Models on various levels of detail

2.4 Case study: the countercurrent heat exchanger

2.4.1 Models on various levels of detail

Oversimplied model

On the top levelthe dynamics ismodeledbya singlepair of perfectlystirred lumps

forming a so called heat exchanger cell. Each cell consists of two perfectly stirred

tanks with in- and outows. The two tanks are connected by a heat transfer area

between them.

Themodelequationsforasinglecellconsistofthetotalmassandenergybalances

for the hot and cold sides respectively. In order to connect the top level model to

theotheronesthecellindexj (j =1)isusedforthesinglecell. Constantmass (and

thusconstantvolumeV

with constantdensities

jc ,

)isassumed inthehotside

thereforethe totalmass balanceispresent onlyforthe cold side. The equationsare

asfollows:

where the subscripts h and c denote the hot and cold sides respectively. T

are the inlet and outlet temperatures, v

and v

are owrates, v

jl is

the leaking owrate, V

and V

are the volumes,A the heat transfer surface area,

and c

the specic heats,

and

the densities and U

is the heat transfer

coecient (abbreviated as HTC). Note that all of the variables and parameters

above are positive by denition.

With only a single cell in g. 2.1.a. describing the heat exchanger the outlet

temperatures T

ho and T

of the heat exchangerare the same asthat of the cell. In

this case only two state variables fT

g are considered with a single triplet of

equations(2.1)-(2.3).

In case of the simplest top level model all the model parameters including the

owrates v

and v

are constants and the inlet temperatures T

and T

are the

input ordisturbance variablesonly.

T _ci

Figure 2.1: a. Single heat exchanger cell; b. Cascade model of a countercurrent

heat exchanger

Linear cascade model

Thecountercurrentheatexchangerismodeledasasequence ofcascadedcellsabove.

Athreecellmodelisused(seeing. 2.1.b)asamiddlelevelcascade modelbecause

it can describe the dynamics of the heat exchanger well enough for fault diagnosis

purposes for majority of the industrialequipments.

With the three-cell model three set of modelequations (2.1)-(2.3)are usedwith

the followingvariablerelationsinstead of Eqs. (2.4):

j =1;2;3; T

The modelabovecan beregarded asalumped version ofthe distributed

param-eter model of the heat exchanger, therefore the cells are associated with particular

spatiallocationsalongtheequipmentlength. Thestatevariablesoftheabovemodel

arethe temperatures ofthe coldand hot sidesand the coldside volumeinthe cells,

i.e.fT

g. Ifanequidistantinspacelumpingis

per-formed, i.e. the parameters A

and V

are the same for allcells then heat transfer

coecients U

carry spatialinformation.

Here again, allthe modelparameters includingthe owrates v

h and v

are

con-stantsand the inlettemperatures T

hi and T

are the inputordisturbance variables

only.

Bilinear cascade model (Nonlinear model)

The nonlinearity of process models is caused either by the convective ow when

the owrate is alsoa dynamic variable (bilineartype nonlinearity) or by the

chem-ical reactions and other mechanisms appearing in the source term of the balance

equations. In our case only the bilinear type nonlinearity appears caused by the

convective ow.

Inthebottomlevelthree-cellbilinearcascademodelthreesetofmodelequations

(2.1)-(2.3)canbeusedwiththe variablerelationsEqs. (2.5). Butnowthe owrates

h andv

are time varying variablesand areinput ordisturbancevariablestogether

with the inlet temperatures T

and T

ci .

There are two types of faults considered: the deterioration of the heat exchanger

surfaceand leakingintheoutercontainer. Empiricalgrey-boxmodelisusedforthe

deteriorationand physical white-box modelis set up for the leakage.

Deterioration of the heat transfer area

Under normal operating conditions the HTC is constant or slowly decreasing due

toalayerof settledmaterialbuildingup on theheat transfer surface (see g.2.2.a).

In oldheat exchangers large pieces of the settled materialcan break away fromthe

surface, causing damage in the equipment. When the settled material breaks o,

the HTCs will undergo abrupt positive jumps. These jumps can be grouped into

two qualitativelydierent types.

1. Inthe beginningwhenthe jumpsstart tooccur,they aresmall andinfrequent

becauseof the smallpiecesof CaCO

that leave theheat transfer surface(see

g.2.2.b).

2. Then in the later stages the jumps become large and frequent when bigger

'stones'of CaCO

comeo (see g.2.2.c).

The usual industrial practice is to wash heat exchangers with some kind of acid

between certain time intervals. However, this washing procedure requires the heat

exchangerandpossibly otheroperatingunitstobestoppedand thusitmay become

a rather expensive operation. On the other hand, if some information is available

abouttheprocessofthechangingoftheHTC,thenweareabletoavoidunnecessary

stoppings and damage caused by the settled material coming o the heat transfer

surface, too. Thus, one of the main ideas in this work is to track the HTC in the

heat exchanger and detect abruptpositivejumps in it.

Leakage

When the outer container is leaking, there is an unmeasurable outow v

on the

cold side from the heat exchanger (see Eq.2.3 and g.2.3) which may result in a

slowdecrease of the mass inthe outerphase, or- if the levelsare controlled, inthe

decrease of the outow from the equipment. Here it is assumed that the volume

ow rates v

h and v

are known and constant or measured, leaking can therefore be

detected via detection of a slow decrease in the cold side volume V

. Note that we

want to detect smallchanges in the cold side volume compared tothe initialvalue

ofthe volumeinthe heatexchanger. Thisisareasonableaimbecausethedetection

oflargeandquick changes(i.e. bigleaks)can beasimplertaskwithothermethods.

Moreover, there is no possibility to get spatial informationabout the leakage even

with our cascade model, since the liquid level on the cold side decreases uniformly

inthe whole heat exchanger. Thatis why itis importanttodevelop such a leakage

detectionalgorithmthatrequiresasfewtemperaturemeasurementdataaspossible.

H T C

t

H T C

t

H T C

t

a b

c

Figure 2.2: Three consecutive stages of the change of the heat transfer coecient:

a.Constant or slowly decreasing HTC; b.Smaller and infrequent jumps; c.Large

and frequent jumps

T _hi T _{h o}

T _ci T _{c o}

v _h v _h

v _c v _c

v _cl T _{c o}

Figure2.3: Leakage modelledas anunmeasurable outowfrom the cold side

Filteringis needed toavoidnumericaldierentiationof the noisytemperature

mea-surements, because the forthcoming parameter estimation methodsrequire the

ap-proximationof the rst and secondderivatives of certainmeasured signals.

Numerical dierentiation is sensitive to noise and round-o errors (particularly

whenthesamplingintervalisshort),andlargeerrorsinthedierentiationcanoccur,

whichmayleadtonon-robustidenticationmethods. Thatis,themethodsmayfail

or have a very slow convergence in the presence of noise. The question is how can

we avoidtaking derivatives.

For adiscrete time signaly(k) the so-called Æ operatoris dened as[55]

Æy(k)=

y(k+1) y(k)

(2.6)

where t

is the samplingtime.

Assume we have an nth order discrete time system in the delta-operator

(ob-tained fromacontinuous time system), i.e.

A(Æ)y(k)=B(Æ)u(k) (2.7)

where the highest order occurring in the A and B polynomials is n. This means

thatthe nthorderderivativeofthesignalsisinvolved. Oneobviouswayof avoiding

taking this derivative is to integrate both sides of the equation n times, that is we

lter the signals through the lter 1

Æ n

. The system equation(2.7) then reads

Æ n

A(Æ)y(k)= 1

Æ n

B(Æ)u(k) (2.8)

and we see that 1

Æ n

A(Æ) and 1

Æ n

B(Æ) are now polynomials in 1

, i.e. we are now

integratingthesignalsinsteadoftakingderivatives. Thelter 1

Æ n

isjustoneexample

of a possible lter, but we have to integrate at least n times in order to avoid the

derivatives which means that the lter must be at least of order n (see [55]). This

means that if a particular parameter estimation algorithmrequires the rst or the

second derivative of certain signals then a rst or second order lter is applied on

the measurement data respectively.

The ideaofthe InniteImpulseResponse (IIR) lterdesigninour case isthata

continuous time lter is designed rst using the continuous time model of the heat

exchanger andthen the obtained lter istransformed into discretetime usingthe Æ

operator.

The generalmethodforconstructingsucha lterisasfollows[55]. Assumethat

acontinuous time system isgiven by the following input-outputmodel

Y(s)

U(s)

= B(s)

A(s)

(2.9)

the orderof B(s)). Thenthe lter thatshould beused is 1

A(s)

. After applyingthis

lter onthe system weget

Y(s)= B(s)

A(s)

U(s) (2.10)

To obtainthe discretetime lter, we substitute s in(2.10) for the Æ operator.

On the basis of the guidelines above, the lters that will be used in the fault

detection methods are the following.

First order lter. In this case we want to use a rst order lter with cuto

frequency !

. The transfer function describing sucha lter is given by

(s)are theLaplacetransformsofthe signaltobelteredandthe

lteredsignalitselfrespectively. The discretetimeversionof thislter iswrittenas

fromwhichweget

Æf

Tocalculatetheappropriateltercoecientinthiscase,letusselectfromEqs.(2.1)

and(2.2)theonethatdeterminesthedominanttimeconstantofoneheatexchanger

cell. Let us assume that this equation is Eq.(2.1), and that the cold side liquid

volume and the HTC in the heat exchanger are constant (i.e. we assume normal

operation). Let uswrite the Laplace transformof Eq.(2.1) inthe followingform.

According to the previously discussed lter design method, the rst order lter

coecient isas follows.

where V

and U

denote the nominal values of the cold side liquidvolume and the

HTC in the jth cellof the heat exchanger respectively.

Second order lter. Again we start from the transfer function of a continuous

time second orderlinear lter whichreads

Converting it intodiscrete time gives

Thecalculationof thesecondorderlter'sparametersfromthe heatexchanger's

parameters willbe shown inSection 2.6.2.

2.6 Parameter estimation

2.6.1 Linear cascade model - all measurements available

In this ideal and not too realistic case we assume that we have access to both the

hot andcold side temperatures ineachcellof the heatexchangerinadditiontothe

inlet temperatures. In other words, we know the temperature distribution along

the heat exchanger on both the cold and the hot sides. Furthermore, itis assumed

that A, c

are known. It must be admitted that it is

not often possible to measure the cold and especially the hot side temperature on

several points alongthe heat exchanger. On the other hand,we have agoodreason

toexpect thatweobtainthe mosteasilyimplementablealgorithmsinthis casethat

produce the most reliableestimates forthe parameters.

Estimating the HTC

An additionalassumption inthis case is that the cold side volume is constant and

known in the heat exchanger since the purpose of the algorithm discussed here is

onlyto give anestimate forthe HTC.

Using the gradient method with a forgetting factor 0 < < 1, it is possible to

obtain estimates directly for both U

and U

using the discrete time modelof the

heat exchanger. The forgetting factor means that old measurements are weighted

at an exponentially decreasing rate. (see [51]). Usingthe ltered temperatures we

can easilywrite the discretetime modelof the jth cellin the heat exchanger.

ÆT

One can see from(2.19)and (2.20) thatthe only unknown parameter inthis model

is the HTC, i.e. U

that carries the diagnostic informationabout the deterioration

of the heat transfer surface in the jth cell.

The prediction errors for (2.19) and (2.20)in the jth cellare given by

(k) has tobecalculated

Since we don't know the real value of U

we can approximate

and

in the

following way (see [51]).

The standardrecursivescheme for estimatingU

fromthe cold and hot side

respec-tively is asfollows

Thechoiceof in(2.28)and (2.30) isatradeo between goodtracking capabilities

of the HTCs and sensitivity tonoise and unmodeled dynamics. A small value of

meansthat the estimatewilltrack variations inthe HTC quicker, but onthe other

hand,the itwillalsobecomemoresensitivetonoiseand unmodeleddynamics. The

actual estimates of the HTCs in the jth cell can be computed by using a convex

combination:

where shouldreect the relative condence wehave inthe two estimates.

Estimating the cold side volume

Theonlydierencebetweentheassumptionsinthisandthepreviouscaseisthatnow

theHTC ineachcellisassumed tobeknownand constantandthecold sidevolume

becomes a time-varying parameter. In most cases this is a reasonable assumption,

sincethechangeoftheHTCisusuallymuchslowerthanthatofthecoldsidevolume

during leakage. According to these assumptions the discrete time form of the cold

side energy balanceequation of the jth cellis given by

ÆT

of the same size

where V

(k) denotes the overall cold side uid volume inthe heat exchanger and n

isthe number of cells the heat exchanger is divided into.

For the purpose of tracking the cold side uid volume let us introduce the

fol-lowing variable

Thuswe can rewrite (2.32)as

ÆT

Starting from (2.35) let us now present a possible identication scheme for the

trackingof the cold side uid volume.

Tracking with the gradient algorithm. Using (2.35) the prediction error is

writtenas

Its negative gradient with respect toV

jcr

The approximation of

(k) is writtenas

Using(2.38)and (2.37)therecursiveschemeforthe estimationofthe coldsideuid

volumecan bewritten as

is again a tradeo between quick tracking capabilities of the parameter

and sensitivity tonoise.

In this case both the HTC and the cold side volume are unknown parameters and

wewould liketoestimatethem simultaneously. An obviousstrategyto achievethis

goal is to combine the two algorithms described in Eqns.(2.23)-(2.31) and

(2.37)-(2.41) respectively. Within the jth heat exchanger cell we have two parameters

to be estimated in this case, namely V

and U

. The idea of the solution is to

let the two gradient algorithms use each other's estimations. This approach works

very well if both V

and U

vary slowly in time, but when abrupt jumps occur in

the HTC, problems arise with the estimation of the cold side liquidvolume. These

problems can be handled successfullyif weutilize allthe informationwehave from

the complete availabilityof measurement data.

The suitable form of the discrete time energy balance equation of the cold side

inthis case iswritten as

ÆT

while Eqn.(2.20) describing the hot side can be used here without change. After

combining the algorithms described in Eqns.(2.23)-(2.31) and (2.37)-(2.41) we get

thefollowingprocedureforthesimultaneousestimationofthecoldsideuidvolume

and the HTC inthe jthcell

The estimate for the cold side liquid volume in the whole heat exchanger is given

by the sum of the estimates of the cells, i.e.

wheren denotesthe numberofcells intheheatexchanger. Thetuningknobs ofthe

algorithmare

jUcc ,

jUch and

jVc

respectively.

2.6.2 Linear cascade model - only cold side measurements

available

Estimating the HTC

The algorithm used in this case is based on exactly the same idea as the one for

estimating the cold side liquid volume described in the next subsection. However,

only one of them can be presented in a detailed way due to the space limitations.

Usingthisapproachwecanapproximatelycalculatethehotsidetemperaturesalong

theheatexchangerandestimatetheHTCbyprocessingthemeasuredandcalculated

temperature data.

Estimating the cold side volume

In this section the identicationalgorithmwillbe extended to the case where only

measurementsofthecoldside temperaturesand thehotinlettemperatureare

avail-able. It is assumed that the HTC is constant. Furthermore, we assume that we

know the constant parameters of the heat exchanger (U, A, V

). Thus the only time-varying parameter which is to be estimated in order to

detectleaking isthe cold side liquidvolume,V

c .

Wecanconsideracascademodelconsistingofthreecellsforthesakeofsimplicity,

without the loss of generality. Let T

denote the cold side output temperature in

cellno. n. Let usintroducethe following notations

Thusthe energy balanceequations take the form

The system can approximately be described by a linear time invariant model by

assumingthattheowratesandtheHTCareapproximatelyconstant. AfterLaplace

transforming (2.59) we get (for the sake of simplicity, variable s in the argument

of the Laplace transforms of v

will be suppressed in the following

equations).

Since, according to our assumptions, we do not have access to the hot side

tem-perature measurements along the heat exchanger (we can only measure the hot

inputandoutput), weneedtoeliminatethehot sidetemperaturesT

1h andT

2h from

Eqns.(2.59)-(2.64). In order todoit we express T

The Laplacetransform of (2.60) isgiven by

Writing (2.68) into (2.69)gives

fromwhichit follows that

Similarly to (2.68) we can express T

from the Laplace transform of (2.61) in the

following way

(s+v

Substituting(2.72) into (2.73)givesthe transfer functionfor T

which reads

AftertakingtheLaplacetransformof(2.63)and(2.64),andexpressingT

fromthe

Laplacetransformof (2.63) andsubstituting itinto theLaplace transformof (2.64)

we obtainthe following transfer functionfor T

Notice the pattern inthe transfer functions (2.71),(2.74) and (2.75) as the number

ofcells grows. Next,itwillbe shown that (2.74)and (2.75)canbebroughtintothe

form(2.71) by introducingltered variables.

Let usintroduce the signal

Then (2.74)can be writtenas

which has exactly the same form as (2.71). The following ltered variable is

intro-duced

Now Eqn.(2.75)reads

which againhas the sameformas(2.71), andthe patterncontinues if weconsider a

model with more than three cells.

Our identication strategy is now to use a discrete time version of (2.71) to

obtainestimatesforV

to(2.76) usingthe estimates of v

c and

inthe implementationof the lter. Then

we use a discrete time version of (2.77) with the newly obtained ltered signal to

getanestimateof

. Werepeatthe lteringandidenticationprocedureon(2.78)

and(2.79)toobtainestimateof

. Becauseofthepatterninthetransferfunctions,

anarbitrary number of cells.

Inorder toobtainadiscretetimeversionof(2.71)wesubstitute swiththe delta

operator. Wethenobtain (forthe sakeof simplicitywe suppress thetime argument

of v

Toavoidnumericaldierentiationwelterallthetemperaturemeasurementsthrough

a second order linear lter as it was described in Section2.5. Recall that the lter

In document Nemlineáris folyamatrendszerek diagnosztikája, analízise és irányítása szürke doboz módszerek segítségével (Pldal 24-0)

2.4 Case study: the countercurrent heat exchanger

2.4.1 Models on various levels of detail

T ci

H T C

t

H T C

t

H T C

t

a b

c

T hi T h o

T ci T c o

v h v h

v c v c

v cl T c o

T _ci

T _hi T _{h o}

T _ci T _{c o}

v _h v _h

v _c v _c

v _cl T _{c o}