5 Input, output and process dynamics

134 Elements of Process Control



A

Q



h

Qo

Figure 5.8 Water tank as a first order system



In the system described, the input is the feed flow rate. The output (i.e. the change

in the system caused by the input) is the height h of the liquid in the tank. Let us analyze the relationship between Qi and h. A material balance can be written as follows:

A



dh

ϭ Qi Ϫ Qo

dt



(5.1)



Assume laminar flow in the discharge pipe. Then, the Hagen–Poiseuille rule, given

in Eq. (2.12) applies. The discharge flow rate at any moment is proportional to the

height of the liquid in the tank:

Qo ϭ



h

R



(5.2)



R is the resistance to flow at the discharge pipe, assumed constant.

Substitution in Eq. (5.1) gives:

RA



dh

ϩ h ϭ RQi

dt



(5.3)



Recalling that Qi is the input Si and h is the output, So, Eq. (5.3) can be written in

the following general form:

KSi ϭ τ



dSo

ϩ So

dt



(5.4)



Eq. (5.4) represents the response of systems known as first order systems. The constant K is called the system gain and τ is the time constant of the system. In the tank

example, R is the gain and RA is the time constant.

Now, let us examine the response of a first order system to a step-function ( jump)

change in input. We indicate the jump in input as boundary conditions:

Si ϭ 0

Si ϭ 1



for t Ͻ 0

for t Ն 0



With these boundary conditions, the solution of Eq. (5.4) is:

Ϫt



So ϭ 1 Ϫ e



τ



(5.5)



Input, Output and Process Dynamics 135



Si



S

1

0.63

So

τ



0



t



Figure 5.9 Response of a first order system



It can be seen (Figure 5.9) that the output lags after the input. In system dynamics,

this is known as a first order lag. Many real systems fit well the first order model.



EXAMPLE 5.1

Show that a steam-jacketed stirred tank used to heat a batch of water behaves

like a first order system. Assume that the temperature of the steam and the overall

heat transfer coefficient are constant. Neglect heat losses. Find the gain and the

time constant of the system.



Solution:

Let:

mϭ

Tϭ

Cp ϭ

Ts ϭ

Uϭ

Aϭ



mass of the water in the tank

instantaneous temperature of the water, assumed uniform

specific heat of the water, assumed constant

temperature of the steam, assumed constant

overall heat transfer coefficient, assumed constant

heat transfer area, assumed constant.

m.C p



dT

ϭ UA(Ts Ϫ T )

dt



⇒



⎡ mC p ⎤ dT

⎢

⎥

⎢ UA ⎥ dt ϩ T ϭ Ts

⎣

⎦



Remembering that the temperature of the steam is the input and the temperature of the water is the consequence (output), the result fits the first order model,

with:

τϭ



mC p

UA



and



K ϭ1



5.5.2 Second order systems

The response of many real systems approximates to a behavior, represented by a second order differential equation like Eq. (5.6):

τ2



d 2 So

dS

ϩ 2ζ τ o ϩ So ϭ KSi

2

dt

dt



(5.6)



136 Elements of Process Control



The additional constant ζ is called the damping factor because of its effect on the

oscillatory character of the response of the system to step changes in input. At low

ζ, the response is highly oscillatory (Figure 5.10). At high ζ (ζ Ͼ 1), the response is

non-oscillatory or over-damped.

Second order response characterizes systems consisting of two first order lags

connected in series, mechanical devices containing a spring-loaded element together

with a viscous damping component, thermometers in protective wells etc.



5.6 Control Modes (Control Algorithms)

Control modes (control algorithms) define the relationship between the error e and

the correction signal m issued by the controller. Following are some of the frequently

applied types of control strategy:

●

●

●

●

●



On-off control

Proportional (P) control

Integral (I) control

Proportional-integral (PI) control

Proportional-integral-differential (PID) control.



5.6.1 On-off (binary) control

In this mode of control, the actuator can have only one of two positions: either on or

off (open or closed, all or nothing, 1 or 0). There are no intermediate positions such

as partially open. This is the simplest and the least costly of the controllers. On-off

control is very common in household devices, laboratory instruments and in some

cases of industrial control systems (air compressors, refrigeration engines, limit

switches and alarms etc.).

In on-off control, the controlled variable must be allowed to fluctuate about the

set point, within a range known as the differential band (or dead zone). For example,



S



t

Figure 5.10 Response of a second order system



Control Modes (Control Algorithms) 137



in a thermostat set to 25°C, the switch turns the heater on at 24°C and off at 26°C.

Between these two values, the heater would be either on or off, depending on the

direction of change. Without a differential band, the controller would cycle between

two contradictory positions at very high frequency (chatter), causing damage to

contactors and valves or causing the actuator to ‘freeze’ altogether. The differential

band or ‘hysteresis’ is either stored in the controller as an analog or digital datum or

built-in in the hardware as the physical lag between the controller and the actuator.

In practice, the width of the differential band is usually set between 0.5% and 2% of

the control range. A narrow band results in more precise control but high frequency

of change and vice versa.



EXAMPLE 5.2

An electrically heated fryer is on-off controlled. The set point is 170°C. When the

heater is on the temperature of the oil rises at the rate of 4°C/min. When the

heater is off, the temperature drops at the rate of 4°C/min. Calculate the duration of one cycle of the heater switch if the differential band on each side of the

set point is:

a. 10°C.

b. 1°C.



Solution:

a.



The switch is turned off when the oil temperature reaches 180°C and on

again when the temperature drops to 160°C. The gap is 20°C. The duration of the cycle is:

t ϭ 20/4 ϭ 5 minutes.



b.



The gap is now 2°C. The duration of the cycle is: t ϭ 2/4 ϭ 0.5 min.



The response of a first order system to on-off control with hysteresis is shown in

Figure 5.11.



c

r



1

m

0

Figure 5.11 On-off control with differential band (dead zone)



‘Dead

zone’



138 Elements of Process Control



5.6.2 Proportional (P) control

In proportional (P) control, the magnitude of the correction signal m is proportional

to the error e.

m ϭ Ke ϩ M

The proportionality factor K is called proportional gain. The constant M is known

as the controller bias, because it represents the magnitude of the correction signal

when no correction is needed (e ϭ 0). K is dimensionless. M, e and m are usually

expressed as percentage.

Proportional control serves to eliminate the oscillation associated with on-off controllers. The magnitude of the corrective action is reduced as the controlled variable

approaches the set point. In control systems of the proportional type, the actuator can

assume intermediate positions between two extremes, depending on the amplitude

of the correction signal from the controller. In some simpler systems, the actuator

works in on-off fashion and the proportionality is achieved by regulating the ratio of

on-time/off-time. At the set point that ratio is equal to 1.

The proportional gain K is usually a fixed property of the controller but, in some

proportional controllers, K is manually adjustable. If K is increased, the sensitivity of the controller to error is increased but the stability is impaired. The system

approaches the behavior of on-off controlled systems and its response becomes oscillatory. The bias M is usually adjustable. As a rule, it is customary to adjust M so as

to stabilize the system at a state slightly different from the set point. The difference

between the measurement c and the set point r at steady (stable) state is called the

offset. It is possible to eliminate the offset by adjusting the bias. This action is called

reset. However, adjusting the controller to zero offset under a given set of process

conditions would require re-adjusting every time the load or any other process condition changes. In automatic control, this would be highly problematic.



EXAMPLE 5.3

A steam heated plate heat exchanger serves to pasteurize orange juice at the nominal rate of 2000 kg/h. A proportional controller regulates the flow rate, according

to the exit temperature of the juice. The juice enters the pasteurizer at 20°C. The

set point is 90°C. The steam valve is linear and normally closed. When the valve is

fully open the steam flow rate is 400 kg/h.

The controller has been tuned according to the following equation:

m (%) ϭ 0.8e (%) ϩ 50

The temperature measurement range is 50–150°C.

Assume that the heat transferred to the juice is exactly equal to the heat of

condensation of the steam, which is 2200 kj/kg. The specific heat of the juice is

3.8 kJ/kg.K.

a. What is the exit temperature of the juice and the offset at steady state?

b. It is desired to reduce the offset to half its value, by changing the bias, without changing the gain. What should be the bias?



Control Modes (Control Algorithms) 139



Solution:

Let J be the mass flow rate of the juice and S the mass flow rate of the steam. Let

Cp be the specific heat of the juice and hfg, the latent heat of condensation of the

steam. The notations of Section 5.4 will be used for all other parameters.

a. The system equation is:

J.C p (Tout Ϫ Tin ) ϭ S.hfg



⇒



2000 ϫ 3.8(Tout Ϫ 20) ϭ S ϫ 2200



The controller equation is:

(r Ϫ Tout ) ϫ 100

ϩ 50

Δ

(90 Ϫ Tout ) ϫ 100

m ϭ 0.8 ϫ

ϩ 50

100

m ϭ 0.8e ϩ 50 ϭ 0.8



The steam valve equation is:

S ϭ Smax ϫ



m

100



⇒



S ϭ 400 ϫ



m

100



Solving the 3 equations for Tout we find:

Tout ϭ 83.7°C.

The offset is 90 Ϫ 83.7 ϭ 6.3°C.

b.



The gain m is:

m ϭ 0.8 ϫ



(90 Ϫ 83.7) × 100

ϩ 50 ϭ 55

100



Let the new bias be M. Now the offset is 6.3/2 ϭ 3.15°C.

55 ϭ 0.8 ϫ



3.15 ϫ 100

ϩM

100



⇒



M ϭ 52.48



In order to reduce the offset to half its value, the bias has to be changed to 52.48%.



5.6.3 Integral (I) control

In integral control, the rate of change of the correction signal (and not the actual

value of that signal) is proportional to the error.

dm

ϭ Re

dt



(5.7)



m ϭ R ∫ e dt ϩ Mi



(5.8)



Integration gives:



140 Elements of Process Control



The corrective signal m depends not on the actual value of the error but on its time

interval, i.e. on the past history of the system. By virtue of Eq. (5.7), the corrective

signal to the actuator continues to grow as long as the error persists. This results in

the elimination of the offset.

Removal of the offset is the main advantage of I control. However, the response of

I-controlled systems is slow and may be highly oscillatory. The response of such systems

to a step change in the controlled variable is a ramp, with a slope proportional to the proportionality constant R.



5.6.4 Proportional-integral (PI) control

Because of its shortcomings, integral control is usually applied in combination with

other modes. In the PI mode, proportional control is combined with integral control.

The combined algorithm is represented in Eq. (5.9):

m ϭ K (e ϩ ∫ e dt) ϩ M



(5.9)



The integral term contributes the feature of automatic reset (removal of the offset).

The proportional term increases the stability.



5.6.5 Proportional-integral-differential (PID) control

A third mode is incorporated in this popular type of control, known also as the threeterm control. The third term is that of differential control, defined in Eq. (5.10):

mϭT



de

dt



(5.10)



In the differential mode, the correction signal is proportional not to the instantaneous error but to the rate of change of the error. If the error increases rapidly, the

correction signal is larger. It can be said that the differential action is predictive. The

response of differential-controlled systems is fast, but highly unstable. The differential mode is, therefore, not applied alone but in combination with other control types.

The PID mode provides proportional control with the automatic reset feature of integral control and the rapidity and predictive action of differential control. The behavior

of the PID control is represented by Eq. (5.11), containing the three terms, P, I and D:

⎛

de ⎞

⎟ϩ M

m ϭ K ⎜ e ϩ R ∫ e dt ϩ T

⎟

⎜

⎜

⎠

⎝

dt ⎟



(5.11)



The PID algorithm is extensively used in industrial process control. For optimal performance, the controller must be ‘tuned’ by adjusting the parameters of the three terms:

K, R and T. The notion of ‘optimal performance’ will be defined in the following paragraph. Several methods exist for the manual tuning of PID controllers. Automatic tuning (self-tuning) systems are also available. Figure 5.12 shows the response of three

types of control to a disturbance. Note the large offset in the case of a P controller. In

the PI controller, the offset has been removed but the response is oscillatory. In the PID

system, the offset has been eliminated and the system is more stable.



Process variable



Control Modes (Control Algorithms) 141



P

PI



PID

Set-point



t



Figure 5.12 Response of P, PI and PID controlled systems to disturbance



5.6.6 Optimization of control

The definition of ‘quality of control’ is not unequivocal. Consider the response of a

controlled system to a disturbance. There are several ways to define the quality of the

control and the objective of optimization:

●



●



●



Since the fundamental objective of process control is to eliminate error as much

as possible, the objective of optimization might be to minimize the maximum

deviation emax. We have seen that this can be done by increasing the controller

gain K. However, if K is increased beyond a critical value, stability is impaired

and the response shows weakly damped oscillations. The time to reach stability

(steady state) may be too long

The time to reach steady state, known as recovery time is an important factor.

The objective of optimization could then be to shorten recovery time as much

as possible. However, this would require decreasing the gain K, which would

result in increased emax, i.e. in loss of accuracy

The previous criteria may be both incorporated in a parameter known as the

integral absolute error (IAE), defined in Eq. (5.12):

IAE ϭ



∞



∫



e dt



(5.12)



0



●



In this case, optimization would mean minimizing the IAE. Obviously, this criterion can be applied only to controllers comprising an integral (I) component,

because otherwise the IAE would always be infinite because of the offset.

Another measure, sometimes used as a criterion for control quality is the integral squared error (ISE), defined in Eq. (5.13):

ISE ϭ



∞



∫ e2 dt



(5.13)



0



ISE is particularly sensitive to large deviations and therefore, optimization with

respect to ISE enhances the protective action of the control against damage to the

product or to the equipment. Like IAE, ISE can be applied only to controllers comprising an integral (I) feature.



142 Elements of Process Control



5.7 The Physical Elements of the Control System

5.7.1 The sensors (measuring elements)

The number of controlled variables in food processing is quite large and often a considerable number of different sensors are available for measuring each parameter

(Bimbenet et al., 1994; Webster, 1999). Some of the most common types of measuring instruments will be described in this section. For a detailed survey of this highly

specialized topic, please consult Kress-Rogers and Brimelow (2001).

A number of general principles apply to all the sensors used in food process control, regardless of the measured variable:

●



●



●



●



●



The output signal must be of a kind that can be transmitted to and read by the

controller. Very often, a ‘converter’ has to be used to transform the measurement signal to the desired type without distortion. In modern control systems,

the desired format is a digital signal. In transmission, the signal must be protected against electrical and electromagnetic disturbances

The range of measurement (scale) and the sensitivity must meet the process

requirements. The measurement has significance only if the measured parameter falls within the range of the sensor. Thus, if the measurement range of a

thermometer is 20 to 100°C, a temperature of 120°C will be read as 100°C and

a temperature of 10°C as 20°C

Sensors can be on-line, at-line or off-line. On-line measurements, whenever

feasible, are preferable. At-line measurement refers to rapid tests that can be

performed on samples, near the production line

Sensors can be contacting or non-contacting (remote). Remote sensing, whenever possible, is the generally preferred type

On-line sensors are physically inserted in the process line and often come in

contact with the food. In this case, the sensor and its position in the line must

comply with the strict rules of food safety. Remote sensors eliminate this problem, but the number of commercially available remote sensing elements is still

limited.



A. Temperature

The most common temperature sensing devices are: filled thermometers, bimetals,

thermocouples, resistance thermometers, thermistors and infrared thermometers.

Filled thermometers measure temperatures either through the thermal expansion

of a liquid or through changes in the vapor pressure of a relatively volatile substance.

The thermal expansion thermometers are the most common type. The fluid in the

thermometer is usually mercury or colored alcohol. Although the sturdy construction

of industrial filled thermometers protects the product from contamination with glass,

mercury or spirit in the case of breakage, filled thermometers are being replaced with

other types that do not present that kind of risk. For traditional reasons, however, the

use of mercury-in-glass thermometers as a temperature reference in food canning is

still mandatory.



The Physical Elements of the Control System 143



Bimetal thermometers consist of strips of two different metals, joined together.

Due to the difference in the thermal expansion coefficient of the metals, a change in

the temperature causes the strip to bend or twist. The displacement is usually read on

a dial. They can serve as on-off actuators in simple thermostats in ovens, frying pans

etc. Bimetal thermometers are not accurate and they lack stability.

Thermocouples (Reed, 1999) are among the most common industrial temperature measuring devices. They are based on a phenomenon discovered by the German

physicist Thomas Johann Seebeck in 1821. Seebeck discovered that a voltage is generated in a conductor subjected to a temperature difference between its extremities.

The value of the voltage generated varies from one metal to another. Consequently,

an electric current flows in a closed circuit made of two different metals when their

two junctions are held at different temperatures. The EMF created is a measure of

the temperature difference between the junctions. Thus, a thermocouple measures a

temperature difference, hence the temperature of one junction if the temperature of

the other (reference) junction is known.

The voltage V generated as a result of the thermoelectric effect is given approximately by:

V ϭ S ΔT



(5.14)



where S is the Seebeck coefficient of the material. The coefficient S is temperature

dependent. If two electrothermally dissimilar conductors A and B are joined at points

1 and 2, then the voltage generated between 1 and 2 is approximately:

V1Ϫ2 ϭ (S A Ϫ SB )(T1 Ϫ T2 )



(5.15)



If the Seebeck coefficients do not change much within the temperature range in

question, then:

V1Ϫ2 ϭ k.ΔT



(5.16)



Indeed, within a known rage of temperatures, the response of a thermocouple is

fairly linear, i.e. the EMF generated is proportional to the temperature difference.

This EMF is generally in the order of a few mV per 100°C. The most common pairs

used are copper/constantan and iron/constantan (Constantan is a copper-nickel alloy).

The Seebeck coefficients of copper, iron and constantan are ϩ6.5, ϩ19 and Ϫ35 μV/K,

respectively. The measuring junction of a thermocouple may be very small, thus permitting measurement of the temperature in a precise location. Thermocouples with

different kinds of tips (measuring junctions) are available for different applications

(e.g. thermocouples for measuring the temperature inside a can).

Resistance thermometers (Burns, 1999) or resistance temperature detectors (RTD)

are based on the effect of temperature on the electric resistance of metals. Due to their

accuracy and robustness, they are extensively used as in-line thermometers in the

food industry. Within a wide range of temperatures the resistance of metals increases

linearly with temperature. The measuring element is usually made of platinum. The

resistance of platinum changes by approximately 0.4% per K. Since electrical current

flows through the measuring element during the measurement, there is some degree

of self-heating of the thermometer, causing a slight error in the readings.



144 Elements of Process Control



Thermistors (Sapoff, 1999) are also resistance thermometers but the resistance of

the measuring element, a ceramic semiconductor, decreases with the temperature.

Thermistors are very accurate but highly non-linear. They are used where very high

accuracy is a requisite.

Infrared thermometry (Fraden, 1999) measures temperatures by measuring the

infrared emission of the object. Infrared thermometers may be remote (non-contact)

or on-line (contact). In the contact type a small black-body chamber is in contact with

the object. A small infrared sensor installed inside the chamber measures the emission

of the black-body walls. In the non-contact type, the lens is directed to the object. In

infrared thermometry, it is important to consider the emissivity of the object.

Furthermore, in non-contact applications, the instrument reads the average temperature of what it sees, i.e. the object and its surroundings. To overcome this problem,

instruments can ‘crop’ the image so as to consider only the part where the object is

present. Remote infrared thermometry is extremely useful in measuring objects that

cannot be accessed for contact (e.g. in microwave heating) and moving objects.

B. Pressure

Pressure is measured either as a variable by itself or as an indicator of another variable. Examples of the second case are the use of pressure measurement as an indicator of level (hydrostatic level measurement), flow rate (orifice or Venturi flow meter),

temperature (autoclave), shear (homogenizer) etc. The measured value is expressed as

absolute pressure, gage pressure (absolute pressure minus atmospheric pressure) or

differential pressure (the difference between the pressures at two points of the system).

Pressure measuring instruments may be manometric, mechanical or electrical/

electronic. Manometers measure pressure or pressure difference through the level of

a fluid in a tube. This type of instrument is used in the laboratory but not in industry. The MacLeod Gage, used for the measurement of high vacuum, e.g. in a freeze

dryer, is a special type of mercury manometer.

In mechanical devices, the pressure signal is converted to displacement. The process pressure is applied to a flexible surface (a Bourdon tube, a membrane or bellows), causing the surface to move. The movement may be transmitted to a pointer

for direct reading or converted to an analog or digital signal sent to a distant data

acquisition element. A Bourdon gage is shown schematically in Figure 5.13.

Electrical/electronic pressure transducers are devices that emit an analog or digital

electrical signal as response to changes in applied pressure. There are many types of

pressure transducers, each based on a different physical effect. Strain gages generate

an electric signal (change in resistance) as a response to deformation (strain). The

most common configuration is a metallic foil pattern bound to an elastic backing. If

the gage is subjected to deformation under the effect of applied force, the resistance

of the metal pattern changes according to the following expression:



ΔR /R ϭ (GF )ε



⎛

ΔL ⎞

⎟

ε ϭ strain ⎜ e. g .

⎟

⎜

⎜

⎠

⎝

L ⎟



where:

R ϭ resistance of the unstrained gage

ΔR ϭ change in the resistance due to strain



(5.17)