Elementary process dynamics and control: Part one

(1)

(2)

E

p

C 539706

C10053

91060

BIBLIOTHEEK TU Delft p 1799 6144

11111111 11111

- = =-= ~§§;

-P1799

6144

---= ---= ---= ---= = ~-= = . = . = = = =

====-:=

= = == = = --== = - - =

(3)

part one

A.Johnson

ELEMENTARY

(4)

p

s:

o b \I h p a s t f v s c v i a r !< j

(5)

PREFACE

The term "chemica! process contro!" has been defined by Gou!d as that branch of industria! process contro! which is concerned with the dynamic behaviour and contro! of chemica! processes. In this book our horizons are limited to investigating chemica! process contro!, although, unlike Gould in his book

*

a few biological systems are presented as examples of living chemical processes.It goes almost with-out saying that processes such as steelmaking, water-purification , space flight, human behaviour etc. etc. could be analysed by the techniques presented here: in this book we are dealing with only a part of control- or systems engineering. It is the perversity ho wever of the dynamical behaviour of chemical processes which makes chemical process control a worth while study on its own.

The author's approach is somewhat mathematical in character, having evolved from an interest, rather than an ability, in applied mathematics. Frequently in complex situations the shorthand of mathematics is invaluable for seeing the "wood for the trees", aithough ad mittedly often the processing of equations towards a solution for the problem is a discouraging venture. The use of mathematics is not, of course, without its own risks and possible pitfalls, and to reduce these to a minimum a somewhat pedantic style is encouraged , at varianee with the traditional approach of the real mathematician, who will doubtless deern much as superfluous.

It is to be hoped that with a good command of the material presented the reader will be able to communicate intelligently with practising con trol engineers.As an introduetion to some of the more difficuit concepts such as freq uency response and stochastic processes (chapters 6 and 7) it is hoped that the book might reach a wider audience from biological, biochemical and chemical backgrounds.

Unfortunately it has proved necessary to split the topics (which have been kept to an absolute minimum consistent with providing an outline of the sub-ject) into two volumes. Part II contains an introduetion to system decornposit-ion, the state space approach, optimal control, chemical reactors, distillatdecornposit-ion, simulation and computer contro!.

My sineere thanks go to the students and staff of the Laboratory for Physical Technology in Delft for help, criticism and encouragement and in particular the toleranee and patience shown by Profs. J.M. Smith and N.W.F. Kossen,

S. Reurich, A. Schinkel and A. van Maarseveen.

(6)

(7)

la

IQ

la

II 12 13 17 19 22 23 24 24 25 25 27 27 28 28 30 30 30 34 36 38 40

(8)

4. TYPICAL ELEMENTS AND MORE THEORETICAL PROCESS

DYNAMICS 41

4.1. Introduetion 41

4.2. The typical process elements 41

4.2.1. Pure capacitive (or integrator) element 41

4.2.2. First-order (or capacity) element 43

4.2.3. Cascaded capacity element 43

4.2.4. Dead-time element 45

4.2.5. Nonminimum phase element 45

4.2.6. The second-order element 46

4.3. A drum-boiler model 48

4.4. A model for the growth of algae 50

4.5. A model for liquid to solid heat transfer, percolation 51

4.6. Batch processes 52

4.7. References and bibliographical notes 54

5. PROCESS IDENTIFICATION AND PARAMETERESTI MATI O N 56

5.2. Step testing 56

5.3. Impulse testing and the time domain response 57

5.4. Pulse testing 60

6. RESPONSES AND RELATIONSHlPS IN THE FREQUENCY DOM AlN 63

6.2. The magnitude ratio and the phase angle 63

6.3. The Fourier transfer function, GpUw) 65

6.4. Nyquist plots 66

6.5. Bode diagrams 66

7. STOCHASTIC SIGNALS AND PROCESSES 70

7.2. Probability density functions and disturbances 70 7.3. Disturbance dynamics and corre1ation functions 71

7.4. Power density spectrum, <P_x(jw) 75

7.5. Stationary stochastic signals applied to linear processes 77

7.5.1. The expected value of Y(t) 77

7.5.2. The density function of Y(t) 77

7.5.3. The power density spectrum of Yït) 77

8. THE DYNAMICS OF SENSING,TRANSMISSION AND FINAL

CONTROL ELEMENTS 82

8.1. Introd uction 82

8.2. Transmitters 83

8.3. Measurement noise 83

8.4. Pneumatic transmission lines 87

(9)

8.5. _{The final con trol element}

88 8.6. _{References and bibliographical notes}

91 9. _{BASIC CONTROLLERS, CONTROL ACTIONS AND CONTROL}

SCHEMES ₉₃

9.1. Introduetion ₉₃

9.2. _{Performance criteria}

93 9.3. _{Basic linear controllers and control actions}

94 9.3.1. _{Proportional action controller}

94 9.3.2. _{Proportional-integral action controller}

95 9.3.3. Proportional-integral_-derivative_{action controller}

95 9.4. _{Feedback control using P, P-I and P-I-D controllers}

96 9.4.1. _{The effect of proportional con trol action}

96 9.4.2. _{The effect of P-I control action}

97 9.4.3. _{The effect of P-I-D control action}

99 9.5. _{Feedforward control}

99 9.6. _{Adjustment of controller settings}

lOl 9.7. _{Nonlinear controllers}

103 10. _{MORE COMPLICATED CONTROL SCHEMES}

lOS

10.1. Introduetion _lOS

10.2. _{Feedforward -feedback control} _lOS

10.3. _{Decoupling interacting control systems} ₁₀₆

10.4. Cascade control ₁₀₈

10.5. Ratio control ₁₀₉

10.6. Override control ₁₁₀

10.7. _{Split-range control}

113 11. _{ELEMENTARY STABILITY CONSIDERATIONS}

114

11.1. Introd uction ₁₁₄

11.2. _{Factors influencing stability and types of stability} ₁₁₄

11.3. _{Poles and zeros}

_{n s}

11.4. _{The s-plane representation of stability}

117 11.5. _{The Hurwitz- Routh stability criterion}

118 11.6. _{The root-locus technique}

119

11.7. _{The Nyquist stability criterion} ₁₂₀

11.8. _{Cancellation of common factors} ₁₂₂

11.9. _{References and bibliographical notes} ₁₂₃

APPENDIX A ₁₂₄

A.1. _{Comparison of nonlinear equation (2-3) and the} linearised approximation obtained by application of Taylor's series expansion, eq. (2-8)

APPENDIX B ₁₂₄

BASIC DEFINITIONS AND RELATIONSHIPS OF STOCHASTIC PROCESSES

(10)

B.2. Distribution function,F{a,J3} B.3. The density function,p{a ,J3}

BA. The expected value E {X({3)} B.5. Stationary processes B.6. The varianee,a2

B.7. Normally distributed process

125

1

125

j

126

c

1

26

c

127

C d d D D D, ID E E

f

F F( F{ g g Ge Ge I Go G_p Gr( G_t( Gy{ h h H HA H

1

₀

(11)

NOMENCLATURE

A c d, n d D O(s) °o(s) 10 E E{X}

f

F F(s) F{a,f3} g g Ge(s) Ge (s) o Go(s) Gp(s) Gr(s) Gt(s) Gy(s) h

a constant; heat transfer area (rn"); cross sectional area (m2 ) concentration of a species (rnoles/kg): controller output signal specific heat per unit mass at constant pressure (J/kgOC) concentration of a species(moles/rn"): _{capacitance per unit} length (m4

/ N) a disturbance variabie set of disturbance varia bles diameter (m)

denominator of the closed loop transfer function denominator of the open loop transfer function mass diffusitivity (m2_Is)

activation energy (J/mole)

the expected value of a stochastic variabIe set of functions describingy

set of functions describing the chemical process; mass flowrate (kg/s) filter or estimator function

the distribution function of a stochastic process accelaration due to gravity (m/s2

) set of functions describing y controller transfer function

optimal controller transfer function open loop transfer function process transfer function ratio controller transfer function

transducer or transmitter transfer function final control element transfer function

a function describing the controller; height of liquid level (m); or heat transfer coefficient (J/m2s °C)

liq uid enthalpy (J/kg) heat flux (J Is)

heat flux per unit area (W/m2 ) vapour enthalpy (J/kg)

intensity of noise; inertance per unit length(kg/rn'") intensity of incident light

(12)

J performance index

k constant; reaction velocity (S-1) km gain margin

L length (m)

m mass of object (kg); transmitter output signal

M magnitude ratio; mass of fluid (kg); molecular weight (kg/rnole) n number of moles; measurement noise

NA N(s) No(s) P

PB

Pct

Ij

p.

o

J

P

_u P{a,{3} q

Q

r R S u u

u

v . v

v

x X(t) y x

mass flowrate ofA (moles/rni s)

numerator of the closed loop transfer function numerator of the open loop transfer function pressure (N/m2)

proportional band of a controller pressure downstream (N/m2)

partial pressure of component j (N/m2)

vapour pressure of the pure j-th component (N /m2)

pressure upstream (N/m2)

the probability of a stochastic process controller actuating signal

volumetrie flowrate (m3/s)

volumetrie flowrate at standard conditions (SCM/s) a setpoint signal

gas constant U/mol.K);resistance to flow per unit length (Ns/rn") perimeter of pipe (m)

dead time (s) an adjustable variable a set of adjustable variables unit step change

an unmeasured output variable; velo city (m/s) set of unmeasured variables

volume (m3)

an output or state variable; stem travel (m) mole fraction of j in liquid

set of all output variables a stochastic input variable

an observed or measured output variable

Î

s

n

e

_,

À.,

u

P T S) a.b Al XE Áx L(x x(s)

(13)

SYMBOLS

the desired value of the measured process output, Y mole fraction of j in vapour

set of measured variables a stochastic output variabIe

correction variabIe; coordinate direction thermal diffusitivity (m2/s)

relative volatility of component i to component j activity coefficient of the j-th component

unit impulse change; incremental change in magnitude the mean value of a stationary stochastic process; dynamic viscosity (Ns/ m2

) temperature

t

C)

absolute temperature (K) heat of vaporisation(J/kg)

an artificial parameter; kinematic viscosity (m2/s) density of fluid (kg/m")

density at standard conditions(kg/rn")

the density function of a stochastic process the varianee of a stationary stochastic process time constant (s); momentum flux (N/m2₎ phase angle (radians)

phase margin

(autojpower density spectrum of X(t) autocorrelation function of X(t)

crosscorrelation function of X(t) and Y(t) angular velo city(rad/s)

a.b, .. .,Z _{Italie type denotes the set of elements} A U B _{denotes the union of two sets,} _A _and _B x€a _{the variabIe x}_is_{an element of the set}_a

a.. IJ z

.! _{the column vector containing elements xi}

~x _{perturbation of x from a reference value; finite difference}

L(x) _{the Laplace transform of x} :K(s) _{the Laplace transform of x(t)}

a Yd Yj y Y(t) Ps P{a,,B} 7 J1 P

(14)

det

I I

Re Im lim Sc

the value of the function just after time zero equal to by definition

determinant modulus

the real part of a complex number the irnaginary part of a complex number .the limit

(15)

Let's start at the very beginning -that's a very good place ta start.

with apologies to The Soundaf Music

1 OPEN AND CLOSED-LOOP SYSTEMS

1.1. introduetion

Q._{1: What considerations might}play_{a role in determining which af the output variablesare} fa

be measured?

unrncasur-ed

out.p ut. variables V

Fig. _{1.1. The basic block diagram}_.

Figure 1.1 is of ablock diagram _{representing a typicaJ chemical process}_._Following normal conventions, the inflowing material, energy or information enters the process bJock either from above or from the left hand side.

It_{becomes clear from the diagram th at the chemical plant of process rnay be viewed} as the relationships:

It_{is customary to distinguish between processes having a controller(s) between the} process output(s) and its adjustable input(s) and those which do not. The chemical process is usually sufficiently characterised by a chemical plant having:

adjustable inputs,_{such as flowrates or pressures, which are denoted henceforth*} by the set u,

uncontrollableordisturbance inputs,_{for example the ambient temperature or} catalyst ageing,denoted by d,

measured output variables,_{y, which rnay be sensed either continuously or discretely} in time and

unmeasured output variables, v.

~~p:::;l~a::n-;-t

---'-

1--- -

measured

é'ldjustc::ble

==:J

or out p ut inputs - - process ver-Iables

U

_y

*

A list of the nomenc1ature usedis given on page ix. ltalic script is used to denote a set or subset.

dtstur-bance inputs

d

(16)

y

=

f(u,d,v,space,time) v=g(u,d,y,space,time) or defining x =

yu

v

t

x

=

F(u,d,space,time) (l-I a) (I-Ib) (I-I) where we have used our intuition that the output may be depend upon position (for example the temperature inside a refrigerator) and time (such as the temperature of washing-up water). From experience we also know th at either u (in the case of an explosive reaction) or d (as for the production of an ice block in a freezing corn-part ment) rnay, for all practical purposes, be absent from (I-I) - u and d are,in such cases, ernpty sets. It should be kept in mind that the relationship (l-l) can change with time, as do processes such as electric ketties, lamps and automobiles, where deposits of material and the reaction of chemieals cause measured perform-ance to deteriorate as the months go by,

1.2. the open-loop .system

When there are no controllers present between the measured output variables(Ey) and the adjustable inputs(EU) the systern is said to be an open-loop system.

Q2: The process can still be controlled. How?

Possibilities for control depend upon the severity of the disturbances. Ifthere are no internalor external disturbances of appreciabie magnitude or duration ex-pected, then the controller may operate on an time basis. As an example of an open-loop-control systern Ogata [1] quotes the washing machine, where soaking,washing and rinsing are performed at fixed positions of a time cyc1e.

Q3: Give a second example of open-loopcontrol on a time basis. Can the possibility of disturbances be neglected?

Ifthe disturbances must be accounted for, then the possibility of feedforward con trol can be considered.Here, estimated or measured values of d Ed are used by a controller to regulate the inputs u Eu of the process. Feedforward control, to be discussed and illustrated later, is also used in situations where no output variabie can be measured (i.e. y an ernpty set).

1.3. the closed-Ioop system

The result of placing a controller between y EY and u Euis the closing of a loop. shown diagrammatically in Fig. 1.2.

Notice that just as the process needs energy to perform its task, so does the con-troller, although we shall not return to this point hereafter. More fundarnentally important is the concept of a feedback of information which has been introduced by closing the loop. The.great advantage that such feedback controloffers is that it operates independent of the cause of any deviation of the measured output

varia-t

An explanation of symbols uscd isgiven on page xi.

(17)

bie, y. That is to say that no matter which d

e

d _{caused y to change its value, a} correction will be applied to the process in an attempt to nullify the deviation. Q_{4: Is this always so for feedforward control?}

d

In Fig. 1.2 the various physical elements which make up the closed loop have not been individually shown.

e U- --"-r-r--r---r---

Y

v

V u _y (1-2) ,-A ,-,-'-

'"

, /

power- setpolnt vatue

u = her - y,time) Fig. 1.2. The basicclosed-Ioopsystern.

Notice that in writing function (1-2) this way,we _{have introduced an extremely} important sign reversal and thereby foreshadowed the use of negative feedback. Just how the sign reversal comes about is best seen from Fig. 1.3, where positive and negative feedback are contrasted;_{we shall show later that negative feedback} can be proved to be the best form of control in many situations.

In general seven elements are distinguishable: I. _{The chemica I process}

2. Measuring instrurnents or sensors 3. Transducers or transmitters 4. Transmission lines

5. The controller 6. Recording apparatus

7. _{Control element (valve, switch, pump)}

The controller itself is usually constructed to compare a pneumatic or electrical signal analogous to the measured output signal y with a "desired value" or setpoint signal, r. _{Similar to our brief analysis of the chemical process we can often describe} the controller block in terms of

Q.5: Reconstruct Fig._{1.2 showing the relative positions of the seven e/ements}_.

(18)

1-d d input to devtauen reduced in rneasur-ed variabie r r

-rr

_F1

=f'=,

u _process

I~I

pocltlve input to pr-oce ss deviatlon

error pr-ocess reacts to .n c r-e e sed

signalln Incr-eased Increased in

controller input measured

variabie devieuon in d - i

4:C

devlaucn in rneasur-ed variabie d process (b) Positive feedback ~

Fig. 1.3. Negative and positivefeedb ack.

process rea cts to disturbance "~tu,b IIIfG CI ~ U: i "

l

val~iable IdA -

Y~

tt

u f \

--~

y

u - q

_-

_u+"

_pr-ocess

1

r-y

l

process . I

I

'

- t

pr-ocess re"3cts negative process

te dist.urbance error reacts

iative feedback

t

signal in to reduced controller- Input

J.:./\.

_r-X.

~

l-d:st.urbancc d occurs I

I

1~lli

LJ

,u

~

(19)

The physical construction of the elements 1 to 7 of the con trol loop is described elsewhere [3].

1.4. four reasons for control

Controllers rnay be applied to chemical processes in order to

enhance_{and ensure process stability} exploit process instability

suppressthe influence of disturbances optimize process performance

V.d ._{Grinten [2] lists three of the above reasons, to which has been added the expl}

oit-ation of process instability. _{A brief explanation of the terms and ideas introduced} is given below.

1.4.1. control1ing the processstability

Figure 1.4 shows the output variables of two different processes which have under -gone asmall input change.

Fig. lAa. pHoseillation (After C.C.Prasad and P.R._{Krishnaswarny, Chern. Eng. Sci.,}

30,214(1975)).

4 000~

_i

30oJ

coun"

plZr rnlnut.e

/

°

/

/0

0/0

_0"

0 - 0 0 _ 0 0 0

-1

J: 9.0

,0-"

0..

_°

_~ 8.6

,

°

1

I \

°

8.2 I ~

°

_I

\

°

7.8 I

°

,

I

°

7.4 I

°

7.0, ~

r

200L..- _O-_{=-== ....:...} _ 1000 2000

Nurnber-ofcornplet.ed Iayer-s -F_{ig. lAb. Radioaetivity of Atomie Reaetion} _(Af_{ter H.D.}_Smyth_,

(20)

The first process has astable response (Fig. l.4a) in contrast to the second, un-stabIe response (Fig. l.4b). In both cases, we are considering the process alone -there are no external controllers present - so that it is possible to deduce that some (indee d, most) chemical processes have a certain inbuilt control action in play. These so-called self-regulating processes make the use of external stabilizing control less vital and sometimes even unnecessary.

Q.6: Mosthouseholdsused to possess an exampleof a chemical process with a large degree of self-regulation.Which process?

A common industrial self-regulating process is a buffer tank. An analysis of

the buffer tan k isleft until lat er, but in this context we can get a feel for the behaviour of the process as follows. The pressure above the valve in the outlet pipe, and hence the flow of liquid through the valve, corresponds to the height of liquid in the tank .Thus, an increase in the liquid level causes more liquid to flow out

throug h the valve - thisleading to self-regulation of the level.

Q. 7:Dy similar intuitive reasoning it is possible to explain the behaviour of processes with

little or no selfregulation. Describethe sequences of events, for example,occurring during a

strongly exothermicreactiontaking place in aninsulated environment.

Whatever the process, stabilizing control is always accomplished by closing the loo p, asdiscu ssed in the last section, withnegative feedback. Such a c1osed-Ioop

syst ern (process+ stabilizing contro ller) is not, unfortunately, always stabIe; indeed the addition of a controller to a stabIe process can lead to an unstable systern,a

point whic hwe will return to in our discussion of stability (Chapter 11).

'1.4.2. exploitation of process instab ilit y

Depend ing upon the definition of stability chosen (see Himmelblau and Bischoff

[5) and Chapte r 11) the third process shown in Fig. 1.5 can be c1assified as either

unstabl e or stabIe. Here the process response hun ts, or oscillates, between a maximum

and a minimum value, in much the same way as the pendulum of a grandfather c1ock. Denn [4 ) gives examples of such oscillat ing responses of processes from many fields

of interest, and points ou t tha t since hunti ng is usually asymmetrie about the design value,the average output over aperiod will be different from the nominal design

16 L.--'-- -'----'-- -'----"

o

10 20 30 .tiC) 50

- lim e (m in ) Fig. 1.5. Temperature oscillations in the acetylchloride

- acetic acid reaction (After G.? Baccaro, N.Y. Gaito nde and J.M. Douglas,AIChEJ.,

249 (1970» .

~2

8~

• tlI •• 526

i

\.

~

_{.. H}

2 4[

I \

_.

'"

, .

E

2 2

'----1

~

20

\ / -' \ . C ' . tlI 18 '.... ::J ;:::

....

tlI

(21)

ne

tese

.d

value. Under some conditions this difference might be favourable, inwhich case the control system might be required to enhance the oscillations, rather than to pre vent or damp them out. It _{must immediately be said, ho wever , that such control} _action has yet to find general application in industry.

1.4.3. suppressionof disturbances

The term "disturbance" is used here to include all those variableswhose values cannot be manipulated (see Fig. _{1.1), so that, for example, the inflow to t}_he _buffer tank mentioned previously would be considered as a disturbance variable, Usually we will find it necessary to handle disturbances as stochastic variables where, rather than speaking in terms of an amplitude or particular location with respect to time, we would consider the probability of the disturbance being found bet ween range of values in magnitude and time. The concentration of a slurry is a good example of a disturbance best treated as a stochastic variabie [6] .

Control of disturbances themselves is sometimes possible.FOT example, a

fluctuat-ing disturbance variabie can often be rendered harmiess by the use of inter mediate buffer tanks [2] _{although the cost of the extra capacity may be high}_._Th_{e a}

lte rnative is to use feedforward, feedback or combined feedforward -feedback control on the process itself. Pure feedforward control (Fig. 1.6) is rarely used, due to difficulties

or:

y

V

,

d

I

unrneasur-ed dist u r b a nces

T

~

t

V d - -..., pro cczss I--~

u

r

I

₁ con t rolle r u d mea sur-ed disturb ance

Fig.1.6.Feedforward control. mum .lock -elds sign er iff ~d

(22)

in measurernent and the fact that only deviations occurring in the measured disturb-ance can be compensated for.

Q. 8: Wh at are the "difficulties in measurement " referred to above?

1.4.4. optimizing process performance

The last reason for con trol is to optimize the process performance. The criterium for optimization is usually short term financial profit [2] although possible scarcity of material resources may enduce a longer-term approach in the future. The important thing here is to realise that, no matter what criterium adopted, it is highly unlikely that the actual performance of the process will match the design performance or that.the optimum performance judged by one market situation will prevail for more than a few months at the most. Thus, as knowledge of the process increases with usage or the markets for products alter, it becomes necessary to change the adjust-able process inputs in such a way as to effect a new (more profitable) performance.

Q.9: What other changes can necessitate a review of the profitabilityof the present perform-ance of aprocess?

For processes in a process-train it is necessary, of course, to take the global view -point and maximize a collection of processes together. This can mean running one or more processes with a sub-optirnal performance, a point which we shall turn to later in Part II of this book.

l.S. concluding remarks

It is hoped that the reader has now an idea of open- and closed-loops, feedfor-ward and negative feedback and the four reasons for applying external control action. The following chapters become progressively less chatty, although a few more terms and periferal topics have still to be introduced in the next chapter.

1 .6.

references and bibliographical notes

A readable introduetion to open- and closed-loop systemscan be found in

[I]: OGATA, K. Modern Con trol Engineering, Prentice-Hall, Eaglewood Cliffs, 1970. No refercnce is made, however,10 feedforward control as an example of open-loop contro!.

The c1assification used here follows

[2]: VAN DER GRINTEN, P.M.E.M. Regeltechniek en automatisering in de procesindustrie, Prisma Tecnica 9, Het Spectrum,Utrecht, 1968.

Details of the construction of measuring devices, chemical plant, transducers and controllers can be found in

[3]: PERRY, J.H.The Chemical Engineers' Handbook, sth Edition, McGraw Hill, New Vork, 1974.

An excellent book concemed with process stability is

[4]: DENN, M.M. Stability of reaction and Transport Processes,Prentice-Hall, Englewood CHffs, 1975.

(23)

rrb-urn ity rrtant ely .o re 1 st-.ce . 1

-[5]: _{HIMMELBLAU, O}_.M._{and BISCHOFF}_,_{K.B. Process Analysis and Simu}

lation:Deterministic

Systerns,John Wiley, New York, 1968.

The power of a stochasticas opposed to deterministicapproach is shown in a paper on the

on-line digital computer con trol of slurry-conditioning in mineralf1otation: [6]: KING, R.P.Automatica , 10,p.5- 14.(1974)

which is not,_{however, for the mathematically faint-hearted.}

iew-ne to r -ction -.r ms .trie, ·s 'ork,

(24)

It ain't what you do, it's the way that you do it: That's what gets results.

2 MATHEMATICS FOR DYNAMICS

2.1. introduetion

The purpose of this chapter is to present the necessary mathematical background and definitions so that later the reader wil! be able to follow what is written with-out undue recourse to mathematical texts.

2.2. c1assification of systems

The elements of the open or closed-loop systern, introduced in the previous chapter, do not respond instanteneously when perturbed. That is to say that because each element has its own dynamic characteristics - or dynamics - the response of an output variabie to a perturbation of an input will have its own time dependence.

Ifthe systern variables are uninterrupted in time (Fig. 2.la) we talk of a

contin-uous system, whereas variables in the form of pulses are characteristic of a discrete

or sampled data systern (Fig. 2.1b). The lat ter occur for example in digital computer control systems, discussed in Part 11 of this book.

y,u ord

• _I

Fig. 2.la. Continuous system response. - t i m e JO y,ucrd .1

i

_I

,,\

, \ \ \

,

\ - t i m e

(25)

und .h - tin-ete puter \ \

A distinction is also made between single input - single output (s.i.s.o.) systems and multivariable systerns. The so-called classical con trol theory approach, presented in the first part of this book, is concerned with reducing all problems to the s.i.s.o, form. Most real processes are complex enough, however, to warrant consideration as interacting multi input - multi output (multivariable) systerns, and can benefit more from the state-space approach of modern con trol theory.

Returning for a moment to equation (I-I) we reeall that we had included in the response not only a time dependenee but also a spatial dependence. This spatial dependenee is frequently so small that it _{can be neglected in our analysis, so that} for a continuous systern it is possible to describe the dynamics by means of one or more ordinary differenttal equations (o.d.e.). Systerns described by o.d.e.'s are refer-red to as lumped parameter systems. If _{the spatial dependenee is important, the} dynamics must be formulated in terms of partial differential equations (p.d.e.) and such a system is said to have distributed parameters. Examples of both lumped and distributed parameter, continuous systerns are the basis of the next chapter.

2.3. the

_{analysis of dynamical systems}

Let us initially confine our discussion to the behaviour of lumped parameter systems. Anticipating the analysis presented in the next chapter, we state here that the general form of such systems is an n-th order o.d.e. The order of the

sy stem is the same as the order of the o.d.e. describing the dynamics.

Q. 10: whatis_{the order of the two systems described}_by _the _following_o.d_.e._'s:

d2x (dx )3 A)

df2

+

t

di- +10x=9u. { dXt

=

5x +4u dt 1 E) dx,- =3x

+

2x dt 2 1

ft _{is a simple matter to convert an n-th order o.d.e. to n simultaneous first} order o.d.e.'s, and thus give another meaning to the order of the system. The conversion is acheived by defining (n - I) new variables [1,2):

Xl

==

x X - dx 2

=

Of d2

x

x3

==

dt2 etc. dn-I X Xn

==

dtn-I

(26)

dx

replaced by

dt.

In Part 11 of thisbook we sha ll conside r how to modify the

above transformation to account for mor e complicated in put ter ms.

Q. 11:Express the first equation afQ.10 as first ordero.d.e.'s and thesecond equation

as an n-tn order o.d.e.

Thus we can write the general form of the dynamic behaviouroflump ed

parameter systems as [3]:

(2-la)

Here,x_J,x

2'. . .,xn(Ex, see Chapter I,eqn. (I-I» define the stateof the system at any time, t. We can also write (2-la) in vect or notati on, letting~

be the column vector, whose elements are xi' and so forth:

• dx

X = dt =

f

(IÇ,)J,Q,t) (2-lb)

where F is a vector valued function of X,getc. A third possibil it y isto wr it e eqs.(2-la) and (2-lb) in terms of a typical co mpo ne nt:

• dx.

\ ==

ctt

=

fi(~,!!,Q,t) , i

=

1,2,3 ,. .. ,n (2- lc)

All three forms are equivalent.

An analysis of dynamical systems depends on the adheren ce to the principle

af causality: 1fX is known at time t = t_o (say the vec tor ~o)' then the response of the system at any later time t is uniquely defined by the solu t ion to eqs.

(2-1) which passes through X

o

at t

o'

A sufficient condition for uniqueness is

derived by Denn [3],who also provides referenceson page 31 for those inte

r-ested in persuing this topic.

Dynamical systerns where

f

(of eqs. 2- 1b) does not depend ex plicit lyo n

the time, t:

(2-2) are calied stationary systems.

2.4. linear, nonlinear and linearised systems

Linearsysterns are those systems whose dynamic behaviour is described by eqs.(2-1) where the coefficients of~,Qand !:!are constan ts or fun ct ion s only of the time t [2].

(27)

Q.12: Are the fol/owing systems linear?

A)

X

=_ax2 +bu d2x

rf!X] 3

B)

d?

+i.dt

-

ax=bu C) ; =-sgn (t).x+b2u

Linear systerns are mathematically desirabIe since the principle of

super-position can be applied to facilitate analysis. This principle states that the

response to several inputs can be calculated by applying one input at a time and adding the results. The converse has important implications for the exp-erimental investigation of dynamics since if it can be shown that cause and effect are proportional then the system can (by the principle) be considered linear [2].

Nonlinear systerns can be classified according to whether the nonlinearity

is continuous or discontinuous [4]. Figure 2.2.shows examples of both types and the reader is referred to [5] for details of methods available for analysis of discontinuous nonlinearities. - u DIscontinuous nonllnearlty

x

t

~u ContInuous nonlinearity

2.5. linearization

Q. 13: Sketch the graph of; versus u from Fig.2.2.Is the significaneeof the terms

continuous and discontinuous apparent?

We shall not consider discontinuous nonlinear systerns further. Some contin-uous nonlinearities can be handled, if not too pronounced, by approximation Over a limited range to a linear relationship. The technique used is calied

linear-ization and is discussed in the following section.

Most real chemical processes are only accurately described by nonlinear

eq uations. When confronted with a continuously nonlinear system the engineer may attempt to

transform, by means of a substitution, the nonlinear equation into a linear equation;

Fig. 2.2. Nonlinear responses. X

A

I

(28)

simulate the system on a computer;

linearise the nonlinear terms.

Unfortunately the transformation method is limited to the few simple non

-linear o.d.e.'s having a known closed form solution [6]. Amo ng these the forms of the Bernoulli, Riccati and Abel equations seem to be the most recurrent. Example2.1.

Suppose that allopen tank (Fig. 2.3) isbeing emptiedaf its contents. The height af

liquid h in thetank is known [7] to be related the valve ste m travel, u, by the equation

dh_dt +au_VII

r,;

=0

Is it possible to modelthe processexactlybymeansaf alinear system?

h

J

Fig. 2.3.Emptying a vessel.

Setting x

=

Vh

the original nonlinear relationship can beexpressed as

2x

.~~

+ aux= 0

(2-3)

or

x(2~~

+au)

=

0 (2-4)

Neglecti ng the trivial solution h= 0 [8] we find the linear o.d .e.

dx _ au dt - -2

describes the dynamic behaviour exactly.

(2-5)

Q.14:If the valvein the above exampleisnat moved but acontinuous flow,Ft' enters thetank the eq. (2-3) must be replaced by

dh_dt+b

r,;

=cf'

VII t

Is it possible to transfarm this equationto a linear farm ?

A discussion of computer simulation techniques is deferred until Part 1Iof

this book. Most important for the analysisofdyna mic systerns is the techniq ue

(29)

(2-6) of linearization. Two methods [6] _{are noteworthy although both only provide}

alinear _{equation applicable over a limit}_ed _{range of inter}

est. lust how large the

range is wiII bediscussed later. Firstly,_{then, is the method based upon a Taylor}

series expansion about a point of reference.To recaIl, the Taylor's series for

afunction of one variabie,f(x), _{about a point of reference x}₌ _x, _{is [I]}_:

f(x) = _{f(x ) + (x _ x )df(\) + (x} _- _{x / d}2f(x_{r )} ₊ _...

r r dx 2! dx?

Neglecting second and higher-order terms gives a relationship on which the linear-ization procedure is based._{For functions of two variables, neg}_lecting_seco_nd _a_nd higher-order terms again, the relationship becomes:

and so on for functions of still more variables. Example 2.2.

Using the above method,whatisthe_{approximate linear form obtained for eq. (2}_.3)

afthe previous example?

Since the term

auVh

_{is nonlinear we set} f(u,h) = auy'h

Introducing the new variables

(2-8)

r,- .r- .Il:: au

auyh~auyh + _{(u - u ) ay h. + (h - h )}

!t:-r r r r r 2y h,

r

that, when substituted in eq.(2-3), yields:

dh .rt: au au Vh

- +

ay h•.u

+

__

r_.h

=

r r

dt r 2v'h: 2

r

Example 2.3.

Using perturbation variables. what approximate,linear farm isobtained for eq.(2-3)? the required, linear, approximation. Equations (2-5) and (2-8) are compared

graphically in Figure 2.4, _{and discussed further in Appendix A}_._{We note}

here only that the agreement between the nonlinear and approximate,linear, e_{q uatio ns is good around the point of reference (h}

₌

_{ho at t}

₌

₀₊

for this

exa mple).

and apply eq. (2.7). The resuIt is

The second method of Iinearization is the so-called perturbation method, where variables representing _{smaII deviations from fixed reference points re} -place the problem varia bles.

(30)

~h= h - hand_r

~u = u - U r

we can replace the nonlinear term

..jh

using eq. (2-6). Thus

(h - h ) ~h v'h=~+ r

=

+~ r

2K

_r

2V'h:

_r r 0.2 liquid level h~ight.h

r

0.1 Eq.(2-51 - - - ; r > tlrne , t O -'--_--'L_.__ ..._.----1 J

o

0.2 OA 0.6

Fig. 2.4.TIJe fall in liquid level predicted by eqs. (2-5) and (2-8).

1.0

In terms of perturbation variables, eq. (2-3) becomes:

d~h

+

a(~u

+

Ur)(2~

+

v'l\)

=

0

since ddhr is zero (differentiation of a constant, h ). NegIecting productsof

t r

perturbation variables (if excursions from the reference point t are smalI) we obtain the linear approximation

(31)

d zsh . rt: aurboh re: -dt + ay h..bou + .~

=

-au y h, r 2yh, r r r (2-9) which is equivalent to eq.(2-8).

2.6. the range of the linear approximation

Douglas [9] has paid attention to the problem of establishing the range over which the linearization leads to satisfactory results. He concludes that the frequency response

*

of nonlinear equations contain higher harmonies, in addition to the fundamental component, and that the time-average value

t

of the system output wil! be different from that of the linear approximation.

The analysis for a single nonlinear equation proceeds as follows (see [9,10] for extension to the general case of eq.(2-1 )). Writing the nonlinear function f in terms of a Taylor series expansion and using the perturbation variables introduced in the previous section we find that

d box af

ar

-dt= f(x ,u ,d ) + box-a + bou-a + r r r X U [ a2f a2f 2 a 2 f]

+

~

box2

'a?

+ 2boxbou axau + bou 'aU2 + ... (2-10) where it has been assumed for simplicity that the disturbance remains constant, and that an artificial parameter Jl may be introduced with the quadratic and higher-order terms.Now suppose that the solution to eq. (2-10) can be expressed in the form:

Substituting this into eq. (2-10) gives:

(2-11 ) (2-12) (2-13) 1 T x =-Tf xdt. av 0 dz af af -=.!!.d= f(x ,u ,d ) + zo'>:": + bou -a t r r r OX U

Equating terms having like coefficients of Jl we obtain the set of equations: dz

O dz 2dz _

(ft+Jl~ + Jl~ _{+ ... - f(xr,ur,d r) +}

2

ar

af

+ (zo + JlZt + Jl z2 + ... ) ax + bouau+ etc.

*

See Chapter 6.

(32)

(2.14)

These equations are the sought-for·link between linearity and accuracy.We know that in some small region around the chosen reference point, r,the linearapproximation is good (i.e. terms containing pand p2 can be neglected in eq. (2-11)).This means that zl ,z2' . .. etc.will be small compared with zo' As we move further away from r the magnitudes of ZI,z2' .. . etc . beco me greater as the accuracy decreases.

Q:15: What relationship has eq (2-11) to the original nonlinear equation ? Example 2.4.

Use Douglas's methad to investigatetheaccuracyaf linearising eq.(2-3)af Example 2.1,ifu is changed in a stepwise mannerta ufat time t=O.

Applying eqs. (2-12),(2-13) and (2-14) to eq. (2-3) we find that:

• 2bh Zo = -2h_r·b - _{bzo -}

7

.Lîu r (2-15) (2- 16) (2-17) etc., aU_r where b = 2Vh:. r

Note how the first of these eq uations, eq. (2-15) is the same as the linear form eq. (2-9) of Example 2.3, if Zo = Lîh.

Integration of eq. (2-15) gives

Z

=

2h (e-bt - 1)

o

0

(2-18)

where the initial condition is Zo

=

0 at t

=

0+. In b we have u,

=

uf and

h_r = ho· Substitution of this result into eq. (2-16), followed by integration, gives

(2-19)

(2-20)

Z

=

h (l - 2bte-bt _ e-2bt)

I 0

The first term is, indeed, the response of the linearised approximation (see Appendix A, eq. (A-3)). The second term, multiplied by the dummy par a-It is usually unnecessary to proceed further than the first correction variabIe, zi' Thus the solution of the nonlinear equation, which is given by eq. (2-1·1) may be written as:

(33)

meter/l (= _{I) enables us to predict immediately that the linear approximation} is best for small initial levels of liquid in the tank (since ho multiplies the bracketed term).

Q. _{16: Compute the response predicted by eq. (2-20) and campare with Fig. 2.4. Is it possible} to access the effect of the magnitude af parameter b on the accuracy af the linearized farm?

2.7 .

_{the use of Laplace transforms}

It_{is most likely that the reader will have already encountered, and perhaps} used,_{Laplace transforms. A good introduetion to the technique is provided in} Jenson and Jeffrey's book [I] _{for example. Here, some observations are made}

whichmay _{prove especially useful in the context of chemicaI}

process analysis. It_{is common to consider first the various forms of [orcing function in general} use,_{since before any analysis of the process dynamic response can be attempted} some indication must be given as to the type of change undergoneby,or expected at an input.

The most important forcing function is the step, shown in Fig. 2.5a and such as might occur, for example, when a valve setting is suddenly changed.Itis a very severe disturbance to apply to a process and is usually described mathem-atically in the time and Laplace domains by

* The syrn bolLis_{used here and henceforth to denote the operation of Laplace transforming}

. (2.22c) (2-21) (2-22b) (2-22a) f(t)

=

A.tU(t) Lf(t)

=:;}

s A.U(t)= {A for t

>

°

o

for t

<

0

A

*

L.U(t)

=

s

'

A.Ö(t)= 0 ifft=f::O

1

+~

J

A.Ö(t).dt

=

A -~ and and and

The last forcing function we shall encounter is the s_{inusoidal input, Fig. 2}.5d.

Depending up on the phase of the forcing function at zero time we have where LA.Ö(t) = _{A, where Ö is the unit impulse, or Dirac delta, function.}

Another possibility is to use the so-called ramp function, Fig. 2.5c. Here, Here, and in the remainder of this section, the input change is assumed to occur when time is zero. U(t) is the unit step input.

The impulse function, Fig. 2.5b, corresponds to an infinitly large excursion frorn the reference value for an infinitely small time. A rectangular pulse is of ten used to approximate the impulse. Mathematically

(34)

-tlm~

2.5a.Th~stepchang~.

_A.Ö(tl

t

Ar~aof' splke

=

A

-tlm~

2.5b.The lrnpulse f'orclng functlon

flu

_Î

l

_Slop~=A

I

2.5c .Theramp func t lo n

flt!=At.U(tl

-tim~

2.5d.Trie coslne input Fig. 2.5. Variouscomrnon forcing functions.

or 20 A fIt ) !

I

.

f(t) = A.U(t).c os wt As Lf(t)

=

2 2 S

+

w f(t)

=

A.U(t).sin wt Aw Lf(t)

=

2 2 S

+

w f(t)=AU(t)cos wt (2-22d) (2-22e)

(35)

Q.17:A shift theorem [9], whieh states that

Lf(t- tol=e-st0 Lf'[t] (2-23)

enables usto investigateforcingfunctionsapplied some time,t0'after zero time.Give

expressions,eorresponding toeqs(2-22a to e),for sueh delayed forcing functions, in both timeandLaplacedomains.

Periodic forcing functions sueh as shown in Figure 2.6 can be transformed using the relationship

[9]:

T

J

f (t ).e- st d t

o

p Lf(t)

=

I -ST - e (2-24)

rltl

t

Fig. 2.6.A perio dicforcing function

(asawtooth signa l). I

1:

:

at at

41 51:

t---1

- ttrne.t, whereT is the period of the function, which isdescribed by f (t) over the

p range 0';;;;; t ,;;;;;T.

Q.18:Sho w that the Laplaeetransfarm of thesawtooth waveof Fig.2.6ean be expressed as

Throughout the remainder of this book the bar notation will be used to denote aLaplace tran sform. Thus

1 e-T S

:;:i2 - s(1_ e-T's) ifthe height of thetooth isunity .

(2-26) (2-25) x(s)

==

L x(t)

L dx(t)

=

sx(s) - xlO") dt

t Difficulties associated with theuse of theone-sidedLaplacetran sformation . , e

Rather than discuss here methods of performing the inverse transformation - such as partial fractions and convolution techniques - which will be needed later on, the reader is directed to Jenson and Jeffreys[I] or Douglas [9].

Wealso omit any discussionof the difficulties arising

t

when transforming functions not premultipliedbyU( t);a practice which hereafter is implicitly assumed in all transfo r ma t io ns.

The great advantage of the Laplace transform technique is that differential equations can be replaced by algebraic equations. The transforms of the first Qd) and second derivatives can be written,for example, as

(36)

(2-27) The form xfO") announces that the value just before the time zero must be taken whenever a discontinuity arises in the dependent variable at t

=

0 [2].

From what we have seen in section (2.3),the general form of lumped par a-meter chemical processes, eq.(2.1) might require application of eq. (2-26). By defining a problem in terms of perturbation variables ÁX,where

it is apparent th at much work can be saved in the calculation stage . For example, (2.28) and dÁx -L

dt

=

sÁx(s) d2Áx -L - -

=

S2ÁX(s) dt (2-29) (2-30) We shall return to this topic in the next chapter.

Regarding distributed parameter systems the following transformatio ns [11] will be needed in dealing with partial differential equations, where the tr ans-formation is in the time domain:

L lim f(x,t)

=

lim f(x,s),

x-+-x

_o

x-+-x

_o

(2-31)

assuming that f(x,t) is transformable w.r.t. time tand that both limits exist. Furthermore and finally êf(x.t) ( ) f( 0-) L-~= sf x,s - x, (2-32)

.

(2-33)

2.8. concluding remark

s

The foundations for a mathematical description of the dynarnics of contin-uous chemical processes have now been covered. A parallel development could be given for discrete systems, where, for example, the z-transform replacesthe Laplace transform, but this is left for the reader to attack asthe need arises. Some techniques and theorems, especially in conneetion with Laplace trans-forms, have been omitted since at some point a sacrifice of mathematical completeness has to be made. Hopefully the reader's mathematical backgro und, or a short reading of Chapter 4 of Douglas's book [9] will enable what follows to be painlessly assimilated.

(37)

)

j,

'5

2 .9.

references and biographical notes

Aninvaluable text in conneetion with engineering mathematics is

[1]: JENSON, V.G. and JEFFREYS, G.V.Mathematical Methods in Chemical Engin-eering. Academie Press.London, 1963.

while

[2]: OGATA, K. Modern Con trol Engineering. Prentice-Hall,Englcwood Cliffs, 1970

is an introduetion to the basic mathematics of control work.

[3]: DENN, M.M. Stability of Reaction and Transport Processes. Prentice-Hall,Englewood

Cliffs,1975

presents a c1ear exposé of the principle of causality (p.18).This book is also to be acknow-ledged as containing many powerful mathematical methods extreme lyweilandsimp ly p res-ented.

[4]:

HEALEY, M. Principles of Automatic Contro!. English Universities Press, London ,

1967.

isa low-pricedbook giving a somewhat elementary introduetion to the branch ofcontrol engineering dealing with servo-mechanisms.

The book by

[5]:

GIBSON, I.E. Nonlinear Automatic Contro!. McGraw-HiIl, New York, 1963 .

presents a treatment of systems containingdiscontinuous non linearitie s.

In

[6]: HIMMELBLAU, D.M.and BISCHOFF, K.B. Process Ana lysisand Simulat ion: DeterministicSyste rns.J.Wilcy, New York, 1968,

on pages92-93a largenumber of mathematical texts are recorded.See alsopage129for

the section on Iinearization.

[7]: HARRIOT, P. Process Contro!. McGraw-Hili, New York,1964

and

[8

]:

DAVIS, H.T.Introduetionto Nonlincar Differential and Integral Equations.

Dover Publications, New York,1962

provide additio na linformation for Example2.1.

Chapter 6 of

[9]: DOUGLAS, J.M.Pro cess Dynamics andControl :Vol1: Analysis of Dyn ami cSyst ern s, Prentice-Hall,Englewood Cliffs, 1972

is the basis of section2.5.The volume is also useful for matrixalgebraand Lapl ace tra

ns-formations.The analysisof section2.5is alsoprcse nted in

[10

]:

RITTER, A.B. and DOUGLAS, J.M.Ind.Eng.Chern.Fundamentals,

2,

21(1970).

For distributedsystems and for thosc with a mathe matica linclina tionthe bookby

[11]: GOULD, L.A. Chemical Process Control;Theory andApplications.Add ison

-Wesley, Reading, Massachusetts,1969.

(38)

"Sir, it's no matter what you teach them first, any more than what leg you shall put into your breeches [irst." Dr.S.Johnson (1709 - 1784)

3 ELEMENTARY THEORETlCAL PROCESS DYNAMICS

3.1. introduetion

An investigation of the dynamic behaviour of a chemical process can prod -uce answers to questions where the more usual steady state or stationary analysis fails. For example, the correct choice and placing of a control1er will depend on the process dynamics together with an assessment of the expected disturbances and a profitability criterion. It can also suggest irnprove-ments in future plant design and lead to a better understanding of existing process operations.

Of course, the picture is not complete if only the process dynamics are investigated. We have seen in Chapter I that typical1y seven elements can be distinguished in a system, and it is important that this does not escape attention. Usually, the process will react much more slowly to changes than any of the other six elements which we can then (as a first approximation) consider to be reacting instantaneously. The concept of instantaneous reaction implies that the dynamic behaviour can be adequately described by the station-ary analysis mentioned earlier.

Two approaches to the estimation of the dynamics of a system can be dist-inguished. The experimental approach, that of identifying a model and estimating the necessary parameters, is discussed in Chapter 5. Here we will work on the equations of conservation of mass, energy and momentum to produce a purely theoretical insight into the dynamics. These laws, in quite general terms state that, inside a given boundary:

Ingoing Outgoing

+

Quantity Quantity Accumulated

quantity quantity of mass, of mass, mass,

of mass, of mass, energyor energy or energyor

energyor energyor momentum momentum momentum (3.1

momentum momentum generated consumed during the

per time per time internal1y internal1y time unit

unit unit per time per time

unit unit

The fol1owing variants of eq.(3-1) can be distinguished [5]:

1. The total continuity equation - the overal1 mass balance 2. The component continuity equations - component balances 3. The energy equation - the so-called heat balance

(39)

4. The equations of motion 5. Molecular transport equations 6. Equations of state

7. Equilibrium laws

- conservation of momentum

- Fourier's, Fick's and Newton's laws - often based on the perfect gas law

-- applying the 2nd law of thermodynamics

*

(3"

8. Chemical kinetics relationships

Together with appropriate initial and boundary conditions eq. (3-1) in one or more of the above forms defines amathematical model of the process, as it is called. Depending upon the degree of detail inroduced in the modelling many model types are possible [2]. Gould [3], for example, begins describing process dynamics at once in terms of distributed parameters (p.d.e.'s), which is the logically correct procedure. A simpier approach, adopted here, is to consider firstly lumped parameter models (without even stating the assumptions which allo wed the lumping in parameters to be made). Another point to be borne in mind is that we wil! be essentially modelling continuous processes by means ofcontinuous variables, which seems the logica I thing to do. Altern-ative formulations are, however, possible in terms of discrete or discrete plus continuous variables, in which case finite difference or difference-differential equations, respectively, are used.

3.2. the energy balance

3.2.1. setting up tbe dynamic model of a simple lumped parameter tbermal process: tbe dynamic model

The process cbosen to illustrate the procedure of mathematical modelling is shown in Fig. 3.1.

environment

lemp.=e~

Fig. 3.1. Heating an object.

An object which has a temperature () is warmed by a heating medium inside a heat-exchanger. The rate of heat liberation in the exchanger is given the SYmbol Hand the temperature of the surroundings of the object, ()e'

The first step is to establish the range of tbe independent variabIe time, t.

*

Therrnodynamics is, it seems, amisnomer. Rumour has it that the adoption of the lettert for temperature in the early days of thermodynamics has prccludcd ever since

(40)

It is usual to examine the dynamics from t

=

0 for some (arbitrary) time until t

=

tI' Now, rather than apply eq. (3-1) directly, or its variant the energy equation, which is applicable here, we seek an answer to the question "How much heat will enter the object during tI seconds?"

3.2a. Variation of rate of heat input to object versus time.

l1

-llmoz

3.2b. The loss of heat to surroundings as a functionof time. Fig. 3.2. The dynamics of input and output variables.

The answer can be deduced from Fig. 3.2a as the area under the curve of H versus time. Similarly, the heat leaving the object during tI seconds, being lost to the surroundings, mayalso be expressed pictorially, Fig. 3.2b. Since there are na other heat sinks or sourees in the object the left hand side of eq. (3-1) is equivalent to

t₁ t1

J

Hdt -

J

hA(e -

e

)dt

o 0 e

As far as the accumulation of heat in the object during the tI seconds is concerned we can say that for small tI the accumulation is the heat content of the object at time t

=

tI minus the heat content of the object at time t

=

O. Thus, the heat balance may be written as

(41)

ti ti

J

H(t)dt -

J

hA(O(t) - B (t) )dt

=

m.c .(O(t

1) - 0(0)) (3-2)

o 0 e p

where a distinction has been made between variables (having a dependenee on time) and parameters (for example h, A)in a somewhat arbitrary way. Differ-entiating eq. (3-2)with respect to the upper limit tI - the mathematica1 back-ground can be found in any elementary text (see

[4]

for example)- gives

_ dO(t)

Htt) - h.A(O(t) - Oe)- m.cp.(ït (3-3) de (O )

since -d-- is zero (e (O) is constant)and since it has been assumed that Oe tI

is invariant.We have a1so replaced the arbitrary time interval tI' Eq. (3-3) is oneequ ation - an energy or heatbalance - which describes the process dyn-amics. For the more genera! form of the energy equation (basically the first law of thermodynamics applied to a flow process) and an altemative approach see Luyben [5].

Q. 19:In Chapter1 mention was madeof block diagrams[6]. Drawabloek diagramof the thermal proeess studied here, in the light of eq.(3-3).

3.2.2. the initial state

Before we can truly speak of a mathematica! model of our thermal process, we must consider the initial state,that is,the situation before and up to t= O. Usually we can say that before zero time the process was at rest, meaning that thetemperatur e of the object,

e

,

was constant and (consequently) that heat input matched heat loss. This description fits that of an initial steady state and if we denote "just before zero time"by 0-,we can write from eq.(3-3) that

(3-4) whichdefinesaninit ial equilibrium state.

3.2.3. the reduced model

Together, eqs. (3-3) and (3-4) contain all the information characterising the behaviour of the process.It would be nice, however, if we needed only to work with one equation, and in the light of remarks made in Chapter 2 regarding the use of perturbation variables as a means of saving computational effo rt , we seek a reduction in the model size as follows.From the differential equation describing the process dynamics we subtract the algebraic equation of the initial steady or equilibrium state. The result in terms of perturbation variables is

where

d.18

(42)

and

.1H= H - H(O-)

.18 = 8 - 8(0-)

(3-Sa) (3-Sb) which is called the reduced form of the model. Notice how the constant term

containing 8e has disappeared from the relationship. Reduced model forms

are so generally used in the literature th at often the ".1" symbol, or its equi-valent, is omitted and no distinction made between perturbed and ordinary variables. The distinction will be retained, however, hereafter since it has been found by the author to be always worthwhile keeping in mind.

3.2.4. the transfer function of the process

Itwas remarked in Chapter 2 th at the Laplace transformation offered the

great advantage of working with algebraic, instead of differential, equations. Anticipating a future saving in computational effort, eq. (3-S) is transformed (see also eq. 2-29) to give:

(m.c .s_p

+

h.A)M(s) = .1H(s) (3-6) I h.A

+

m.cp's h.A .18(s) --,---- .,,.1H(s) -or

This ratio, the Laplace transform of the output perturbation variabie divided by the Laplace transform of the input perturbation variable, is called the transfer function of the process, and is often assigned the letter G (s). Two_p other terms introduced here are the statie gain r or amplification factor, which is the constant term multiplying the right-hand side of eq. (3-6) (thus

=

h~)

and the time constant, which is the constant term multiplying the

Laplace parameter s (i.e. mh·cp). From eq. (3-6) we classify the process as

.A

a first order process, since s is raised only to the first power.

F s:

f,

( 3.2.5. the transient response of the process

Unless we make some assumption as to the form of a disturbance entering

the process we can proceed no further than eq. (3-6). Ifa particular forcing

function is applicable, ho wever, (see Chapter 2), than it is possible to use the transfer function to give us the transient or dynamic response of the model. Let us assume here that the heating rate is likely to change abruptly from

one rate to another at time t = 0 (a step change). Figure 3.3a shows the

situation, and mathematically we have that

.1H(t).U(t)

=

[H(t) - H(O-)]U(t)

=

[H(O+) - H(O-)]U(t)

=

oH.U(t)

*

(3-Sa)

c

*

We use ëH here to denote the change in magnitude of H as a result of the step;

öis thus here not the Dirac or delta function introduced in Chapter 2.

t Foundby placing s=O in the transfer function.

(43)

-timlZ Fig. 3.3a. A step change in heating rate.

e

t

- t i m e

Fig. 3.3b. The temperature of the object as a resuIt of the above disturbance. Fig. 3.3.The transient response of the model.

since we are only considering t;;;' 0 in our model, and thus Laplace trans-fOrming

LlH(s)

=

oH s

from eq.(2-22a).Substituting this result into eq. (3-6) gives oH

LlO(s)

=

ïl.A

s(1

+

mh~rs)

(3-7)

(3-8)

Using any table of Laplace transforms the inverse transformation is readily found to give:

or returning to normal variables,

oH h A t ] )

-O(t)

=

hA

(I -

exp

[-me

+

0(0 ) p

(3-9)

(44)

which is the transient response of the temperature of the object to a step change in the heat flux rate, shown in Fig. 3.3b. Often, it must be admitted, the step from eq. (3-8) to eq. (3-9) iscomplicated and partia1 fraction, convol-ut ion or other techniques must be resorted to.

3.2.6. the fin al state

From eq. (3-10) and Fig. 3.3b it is apparent that the temperature of the object eventually obtains a second, steady state, given by

(3-11) This resu1t can a1so be obtained by using the so-called Fina1 Limit Theorem of the Lap1ace transformation.

3.3. mass and momentum balances

3.3.1. the overall mass ba1ance

The heart of the model just described was an energy or heat ba1ance. More examp1es of energy ba1ances are given in

[7].

Just as on1y one overall energy ba1ance can be formu1ated for a process, so can on1y one total contin-uity equation, which we will call overall mass balance, be written down.

As an examp1e, consider a tank containing liquid, open to the atmosphere, and discharging through a length of piping to atmosphere (Fig. 3.4)

Fj

I

( 1 s e v t 1 p e p ij p

Fig. 3.4. The Gravity Flow Tank.

~ - F

o An overall mass balance, based on eq. (3-1) and the approach outlined in previous examp1e, is that

F (t) - F (t)

=

dM(t)

=

d(P(t).Vet»~

i 0 dt dt

Assuming invariant density p, we have that

F.(t) - F (t)

=

A dh(t)

1 0 p . dt

30

(3-12)

(45)

~~--- -

-Unless more assumptions are made, our process must be modelled by eq. (3-13):

The simplest of these is to consider the outflow, F o(t), as being constant, such as would occurifa constant displacement pum p emptied the tank [7]. Returning to an instant in time (t=0-) when the inflows and outflo ws were equal, the initial steady state equation becomes

(3-14 ) whereupon subtraction of eq. (3-14) from eq. (3-13), for constant F

o yields the reduced model:

(3-15) Taking Laplace transforms of both sides, we have the transfer function of the process: 1 ,óh(s)

PA

M.(s) = -s-1 (3-16)

A1though we still find a static gain (pIA) the time cons tant of the previous

example is missing. The form of eq. (3-16) is characteristic of a pure capacitive process, that is to say that the output variabIe, h, behaves as if there were an integrator betweenit and the input, F.. The response of such a pure capacitive

1

process to a step change in input at time t

=

0 is shown in Fig. 3.5.

mass

inflo w

i

FI(0+)

.1---=1

.--

1 _(0_-_)- - - _. _ -- time

Inp

ut

hel g h t· of liq u ld

r

h(

O)

- time

Outpu

t

F"