Pergamon

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Control Eng. Practice, Vol. 3, No. 7, pp. 1011-1015, 1995 Copyright © 1995 Elsevier Science Ltd

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LONG-TERM WEAR MONITORING AND WEAR PREDICTION BY MEANS OF WEAR MODELS

P. Neuenschwander*, D.E. Maurer** and L. Rychlicki*

*Swiss Federal Institute of Technology, Energy Systems Laboratory, ETH-Zentrum, CH-8092 Zurich, Switzerland **Ulrich Ammann AG, Energiesysteme, CH-4900 Langenthal, Switzerland

(Received September 1994; in final form January 1995)

Abstract: A method for monitoring and predicting fouling of heat exchangers by means of long- term wear models is described. To monitor progressive fouling, the dependencies between the poles of a second-order transfer-function, the fouling and the operation point are examined. The estimation of the poles, together with a wear model, allows the long-term wear monitoring and the prediction of the remaining life-span. Poles are estimated with a generalised Fourier-series, from data calculated by numerical simulation and taking nonlinear behaviour into account. The device properties are taken from an exhaust gas to water heat exchanger.

Keywords: Distributed-parameter systems; heat exchangers; monitoring; process parameter es- timation; prediction; frequency domain; maintenance; wear; failure detection

1. I N T R O D U C T I O N

Maintenance costs are substantially determined by the chosen maintenance strategy. In many cases con- dition-based repair is the key to an effective strategy. The use of this strategy requires information about the state or wear and the remaining life-span of the monitored device.

The calculation of the remaining life-span requires assumptions about the progress of wear I .

Parts performing a mechanical function are often re- paired on the basis of recorded load history. How- ever, it is unusual to base the prediction of the re- maining life-span of parts with thermal (e.g. heat exchanger) or chemical (e.g. catalytic converter) tasks on load history. On the other hand, condition- based repair using linear extrapolation of momentary trends of state or wear is very sensitive to measure- ment inaccuracy and to inadequate wear model struc- ture.

1 In this paper "wear" is used for mechanical wear, ageing, fouling and drift.

This paper presents a method to improve prediction quality by means of specific long-term wear models containing parameters for the individual case. The models for the process itself (e.g. heat exchange) shall be parametric and derived from physics and its equations. It turns out that there are advantages for the type of systems under consideration, to handle the behaviour of the process due to wear and opera- tion point in the frequency domain.

2. W E A R M O D E L S F O R P R E D I C T I O N

The progress of wear can be described by wear mod- els. The model type depends on the wear physics. For the example of heat exchangers which is used to demonstrate the procedure, the fouling resistance Rf can be modelled by equation (1) (Schnell and Slipce- vic, 1990).

R f = R r * ( 1 - e -I~*t) (1)

With values Rf calculated from measurements, the parameters Rf and 13 are estimated. If these parame- ters are known, prediction of wear is possible (Fig. 1).

1011

1012 P. Neuenschwander et al.

trend based on parameters wear from 1. 2. 3. estimation

- t eshold -- - !

, / ~ + + calculated from + measurements 4¢¢ -

i I ll-

Fig. 1.

remaining life-span '~ r . t

Improving wear prediction, on the basis of growing historic database

Based on the historic process parameter database, pa- rameters of the wear models can be estimated over longer time intervals. They become more accurate as the historic database is growing. The computation of these parameters is neither time-critical nor re- stricted concerning applied methods.

Given a wear threshold, the remaining life-span can be estimated. The threshold level should be based on criteria that are valid as generally as possible, e.g. total cost optimisation.

The measured data depends on the operating point and the fouling resistance Rf. To compensate for the influence of the operating point and therefor to iso- late Rf, it is necessary to model plant and process. This task is explained in Section 3. Additional as- pects to the choice and use of wear models are pre- sented in Section 4.

perature gradient in order to overcome the thermal resistance. The resistance can increase due to fouling (see Fig. 2). For heat exchangers the resistance B is calculated by equation (2) for a pipe.

~1 R] mediuml

~i] ~ ~ \ \ \ \ \ ~ , \ \ \ \ \ \ \ \ " ~ \ N ~ ~ fouling f i lml

~ / ~ ¢ J ~ Y ' J w all ,~.r~. N , , ^ N \ \ \ \ \ N N \ N N ~

/ / I ~. R2 ..41......_ fouling film 2 012 w 2 ~ medium 2

Fig. 2. Fouling at a heat exchanger surface

B = 1 + R ~ 5 + *lnd2 +

~l *dl dl ~ , * 2 * n d 1

+ R E + 1 ) , d l d2 tx 2 * d 2

(2)

The heat transfer coefficients ct are functions of the velocity; therefore B depends on the operating point of the heat exchanger.

On the basis of the input and output quantities of a heat exchanger the fouling is not directly deter- minable. It has to be separated from operation point influences.

3. P R O C E S S M O D E L S

A way to monitor a system is to examine its dy- namic behaviour, and in particular to examine the poles of the transfer functions in the frequency do- main. Fouling has an influence upon this dynamic behaviour. I f there are sensitive relations between the position of the poles and wear parameters, e.g. for the fouling of a heat exchanger, then the estima- tion of poles can be used for monitoring the system. Linear behaviour of the system over the whole oper- ating area is not a prerequisite for this method (Lin- ear behaviour is only assumed for the reaction to a generated small disturbance of an input quantity).

The estimation of the poles of the transfer function can be realised by the method of generalised Fourier- series from (Franke et al., 1993) using Legendre- Polynomials. This method allows the estimation di- rectly from measurements with little computational power.

3.1. Fouling at a Heat Exchanger Surface

The transfer of energy in a heat exchanger from a primary medium to a second medium requires a tem-

3.2. Heat Exchanger Input and Output Quantities

There are two ways to obtain the momentary com- ponent and wear model parameter Rf of the heat ex- changers: measuring the steady state or the transient signals.

Steady state. Monitoring of the fouling of a heat ex- changer without phase-change needs at least 5 input quantities: 1 mass flow and 4 temperature sensors, or 2 mass flow and 3 temperature sensors. Addi- tional information about the mean temperature dif- ference between the media is necessary if the heat exchanger is not of the co- or counter-current type.

Tex,i

I T w , o

Tex,o

~w Tw,i

Fig. 3. Heat exchanger input and output quantities

Because of the sensor requirements this kind of monitoring is too expensive in many cases. How- ever it was used during commissioning of the moni- toring system (see Section 3.5).

Long-Term Wear Monitoring and Wear Prediction

Transient signals. The estimation of the relevant pa- rameters by means of artificially introduced tran- sients saves 3 sensors compared with the steady- state method. Therefore, it is a promising way of monitoring the fouling of heat exchangers.

Selection of the quantities to vary and to measure. A heat exchanger has four input quantities. The varia- tion of the input temperatures without changing the appropriate mass flow is difficult in real plants be- cause of the large amount of energy necessary to change the temperature, compared with the quite small amount to change the mass flows. This is the first reason to change the flow speed by decreasing or increasing the rotation velocity of a pump, or by moving a throttle valve. A second reason is the sen- sitivity of the heat exchange behaviour to changes in the mass flow ratio. Therefore, uncertainties in the mass flow must be avoided, i.e. direct command of mass flows is advantageous.

1013

K G(s) = (3)

(s - Pl )(s - P2 )

For a counter-flow exhaust-gas-to-water-heat ex- changer, this simplified model was compared with the more-accurate model using distributed parameters (Law, 1962). Whereas the latter gives the relation between the pole location in the frequency domain and the resistance (including the fouling Rf) directly, the former needs a response curve in the time do- main in order to define the poles; this response (ob- tained by experiment or by simulation, e.g. by fi- nite elements according to Fig. 4) of course also de- pends on the fouling, thus leading to the desired re- lation between pole location and Rf. The nonlinear behaviour of the heat convection coefficients c~ is taken into account.

In the case of an exhaust gas to water heat exchanger it is best to produce a step disturbance in the water mass flow. If this is not possible, another rapid change in mass flow is accepted by the estimation method.

As system result, the output temperature of the ex- haust gas is measured.

3.3. Modelling the Heat Exchanger

Heat exchangers can be described as nonlinear dis- tributed-parameter systems. There are no general so- lutions for their dynamic behaviour in the time do- main. However, using analytical approximation, several authors found transfer functions which de- scribe the behaviour with the help of linearisation around the operating points. For example, Law (1962) found analytical transfer functions in the fre- quency domain for flow changes for shell-and-tube heat exchangers. They represent the relations be- tween physical parameters and poles in the frequency domain and allow one to study the principal rela- tions. For practical use, a simplified transfer func- tion is necessary to be able to estimate the poles from measurements.

~ - - numerical simulation -0.2r ~ x ~ --estimation

-0.4! " ~ ~ . _ ~ R.t = 0 [m2K/W! ...... r~ex- 1=00 %

-0.61 "~-~ Rf= 0 006 [meK/WI

-0.8 i Rf= 0.012 [m2K/W] ATex,o ~ [°C]

.l t . . . . . . . . . . . . . . . . . . . . 0 10 20 30 40 t [s] 50

Fig. 4. With finite elements simulated and with second-order transfer-function estimated curve

Tube geometry, mass flow rates and input tempera- tures of the heat exchanger are taken from real appa- ratus.

Input temperatures are constant.

The mass flow on the water side is increased by a step from 75% to 100% of the nominal value.

The mass flow on the exhaust gas side is varied +50% around the nominal point.

Second-order transfer function. Pfitter (1972) showed that simple transfer functions can well describe the dynamical behaviour of double-pipe and tubular heat exchangers in many cases, including mass flow vari- ations. For the reaction of the temperature on one side to the variation of the mass flow on the other side he proposed a second-order model.

The heat conductivity is varied by introducing a fouling resistance Rf on the exhaust gas side be- cause the water side is less critical to fouling if treated cooling water is used.

Poles and gain are estimated using the method of Franke et al. (1993).

1014 P. Neuenschwander et al.

Pole 2

140 " , " II <

1/B [W/maK] Rf= 0 120 ]////+

100 o actual R$

actual mex ~ / ~ / m, 80 - - ~ - - ~ * ~ . ( /~,, i /

- actual liB _ ~ . . ~ . . . . ~ X / / "~ j / / + . ~ > , ( ~ /x ,.

60 rTlex=150% * / t ~+ ./+ /"" /.. ~/'Xx

40 - ~ . * " * / ' + ' + - . " "" × - × ' ~ ' - < " ~ : ~ " " i + + ~ " . ~ t + + ÷ x x x × o c ° ° o

~-._zz......__-_z rlOlex=50%/ , li ,

Re -1.7 -1.5 -1.3 .2

Pole 1 /N

Rf = -0.002[m2K/W]

Gain ,IX ,"11:

Rf = 0.002[m2K/W]

t\ ,, × I ~ ~. x ,,/.l,X .,,

~'o/ <>< , . " . 2 i , ° o xx×~,, ' .4*** t

ec "x c¢ o

check , II,

-0.15 -0.1 0

' \

0.1 0.2 0.3

Fig. 5. Characteristic curves from the poles and gain of the second-order transfer function versus 1/B

Figure 5 shows the location of the poles and the gain according to the second-order transfer function versus 1/B estimated under the mentioned assump- tions. The thin straight lines with the arrows show the steps for the determination of 1/B if poles and gain are estimated from a real device (see also Sec- tion 3.5).

Figure 6, showing the position of the poles for a model with distributed parameters, indicates that the location of the first pole has a similar characteristic behaviour by varying the fouling resistance and the operating point as pole 1 from the second-order model (compare Fig. 5 and Fig. 6). The pole 2 in second-order model (Fig. 5) represents all the other poles in Fig. 6. The fundamental evolution is there- for identical. The second-order model is unable to represent certain resonance effects, which are how- ever not important in the context of wear prediction.

t re alparts f r o m ~ / / / / / r h e x : h l e " ~ t

0 ~ ~ real pole

~-2 Re 0

resistance Rf and operating point (e.g., variable gas mass flow). The following procedure is proposed: • Determine during commissioning (fouling Rf

equal to 0) the pole location and gain, introduc- ing artificial disturbances and analysing the re- sponses as well as the steady-state values.

• The variation in pole and gain values as a func- tion of the (future) fouling resistance are then obtained by an affine transformation of calcu- lated theoretical curves (as in Fig. 5) in such a manner, that the values for Rf=0 coincide with the real values (see Fig. 7).

140 1/B [W/m2K] 120 ,~~~

I00

80

60 61ex/~-~ 4C i ! ' ~ 2(3

-0.2 Re -6. -0.16

\ a , \ ,~-- Rf = 0

. ~ / ~ _ " ~ ',, after moving and ~"\\ transformation

~ . . x , • , G, ~.~,,-, ,.-,'.'.,'.,.,,

-0.12 -0.08 -0.04

Fig. 7. Positioning of the characteristic curves through reference poles and gain, example pole 1

Fig. 6. Principal characteristic behaviour from poles of the distributed-parameter model transfer function

3.4. Practical Application

In a real installation it is necessary to obtain charts describing the pole location as a function of fouling

3.5. B and Rf Determined from Measurement

During long-term operation, fouling will develop. It can be determined as follows: With one pole and gain estimated from one measurement with an artifi- cial disturbance, the actual heat conductivity is deter- mined exactly. A check is possible with the unused pole (see Fig. 5).

Long-Term Wear Monitoring and Wear Prediction 1015

4. WEAR M O D E L C H O I C E

Sometimes, several contradictory models for the same wear problem are described in literature. In this case all the models can be implemented. The selec- tion of the most confident prediction is performed on the basis of the prediction accuracy of the models in the recent past. Using this method the change from one model to another is always possible, and experience about model quality can be collected.

Heat exchanger fouling depends on the chemical composition, the temperatures and the velocities of the media, the design, the tube materials and the di- mensions of the device. It is difficult to determine the individual influence of those quantities on foul- ing. The prediction of fouling by wear models is es- pecially useful as it reduces the uncertainty of the future behaviour of wear.

6. R E F E R E N C E S

4.1. Classifying the Wear Model Parameter

To detect critical changes in the operating condition of the device the wear model parameters must be analysed by an expert system. If Rf is decreasing significantly and no cleaning was carried out, a heat exchanger leakage is probable. If Rf is increasing significantly, clogging of a part of the heat ex- changer is probable. If Rf exceeds certain thresholds with respect to cleaning, a maintenance order or an alarm message has to be sent. If Rf is within the expected limits, then the previously described wear prediction can be performed.

4.2. Remaining Life-span

Franke, D., Krfiger K., and Knoop M. (1993). Sy- stemdynamik und Reglerentwurf, Ein Zugang fiber verallgemeinerte Fourier-Reihen. R. Olden- bourg, Mfinchen, Wien.

Law, W.M. (1962). The Dynamic Response of Shell-and-Tube Heat Exchangers to Flow Changes. Neue Technik, 4, 34-44.

Ptitter, L. (1972). Einfache dynamische Modelle fiir Doppelrohr- und Rohrbfindel-W~irmeaustauscher. Dissertation, Technische Hochschule Aachen.

Schnell, H. and Slipcevic B. (1990). W~irmeaustau- scher: Rohrbfindel-W~irmeaustauscher; Grundla- gen, Aufbau, Anwendung. expert-Verlag, Ehnin- gen bei B6blingen.

As a result of the wear prediction the calculation of the remaining life-span or the time to the next ser- vice actions will be given. The remaining life-span also depends on the threshold, which must be fixed. This is mainly a question of repair versus operating cost. From the thermodynamic point of view, an ef- ficiency based on the exergy loss by heat transfer may be a suitable criterion.

5. C O N C L U S I O N

The estimation and observation of the poles of a transfer function in the frequency domain allow the determination of actual operating points and the monitoring of wear of systems that cannot be de- scribed in the time domain. The use of wear models makes the prediction more accurate and less sen- sitive to noise. For the example of the heat ex- changer it is necessary to reduce the complexity of the transfer function, at the cost of less direct rela- tions between poles and physical parameters. The determination of these relations for a real installa- tion is the key problem of this method, in the con- text of heat exchanger monitoring. The possible savings in energy consumption and maintenance costs may justify the expense of determining the relations mentioned, especially by taking into ac- count the decreasing costs for a single solution after collecting characteristic diagrams by practical experi- ence.

B;

dl, d2:

G(s):

K:

mex, mw:

Pl, P2:

W, Wl , W 2"

R 1, R 2, Rf:

Rf~o:

Tex,i, Tex,o:

Tw,i, Tw.o:

t"

O~ 1 , 0~2:

8:

L IS T OF S Y M B O L S

heat exchanger resistance

inner, outer tube diameter

transfer function

gain

exhaust gas, cooling water mass flow

poles of second-order transfer function

velocity

fouling resistance

asymptotic fouling resistance

input, output temperature of exhaust gas

input, output temperature of cooling water

time

heat transfer coefficient

constant (wear model)

wall thickness

thermal conductivity

Acknowledgements

The work is sponsored by the Swiss Federal Office for Energy (BEW) in Berne.