JPS6020697B2 - Measuring method of thermophysical properties by arbitrary heating - Google Patents

Description

[Detailed description of the invention]

This invention measures thermophysical property values such as thermal conductivity, temperature conductivity, and heat capacity under arbitrary heating input under arbitrary boundary conditions, and under arbitrary initial conditions with the initial temperature distribution within certain limited conditions. Regarding how to. Thermophysical property values are important in almost every production industry and in the academic field of natural science, including the industrial field where energy countermeasures are urgently needed. The methods used to measure thermophysical properties are roughly divided into steady methods and unsteady methods based on analytical solutions of basic heat conduction equations. Although the steady state method is widely used, there are practical difficulties in maintaining the temperature at the required steady field, and measurement requires a long time and skill at high temperatures. On the other hand, unsteady methods based on analytical solutions have been greatly developed recently, but care must be taken to experimentally realize the ideal boundary conditions used to obtain the analytical solutions, and in general, Equipment becomes complex and expensive. The common feature of the above methods is that the boundary conditions of the sample are set to steady values,
It is necessary to determine the ideal shape such as step shape,
This is a factor that makes it extremely difficult to measure thermophysical properties. Therefore, in order to meet the recent demands for thermophysical property data, it is necessary to propose a method that eases the need to determine boundary conditions. One such method is a method based on numerical calculations. However, this method not only has limited arbitrariness, but also requires quite complex calculations, so it is currently difficult to imagine that this method will become a general method. Furthermore, methods similar to the principle of the present invention that have been proposed in the past have only been shown for semi-infinite solids. Since the heat flow rate is determined from the average temperature change of the object, the boundary conditions cannot be said to be completely arbitrary, and furthermore, the measurement range is limited and there are drawbacks such as large errors such as heat loss. Therefore, in contrast to conventional methods, if thermophysical property values can be obtained in a system where the boundary conditions are completely arbitrary and the heating conditions are completely arbitrary, using the procedures normally performed, it is possible to improve the accuracy of the equipment and method. A significant improvement is expected. The present invention relates to such a method, which has a wide range of applicable conditions, a simple device, and an easy method, and also guarantees high accuracy. The basic principle of simultaneous measurement of thermophysical properties using the flat sample shown in FIG. 1 will be explained as follows. According to Figure 1, standard sample [1] with thickness L (thermal conductivity ^
1. The temperature conductivity al, heat capacity plcl known) and the thickness of the test sample [b] (^0, ao, po, co unknown) are in contact, the heat flow is perpendicular to the plane, and the coordinates are Do as shown. The basic equation for heat conduction is, where the temperature is T and the time is t, 6Tji・ri=a6thing kiln, t) --...----'118tThe initial temperature distribution is T(x, o), and the formula {2} Considering the temperature difference, equation {1} becomes equation {3}. 8(x,t)=T(x,t)−T(x,o).・・・．・
...{2) 68 Furi = a62 Yume, t) 10a62 Morikasa's ......'3} 6T type lake L = T (o, o)
= constant, T(X,.) nim is 19......{4) In other words, if the initial temperature distribution is uniform (m=o) or linear, then 6
Kagoto, t) = a62 Hair Kasari - - - - - - - - ■ Applying the Laplace transformation by formula t, substituting 8 (x, o) = 0, and converting it into an ordinary differential equation, we get d28 1≦s=o
......{6}dX2 aHere,
s is a Laplace parameter and a Laplace integral defined by equation (7). 8 Nino skin e vs. 8 (x, t) dt ......{7) Formula (
6} is given by the formula ■. 8=AezoS'aY&~zoryo A×+B×-1...
・・・・・・(8}However, X3ezoSnoaX
………{9) Furthermore, position i (i=
-1, 0, 1, 2), the Laplace integral 8i of the temperature response ai(t) is calculated using equation {10}. 8 i ni′ togethereNa i(t)dt …,. ．． ．． ，
．． {i0) On the other hand, the heat flow east q(x, t) is the Fourier equation q
ku
) is Labrass transformed, q=-＾Shiba...
...... (12) Substituting the formula shelf into equation (12), q = entry no s/a (1 Aef A do technique - Gua...)
(13) The above are the basic relationships of the method. First, let us consider the inside of the standard sample [1], and position i:1
, 0 using equation 00 and substituting it into equation (■), the constants of integration AI and BI are determined. Substituting them into equation (13), the position i=0, that is, (qo) 1 at the boundary surface, is determined. Find (q.) 1:1 times {80(X-・11/X-・)-28-・
} ,． ．． 、． ．． X-'1/X-. Here, ・×"=e-rF2X-1.Furthermore, for [B], find (qo)0 in the same way as above for i=0 and 1 (or 2), then (q.)0= River Samurai Nozomi {Misaki・Jikō} - Kyou} X.-1/X. . . . (15) Here, = (qo)0, so equation (14)
If we organize ``and'' (15) with the same eyes, we can see that tsuba=dōpāyōshi,）Metaizō sensō kenko.・・・． (16) Next, aD' is determined from the temperature response at three positions within the test sample, regardless of the presence or absence of a standard sample. That is, in this case, 8o, 8,, a2 are each x=0
, x, , by substituting it into Equation (8) in correspondence with the sea and rearranging it, Equation (17) can be easily obtained. vinegar. (Nursing - receiving) 10s. (X2-house) 10s 2ku1-X. )
=○・・・・・・・・・(17) Formula XI (
In 16) and (17), the unknown quantities are the arrival of the sun, ao, and s. But rough. Due to the nature of the lath transformation, s can take any finite positive value as long as the equation converges, and in practice it is determined as described below, so the inputs ao and ao are determined. Further, the heat capacity pD,co can be found from equation (18). P port C port = entrance/a port. ．． ．． ．． ．． ．． ．． ．． ．．
(18) Although the principle has been explained above for an infinite flat sample, the same method can be applied to other one-dimensional systems, such as semi-elutrionic solid samples, infinite cylindrical or tubular samples, and spherical samples. can. However, in contrast to the exponential function in a rectangular coordinate system, the Bessel function is used in a cylindrical coordinate system, and the Lujendre function is used in a spherical coordinate system. For example, the basic principle of measuring thermophysical properties of a cylindrical or hollow cylindrical sample is the same as for the flat sample described above for the system shown in Figure 2. ] is used as a standard sample, the formula for calculating the temperature conductivity a is 8,1 from formula (19). (nos/aDr2)-821. (Zotesu bothro r,) = 0
・ ・・・・・・・・・(19) Also, from equation (20), the formula for calculating the thermal conductivity input is as follows: (Nos/a1
r2)}.・・'． ...(20) It is derived that. Here, lo is the zero-order modified Bessel function of the first kind, 1, ,
K, is a linearly deformed Betzsell function of the first kind. AI,B
I is an integral constant determined separately. As is clear from the above explanation of the principle, as long as the Laplace integral of the temperature response at the required location is obtained, it is possible in principle to accurately measure thermophysical property values no matter how the boundary conditions change due to arbitrary heating conditions. . As described above, the Laplace integral in equation {10) requires integration up to t. However, e-SL in the equation is a function that converges to 0 as t becomes large. Further, 8i(t), that is, the magnitude of the temperature response necessary for measurement, needs to be kept within a certain finite value without losing the meaning of the thermophysical property value of the temperature. Therefore, e-St8i(t) is a function that inevitably converges to 0 as t becomes larger. Therefore, in measuring thermophysical properties, there is a measurement time interval ax in which equation (21) approximately holds true. Noga e-Sto: (t) dtNino is aXe heart 8i (t) dt1・1.・'...(21) As an ideal example, if we consider a step-like temperature response where t>0 and 8i(t)=0oo, it is clear that ′ is Xe−St0i(t
)dtnono〆e-St8i(t)dt=11e-Stm
aX ......(22),
It is shown that the degree of approximation of equation (21) depends on the size of s skin ax. Therefore, many numerical experiments were conducted to determine the Stm broadside x value and to verify the method, but an example will be described here in order to explain the content of the method in a simple manner. now,
In Figure 1, the temperature is constant at x=〆〆plane, that is, 6 days (t);
0, and a-L(t) has a step-like temperature response 8-L,o
Let's consider the case where we give Standard sample [1], test sample

Each thermophysical property value of [0] and x
-,,x. ,Ike,L,so is set in advance and from the analytical solution,8-,(t),8. (t), 8, (t), a2(t)
The results obtained are as shown by the solid lines in Figure 3. The thermophysical property values were obtained using this method assuming that each solid line was obtained experimentally. In addition, in FIG. 3, e-St, eMoo river is shown as an example, and 8o is shown as the area of the shaded area. In Figure 4, each 8 is obtained from each 8(t) in Figure 3 by numerical integration (sampling number N) using Simpson's method, and input 0 and aro are obtained from formula (16) and formula (17), respectively. The obtained results are shown relative to stmax. As shown in the figure, if the number of samplings N=200 and the accuracy of numerical calculation is good, if stmax is a certain value or more (here, about 7 or more), a result that sufficiently matches the set value is obtained. stm
The reason why ax differs from the set value in a small range is because the approximation of equation (21) does not hold. On the other hand, not much stma
If x is increased, e-St will converge to 0 in a small range of t, and data will be evaluated based on temperature response only within a short period of time, so it is desirable to set an upper limit. Similar results to the above were confirmed by many other numerical experiments and actual measurement experiments, regardless of whether the coordinate system was rectangular or cylindrical. As a result, it was found that if stmax is set within the range shown in Equation (23), the approximation of Equation (21) holds true in all cases, which is convenient for measurement. 8S
stmaxS12 (23) s may be arbitrarily selected as long as it is within the range of equation (23). stmax may be arbitrary as long as unsteady behavior is noticeable during that time and there is a change enough to compare each temperature response. In addition,
If it is within the range of Equation (23), it can be easily obtained with relatively good accuracy by panogram integration. Furthermore, the temperature response can be AD converted, and a microcomputer or the like can automatically perform Laplace integral and calculation. Also, stma
If it is necessary to determine x, equation (24) is recommended. Stmax=8 (24
) As described above, according to the present invention, it is not necessary to perform Laplace integral until an infinite time in practice, and the measurement time tmax
This makes it possible in principle to perform accurate measurements no matter how the boundary conditions change due to arbitrary heating. According to this invention, the thermophysical property values of thermal conductivity, temperature conductivity, and heat capacity can be measured simultaneously. In addition, the temperature conductivity ao can be determined from the basic relational expressions (17) for flat samples and (19) for cylindrical samples, but in this case, even if the standard sample is only the test sample, the required It can be determined from the temperature response of the position. In addition, if one boundary condition is fixed for samples of various shapes, such as isothermal, adiabatic, or constant heat input, the method of the present invention can be improved, such as by reducing the number of sideflow positions. Easy to apply. In addition, each physical property value can be obtained from the equation obtained by installing a heat flow meter on the boundary surface of the standard sample and isometrically calculating the Laplace integral of the heat flow east with equation (15). Furthermore, even if the initial temperature distribution is a quadratic curve, measurement is possible with simple correction. Furthermore, if the mode is appropriate, not only bodies but also liquids and gases can be used as test samples. In addition, examples of various easier and more practical measurement systems that are possible by setting measurement points on the sample surface and selecting boundary conditions in contrast to the general systems shown in FIGS. 1 and 2 will be shown. Table 1 lists examples of practical measurement systems that apply basic principles to flat samples, and also lists each relational expression. Table 1 Examples of Practical Measurement Systems Various applied measurement systems can also be considered for cylindrical and tubular samples. For example, in Figure 2, the outer periphery of sample [1] may be kept at a constant temperature or heat insulated, the central cylindrical portion may be used as a standard sample and the hollow cylindrical portion may be used as a test sample, or a cylindrical heating portion may be provided in the center and the surroundings may be Both can be made into hollow cylindrical test samples and standard samples. Furthermore, it can be applied to many cases such as providing a cylindrical heating section in an infinite body test sample. It should be noted that the number of side warming bodies varies depending on the needs in each case. Next, an example of the present invention will be described with reference to the drawings regarding a flat sample. FIG. 5 is a schematic diagram of the test section, which is the embodiment shown in the first column of Table 1. As test sample 1, two acrylic resin disks and a soda glass disk, each having a thickness of approximately 5 ribs and a diameter of 15 ribs, were closely attached. As standard sample 2, 3-yanagi-thick Pyrex glass was used. As the side heating element 3, hot junctions of 0.1 figure alumel/chromel thermocouples were set at various diamond positions in the center of the disk. On both sides of the sample, there is a Teflon plate 4 with a thickness of 1 inch for uniform adhesion.
3-legged thick copper plate 5 to make the temperature field uniform in the radial direction; 3-sided thick brass plate 6 with a slightly convex curvature toward the sample and a spring action in order to uniformly press the entire body; 6
Furthermore, an acrylic resin plate 7 with a specific thickness of 1 is provided to slow down the temperature response of the sample to a required degree, and these are attached to a fastener 8.
Tighten it evenly and set it. FIG. 6 is a system diagram of the measuring device. The test section a set as described above is set in the center of a permanent gutter box 10 having glass windows 9 on the upper and lower surfaces. The temperature inside the fence warming box 10 is maintained at a predetermined temperature by the slider 111. After confirming that the indicated values of the side feed bodies in the test section a are uniform, the sliders 12 and 13 are operated to supply energy from the infrared lamps 14 and 15 to the test section a.
amount. The thermoelectromotive force from the hot body thermocouple is recorded by a balanced recorder 18 via a cut-off switch 16 and a voltage compensator 17. Note that the indicated value of the digital voltmeter 19 may be recorded by the printer 20. FIG. 7 shows each temperature response 8,(t), obtained from the four leakage temperature positions marked with an x in FIG.
These are recording examples of 82(t), 83(t), and 84(t). The Laplace integral value 8i at each layer i (i = 1, 2, 3, 4) is calculated using equation (21), but for this numerical calculation, the analog recording amount in Figure 7 is converted to a digital amount. ,
We used the Simpson method, which is widely used for numerical integration. The thus determined oi (i=1, 2, 3, 4) was substituted together with the corresponding distance x, and integration constants A and B were determined for each sample. (qo) Town: The above integral constants A and B for each sample were substituted into the relational expression for determining the thermal conductivity input D obtained from (qo)1. Furthermore, an integral constant was substituted into the relational expression that determines the temperature conductivity ao. With the above, two equations are obtained, in which the unknowns are ^0,
There are three, a0 and s, but based on the properties of Laplace transform, s can be any number, so we determined the thermal conductivity 0 and the thermal conductivity ao. FIG. 8 shows the results of measurements on acrylic resin plates, where the experimental points are the results of experiments according to the present invention, and the shaded areas are "experimental values by Okada et al."
79 (1976) 247]. FIG. 9 shows the measurement results for a soda glass plate, and the solid line and dotted line indicate experimental values by Katayama et al. [Kiron, 34 (1968) 2012]. These results show good agreement with other experimental values and good reproducibility. On the other hand, when showing measurement examples using cylindrical and hollow cylindrical samples, the sample portion is schematically shown in FIG. 10, and FIG. 11 is a schematic diagram of the measuring device. The side heating body (marked with *) is on the central axis of the sample 1, at the interface between the sample 1 and the standard sample 2, and on the outer periphery of the standard sample 2 is a brass tube 2.
1, and a heating heater 22 is provided on the outer periphery of the brass tube 21.
and a heater 23 for heating during testing. Powdered alumina 24 is placed between the brass tube 21 and the standard sample 2.
to improve heat conductivity. In the experiment, 18-8 stainless steel was first measured as an example of measuring only the thermal conductivity. FIG. 12 is an example of temperature response recording in this case.
Figure 13. shows the results, which were in very good agreement with the literature value (TPRC) shown by the solid line, and the accuracy was within about 1%. Further, as an example of simultaneous measurement of thermal conductivity, temperature conductivity, and heat capacity, the measurement results of the thermophysical properties of alumina powder are shown in FIG. 14. Comparisons with literature values from the TPRC data book can only be made on the thermal conductivity for which data is available, but this is also in relatively good agreement. FIG. 15 shows the temperature conductivity of an acrylic resin plate in comparison with the temperature conductivity of a flat plate sample of the same material and the experimental value of Okada et al. [Kiken, 79 (1976) 247]. As a result, the experimental results obtained in this example were in good agreement with both. As is clear from the above explanation, according to the present invention, thermophysical property values can be measured simply by measuring the temperature response of each required part under any heating conditions and any heating conditions. Compared to conventional methods, this method not only requires simpler experimental equipment, but also requires no skill. In addition, compared to the conventional method, the results are evaluated using all the data within the measurement time, so it is possible to guarantee highly accurate values. This is an innovative method that makes it easier.

[Brief explanation of drawings]

Figure 1 is an explanatory diagram of the principle of simultaneous measurement of thermophysical properties in the case of an infinite body sample, Figure 2 is an explanatory diagram of the principle of simultaneous measurement of thermophysical properties of a cylindrical or hollow cylindrical sample, and Figure 3 is a diagram of temperature changes. Figure 4 is a diagram showing the relationship between s/tmax values and the number of samplings; Figure 5 is a schematic diagram showing an example of a test section for a flat sample; Figure 5 is a schematic diagram of the measuring device in the test section, Figure 8 is a diagram showing the temporal change curve of temperature (thermoelectromotive force) at the measurement point, Figure 8 is a diagram showing the thermophysical property values of the acrylic resin plate, Figure 9 10 is a schematic diagram showing an example of a test section for cylindrical and hollow cylindrical samples, and FIG. 11 is an example of a measuring device for the test section shown in FIG. 10. Figure 12 is a diagram showing an example of recording the temperature response of 18-8 stainless steel, Figure 13 is a diagram showing the temperature conductivity measurement results of the same as above, and Figure 14 is a simultaneous measurement of alumina powder. Figure 15 shows the results, comparing the thermal conductivity of an acrylic resin plate with the thermal conductivity of a flat plate sample of the same material and the experimental values of other researchers. Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 7 Figure 6 Figure 8 Figure 10 Figure 11 Figure 12 Figure 13 Figure 9 Figure 14 Figure 15

Claims (1)

[Scope of Claims] 1 (a) The standard sample [I] and the test sample [II] are brought into contact, and at least one point on the contact surface, the standard sample (including the surface opposite to the above-mentioned contact surface), Temperature response θi(t) at one or two points within the sample (including the surface opposite to the above contact surface)
(b) Determine the Laplace integral of the temperature response θi(t) as ≒∫^t^m^a^x_0θi(t)・
e^-^s^tdt...Calculated by the Simpson method from the approximate formula (1) (where s: Laplace parameter,
tmax: Measurement time, s and tmax are selected from values that satisfy the relationship 8≦s・tmax≦12) (c
) Next, the temperature distribution T(x, t
) and the initial temperature distribution T(x, o), the temperature difference θ(x, t)
Find the following general solution from the ordinary differential equation that is a positive conversion of the basic thermal conductor equation using , ■■=AX+B/X...
2) (However, A and B are constants, X=e√(s/(ax))(
d) The Laplace integral calculated in (b) above is expressed as (2)
Find the integral constants A and B for each sample by substituting into the equation, (e) On the other hand, the following equation (3) obtained by Laplace transform of the Fourier equation expressing the heat flux, q = -λ (dθ)
/(dx)...(3) (where λ is thermal conductivity) Substitute the above equation (2) into the equation (3) to obtain the following equation (4), q=λ√(s/a) (AX+B/X)...(4
) (f) Next, by substituting the integral constants A and B of each sample obtained in (d) above into equation (4), we can calculate (q_0 ) II and (q
_0) Find I, (g) (q_0) II = (q_0)
I... From the relationship (5), the thermal conductivity λ of the test sample [II]
Obtain the relational expression that defines II, (h) Furthermore, the above (a), (
Substitute the Laplace integral value of the temperature response at 2 or 3 points within the test sample [II] and the contact surface measured in b) into the above equation (2), rearrange it, and calculate the temperature conduction of the test sample [II]. Obtain the relational expression that determines the rate aII, (i) Determine the thermal conductivity λII and the temperature conductivity aII from both the relational expressions obtained in (g) and (h) above,
(j) pIIcII=(λII)/(aII)...(6) Substitute λII and aII obtained in (i) above into the above equation (6) to find the heat capacity pIIcII of the test sample [II]. A method for measuring thermophysical property values by arbitrary heating, which is characterized by determining thermophysical property values such as thermal conductivity, temperature conductivity, and heat capacity using the following procedure.