IEEE International Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications 21-23 September 2009, Rende (Cosenza), Italy

978-1-4244-4881-4/09/$25.00 ©2009 IEEE 201

The Control System of the Profile of Temperature Field Nadiya Vasylkiv, Orest Kochan, Roman Kochan, Mykhaylo Chyrka

Research Institute of Intelligent Computer Systems Ternopil National Economic University

nva@tneu.edu.ua, oko@tneu.edu.ua, roman.kochan@gmail.com

Abstract - There is proposed method of control which could be implemented for heaters situated across electrodes of the main thermocouple and provides stabilization profile of temperature field. The proposed method has small requirements to memory space and computing power so it could be implemented using 8-bit microcontrollers.

Keywords - temperature control, multi zone furnace,

profile of the temperature field, microcontroller based system

I. INTRODUCTION

As it was presented in [1, 2], one of most dangerous and the little investigated (at the present time) component of thermocouple’s errors is error caused by heterogeneity of its thermoelectrodes obtained during exploitation. This error is related with degradation processes in thermoelectrodes caused by the influence of temperature and time of exploitation. There are two characters of display of the heterogeneity error [2]: as a conversion characteristic’s (CC) drift of thermocouples and as dependence of electro motion force (EMF) thermocouple not only from the difference of temperatures of hot and cold junctions but also from the profile of the temperature field between them – considerably bother a fight against influence of this error. In [3, 5] the method influence removal of the acquired heterogeneity is offered by creation it’s own profile of the temperature field, independent of external along the electrodes of main thermocouple of thermocouple with controlled profile of the temperature field (TCPTF). By the set of the heaters and sensors displaced along the axes of thermocouple and which are included in the additional temperature control subsystems. In this case an error caused the purchased heterogeneity of electrodes of thermocouples does not can directly to prove. The however offered method requires complication construction of thermocouple together with additional providing of his capacity due to work of the mentioned temperature control subsystems for all separate areas which create the set the profile of the temperature field along the electrodes of main thermocouple. Basic difficulty for their functionality is dependence of temperature for each zone of the temperature of other zones through the presence of process of transmission of heat along. This dependence creates the danger of excitation of the systems of adjusting (through the considerable delay of process of transmission of thermal influence between areas) and generation of heat-waves

along the electrodes of main thermocouple. Therefore control influences of every area must be formed in a complex with other zones [6, 7]. Thus, task of temperature setting in TCPTF is enough difficult.

The purpose of this paper is in development control method of profile of the temperature field along the electrodes of main thermocouple, which will allow to provide firmness and necessary exactness of establishment of the set profile without the necessity of difficult theoretical and experimental researches of adjusting object and will have low calculable complication, which will allow him to implement in the systems which are based on microcontrollers.

II. BASIC IDEA OF THE PROPOSED METHOD

The structure of TCPTF’s heaters is presented on fig.1. There are sections of heaters areas H1…N…K situated on case of TCPTF. The electrodes of main thermocouple - MTC and thermocouples of temperature control subsystems Ts1…ТСN…ТСK are placed inside a case. All thermocouples are isolated between each other. Their hot junctions are situated in the centers of the proper heaters. Heaters H1…K are isolated from environment by the layer of heat-insulation. Such construction allows to retain the set profile of the temperature field along the electrodes of MTC, which covers the possible changes of external profile of the temperature field. This profile is formed by the subsystem’s control commands and contained during exploitation and testing of main thermocouple. Thus, heterogeneity of main thermocouple can appear exceptionally as a drift of it CC.

However much the same structure creates problems at control (establishment) of areas temperature. Fig. 1 show, that every heater generates four thermal flows: • 1q oriented to the center of cover. In this case 1q is

useful, just he creates the set profile of the temperature field along electrodes of MTC;

• 3q opposite to 1q , oriented outside of TBS. 3q determines thermal losses, that is why his intensity is diminished by heat-insulation;

• 2q oriented to the next area of heater; • 4q oriented to the previous area of heater.

Presence of thermal flows 2q and 4q (and also them relatively high intensity, predefined that they go along metallic cover of TCPTF) determines the presence of good thermal connection between areas,

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that mutual dependence of temperatures, created by separate heaters. Therefore thermal flows 2q and 4q create the danger of to self-excitation of control system. Control actions for each zone must be generated taking into consideration the influence of other zones and the time delay of this influence.

q1 q2

q3

q4

Heater НN

TCN+1

Case

Isolation

q2 q4 q1

q3

TCN

TCN-1

MTC

Fig. 1. A model structure of heaters.

Thermal flows 4...1 qq can be evaluated using the following equation [8, 9]:

LTSq N∆

=λ

(1)

where λ is a coefficient of heat-conducting of the proper material; ∆T is a difference of temperatures, which causes this thermal flow; NS it is an area of

surface which a thermal flow is passed through; L it is distance between a heater and area which this thermal flow passes through.

In general thermal processes which pass in the areas of heaters can be described differential equation according to Newton-Rikhman law of cooling [8]:

dtTTSdTcVqdt S )( −+= αρ (2)

where q is a thermal flow, caused the difference of temperatures; c it is a heat-capacity of material which the heat-transfer passes through; V it is a volume of this material; ρ it is a density of this material; S it is an area of heat emission; α it is a coefficient of heat emission. It is possible to use principle of superposition to the thermal flows. So the resulting temperature of every area will be determined a total thermal flow which is created by all heaters, independently each of other. The thermal flows 4...1 qq created by all heaters, are determined by their electric powers. Therefore, according to [6, 7], it is possible to define required control influence for current time by the decision of the equations system of kind (2). Each of

these equations describes the process of the thermal flows composition from heaters of all zones for one separate zone. The real-time decision of such differential equations system requires the high computing performance. And this task is to complex for widely used microcontrollers.

Thermocouple with controlled profile of the temperature field does not require high accuracy of temperature field stabilization along the electrodes of main thermocouple. It is related with the fact that in this thermocouple the error of temperature measurement, caused by main thermocouple’s electrodes heterogeneity, is negligible quantity. It is mainly defined by imperfect of stabilizing process for own temperature field profile of the main thermocouple. Therefore it is proposed to implement simple method for temperature control for thermocouple with controlled profile of temperature field. The basis for simplification the estimation of necessary exactness of profile support of the temperature field can serve along the electrodes of main thermocouple. As it was presented in [10, 11], the error 5 °С causes the residual error of heterogeneity less than 0,2 °С. Thus, the allowed error of establishment of the profile of the temperature field can exceed 5%. It allow using simplified model.

The proposed simplifications of model consists of: (i) linearization of temperature dependence from power and (ii) transition from thermal flows composition to temperatures composition, caused by these thermal flows. In this case it is possible to write the following system of linear equations, which define the every increase temperature of each zone iT∆ as total action

of power increase of all heaters iP∆ :

⎪⎪⎩

⎪⎪⎨

⎧

∆×++∆×+∆×=∆

∆×++∆×+∆×=∆∆×++∆×+∆×=∆

−−−−−−

−−

−−

)1()1)(1(11)1(00)1(1

)1()1(11110101

)1()1(01010000

......

...

...

KKKKKK

KK

KK

PkPkPkT

PkPkPkTPkPkPkT

(3)

So it is necessary to do the following steps for setting profile of temperature field according to proposed method: (i) measure the real temperature of all zones and calculate temperature increases iT∆ for each zone; (ii) calculate

using (3) the required power increases iP∆ of each heater; (iii) calculate total power and pass it to heaters.

Control object simplification, errors of temperature measurement, incomplete conditions equivalence of coefficients )1)(1(00... −− KKkk definition, and other influences brings that the real temperature increases will not correspond to the computed. Therefore the proposed method of control demand its cyclic iterative implementation for the predefined profile.

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III. IMPLEMENTATION OF PROPOSED METHOD

The system of linear algebraic equations solving demands less computing resources in comparison with solving the same system of differential equations but it is also difficult task for microcontrollers, mainly because of the limited memory. Therefore it is proposed to use, Gauss method [12], for decision the system of equations (1). The main advantage of this method is possibility of making all computing operations directly above the elements of equations. Equation (1) can be presented as:

TPk ∆=∆× (4)

where k - „square” matrix of model coefficients;

P∆ - vector of power increases; T∆ - vector of temperature increases.

The Gauss method could be reduced to the two-motion algorithm:

1. forward motion at which, using elementary transformations of lines, which includes: adding of one line to another, which is multiplexed on the appropriate coefficient, and exchanging the lines and columns matrix (4) is transformed to the triangular kind;

2. backward motion at which, beginning from the last equation we consistently determine the element of unknown vector and put it in previous equations.

There are two key operations for implementation forward motion:

1. multiplying of all elements of coefficients matrix and one element of result vector on such constant, that after subtraction of two lines the result in certain positions was zero;

2. subtraction process. It is necessary to execute a following calculation for

implementation of the first stage first operation of the forward motion:

1,,2,0,

1,,1,,2,0,

−=−=∆

=∆

−=−=−==

KijKikT

T

KilKijKikk

k

ji

jj

ji

jljl

(5)

It is necessary to execute a following calculation for implementation the second stage of the first operation of the forward motion:

1,,2,0,

1,,1,1,2,0,

−=−=∆−∆=∆

−=−+=−=−=

KijKiTTT

KilKijKikkk

jij

jliljl (6)

As a result of implementation (5), (6) will get the matrix of coefficients )1)(1(00... −− KKkk with zero

values of coefficients 1,0,1,1,0 −=−== ijKikij . It is necessary to execute the following calculation

for implementation of the backward motion:

0,1,

1,0,0,1,

−=∆

=∆

−=−=×∆−∆=∆

KikTP

ijKikPTT

ii

ii

jiiii (7)

As a result elements of vector P∆ are increases powers which must be additionally transmitted on a heater. It is necessary to use only three additional variables as indexation elements of arrays for such implementation of Gauss method, because all results of calculations are stored directly in the elements of arrays and vectors. It allows to implement a method using computing devices which have minimum resources, for example 8-bits microcontrollers.

It is necessary to note that definition of model coefficients )1)(1(00... −− KKkk of the system of equations (3), based on result of experimental researches is also not a simple task which demand rational approach. In general case these coefficients could be determined, measuring the changes of temperature for every zone iT∆ under the action of the

known increases of power iP∆ . But such method

requires form the system KK × of equations, and then its decision. Although decision of such system of equations, using specialized software tools (for example, MATLAB) is not so complex, but it requires the making KK × experiments which are practically unacceptable. Rational approach in this case consists in the experimental finding of change of temperature for every zone iT∆ under the action of the known

increases of power iP∆ separate zone (when power of other zones is equal to the zero). In such case, for every experiment the system of equations (3) degenerates at K equations like:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

∆×=∆∆×=∆

∆×=∆∆×=∆

−−

iKiK

iiKK

ii

ii

PkTPkT

PkTPkT

)1(1

22

11

... (8)

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it allow to define coefficients iKi kk )1(0 ... − directly from every equation separately. Thus it is necessary and enough to provide only K experiments for definition all coefficients )1)(1(00... −− KKkk , and it is preferable.

The experimental researches method consists of the following steps: 1. Power off of all of heaters; 2. Waiting the end of transient process of cooling all

zones and measurement temperatures of all zones; 3. Set the maximal power of one heater and power off

all of other heaters (for getting the maximal change of temperature of all of zones and minimization influence additive error of temperature measurement);

4. Waiting the end of transient process of heating and measurement temperatures of all zones;

5. Computing the change of temperatures of every zones

iT∆ as difference of measuring results p.4 and p.2; 6. Iteration of operations of p. 1…5 for all heaters

separately. It should be noted that the considerable errors of coefficients )1)(1(00... −− KKkk arise up at their determination by the rapid passing to measuring of changes of temperature iT∆ from a next heater, without the complete removal of changes from previous (complete cooling of all of areas);

7. Drafting and solving the systems of equalizations (8).

It should be noted that differences in passing of process of heat-transfer toward hot junction thermocouple and toward could junction will lead to that coefficients

)1)(1(00... −− KKkk at experimental research of separate areas will be asymmetrical in relation to the heater of area. The example of distributing of temperature (it is indicated relative to the maximal heating of the included area of value) for the model standard TCPTF is presented on fig. 2. Additional verification of rightness of leadthrough of experiment at determination of coefficients

)1)(1(00... −− KKkk it is possible to do using equation (1)

– at the fixed valuesλ , ∆T and NS relative change of temperature within the limits of middle areas there will be proportional 1/ L , that reverse relative distance of area which we analyze. This dependence is well enough illustrated by presented on fig. 2 results of experimental researches of model of TCPTF – distributing of temperatures of areas in relation to central well enough answers a hyperbolical law.

The basic control algorithm of the proposed method is presented on fig. 3. In the process of its executing is calculated the results of control action of the system using equalizations (3) (power for each heater). Obviously, the errors of computing of coefficients )1)(1(00 ... −− KKkk will not allow to get the required temperatures of all zones

using single step. Therefore after some time the new control action should be defined on the basis of new measuring results, which will better approach the temperatures of zones to the specified set. Because a difference of temperatures in the next control cycles will be less than previous, and the error of approaching will fall.

0

0,2

0,4

0,6

0,8

1

1,2

1 2 3 4 5 6 7 8 9

# of zone

Rel

ativ

e Te

mpe

ratu

reFig. 2. Distributing of temperature is relatively to the maximal

heating of fifth area.

Begin

Loading the start parameters

Measurement the temperature of all zones

Computing the temperature shift of all zones

Computing power shift of all zones

Setting power of all zones

Delay

Fig. 3. A control algorithm is on the base of the offered method.

A new control action are generated, for avoidance of excitation of the systems there are adjusting, guilty to remain unchanging during the interval of time which substantially exceeds permanent time of transient of establishment of temperature in the areas of TCPTF. It is possible to find value of permanent time during experimental determination of coefficients

)1)(1(00... −− KKkk . It should be noted that reason of autogeneration of

small amplitude (systematic oscillation of temperatures of separate areas in relation to a mean value with a period a few hours) can be an error of determination of

205

coefficients )1)(1(00... −− KKkk . If calculated

concordantly (5)…(8) changes of power of areas iP∆ will be perceptibly higher from necessary for achievement of the set temperature of area, in every control loop temperature will “get” through set value which will bring temperatures over to the vibrations. For avoidance of this phenomenon offered after calculations concordantly (5)…(8) artificially to decrease the value of change of power on a few percents. In addition, for avoidance of “loosening” of the system because of act of random error term which is predefined noise of measuring channel and action of remaining hindrances of normal and general kind, it is expedient to enter hysteresis in the control channel. For this purpose beside the purpose to change power of heaters at small deviations of temperatures of areas from set – in obedience to [10, 11] a rejection, that does not exceed 1°С, cause such small residual errors of heterogeneity, that it is possible to ignore them.

IV. EXPERIMENTAL RESEARCHES OF METHOD

It is necessary to investigate the following points during experimental researches of subsystem of generating the predefined temperature field of TCPTF: 1. Whether the error of temperature field stabilization

does not exceed predefined limits; 2. Time of error decreasing to the defined limits; 3. Whether there are not fluctuations in a temperature

(self-excited oscillator) during exploitation of TCPTF.

It was used the testing system [13, 14] for the experimental researches of proposed approach. This system could be presented as pipeline electrical furnace. Its temperature is regulated by the separate controller. The termo-emf. of MTC and thermocouples of all zones is measured by the 16-point system based on microconverter ADuC-834 and hermetically scaled relays RGK-15 with additional thermo shunts [10, 11]. Also this system measures resistance of thermometer for the accounting of temperature of could junctions of all thermocouples. Power of heaters is set with discreteness 1% by to 9-channel PWM based on microcontroller At89s2051 with the external power switches. All of nodes are optically isolated between each other. The total error of voltage measurement on the range of 80 mV at the relative measurements does not exceed 2 uV.

There was investigated TCPTF based on type K thermocouple with nine heaters placed along this thermocouple [5]. Thus, the presented method of definition of coefficients )1)(1(00... −− KKkk requires the leadthrough of 9-th experiments instead of 81. The profiles of the temperature fields are experimentally investigational from heating of model of TCPTF during determination of coefficients )1)(1(00... −− KKkk resulted

separate heaters )1)(1(00... −− KKkk on the graph of lines. 4.

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9

# of zone

Tem

pera

ture

shi

ft, K

Fig. 4. The profiles of the temperature fields are resulted from heating of model by separate heaters.

According to the results of experimental researches, presented on the fig. 4 lines, a 81 coefficient

)1)(1(00... −− KKkk , which were plugged in the program of control system of a profile of the temperature field of model standard. The results of experimental researches of prototype of TCPTF [5] are presented on fig. 5 and fig. 6. The maximal rejection profile of the temperature field of prototype of TCPTF from an external profile made 16°С. On fig. 5 the graph of error of profile establishment of the temperature field is presented according to the number of zone by first cycle of power estimation (5)…(7). As it is presented on fig. 5, this error does not exceed 2,5°С, that causes the residual error of heterogeneity which, concordantly [10, 11], it is possible to consider as negligible. The process of profile establishment of the temperature field in accordance to all zones is presented on fig. 6. As it could be visible from fig. 6, the error of establishment of the temperature field decreases, that is fading of transient. As a result the error of profile of the temperature field of TCPTF to approximately 1°С. However, this error does not fully characterizes possibilities of the proposed method, because according to section 2, power of heaters changed only at deviations of profile from set greater 1°С (hysteresis).

It should be noted that time on fig. 6 is the relative and this intervals corresponds to two permanent time of transient process. And this time is rather large, for a prototype it was approximately 1,9 hours. As it visible on fig. 6 there was not significant self-excited oscillation of noticeable amplitude did not arise up. The oscillation of the lines on fig. 6 with amplitude 0,5°С could be explained both the remaining self-excited oscillation of small amplitude and influence of hysteresis, quantization errors of PWM (1%) and error of measurement system (1 µV, that corresponds 0,025°С), and also influence of noises of measurement system (also about 1 µV) and remaining normal and common noise.

206

-3-2,5

-2-1,5

-1-0,5

00,5

11,5

1 2 3 4 5 6 7 8 9

# of zone

Tem

pera

ture

shi

ft, K

Fig. 5. An error of profile establishment of the temperature field is by first estimation of power.

-3

-2,5

-2

-1,5

-1

-0,5

0

0,5

1

1,5

1 2 3 4 5 6Time (relative)

Tem

pera

ture

shi

ft, K

Fig. 6. Profile establishment of the temperature field is during the protracted exploitation.

V. CONCLUSIONS

There was proposed method of temperature profile control along the electrodes of thermocouple with controlled profile of the temperature field and presented some results of its investigations. The proposed a method provides sufficient accuracy of establishment and maintenance of temperature profile.

Method is based on suppositions that contradict to thermodynamics. That is why it have relation to heuristic methods.[15,16] The proposed method does not require large volume or complication experimental researches for identification parameters of simulation model. In the same time implementation of the proposed method require low

computer power and it could be implemented using 8-bit microcontrollers. For example, for MCS-51 compatible microcontrollers, using C51 programming language (Keil uVision), it requires not more than 4 кB program memory and only 380 bytes of data memory.

REFERENCES [1] I.I. Kirenkov, “Some laws of thermoelectrical heterogeneity,”

Transaction of VNIIM: Investigation in the field of temperature measuremen, 1976, pp.11-15.

[2] O. Kochan, N. Vasylkiv, R. Kochan, V. Yaskilka, “Evaluation the maximal error of thermocouples heterogeneity,” Transaction of Ternopil State Technical University, №1, 2007, pp. 122-129.

[3] O.Kochan, R.Kochan, G01K 7/02, Thermoelectric transducer. Application for patent of Ukraine № a200701855.

[4] О. Kochan, “Thermocouple with compensation the heterogeneity error,” Measuremet technology and metrology, №. 68, 2008, pp.144-153.

[5] О. Kochan, “Thermocouple with controlled profile of temperature field,” Transaction of Ternopil State Technical University, №2, 2008, pp.102-108.

[6] А. V. Sobolev, “Accuracy improvement of temperature field control by improvement control algorithm for multi-zone thermal object,” Ph.D. thesis on specialty: 05.13.06, Rybinsk, 2004, 159 p.

[7] A.G. Butkovsky, “Control methods for systems with distributed parameters,” Print house “Nauka”, Moscow, 1975, 568 p.

[8] S.E. Kryjanivsky, “Differential equations” Kharkiv, State scientific&technical print house of Ukraine, 1938, 398 p.

[9] I.А. Nedujyj, А.N. Alabovskyj, “Technical thermodynamic and heat transfer,” Textbook, 2-nd edition, Print house “Vyscha shkola”, 1981, 248 p.

[10] O. Kochan, R. Kochan, “Evaluation temperature measurement error using thermocouple with controlled temperature profile,” Transaction of Khmelnytsk National University, no. 2, vol. 1, Section Technical science, 2007, pp. 237-241.

[11] О. Kochan, R. Kochan, O.Bojko, M. Chyrka, “Temperature Measurement System Based on Thermocouple With Controlled Temperature Field,” Proc. of the IEEE international workshop on Intelligent Data Acquisition and Advancing Computing Systems (IDAACS’2007), Dortmund, Germany, September 6 – 8, 2007, pp. 47-51.

[12] http://ru.wikipedia.org/wiki/Метод_Гаусса [13] О.Kochan, N.Vasylkiv, V.Yaskilka, “Furnace for investigation

influence of temperature field on thermocouples,” Proc. of XIІ scientific conference of Ternopil State Technical University, Ternopil, 15 May 2008, p. 150.

[14] О.Kochan, N.Vasylkiv, V.Jaskilka, “Testing bench for investigation thermocouples with recontrolled profile of temperature field,” Transaction of Ternopil State Technical University, no. 1, 2009, pp.122-130.

[15] A.M. Andrew, “Artifical intelligence,” Viable Systems Chillaton, Devon(U.K.), Abacus Press, 1983, p. 264.

[16] M.L. Minsky, “Some methods of artifical intelligence and heuristic programming, In Mehanisation of Thougt Processes,” London: HMSO, 1959, pp. 3-36.