JP5506624B2 - Thermal property identification method of multilayer laminates by unsteady heating - Google Patents

Description

  The present invention relates to a method for identifying thermal properties such as thermal conductivity and thermal diffusivity of a sample, and more particularly to a method for identifying thermal properties such as thermal conductivity and thermal diffusivity of each layer of a multilayer laminate.

  A flash heating method is a method for accurately measuring the thermal characteristics of solid materials such as metals, ceramics, glass, carbon, and plastics. In this flash heating method, the surface of a sample is irradiated with laser light instantaneously (flash) to heat, and the heat transmitted to a place away from a certain distance is measured from a temperature change. For example, there is a device that instantaneously heats one side of a small disk-shaped sample having a diameter of 10 mm and a thickness of 1 to 3 mm with a laser beam to measure heat from the temperature behavior on the opposite side. If the density is known, the thermal conductivity can be calculated.

  On the other hand, as a trend of new material development in recent years, materials with new functions that have never existed before have been developed by combining multiple materials, especially high thermal conductivity. Substrates are drawing attention. However, in the department of evaluation and measurement required in this field, a method for directly measuring the thermophysical properties of each layer of the completed new functional material has not yet been established.

  In the case of measuring the thermophysical properties of such a multilayer sample, conventionally, the thermal conductivity of the entire multilayer sample is measured by measuring the thermal conductivity by a steady method. In this measurement method, one side of the sample plate is heated, the opposite side is cooled, a temperature gradient is provided in the thickness direction of the sample plate, and the thermal conductivity is calculated from the temperature difference, the sample thickness, and the heat flux penetrating the sample. .

  As an attempt to measure the thermophysical properties of each layer, it has been proposed to use the above flash heating method. That is, the evaluation error (potential) is a square error between the temperature response on the back side obtained by irradiating the laser flash from the surface side of the multilayer material including the unknown layer and the theoretical temperature response due to the thermal properties of the multilayer material. By obtaining a parameter that minimizes the evaluation function, a thermal constant such as an unknown thermal diffusivity and specific heat in one layer of the multilayer material is determined with high accuracy and efficiency (for example, Patent Document 1). reference).

JP-A-8-261967

  As described above, the temperature response on the back side obtained by irradiating the laser flash from the front side of the multilayer material including one layer whose thermal diffusivity and specific heat are unknown, and the thermal diffusivity, specific heat, and biot number are included as variables. It has been proposed to identify the thermophysical properties of multilayer laminates by comparing the theoretical temperature response, but with this method, the thermophysical properties can only be determined when the thermal diffusivity and specific heat of the multilayer material are unknown. It can only be identified, and it is difficult to identify when all the thermophysical values of the substances constituting the laminated sample are unknown.

Hereinafter, it is possible to determine each parameter, that is, the thermophysical property value by the least square method using the flash heating method for each combination of unknown thermophysical property values, with the shape (thickness) of each sample constituting the two-layer laminated sample being known. Will be explained.
Define evaluation function as independent variables the unknown parameters (potential) in the following J _P, if this potential has a single extreme, fitting an analytical solution curves to raise the temperature curve observed in the downhill simplex method By doing so, a parameter, that is, a thermophysical property value can be determined.
J _P (λ ₁ , α ₁ , λ ₂ , α ₂ ) = ∫ {Y _P (t) −y _P (t, λ ₁ , α ₁ , λ ₂ , α ₂ )} ² dt
However, Y _P (t) represents observation signal data obtained by flash heating, and y _P (t, λ ₁ , α ₁ , λ ₂ , α ₂ ) represents an analytical solution to be fitted.

The analytical solution y _P (t, λ ₁ , α ₁ , λ ₂ , α ₂ ) is expressed by θ _r (t) in the following Equation 1. In Equation 1, where λ, α, and d are the thermal conductivity, thermal diffusivity, and sample thickness, b is the thermal permeability, b = λ / √α, τ is the characteristic time, and τ = d ² / α. Subscripts 1 and 2 represent physical property values of the first and second layers.

However, s _k is the kth solution of Equation 2 below, _where s is a complex number.

Here, the first layer (sample 1) is copper (λ = 398 W / mK, α = 117 mm ² / s, thickness 1 mm), and the second layer (sample 2) is aluminum (λ = 237 W / mK, α = 96.8 mm ² / s, thickness 1 mm), and the maximum value of the temperature rise waveform is assumed that there is no heat dissipation from the sample (that is, the biot number is zero), and is known from the observation data Y _P (t) The analytical solution calculated by the above equation 1 when λ ₁ = 398 mK, α ₁ = 117 mm ² / s, λ ₂ = 237 W / mK, α ₂ = 96.8 mm ² / s is observed data Y _P (t), the thermal conductivity λ ₁ of sample ₁ and the thermal conductivity λ ₂ of sample 2 are unknown parameters, and λ ₁ is changed in the range of 398 ± 50 W / mK and λ ₂ is changed in the range of 237 ± 50 W / mK. In this case, the potential _JP is as shown in FIG. 1, and it is difficult to identify the extreme value. Note that the number of series sums of the analytical solution of Equation 1 is 40.

The thermal diffusivity alpha ₁ of the sample 1, the thermal diffusivity alpha ₂ of the sample 2 as unknown parameters, alpha ₁ and 117 ± 20 mm ² / s, varied between alpha ₂ to 96.8 ± 20 mm ² / s In this case, the potential _JP is as shown in FIG. The number of series sum of analytical solutions was 40.

In this way, when the potential _JP is used as an evaluation function, it is expected that the minimum value is not seen and the solution becomes indefinite. In addition, the landscape of the evaluation function differs depending on how to take the series sum of the analytical solution, the multimodality is seen in the minimum value, and the problem that the thermal physical property identification method cannot be handled by the downhill simplex method alone arises.

  The present invention was devised to solve the above-described problems, and unsteady heating capable of identifying thermophysical values even when all thermophysical values of substances constituting the laminated sample are unknown. An object of the present invention is to provide a thermophysical property identification method for a multilayer laminated material.

  The method for identifying thermophysical properties of a multi-layer laminate by unsteady heating according to claim 1 is a deviation between a temperature rise signal on the back side of a sample obtained by flash heating and an analytical solution of flash heating using the thermophysical properties of the sample as a parameter. The deviation between the temperature rise signal on the back of the sample obtained by step heating and the analytical solution of step heating using the thermophysical properties of the sample as an evaluation function is used as an evaluation function. The thermophysical property of each layer is determined.

  In addition, the method for identifying thermophysical properties of a multilayer laminate material by non-stationary heating according to the invention of claim 2 is the method for identifying the thermophysical properties of a multilayer laminate material by non-stationary heating of the invention according to claim 1, The downhill simplex method is used for the algorithm, and a particle filter that generates a particle distribution that includes multiple particles that are candidates for the thermophysical properties of each layer was used to support the extreme value of the evaluation function having multimodality. It is characterized by that. The particle filter prepares a plurality of prediction models, compares observation data with likelihood, and selects an appropriate prediction model to identify a highly accurate state.

  In the present invention, since the solution becomes indefinite only by the flash heating information, step heating information is further added as an incidental condition. That is, if the evaluation function (potential) with four thermophysical values as parameters has a single minimum value, it is a downhill simplex method, and if it has a multi-peak extreme value, a random element is added, for example, a particle filter, etc. The parameter, that is, the thermophysical property value can be determined by fitting the analytical solution curve group to the observed temperature rise curve so that the evaluation function is minimized.

Thermal conductivity lambda ₁ of the sample 1 as the unknown parameter _is a graph showing the potential J _P when you choose a thermal conductivity lambda ₂ of the sample 2. Thermal diffusivity alpha ₁ of the sample 1 as the unknown parameter _is a graph showing the potential J _P when selected thermal diffusivity alpha ₂ of the sample 2. It is a figure which shows the structure of the apparatus which implements the thermophysical property identification method of the multilayer laminated material by unsteady heating of this invention. It is a figure which shows an example of the observation signal data and analytical solution which measured the temperature rise of the sample back surface at the time of performing flash heating. It is a figure which shows an example of the observation signal data which measured the temperature rise of the sample back surface at the time of performing step heating, and an analytical solution. Thermal conductivity lambda ₁ of the sample 1 as the unknown parameter _is a graph showing the potential J when you choose a thermal conductivity lambda ₂ of the sample 2. It is a graph which shows the potential J at the time of selecting the thermal diffusivity (alpha) _{1 of} the sample ₁ and the thermal diffusivity (alpha) ₂ of the sample 2 as an unknown parameter. It is a flowchart which shows the algorithm which implements the thermophysical property identification method of the multilayer laminated material of this invention.

FIG. 3 is a diagram showing a configuration of an apparatus for carrying out the thermal property identification method for a multilayer laminate material by unsteady heating according to the present invention, and measures the temperature of the sample 10 to be measured which is flash-heated or step-heated and the back surface of the sample 10. The radiation thermometer 20 is configured. Y _P (t) shown in FIG. 4 is a temperature rise curve when a laser pulse generator (not shown) is irradiated and heated with a laser beam having a pulse width of about 0.5 ms and the temperature rise on the back surface of the sample is measured. Further, Y _S (t) shown in FIG. 5 is an example of a temperature rise curve when the sample 10 is step-heated and the temperature rise on the back surface of the sample is measured.

Hereinafter, taking thermal property identification of a two-layer laminated material sample as an example, the thermal property values such as the thermal conductivity and thermal diffusivity of the sample are obtained from the information on the temperature rise signal on the back of the sample obtained by the flash heating method and the step heating method A case of identification will be described.
The analytical solution of the sample back surface temperature rise signal at the time of flash heating of the two-layer laminate is expressed by Equation 1 as described above, but the analytical solution of the back surface temperature rise signal due to step heating of the two-layer laminate is represented by Equation 3 below. It is represented by However, Q is the magnitude of step heating, and x _k is the kth solution of Equation 4 below.

Next, a method for determining the thermophysical property values (parameters) by the least square method for a combination of unknown thermophysical property values, assuming that the shape (thickness) of each sample constituting the two-layer laminated material sample is known, will be described.
That is, an evaluation function (potential) J with four thermophysical values as parameters is defined as follows. If this potential J has a single minimum value, it has a multimodal extreme value by the downhill simplex method. Then, taking the probability factor into account, for example, adopting a method such as a particle filter, and fitting the analytical solution curve to the observed temperature rise curve so that the evaluation function is minimized, the parameter, that is, the thermophysical property value Can be determined.
J = W _S J _S + W _P J _P
W _S and W _P are weighting factors.

The above J _S is represented by the following formula.
J _S (λ ₁ , α ₁ , λ ₂ , α ₂ ) = ∫ {Y _S (t) −y _S (t, λ ₁ , α ₁ , λ ₂ , α ₂ )} ² dt
Y _S (t) is the temperature rise curve observation data on the back surface of the step-heated sample, and y _S (t, λ ₁ , α ₁ , λ ₂ , α ₂ ) is the analysis in the case of step heating expressed by the above equation 3. The solution is indicated by y _S (t) in FIG. 5, and J _S is the integral value of the square error between the temperature rise curve observation data Y _S (t) and the analytical solution y _S (t).
Further, J _P is expressed by the following equation.
J _P (λ ₁ , α ₁ , λ ₂ , α ₂ ) = ∫ {Y _P (t) −y _P (t, λ ₁ , α ₁ , λ ₂ , α ₂ )} ² dt
As described above, Y _P (t) is the temperature rise curve observation data on the back surface of the sample heated by flash heating, and y _P (t, λ ₁ , α ₁ , λ ₂ , α ₂ ) is expressed by Equation 1 above. 4, which is indicated by y _P (t) in FIG. 4, and J _P is 2 between the temperature rise curve observation data Y _P (t) and the analytical solution y _P (t). This is the integral value of the power error.

Here, the first layer is copper (λ = 398 W / mK, α = 117 mm2 / s, thickness 1 mm), and the second layer is aluminum (λ = 237 W / mK, α = 96.8 mm2 / s, thickness 1 mm). And using the analytical solutions calculated by Equation 3 and Equation 1 using the true values of copper and aluminum as observation data Y _S (t) and Y _P (t), λ ₁ is 398 ± 50 W / mK, λ ₂ Is changed in a range of 237 ± 50 W / mK, the evaluation function J is as shown in FIG. 6 and multimodality is recognized, but an extreme value appears.

The evaluation function J using the thermal diffusivity α ₁ of the sample ₁ and the thermal diffusivity α ₂ of the sample 2 as parameters is as follows: α ₁ is 117 ± 20 mm ² / s, and α ₂ is 96.8 ± 20 mm ² / s. When changed in the range, it becomes as shown in FIG. 7, and the multimodality of potential seems to disappear.

  As shown above, including not only flash heating, but also step heating signals, it can be seen that the instability of thermal conductivity is eliminated, and the multimodality of extreme values adds a probabilistic element to the search algorithm. It is thought that the problem can be solved by bringing it in. The following is an example of using an algorithm in which a particle filter technique is added to the downhill simplex method as an algorithm for identifying thermophysical properties by actual measurement. explain.

In the apparatus shown in FIG. 3, the temperature rise curve observation data YP (Y _P ( Next, after the sample 10 is step-heated to obtain the temperature rise curve observation data Y _S (t) obtained by measuring the temperature rise on the back side of the sample, the program of the flowchart of FIG. 8 is executed.

When the program of the flowchart of FIG. 8 is started, first, an appropriate state vector x ₀ is set as an initial value of the thermophysical properties of the two-layer laminated material (step 101). The state vector referred to here is a quaternary vector representing λ ₁ , α ₁ , λ ₂ , α ₂ .
Thereafter, an evaluation function (potential) J at the state vector x ₀ is calculated from the temperature rise curve observation data and the analytical solution, and stored as J ₀ (step 102).

Next, m particles having a dispersion centered on the state vector x ₀ are uniformly scattered (step 103). Each particle represents various state vectors.
Thereafter, the evaluation function J is calculated using the state vector and the observation data of each particle, and the extreme value of the evaluation function of each particle is traced by the downhill simplex method (step 104). The downhill simplex method is a method widely used as a method for obtaining a local minimum value of a multidimensional function. Simplex (if described in N dimensions, N + 1 vertices and sides, surfaces connecting them) The local minimum value of the multidimensional function can be obtained by sequentially updating the polyhedron formed from

After each particle reaches the extreme value of the evaluation function, each particle is weighted by the reciprocal of the extreme value (potential) (step 105), and then the center of gravity of all the weighted particles is obtained, and a new state vector is obtained. x is set (step 106).
Was then calculated evaluation function J ₁ in the new state vector x (Step 107) after determining whether the difference between the J ₀ and J ₁ is sufficiently small (step 108), the J ₀ and J ₁ If the difference is large, J ₁ is replaced with J ₀ (step 109), and then the process returns to step 103 and m particles having dispersion centered on the state vector x are uniformly scattered.
On the other hand, if it is determined in step 108 that the difference between J ₀ and J ₁ is sufficiently small, it is determined that the state vector has converged, the current state vector x is taken as a solution (step 110), and the program is terminated.

Next, the thicknesses d ₁ and d _{2 of} the samples 1 and ₂ are known, λ ₁ , α ₁ , λ ₂ , and α ₂ are unknown, and the analytical solution calculated with the target value is used as the temperature rise curve observation data. The result of calculating the evaluation function from the analytical solution based on each parameter and searching for the optimal solution using the search algorithm will be described.
The following soft algorithm was executed under the conditions shown in Table 1 below.
1. Gives the initial value of the state vector x.
2. 100 particles distributed around the state vector x are dispersed by the dispersion SigmaW.
3. The evaluation function of each particle is calculated, and the extreme value of the evaluation function is tracked by the downhill simplex method.
4). Each particle is weighted by the inverse of the ultimate value.
5. The center of gravity of all particles is obtained and set as a new state vector x.
6). Repeat steps 2-5 until convergence.
7). After convergence determination, the state vector that gives the minimum value is taken as the solution.

  The search result of the optimum solution by the above search algorithm is as shown in Table 2 below, and although it depends on the initial value, it has reached the correct solution in three trials. Note that the minimum value X in Table 2 is a solution of the state vector indicating the minimum value of the extreme value that appears during the execution of the calculation. Thus, by obtaining the state vector indicating the minimum value, It is also possible to deal with a case where there is a solution at a position away from the set state vector.

Further, as a common example, when the thermophysical value of sample 1 and the thickness of the entire sample are known, and the thickness d ₂ , thermal conductivity λ ₂ , and thermal diffusivity α _{2 of} sample ₂ are unknown, the following table The following soft algorithm was executed under the condition of 3.
1. Gives the initial value of the state vector x.
2. Ten particles distributed around the state vector x are scattered by the dispersion SigmaW.
3. The evaluation function of each particle is calculated, and the extreme value of the evaluation function is tracked by the downhill simplex method.
4). Each particle is weighted by the inverse of the ultimate value.
5. The center of gravity of all particles is obtained and set as a new state vector x.
6). Repeat steps 2-5 until convergence.
7). After convergence determination, the state vector that gives the minimum value is taken as the solution.

  The search result of the optimum solution by the above search algorithm is as shown in Table 4 below, and it is very convergent and has no stopping point, and can reach a sufficiently correct solution with about 10 particles. As described above, the minimum value X in Table 4 is a solution of a state vector indicating the minimum value of the extreme value that appears during the execution of the calculation.

Further, the following soft algorithm was executed under the conditions of Table 5 below, taking as an example the case where the four thermophysical values of the two samples are unknown and the sample shape is only known as the total thickness L.
1. Gives the initial value of the state vector x.
2. 100 particles distributed around the state vector x are dispersed by the dispersion SigmaW.
3. The evaluation function of each particle is calculated, and the extreme value of the evaluation function is tracked by the downhill simplex method.
4). Each particle is weighted by the inverse of the ultimate value.
5. The center of gravity of all particles is obtained and set as a new state vector x.
6). Repeat steps 2-5 until convergence.
7). After convergence determination, the state vector that gives the minimum value is taken as the solution.

  The search result of the optimum solution by the search algorithm is as shown in Table 6 below, and it can be seen that the accuracy can be identified although the accuracy is lowered. Similarly to the above, the minimum value X in Table 6 is a solution of the state vector indicating the minimum value of the extreme value that appeared during the execution of the calculation.

  One of the reasons why it was difficult to identify the thermophysical properties of laminated materials is that the potential does not have an extreme value and may become indeterminate, and the other is that the extreme value is not unimodal but multimodal. However, according to the thermal property identification method for multilayer laminates by non-stationary heating according to the present invention, multi-modality can be obtained by adding step heating information as an additional condition to the objective function for indefiniteness. Can be solved by bringing probability into the minimum search algorithm.

In the algorithm of the above embodiment, the center of gravity of the weighted particle group is obtained and used as a new state vector. However, a state vector that gives the minimum value of the evaluation function can be used as a new state vector.
In the above embodiment, the downhill simplex method is used to search for extreme values, but a hill-climbing method or the like can also be used. Further, although the particle filter algorithm is used as the stochastic element, it can be replaced with a genetic algorithm.
Furthermore, in the above embodiment, the square error is used to obtain the deviation between the observation data and the analytical solution. However, the sum of the absolute values of the errors can also be used as the deviation.

  In the above example, an analytical solution was used for comparison with the observation data. However, when an analytical solution is not obtained, many of these types of problem solving methods use Laplace transform to solve the problem. It is also possible to use a solution obtained from Laplace inversion by numerical analysis.

  Further, in the above embodiment, the step heating signal is used as an additional condition. However, since the reciprocal of the slope of the straight line in the graph shown in FIG. 5 indicates the total heat capacity of the sample, the step heating signal is measured separately from the total heat capacity of the sample. It may be replaced. However, in this case, since the information amount of the additional condition is reduced, the number of unknown variables that can be naturally determined is reduced as in the examples shown in Tables 3 and 4, for example.

10 samples 20 radiation thermometer

Claims (2)

  Deviation between the temperature rise signal on the back side of the sample obtained by flash heating and the analytical solution of flash heating using the thermal properties of the sample as a parameter, and the temperature rise signal on the back side of the sample obtained by step heating and the thermal properties of the sample The deviation from the analytical solution of step heating as a parameter is used as an evaluation function, and the thermal properties of each layer of the multilayer laminate are determined from the parameter that minimizes this evaluation function. Thermophysical property identification method.   In the thermophysical property identification method for multi-layer laminates by unsteady heating according to claim 1, the downhill simplex method is used for the extremum search algorithm of the evaluation function, and the extremum of the evaluation function has multimodality. Therefore, a thermophysical property identification method for a multilayer laminated material by unsteady heating, which uses a particle filter that generates a particle distribution including a plurality of particles that are candidates for thermophysical properties of each layer.

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