Experimental Estimation of Thermophysical Properties of Materials

International Scientific Colloquium Modelling for Material Processing Riga, June 8-9, 2006 Experimental Estimation of Thermophysical Properties of Materials A. Timuhins, S. Gendelis, A. Jakovi s Abstract Analysis of potential of the guarded hot plate method and non-stationary temperature regime for experimental estimation of the thermal diffusivity and specific heat capacity of homogenous building materials is described in this paper. The advantages/disadvantages of different measurement schemes are analysed. Numerical estimation of accuracy influence of the process parameters is presented. Introduction Experimental device for measurement of heat conductivity of materials has been presented in work [1]. The device with a hot plate allowed to set up the constant power of the heater and to maintain constant temperature on surfaces of an investigated material. Such stationary temperature mode is used for standard measurement [2] of heat conductivity of a material. For the complete description of the thermophysical properties of a material in the non-stationary conditions it is necessary to know value of thermal diffusivity, therefore the electric scheme of the device and the software have been advanced in such away, that power of the heater could be changed harmonically (sine in time). Further, several methods for estimation of thermophysical parameters of materials employing sinusoidal mode of heating are considered. 1. Mathematical description of heat conductivity Process of the heat transfer can be assumed being one-dimensional because of the design features of the device. Governing equation of heat conductivity is heat diffusion / c is the thermal diffusivity, c is the specific heat capacity, equation Tt aTxx , where a is the heat conductivity and is the density of substance. Material properties are assumed not dependent on temperatures, because the range of change of working temperature is relatively small (about 50° ). The solution of the heat diffusion equation in case of the heater power represented by a sine wave is searched in a following particular form: T TC x T x e i t , where i 1 and is the cyclic frequency. After substitution of temperature with the particular solution in the heat conductivity equation we get: i T aTxx ei t aTxx . The equation splits up into two independent equations 2 TC T 0 , 0. (1.1), (1.2) 2 x x2 The equation (1.1) contains only amplitude and phase of the fluctuations of temperature, and the equation (1.2) describes the stationary mode of the heat transfer. Thus, the sine wave operating mode can be used also for measurement of heat conductivity of a material with the minor difference with the usual stationary mode that the temperature used in 2 i T a 227 calculations should be averaged over the time period. Let's introduce A2 A 1 i / 2a . Then the equation of the heat transfer (1.1) becomes: i / a or 2 T A2T 0 . (2) 2 x The solution of the equation (2) is the function T x C1e Ax C2 e Ax . (3) Our purpose is to estimate value of A, therefore it is necessary to formulate three boundary conditions for definition of all unknown variables C1 , C 2 , A . We shall consider the basic possible cases of the measurement of the parameter A 1) Lumped capacity case (Fig. 1a). The heater is switched on for a short period of time. At the same time temperature T1 and heater power is measured. 2) Periodic process in case of one slab of the tested material. The measured values of temperature on boundaries of the testes material (T0, T1) and the density of heat flux (Fig. 1b) is used as initial data. 3) Periodic process in case of two slabs of investigated materials. In this case temperatures T0, T1 and T2 (Fig. 1c) are measured. It is assumed that the value of the heat conductivity is known. It can be measured by the stationary measurement mode [1]. a) b) m1 Q heat source c) m1 Q m1 Q T1 T0 T1 insulation material m2 T1 T0 T2 slab contact Fig. 1. Schemes of experiments for measurement of coefficient A 2. Lumped capacity case T This is the simplest method based on the energy conservation law. Let's assume that heat losses Qout K Tmin T are proportional to the difference of the minimal temperature in the sample and the reference temperature. Then the heat capacity of a material C m m c , where m is density of the material, is easy to estimate by using recursive procedure Eh ( i 1) E out T Tmax TL Tmin tL t tmax Fig. 2. The schematically representation of process temperature in time t max tL C mi ; K ( i 1) C mi Tmax TL / Tmin t mas (i ) T dt ; E out K ( i 1) Tmin T dt , 0 where Eh is energy emitted by the heater, Eout is the heat losses, index in the brackets point to iteration number. 228 3. One layer case Periodic process in the case of one layer of the material is described by following system of equations: T x T x C1e Ax C 2 e Ax , T 0 T0 , T L T1 , Q. (4) x x0 From here follows the equation for determination of A: Q sinh AL T1 0 (5) T0 cosh AL A It is necessary to note, that in real conditions the equation (5) is not exact. The left hand size part of the equation should be understood as the function which should be minimized. Implicitly differentiating expression (5) we can find sensitivity of A to the change of the parameters. All values of sensitivity are shown in Table 1. Table 1. Sensitivity of A to the change of the parameters T0, T1, Q and L A/ Q A/ L A / T0 A / T1 A2 cosh AL / Z * *- Z T0 LA2 sinh AL A2 / Z A sinh AL / Z Q sinh AL A 2 T 0 A sinh AL Q cosh AL /Z QLA cosh AL 4. Two layers case In the case of several layers of material thermal contact resistance R l / can influence distribution of temperature. For the estimation of the contact resistance characteristic properties of the tested materials T2 T1 T0 T3 and air are used (see Table 2). For a layer of air l 1 mm resistance is between materials R 0.04 [(m2×K)/W], for a layer of foam m1 m2 l 5 cm resistance is R 0.4 concrete [(m2×K)/W]. Thus, the jump of temperatures on contact of investigated materials can reach 10 % of the common temperatures difference in the T(1) T(2) stationary mode. The contact resistance between ceramic surfaces given in literature (for example [3]) varies within the interval 3.3·10-4-2·10-3 [(m2×K)/W]. Such values of the parameters give Fig. 3. The schematic representation of smaller jump of temperature. The periodic mode spatial distribution of temperature is characterized by the following ratio of thermal A l la m a ca 8 10 5 , where index a corresponds to air and m to resistances: a a Am l m c l a m m m tested material. Thus, the influence of the contact resistance to the periodic mode is Table 2. Properties of tested materials negligible comparing to the stationary case material kg/m3 W/(m K) J/(kg K) resistance. If it is possible to use different foam contact materials, then material should be 400 1000 0.120 concrete considered satisfying: a ca m cm , a m. air 0.026 1.293 1006 Let's introduce the system of 229 coordinates with the origin on the interface of materials. Temperature in the left material is T 1 x , and temperature in the material to the right is T 2 x (see Fig. 3). Temperatures on the boundaries of slabs are known, and spatial distribution of the temperature is in the form T 1 x C1e A1x C 2 e A1x , T 2 x C3e A2 x C4 e A2 x , therefore periodic process is described by the following system of equations: T1 L1 T0 , T 2 L2 T3 1 T1 x x 0 T2 x , R 1 2 x 0 T1 x x 0 T1 0 T2 0 (6) Neglecting contact resistance R=0 and requiring that the average temperature between materials is equal to the measured temperature T 1 0 T 2 0 2T2 , we have the equation for the determination of A2 : Q1 2 A2 T3 T2 cosh A2 L2 , sinh A2 L2 (7) where the heat flux through the interface between layers is Q1 T0 / sinh A1 L1 . Assuming, that parameters of the material 1 are 1 A1 T2 cosh A1 L1 known, sensitivity to the change of the parameters is defined in Table 3, but accuracy of definition of the heat flux can be found by differentiating Q1 A1 , L1 , 1 , T0 , T2 by all of the parameters. In case of the identical slabs A1 L1 simplifies: 2T2 cosh AL T0 T3 . A2 L2 AL the equation (7) essentially (8) Table 3. Sensitivity of A to the change of the parameters L, T0, T1 and T3 in the case of two equal layers A/ L A / T1 A / T2 A / T3 A/ L 1 / 2T2 L cosh AL 1 / 2T2 L cosh AL sinh AL / T2 L cosh AL 5. Estimation of the properties of the measuring system We have employed the lumped capacity model for determination of effective characteristics of the heater (Fig. 1 ). Applying the recursive procedure for the received data, we obtain Cm=1047 [J/kg] and the effective specific heat capacity of the heater ch=421[J/(kg K)]. Complexity of the estimation of the heat conductivity is in fact that the maximal heater temperature is unknown. To overcome the problem we employ solution for semi-infinite region [3] of the heat conductivity equation. Initial temperature of material is equal to Tmin, and the heat flux from the heater is equal to Q: T x, t Tmin Q 2 at e x2 4 at x erfc x 2 at (9) The given assumption will hold only in the case of small deviations of temperature from Tmin. Using data of experiment (Fig. 4a) and the least squares method, we find the coefficient of heat conductivity. Calculated effective heater parameters are: weight of the heater mh=5.6 [kg], its density h=5185 [kg/m3], specific heat capacity ch=421 [J/(kg·K)], heat conductivity 230 and heat diffusion coefficient ah=1.45·10-7 [m2/s]. The approximate calculation for the layered heater (width of copper is 6mm, ceramics - 3mm and air - 3mm) gives the following values: h=5217 [kg/m3], ch=415 [J/(kg·K)], h=0.1 [W/(m·K)]. These results are in good agreement with the experimental data. Thus, the method applied here can serve also as the alternative method for calculation of the heat diffusion coefficient. h=0.32[W/(m·K)] 42 b) 1.2 28 35 1 26 28 24 21 22 14 20 7 18 0 100 Heater power 200 300 400 Temperature Temperature shift, °C. 30 Power, [W] Temperature, °C a) 0.8 0.6 0.4 0.2 0 -0.2 0 500 600 Time, [s] 0 20 40 Theoretical Experemental 60 80 Time, [s] Fig. 4. Heater parameters estimation experiment: a) heater power and temperature, b) theoretical and experimental temperatures close to Tmin. 4. Results of thermal diffusivity calculations Results of calculation of parameters of plasterboard (which density is m=773 [kg/m3]) are presented in Table 4. The experiment set up corresponds to Fig. 1c. Two layers were made with two equal 12.5 mm width plasterboard slabs. First slab was separated from heater by soft rubber substrate, in which the temperature sensors have been installed. Considering design features of the device we shall note some important features of calculations. The phase of the heat flux entering the test material does not coincide with the phase of the heater power. It is possible to compensate for phase deviation using equation (5) as well as using effective heater parameters. Phase deviation should be equal to =25.4° accordingly to the calculation in chapter 3. Fig. 5 corresponds to the case of one layer when the phase of heat flux was assumed equal to heater power phase =0° (an advancing of temperature by phase to heater power phase is 69.2°). The Fig. 5b displays a case when the phase is compensated by the inertia of the heater (phase delay =25.4°). In Fig. 5 the phase is compensated so that the =36.1°). The scheme with two layers does not equation (5) holds precisely (phase delay have drawback because the heat flux is defined by temperatures. Table 4. Calculated properties of measured plasterboard c J kg K Method W m K a m2 s Lumped capacity case 1621 - - One layer case (one slab) 1326 0.25 2.438·10-7 One layer case (two slabs together) 1211 0.25 2.672·10-7 Two layer case 1229 0.25 2.632·10-7 231 a) b) 0 25.4 a 2.67 10 7 c 1211 c) 36.1 a 2.98 10 7 c 1085 a 1.94 10 7 c 1352 Fig. 5. Phase (curve crossing the abscissa axis) and amplitude of the left hand size part of the equation (5) at different phase delay . On abscissa axis A log a Table 5. Numerical values of sensitivity of to the change of parameters A/ Q A / T0 A / T1 A / T2 Method One layer case 10.00 e i 2.73 9.70 ei 0.11 0.49 ei 0.21 (one slab) One layer case (two slabs 2.21e i1.40 1.60 ei 0.76 0.16 ei1.14 together) Two layer case 8.06 e i 0.33 25.22 e i 2.30 8.06 e i 0.33 - A/ L 25936 ei1.01 4727 ei 2.39 2432 e i 2.36 Conclusions Comparison of different experimental schemes of thermal diffusivity measurements using guarded hot plate shows that: it is possible to accurately estimate the thermal inertia of the heater, as well as the impact of the contact resistance using simplified calculation models; functional minimisation method can be successfully used for the estimation of the thermal diffusivity of materials on the basis of the collected measurement data. References [1] I. Javaitis, K. rglis, A. Jakovi s. Determination of Heat Conductivity of Materials by Guarded Hot Plate Method. Proceedings of the international scientific conference Civil Engeneering ’05. Jelgava, 2005. [2] EN ISO 8302:1991.Thermal insulation. Determination of steady-state thermal resistance and related properties. Guarded hot plate apparatus. [3] J. H. Lienhardt IV, J. H. Lienhardt V. A heat transfer textbook. 3rd Edition. Phlogiston Press, 2003 th [4] F.P. Incropera, D.P. DeWitt. Fundamentals of Heat and Mass transfer. 5 Edition. John Wiley&Sons, 2002. Authors Timuhins, Andrejs Gendelis, Sta islavs Dr. Phys. Jakovi s, Andris University of Latvia Zellu Str. 8 LV-1002 Riga, Latvia E-mail: [email protected] 232