DESIGN OF EXPERIMENTS TO ESTIMATE TEMPERATURE DEPENDENT THERMAL PROPERTIES *

Transcription

1 DESIGN OF EXPERIMENTS TO ESTIMATE TEMPERATURE DEPENDENT THERMAL PROPERTIES * K. J. Dowding and B. F. Blackwell Sandia National Laboratories P.O. Box 5800 Mail Stop 0835 Albuquerque, NM 8785 (505) , FAX (505) kjdowdi@sandia.gov and bfblack@sandia.gov.0 Abstract Experimental conditions are studied to optimize transient experiments for estimating temperature dependent thermal conductivity and volumetric heat capacity. Experimental conditions refer to the test specimen geometry, duration of the experiment, duration of dynamic boundary conditions such as heat flux, and magnitude of boundary conditions. A numerical model of experimental configurations is studied to determine the optimum conditions to conduct the experiment. Conditions for two configurations are studied to quantify the optimal experiment to estimate temperature dependent thermal properties. The two configurations are a finite and semi-infinite test specimen. Both configurations have an applied heat flux on one surface. The finite configuration prescribes an isothermal boundary on the opposite surface. Both configurations can be realized in the laboratory, in particular for low conductivity materials. The criteria D-optimality is used to identify the optimal sensor locations, heating duration and magnitude, and experiment duration. Thermal properties are assumed to vary linearly with temperature; a total of four parameters describe linearly varying thermal conductivity and volumetric heat capacity. Results indicate that the finite configuration is better than the semi infinite, assuming a single experiment is conducted to estimate the linearly varying properties. An alternative to estimate temperature dependent properties is to combine several experiments. Each individual experiment has a temperature range covering some or all of the temperature range of the properties. Combining two experiments is studied. In this case, two experiments covering the entire temperature range of the properties is better than two experiments covering a subset of the temperature range. *. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL Postdoctoral Appointee, Member ASME.. Distinguished Member of Technical Staff, Fellow ASME. /home/kjdowdi/foam/doc/inverse_99.fm

2 .0 Nomenclature c C k L N s N t N p q'' t T T max T 0 T i T β T X specific heat, J/kg- o C volumetric heat capacity J/m 3 - o C thermal conductivity, W/m- o C thickness, m number of sensor locations number of discrete time measurements number of parameters heat flux, W/m time, sec temperature, o C maximum temperature, o C boundary temperature, o C initial temperature, o C temperature sensitivity coefficient for parameter temperature vector, o C sensitivity matrix β, o C x 0 characteristic dimension, m Greek β parameter, optimality criteria t optimality criteria for combining two experiments time step, sec ρ density, kg/ m 3 Superscript dimensionless

3 3.0 Introduction It is becoming common practice to combine transient experiments with parameter estimation techniques to estimate thermal properties of solids. A sample of investigations using this approach are Jin et al., (998); Maddren et al., (998); Dowding et al., (998); Dowding et al., (996); Dowding et al., (995); Scott and Beck (99a and 99b); Beck and Osman (99). Properly designing the experiment strongly influences the accuracy of the parameters estimated from it. Considering the duration of the transient experiment, boundary conditions, and the impact of ancillary materials in the model are issues that can have a significant impact on the accuracy of the estimated parameters. Forethought, in particular using analysis to investigate and design the experiment, is paramount to the final success of the experiment. This statement is true for any experimental endeavor, not only those intended for parameter estimation. Conditions to estimate one or more thermal properties have been studied for several one dimensional configurations; Beck and Arnold (977), Taktak et al., (993), and Emery and Nenarokomov (998) are examples. In cases where the goal is to estimate k and ρc simultaneously, a known heat flux boundary conditions is required. Many of the configurations addressed by Beck and Arnold (977) and Taktak et al. (993) have a known heat flux on one boundary. Emery and Nenarokomov (998) studied other prescribed boundary conditions, such as convection. A two dimensional geometry has been studied by Emery and Nenarokomov (998) and Taktak (99). Taktak (99) investigated conditions to estimate orthotropic thermal conductivity. Emery and Fadale (996) studied the design of experiments while including the uncertainty of model parameters that are not estimated; in all previous studies additional parameters in the model are assumed to be exactly known. In all cited investigations, however, the thermal properties do not depend on temperature (constant properties). 3

4 Several investigators have presented formulations to estimate variable (temperature or spatially dependent) thermal properties; see Lesnic et al. (996); Woodbury and Boohaker (996); Sawaf et al. (995); Huang and Yuan (995); Huang and Ozisik (99); Flach and Ozisik (989). Experimental design to maximize the information was not addressed in these studies. Temperature dependent properties can also be obtained by conducting several experiments at different initial temperatures that are analyzed independently assuming constant properties for each independent experiment; this approach requires that the individual experiments cover a small temperature range. Then by combining the estimated properties from the individual experiments temperature dependent properties are obtained. Several investigations have used this approach: Dowding et al. (995); Dowding, et al. (996); and Loh and Beck (99). Experiments can be combined in a sequential fashion during the analysis to estimate temperature dependent properties, Beck and Osman (99) and Dowding et al. (998a). Dowding et al. (998a) discuss approaches to estimate temperature dependent properties without quantifying the best approach. An open question is what type of experiment will result in the most accurate estimated properties when properties are temperature dependent. Given that we want to estimate properties for a given temperature range, should a single experiment be conducted to cover the range? Or, should several experiments, each covering a portion of the temperature range, be conducted. This question is addressed in this study. Remarkably, the only known investigations to design an experiment to estimate temperature dependent properties are attributed to Beck (964, 966 and 969) over thirty years ago. Beck represented the thermal properties as varying linearly with temperature kt ( ) k T T T T = k T T T T () 4

5 where k and k are the values of thermal conductivity at temperatures T and T, respectively. Volumetric heat capacity would have a similar temperature dependence with parameters C and C. The parameter values Beck investigated are such that the low temperature and high temperature parameters describing the temperature variation in Eq. () are both equal to the average value, i.e., k = k = k ave and C = C = C ave. For a semi-infinite geometry with a prescribed surface temperature history, Beck (969) proved the sensitivity coefficients are not linearly independent. Hence, all parameters describing the temperature dependence cannot be simultaneously estimated. He suggested that certain parameters should be set while others are others estimated. We address different conditions and show later in this study that linearly varying k and C can be simultaneously estimated. The objectives of this study are to investigate the optimal experimental conditions to estimate temperature dependent thermal properties. We are presently interested in an experimental design to estimate temperature dependent properties of relatively low conductivity materials, e.g., polyurethane foams. Experiments are considered that utilize electric heaters providing a known heat flux on one surface of a specimen. The opposite surface is isothermal for a finite body or semiinfinite, * both of which can be realized in the laboratory for low conductivity materials. The two cases (finite or semi-infinite) are studied to determine which is optimum. A single experiment is assumed for estimating all parameter simultaneously to compare these two cases. Then we will combine two experiments to estimate temperature dependent properties and investigate what combination of two experiments is optimum. All references to experimental conditions are in the context of conditions in a numerical model of the experimental configuration; no experimental data are discussed. *. semi-infinite means that the specimen is sufficiently thick such that the face opposite the heated surface does not experience any temperature rise 5

6 4.0 Optimality Criteria A criteria is needed to identify an optimum design. The criteria selected for this study is D- optimality, Beck and Arnold (977). It is selected because it relates to the volume of the confidence region for the estimated parameters. Emery and Nenarokomov (998) discuss several other optimality criteria. All cited optimality criteria are related to the sensitivity coefficient matrix X X T = T β T β T β N p, () which is the partial derivative of temperature with respect to the parameter. T in Eq. () is a β i vector (length N t N s ) of temperatures for each time and sensor location. Assuming there are N t discrete measurements in time, measurement locations, and parameters, the dimensions N s of X are N s N t N p. D-optimality maximizes the determinant of the information matrix N p X T X. (3) In maximizing this determinant, it can be shown that this minimizes the hypervolume of the confidence region under certain assumptions. These assumptions are summarized as additive uncorrelated normal errors with zero mean and constant variance, with errorless independent variables, and no prior information, Beck and Arnold (977). Additionally, we impose constraints on the criteria. It is subject to a maximum temperature rise and fixed number of measurements. The maximum temperature rise of the experiment is specified as T max. To introduce these constraints the optimality criteria is modified as Each entry in the ( X ) T X ( X ) T X (4) ( T max ) N t N s is normalized as shown in Eq. (4). The dimensionless sensitivity matrix is 6

7 T 0 L x q 0'' Fig. One dimensional thermal model ( X ) T β = T q 0''x ( 0 /k ) β and the maximum dimensionless temperature rise is β β p T T ( q 0''x /k ) β q 0''x ( 0 /k ) β P, (5) T max T max T = i q 0''x. (6) ( 0 /k ) In Eq. (5) and Eq. (6), T is the initial temperature, q 0'' i is a constant heat flux, x 0 is a characteristic dimension, and k is a characteristic thermal conductivity value. D-optimality used in this study is essentially a discrete version of the criteria applied in Taktak et al. (993). Prescribing a maximum temperature constraint provides consistency when comparing experiments that may otherwise have different temperature ranges. Normalizing by the maximum temperature will give all experiments the same maximum value, namely. Fixing the number of measurements by normalizing in Eq. (4) by N t N s means the criteria is not increased by taking smaller time steps (increasing should not reasonably produce a better experiment. N t ) or having multiple sensors at the same location. Both situations 5.0 Experimental Configuration The configuration studied in this paper is shown in Fig.. Two variations of the configuration are addressed. The first variation is a finite slab of thickness L heated with a constant heat flux 7

8 ( q 0'' ) on one surface while the opposite surface is isothermal ( T 0 ). Although Fig. is an idealized case, the configuration can be closely approximated in the laboratory for relatively low thermal conductivity materials. For low conductivity materials the isothermal boundary conditions is achieved by placing a high conductivity material in intimate contact with the surface. The second variation approximates a semi-infinite specimen, the length L is taken to be large enough such that thermal effects do not reach x = L for the experimental duration, i.e., T 0 = T i. Again, for low conductivity materials this case can be closely approximated in the laboratory. The applied heat flux can be supplied and quantified with an electric heater and symmetric design of the experiment, Dowding et al. (995). For the most accurate estimated properties, the thermal effects of the heater should be included in the model. However, for generality in this study the thermal effects of the heater are not included. Because the heater s effect should be secondary compared to the specimen for a well-designed experiment, neglecting the heater should not significantly change the optimum design. It may be necessary to run secondary experiments to characterize thermal properties of the heater, as discussed in Dowding, et al. (995). TABLE. Optimal experimental conditions to estimate constant thermal properties k, C Case Geometry Boundary Conditions x = 0 x = L Sensor Locations Heating Max Duration t h Optimal Duration t f Semi-infinite a Semi-infinite b q 0'' q 0'' - - (x = 0) and (x > 0) (x = 0) and (x > 0) Finite b q 0'' T 0 x = Finite b q 0'' T 0 x = a. Beck and Arnold (977) b. Taktak et al. (993) 8

9 Optimal experimental conditions to estimate constant thermal properties are well known and given in Table. Two cases are listed. The two cases are for a semi-infinite and finite geometry, which is shown in the second column and refers to the thickness of the slab. Boundary conditions are identified in the third and fourth columns. For the semi-infinite case the sensors (column five) are located at the surface of the applied heat flux and below the surface; the finite case is optimal with sensors only at the heated surface. It is not possible to estimate both k and C with measurements only at the surface of a semi-infinite body. The value of D-optimality with constraints is given in sixth column for the dimensionless experimental durations listed in the seventh and eighth columns. Dimensionless heating duration is t, the applied heat flux is equal to q 0'' h up to this time, after which the applied heat flux is zero. Dimensionless optimal experiment duration is the time that the maximum value of D-optimality is achieved. These dimensionless times are defined as t f t h = ( k/c)t h x 0 (7) where is the duration that the heating is applied, is the optimal duration of the experiment, t h t f ( k/c)t = f x 0 t f (8) ( k, C) are the constant thermal properties, and x 0 is a characteristic dimension. For the configuration shown in Fig. the characteristic dimension is the thickness for the finite body x 0 = L and the depth of the sensor from the heated surface for the semi-infinite body. Temperature can be made dimensionless by a group containing the heat flux q 0''x 0 /k to remove dependence on the heat flux magnitude. Consequently, the optimal conditions depend only on the dimensionless heating time and experiment duration. 9

10 Two variations of each case (semi-infinite and finite) are shown. In the first variation, the heating duration extends for the entire experiment duration t h ( = ). The second variation continues t f the experiment after the heating duration ( t f > t h ). By comparing the two variations we see that continuing the experiment after the heat flux ends ( t f > t h ) increases the value of for both cases. The value of indicates that the finite case is better than the semi-infinite for constant properties. The ratio of for the two cases is approximately 4. Optimum conditions to estimate temperature dependent properties are less clear than those for constant properties. Beck (964, 966, 969) studied linear property variation such that k = k = k ave and C = C = C ave. In this study we will address estimating linearly varying temperature dependent properties without this restriction. We assume the properties are described as shown in Eq. (). Hence there are two parameters describing the temperature dependence of thermal conductivity ( k, k ) and two for volumetric heat capacity ( C, C ); where ( k i, C i ) are values of the properties at temperature T i. Linear variation is the simplest description of temperature variable properties, but is commonly used in general purpose thermal analysis codes. Two cases (semi-infinite and finite) considered previously for constant properties are studied. One can show for thermal properties varying linearly with temperature that the following three parameter groups mathematically describe the thermal properties: k k C C T T ,, (9) k C T T The dependence is on a normalized change in the property over the linear segment. We also see a temperature dependence; the group contains the temperature change of the experiment divided by the temperature variation of the properties ( T T ). In addition to the groups shown in Eq. (9), there is a dependence on dimensionless spatial location ( T T ) 0

11 and dimensionless time x = ( x/x 0 ) (0) t ( k /C )t = () x 0 Property values at the lower temperature ( ) are used in the dimensionless time. T Because the properties depend on the magnitude of the temperature (actually the temperature rise), the following group is used to define a characteristic temperature q 0''x 0. () k This group is indicative of the temperature magnitude. Ideally, we would like to explicitly specify all parameter groups in Eq. (9) and study sensor locations and durations (heating and experimental) that are optimum. But the last parameter group n Eq. (9) can not be explicitly set. Consequently, by varying the group in Eq. (), we can adjust the temperature magnitude so that the last parameter group in Eq. (9) can be implicitly set. The optimal conditions to estimate temperature dependent properties for the configuration in Fig. will be characterized by the following parameter groups k k C C x. (3) k C t q 0 ''x 0,,,, k The relative magnitudes of the properties, the sensor location, duration, and magnitude of the heating are all needed to describe conditions when properties vary linearly with temperature. For constant thermal properties we can describe the optimal conditions independent of the properties and heating magnitude; the design only depends on dimensionless sensor location and times. In the results that follow, property variation is taken as ( k k )/k = 0.66 and ( C C )/C =.. The variation is typical of polyurethane foam from room temperature up to 50 C, Jin et al. (998). We will investigate the sensor locations, heating time and magnitude,

12 and experiment duration that maximize D-optimality for estimating the four parameters describing temperature dependent conductivity and volumetric heat capacity. 6.0 Experimental Design We consider two experimental approaches. In the first approach all properties are estimated from a single transient experiment. For convenience the experiment will have an initial temperature and boundary temperature equal to the low temperature at which properties are defined T i = T 0 = T. The second approach considers combining two transient experiments with different temperature ranges for estimating the temperature dependent properties. Calculations are performed with a control-volume finite element based code. In addition to solving the heat conduction equation for the temperature, sensitivity calculations have been incorporated directly into the code, Blackwell et al. (998) and Dowding and Blackwell (999). Sensitivity equations are derived by differentiating the describing heat conduction equation. These sensitivity equations are numerically solved with the control-volume, finite element procedure. The sensitivity calculations are used to calculate the optimality criteria in Eq. (4). 6. Single Experiment To investigate whether a finite or semi-infinite geometry is better to estimate linearly varying thermal properties the optimality criteria is studied. In the search for the optimal conditions, sensor location, duration of heating, magnitude of the heating, and experiment duration are varied to determine which combination produces the largest magnitude of the optimality criteria. By presenting the results in dimensionless terms the specific size (thickness) of the specimen and properties are not important. However, the optimum design will depend on the magnitudes of ( k k )/k and ( C C )/C ; as stated previously, these values are fixed at 0.66 and., respectively.

13 TABLE. Optimal conditions to estimate linearly varying properties of a semi-infinite body Case Sensor Locations (cm) Heating t q 0''x 0 /k h ( C) Optimal T max T time T T t f ( ) E E , E E E E E , E E E E , E E Semi-infinite Case The experimental conditions investigated and optimum information are shown in Table for the semi-infinite geometry. Reported in column two of the table are the assumed sensor locations. All cases have a sensor at the surface where the heat flux is applied and internal to the body. We assume there are only two sensors. Conditions describing the applied heat flux are listed in columns three and four. Heating time t h is the time that the applied heat flux goes to zero. Prior to this time it is equal to the magnitude listed in column four. Conditions in columns two to four are prescribed in the analysis; the remaining information in the table is data resulting from the analysis. The resulting maximum temperature of the experiment is listed in column five. The maximum 3

14 of the optimality criteria and dimensionless time when maximum is achieved are listed in the remaining columns. The search for a maximum in the optimality criteria is easily done by trialand-error. An optimization code could be used in the search, but is not used in this study. The transient variation of the optimality criteria for the semi-infinite geometry are shown in Fig.. Numbers corresponding to the case number in Table are identified in Fig.. Because temperature variable properties depend on ( T T )/( T T ), we cannot completely remove the dependence on the sensor location by nondimensionalizing. Consequently, three combinations of sensor locations are selected and their results are shown in Fig.. All have one sensor located at the surface of the applied heat flux x = 0. The second sensor is located internal to the body; top figure is for the sensor located at x 0 =.54cm, middle figure is for the sensor located at x 0 =.7cm, and bottom figure is for the sensor located at x 0 = 3.8cm. Shaded in Table are the conditions that maximize for each of the three sensor locations investigated. Clearly, a sensor closer to the applied heat flux would respond at an earlier time and require a shorter experiment than a sensor farther away. However, when time is made dimensionless using the distance of the sensor from the heated surface ( x 0 ) there is only a small change in the optimal time; optimal experiment duration is =.84,.86, and.89 for sensors at t f x 0 =.7,.54, and 3.8cm, respectively. Furthermore, the magnitude of is not sensitive to the sensor location; moving the in-depth sensor from a location of x 0 =.54cm to.7 or 3.8cm changes -7.0 and. percent, respectively. The transient plots in Fig. demonstrate that the maximum of is attained after the heat flux ends. At the time when the heat flux ends the value of increases and the curve changes shape. This change is due to the sensitivity coefficients changing shape after the heat flux ends, Dowding and Blackwell (999). ( t h ) 4

15 x Semi-infinite Case x 0 =.54cm case = t x Semi-infinite Case x 0 =.7cm t x Semi-infinite Case x 0 = 3.8cm t Fig. Optimality criteria for estimating linearly varying thermal properties for a semi-infinte case. Sensors are located at ( x = 0, x 0 ) 5

16 The approach taken to investigate the optimal conditions is to select a heating duration and magnitude ( q 0''x 0 /k ) such that the temperature variation during the experiment would cover the temperature range of the property variation. The parameter group in the fifth column of Table indicates the resulting temperature variation of the case. The maximum allowable value of this group is unity; temperature can not exceed the temperature range for which the properties are ( t h ) defined. The maximum normalized temperature varies from 0.95 to for the cases shown. In general, the optimality criteria increases as ( T T )/( T T ) increases. The effect of this parameter group is studied in greater detail in the next section. 6.. Finite Case The conditions investigated and optimum conditions for a finite geometry are listed in Table 3; data presented in Table 3 parallels that of Table. The characteristic dimension for the finite body is taken as the thickness of the one dimensional body x 0 = L. A sensor is located at the surface ( x = 0) of the applied heat flux and internal to the body. Three locations are investigated for the internal sensor: x/l = 0.5, 0.5, The transient variation of is shown in Fig. 3. The top figure is for sensors located at the heated surface and at midplane of the slab ( x/l = 0, 0.5) while the bottom figure is for locations ( x/l = 0, 0.5 and 0, 0.75). Comparing continuous heating (case ) with cases -6 demonstrates the significance of continuing the experiment after ending the heat flux ( t f > t h ). We get over a 600 percent increase in by terminating the applied heat flux but continuing the experiment. The reason for this significant increase is that the sensitivity coefficients dramatically change shape after the heat flux ends; see Dowding and Blackwell (999). We note that the increase in case. after heat flux ends is greater for the finite case (Fig. 3) than it is for the semi-infinite (Fig. ) 6

17 TABLE 3. Optimal conditions to estimate linearly varying properties for a finite body Case Sensor Locations ( x/x 0 ) Heating t q 0''x 0 /k h ( C) Optimal T max T time T T t f ( ) E E E , E E E E , E E E E , E E E E , E E

18 9 x Finite Case x/l = 0, t 9 x Finite Case x/l = 0, 0.5 x/l = 0, t Fig. 3 Optimality criteria for estimating linearly varying thermal properties for finite case Moving the internal sensor closer to the heated surface does not significantly change. Its value decreases about 4 percent when the sensor is moved from x/l = 0.5 to 0.5. We note that the optimal time listed in Table 3 is different for these two sensor location. The maximums for sensors located at ( x/l = 0, 0.5) and ( x/l = 0, 0.5) in Fig. 3 are quite broad, however. Similar magnitudes of are obtained for heating durations between t h =.33 (case 3) and t h =.79 (case 4) for ( x/l = 0, 0.5) with the optimal time being t f t h.05. Likewise, heating dura- 8

19 tions between t h =.33 (case 8) and t h =.79 (case 9) produce magnitudes of near the maximum for ( x/l = 0, 0.5) with the optimal time being t f t h.3. Moving the internal sensor farther away from the heated surface reduces the magnitude of. It decreases 50 percent when the internal sensor is moved from x/l = 0.5 to The location of the maximum does not change, however. This decrease in is caused by moving the sensor closer to the isothermal boundary where the sensitivity is zero for all parameters. As stated earlier, the thermal properties depend on the parameter groups given in Eq. (9) and one of these group depends on the temperature change of the experiment ( T T )/( T T ). Consequently, the optimal conditions depend on the magnitude of the applied heat flux in addition to the heating and experimental duration. Most of the cases studied used a heating magnitude such that ( T max T )/( T T ) in Table 3 is near unity. For comparison, the value of this group is reduced to 0.5 for sensors located at x/l = 0, 0.5. The results are shown in the bottom of Table 3 (cases 4 to 7). Although the magnitudes of for these cases are considerably smaller than cases 3 to 7 where the parameter group is 0.98, the conditions where the maximum occurs are similar. Consequently, increases as ( T max T )/( T T ) increases. In other words, selecting heating magnitudes that result in the temperature covering the entire temperature range of the properties produces larger magnitudes of and is desirable. However, the optimal heating times are similar for heating magnitudes that cover different temperature ranges, e.g., compare heating duration and optimal time for cases 4 and Summary If we compare the magnitudes of for the semi-infinite (Table ) and finite case (Table 3) we can conclude that the finite case would provide estimates with a smaller confidence region. The ratio of for the finite and semi-infinite cases is nearly 0. Furthermore, selecting heating 9

20 magnitudes so that the temperature covers the entire temperature range of the properties ( T max T )/( T T ) produces larger magnitudes of. These conclusions are based on linear property variation such that ( k k )/k = 0.66 and ( C C )/C =.. However, we believe that these conclusions (finite body is better than semi-infinite and larger temperature variation is better) will not change for other magnitudes of the parameter variation. The specific optimum conditions would be expected to change for different magnitudes of the parameter variation. 6. Multiple Experiments A potentially powerful way to estimate many parameters is to combine experiments. Frequently when estimating many parameters from a single experiment, as the number of parameters gets large, it becomes more difficult to estimate all the parameters due to correlation in the sensitivity coefficients. However, even when the number of parameters is not large, such as the case here, it may be advantageous to combine experimental. A central issue addressed in this section is the following. To estimate linearly varying temperature dependent properties, is it better to conduct experiments that covers the entire temperature range of the properties, or conduct several experiments, each covering a portion of the temperature range? We will study the optimality criteria for these two approaches to answer this question. Consider that we will conduct only two experiments to estimate thermal properties varying linearly with temperature; four parameters describing thermal conductivity and volumetric heat capacity are to be estimated. In the first approach, the two experiments will each cover the entire temperature range corresponding to the linear property variation. The second approach will conduct two experiments: one experiment will have a temperature range that is near the minimum temperature that the properties are defined ( ); the second experiment will have a range near the T maximum temperature ( ). The total number of combined measurements from the two experi- T 0

21 ments is assumed to be fixed. While we could compare the second approach with a single experiment from the first approach, this comparison is not appropriate. Because the second approach would have double the number of measurements in two experiments that the first approach has with one experiment. Conducting two experiments for both approaches fixes the total number measurements and provides a fair comparison of the two approaches. We will assume the two experiments are combined during the analysis. This can be accomplished by simultaneously considering the two experiments or through a sequential approach (Beck and Osman, 99). For two experiments covering the entire range we could estimate the properties independently for each experiment and not combine them. The independently estimated properties would provide evidence of consistency between the experiments. However, by combining the experiments we include all the measurements to estimate a single set of properties. By assuming that the experiments are combined during the analysis, we can simply add the contributions from each experiment in the optimality criteria. Equation (4) is modified as follows ( X a ) T X a ( X b ) T X = (4) ( T max ) b N t, a N sa, ( T max ) N t, b N sb, where the subscripts a and b refer to data from experiments a and b. Because experiments a and b potentially have different temperature ranges, the temperature range of the properties is used as the maximum temperature, T max = ( T T )/( q 0''x 0 /k ), in Eq. (4). When two identical experiments are combined for a single experiment X a ( = ) X b the optimality criteria in Eq. (4) is related to the criteria = N p. (5)

22 TABLE 4. Optimal experiment to estimate linearly varying properties of a finite body by combining two experiments Case Exp Sensor Locations ( x/x 0 ) Heating t q 0''x 0 /k h ( C) T i T T T T max T T T t f ( ) a 0, b 0, a 0, b 0, a 0, b 0, a 0, b 0, a 0, b 0, a 0, b 0, E E E E E E Table 4 presents the optimal conditions for combining two experiments to estimate linearly varying thermal properties for sensors located at x/l = 0, 0.5 for a finite geometry. The two experiments are identified as experiment a and b for each case. The information in Table 4 parallels the data presented in the Tables and 3, except that the initial temperature of the experiment is given in column six. The isothermal boundary in Fig. is prescribed to be equal to the initial temperature as well. Cases to 4 are variations using the optimal heating duration identified for a single experiment with a finite geometry in Table 3. The magnitude of the heating and initial temperature are varied, which in turn varies the temperature range of the experiment (column seven). Cases to 4 combine two experiments covering an increasing percentage of the temperature range of the prop-

23 erties; cases,, 3, and 4 combine experiments covering roughly 5, 50, 75, and 00 percent of the temperature range, respectively. Cases 5 and 6 combine two experiments covering the entire temperature range, but use a different heating duration than the previous cases. The tabulated values of and transient variation shown in Fig. 4 demonstrate that two experiments covering the entire temperature range produce a larger magnitude of than do two experiment with each covering only a portion of the range. As the temperature range of the experiment increases from 5 to 98 percent the value of increases almost three order of magnitudes. The optimal duration of experiment ( ) increases only slightly from 3.74 to t f Presuming the sensitivity coefficients are not correlated, one can see from the optimality condition in Eq. (4) that to maximize the combination of two experiments one needs to maximize the individual contributions from each experiment. As demonstrated for a single experiment, conducting an experiment that covers the entire temperature range of the properties is better than an experiment that only covers a portion of the temperature range. Hence, combining two experiments that cover the entire temperature range of the properties will give larger magnitudes of 3

24 x t Fig. 4 Optimality criteria to estimate linearly varying thermal properties for finite case while combining two experiments than two experiments that cover less of the temperature range. As the portion of the temperature range that the experiments cover increases, the value of increases. Because the optimum for combining two experiments is the optimum of the individual experiments, the heating durations used in cases 5 and 6 do not give as large of magnitude for as case 4. Heating duration and magnitude for case 4 is shown to produce larger values of (single experiment) in Table 3 than the conditions used for cases 5 and Summary and Conclusions A study of conditions to optimize transient experiments for estimating linearly varying temperature dependent thermal properties was presented. Four parameters describing linearly varying thermal conductivity and volumetric heat capacity are to be estimated. Thermal property variation was assumed to be ( k k )/k = 0.66 and ( C C )/C =.. An optimality criteria was presented and optimal conditions, including sensor location, heating duration, and heating magnitude (or temperature range), were derived for a one dimensional finite and semi-infinite body. Two 4

25 approaches were considered. In the first approach a single experiment was used to estimate all four parameters. The second approach studied only the finite case, combining two experiments with different temperature ranges. The finite case produced larger values of D-optimality than the semi-infinite case when estimating all four parameters from a singe experiment. Sensors were located at the surface of the applied heat flux and below the heated surface. The semi-infinite case was not sensitive to the location of the sensor below the heated surface. In the finite case, optimal conditions were comparable for sensors located below the heated surface at one-half and one-fourth the specimen thickness; a sensor located at three-quarters of the specimen thickness was less optimal. For both the finite and semi-infinite cases the optimality criteria increases as the temperature range of the experiment increases relative to the temperature range of the property variation. The finite case was further studied to determine the combination of two experiments that was optimum. Two experiments that cover the entire temperature range of the properties are more optimum than experiments covering a subset of the temperature range. 8.0 References Beck, J. V., 964, The Optimum Analytical Design of Transient Experiments for Simultaneous Determinations of Thermal Conductivity and Specific Heat, Ph.D Dissertation, Michigan State University, East Lansing, MI. Beck, J. V., 966, Analytical Determination of Optimum, Transient Experiments for Measurement of Thermal Properties, Proceedings of Third International Heat Transfer Conference, Chicago, IL, American Institute of Chemical Engineers, New York, Vol. IV, paper number, pp Beck, J. V., 969, Analytical Determination of High Temperature Thermal Properties of Solids Using Plasma Arcs, Thermal Conductivity, Proceedings of the Eighth Conference, Purdue University, West Lafayette, IN, Plenum Publishing, New York, pp Beck, J.V. and Arnold, K.,977, Parameter Estimation in Engineering and Science, Wiley, New York. 5

26 Beck, J.V., and Osman, A.M., 99, Sequential Estimation of Temperature-Dependent Thermal Properties, High Temperature-High Pressure, Vol. 3, pp Blackwell, B. F., Cochran, R. J., and Dowding, K. J., 998, Development and Implementation of Sensitivity Coefficient Equations For Heat Conduction Problems, ASME proceedings of the 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, Vol., eds. A. Emery et al., ASME-HTD-357-, pp Dowding, K. J. and Blackwell, B. F., 999, Sensitivity Analysis for Nonlinear Heat Conduction, to be presented at 33rd National Heat Transfer Conference, Session 30, August 5-7, 999, Albuquerque, NM. Dowding, K. J., and Beck, J. V., and Blackwell, B. F., 998, Estimating Temperature-Dependent Thermal Properties of Carbon-Carbon Composite, 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, AIAA paper , June 5-8, 998, Albuquerque, NM. Dowding, K., Beck, J., Ulbrich, A., Blackwell, B., and Hayes, J., 995, Estimation of Thermal Properties and Surface Heat Flux in a Carbon-carbon Composite Material, Journal of Thermophysics and Heat Transfer, Vol. 9, No., pp Dowding, K. J., Beck, J., V., and Blackwell, B. F., 996, Estimation of Directional-Dependent Thermal Properties in a Carbon-Carbon Composite, International Journal of Heat and Mass Transfer, Vol. 39, No. 5, pp Emery, A. F., and Nenarokomov, A. V., 998, Optimal Experimental Design, Measurement Science and Technology, Vol. 9, pp Emery, A.F. and Fadale, T. D., 997, Handling Temperature Dependent Properties and Boundary Conditions in Stochastic Finite Element Analysis, Numerical Heat Transfer, Part A, Vol. 3, pp Emery, A.F. and Fadale, T. D., 996, Design of Experiments Using Uncertainty Information, ASME Journal of Heat Transfer, Vol. 8, pp Huang, C. H., and Ozisik, M. N., 99, Direct Integration Approach for Simultaneously Estimating Temperature Dependent Thermal Conductivity and Heat Capacity, Numerical Heat Transfer, Part A, Vol. 0, pp Huang, C. H., and Yuan, Y. Y., 995, An Inverse Problem in Simultaneously Measuring Temperature-Dependent Thermal Conductivity and Heat Capacity, International Journal of Heat and Mass Transfer, Vol. 38, No. 8, pp Flach, G. P., and Ozisik, M. N., 989, Inverse Heat Conduction Problem of Simultaneously Estimating Spatially Varying Thermal Conductivity and Heat Capacity per Unit Volume, Numerical Heat Transfer, Part A, Vol. 6, pp Incropera, F. P., and Dewitt, D. P, 990, Introduction to Heat Transfer, second edition, Wiley, New York. 6

27 Jin, G., Venkatesan, G., Chyu, M.-C., and, Zheng, J.-X., 998, Measurement of Thermal Conductivity and Heat Capacity of Foam Insulation During Transient Heating, AV-7569, Final Report, Department of Mechanical Engineering, Texas Tech University, May 0, 998. Loh, M. and Beck, J. V., 99, Simultaneous Estimation of Two Thermal Conductivity Components from Transient Two-Dimensional Experiments, paper 9-WA/HT- presented at ASME Winter annual meeting, Atlanta, GA, December -6, 99. Lesnic, D., Elliott, L., Ingham, D. B,. 996, Identification of the Thermal Conductivity and Heat Capacity in Unsteady Nonlinear Heat Conduction Problems Using the Boundary Element Method, Journal of Computational Physics, Vol. 6, pp Maddren, J., Parks, J., Marschall, J., and Milos, F. S., 998, Thermal Property Parameter Estimation from Lagged Temperature Data, 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, paper 98-93, June 5-8, 998, Albuquerque, NM. Sawaf, B., Ozisik, M. N., and Jarny, Y., 995, An Inverse Analysis to Estimate Linearly Temperature Dependent Thermal Conductivity Components and Heat Capacity of an Orthotropic Medium, International Journal of Heat and Mass Transfer, Vol. 38, No. 6, pp Scott, E. P., and Beck, J. V., 99a, Estimation of Thermal Properties in Epoxy Matrix/Carbon Fiber Composite Materials, Journal of Composite Material, Vol. 6, No., pp Scott, E. P., and Beck, J. V., 99b, Estimation of Thermal Properties in Epoxy Matrix/Carbon Fiber Composite Materials During Curing, Journal of Composite Material, Vol. 6, No., pp Taktak, R., Beck, J.V., and Scott, E., 993, Optimal Experimental Design for Estimating Thermal Properties of Composite Materials, International Journal of Heat Mass Transfer, Vol. 36, No., pp Taktak, R., 99, Design and Validation of Optimal Experiments for Estimating Thermal Properties of Composite Materials, Ph.D thesis, Michigan State University, East Lansing, Michigan. Woodbury, K. A., and Boohaker, C. G., 996, Simultaneous Determination of Temperature- Dependent Thermal Conductivity and Volumetric Heat Capacity by an Inverse Technique, Proceedings of the ASME Heat Transfer Division, HTD-Vol. 33, Volume, pp