ExtendedKalmanFilter

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Use of extended Kalman ltering in detecting fouling in heat exchangers

Gudmundur R. Jonsson a , Sylvain Lalot b, * , Olafur P. Palsson a , Bernard Desmet b

a Department of Mechanical and Industrial Engineering, University of Iceland, Hjardarhaga 2-4, 107 Reykjavik, Iceland b LME, Universite de Valenciennes et du Hainaut Cambre sis, Le Mont Houy, 59313 Valenciennes Cedex, France

Received 31 August 2006; received in revised form 23 November 2006Available online 17 January 2007

Abstract

This paper is concerned with how non-linear physical state space models can be applied to on-line detection of fouling in heatexchangers. The model parameters are estimated by using an extended Kalman lter and measurements of inlet and outlet temperaturesand mass ow rates. In contrast to most conventional methods, fouling can be detected when the heat exchanger operates in transientstates. Measurements from a clean counterow heat exchanger are rst used to optimize the Kalman lter. Then fouling is considered.The results show that the proposed method is very sensitive, hence well suited for fouling detection. 2006 Elsevier Ltd. All rights reserved.

Keywords: Heat exchangers; Fouling; Extended Kalman lter; Parameter estimation; On-line detection

1. Introduction

The heat transfer between two uids will inevitablyresult in fouling. For a district heating system, where theheat transfer is great, it is important to minimise foulingin the heat exchangers. The expense for achieving a desiredheat transfer is increased with the decrease of the heattransfer coefficient. For example, at the VestegnensKraftvarmeselskab in Denmark (VEKS), it is said, thatfor every 5 temperature increase for which the hot waterowing into the heat exchangers must be heated becauseof fouling, is an added 800940 k /year cost for the con-sumers, see Jakobsen and Stampe [1]. The installed power

at VEKS is 770 MW which is similar to that of the DistrictHeating Company of Reykjavik, Iceland. From the envi-ronmental point of view, fouling is also harmful. For exam-ple, Casanueva-Robles and Bott [2] report that a foulingbiolm thickness of 200 l m in a 550 MW coal-red power

station leads to about an increase by 12 tons of CO 2 perday.

Causes for fouling, common methods to avoid it, or atleast to mitigate it, have long been studied, see for examplePoulsen [3], Thonon et al. [4], Ramachandra et al. [5], andAbd-Elhady et al. [6]. The detection of the presence of foul-ing is also an active research area, see for example Jeronim-o et al. [7], Bott [8], Riverol and Napolitano [9,10], Chenet al. [11], Nema and Datta [12] and is still a challengeand conferences are regularly organized (see for examplehttp://www.engconntl.org/ ).

The classical detection methods are based on e.g.

1. Examination of the heat transfer coefficient (or theeffectiveness),2. simultaneous observations of pressure drops and mass

ow rates,3. temperature measurements, e.g. T h;in T h;out =T h;in

T h;out design ,4. ultrasonic or electrical measurements,5. weighing of heat exchanger plates.

To be very accurate, these methods require either thatthe system presents successive steady states (13), i.e. the

0017-9310/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2006.11.025

* Corresponding author. Tel.: +33 327 511 973; fax: +33 327 511 960.E-mail addresses: [email protected] (G.R. Jonsson), sylvain.lalot@univ-valen-

ciennes.fr (S. Lalot), [email protected] (O.P. Palsson), [email protected] (B. Desmet).

www.elsevier.com/locate/ijhmt

International Journal of Heat and Mass Transfer 50 (2007) 26432655
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inlet temperatures and ows must be stable for a periodlong enough to be able to compute or measure the valuesof interest, or are local (4), or require to stop the process(5). This is far too restrictive or costly.

Another approach is based on modelling the heatexchanger and then looking for any discrepancy betweenwhat is predicted by the model and what actually occurs.The method proposed by Prieto et al. [13,14] is based onan adaptation of the model when necessary (after the detec-tion of the discrepancy), and on the analysis of the differ-ences between two consecutive models. To pursue thisapproach, the aim of the present study is to show hownon-linear physical state space models, recursively deter-mined, can be applied to detect fouling in heat exchangers.

In Jonsson and Holst [15] and Jonsson [16] heat exchanger

models and the statistical estimation of their parametersare discussed. The models are based on the physical prop-erties of the heat exchanger. An improvement has beenbrought so that the estimation is continuous, and the mainfocus here is to determine how sensitive the proposedmethod is to changes in the parameters of the model dueto fouling.

Being based on the measurements of the mass ow ratesand on the inlet/outlet temperatures, the estimation couldbe done while the heat exchanger is in use, even in a nonsteady state.

In the rst part, the heat exchanger model is developed.Then the estimation algorithm is presented. This estimationis carried out using the extended Kalman lter. The last

section is split into two subsections. The rst subsection

Nomenclature

A state matrix or surface area for heat transferwhen used with a subscript

B state matrix

Cus cumulative sum functionc constant or specic heat when used with a sub-script

ddt derivative with respect to timeo

o v partial derivative with respect to variable v E v mean value of variable vF state matrix f state space function g threshold coefficientH measurement matrixh threshold coefficient or convection coefficient

when used with a subscriptI identity matrixK Kalman gain matrix or constant when used with

subscripts c or hk discrete time indexM mass of uid in one section

_

m mass ow rate_

m mass ow rate state vectorN normal (Gaussian) distributionns number of sections in the heat exchangerP covariance error matrixQ covariance matrix Rf fouling factor Rth thermal resistance of the tube between the two

uidsT temperatureT temperature state vectort timeU overall heat transfer coefficientv dummy variablew white Gaussian noise sequence with zero mean

and covariance matrix Q

w noise state vector x overall state vector y exponent of the Reynolds number in a convec-

tion correlation z measurements vector0m n null matrix having m lines and n columns0 null state vector

Greek symbolsa model parameterb model parameterDt time steph parameter state vectorr standard deviations model parameter

Subscriptsc cold sidedesign as should be obtained according to the design

computationsf relative to the state space functionh hot sidei in section # i in inletk at discrete time index k out outletref referencez relative to the measurements

a relative to the model parameter ab relative to the model parameter bh relative to the parameter state vector

Superscripts* relative to the reference stateT transpose

2644 G.R. Jonsson et al. / International Journal of Heat and Mass Transfer 50 (2007) 26432655


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is devoted to the tuning of the Kalman lter, to the study of the inuence of the initial random conditions, and theinuence of the sampling period. This is carried out on aclean heat exchanger. The second subsection deals withthe application of the model, coupled to a drift detectiontool to fouling detection.

2. The heat exchanger model

2.1. The model

The heat exchanger model is based on the direct lump-ing of the process. The heat exchanger is split up into sec-tions (in the axial direction of the exchanger), each sectionbeing a set of two cells, one per uid. The temperature ineach section is a model state. An extensive discussion of the derivation and assumptions of this model is given inJonsson and Palsson [17].

It can be shown that two sections are sufficient, eventhough the number of cells in the model may of coursebe increased, cf. Jonsson [16]. In this case, the equation rep-resenting the model is

ddt

T h;1

T h;2

T c;1

T c;2

266664377775

1 a2 =sh 0a

2sha

2sh

1 a2 =sh 1 a2 =sh

a2sh

0b

2scb

2sc 1 b2 =sc 0b

2sc 0 1 b2 =sc 1

b2 =sc

26666664

37777775

T h;1

T h;2

T c;1

T c;2

266664

377775

1 a2 =sh 00 a2sh0 1 b2 =scb2sc 0

2666664

3777775

T h;in

T c;in" #1

where T h;1, temperature in hot section #1; T h;2, temperaturein hot section #2 = outlet temperature of the hot uid; T c;1,temperature in cold section #1; T c;2, temperature in coldsection #2 = outlet temperature of the cold uid; a ; b ; shand s c are model parameters. They are time dependent asthey are functions of the mass ow rates. In addition aand b may also be temperature dependent, but this willnot be considered here, due to the fact that the thermo-physical characteristics of the uids are considered con-stant in the CFD code.

Generally the models may be written as

ddt

T A_

m; T ; hT B_

m; T ; hT in 2

or

ddt

T f _

m; T ; h; T in 3

where _m _

mh_

mcT, T in T h;in T c;in

T, T contains the tem-perature in each section, and the model parameters are

contained in the state vector h.

2.2. The parameterization

The models contain four basic parameters which aregiven by

at AhU

_

mht ch; sht

M h_

mht ;

bt AcU

_

mct cc; sct

M c_

mct 4

a and b are dimensionless, they are the instantaneous val-ues of the number of transfer units, for the hot and coldside respectively, but sh and sc are given in seconds. In arst step, and to introduce reference values, it is assumedthat U in Eq. (4) is a constant and hence the parametersare only ow dependent as shown in Eq. (5). To link theparameters to some reference state ( ) they are scaled asfollows (U now written as U ).

at U Ah

_

mhch

_

mh_mht a

_

mh_mht ;

sht M h

_

mh

_

mh_

mht sh

_

mh_

mht ;

bt U Ac

_

mccc

_

mc_

mct b

_

mc_

mct ;

sct M c

_

mc

_

mc_

mct sc

_

mc_

mct

5

The reference state is arbitrary, i.e. the parameters

h a; sh; b; sc

T

6

are constants.In Jonsson and Palsson [17] it is shown that considerable

enhancements in modelling are obtained when U isassumed mass ow dependent. Further improvements canbe made when U is also temperature dependent. As alreadymentioned, the uid characteristics have been consideredconstant in the CFD code, so this temperature dependencyof U will not be used here.

For both sides of the heat exchanger used here, the con-vection coefficients, hh and hc are (see e.g. [18] or [19]):

hht K h_

m y hh t ; hct K c_

m y cc t 7

where K h , K c, y h and y c are constants. In the present study,the uids ow in either a circular tube or in the annulus, theow regime is always turbulent in all experiments for bothuids, and the entrance effect length is negligible comparedto the length of the heat exchanger. Hence, it is consideredhere that y h and y c can be taken as y h y c y 0:8 (seee.g. [20]).

In this study, the thermal conductive resistance as wellas the thermal inertia of the tube are neglected (but couldhave been taken into account in the model [21]), but theheat exchange areas are considered different (as should benecessary for nned tube heat exchangers), and are respec-

tively based on the inner and outer radius of the tube. So,

G.R. Jonsson et al. / International Journal of Heat and Mass Transfer 50 (2007) 26432655 2645


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the overall heat transfer coefficient can be computed using1=UAref ffi1=hh Ah 1=hc Ac, where Aref can be arbi-trary chosen as the heat exchange area of the cold side orof the hot side. This leads to:

Aref U t Ahhht Achct

Ahhht Achct

Ah Ac K h K c_

mht _

mct y

Ac K c_

myct Ah K h

_

myht

8

The overall heat transfer coefficient at the reference massows, _mh and

_

mc , is similarly

Aref U Ah Ac K h K c

_

mh_

mc y

Ac K c_

mc y Ah K h

_

mh y 9

The parameters set from Eq. (5) now becomes

at a_

mh_

mht U t U

; sht sh_

mh_

mht

bt b_

mc_mct

U t U ; sct sc

_

mc_mct

10

where it is observed that only a and b are affected by themass ow dependence of U , and that the reference areahas cancelled out. Note that in an unknown geometry,there could be two additional parameters to be estimated,the exponents y giving

h a; sh; b; sc ; y h; y c

T11

3. Parameter estimation

As shown in Jonsson and Palsson [17], the extendedKalman lter is employed to estimate the parameters. Asit is desirable to keep as many parameters constant as pos-sible, only a and b will be estimated, the last two param-eters are computed considering that they represent themedian residence time of the uids in each section duringa xed period of time (here the period when the heatexchanger is not fouled).

By letting a and b be model states it is possible toobserve the changes in U . Thus, two differential equationsare added to the heat exchanger model. Here a and b are

described as purely random processes, i.e.ddt

at bt ! ddt h wa t wb t ! wht 12

where wht is an independent normal distributed whitenoise process with zero mean and covariance matrixQht , i.e. wht 2 N 0; Qht .

The heat exchanger model then becomes (see Eqs. (3)and (12) )

ddt

h

T !

0 f

_

m; T ; h; T in

" #

wh

wf !13

Here wf % N 0; Qf has been added to Eq. (3) to compen-sate for deviations from the correct temperature values.

3.1. The extended Kalman lter (EKF)

The type of Kalman lter used (see for example Welch

and Bishop [22] for an introduction to Kalman ltering)is the continuous-discrete version of the EKF (e.g. [23]).The model is continuous in time but the measurementsare discrete. With x hT T T

T and w wTh wTf T

the Kalman lter equations are (see [23] for more detailson the Kalman lter equations):

Model (see Eq. (13))

ddt

xt 0

f _

m; x; T in" # wt w % N 0;

Qh 0

0 Qf " # N 0; Q

Measurement model (extracts the sampled states (the out-puts) from the state vector. Measurement noise is added.See H in Eq. (14))

z k Hxk w z ;k ; k 1; 2; ::: w z ;k % N 0; Q z ;k

Initial conditions (assume that the initial states are normaldistributed. See ^ x0 and P 0 in Section 4.2)

x0 %N ^

x0; P 0

Assumptions (i.e. the state noise and the measurement noiseare not correlated)

E wt wT z ;k 0 for all k and all t

State estimate extrapolation (shows how the estimatedstates evolve in time)

ddt

^

xt 0

f _

m;^

x; T in" #Error covariance extrapolation (shows how the estimatedcovariance matrix evolves in time)

ddt

P t F ^

xt P t P t F ^

xt T Qt

State estimate update^

xk ^

xk K k z k H ^

xk

Error covariance update

P k I K k H P k

Kalman gain matrix

K k P k H T

HP k H T

Q z ;k 1



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Denition

F ^

xt

o0

f _

m; x; T in" #o x

xt ^

xt

where t stands for continuous time, k stands for thediscrete time index, () indicates the correspondingvalues after propagation from t to t Dt , (+) denotes val-ues of estimates and covariance of the state estimationerrors at time t Dt after measurement and ^ denotesestimates).

If the measurement matrix ( H ) is a function of the statevariables, a comparable denition to F is used instead.Here however H is constant, for example for the modelgiven by Eq. (13)

H 0 0 0 1 0 0

0 0 0 0 0 1 !14

That is to say, the measurement matrix extracts the statevector components in x which are sampled, i.e. T h;2 andT c;2 (see Eqs. (13) and (2)).

The extrapolation equations for x and P are solved byusing 4th and 5th order Runge-Kutta formulas. The selec-tion of values for the system noise covariance matrix Q andmeasurement noise covariance matrix is in fact part of theproblem of estimating the model parameters. Algorithmsfor estimating the values of Q given Q z can be found inJonsson [16]. However, the approach taken here is to trydifferent values for Q with respect to a and b and studyhow sensitive the parameter estimates are.

3.2. Calculation of F ^

xt

In this section the calculation of the F matrix is shown.This is in fact a rst order Taylor series expansion of thenon-linear model about the current estimate of the statevector. Here the heat exchanger model is given by

ddt

a

b

T h;1T h;2T c;1T c;2

2666666664

3777777775

00

1 a=2sh T h;1 a

2shT c;1 a2sh T c;2

1a=2sh

T h;in

1a=2sh

T h;1 1 a=2sh T h;2 a

2shT c;1 a2sh T c;in

b2sc T h;1

b2sc T h;2

1 b=2sc

T c;1 1b=2sc T c;inb

2sc T h;1 1b=2

scT c;1 1 b=2sc T c;2

b2sc T h;in

266666666664

377777777775

whwf !

00

f 1 f 2 f 3 f 4

2666666664

3777777775

whwf !

then F is the following 6 by 6 matrix

F

0 0 0 0 0 00 0 0 0 0 0

o f 1o a

o f 1o b

o f 1o T h;1

o f 1o T h;2

o f 1o T c;1

o f 1o T c;2

o f 2o a

o f 2o b

o f 2o T h;1

o f 2o T h;2

o f 2o T c;1

o f 2o T c;2

o f 3o a

o f 3o b

o f 3o T

h;1

o f 3o T

h;2

o f 3o T

c;1

o f 3o T

c;2o f 4o a

o f 4o b

o f 4o T h;1

o f 4o T h;2

o f 4o T c;1

o f 4o T c;2

266666666664

377777777775

whereo f 1o a

1

2shT c;1 T c;2 T h;1 T h;in;

o f 1o T h;1

1 a=2

sh;

o f 1o T c;1

o f 1

o T c;2

a2sh

;

o f 2o a

1

2shT c;1 T c;in T h;1 T h;2;

o f 2

o T h;1

1 a=2

sh;

o f 2

o T h;2

1 a=2

sh;

o f 2

o T c;1

a

2sh;

o f 3o b

1

2scT h;1 T h;2 T c;1 T c;in;

o f 3o T h;1

o f 3

o T h;2

b2sc

;o f 3

o T c;1

1 b=2sc

;

o f 4o b

1

2scT h;1 T h;in T c;1 T c;2;

o f 4o T h;1

b

2sc;

o f 4o T c;1

1 b=2

sc;

o f 4o T c;2

1 b=2

sc;

o f 1

o b

o f 1

o T h;2

o f 2

o b

o f 2

o T c;2

o f 3

o a

o f 3

o T c;2

o f 4

o a

o f 4

o T h;2 0

4. Results

In the rst part, a clean heat exchanger is studied, todetermine the inuence of the initial values of the Kalmanlter, of the sampling period, and of the Q matrix. In thesecond part, a continuously fouling heat exchanger is stud-ied, and the detection tool is presented.

4.1. The data

The heat exchanger used in the case study is a water towater counterow tube-in-tube heat exchanger. The innerdiameter of the tube (made of stainless steel) separatingthe cold uid and the hot uid is 14 mm. Its outer diameteris 18 mm. The outer diameter of the annulus is 26 mm. Thehot uid is owing in the inner tube (the cold uid is thenowing in the annulus). The length of the exchanger is11 m. The experiments are simulated using a CFD software(namely uent), in an axisymmetric conguration. Thestandard j e model [24] is used, as the ow is turbulentin all the studied congurations for both uids. It has beenveried that the time step is short enough with regards to

convergence and accuracy. It has been found that 0.5



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second is suitable; so that actual time is the sample numberdivided by two. It has also been veried that the size of thecells is correct (in terms of boundary layer description, andin terms of length/width ratio), and of course that the out-let temperatures during steady states are correctlydetermined.

Fig. 1 shows the mass ow rates and the inlet tempera-tures that have been considered for the rst part of thestudy (clean heat exchanger) as they were introduced in

the software (using subroutines written in C). Fig. 2 showsthe corresponding outlet temperatures as they wereobtained by the software. It can be observed that the steadystate is never reached.

It has been considered necessary to simulate measure-ment noise and inaccuracy of the sensors. To do so, a

2% normally distributed random noise has been addedto all values (inputs and outputs). Figs. 3 and 4 show thedata that are actually used.

Fig. 2. Outlet temperatures for hot and cold sides (computed by the CFD code).

Fig. 1. Inlet temperatures and mass ow rates for hot and cold sides (introduced in the CFD code).



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4.2. Starting values for the Kalman lter and model parameters

The following values are common for the Kalman lterand models presented in Sections 3.1 and 3.2

Q z 0:1 0

0 0:1" #; _mh 0:1083 kg=s; _mc 0:1803 kg=s

Furthermore, the initial values and parameter values in thedifferent models are

P 0

0:022 0 0 0 0 00 0:022 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 0

0 0 0 0 0 1

266666664

377777775

Fig. 3. Inlet temperatures and mass ow rates for hot and cold sides (used for the estimation of the parameters) (partial view).

Fig. 4. Outlet temperatures for hot and cold sides (used for the estimation of the parameters) (partial view).



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^

x0

randomrandom74:63 C60:07 C54:42 C

72:44 C

2666666664

3777777775

; sh 7:7392s; sc 8:3502s; y 0:8

The rst two values of ^ x0 are random values rangingfrom 0 to 2. The above values (for Q z and P ) are basedon the results in Jonsson and Palsson [17]. The only unde-ned value is the Q matrix for the Kalman lter. GenerallyQ is dened as the diagonal matrix

Q cI 2 2 02 ns0ns 2 0:1 I ns ns !

where I is the identity matrix, ns is the total number of sec-tions in the model, 0 is the null matrix and c is a variable.

4.3. Clean exchanger

In this rst part, a clean heat exchanger is studied. Thismakes possible the determination of the quasi optimal val-ues of the time step and of the c value. It also makes pos-

sible the analysis of the effect of the initial conditions of theKalman lter.

4.3.1. Role of the Q matrixIn this subsection, the actual time period is 5 s (one out

of ten samples is taken into account). Fig. 5 shows the esti-

mation of a and b

for various values of c in the Q matrix.The effects of changing c are obvious. If c 1010 then theestimation of a and b is slow and the new equilibriumposition is barely reached before the next outlet tempera-ture is reached (e.g. about 2500 samples to reach a correctvalue for b). As c increases, the estimates will result ingreater uctuations about the equilibrium position. Forthis reason, it is necessary to select c in such a way to insurea sufficiently prompt response while keeping uctuations toa minimum. It has been chosen to select 10 8 as a correctvalue, so that the lter response is fast enough (about500 samples), while the uctuations are small enough.

4.3.2. Inuence of the starting values of the lterFig. 6 shows the values of a and b (whole set and the

rst thousand ones) for a sampling period of 5 s and for100 random initial conditions ( c 108). Fig. 7 showsthe evolution of the standard deviation of the estimated

Fig. 5. Estimation of a and b when c varies from 1010 to 105 . (Sampling period = 5 s random initial values.)



9/13

values of a and b for the same experiment (rst 3000 val-ues and for the whole range).

On the one hand, it can be concluded that any random

initial values are suitable. On the other hand, it can be con-

cluded that 2000 values have to be estimated (then the stan-dard deviation both for a and b is less than one percent of the mean value) before the initial values play no longer anysignicant role (for a sampling period of 5 s, see Section4.3.3 for the selection of the sampling period). This corre-sponds to a functioning period of less than three hours.If the user wants a shorter adaptation duration, it is possi-ble to use a shorter sampling rate, as shown in the nextsection.

4.3.3. Inuence of the sampling period Fig. 8 shows, for random initial values, the mean values

of the last twentieth of the values of a and b for a sam-pling period ranging from 1 sec (1 sample out of 2 com-puted samples) to 30 s (1 sample out of 60 computedsamples); c is 10-8 . It can be concluded that the procedurefor the determination of the parameters is very robust.Even though the uctuations are quite small, it has beenchosen to keep a sampling period of 5 s as a good compro-mise between reactivity and amount of data to be analysed.

4.4. Fouling

In this second part, a continuously fouling heat exchan-

ger is studied. To get the data, the thermal conductivity of

Fig. 6. Values of a and b for a sampling period of 5 s and for 100 random initial conditions ( c is 108). (Whole set and the rst thousand ones.)

Fig. 7. Evolution of the standard deviation of the estimated values of aand b for a sampling period of 5 s and for 100 random initial conditions(c is 108).



10/13

the inner tube has been continuously decreased (a subrou-tine is used in the CFD software). The resulting equivalentfouling factor is then computed using:

Rf Ah Rth t Rth 0: 15

This fouling factor is plotted ( Fig. 9) versus dimensionlesstime; the latter is dened as the ratio of the sample numberto the total number of samples during the fouling period. Itcan be seen that the variation is similar to the variation ob-served (after a slight decrease which is not taken into ac-count here) in e.g. Hays et al. [25] or Fahiminia et al.[26]. Typical values of the fouling factor can be found inCengel [27], in Incropera and De Witt [20] or at the engi-neering webpage [28]. For water, the fouling factor is typ-ically comprised in the range [0.0001 0.0007]. This interval

is smaller than the interval covered by the variation of thethermal conductivity, and corresponds to the [0.663 0.949]interval of the dimensionless time ( Fig. 9).

Fig. 10 shows the values of a and b for two similardrifts when the ow rates and the inlet temperature stillrandomly vary. The rst drift is eight hours long, and thesecond drift is four hours long; but they both have the samevalue of the fouling factor for a given dimensionless time.Note that the same clean period is considered before thefouling period to allow for the convergence of the lter

(initial values are still random). It can be seen that the evo-lutions of the parameters are quite similar, having the samerange.

A Cusum test, see e.g. Navidi [29], is carried out todetect the drift. This test has been preferred to the Shew-hart test due to the fact that it is more sensitive to smallshifts, as stated in the NIST online engineering statisticshandbook [30]. Due to the fact that fouling leads to thedecrease of the estimated parameters, a one-sided test issufficient (the second side - min operator- of the test wouldbe necessary to detect an increase of the estimated para-meters). This test is carried out in two steps:

1) Compute thecumulative sum: Cus k max0; E vref E v g r Cusk 1,

2) Check if Cusk > hr

; if so, then the drift is detected.The moving mean value of the estimated parameters is

computed using a number of samples equal to one twenti-eth of the clean period (see Section 4.3.2). The standarddeviation is either the standard deviation of the last twen-tieth of the clean period or the moving standard deviation.

If g and/or h are too low, false alarms are encountered.If they are too high, the drift is not detected. It has beenfound that g 1:96 and h 1:96 are a good compromise

Fig. 8. Mean values of the last tenth of the values of a and b (c is 108).

Fig. 9. Fouling factor versus dimensionless time.



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(note that 1.96 corresponds to the upper 2.5% tail of thenormal N 0; 1 distribution, and that, in our case, the testis not very sensitive to the threshold coefficient h).

Fig. 11 shows the values of the cumulative sums for aand b. The vertical lines are located at the detectiondimensionless time. It can be concluded that the drift isdetected quite soon, and that the standard deviation of

the clean period should be used in the Cusum test. FromFig. 11 it can be seen that a fast drift (second drift) isdetected later than a slow drift (rst drift). This is due tothe fact that the value (see Fig. 5) should be chosen inaccordance with the sampling period; itself being chosenin accordance with the speed of the drift. A too small valueleads to a longer adaptation time; this is clearly seen on

Fig. 10. Values of a and b for two similar drifts.

Fig. 11. Cumulative sum tests for a and b for two similar drifts.



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Fig. 10. From what has been done so far, it can be con-cluded that a minimum of 2000 values are necessary duringa clean period and that a sampling period of 1/12,000th of the duration of the drift leads to an early detection.

5. Conclusion

It has been shown how state space models can beapplied to detect fouling in heat exchangers. The modelsare physically based which means that some of theirparameters contain the heat transfer coefficients of the heatexchanger. By letting those parameters be time varying, it ispossible to follow changes in the heat transfer with time.The model parameters are estimated by using an extendedKalman lter and measurements of inlet and outlet temper-atures and the mass ows rates.

In contrast to conventional methods for detecting foul-ing which require measurements of the heat exchangeroperating in steady state this is not the case here. It is suf-cient to use measurements from ordinary operation wherethe load may be changing considerably. It has been shownthat the method is not sensitive to the sampling period.

Furthermore, it turns out that the estimates of the modelparameters drastically change when quite moderate foulingis introduced. This shows that the method is quite sensitiveand in that respect well suited for on-line detection of foul-ing in heat exchangers.

From the theoretical point of view, future studies willaddress other congurations such as crossow heatexchangers, comparisons with other models such as neuralnetworks, and systems such as heat exchanger networks.

From the practical point of view, experimental data willbe used to conrm the robustness of the method presentedhere.

Acknowledgements

The authors would like to thank the Research Fund of the University of Iceland for its nancial support in thisproject. This work would have not been carried out with-out the French/Icelandic Jules Verne program. Hence,the support of Rann s The Icelandic Centre for Research and the French Ministry of Foreign Affairs (under Con-tract EGIDE 12331SJ) is greatly acknowledged.

References

[1] P. Jakobsen, O.B. Stampe, Controlling the Effectiveness of HeatExchangers. In Danish: Kontrol af varmeveksleres effektivitet). VVSno. 1, vol. 26, 8 January, 1990. ISSN 0042-1944.

[2] T. Casanueva-Robles, T.R. Bott, The environmental effect of heatexchanger fouling: a case study, in: Hans Mu ller-Steinhagen, M. RezaMalayeri, A. Paul Watkinson (Eds), Proceedings of 6th InternationalConference on Heat Exchanger Fouling and Cleaning Challengesand Opportunities, Engineering Conferences International, KlosterIrsee, Germany, June 510, 2005, .

[3] H. Poulsen, Fouling in Heat Exchangers in Hot Tap Water Systems.

In Danish: Tilsmudsing af varmevekslere i varmt vandssystemer).

Department of Heat and Power Engineering, Lund Institute of Technology, Lund, Sweden, 1992. ISBN 91-971587-3-4.

[4] B. Thonon, S. Grandgeorge, C. Jallut, Effect of geometry and owconditions on particulate fouling in heat exchangers, Heat TransferEng. 20 (3) (1999) 1224.

[5] S.S. Ramachandra, S. Wiehe, M.M. Hyland, X.D. Chen, B. Bansal, Apreliminary study of the effect of surface coating on the initialdeposition mechanisms of dairy fouling, in: Hans Mu ller-Steinhagen,M. Reza Malayeri, A. Paul Watkinson (Eds.), Proceedings of 6thInternational Conference on Heat Exchanger Fouling and Cleaning Challenges and Opportunities, Editors Engineering ConferencesInternational, Kloster Irsee, Germany, June 510, 2005, ( ).

[6] M.S. Abd-Elhady, C.C.M. Rindt, J.G. Wijers, A.A. van Steenhoven,E.A. Bramer, Th.H. van der Meer, Minimum gas speed in heatexchangers to avoidparticulate fouling, Int. J. Heat Mass Trans. 47(2004) 9433955.

[7] M.A.S. Jeronimo, L.F. Melo, A. Sousa Braga, P.J.B.F. Ferreira, C.Martins, Monitoring the thermal efficiency of fouled heat exchangers:a simplied method, Exp. Therm. Fluid Sci. 14 (4) (1997) 455463.

[8] T.R. Bott, Biofouling control with ultrasound, Heat Transfer Eng. 21(3) (2000) 4359.

[9] C. Riverol, V. Naopolitano, Estimation of the overall heat transfercoefficient in a tubular heat exchanger under fouling using neuralnetworks. Application in a ash pasteurizer, Int. Commun. HeatMass Trans. 29 (4) (2002) 453457.

[10] C. Riverol, V. Naopolitano, Estimation of fouling in a plate heatexchanger through the application of neural networks, J. Chem.Technol. Biotechnol. 80 (2005) 594600.

[11] X.D. Chen, X.Y.L. Lin, S.X.Q. Lin, N. O zkan, On-line fouling/cleaning detection by measuring electrical resistance, in: Proceedingsof the International Meeting Fouling, Cleaning and Disinfection inFood Processing, Jesus College, Cambridge, UK, 35 April 2002, pp.235244.

[12] P.K. Nema, A.K. Datta, A computer based solution to check thedrop in milk outlet temperature due to fouling in a tubular heatexchanger, J. Food Eng. 71 (2005) 133142.

[13] M.M. Prieto, J. Miranda, B. Sigales, Application of a step-wisemethod for analysing fouling in shell-and-tube exchangers, HeatTransfer Eng. 20 (4) (1999) 1926.

[14] M.M. Prieto, J.M. Vallina, I. Suarez, I. Martin, Application of adesign code for estimating fouling on-line in a power plant condenserrefrigerated by seawater, in: Proceedings of the ASME-ZSITSInternational Thermal Science Seminar, Bled, Slovenia, on CD-ROM, 2000.

[15] G.R. Jonsson, J. Holst, Statistical parameter estimation of a counterow heat exchanger, International Symposium on District HeatSimulation, Reykjavk, Iceland, 1316. April 1989.

[16] G.R. Jonsson, Parameter estimation in models of heat exchangers andgeothermal reservoirs. PhD thesis, Department of MathematicalStatistics, Lund Institute of Technology, Lund, Sweden, 1990.

[17] G.R. Jonsson, O.P. Palsson, An Application of Extended Kalman

Filtering to Heat Exchanger Models, J. Dyn. Sys., Measure. Contr.116 (June) (1994) 257264.[18] J.W. Palen, Heat Exchanger Sourcebook, Hemisphere Publishing

Corporation, Washington, 1986.[19] J. Padet, Echangeurs thermiques, methods globales de calcul avce 11

proble mes resolus, Masson, Paris, 1994.[20] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass

Transfer, fth ed., John Wiley & Sons, 2001.[21] G. Jonsson, O.P. Palsson, K. Sejling, Modeling and parameter

estimation of heat exchangers A statistical approach, J. Dyn. Sys.,Measure. Contr. 114 (December) (1992) 673679.

[22] G. Welch, G. Bishop, An introduction to the Kalman lter, , updated July24, 2006.

[23] A. Gelb, Applied Optimal Estimation, MIT Press, Cambridge,

Massachusetts, USA, 1974.

http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdfhttp://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdfhttp://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdfhttp://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdfhttp://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/


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[24] B.E. Launder, D.B. Spalding, Lectures in Mathematical Models of Turbulence, Academic Press, London, 1972.

[25] G.F. Hays, E.S. Beardwood, S.J. Colby, Enhanced heat exchangetubes: their fouling tendency and potential cleanup, in: Hans Mu ller-Steinhagen, M. Reza Malayeri, A. Paul Watkinson (Eds.), Proceed-ings of 6th International Conference on Heat Exchanger Fouling andCleaning Challenges and Opportunities, Engineering ConferencesInternational, Kloster Irsee, Germany, June 510, 2005, .

[26] F. Fahiminia, A.P. Watkinson, N. Epstein, Calcium sulphate scalingdelay times under sensible heating conditions, in: Hans Muller-Steinhagen, M. Reza Malayeri, A. Paul Watkinson (Eds.), Proceed-

ings of 6th International Conference on Heat Exchanger Fouling andCleaning Challenges and Opportunities, Engineering ConferencesInternational, Kloster Irsee, Germany, June 510, 2005, .

[27] Y.A. Cengel, Introduction to Thermodynamics and Heat Transfer,Irwin/McGraw-Hill, 1997.

[28] .

[29] W. Navidi, Statistics for Engineers and Scientists, McGraw-HillInternational Edition, New York, USA, 2006.

[30] NIST/SEMATECH, 2006, e-Handbook of Statistical Methods, 2006, .

http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://www.engineeringpage.com/technology/thermal/fouling_factors.htmlhttp://www.engineeringpage.com/technology/thermal/fouling_factors.htmlhttp://www.itl.nist.gov/div898/handbook/http://www.itl.nist.gov/div898/handbook/http://www.engineeringpage.com/technology/thermal/fouling_factors.htmlhttp://www.engineeringpage.com/technology/thermal/fouling_factors.htmlhttp://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/http://services.bepress.com/eci/heatexchanger2005/

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