t liquid height multiplied by cross sectional area that is, volume.

Volume multiplied by density that is mass, h is the liquid height multiplied by cross

sectional area, this is volume. Now, volume multiplied by density that is mass so, this is

mass flow rate I mean, this is the rate of accumulation. What is the inflow rate, F i rho

minus F rho, this is a outflow rate so, we can write this equation A d h d t equals to F i

minus F. This is the total mass balance equations since rho is constant so, we can get this


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from this equation, you give some equation number for this suppose, this is equation

number 1.

In the next, you will go for energy balance, what is the accumulation term, height

multiplied by area that is volume, volume multiplied by rho, mass. Mass, C p,

temperature difference is T minus suppose, T reference so, d d t of this is the

accumulation term, h multiplied by rho that is volume, multiplied by rho, h multiplied by

A that is volume, multiplied by rho that is mass. So, if this is the mass C p and this is

temperature difference, what is the energy input rate, F i rho this is mass flow rate, C p T

i minus T reference this is the energy input rate.

What will be the energy output rate, F rho C p T minus T reference and another term is

energy supplied by steam per unit time. Energy supplied by steam per unit time that is, Q

basically, the unit of Q is here, energy per unit time say for example, ((Refer Time:

47:47)) thermal unit per minute, the unit of this is energy per unit time. So, this is the

energy added or energy supplied by steam per unit time.


Now, if we simplified this considering T reference equals to 0 and if we simplify the

energy balance equation, we will get A h d T d t equals to F i T i minus T plus Q divided

by rho C p, this is a energy balance equation. If we consider T reference equals to 0 and

if we simplify finally, we will get this energy balance equation. So, there are basically 2


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equations, one is based on total mass balance and another one is based on energy

balance.

So, these two equations are, first one is A d h d t equals to F i minus F and this is A h d T

d t equals to F i T i minus T plus Q divided by rho C p, you give some equation number

to this suppose, this is equation number 2 so, this is the model structure. Now, you will

go for the degrees of freedom analysis, how many variables are involved in this equation

h, F i, F, T, T i and Q so, these are the variables. We can write v equals to 6 agree or not,

agree so, there are 6 unknown variables then, how many equations are involved there,

one is equation 1 and another one is equation 2.

So, we can write E equals to 2 so, what is F, 6 minus 2 that is equals to 4 so, degrees of

freedom for the example system is 4. We have discussed, there are two ways to reduce

the degrees of freedom, first option is we can specifies some load variables, what are the

lowered variables in this system, one is F i, another load variables is T i. So, if we can

specify these two load variables then, the degrees of freedom reduces to 4 minus 2 that

is, 2 initially it was 4.

Now, two lowered variables we are specifying, how we can specify, by the direct

measurement. We can measure this flow rate, we can measure this temperature then, we

can get the information of flow rate and temperature that means, F i and T i are known

that means, degrees of freedom we can write 4 minus 2 that is, 2. Another option, I told

by including some controlled equations, for the example liquid heating tank system, what

are the controlled variable and manipulated variable pairs to be considered, one is height,

another one is temperature.

So, we can manipulate this height, we can control this height by the manipulation of

suppose F, we can control this temperature by the manipulation of Q. So, we can develop

two control equations, although we did not study the control equations, I am just

mentioning the simplest control equations for these two control pairs. If F is the

manipulated variable, the control equation we can write like this, F equals to F s plus k c

F multiplied by h d minus h.

F s is the steady state value of F, k c F is one tuning parameter, which the value of that

tuning parameter we need to determine, that is constant k c F basically, h d is the desiredvalue and h is the liquid height. So, this is one additional equation similarly, if we


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consider another control scheme for temperature, in which Q is the manipulated variable,

we can add another equation Q equals to Q s plus k c Q T d minus T. Here, Q s is the

steady state value of Q, k c Q is the control at tuning parameter, that is a constant term

then, this is desired temperature.

So, additionally, we are getting two equations, we had F equals to 2 now, if we can add

two equations then, the degrees of freedom becomes 0. So, we had basically degrees of

freedom 4 additionally, we have specified two load variables through direct

measurement. Then, we have just paired control variable manipulated variable then, we

got to additional equations. And finally, the degrees of freedom becomes 0 that means,

the system is exactly specified.

Thank you.


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Prof. A. K. Jana



Lecture - 4Mathematical Modeling (Contd.)

Today, we will continue our discussion on Mathematical Modeling and we will develop

today the mathematical modeling of a distillation column.


Today, we will develop the model of a distillation column previously, we have

configured three control scheme for a distillation column but, we did not discuss the

modeling. So, first the configuration of a distillation column, this is the tower column

section, this is the top tray, this is the feed tray and this is suppose, the bottom tray. Now,

feed is introduced to this feed tray, this is the feed stage or feed tray, feed has the flow

rate of F.

Suppose, this feed mixture contains only two components they are A and B, feed mixture

contains two components namely A and B. The composition of this feed mixture is z

now, the vapor which is living this top tray, this top tray is denoted by N. A vapor steam

which is living this top tray has the flow rate of V N, the vapor steam which is living this

top stage has the flow rate of V N. Now, this vapor steam is condensed in an over hit

condenser, this is a condenser.


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After condensation, the produced liquid, condensed liquid is accumulated in a drum, this

is called reflux drum. The condensed liquid is accumulated in this reflux drum suppose, a

holdup of this condensed liquid in the reflux drum is m D. If part of this accumulated

liquid is withdrawn as top product, this is top product, this is also called distillate, top

product is also called distillate and definitely, this is a liquid steam. Now, we will assume

the flow rate of this distillate is D and composition is x suffix D.

The top product flow rate we are representing by capital D and composition is by x D, a

fraction of this accumulated liquid is recycled back to the top tray, this stream is called

reflux stream. This reflux stream has the flow rate of R and composition is same with the

distillate steam I mean, the composition is x D. Similarly, at the bottom section, the

liquid which left this bottom tray, this is we can say first tray, this is first stage, the liquid

which is living this first stage is accumulated in this column base.

Suppose, the holdup in the column base is m suffix B now, this liquid then goes to a

bottom reboiler and vaporization of liquid occurs in this reboiler. Then, the vaporized

stream is recycle back to the bottom stage suppose, this vapor flow rate is V suffix B, the

flow rate of this recycle vapor is suppose V B and the part of that accumulated liquid in

the column base is withdrawn as bottom product. This is bottom product, this is also a

liquid stream, it has the flow rate of suppose B and composition is x B, the flow rate of

this bottom product is suppose B and composition is x B.

So, we are basically introducing to this column is single feed stream and we are getting

two products, top product and bottom product. Now, this condenser is basically a total

condenser, the operate condenser is actually a total condenser because, the operate vapor

is totally condensed, the operate vapor which is entering the condenser is totally

condensed. We can call this condenser as a partial condenser, there is a operate vapordistillate in forward.

When there is operate vapor distillate involve then, we can only say this is a partial

condenser but here, we are considering, it is a total condenser so, there is no operate

vapor withdrawn. So, this is all about the description of the distillation operation next,

will develop

Process Control and Instrumentation by Prof. a. K. Jana Department of Chemical Engineering Indian...

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