JP4743781B2 - Method, apparatus, and computer program for estimating temperature and heat flux of inner wall surface of container - Google Patents

Description

  The present invention relates to a method, an apparatus, and a computer program for estimating the temperature and heat flux of an inner wall surface in a container having a temperature difference between an inner wall surface and an outer wall surface.

  Reaction vessels with high temperature gas reaction or liquid reaction, such as blast furnaces, converters, degassing furnaces, steel heating furnaces by combustion, coal gasification reactors, and containers for transporting molten iron, such as kneading cars, hot metal ladle, molten steel pans, etc. In the case of managing the operation using the container, it is necessary to observe and manage the state of the wall of the container that handles these high-temperature substances (for example, the worn state of the inner wall surface).

  Conventionally, the wear state of the container wall has been managed by visually observing the state of the inner wall surface when there is no high-temperature material such as molten steel in the container.

  However, even when there is no high temperature substance in the container, the surface of the refractory is heated to a high temperature of 500 ° C. or more immediately after the discharge of the high temperature substance. In the visual estimation as described above, it is extremely difficult to grasp the wear state as a quantitative numerical value, and qualitative management is unavoidable. In addition, since the management is forced on the condition that no high-temperature material is present in the container, the situation of the outflow of the high-temperature material during operation cannot be managed.

Here, the worn state of the container wall can be determined by the thickness of the container wall. For example, when the wear can be approximated to a one-dimensional shape with a uniform shape, if the container wall is in a thermally steady state, the temperature distribution inside the container wall is linear. Therefore, the thickness L of the container wall uses the heat flux Q measured on the outer wall surface of the container, the thermal conductivity k _x in the thickness direction of the container wall, the inner wall surface temperature T _{in of} the container, and the outer wall surface temperature T _out of the container. And can be estimated from the following equation (51).

  However, the actual temperature of the container wall shows different values depending on the time cycle of operation (high-temperature material is present in the container) and non-operation (no high-temperature material is present in the container). The measured heat flux Q also changes unsteadily. In addition to this, since the temperature distribution inside the container wall changes unsteadily in a curved shape, estimating the thickness of the container wall with the above equation causes a large error.

  On the other hand, considering the heat conduction phenomenon inside the container wall as an unsteady heat conduction inverse problem, based on the temperature data measured by the temperature measurement means installed on the container wall, the temperature inside the container wall by the unsteady heat conduction problem And a method of estimating the thickness of the container wall by searching for a position where the temperature of the container wall matches the solidification temperature of the molten iron has been proposed (see Patent Document 1).

JP 2001-234217 A

  However, the inverse problem analysis disclosed in Patent Document 1 is performed by assuming the initial temperature inside the container wall when estimating the thickness of the container wall. For this reason, if the initial temperature setting is not appropriate, a large error will occur in the temperature calculation result, causing a significant decrease in calculation accuracy. In some cases, the calculation may diverge and the calculation may be interrupted. .

  The present invention has been made in view of the above points, and in the case of estimating the container wall thickness in a state where a high-temperature substance exists in the container, the initial temperature inside the container is assumed as in the conventional method. Even if it is not, the aim is to improve the estimation accuracy by uniquely determining the initial temperature distribution inside the material of the container wall, and accurately calculate the temperature and heat flux of the inner wall surface of the container, which is necessary information for that purpose. The purpose is to be able to.

The method for estimating the temperature and heat flux of the inner wall surface of the container according to the present invention is to estimate the temperature and heat flux of the inner wall surface of the container in a state where a high-temperature substance is present in the container ,
(1). Obtaining a temperature h (t) at the outer wall surface of the container and a physical quantity g (t) obtained by dividing the heat flux by the thermal conductivity of the material of the container wall;
An equation (102) in which a variable v (x, t) is defined and introduced as an alternative to the container wall temperature u (x, t) in equation (101) with the x direction from the inner wall surface (x = 0) to the outer wall surface direction. ) To calculate the variable v (l, t _i )
The variable w (x, t) is expressed by the formula (104) from the formula (103) introduced by defining the variable w (x, t). Based on the formula (104), MA × V _x of the formula (105) = Determine V _x by solving for V _w ,
Using physical quantities f (t ₁ ), f (t ₂ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall determined from V _x by the thermal conductivity of the material of the container wall, Based on (106), the temperatures p (t ₁ ), p (t ₂ ),..., P (t _M ) of the inner wall surface are determined.

(2). Obtaining a temperature h (t) at the outer wall surface of the container and a physical quantity g (t) obtained by dividing the heat flux by the thermal conductivity of the material of the container wall;
An equation (102) in which a variable v (x, t) is defined and introduced as an alternative to the container wall temperature u (x, t) in equation (101) with the x direction from the inner wall surface (x = 0) to the outer wall surface direction. ) To calculate the variable v (l, t _i )
The variable w (x, t) is expressed by the formula (104) from the formula (103) introduced by defining the variable w (x, t). Based on the formula (104), MA × V _x of the formula (105) = Determine V _x by solving for V _w ,
Using physical quantities f (t ₂ ), f (t ₄ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall determined from V _x by the thermal conductivity of the container wall material, Based on (106), the temperatures p (t ₂ ), p (t ₄ ),..., P (t _M ) of the inner wall surface may be determined.

  Here, as a container having a temperature difference between the inner wall surface and the outer wall surface, a reaction involving a high temperature gas reaction or liquid reaction such as a blast furnace, a converter, a degassing furnace, a steel heating furnace by combustion, a coal gasification reactor, etc. There are containers, containers for transporting molten iron, such as kneading cars, hot metal ladle, and molten steel ladle. This also applies to the case where one side of the material is rapidly heated or cooled in order to investigate defects inside the material.

  According to the present invention, it is necessary information to estimate the thickness of the container wall in a state where a high-temperature substance exists in the container without assuming the initial temperature distribution inside the container wall as in the conventional method. It is possible to accurately estimate the temperature and heat flux of the wall surface.

Preferred embodiments of the present invention will be described below with reference to the accompanying drawings.
First, the basic concept of the method for estimating the temperature and heat flux of the inner wall surface of the container according to the present invention will be described. FIG. 1 is a diagram showing a part of the container wall, where x = 0 is the position of the inner wall surface of the container. In the figure, the thickness of the container wall is l, the temperature of the container wall is u (x, t), the temperature measured at the temperature measurement point on the outer wall surface is calculated based on h (t) and the temperature h (t). Let g (t) be the physical quantity obtained by dividing the heat flux (or the heat flux measured at the temperature measurement point on the outer wall surface) by the thermal conductivity of the material.

(Formulation)
Equation (1) represents an unsteady heat conduction equation. Note that. u _t represents ∂u / ∂t, and u _xx represents ∂ ² u / ∂x ² . In equation (1), α is the thermal diffusion coefficient (also called temperature conductivity, defined by thermal conductivity / (density × specific heat)), u (x, 0) = u ₀ (x) is the temperature of the container wall Is the initial value of. In this case, the initial value u (x, 0) = u ₀ (x) of the temperature of the container wall is unknown.

  Here, by introducing the formula (2) and performing Fourier expansion, the temperature u (x, t) of the container wall of the formula (1) is obtained as the formula (3).

  If x = 1, equation (4) is obtained.

  However, when performing arithmetic processing by a computer, there is a problem that the calculation of the second term and the third term in particular on the right side of Equation (4) tends to cause a truncation error.

Therefore, in the present invention, the variable v (x, t) is defined as an alternative to the container wall temperature u (x, t), and the equation (5) is introduced. In equation (5), since the initial value g (0) of the physical quantity obtained by dividing the heat flux of the inner wall surface by the thermal conductivity of the material can be known, the initial value of the variable v (x, 0) = (x ² / 2l) g (0) can also be known. Therefore, the variable v (x, t) can be directly calculated by the backward difference method, for example.

  Further, when the difference between the temperature u (x, t) of the container wall and the variable v (x, t) is defined as w (x, t), the equation (6) is obtained.

  Here, by performing Fourier expansion in the same manner as described above, the variable w (x, t) in Equation (6) is obtained as in Equation (7). In formula (7), as is clear from comparison with formula (3), a simple formula without the third term can be obtained.

When x = 1 and t = t _j and the infinite series is taken up to a finite N term, Expression (8) is obtained.

Instead of calculating the second and third terms of equation (8), define equation (9) and solve equation (10) that satisfies the defined G ^ (x, t), ) W (l, t _j ) can be expressed by equation (11). In the present specification, the hat mark of the symbol G described in the equation (9) or the like is expressed as G ^.

Further, w (l, t) = u (l, t) −v (l, t). U (l, t) is a known h (t), and v (l, t) can be directly calculated from the equation (5) by the backward difference method. Therefore, assuming that w (l, t) is known, approximate values of B _n (l) and f (t _i ) can be obtained from the equation (11).

(Inverse problem to find the temperature and heat flux of the inner wall of the container)
In the inverse problem, the heat flux (the physical quantity f (t) obtained by dividing the heat flux by the thermal conductivity of the material) and the temperature p (t) on the inner wall surface of the container are unknown. The observation time is set as (T _st , T _end ), and T _st <T ₁ <T ₂ <T _end . Then, T ₁ = t ₁ <t ₂ <... <T _M = T ₂ and a uniform lattice.

Here, the temperature h (t) measured at the temperature measurement point of the outer wall surface is known, and the variable v (l, t _i ) is obtained from the equation (5) by the backward difference method. As shown in Expression (12), MA ₁ is an M × (N + 1) matrix, MA ₂ is an M × M matrix, and MA is an M × (N + M + 1) matrix in which these are arranged side by side. V _x is (N + M + 1) × 1 vector, Vw is a M × 1 vector, by solving the MA × V _x = V _w, the temperature measuring points t _1, t _2, · · ·, inner in t _M Physical quantities f (t ₁ ), f (t ₂ ),..., F (t _M ) obtained by dividing the heat flux of the wall surface by the thermal conductivity of the material are obtained.

  However, since MA in Expression (12) is an M × (N + M + 1) matrix, the number of unknowns is larger than the number of equations, and a solution cannot be obtained in this state. N + 1 rows are inserted, and MA is a square matrix, so that a solution can be obtained.

Further, as a different solution, Equation (13) is newly defined. In Equation (13), MA ₁ is an M × (N + 1) matrix, MA ₂ is an M × M / 2 matrix, and MA is an M × (N + M / 2 + 1) matrix as a sum of these. V _x is an (N + M / 2 + 1) × 1 vector and Vw is an M × 1 vector. By solving MA × V _x = V _w , each temperature measurement point t ₂ , t _{4 ,.} The physical quantities f (t ₂ ), f (t ₄ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall surface by the thermal conductivity of the material are obtained. Here, by setting N ≦ M / 2-1, the number of unknowns becomes equal to or smaller than the number of equations, and a solution can be obtained.

Then, using physical quantities f (t ₂ ), f (t ₄ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall surface of the container determined from V _x by the thermal conductivity of the material, W (0, t ₂ ), w (0, t ₄ ),..., W (0, t _M ) and v (0, t ₂ ) directly calculated by the alternating difference method based on the equation (14). v (0, t _4), determined ... from v (0, t _M), the temperature p of the inner wall surface of the vessel _{(t 2), p (t} 4), ···, p a (t _M) To do.

  FIG. 2 is a diagram illustrating a schematic configuration of the temperature and heat flux estimation device for the inner wall surface of the container according to the present embodiment. 3A and 3B are flowcharts for explaining the estimation process of the temperature and heat flux of the inner wall surface of the container in the present embodiment.

In FIG. 2, 101 is an input unit, and the temperature h (t _i ) measured at the temperature measurement point on the outer wall surface of the container is input (step S101).

Reference numeral 102 denotes a heat flux calculator, which calculates a heat flux g (t _i ) based on the temperature h (t _i ) input to the input unit 101 (step S102). Note that the heat flux g (t _i ) measured at the temperature measurement point on the outer wall surface may be input to the input unit 101, and in that case, the heat flux calculation unit 102 is unnecessary.

Reference numeral 103 denotes a calculation unit that calculates the temperature and heat flux of the inner wall surface of the container by the above-described method for estimating the temperature and heat flux of the inner wall surface of the container. That is, the variable v (l, t _i ) is obtained from the equation (5) (step S103). Then, the vector V _x is determined by solving MA × V _x = V _w in the equation (13), and its components B ₀ (l), B ₁ (l),..., B _N (l ) And physical quantities f (t ₂ ), f (t ₄ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall surface by the thermal conductivity of the material (step S104). The temperature p (t ₂ ), p (t ₄ ),..., P (t _M ) of the inner wall surface is calculated from the equation (14). The heat flux of the inner wall surface of the container can be obtained by multiplying the determined physical quantities f (t ₂ ), f (t ₄ ),..., F (t _M ) by the thermal conductivity of the container wall (step S105). ).

  An output unit 104 displays the temperature and heat flux of the inner wall surface of the container calculated and estimated by the calculation unit 103, for example, on a display (not shown).

When a plurality of types of container wall materials overlap, physical quantities f (t ₂ ) and f (t (t) obtained by dividing the heat flux of the inner wall surface of the first-layer container wall material determined by the above method by the thermal conductivity. ₄ ), ..., f (t _M ), physical quantities g (t ₂ ), g (t ₄ ), ... obtained by dividing the heat flux on the outer wall surface of the container wall material of the second layer by the thermal conductivity. , and g (t _M), the temperature p of the inner wall surface of the first layer (t _2), p (t _4), · · ·, based on p (t _M), was calculated by the formula (15) The above calculation is repeated with the temperatures h (t ₂ ), h (t ₄ ),..., H (t _M ) of the outer wall surface of the second-layer container wall material. In Formula (15), R shows the contact heat transfer resistance between the container wall material of the 1st layer and the 2nd layer. For example, when the first layer and the second layer are bonded with an adhesive such as mortar, R represents the thickness ds of the adhesive as the thermal conductivity λs of the adhesive, as shown in Equation (16). The divided value. Further, when there is a gap between the first layer and the second layer, the equation is obtained by using the equation (17) instead of the equation (15), and the temperature h (t ₂ ) of the outer wall surface of the second layer container wall material. ), H (t ₄ ),..., H (t _M ). σ is the Stefan Boltzmann coefficient, ε ₁ is the emissivity of the container wall material of the second layer, and ε ₂ is the emissivity of the container wall material of the second layer. The same applies to the container wall material from the third layer onward in the first layer.

(Example)
The estimation result of the temperature and heat flux of the inner wall surface of the container according to the method of the present invention (this method) will be described. In this example, a method in which the above formulas (13) and (14) were introduced was used. For three types of molten steel pans (thickness of container wall 100 mm, 150 mm, 200 mm) whose thickness is known in advance, the thickness of the container wall is estimated by this method and the conventional method (the method disclosed in Patent Document 1), The results were compared.

In this example, the thermal diffusion coefficient α of the molten steel pan (refractory) is 0.00865 m ² / Hr (thermal conductivity: 8.49 W / m / K, specific heat: 124.2 J / kg / K, density 2910 kg / m). ³ ). The container is filled with molten steel at 1600 ° C., and the temperature of the inner wall surface of the refractory is 1600 ° C. equal to the molten steel temperature.

The data of the temperature and heat flux of the outer wall surface of each molten steel pan are shown in FIGS. In this example, after 1200 seconds from the steel receiving, the temperature was measured 5 times every 5 seconds at the temperature measurement point on the outer wall surface, and the heat flux was calculated based on each temperature. The heat flux was calculated by equation (18) assuming heat release by radiant heat transfer. σ is the Stefan Boltzmann coefficient of radiant heat transfer, ε is the emissivity, and u _a is the ambient air temperature. Here, ε was 1.0.

  7 to 9 show a comparison between the estimated value of the inner wall surface temperature of the container wall (refractory) estimated by this method and the true value (1590 ° C.). As the container wall thickness (refractory thickness) increases, the estimation accuracy decreases, but the error is 1% or less, and high accuracy estimation is possible.

  FIG. 10 shows the result of determining the container wall thickness (refractory thickness) so that the set thickness of the container wall (refractory) is changed and the inner wall surface temperature of the refractory matches the molten steel temperature. In the conventional method, an error is likely to occur in the estimated value of the remaining thickness, particularly in a molten steel pan having a thick container wall. As described above, in the conventional method, it is considered that this error is caused by calculation assuming that the initial temperature of the inner wall surface of the refractory is a constant temperature (300 ° C.).

  On the other hand, in this method, the actual thickness and the estimated thickness almost coincided with each other, and a good estimated value was obtained.

  The object of the present invention described above is to supply a storage medium storing software program codes for realizing the functions of the above-described embodiments to a system or apparatus, and store the computer (or CPU or MPU) of the system or apparatus. Needless to say, this can also be achieved by reading and executing the program code stored in the medium.

  In this case, the program code itself read from the storage medium realizes the functions of the above-described embodiments, and the program code itself and the storage medium storing the program code constitute the present invention. As a storage medium for supplying the program code, for example, a flexible disk, a hard disk, an optical disk, a magneto-optical disk, a CD-ROM, a CD-R, a magnetic tape, a nonvolatile memory card, a ROM, or the like can be used.

It is a figure showing a part of container wall. It is a figure which shows schematic structure of the estimation apparatus of the temperature and heat flux of the inner wall face of the container which concerns on this embodiment. It is a flowchart for demonstrating the estimation process of the temperature and heat flux of the inner wall face of the container in this embodiment. It is a flowchart for demonstrating the estimation process of the temperature and heat flux of the inner wall face of the container in this embodiment. It is a characteristic view which shows the data of the temperature and heat flux of an outer wall surface in case the thickness of a container wall is 100 mm. It is a characteristic view which shows the temperature and heat flux data of an outer wall surface in case the thickness of a container wall is 150 mm. It is a characteristic view which shows the temperature and heat flux data of an outer wall surface in case the thickness of a container wall is 200 mm. It is the figure which showed the comparison with the estimated value and true value of the estimated inner wall surface temperature in case the thickness of the container wall in an Example is 100 mm. It is the figure which showed the comparison with the estimated value and true value of the estimated inner wall surface temperature in case the thickness of the container wall in an Example is 150 mm. It is the figure which showed the comparison with the estimated value and true value of the estimated inner wall surface temperature in case the thickness of the container wall in an Example is 200 mm. It is a figure which shows the estimation result of the thickness of the container wall in an Example and a comparative example.

Explanation of symbols

DESCRIPTION OF SYMBOLS 101 Input part 102 Heat flux calculation part 103 Operation part 104 Output part

Claims (6)

A method for estimating the temperature and heat flux of the inner wall surface of the container in the presence of a high-temperature substance in the container ,
Obtaining a temperature h (t) at the outer wall surface of the container and a physical quantity g (t) obtained by dividing the heat flux by the thermal conductivity of the material of the container wall;
An equation (102) in which a variable v (x, t) is defined and introduced as an alternative to the container wall temperature u (x, t) in equation (101) with the x direction from the inner wall surface (x = 0) to the outer wall surface direction. ) To calculate the variable v (l, t _i ),

The variable w (x, t) is expressed by the formula (104) from the formula (103) introduced by defining the variable w (x, t). Based on the formula (104), MA × V _x of the formula (105) = Determining V _x by solving for V _w ;

Using physical quantities f (t ₁ ), f (t ₂ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall determined from V _x by the thermal conductivity of the material of the container wall, (106) a procedure for determining the temperature p (t ₁ ), p (t ₂ ),..., P (t _M ) of the inner wall surface;

A method for estimating the temperature and heat flux of the inner wall surface of the container. A method for estimating the temperature and heat flux of the inner wall surface of the container in the presence of a high-temperature substance in the container ,
Obtaining a temperature h (t) at the outer wall surface of the container and a physical quantity g (t) obtained by dividing the heat flux by the thermal conductivity of the material of the container wall;
An equation (102) in which a variable v (x, t) is defined and introduced as an alternative to the container wall temperature u (x, t) in equation (101) with the x direction from the inner wall surface (x = 0) to the outer wall surface direction. ) To calculate the variable v (l, t _i ),

The variable w (x, t) is expressed by the formula (104) from the formula (103) introduced by defining the variable w (x, t). Based on the formula (104), MA × V _x of the formula (105) = Determining V _x by solving for V _w ;

Using physical quantities f (t ₂ ), f (t ₄ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall determined from V _x by the thermal conductivity of the container wall material, (106) a procedure for determining the temperature p (t ₂ ), p (t ₄ ),..., P (t _M ) of the inner wall surface;

A method for estimating the temperature and heat flux of the inner wall surface of the container. An apparatus for estimating the temperature and heat flux of the inner wall surface of the container in the presence of a high-temperature substance in the container ,
Means for obtaining a temperature h (t) at the outer wall surface of the container and a physical quantity g (t) obtained by dividing the heat flux by the thermal conductivity of the material;
An equation (102) in which a variable v (x, t) is defined and introduced as an alternative to the container wall temperature u (x, t) in equation (101) with the x direction from the inner wall surface (x = 0) to the outer wall surface direction. ) To calculate the variable v (l, t _i );

The variable w (x, t) is expressed by the formula (104) from the formula (103) introduced by defining the variable w (x, t). Based on the formula (104), MA × V _x of the formula (105) = Means to determine V _x by solving for V _w ;

Using physical quantities f (t ₁ ), f (t ₂ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall surface determined from V _x by the thermal conductivity of the material, equation (106) , P (t _M ) for determining the temperature p (t ₁ ), p (t ₂ ),.

An apparatus for estimating the temperature and heat flux of the inner wall surface of the container, characterized by comprising: An apparatus for estimating the temperature and heat flux of the inner wall surface of the container in the presence of a high-temperature substance in the container ,
Means for obtaining a temperature h (t) at the outer wall surface of the container and a physical quantity g (t) obtained by dividing the heat flux by the thermal conductivity of the material of the container wall;
An equation (102) in which a variable v (x, t) is defined and introduced as an alternative to the container wall temperature u (x, t) in equation (101) with the x direction from the inner wall surface (x = 0) to the outer wall surface direction. ) To calculate the variable v (l, t _i );

The variable w (x, t) is expressed by the formula (104) from the formula (103) introduced by defining the variable w (x, t). Based on the formula (104), MA × V _x of the formula (105) = Means to determine V _x by solving for V _w ;

Using physical quantities f (t ₂ ), f (t ₄ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall surface determined from V _x by the thermal conductivity of the material, equation (106) , P (t _M ) for determining the temperature p (t ₂ ), p (t ₄ ),.

An apparatus for estimating the temperature and heat flux of the inner wall surface of the container, characterized by comprising: A computer program for causing a computer to execute an estimation calculation of the temperature and heat flux of the inner wall surface of the container in a state where a high-temperature substance is present in the container ,
A process of obtaining a temperature h (t) at the outer wall surface of the container and a physical quantity g (t) obtained by dividing the heat flux by the thermal conductivity of the material;
An equation (102) in which a variable v (x, t) is defined and introduced as an alternative to the container wall temperature u (x, t) in equation (101) with the x direction from the inner wall surface (x = 0) to the outer wall surface direction. ) To calculate the variable v (l, t _i ),

The variable w (x, t) is expressed by the formula (104) from the formula (103) introduced by defining the variable w (x, t). Based on the formula (104), MA × V _x of the formula (105) Processing to determine V _x by solving = V _w ;

Using physical quantities f (t ₁ ), f (t ₂ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall surface determined from V _x by the thermal conductivity of the material, equation (106) , P (t _M ) for determining the temperature p (t ₁ ), p (t ₂ ),.

A computer program for causing a computer to execute. A computer program for causing a computer to execute an estimation calculation of the temperature and heat flux of the inner wall surface of the container in a state where a high-temperature substance is present in the container ,
A process of obtaining a physical quantity g (t) obtained by dividing the temperature h (t) and heat flux at the outer wall surface of the container by the thermal conductivity of the material;
An equation (102) in which a variable v (x, t) is defined and introduced as an alternative to the container wall temperature u (x, t) in equation (101) with the x direction from the inner wall surface (x = 0) to the outer wall surface direction. ) To calculate the variable v (l, t _i ),

The variable w (x, t) is expressed by the formula (104) from the formula (103) introduced by defining the variable w (x, t). Based on the formula (104), MA × V _x of the formula (105) Processing to determine V _x by solving = V _w ;

Using physical quantities f (t ₂ ), f (t ₄ ),..., F (t _M ) obtained by dividing the heat flux of the inner wall surface determined from V _x by the thermal conductivity of the material, equation (106) , P (t _M ) for determining the temperature p (t ₂ ), p (t ₄ ),.

A computer program for causing a computer to execute.

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