Computation of heat flux in a circuit-breaker electrode using temperature measurements P. Scarpa, L. Spronck & W. Legros o/,4pp W EWrzc%, [/Tz^er^^?/ o/ ABSTRACT This paper describes a numerical method developed to estimate the heat flux at the surface of a solid, using temperature measurements inside it. This inverse problem is solved while taking advantage of the available analytical solution of the direct problem. Moreover, the efficiency of the proposed algorithm appears when analysing several measurements because most of the calculations are made only once for all the records. INTRODUCTION During a current interruption process in a circuit-breaker, the energy dissipated in the electric arc is transmitted to the surrounding media (quenching gas, insulating nozzle, metallic contacts,...). Specifically this paper deals with the problem of estimating the heat flux flowing from the arc column to the electrodes : this information is indeed very important to characterise the arc-electrode interaction. Unfortunately a direct measurement of that heat flux is practically impossible due to the high temperatures involved by the electric arc. Nevertheless that information can be obtained using temperature measurements inside the electrode. This paper doesn't deal with the measurement problems; but it describes a numerical method to determine the unknown heatfluxwhen the measurements are available. In the practical studied situation, one can assume that: the heat flux is different from zero only during a given period of time (corresponding to the arcing time); the heat flux is uniformly distributed over the electrode surface; the material properties are supposed to be constant (no phase changes).
376 Heat Transfer MATHEMATICAL FORMULATION A rather general approach can be derived using the concept of "transfer function H" of a system. Here the relation between the heat flux ("excitation") and the temperature T(t) ("response") can be noted: Q(t) H T(t) (1) For a given geometry, the operator H depends on the material properties and on the probe position. First, one can choose a set of N linearly independent "excitation" functions E(t) that are different from zero only in an interval [0,t] (outside which the heat flux is a priori equal to zero). The "response" functions R(t) are then defined by : E(t) H R(t) (2) Since the problem is linear, these "response" functions are also linearly independent. They can be linearly combined to define a set of functions S(t), orthonormal over a chosen time interval [ti,t2j : S(t) = A-R(t) (3) This set of orthonormal functions is the "response" of the system to a set of "excitation" functions defined as follows : F(t) = A-E(t) and F(t) ** ) S(t) (4) The elements of the matrix A have to be determined to satisfy : Si (t) Sj (t) dt = 8jj (5ij is the Kronecker symbol) (5) 600 Figure 1: Example of injected heat flux and "measured" temperature.
Heat Transfer 377 Classical orthonormation algorithms could be used. But the matrix A has not to be known explicitly. Instead, if B = A"\ then : R(t) = B-S(t) (g) and, one can define a (symmetrical) matrix C whose elements are : Therefore, Cij = * Ri(t) Rj(t) dt = 2 I B^ Bj^ S (7) 1 m=l n=l c=b B^ «CT^A-A* (8) Now, if T(t) is the actual measured temperature, it can be approximated by a linear combination of the functions S(t) : N % X; Sj(t) The coefficients Xi can be determined using the least square method. According to relation (5), the minimisation of the expression : dt, (10) leads to : J, 1 V J- J-/ So, taking into account the equations (1), (2), (3), (4), (8), (9) and (11), the estimation of the heat flux is : N N N Q(t)= Z Z [}/T(t)Rj(t)dt] [%Ajj A^] Ek(t) (12) j=l k=l ^ ^i=l Finally, if one defines a vector Y whose elements are : Yi =J^T(t)Ri(t)dt, (13) the heat flux estimation is given in matrix form by : Q(t) «Y* <T* E(t) (14)
378 Heat Transfer In summary, to apply the proposed method, one has to : define a set of N linearly independent functions E(t); compute the set of response functions R(t); calculate the matrix C and its inverse according to equation (7). These steps are achieved only once. Then, for each different measurement, the remaining operations consist in : computing the elements Y; according to expression (13); applying the relation (14) to obtain the estimated injected heat flux. NUMERICAL APPROACH The above described mathematical approach is rather general and can be applied to many problems. But the parameters it involves have to be carefully adapted to the considered situation. Solution of the direct problem The studied geometry corresponds to a semi -infinite solid (bounded by the plane z = 0 and extending to infinity in the direction of z positive). The initial temperature is constant and equal to the room temperature TO. If, for t > 0, the heat flux "q" at the surface is constant, the temperature rise inside the semi-infinite solid is given AT(z,t) = T(z,t)-To = q Vat ierfc( ^=) (15) k 2Voct where "k" is the thermal conductivity, "cc" is the thermal diffusivity and "ierfc" is the first repeated integral of the error function [2]. This expression can be rewritten as follows : AT(z,6) = T(z,8)-To =q with 6 = = (16) k 6 2Vat For a given probe position, the adimensional parameter 0 directly governs the temperature evolution. Choice of a set of "excitation" functions E(t) The functions E(t) have to be defined to permit an easy computation of the "response" functions R(t). In this way, step-by-step varying functions seem adequate : one can use the superposition theorem and exploit the solution (15) of the direct problem. Moreover, the functions E(t) should be as "independent" as possible and their combination should allow to represent any kind of injected heat flux. The retained set of functions E(t) has been defined as follows :
Eo(t) =1 if 0 < t < T Heat Transfer 379 (17) Ej(t) = V(2J t - k t) j = 0, 1, 2,..., jmax Y(t) = -l Y(t)= 1 if 0 < t < T/2 if T/2 < t < % The function Eo(t) is intended to represent the mean value of the flux over the time interval [0,%]. The other functions Ej(t) are able to express the mean increase over intervals whose length is equal to T/2\ 0 T Figure 2: The chosen "excitation" functions E(t) Choice of the time interval fti.tol Considering the solution (16) of the direct problem, it appears that the function AT is a continuously increasing function. But, looking at its derivative, one can observe that the rate of rise passes through a maximum when : V2 2 a J ṁax = q J- ^- (is) 71 e k z So, at that characteristic time, the presence of the injected flux is the more evident when measured by a temperature probe. In order not to consider only a single instant, one can select the interval [ti,t2] in which the rate of rise is greater than a given fraction < > of the maximal rate of rise. For instance, if $ is equal to 0.75 : Z 5oc 2 Z a ] (19)
380 Heat Transfer Number of "excitation" functions (parameter jmax) The parameter jmax has to be chosen in order that the effective duration of the excitation functions is longer enough to induce significant temperature rise at the measurement point. Indeed, the temperature measured inside the electrode is only a "filtered" image of the injected flux. Practically, the information contained in the temperature measurement is not strong enough to detect very rapid variations of the flux. A first criterion consists in choosing jmax so that the effective duration of the shortest excitation functions is comparable to the characteristic time appearing in (18). A second possibility arise from the analysis of the amplitude of the "response" functions : jmax should be chosen in order that the maximum amplitudes of the different "response" functions don't differ more than one order of magnitude. APPLICATION Description To analyse the behaviour of the developed mathematical method, a theoretical problem has been studied. First, for a given injected flux, the temperature inside the electrode has been computed using the solution of the direct problem and applying the Duhamel theorem. Then, the flux computed from the temperature signal has been compared to the known flux. The injected flux Q(t) is different from zero during a period of 10 msec (starting at t=0) and is described by : Q(t) = sin(50%t) + 0.06 sin (300%t) - 80 t (GW/m%) (20) The temperature T(t) is "measured" at 0.5 mm from the surface of a copper electrode (thermal diffusivity =116 10"^ m^/s, thermal conductivity = 400 W/(m.K) [3]). Figure 1 shows the evolution of Q(t) and T(t). In this situation, one finds : the characteristic time [z2/(2.a>] = 1.078 ms; ti= 0.43 ms; t2 = 14.31 ms; jmax = 3 (shortest "excitation" functions : length - 1.25 ms relative response amplitude = 15.7 %) Solution in ideal conditions Figure 3 shows the computed flux compared to the known one. This very good agreement proves the validity of the developed method. One should also notice that the flux shape is correctly "extracted" from the temperature signal even if this one rises nearly linearly during the injection duration and, therefore, seems to contain only reduced information.
Heat Transfer 381 Influence of noise In order to test the robustness of the method against noise in the temperature signal, the theoretical temperature signal has been modified by adding a random noise. Figure 4 presents the results, in two different cases, when the noise total amplitude is equal to 40 and 100 degrees (respectively, = 8% and = 20% when compared to the maximum value of the temperature rise : 523 degrees). Despite these very bad conditions (worse than those encountered in real measurements), one can guess the real flux evolution quite easily. Influence of the inaccuracy on the thermal diffusivity Errors in the estimation of the injected flux can also be due to a bad knowledge of the thermal diffusivity of the electrode material. Figure 5 shows the flux computed while over- (or under-} estimating the thermal diffusivity (20 %). The main effect is that the flux is under- (or over-) estimated (respectively, = 15% and = 20%). Moreover a spurious oscillation appears at the end of the injection period. But an interesting observation is that the two oscillations are out of phase. This property can be exploited to improve the value of the thermal diffusivity when this one is not known very accurately. 500 Reference +++ Computation c +J ffi 0 0 5 t (ms) 10 Figure 3: Comparison between reference and computed heat flux. I 500 250-250- Reference ###8% noise XXX 20 % noise 0 0 5 t (ms) 10 Figure 4: Influence of noise on heat flux estimation.
382 Heat Transfer Influence of the inaccuracy on the sensor position An inaccuracy on the sensor position leads also to an erroneous estimation of the injected flux. Looking at the equation (16) and taking into account the results of the previous paragraph, one can deduce that a 10% over- (under-) estimation of "z" leads to a 22% over- (16.5% under-)estimation ofthe injected flux. 500 Reference x x 250- I 0 0 5 t (ms) 10 Figure 5: Effect of over- (or under-) estimating the thermal diffusivity. CONCLUSIONS The different tests carried out show the validity of the developed method. When analysing many temperature records, its efficiency clearly appears : the computation of the injected flux demands not much CPU time (most of the calculations are made only once, for all the different records). The sensitivity analysis shows that: the effect a random noise is not very important; the inaccuracy on the thermal diffusivity leads to an error on the estimated flux, more or less of the same order of magnitude; but, the spurious oscillations appearing at the end of the injection period can give a hint to improve the value of the thermal diffusivity; the accuracy on the sensor position is a really critical factor. REFERENCES 1. Carslaw, H.S. and Jaeger, J.C. Conduction of Heat in Solids Clarendon Press, Oxford, 1959. 2. Abramowitz, M. and Stegun, LA. (Ed.) Handbook of Mathematical Functions Dover Publications, Inc, New York, 1972. 3. Wilson, J. and Hawkes, J.F.B. Lasers : Principles and Applications Prentice Hall International, 1987.