SOLVING THE HEAT EQUATION IN THE EDPS METHOD

Transcription

1 SOLVING THE HEAT EQUATION IN THE EDPS METHOD Peter Krupa 1, Svetozár Malinarič 1, Peter Dieška 2 1 Department of Physics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, Tr. A. Hlinku 1, Nitra, Slovakia 2 Department of Physics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology in Bratislava, Ilkovičova 3, Bratislava 1, Slovakia Correspondig author: peter.krupa@ukf.sk Abstract In the present paper we deal with the solution of temperature function in the Extended Dynamic Plane Source (EDPS) method. In this non stationary, single sensor transient method, one needs to find a time dependent temperature function developing at the heating element during the experiment. The procedure is based on solving the linear differential heat equation of second order with known initial and boundary conditions. We consider this system linear and time invariant and so the Laplace transform can be applied. Convolution theorem as well as other mathematical tools are discussed throughout the solution. Key words: EDPS, transient method, hot disk, temperature function, Laplace transform 1 Introduction In every dynamic method for measuring thermophysical parameters, knowing the temperature development at a certain point on (or inside) the sample is a key task. The EDPS method belongs to the group of contact transient methods, in which the temperature probe is completely surrounded by the sample. By knowing the temperature coefficient of resistance of the heating element, one can measure the temperature directly from the change of its resistivity. So in EDPS heat source serves simultaneously as a resistance thermometer (Karawacki et al., 1992). Sample s Heat sinks PS disc Fig. 1 EDPS method (schematic view) (Krupa, 2012) The planar heat source (PS disc) is clamped between two identical samples as shown on fig. 1. On the rear side the samples are in contact with two isothermal blocks (usually Cu or Al) with high thermal conductivity. At the beginning of the experiment, temperatures of all three elements are stabilized and equal. Then an appropriate DC current (in the form of step wise function) is let flow through the planar source, causing it to heat up. This generates a heat flow into the sample. From the temperature response of the sample (or better, the system) two thermophysical parameters can be evaluated. These are thermal conductivity and thermal diffusivity. EDPS method is suitable for materials with low thermal conductivity ( ). 374

2 2 Mathematical Model λ a λ s a s l l x heat sink specimen PS disc Fig. 2 EDPS method (ideal model) (Krupa, 2012) 2.1 Approximations In order to simplify the mathematical model certain assumptions can be made: The samples are an infinite plates with known thickness. The isothermals are infinite planes, planparallel to the zeroth plane of PS disc (fig. 2). It follows that the heat flow is one dimensional. In addition, symmetry of the method allows us to split the system into two halves and calculate only one of them. Heat flux density (power per unit area) from PS disc can also be split into two halves (i. e. opposite directions). PS disc is homogeneous, of negligible thickness and heat capacity. Thermal contact between PS disc and the sample can be approximated by Newton s cooling law. Heat sink is semi infinite and his thermophysical properties are known. Thermal contact between the specimen and heat sink is ideal. Lateral sides of the sample are not the source of any heat losses. Electric power is constant throughout the whole experiment. To start we write the linear differential heat equation, for each medium respectively (1) (2) where is the thermal diffusivity of heat sink. 2.2 Initial and boundary conditions The first boundary condition is the Newton s cooling law at ( ) (3) where is the temperature, position, heat transfer coefficient, temperature of heat source, temperature of the adjacent surface of the sample and is half of the power per unit area dissipated from the sensor (Malinarič and Dieška, 2009). Here is total power output of PS disc and its area. The remaining boundary conditions are (4) (5) (6) 375

3 Eq. (5) describes the ideal thermal contact at. Similarly, eq. (6) represents the equality of heat flow density at. is the thermal conductivity of heat sink. It is convenient to define the initial condition as (7) 2.3 Laplace transform Laplace transform (LT) is an integral transform which transforms function ( ) with real argument to a function ( ) with complex argument (Spiegel, 1965). { ( )} ( ) ( ) (8) ( ) is called the original and ( ) the image of ( ). The inverse LT is denoted as { ( )}. Integral (8) converges under some conditions placed upon the argument and the function ( ). In general, for linear and time invariant systems { } (Ondráček, 1999). Applying the LT on eq. (2 7) the transformed differential equations are (9) (10) and likewise the initial and boundary conditions (11) (12) (13) (14) ( ) (15) Eq. (9 10) are second order linear differential equations with constant coefficients of the form. General solutions of these equations are ( ) (16) ( ) (17) where are arbitrary constants, and. Combining the condition (12) with eq. (17), then by assuming { } and, expression converges only when. Then from (13) we have (18) Derivating (16) and (17) by Substituing (16) (for and also and then substituing into (14) leads to ( ) (19) ) into (15) gives (20) (21) 376

4 System of equations (18 21) contains variables,, and. Rewritten to the matrix form ( ) ( ) ( ) (22) Here we have defined a new constant Now we need to find an expression for. To solve the system of equations (22) we use the Kramer s rule (Lipschutz and Lipson, 2001). Finally, we get (23) [ ( )] (24) where ( ) ( ). Part of the expression in round brackets can be expanded into McLaurin series Multiplying by ( ) (25) ( ) (26) Therefore we can write [ ( )] (27) 2.4 Inverse Laplace transform Expression (27) is the image of the desired temperature function. Notice that ( ) ( ) ( ) (28) where ( ) is the image of an arbitrary input function and ( ) image of the system s transfer function. Transfer function is the system s response to Dirac s unit impulse ( ) { (29) Convolution theorem states that multiplication of the arbitrary input function by the transfer function in p domain is equivalent to the convolution of these functions in t domain (Ondráček, 1999). Hence { ( ) ( )} ( ) ( ) (30) where ( ) ( ) denotes convolution of ( ) and ( ). Once we find the original of the system s transfer function, we can convolve it with a step wise heating ( ) and get. According to (27) ( ) ( ) (31) We need to find the inverse LT of ( ). Linearity property of LT allows us to split up the eq. (31) into three. Then (Gradshteyn and Ryzhik, 2007) { } ( ) (32) 377

5 because. By defining new constant, { } { } (33) { } (34) And so ( ) ( ) ( ) (35) Convolution Last step is to convolve the transfer function (35) with a constant heating power input. To define ( ), we simply multiply with a unit step also known as Heaviside s unit function. Convolution of two causal signals ( ) and ( ) is defined as ( ) { (36) ( ) ( ) ( ) ( ) (37) where is a dummy variable of time. denotes, that function is flipped over and shifted by amount. Then for every, convolution gives the amount of overlap of over. Then ( ) ( ) ( ) ( ( ) ( )) ( ) (38) because ( ) inside of the interval ( ). By making the substitution, Integral (39) can also be evaluated sequentially. Then ( ) ( ) (39) ( ) ( ) (40) Last integral in (40) is not trivial and the solution can be found in tables (Gradshteyn and Ryzhik, 2007). After some adjustments we finally get the expression of our temperature function ( ) ( ) { ( )} (41) where (Carslaw and Jaeger, 1959) ( ) is the integral of complementary error function. ( ) ( ) ( ) is defined as ( ) (42) (43) 378

6 Eq. (41) is graphically shown on fig. 3. T (t) Fig. 3 Temperature function in the EDPS method (Krupa, 2012) 3 Conclusion Finding the analytical expression of temperature development inside the sample, even for one dimensional case, is not an easy task. Authors succesfully derived the temperature function for EDPS method using Laplace transform. The function matches with (Karawacki et al., 1992). Nevertheless, the curve is a mathematical idealisation and one must be very careful when fitting the experimental data on it (Malinarič, 2004). 4 References Gradshteyn, I. S. and Ryzhik, I. M. Table of integrals, series and products. 7th ed., Zwillinger, D. Jeffrey, A., Academic Press, London ISBN Karawacki, E. et al. An extension to the dynamic plane source technique for measuring thermal conductivity, thermal diffusivity and specific heat of dielectric solids. In Rev. Sci. Instrum. 10, Vol. 63, Am. Inst. of Physics 1992, p ISSN Lipschutz, S. and Lipson, M. Shaum s Outline of Theory and Problems of Linear Algebra. McGraw Hill 2001 Malinarič, S. and Dieška, P. Improvements in the Dynamic Plane Source Method. In Int. J. Thermophys. 2, Vol. 30, Springer 2009, s ISSN Malinarič, S. Parameter estimation in dynamic plane source method. In Meas. Sci. and Tech. 15, 2004, s Ondráček, O. Signály a sústavy. 1st ed., Slovak University of Technology, Bratislava 1999, ISBN X Spiegel, M. R. Shaum s Outline of Theory and Problems of Laplace Transforms. McGraw Hill 1965 t 379