Received: September 29, 2011 / Accepted: November 04, 2011 / Published: February 15, 2012.
　　Abstract: In this paper a specific measurement method for the determination of thermo-physical parameters of materials is described. All kinds of non-stationary thermal processes in a specimen can be used in this method. In practice, convenient thermal processes(taking into account parameters of dimension etc.) are chosen. The repeated measurement does not guarantee that identical non-stationary thermal processes are applied. In the measurement of the thermal conductivity of insulating materials, it is advantageous to use accumulated core. The time-dependence of the temperature in chosen points of the specimen is to be recorded continuously. The analysis of these records yields thermal parameters. In some cases, the variable thermal power of the laboratory oven is also continuously recorded. The method can be applied to the measurement of temperature dependences of thermo-physical parameters in a wide temperature interval.
　　Key words: Thermal measurement, thermo-physical parameters, thermal conductivity, accumulative core.
　　1. Introduction
　　Considering the measurement of thermo-physical parameters, one may state that there is a plethora of measuring methods and arrangements. Many of them were described in Refs. [1-6]. The methods used for measuring thermo-physical parameters of materials can be divided into steady-state and dynamic ones. Krempasky [2] estimated that there are approximately 500 measuring methods.
　　1.1 Measurement of Thermal Conductivity Coefficient by the Accumulation Core Method (ACM)
　　This paper is a follow up to an earlier one [5]:“Method of accumulation core and its use by measuring thermal parameters of porous materials”. This paper includes a proposal and theoretical analysis of a measuring method for thermo-physical parameters of materials using the so-called accumulation core. The accumulation core AC is a body with a very good thermal conductivity (Fig. 1). Heat penetrates into the accumulation core from an outer metal block (MB) through the measured specimen (sample) S. Thermal differences in the volume of the metal block are to be ignored. The work deduces some general relations applicable for the accumulation core method in the case of a permanent temperature increase. The accumulation core method is an integral method. It is suitable for measuring parameters of heat insulator within a wide temperature range from low up to high temperatures. Essentials of the accumulation core AC method are fully displayed in Fig. 1.
　　1.2 Steady Temperature Increase Conditions
　　Under these conditions, the temperature of the outer block increases linearly in time. Such increase of the border temperature is realized over the whole surface surrounding the specimen and the accumulation core steadily creates regular temperature conditions of the system characterized by linear temperature increase of the specimen volume at each point, including the accumulation core.
　　After having reached the steady state, the created profile of the temperature field is “evenly” shifted towards higher temperatures, while the speed of the temperature shift at each point of the system equals the speed of the temperature increase at the outer isotherm. Under the condition of steady temperature increase it can be shown that the coefficient of the thermal conductivity λ of the specimen can be stated (written) as
　　where cj and mj represent a specific thermal capacity of the accumulation core and its mass, respectively. The product mj.cj means the thermal capacity of the accumulation core. The parameter a is the thermal diffusion coefficient of the specimen. The differential dT means infinitesimal temperature increase at any point of the system and thus also of the outer block temperature. The derivative dT/dt indicates the speed of the outer block temperature increase. The constants A and B represent characteristic constants of a specific arrangement independent of the speed of the temperature increase: the thermal capacity of the core and the thermal parameters of the specimen, respectively. In the case of analytical or computer-based (numerical) determination of the constants A, B for a given system, the above mentioned relation enables determination of the coefficient of thermal conductivity λ for any speed of the temperature increase and for any specimen with different temperature parameters. It is sufficient to measure a difference, ΔΤ = T2 - T1 , between the temperature of the outer metal block (outer isotherm) and the temperature of the accumulation core (inner isotherm) and to determine the speed of the temperature increase.
　　As the right side of the previous relation includes the coefficient a of the thermal conductivity of the specimen, it seems impossible to determine the coefficient λ, unless the first coefficient is known. However, the real situation in the case of thermal-insulating porous materials is more favorable. In the article we will also mention a way of measuring coefficient a.
　　2. Accumulation Core Method of Thermal Conductivity Measurement at General Temperature Regime
　　Relation (1) is not valid. With small deviations from the regime with permanent increase of temperature, relation (1) is usable only for approximate estimations of the thermal conductivity coefficient and in order to achieve precise values, we must use another procedure. It will be described as in follows.
　　2.1 Measurement of the Coefficient λ Using ACM
　　Using arrangement in Fig. 1 at general temperature regime, the outer metal block temperature T1(t) varies randomly (but not very fast). Subsequently, the accumulated core temperature T2(t) varies. During the measurement, these two temperature dependences are recorded almost continuously.
　　We have to underline that the recorded dependence of accumulated core temperature T2(t)-exp will be compared to the theoretical dependence T2(t)-theor. This comparison can be accomplished by the theoretical calculation from the time dependence of the outer metal block temperature T1(t). It could only be done if we know the specimen parameters λ and a, but in fact we do not know them. On the contrary, we want to measure them. In spite of that, this procedure can be used. The sample thermal conductivity coefficient λwill be assumed to be a varying parameter, whose value is only estimated at first. Using this estimated value we can calculate the theoretical function T2 (t, λ, a)-teor. It is probable that this function will not be similar to the real temperature course T2(t)-exp. Next, we may change the parameter λ, in order to achieve a better agreement between these two functions. The least-squares method is used in order to achieve a coincidence of these two functions. The sequence is parametric. The substantial step at its estimation describes the symbolic relation
　　[]theorTaT P
　　Both these relations represent the fact that, employing a computer program, we can determine the accumulation core temperature T2-theor, when using the experimental function T2(t)-exp at selected parameters a and λ. The operator P)is actually a computer program, which gives the function T2(t)-theor. The parameter λ may vary in order to achieve a better agreement between the functions T2(t)-theor and T2(t)-exp. We assume that we already know the parameter a, and then also its temperature dependence a(T). Next we show how to find a better temperature dependence of λ(T).
　　(3)
　　where λo is an estimated input value of the temperature conductivity. We take k1, k2, k3 as constants considering them components of a vector kr= (k1, k2, k3), or kr = (k1, k2) in a simplified version of two parameters. The function λ(T) depends on the kr
　　vector. If we want to specify the λ(T) dependence, it is
　　sufficient to change the kr vector in small steps and in the “right” direction. Symbolically, we may realize changes:
　　and so λ(T) → λ?(T). Fig. 2 explains this procedure for the two-dimensional kr = (k1, k2).
　　The initial vector kr in the plane k1, k2 is represented by a point, to which a particular function λ(T) is assigned. The computer program generates the theoretical temperature function T2 theor for this λ(T). The computer evaluates the differences between this function and the real T2-exp course, using the least squares method.
　　Let us denote the difference reached at this point as S(Fig. 2). The program computes differences S’, S” for two points close to the kr vector. A new vector grad S will be computed by a differential method from this three S, S’, S” values. This vector grad S describes the direction of the steepest increase of S in the k1, k2 plane. The opposite direction (-grad S) shows how to change the kr-vector in order to go over into smaller values S. Next, the kr vector will be replaced by kdk where df is a small chosen parameter. This step enables a shift to a smaller value S, thus the new function λ(T) defined by Eq. 2 gives a result with a better correspondence to the experimental one. This procedure has to be repeated until the minimum of the value S will be reached with a sufficient accuracy.
　　Using a more-dimensional krvector, the gradient will also be more-dimensional. The vector grad S will then be set by computing the S value in more points. This procedure of searching for the λ(T) function with k1, k2, k3,.. parameters presents a non-standard use of the least squares method. The λ(T) curve is not superimposed over the measured λ1(T1), λ2(T2),.... points. The suitability of the k1, k2 parameters is set by the differential method computing when the differences between the two temperature dependences(the computed λ(T) and the measured one) are considered “sufficiently identical”.
　　Evaluating the “agreement” between T(t)-theor and T(t)-exp functions, we have to take into account that the functions are not continuous. The both compared functions consist of a set of discrete points, the density of which is different. That demands a non-standard procedure, the idea of which is shown in Fig. 3. A point on one curve does not have its pair on the other curve, because each of them corresponds with a high probability to different time data. A way out of this situation is the point grouping shown in Fig. 3. It is based on the fact that for a group of points detected at a time interval we assign one representative point on one curve and another point on the other. In this way we reach an acceptable number of paired points in chosen time intervals. Based on them, we can determine the differences of both functions and the corresponding value S. A similar sequence can be applied for measurements, at which a varying thermal power P(t) of the heating body is recorded.
　　 2.3 Accumulation Core Reduced to a Point Measurement of the a Coefficient
　　The measurement technique shown in Fig. 4 is a limited case of the accumulation core method. It corresponds to an endless small accumulation core, which was transformed into a point. The temperature sensor is placed in this point. Let us consider this question: How is it with the theoretical Eq. (1) in this specific case? The numerator in Eq. (1) is equal to zero in this case. The denominator must therefore have also the zero value (because 0≠λ) and thus
　　(5)
　　However, this relation is only valid at the constant-temperature increase regime. By simple geometries of sample, e.g., block, cylinder or sphere, the constant B can also be derived analytically.
　　By these measurements in a general regime, we have to juxtapose the real temperature course in the studying point of the sample with the theoretically computed one. This can be achieved from the outer metal-block temperature change data, using the differential method, as well as by setting the a parameter. The a parameter is then changed, so the best coincidence of the two curves may be achieved. In this way, we obtain a value corresponding to the real thermal conductivity
　　coefficient of the specimen.
　　2.4 The Computer Programs Testing
　　The computer programs used for the determination of theoretical curves and for their comparison with experimental ones can be tested in order to estimate their quality and reliability. This can be done e.g., by means of using the constant growth regime on a geometrically simple measuring sets. In this case we can find the precise analytical solution, which can be used in the confrontation to numerical computing results. A good agreement between the computed and analytical results confirms the reliability of both the methodology and the program. Analytical solutions in the constant growth regime are at disposal for the block, cylinder and sphere geometries.
　　3. Conclusions
　　The accumulation core method is an integral method. It uses the thermal capacity of a core inserted into the sample for the detection of heat transported through the sample. It does not require the source power measurement. The accumulation core has a high thermal conductivity, so it can be considered as an isothermal body. The sample with the accumulation core is inserted into an outer metal block, which creates another isothermal body. The measurement records the time-dependence of both temperatures, the core and the outer block temperature. At a general temperature regime, the temperature of the outer metal block varies freely, it increases or decreases. Thereby, an error related to the repeatability of the temperature regime at standard measurements is eliminated. The real outer block temperature course is “input” into the calculation, this regime need not to be repeated in another measurement. Errors in measurements of the accumulation core temperature are eliminated. However, speed temperature changes are inevitable from the point of view of the measurement accuracy. The thermal conductivity is calculated parametrically by comparing the recorded accumulation core temperature with one that can be determined theoretically. The method can be applied to the measurement of the temperature-dependences of the thermo-physical parameters in wide temperature intervals.
　　Acknowledgments
　　The authors would like to thank Prof. Viktor Bezák from the Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava and Ing. Gabriela Pavlendová, Ph.D. from the FCE, Slovak University of Technology in Bratislava for valuable discussions on this topic.
　　References
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　　[2] J. Krempasky, The measurements of the Thermo-Physical Properties, SAV, Bratislava, 1969, p. 288.
　　[3] F. ?ulík, I. Baník, Determination of temperature field created by planar heat source in a solid body consisting of three parts in mutual thermal contact, International Journal of Thermal Sciences 48 (2009) 204-208.
　　[4] G.M. Kondratiev, Teplovyje Izmerenija, Ma?giz., Moskva-Leningrad, 1957.
　　[5] I. Baník, Method of Accumulation Core and its use by measuring thermal parameters of porous materials, Proc. Thermophysics, 2006, p. 103-109.
　　[6] I. Baník, J. Lukovi?ová, Problem of elimination of undesirable thermal flows at thermo-physical measurements, Proc. Thermophysics 2007, p. 107-119.