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Helwan University
Faculty of Engineering
Helwan
Prepared by
Dr. Mohiy Bahgat
2012
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Course Contents
Chapter1:Components and characteristics of industrialprocesses
1.1. What is a process?
1.2. What does a control system do?
1.3. Why is control necessary?
1.4. Why is control possible?
1.5. How it can be done?
1.6. Where it can be implemented?
1.7. What are the control engineers interests?
1.8. How can the process control be documented?
1.9. Control strategies.
1.10. Components of industrial processes.
1.11. Exercises.Chapter 2: Mathematical modeling of industrial
processes
2.1. Modeling Procedure.
2.2. Linearization.
2.3. Numerical Solution of ODE.
2.4. Model Analysis of Processes.
2.5. Exercises.
Chapter 3: Measurement of control system parameters
3.1. Temperature Sensors.
3.2. Position sensors.
3.3. Speed Sensors.
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3.4. Pressure Sensors.
3.5. Force Sensors (strain gauges).
3.6. Fluid Sensors.
3.7. Flow Measurements.
3.8. Exercises
Chapter 4: Industrial controllers
4.1. On-off
4.2. P, I, D, PI, PD, PID
4.3. Temperature control
4.4. Pressure control
4.5. Flow rate control
4.6. Level control
Chapter 5: Forward control, Sequential control &
Multi-circuit
5.1. Feedback Control.
5.2. Multivariable Control.
5.3. Feed Forward Control.
5.4. Feed Forward plus Feedback Control.
5.5. Cascade Control.
5.6. Batch Control.
5.7. Ratio Control.
5.8. Selective Control.
5.9. Fuzzy Control.
Chapter 6: Introduction to process automation
6.1. Introduction.
6.2. PLC Operation Scan.
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6.3. PLC Addressing.
6.4. Relay Ladder Logics (RLL).
6.5. Exercises.
Chapter 7: Application to process automation using
PLCs
7.1. Signal Lamp Simple Process.
7.2. Machine Safety Process.
7.3. Central Heating Process.
7.4. Automatic Mixing Process.
7.5. Automatic Packing Process.
References
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Chapter (1)
Components and
Characteristics of
Industrial Processes
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Chapter (1)
Components and Characteristics
of Industrial Processes
Process control is an engineering discipline that deals with architectures,
mechanisms, and algorithms for controlling the output of a specific
process. This can be simple as making the temperature in a room kept
constant or as complex as manufacturing an integrated circuit.
For example, heating up the temperature in a room is a process that has
the specific, desired outcome to reach and maintain a defined temperature
kept constant over time. Here, the temperature is the controlled variable,
at the same time, it is the input variable since it is measured by a
thermometer and used to decide whether to heat or not. The desired
temperature is called the set-point. The state of the heater, for example,
the setting of the valve allowing hot water to flow through it; is called
the manipulated variable since it is subject to control actions.
A commonly used control device called a programmable logic controller,
or a PLC is used to read a set of digital and analog inputs, apply a set of
logic statements, and generate a set of analog and digital outputs. Using
the previous heating example, the room temperature would be an input to
the PLC. The logical statements would compare the set-point to the input
temperature and determine whether more or less heating was necessary to
keep the temperature constant. A PLC output would then either open or
close the hot water valve, an incremental amount, depending on whether
more or less hot water was needed.
Larger more complex systems can be controlled by a Distributed Control
System (DCS) or SCADA system.
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To make a good introduction to the process control, we should answer
some questions such as :
1. What is a process?2. What does a control system do?3. Why is control necessary?4. Why is control possible?5. How it can be done?6. Where it can be implemented?7. What are the control engineers interests?8. How can the process control be documented?9. What are control strategies?
1.1.What is a process control?Process control is an engineering discipline that deals with
architectures, mechanisms, and algorithms for controlling the output of
a specific process. This can be as simple as making the temperature ina room kept constant or as complex as manufacturing an integrated
circuit.
In practice, the industrial processes are different in behavior,
architecture and characteristics. So, they can be characterized as one or
more of the following forms :
1. Discrete processes.2. Batch processes.3. Continuous processes.4. Hybrid processes.
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Discrete process : it can be found in many manufacturing,motion and packaging applications. Robotic assembly, such as that
found in automotive production, can be characterized as discrete
process control. Most discrete manufacturing involves the
production of discrete pieces of product, such as metal stamping.
Fig (1.1)Robot arm control in a discrete process.
Batch process : where some applications require that specificquantities of raw materials be combined in specific ways for
particular durations to produce an intermediate or end result. One
example is the production of adhesives and glues, which normally
require the mixing of raw materials in a heated vessel for a period
of time to form a quantity of end product. Other important
examples are the production of food, beverages and medicine.
Batch processes are generally used to produce a relatively low to
intermediate quantity of product per year (a few pounds to millions
of pounds).
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Fig (1.2)Batching processes.
Continuous process : often, a physical system is representedthough variables those are smooth and uninterrupted in time. The
control of the water temperature in a heating jacket, for example, is
a form of continuous process control. Some important continuous
processes are the production of fuels, chemicals and plastics.
Continuous processes, in manufacturing, are used to produce very
large quantities of product per year, millions to billions of pounds.
Fig (1.3)Continuous reject process.
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Hybrid processes : Applications having elements of discrete,batch and continuous process control are often called hybrid
applications.
Fig (1.4)Product line processes as a hybrid process.
1.2.What does a control system do?A control system normally performs three main steps :
1.Measurement process for the variable to be controlled, orcollecting data from the controlled plant. This is done by sensors
or data acquisition cards.
2.Comparison between the measured variable and a referencevalue, doing some calculations to get the change in the variable,
or data processing for the collected data. This is done by
comparators, or through running of an algorithm or program.
3.Making a final decision in order to maintain the sensed variablewithin a desired range, or sending some control signals to the
controlled plant. This is done via the system actuators or final
control elements.
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The following sample of examples illustrates the process manual control
steps and the corresponding automatic process control scheme.
Level process control :
Fig (1.5)Manual level process control steps.
Fig (1).6Automatic level process control system.
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Heating process control :
Fig (1.7)Manual heating process control steps.
Fig (1.8)Automatic heating control system
So, the final goal of the control is to maintain or adapt desired conditions
in a physical system or an industrial process by adjusting selected
variables in that system. This can be done by making a use of an output
signal of the system to influence an input signal of the same system,
which called feedback control.
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1.3.Why is control necessary?The control and dynamic operation is an important factor in an
industrial process design. In other words, the industrial processes need
some degree of control for two main reasons :
1.The first one is to maintain the controlled conditions orvariables in a physical system or an industrial process at the
desired values when small or large disturbances occur.
2.The second reason is to respond to changes in the desiredvalues by adjusting the selected variables in the process. The
response is based on the analysis of the process operation and
objectives.
Finally, the process control will assure the following issues :
a. Safety.b.
Environmental protection.
c. Equipment protection.d. Smooth plant operation.e. Product quality.f. Profit optimization.g. Monitoring and diagnosis.
These issues are usually translated into values of the system or process
variables such as temperature, pressure, flow rate, liquid level, speed of
a motor or conveyor, displacement and so forth which are to be
controlled.
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1.4.Why is control possible?If the plant or the industrial process equipment is not properly
designed, the control system will perform poorly, inadequately or
might be impossible. Therefore, when designing an industrial process
or a plant, several considerations must be accounted such as :
a. Providing adequate equipment : which means includingadequate rapidly responded sensors for the process variable
and appropriate final control elements so that the control
actions can be taken in real time? Moreover, such sensors and
final control elements should be shielded and protected
against the surrounding effects due to the process operation.
b. Expected changes in the plant variables : which concernsabout the anticipation of the expected changes in the process
disturbances or the desired values of the controlling variables
and providing or adding adequate equipment during the plant
design? So, the adequate design calculations must be based on
the expected changes.
c. Adding a percentage extra capacity for the equipmentsizing : this is to allow the plant equipment to respond to all
expected disturbances or system variables by merely adding a
percentage extra capacity in accordance to the anticipated
changes.
If the previous considerations are not correct, or the plant design is not
accurate, the control may not be possible and the plant operation
through manipulating the final control elements may not be achieved.
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1.5.How can control be done?1.In simple process control, it can be done using the human
feedback.
2.In complex processes the feedback actions are automated bysensing, calculating, manipulating the controlled variables by
communicated parts of the control system. Currently, most
automatic control is implemented using electronic equipment at
some levels of current or voltage to represent the values to be
communicated.
3.So, one can say that, the process control is done automaticallyusing instrumentation and computation that perform all the
features of feedback control without requiring or allowing the
human intervention.
1.6. Where can control be implemented?
In order to operate an industrial process on a minute-to-minute basis, a
lot of information from much of the process has to be available at a
central location which known as the control room or control center.
Such control scheme is generally known as SCADA system where :
Sensors and control elements are located in the process.Signals which are mostly electronic or communications with the
control center to be viewed to the operator.
Distances between the process and the control center rangesfrom few hundred feet to a mile or more.
In some processes, small control panels are used nearby theequipment to allow access to them.
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Fig (1.9)Local and centralized control equipment
1.7. What are the control engineers interests?
The main interests of the process control engineers are :
a. Process design : where the process must be designed suchthat being with rapid response and minimal disturbances.
b. Measurements : where the sensors has to be selected withrapid response and high accuracy.
c. Final elements : where the final control elements must beprovided and handled so that the manipulated variables can be
adjusted by the control calculation.
d. Control structure : where the basic issues in designing thecontroller must be considered such as which control element
should be manipulated to control which measurement.
e. Control calculations : where equations are used to handle themeasurements and the desired values in calculating the
manipulated variables.
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1.8. How can the process control be documented?
The process control can be documented in many forms :
a. Equipment specifications and sizing.b. Operating manuals.c. Technical experiments and control equations.d. Engineering drawings.e. ROMs for storing the control algorithms.f. Additional EPROMs.
Fig (1.10)Stirred-tank with composite control
Fig (1.11)Flow controller
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The process drawings include some symbols such as :
A analyzer.
F flow rate.
L level of liquid or solids in a vessel. P pressure. T temperature.
and so on
1.9. What are the control strategies?
The following diagram displays a sample of the most commonly used
control strategies. Of course, the control strategy is different from one
process to another in accordance to its topology, complexity and
objectives.
Classical
ControlModern
Control
Industrial
Controllers
+
PLCs
Adaptive
Control
Optimal
Control
Robust
Control
A.I
Control
Computer Control
Control Strategies
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1.10. Components of industrial process
The industrial processes comprise several types of components :
1. Process : which in general can consist of a complexassembly related to some manufacturing sequence. The
process involves some variables needed to be controlled in
order to accomplish the desired goal of it. So, one can say that
the process may be single variable process or multivariable
process according to the number of variables to be controlled.
2. Measuring elements (sensors) : which representsthe devise that transforms or converts the measured variables
into some forms required by the other elements in the process
control operation. Signal conditioning may be required to
complete the measurement function in some cases.
3.Error detectors (comparators) : which is a physical
part of the controlling circuit that determines the difference
between the actual variable and the set-point value before
taking any control action.
4. Controllers (industrial or computer) : whichperforms the action should be taken in accordance to the
determined error and regulates or compensates the controlled
variable to bring it to the desired set-point or reference value.
5. Final control elements (actuators) : which is thedevice that exerts a direct influence on the process or provides
the required changes in the controlled variable to bring it to
the set-point.
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1.11. Exercises :
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Chapter (2)
Mathematical
Modeling of
Industrial Processes
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Chapter (2)
Mathematical Modeling of
Industrial Processes
2.1.Modeling Procedure :The general steps for building a mathematical model of a process can
be summarized as follows :
1. Define goals :a. Specific design decisions, which means that the goal should be
specific and clear.
b. Numerical values, where the goal is sometimes beingrepresented by numerical values.
c. Functional relationship, where in some cases the systemsbehavior is the goal.
d. Required accuracy, the model accuracy should also beincluded in the goal definition.
2. Prepare information :The information needed to be prepared are :
a. Sketch process and identify the system : identifying theprocess, the key variables and the system boundaries.
b. Identify variables of interest : the data regarding the physicalprocess components and the external inputs to the process.
c. State assumptions and data : the assumptions on which themodel will be built on.
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3. Formulate the model:When formulating a model of an industrial process, the first step is to
select the variables whose behavior is predicted and then deriving the
equations based on the conservation balance in mass and energy in
addition to the accumulation as follows :
a. Formulate the conservation balances (Energy balance Eqns.)b. Formulate the constitutive equations.c. Combine equations and collect terms.d. Check degrees of freedom.e. Convert to the dimensionless form.
Material Balance :
Accumulation of mass = (Mass)in(Mass)out
Energy Balance :
Accumulation of energy = (H + PE +KE)in
(H + PE + KE)out + QWs
where :
H : enthalpy = E + pv
PE : potential energy
KE : kinetic energy
Ws : work done by the process on the surroundings
Q : heat transferred to the process from the surroundings.Q = h . A . T
E : internal energy
pv : flow work
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Fig (2.1)Energy balance in a process.
4. Determine the solution :Determining the process model solution is very important. This can
be made either analytically or numerically. The analytical solutionunder some approximations is usually sought first. If such solution
results in unacceptable errors, numerical solutions are then sought.
Despite they are not exact but errors can be made less. The
analytical solution steps are :
Calculate the required specific numerical values. Determine the important functional relationships among the
process model, variables and system behavior.
Make a sensitivity study of the results associated with datachanges.
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5. Analyze the results :a.The first step of result analysis is to evaluate whether the
solution is correct or not, this can be done by ensuring the
following :
The results satisfy initial and final conditions. Obey the process bounds. Contains negligible errors associated with numerical
solutions.
Obey the process semi-quantitative expectations suchas output change sign.
b.The second step of result analysis is to analyze the processbehavior, this can be done by :
Determining the numerical results quantitavelly to helpin making decisions regarding the equipment operation
and sizing.
Plotting the results. Observing the process characteristic behavior like
oscillations in case of max or min oscillations.
Evaluate sensitivity, which means studying the processbehavior associated with change in data or important
variables.
Relate results to data and assumptions. Answer what if questions.
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6. Validate the model :The model validation involves determining whether the results
obtained in the previous steps are truly representing the physical
process.
This can be done by comparing the obtained results with some
experimental results taken from the process at different operating
points to assure the model validity in representing the process. In
other words, the steps of model validity are :
a. Select key values for validation.b. Compare with experimental results.c. Compare with results from more complex model.
The previous procedure can be divided into two main sections :
a.Model development steps (steps 1 to 3).b.Model solution and simulation (steps 4 to 6).
Example (1) :For the mixing tank shown in figure :
Fig (2.2)Mixing process configuration.
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1. The goal is to determine the dynamic response due to a stepchange in the inlet concentration. In other words, determining the
time needed for the outlet reaches 90% of change in concentration
after the step change in the inlet.
2. The information :a. The process is the tank with its fluid in it, its design and shape
and the speed of making the fluid uniform.
b. Assumptions : well-mixed vessel, density is the same for A andsolvent S in addition the flow in is constant.
c. Data : F0 = 0.085 m3/min , CAinit = 0.925 mole/m3 , CA0 =0.925 mole/m
3and CA0 = 1.85 mole/m
3after the step. The
system is initially at steady state.
3.The model formulation :Since the problem involves concentration, hence using the material
balance equation we can get :
Accumulation of mass = Mass inMass out
(V)(t+t) (V)(t) = Fo .t F1 .t
Dividing by t and taking the limit as t 0
Assuming that the level in the tank is almost constant, which means
that the flows in and out are equal, i.e : Fo = F1 = F
or : dV/dt = Fo F1 = 0 , i.e : V = constant (1)
Applying the same material balance for component A :
Accumulation of comp A = Comp AinComp Aout
10 F-FdtdV
dt)V(d
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(M WA V CA)(t+t)(M WA V CA)(t) = ( M WA F CAo
M WA F CA )t
Dividing by t and taking the limit as t 0
(2)
Applying the same material balance for component S :
Accordingly :
The process variables are :CA and F1
The external variables are :F0 and CA0
The process model is represented by equations (1) & (2) .4.Determine the solution :
As it can been seen from eqn.(2) the process model is a linear 1st
order
ordinary differential equation that can be transformed to the separable
form using an integral factor as follows :
, with V/F = = time constant
Use the integrating factor I.F =e( (1/)dt
= et/
)C-(CFWMdt
dCVWM AA0A
AA
)C-(CFWMdt
dCVWM ss0s
ss
)C-(CFWMdt
dCVWM AA0A
AA
)C-(CF
dt
dCV AA0
A
A0AA C
1C
1
dt
dC
t/A0AAt/ eC
)C1
dt
dC(e
t/A0At/t/
A
At/
e
C
dt
)Ce(d
dt
ed
Cdt
dC
e
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Using the initial conditions, we get : K = CAinitCA0
Substituting with the given numerical values :
(3)
Two aspects of the process dynamic response have to be considered :
The speed of response which is characterized by the timeconstant .
The steady state gain which is :
1. Result analysis :The solution of the process model described in eqn. (3) is an
exponential curve as displayed in the Fig (2.3). The process response
from the change beginning to the end is affected by the time constant
(), where the large time constant the slow process response and vice
versa. According to the goal, it is needed to know the time taken to get
90% of the change in outlet concentration. This time can be calculated
from eqn. (3).
dteC
dteC
)eCd( t/A0t/
A0t/A
KeC
eC t/A0t/A
-t/A0A eKCC
)e-(1])(C-[CC-C
e.)C-(CCC
t/-initA0A0AinitA
-t/A0AinitA0A
)e-(1].9250-[C0.925-C -t/24.7A0A
1C
C
input
outputK
A0
Ap
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Fig (2.3) - Dynamic response of the process.
6.Validation :By performing an experiment on a stirred tank as described in the
controlled process and taking samples of the outlet material,
analyzing the obtained samples and drawing the data points, one can
get the shown Fig (2.4). By visual evaluation, one can say the model
is valid in representing the process.
Fig (2.4)Model validation of the process.
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Example (2) : for the On-Off room heating processshown in figure :
Fig (2.5)On-Off room heating process configuration.
1. The goal : is to determine the dynamic response of the roomtemperature. Also, ensure that the furnace does not switch on or off
more than once per 3 minutes.
2. The information :a. The process is the air inside the room. The important variables
are the room temperature and the furnace on-off status.
b. Assumptions : the air in the room is well mixed, no transfer ofmaterial to or from the room, the heat transferred depends only
on the temperature difference between the room and the outside
environment, no heat is transferred from the floor to the ceiling
and effects of kinetic and potential energies are negligible.
c. Data : the heat capacity of the air CV = 0.17 cal/g C, the overallheat transfer coefficient UA = 45 x 10
3
cal/C hr, the size of the
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room is 5 m by 5 m by 3 m high, the furnace heating capacity Qh
is 0 (of) or 1.5 x 103 cal/hr (on), the furnace switches inst. At 17
C (on) and at 23 C (off), the initial room temp. is 20 C, and the
outside temp. is 10 C.
3.The model formulation :Since the process is defined as the air inside the room, hence using the
energy balance equation one can get :
dE/dt = KE + PE + QWs
KE = PE = Ws = 0 from assumptions
dE/dt = Q .. (1)
but dE/dt = V CV dT/dt
and Q = - UA (TTa) + Qh
and Qh is represented by :
0 when T > 23 C
Qh = 1.5 x 106 when T < 17 C
unchanged when T < 23 C
Finally, the process model is :
.. (2)
Accordingly :
The process variables are :T and Qh
The external variables are :Ta
The process parameters are :UA , CV , V and
haV Q)T-T(UA-dt
dTCV
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4.Determine the solution :Rearranging eqn. (2) gives the process model which is a linear 1 st
order ordinary differential equation that can be transformed to the
separable form using an integral factor as follows :
Use the integrating factor I.F =e( (1/)dt
= et/
Using the initial conditions, we get :
.. (3)
where : t = time from step in Qh
= time constant = 0.34 hr
Tfinal = final value of T as t = Ta + Qh / UA
= 10 C when Qh = 0
= 43.3 C when Qh = 1.5 x 106
Tinit = the value of T when a step in Qh occurs.
UA
CVwith,
CV
QUA TT
1
dt
dT v
v
ha
v
hat/t/
CV
QUA T.e)T
1
dt
dT(e
v
hat/t/t/
t/
CV
QUA T.e
dt
)T.d(e
dt
deT.
dt
dT.e
dtCV QUA T.e)T.d(e v hat/t/ dteCV QUA T)T.d(e t/v hat/
KeCV
QUA T.T.e t/
v
hat/
t/-
v
ha e.KCV
QUA T.T
)e-1()T-T(T-T -t/initfinalinit
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Fig (2.6) - Dynamic response of the process.
5.Result analysis :From the previous figure (2.6), it can noticed that the room
temperature decreases until it reaches 17 C, the furnace will start
heating and the temperature increases until it reaches 23 C. This
process will be repeated with the heater On and Off periodically.
2.2.LinearizationIf the developed process model is linear, analytical solutions can be
obtained easily. Most of the physical system models are nonlinear. The
analytical solutions of the nonlinear models are not available, thus the
numerical simulations are sought. Instead of obtaining non-
understandable solutions for the nonlinear models by numerical
simulations, approximate linearized solutions can be used for
representing realistic processes.
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A model is to be linear if it satisfies the properties of additivity,proportionality and superposition which mean that when it is
subjected to a sum of inputs, a sum of outputs will result. For
example, if there is a system with an input of :
x3(t) = x1(t) + x2(t)
it should result in an output of :
y3(t) = y1(t) + y2(t)
A system satisfies the property of superposition, if a sum of scaledinputs results in a sum of scaled outputs. i.e :
f(Ax + By) = f(Ax) + f(By) = A . f(x) + B . f(y)
If the system has the following performance equation :f(x) = k . X
f(Ax1 + Bx2) = k . (Ax1 + Bx2)
k . (Ax1) + k . (Bx2)
Thus the above system in not linear, it is nonlinear system, and so on.
To illustrate the dynamic behavior of a process, consider the following
example stirred tank heat exchanger when being subjected to a change in
the feed temperature and cooling fluid flow rate.
Fig (2.7) - Stirred tank heat exchanger
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Fig (2.8) - Process response due Fig (2.9) - Process response due to
to a change in feed temperature a change in cooling fluid flow rate
Fig (2.10) - The total process response due to a change in both feed
temperature and cooling fluid flow rate
According to the dynamic behavior of the process, one can say that thisprocess is linear because it obeys the superposition principle. In general a
nonlinear process model can be linearized and approximated by a linear
model using Taylor series expansion. For example, a process nonlinear
model with one variable can be linearized around its S.S point as :
R)xx(
dx
dF
2
1)xx(
dx
dF)F(x)x(F 2s
x
s
x
s
ss
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A process nonlinear model with two variables can be linearized around its
S.S point as follows :
Fig (2.11) - Comparison between linear and exact nonlinear models.
Function examples :
1. F(x) = x xs + xs- (xxs)2. F(x) =
R)xx)(xx(
xx
F
2
1)xx(
x
F
2
1
)xx(x
F
2
1)xx(
x
F
)xx(
x
F)x,xF()x,x(F
s22s11
x,x21
22
s22
x,x
2
2
2s11
x,x
2
2
s22
x,x2
s11
x,x1
s2s121
s2s1s2s12
s2s11s2s1
s2s1
xa1
x
)xx()xa1(21
xa1
x s2
ss
s
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Example (3) : for the tank draining process shown infigure :
Fig (2.12)Tank drainingprocess configuration.
1.The goal : is to determine the model of this tank process. Also,evaluate the accuracies of the linearized model at small (10 m3/hr) and
large (60 m3/hr) step changes in the inlet flow rate.
2. The information :a.The process is the liquid in the tank. The important variables are
the level and the flow out.
b.Assumptions : the density is constant, the cross sectional area ofthe tank A does not change with height, the system is at quasi-
steady state because the pipe dynamics is fast with respect to that
of the tank level, the pressure is constant at inlet and outlet,
c.Data : the initial steady state conditions are : Flows F0 = F1 =100 m3/hr, Level L = 7 m, the cross sectional area A = 7 m2 .
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3.The model formulation :Since the process is defined as the liquid in the tank and the level
depends on the total amount of liquid, thus using the material balance
equation one can get ::
. (1)
Another equation is required, one can relate the outlet flow to the head
as follows :
. (2)
Finally, combining the two eqns., the process model will be :
. (3)
This model has a nonlinear term which can be linearized as :
... (4)
Replacing the nonlinear term in Eqn.(3) and Subtracting the S.S
conditions and putting the input as a constant step : Fo= Fo , one can
get the final process model as :
.. (5)
Accordingly :
The process variables are :L
The external variables are :Fo
The process parameters are :A and kF1
F-Fdt
dLA 1o
Lk)P-L-(PkF0.5
F1
0.5
aaF11 0.5
Fo Lk-Fdt
dLA
1
)L-(LL0.5LL s-0.50.50.5ss
L')L(0.5k-F=dt
dL'A 5.0Fo s1
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5.Determine the solution :Rearranging eqn. (5) gives the process model which is a linear order
ordinary differential equation that can be transformed to the separable
form using an integral factor as follows :
Use the integrating factor I.F =e( (1/)dt
= et/
Using the initial conditions, we get :
where : t = time from step in Fo
= time constant = 0.98 hr
kp = 0.14 hr/m2
0.5-sF1
oLk5.0
Awith,F
A
1L'
1
dt
'dL
ot/t/ F
A
1.e)L'
1
dt
'dL(e
ot/
t/t/t/ F
A
1e
dt
)'Ld(e
dt
deL'
dt
'dLe
dtFA1e)'Ld(e ot/t/ dteAF)'Ld(e t/ot/
KeA
F'Le t/ot/
t/-o e.KA
F'L
A
FK o
)e-1(
A
F'L t/-o
5.0s1F
pt/-
poLk5.0
1
AKithw)e-1(KF'L
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Fig (2.13) - Process response to a small change in inlet flow rate.
Fig (2.14) - Process response to a large change in inlet flow rate.
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6.Result analysis :From the previous figures (2.13) and (2.14), it can be noticed that the
solution of the linearized model is quite accurate with the small
change in the inlet flow rate.
On the other hand, it is inaccurate with the large change in the inlet
flow rate and it gives impossible negative level at S.S.
The general trend is the linearized model is more accurate with small
changes than with the large ones.
2.3.Numerical Solutions of O.D.EAs seen before, most of the practical modeling of process and process
control would result in nonlinear models. The nonlinear algebraic and
differential models cannot be solved analytically. Such models are
solved using different methods of numerical solutions.
The numerical solutions do not give expressions as before, but they
give points close to the exact solutions of the process models. The
concept of the numerical methods is to use initial values and an
approximation of the derivatives over a step of integration, and hence
calculate the variables after that step. Most of the numerical methods
for solving differential equations consider the Taylor series expansion
and make approximations by choosing specified terms of the series.
The most commonly used methods are :
1.Eulers method : which considers the first two terms of theTaylor series :
yi+1 = yi + f(yi, t).t
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2.Heuns method : which considers the first three terms of theTaylor series :
k1 = f(yi, t).tyi+1 = yi + f(yi+k1/2 , t+t/2).t
3.Runge-Kutta fourth order method : which considers thefirst four terms of the Taylor series?
yi+1
= yi+t/6 (k
1+2k
2+2k
3+k
4)
where :
k1 =
k2 =
k3 =
k4 =
The selection of the step size t is very important in reaching theapproximate solution of the process model using the numerical
solutions.
In Eulers method the error is proportional to the step size t,however, in Runge-Kutta method the error is proportional to
(t)4.
In most engineering applications, the appropriate step size ist = 0.01 sec.
)t,y(f ii
)2
tt,
2
k.ty(f i
1i
)
2
tt,
2
k.ty(f i
2i
)
2tt,
2k.ty(f i
3i
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2.4.Model Analysis of ProcessesThe process model which comprises a linear differential equation can
be solved by analytical solution. On the other hand, the process model
which comprises a set of linear differential equations with constant
coefficients can be solved by Laplace transform method.
A control system involves several simultaneous processes and control
calculations can be modeled using input and output variables with the
aid of block diagrams and transfer functions. The process behavior to
sine inputs can be carried out easily using the frequency response
method to illustrate the influence of input frequency.
When a process is subjected to a step disturbance, it is required to
determine whether its behavior is stable or not.
2.4.1.Laplace transformThe Laplace transform is a very powerful method for engineers to
analyze the process control and control systems. It converts the constant
coefficient differential equations to algebraic equations which can be
solved easily. It replaces the time domain by a frequency domain or
complex domain. The Laplace transform is defined as :
The Laplace transform is linear operator, i.e :L[a.f1(t) + b.f2(t)] = a.L[f1(t)] + b.L[f2(t)]
Inverse Laplace can be achieves as :L-1[F(s)] = f(t) for t 0
Laplace transform for a constant :L(C) =
0
st- dtf(t).eF(s)))t(f(L
s
C
es
C
-dteC
st-
0
st-
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Laplace transform for an exponential :L(eat) =
Laplace transform for a step function :
The Laplace Transform properties can be summarized as follows :ExampleProperty
L[a.f (t) + b.g(t)] = a.F(s) + b.G(s)Linearity
Time Change
L( f (t T )) = esT F(s)Time axis displacement
L(eatf (t)) = F(s a)S axis displacement
Initial Value Theorem
Final Value Theorem
a-s
1e
s-a
1dtee a)t-(s-
0
st-at
0t0
0t;A)t(x
s
Ae
s
A-)s(X
dtA.edtx(t).eX(s)
0
st-
0
st-
0
st-
)a
s(F
a
1)]at(f[L
)s(FsLim)t(fLim s0t )s(FsLim)t(fLim 0st
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Tables of Laplace transform and its inverse transform are availablefor most commonly used functions as follows :
Laplace F(s)Function f(t)
1 unit impulse
1 / sU(t) unit step or constant
A / s
A / s2
n! / sn+1tn
1 / (s-a)eat
1 / (s+1)
/ (s2 + 2)sin (t)
s / (s2 + 2)cos (t)
s F(s)
sn F(s)
F(s) / s
0t0
0t;A)t(u
0t0
0t;At)t(r
at0
at)at(f)t(f
/te
1
F(s)e as
dt
)t(df
n
n
dt
)t(fd
dt)t(ft
0
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2.4.2. Partial FractionsIn order to get the time response for any plant or process, the dynamical
model of such plant should be solved. When the model in derived in
Laplace domain, the inverse Laplace can be utilized as follows :
Taking the inverse Laplace for both sides, one can get :
where Ci are constants and Hi(s) are the factors of the characteristic
polynomial D(s) = 0.
If H(s) is a 1st order term, then :
If H(s) is a 2nd order term, then :If there are repeated factors :
Example (4) :For the stirred-tank mixing model in deviationvariables is :
using Laplace transform :
V s CA(s) = F [ CAO(s)CA(s) ]
s CA(s) + CA(s) = CAO(s)
.....)s(H
C
)s(H
C
)s(D
)s(NY(s)
2
2
1
1
....])s(H
1[LC]
)s(H
1[LCY(t)
2
1-2
1
1-1
.....)s(H
B
)s(H
A
)s(D
)s(NY(s)
21
.....)s(H
C)s(HBsA
)s(D)s(NY(s)
21
1n1n
221
n )as(
C.....
)as(
C
)as(
C
)as(
M(s)Y(s)
)C-(CFdt
dCV 'A
'A0
'
A
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CA(s) ( s + 1) = CAO(s)
Considering a step change in the inlet concentration, i.e :
CA0(s) = CA0/s
Using the inverse Laplace, one can get :
Example (5) :Consider an industrial process having a modelin deviation variables as:
Using Laplace transform :
Take the input x(t) as a step function, then its Laplace transform will
be : and the output final equation is :
By then, we have four conditions described as follows :
1. if > o the process characteristic equation will have two realdistinct roots as :
1)s(s
1C(s)C A0
'A
)1ss
1(C(s)C A0
'A
)e-(1C(t)C-t/
A0
'
A
G.x(t)y(t)(t)y'2(t)y"1
20
20
G.x(s)Y(s)sY(s)2Y(s)s1
20
2
20
x(s).
s2s
.GY(s)
20
2
20
s
AX(s)
]s2s.[s
.GAY(s)
20
2
20
222,1 --r
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Finally, the process output y(t) will be :
This condition is called the over damping condition where the
output response does not have any oscillations as shown :
Fig (2.15) - Output response of an over
damped controlled process.
2.if = o the process characteristic equation will have two realequal roots as :
Finally, the process output y(t) will be :
This condition is called the critically damped where the output
also does not have oscillations as shown :
)r(s
k
)r(s
k
s
GA
)rs()r(ss
.GAY(s)
2
2
1
1
21
20
tr
2
tr
121 ekekG.A)t(y
-rr 2,1
221
2
20
)r(s
k
)r(s
k
s
GA
)r(ss.GAY(s)
t21 e)k(kG.A)t(y
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Fig (2.16) - Output response of an critically
damped controlled process.
3.if < o the process characteristic equation will have twocomplex conjugate roots as :
Finally, the process output y(t) will be :
This condition is called the under damped where the output has
decayed oscillations as shown :
Fig (2.17) - Output response of an under
damped controlled process.
j-r 222,1
)r(s
k
)r(s
k
s
GA
)rs()r(ss
.GAY(s)
2
2
1
1
21
20
)tsin(ekG.A)t(y dt
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4.if = 0 the process characteristic equation will have twocomplex conjugate roots as :
Finally, the process output y(t) will be :
This condition is called the oscillatory condition where the output
will have continuous oscillations as shown :
Fig (2.18) - Output response of an oscillatory process.
2.4.3.Transfer FunctionThe process transfer function is defined as the Laplace transform of the
output Y(s) divided by the Laplace transform of the input X(s).
Transfer Function = T.F = G(s) =
jrr 2,1
)j(s
k
)j(s
k
s
GA
)j(s)j(ss
.GA
Y(s)
21
20
t)(scokG.A)t(y 0
)s(X
)s(Y
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Basic definitions :
Order : the system order is the highest power of s in thedenominator of the T.F.
Pole : it is a root of the denominator of the T.F or a root of thesystem characteristic equation.
Zero : it is a root of the numerator of the T.F.Steady state gain : it is the ratio Y/X at steady state or
yss/xin and usually denoted by K.
Example : For a system or a process whose T.F is :
The system is 2nd order The poles are : s = - 8.95 j 5.92 The zero is : s = 7.64 The S.S gain is : K = 45.83/35.8 = 1.28
2.4.4.Block diagramThe block diagram method is a powerful graphical representation for
the system or process individual components based on their T.F. It has
some advantages :
1.It retains individual systems and allows model simplificationand changes.
2.It provides a visual representation of the relationshipsbetween the system components.
3.It gives insight into the effect of components on the overallsystem performance.
8.35s789.1s
45.83-s6
)s(X
)s(Y2
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Fig (2.19) - Block diagram for a general control system.
Fig (2.20) - Physical process control.
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Fig (2.21) - Block diagram of the process control.
Fig (2.22) - OnOff control of a heating or cooling process.
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Fig (2.23) - Analog control of the heating process.
Fig (2.24) - Digital control of the heating process.
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Fig (2.25) - PLC control of the heating process.
Fig (2.26)Multi-loop industrial process control system.
315 psi
L
Flow
valve
Flow
valve
Flow
valve
315 psi
315 psi
420 mA
420 mA
420 mA
Flow
sensor
Temp.
sensor
Level
sensor
Steam
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Block diagram Notations :
There are some standard symbols to be used in developing the block
diagram of any system plant or process. The following is a sample of
such symbols :
X3(s) = x
1(s) + x
2(s) X
3(s) = x
1(s) = x
2(s)
G(s) = Y(s) / X(s)
Fig (2.27)Block diagram notations.
Block diagram Algebra :
1. Series or Cascaded Blocks :
Y1(s) = G1(s) . X1(s)
Y1(s) = X2(s)
Y2(s) = G2(s) . X2(s) = G2(s) . Y1(s)
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= G2(s) . G1(s) . X1(s)
= G(s) . X1(s)
G(s) = G1(s) . G2(s)
2. Parallel Blocks :
Y1(s) = G1(s) . X(s)
Y2(s) = G2(s) . X(s)
Yt(s) = Y1(s) + Y2(s) = G1(s) . X(s) + G2(s) . X(s)
Yt(s) = ( G2(s) + G1(s) ) . X(s)
= G(s) . X(s)
G(s) = G1(s) + G2(s)
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3. Feedback System Blocks :
Example(6):
Apply the block diagram reduction rules to obtain the overall transfer
function Y(s) / X(s).
H(s)G(s)1
G(s)
R(s)
C(s))s(Gt
(s)RC(s).H(s)G(s)C(s) (s)R)
G(s)
G(s)H(s)
G(s)
1(C(s)
(s)R)G(s)
G(s)H(s)1(C(s)
(s)RB(s)E(s)
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Fig (2.28)Block diagram reduction example.
2.4.5.Frequency response :The frequency response is very important when studying the systemdynamic behavior associated with sinusoidal input at different
frequencies. A stable linear system subjected to a sinusoidal input X(t)
will, at steady state, have a sinusoidal output Y(t) of the same
frequencyas the input. But the amplitude and phase of output will be
different from those of the input. The relationship between input and
output can be characterized by :
))Re(G(j
))(G(jIm
tan=
)G(j==Pahseangle
))(G(jIm+))Re(G(j=
)G(j=)t('X
)t('Y=
magnitudeInput
magnitudeOutput=atioAmplituder
1-
22
max
max
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Example (7) : For the mixing process shown in Fig (2.29) :
Fig (2.29)The mixing process.
The model of the stirred tank was written before as :
Since the inlet will be used as a sinusoidal input, i.e : CA0 = A sin(t) ,
one can rewrite the system model as :
Using = V/F and taking the Laplace transform, the system model
becomes as :
Using the partial fraction method, the system model becomes as :
)C-(CFdt
dCV AA0
A
AA C.F-t)sin(AF.
dtdCV
)s(C-s
.A(s)Cs A22A
s
.As)(1(s)C22A
)js(
k
)js(
k
)1
(s
k
)s()1
(s
/.A(s)C
321
22A
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where :
Finally, the system model becomes as :
which can be rewritten in the form :
Note that the input was : CA0(t) = A sin(t)
)js(
k
)js(
k
)1
(s
k(s)C 321A
221 1
Ak
)(tan
e1
1
j2
Ak
e1
1
j2
Ak
1
j
223
j
222
)t-j(3
)tj(2
-t/1A ekekek(t)C
)tsin(1
Ae
1
A(t)C
22
t/-
22A
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2.5.Exercises :
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Chapter (3)
Measurement of
Control System
Parameters
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Chapter (3)
Measurement of Control
System ParametersIndustry and industrial processes use a variety of sensors to control its
operations; the most familiar devices include thermocouples, pressure
gauges, encoders and others , which measure a single variable at a single
point in the process. As manufacturing processes have become more
complex, additional types of information and measurements are required.
Some of industrial processes now need measurements of film thickness,
particle size, solids concentration, and contamination detection. Most of
the used sensors operate on relatively simple principles that are based on
the interaction between matter and sound, light, or electric fields. These
devices or sensors are used in process control to measure some parameters
and the resulting data is used to control the process. In addition, such
measurements enable better process understanding, which often drives
process improvement such as improving the productivity or achieving the
uniformity of a product. Figure (3.1) displays that the process
measurements are very important and representing the basic step leading to
several aspects which finally maximize the process profit and process
improvement. There is often more than one type of sensors that will
function adequately in a given application. For instance:
1. Temperature Sensors.2. Position Sensors.3. Pressure Sensors.4. Force Sensors.5. Fluid (Flow rate) Sensors.
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Fig (3.1) - Process measurement as a crucial step to plantoperation and profitability.
The choice of a sensor always depends on the specific details of the
application, so it is imperative to understand the operation and limitations
of each measuring or sensing device. Clearly, other factors such as cost or
vendor issues have to be considered, but from a purely technical point of
view the best choice of sensor for a given application ultimately depends
on the details of the measurement process.
The performance of any process sensor, new or old, can be summarized as
follows :
The sensor must be completely reliable under continuous operationand ideally require no preventive maintenance.
The sensor should be installed in such a way that it can be replacedquickly in case it does eventually malfunction.
The sensor is easy to use and does not require a complicatedcalibration sequence.
The data it provides should be directly related to the physicalproperties of interest. For example, a sensor that purports to measure
viscosity but in reality is also sensitive to changes in pressure is of
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limited usefulness.
The data it provides should be easily interpreted. For example,sensors that measure scalar quantities such as temperature or flow
rate generally output a signal that is proportional to these quantities.
If the sensor interface is digital, the readout should be provided in the
correct units.
The sensor should be compatible with other sensors and with theexisting distributed control system.
The sensor must provide an immediate pay-off relative to its cost ofpurchase and installation, which is usually many times more than the
purchase price.
3.1. Temperature sensors :
Temperature is a basic measurement used throughout many processes. It
is a measure of the thermal energy in a body, which is the relative
hotness or coldness of a medium and normally measured in degrees
using one of the following scales; traditional Fahrenheit scale (F),
Celsius which is originally called centigrade scale (C) or the absolute
Kelvin scale (K) as standard units of measurement. Temperature
sensors are used in the process control that concerns with temperature
regulation. It depends on the electrical methods of measuring
temperature. The basic types of temperature sensors are :
1. Bimetallic temperature sensor.2. ResistanceTemperature Detectors (RTD) sensors.3. Thermistors or semiconductor usage in measuring temperature.4. Thermocouples.
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3.1.1. Bimetallic temperature sensors :The bimetallic temperature sensors have some advantages of simplicity
and low cost, on the other hand, its main disadvantages are the
existence of hysteresis, inaccuracy and the slow time response. Such
sensors are used in numerous applications, particularly where an
ON/OFF cyclic operation is needed rather than smooth or continuous
control.
The sensor basic operation is built on the thermal linear expansion
which is the change in dimensions of a material due to temperature
changes. The change in dimensions of a material is due to its
coefficient of thermal expansion that is expressed as the change in
linear dimension () per degree temperature change.
L = Lo ( 1 + . t )
where : L = the final length.Lo = the initial length.
t = T To = temperature difference.
= the linear thermal expansion coefficient.
The bimetallic sensor consists of two materials with grossly different
thermal expansion coefficients bounded together. When the sensor is
being subjected to heating, the different expansion rates of the two
materials will cause the sensor assembly to be curved as shown in
Fig(3.2). This effect can be used to close switch contacts or to actuate
an ON/OFF mechanism when the temperature increases to some
appropriate set-point.
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Fig (3.2)Bimetallic strip sensor.
3.1.2. Resistance temperature sensors :One of the most important methods for electrical measurement of
temperature is based on the electrical resistance change of a conducting
material. So, the principle of measuring or sensing temperature is to
place a conducting material with sensitive change of resistance with
respect to temperature in contact with the environment whose
temperature is to be measured or sensed. By then the device will take
the temperature of the environment. Thus a measure of the conducting
material resistance will indicate the temperature of the sensor and the
environment. The resistance of a conductor varies according to the
following factors :
The resistance is directly proportional to the conductor length :Rl
The resistance is inverse proportional to the conductor crosssection area :R1/a
The resistance depends on the type of the conductor material :R= .l / a
The resistance is affected by the surrounding resistance such that :
1
2 < 1
at To
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RT = RTo ( 1 + . t )
where : RT = the conductor resistance at a temperature T.
RTo = the conductor resistance at a temperature To.
t = TTo = temperature difference.
= the linear change coefficient in resistance with
respect to temperature.
The resistance-temperature detector (RTD) is a temperature sensor
whose operation is based on the resistance variation of a metal
conductor with temperature. Metal used in such a sensor vary from
platinum which is quit sensitive and expensive to nickel which is more
sensitive and less expensive.
The sensitivity of the RTD sensor depends on the value of the linear
change coefficient in resistance with respect to temperature ().
Typical values of such coefficient for different materials are :
= 0.004 /C for platinum.
= 0.005 /C for nickel.
In general the RTD has a time response ranges between 0.5 to 5
seconds or more. This slow response is due to the slow thermal
conductivity in bringing the sensor into thermal equilibrium with the
surrounding environment.
The RTD sensor construction is basically in the form of a wire wound
as a coil to achieve small size, improved thermal conductivity and
decreased time response. This coil is protected by a sheath or a tube.
The resistance of the coil will be monitored as a function of
temperature. Fig (3.3) shows the internal construction of an RTD.
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Fig (3.3)Internal construction of a typical RTD.
This design has a platinum element surrounded by a porcelain
insulator. The insulator prevents a short circuit between the wire and
the metal sheath. A nickel-iron-chromium alloy is normally used inmanufacturing the RTD sheath. When placed in a liquid or gas
medium, the sheath quickly reaches the temperature of the medium and
the change in temperature will cause the platinum wire to heat or cool,
resulting in a proportional change in resistance. This device is normally
used in a bridge circuit. Fig (3.4) shows an RTD protective well and
terminal head, which can be used for temperatures up to 1100C.
Fig (3.4)RTD protection and terminal head.
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Since the variation of the RTD resistance is relatively small, the RTD
is usually used a branch of a bridge as shown in Fig (3.5) :
Fig (3.5)RTD sensor with signal conditioning.
The effective range of RTD sensors basically depends on the type of
the effective element wire. For example :
Platinum RTD has the range of : - 100 to 650 C
Nickel RTD has the range of : - 180 to 300 C
3.1.3. Thermistor sensors :The thermistor represents another class of temperature sensor that
measures temperature through changes of material resistance. The
characteristics of such devices are very different from those of the
RTDs and depend on the behavior of semiconductor resistance with
temperature.
So, one can say that the word thermistor comes from a contraction of
thermal resistor. The resistance of a thermistor is a function of the
ambient temperature.
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The change in resistance R of the thermistor is proportional to the
change in temperature T when using a first-order approximation over
limited temperature ranges where () is the characteristic temperature
coefficient of the thermistor.
The thermistor construction may take several forms including discs,
beads and rods varying in size from a bead 1 mm in diameter to a disc
of several cm in diameter and thick. This can provide a wide range of
resistance values at any particular temperature.
The effective range of thermistor sensors depends on thesemiconductor material used in constructing the thermistor. The
thermistor practical range is : - 80 to 300 C.
3.1.4. Thermocouple sensors :The thermocouple is a device that converts thermal energy into
electrical energy. A thermocouple is constructed of two dissimilar
metal wires joined at one end. The most important factor to be
considered when selecting a pair of materials is the thermoelectric
difference between the two materials. A significant difference between
the two materials will result in better thermocouple performance.
Figure (3.6) displays the constructions of a typical thermocouple. The
leads of the thermocouple are encased in a rigid metal sheath. The
measuring junction is normally formed at the bottom of the
thermocouple housing. Magnesium oxide surrounds the thermocouple
wires to prevent vibration that could damage the fine wires and to
enhance heat transfer between the measuring junction and the medium
surrounding the thermocouple.
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Fig (3.6) - Internal construction of a typical thermocouple.
Other materials may be used in addition to those shown in figure, for
example: Chromel-Constantan is excellent for temperatures up to 2000
F and Tungsten-Rhenium is used for temperatures up to 5000 F.
When a thermocouple is subjected to changes in temperature, it will
cause an electric current to flow in the attached circuit. The amount of
produced current depends on the temperature difference between the
measurement and reference junction; the characteristics of the two
metals used; and the characteristics of the attached circuit. Fig (3.7)
illustrates a simple thermocouple circuit.
Fig (3.7) - Simple thermocouple circuit.
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Fig (3.8) - Thermocouple circuit with temperature
control and signal conditioning.
Heating the measuring junction of the thermocouple produces a voltage
which is greater than the voltage across the reference junction. The
difference between the two voltages is proportional to the difference in
temperature and can be measured on the voltmeter (in milli-volts) or
amplified and then sent to operate a control circuit.
3.2. Position sensors :The measurement of displacement, position, or location is important in
the process industries. The requirements of measuring such variables
and the used sensors are varied in the industries. For examples :
Location and position on conveyor systems.Orientation of steel plates in a rolling mill.Liquid or solid level monitoring, etc
The most commonly used sensors for displacement, position or
location are :
3.2.1. Potentiometers :The simplest type of displacement sensor involves the action of
moving the wiper of a potentiometer. This device converts the linear or
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angular motion into a changing in resistance that may converted
directly into a voltage and/or current signals as shown in Fig (3.9).
Fig (3.9)Linear potentiometer displacement sensor.
The output voltage of the sensor can be calculated from the following
formula :
3.2.2. Capacitive sensor :The basic operation of a capacitive sensor can be derived from the
capacitance equation of the parallel plate capacitor :
where :
o
is the air permittivity = 8.85 pF/m
r is the dielectric constant.
A is the plate common area.
d is the plate separation.
The capacitance of the capacitor can be changed by varying the
distance between the plates (d), or by varying the shared area of the
plates (A) as shown in Fig (3.10).
Motion
Wiper
R
r
VinVout
inout V.R
rV
d
AC ro
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Fig (3.10)Variation of capacitance with the distance
or the area between the plates
An A.C bridge or other active electronic circuit is employed to convert
the capacity change to a current or voltage signal.
3.2.3. Inductive sensor :The inductance type transducer consists of three parts : a coil, a
movable magnetic core, and a position sensing element. The element is
attached to the core, and, as position varies, the element causes the core
to move inside the coil. An A.C voltage is applied to the coil, and, as
the core moves, the inductance of the coil changes. The current through
the coil will increase as the inductance decreases.
3.2.4. Linear variable differential transformer (LVDT) :The LVDT is an important displacement sensor in industrial
environment with an operation depending on the inductive sensor
principle and utilizing single core and two coils wound on a single tube
as illustrated in Fig (3.11). The primary coil is wound around the
center of the tube. The secondary coil is divided with one half wound
around each end of the tube.
d
Capacity C
Capacity C
A
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Each end is wound in the opposite direction, which causes the voltages
induced to oppose one another. A core, positioned by a displacement
element, is movable within the tube. When the core is in the lower
position, the lower half of the secondary coil provides the output.
When the core is in the upper position, the upper half of the secondary
coil provides the output. The magnitude and direction of the output
depends on the amount the core is displaced from its center position.
When the core is in the mid-position, there is no secondary output.
Fig (3.10)Construction of the LVDT sensor.
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Fig (3.11)Using the LVDT sensor to produce a bipolar D.C
voltage that varies with core displacement.
3.3. Speed sensors :
The linear speed or position of a translational moving mechanism such
as a conveyor system can be measured or monitored for the purpose of
control by means of a linear optical encoder.
On the other hand, the angular speed or the angular position of a
rotational mechanism such as a motor can be measured or monitored
for the purpose of control by means of either a tachometer or a digital
encoder.
3.3.1. Tachometer as a speed sensor :The tachometer is a permanent magnet D.C generator, when driven
mechanically; it generates an output voltage that is proportional to
shaft speed.
Since : E . N
For permanent magnet machine : = constant
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Hence : E N
Therefore, the tachometer will generate an output voltage proportional
to shaft speed as displayed in Fig (3.12).
The other main requirements for a tachometer are :
1. The output voltage should be smooth over the operating range.2. The output should be stabilized against temperature variations.
Fig (3.12)Tachometer output characteristics.
Small permanent magnet D.C tachometers are frequently used in servosystems as speed sensing devices. These systems usually incorporate
thermistor temperature compensation and make use of a silver
commutator and silver loaded brushes to improve commutation
reliability at low speeds and at the low currents, which are typical of
this application. The tachometer is mounted on the motor shaft and
enclosed within the motor housing as illustrated in Fig (3.13).
Fig (3.13) - Motor with integrated tachometer.
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3.3.2. Optical Encoder sensor :In servo control systems, where mechanical position is required to be
controlled, some form of position sensing device is needed. For
accurate position control, the most commonly used device is the optical
encoder. Optical encoders are devices that convert a mechanical
position or speed into a representative electrical signal by means of a
patterned disk or scale, a light source and photosensitive elements.
Their principle of operation is achieved by moving disk between the
light source and the photosensitive element. The light source may be a
light emitting diode or an incandescent lamp, and the detector is
usually a phototransistor or more commonly a photo-voltaic diode.
When light passes through the transparent areas or the holes of the disk
an output is seen from the detector. There are two forms of the
encoders namely :
Absolute encoders.Incremental encoders.
An incremental encoder : which generates a pulse for a given
increment of the shaft rotation in case of rotary encoder, or a pulse for
a given linear distance travelled in case of linear encoder. Furthermore,
the total distance travelled or shaft angular rotation is determined by
counting the encoder output pulses or with proper interface electronics,
position and speed information can be derived. Rotary encoders are
available as housed units with shaft and ball-bearings or as modular
encoders which are usually mounted on a host shaft at the end of a
motor. The disk count is defined as the number of dark/light line-pairs
that occur per revolution in terms of cycles/revolution or c/r.
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Fig (3.14)Incremental encoders and the resulting signal.
The disadvantage of incremental encoders is the loss of position data at
power-down. This is not a problem if the system can be re-initialized
on power up by searching for the index and re-setting the position
counters.
An absolute encoder : has a number of output channels, suchthat every shaft position may be described by its own unique code. The
higher the resolution the more output channels are required. With this
type of encoders, position information is instantly available as a digital
word on power-up. The disk of an absolute encoder is patterned with a
number of discrete tracks, corresponding to the word-length. Fig (3.15)
illustrates a 3 bit and a 4 bit encoder pattern whose output is reflected
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in binary or Gray code. The advantage of this pattern is that from
position to position only one bit changes its state.
(a)3 bit absolute encoder disk pattern.
(b)4 bit absolute encoder disk pattern.
Fig (3.15)Absolute encoders and the resulting signal patterns.
3.4. Pressure sensors :
The measurement and control of liquid or gas pressure is one of the
most common in most of the industrial processes. The pressure
measurements are very important in order to keep the material under
safe operation.
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The pressure may be either static pressure where the fluid is not
moving or dynamic pressure where the fluid is moving and exerting a
pressure on the surroundings.
Pressure sensors are available in different designs depending on the
pressure to be measured or controlled.
3.4.1. Bellow type sensor :
The metallic bellows pressure sensors are used when it is needed to
sensing low pressures and providing power for activating recording
and indicating mechanisms. Such sensors are most accurate when
measuring pressures from 0.5 to 75 psi. However, when used in
conjunction with a heavy range spring, they can be used to measure
pressures of over 1000 psi. Figure (3.16) shows a basic construction of
the metallic bellows pressure sensing element.
Fig (3.16) - Basic construction of the metallic bellow sensor.
The system pressure is applied to the internal volume of the bellows
where by varying the inlet pressure to the instrument, the bellows will
expand or contract. The moving end of the bellows is connected to a
mechanical linkage assembly.
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As the bellows and linkage assembly moves, either an electrical signal
is generated or a direct pressure indication is provided. The relation
between increments of load and deflection is linear in the range of the
elastic limit of the bellows. This relationship exists only when the
bellows is under compression. So, it is necessary to construct the
bellows such that all of the travel occurs on the compression side of the
point of equilibrium.
3.4.2. Bourdon Tube Pressure sensor :The bourdon tube pressure instrument is one of the oldest pressure
sensing instruments in use today. The bourdon tube consists of a thin-
walled tube that is flattened diametrically on opposite sides to produce
a cross-sectional area elliptical in shape, having two long flat sides and
two short round sides. The tube is bent lengthwise into an arc of a
circle from 270 to 300 degrees.
The pressure is applied to the inside of the tube causing distention of
the flat sections and tends to restore its original round cross-section.
This change in cross-section causes the tube to straighten slightly.
Since the tube is permanently fastened at one end, the tip of the tube
traces a curve that is the result of the change in angular position with
respect to the center. The movement of the tip of the tube can then be
used to position a pointer or to develop an equivalent electrical signalto indicate the value of the applied internal pressure.
Figure (3.17) illustrates the basic construction of bourdon tube pressure
sensor when replacing the pointers by an electronic circuit.
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Fig (3.17) - Basic construction of the bourdon pressure sensor.
3.5. Force sensors (Strain gauge) :
One of the most important force sensors or transducers is the strain
gauge. Figure (3.18) illustrates a simple strain gauge where it is used
for measuring the external force or pressure applied to a fine wire. The
fine wire is usually arranged in the form of a grid or a folded wire.
Fig (3.18) - Basic construction of the strain gauge.
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The pressure change causes a resistance change due to the distortion of
the wire. The value of the pressure can be found by measuring the
change in resistance of the wire grid.
Since : R= .l / awhere :
R = resistance of the wire grid in ohms.
= resistivity constant for the particular type of wire grid.
l = length of wire grid.a = cross sectional area of wire grid.
Therefore, as the wire grid is distorted by elastic deformation, its
length is increased, and its cross-sectional area decreases. These
changes cause an increase in the resistance of the wire of the strain
gauge. This change in resistance is used as the variable resistance in a
bridge circuit that provides an electrical signal for indication of force
or the pressure. Figure (3.19) illustrates a strain gauge pressure
transducer.
Fig (3.19) - The strain gauge as a force sensor.
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Fig (3.20)Using four strain gauges as a bridge force sensor.
Fig (3.21) - The strain gauge as a pressure sensor.
In Fig (3.21), an increase in pressure at the inlet of the bellows causes thebellows to expand and moving a flexible beam to which a strain gauge
has been attached. The movement of the beam causes the resistance of the
strain gauge to change. The temperature compensating gauge
compensates the heat produced by current flowing through the fine wire
of the strain gauge. Strain gauges, which are nothing more than resistors,
are used with bridge circuits as shown in Fig (3.22).
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Fig (3.22) - Strain gauge in a bridge circuit.
Alternating current is provided by an exciter that is used in place of a
battery to eliminate the need for a galvanometer. When a change in
resistance in the strain gauge causes an unbalanced condition, an error
signal enters the amplifier and actuates the balancing motor which moves
the slider along the slide wire, restoring the bridge to the balanced
condition; the sliders position is noted on a scale marked in units of
pressure.
3.6. Fluid sensors :
Liquid level measuring devices are classified into two groups :
a) Direct method. b) Inferred method.
An example of the direct method is the dipstick in the car which
measures the height of the oil in the oil pan. On the other hand, an
example of the inferred method is a pressure gauge at the bottom of a
tank which measures the hydrostatic head pressure from the height of
the liquid.
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The level sensors can be classified as follows :
1. Mechanical sensors Float methods Buoyancy method Vibrating level systems
2. Hydrostatic pressure methods Differential pressure level detectors Bubbler systems
3. Electrical methods Conductivity probes Capacitance probes Optical level switches Ultrasonic level detectors Microwave level systemsNuclear level systems
3.6.1. Floats level sensor :The basic float arm indicator comprises very simply a float connectedto a pivoted arm that drives pointer or a switch. The unit can be made
for either side or top entry. The main disadvantage of such a sensor is
the presence of the moving parts in the liquid which causes corrosion
and seizing to such parts. Methods of providing indication are by using
linkage to a pointer, a potentiometer and magnetic or inductive
switches as displayed in Fig (3.23).
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Fig (3.23)The float level sensor.
3.6.2. Buoyancy level sensor :These devices use Archimedess principle where the mechanical level
indicator consists of the immersion body with calibrated measuring
spring which transmits the change of level to the mechanical or
electrical indicator according to the following equation :
r2 ( h - L ) g = k . L
Fig (3.23)The buoyancy level sensor.
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3.6.3. Ultrasonic level measurement :The measuring equipment consists of the following elements:
A transmitter : which periodically sends an ultrasonic
pulse to the surface of the liquid
A receiver : which receives and amplifies the returningpulse.
A time interval counter : which measures the timeelapsing between the transmission of a pulse and receiption of
the corresponding pulse echo.
The travelling distance can be calculated as :L = c . t / 2
And consequently, the head can be as :
h = LmaxL = Lmaxc . t / 2
where :
c = sonar pulse velocity (m/sec).
t = time in sec.
L = travelling distance (m).
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Fig (3.24)The ultrasonic level sensor.
a) Solid or liquid above b) Liquid material below
surface measurement surface measurement
Fig (3.25)Th