RU2523090C1 - Method of determining specific heat capacity of materials - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: method includes forming first and second identical test samples from granular or porous materials; bringing said samples into thermal contact on a plane with a heat source; bringing the outer surface of the samples into thermal contact with reference samples and bringing the outer surface of reference samples into thermal contact with the heat sources; supplying heat to the samples and measuring specific power of the heat sources; measuring, with constant time spacing, temperature, specific volume of the solid phase of the samples, heat flux from those surfaces of flat heat sources that were not brought into thermal contact with reference samples; determining heat flux through the test samples and calculating specific heat capacity.

EFFECT: high accuracy of determining specific heat capacity of porous, fibrous and granular materials.

2 dwg

Description

The invention relates to the field of technical physics, in particular to thermal methods for studying the properties of materials, namely to determine the specific heat.

A known method for the complex determination of the thermophysical characteristics of materials is that the thickness of the test sample is measured, heat is supplied to the samples, the temperature on the external surfaces of the samples is maintained at a predetermined temperature, the specific power of the heat source is recorded, and the temperature is measured with a constant step in time throughout the experiment , determine at each step the value of the dynamic parameter and compare with the maximum value lying in a given range, determine some thermophysical characteristics by formulas, a second sample is used that is identical to the first sample, these samples are brought into thermal contact on two sides with a volume source of heat, the experiment is carried out in two stages, and at the first stage, constant power is supplied to the volume source of heat, at each time step, the average integral temperature of the volumetric heat source is recorded and the value of the dynamic parameter is calculated as the ratio of the difference in the rates of change of the measured average integ temperature at the first and current steps to the rate of change of the average integral temperature at the first step, the first stage of the experiment is completed when the specified maximum value of the dynamic parameter is exceeded, and at the next time step, the second stage of the experiment is started, namely, the power supply to the volumetric heat source is stopped , at each step of the second stage, the dimensionless temperature and Fourier number are calculated; the second stage of the experiment is stopped at the step at which ix dimensionless temperature becomes smaller than the predetermined value, wherein at the first stage registered experimental data calculated desired heat conductivity, and registered according to the second phase of the experiment determining the desired thermal diffusivity (RF Patent №2243543, IPC ⁷ G01N 25/18).

The disadvantages of this method are the long measurement duration, due to the need for two stages of the experiment, as well as the ability to measure thermophysical properties with a given accuracy only in those ranges of thermal conductivity and thermal diffusivity for which the limiting values of the dynamic parameters are determined.

The closest technical solution adopted for the prototype is a method for comprehensively determining the thermophysical characteristics of materials, which consists in measuring the thickness of the test sample, using a sample identical to the first sample as the second sample, these samples are brought into thermal contact with a volume source on both sides heat, bring heat to the samples, record the specific power of the heat source, measure the temperature with a constant step in time throughout the experiment, determine the required thermophysical characteristics by formulas, the external surfaces of the test samples are brought into thermal contact with the reference samples, the external surfaces of the reference samples are brought into thermal contact with flat heaters, the constant power is supplied to the heaters and the temperature difference is recorded on the surfaces of the reference samples in the contact plane of the reference sample with a flat the heater and in the plane of contact of the reference sample with the test sample, calculate the heat fluxes through the first and second studies blowing samples, calculate the Fourier number and the dimensionless average integral temperature of the volumetric heat source, calculate the slope of the rectilinear portion of the graph of the dependence of the dimensionless average integral temperature of the volumetric heat source on the Fourier number from experimental data calculate the desired volumetric heat capacity and thermal conductivity (Patent No. 2387981 RF, G01N 25/18. A method for the comprehensive determination of the thermophysical characteristics of materials).

The disadvantage of this method is the large error in determining the specific heat of porous and bulk materials, due to the fact that when the temperature changes during the experiment, there is an increase in the specific volume of the solid phase of the studied material, which is not taken into account in the prototype.

Such features of the prototype as the use of a second sample identical to the first sample, bringing samples from two sides into thermal contact with a heat source, bringing external surfaces of samples into thermal contact with reference samples, bringing external surfaces of reference samples into thermal contact with heat sources, summing up heat to samples, registration of the specific power of heat sources, measurement with a constant step in time of temperature throughout the experiment, calculation of heat fluxes through h samples, determination of the desired thermophysical characteristics by formulas, coincide with the essential features of the claimed invention.

The technical task is to increase the information content of the method and increase the accuracy of determining the specific heat of porous, fibrous and bulk materials.

This technical problem is solved in that in the method for determining the specific heat of materials, the first and second identical samples are brought into thermal contact along the plane with the heat source, the external surfaces of the samples are brought into thermal contact with the reference samples, the external surfaces of the reference samples are brought into thermal contact with the heat sources heat is brought to the samples under investigation, the specific power of heat sources is recorded, and the temperature is measured with a constant step in time throughout the experiment that, heat fluxes through the studied samples are calculated, the desired thermophysical characteristics are determined by the formulas, they additionally isolate the volume with the formed samples of known mass from granular or porous material, change the volume of their gas space by a constant value, measure the pressure change, determine the specific volume of the solid phase of the samples, as well as heat fluxes from those surfaces of planar heat sources that are not brought into thermal contact with reference samples, heat fluxes are calculated through the first q ₁ = U ² / (RS) -q _1u and the second q ₂ = U ² / (RS) -q _{2u the} test samples, calculate the specific heat by the formula

, $$c=c_n{p}_n\frac{\left(Q+G_{\mathrm{one}}-\mathrm{one}\right)}{G_{\mathrm{one}}}\cdot {\nu }_{tf}$$

,

- удельный объем твердой фазы образцов; V_тф - объем твердой фазы, m - масса образцов, с_нр_н - объемная теплоемкость материала, из которого изготовлен источник теплоты; Г₁=1+h₀/(2h₀+h_н)-(h₀+h_н)/(2h₀+h_н) - коэффициент толщины образца h₀ и источника теплоты h_н; U - напряжение, подводимое к источнику теплоты, R, S - сопротивление и площадь поверхности источника теплоты, q_1u, q_2u - измеряемые тепловые потоки с поверхностей первого и второго плоских источников теплоты, $$Q=\frac{1+\frac{U^2/\left(RS{h}_{н}\right)}{q_1}{h}_{н}+\frac{q_2}{q_1}}{A}$$

,Where $${\nu }_{tf}=\frac{V_{tf}}{m}$$

- specific volume of the solid phase of the samples; V _tf is the volume of the solid phase, m is the mass of the samples, s _N r _n is the volumetric heat capacity of the material from which the heat source is made; G ₁ = 1 + h ₀ / (2h ₀ + h _n ) - (h ₀ + h _n ) / (2h ₀ + h _n ) - coefficient of thickness of the sample h ₀ and heat source h _n ; U is the voltage supplied to the heat source, R, S is the resistance and surface area of the heat source, q _1u , q _2u are the measured heat fluxes from the surfaces of the first and second plane heat sources, $$Q=\frac{\mathrm{one}+\frac{U^2/\left(RS{h}_n\right)}{q_{\mathrm{one}}}{h}_n+\frac{q_2}{q_{\mathrm{one}}}}{A}$$

,

; $${\overline{T}}_2\left(\tau \right)$$

- температура, измеряемая в плоскости контакта исследуемых образцов; Т₀ - начальная температура.A is the tangent of the slope of the rectilinear plot of dependence $$\frac{{\overline{T}}_2\left(\tau \right)-T_0}{q_{\mathrm{one}}\left(2h_0+h_n\right)/{\lambda }_n}=f\left(\frac{a_n\tau }{{\left(2h_0+h_n\right)}^2}\right)$$

; $${\overline{T}}_2\left(\tau \right)$$

- temperature measured in the plane of contact of the studied samples; T ₀ - initial temperature.

By the method described in the prototype, the determination of specific heat is only possible by calculation by dividing the volumetric heat by the density of the material. However, in the process of heating the test material due to the expansion of the particles, the ratio of the volumes of the solid and gas phases changes. Therefore, the determination of specific heat in the prototype is fraught with large errors.

Compared with the prototype, the proposed method, it is possible to determine not only the volumetric heat capacity of the material, but also the specific heat due to the continuous determination of the specific volume of the solid phase during the experiment. This extends the information content of the method and increases the accuracy of determining the specific heat.

Figure 1 shows the physical model of the measuring cell that implements the proposed method. Figure 2 shows the design diagram of the measuring cell.

The physical model of the measuring cell (figure 1) is a flat three-layer system. Layers 1 and 3 of the system are formed by the studied samples, identical in properties and sizes, between which layer 2 is located, consisting of a heater and a resistance thermometer made of manganin and copper wires. On the external surfaces of the samples under study, heat fluxes q ₁ and q _{2 are} specified.

The mathematical model describing the temperature field in the measuring device was formulated under the following assumptions: 1) there is no heat transfer by radiation in the samples under study; 2) the temperature field inside the system is considered one-dimensional; 3) there are no thermal resistances at the contact boundaries of the layers; 4) at the external borders of the samples under study, constant heat fluxes are specified; 5) during the experiment, the temperature of the layers of the system changes slightly, therefore, the thermophysical properties of the layers are constant; 6) the power allocated to the heater is evenly distributed throughout the entire volume of layer 2. Subject to assumptions, the mathematical model is written in the form of a system of differential heat equations

$$\frac{\partial {\Theta }_i\left(\overline{x},Fo\right)}{\partial Fo}={\overline{a}}_i\frac{{\partial }^2{\Theta }_i\left(\overline{x},Fo\right)}{\partial {\overline{x}}^2}+{\overline{W}}_i,0<\overline{x}<\mathrm{one,}Fo>0i=\overline{1.3},\left(\mathrm{one}\right)$$

with initial conditions $${\Theta }_i\left(\overline{x}0\right)=0\left(2\right)$$

and boundary conditions

$$\frac{\partial {\Theta }_i\left(0Fo\right)}{\partial \overline{x}}=-\frac{{\lambda }_2}{{\lambda }_{\mathrm{one}}},\left(3\right)$$

$${\Theta }_i\left(\frac{l_i}{l_3}-0Fo\right)={\Theta }_{i+\mathrm{one}}\left(\frac{l_i}{l_3}+0Fo\right),i=\overline{\mathrm{1,2}},\left(\mathrm{four}\right)$$

$${\lambda }_i\frac{\partial {\Theta }_i\left(\frac{l_i}{l_3}-0Fo\right)}{\partial \overline{x}}={\lambda }_{i+\mathrm{one}}\frac{\partial {\Theta }_{i+\mathrm{one}}\left(\frac{l_i}{l_3}+0Fo\right)}{\partial \overline{x}},i=\overline{\mathrm{1,2}},\left(5\right)$$

$$\frac{\partial {\Theta }_3\left(\mathrm{one,}Fo\right)}{\partial \overline{x}}=\frac{q_2}{q_{\mathrm{one}}}\frac{{\lambda }_2}{{\lambda }_3},\left(6\right)$$

- безразмерная температуропроводность i-го слоя, определяемая из выражения $${\overline{a}}_i=a_i/a_2$$

; λ_i - теплопроводность; $${\overline{W}}_i$$

- безразмерная объемная мощность внутренних источников теплоты, определяемая из выражения $${\overline{W}}_i=W_i{l}_3/q_1$$

, причем объемная мощность внутренних источников теплоты первого и третьего слоев равны W₁=W₃=0, а объемная мощность внутренних источников теплоты второго слоя определяется как отношение мощности P нагревателя к объему V₂ второго слоя, т.е. W₂=P/V₂; $${\Theta }_i\left(\overline{x},Fo\right)$$

- безразмерная температура, определяемая из выраженияWhere $${\overline{a}}_i$$

- dimensionless thermal diffusivity of the i-th layer, determined from the expression $${\overline{a}}_i=a_i/a_2$$

; λ _i - thermal conductivity; $${\overline{W}}_i$$

- dimensionless volumetric power of internal heat sources, determined from the expression $${\overline{W}}_i=W_i{l}_3/q_{\mathrm{one}}$$

moreover, the volumetric power of the internal heat sources of the first and third layers is W ₁ = W ₃ = 0, and the volumetric power of the internal heat sources of the second layer is determined as the ratio of the power P of the heater to the volume V _{2 of the} second layer, i.e. W ₂ = P / V ₂ ; $${\Theta }_i\left(\overline{x},Fo\right)$$

- dimensionless temperature, determined from the expression

, $${\Theta }_i\left(\overline{x},Fo\right)=\frac{T_i\left(x,\tau \right)-T_0}{q_{\mathrm{one}}{l}_3/{\lambda }_2}$$

,

- безразмерная пространственная координата; $$Fo=\frac{a_2\tau }{l_3^2}$$

- число Фурье, x,τ - пространственная координата и время.where T ₁ (x, τ) is the temperature field of the i-th layer; T ₀ - initial temperature; $$\overline{x}=x/l_3$$

- dimensionless spatial coordinate; $$Fo=\frac{a_2\tau }{l_3^2}$$

is the Fourier number, x, τ is the spatial coordinate and time.

будет автомодельным относительно координаты Fo. Решение задачи (1)-(6) имеет следующий видFrom the theory of thermal conductivity it is known that the temperature field $${\Theta }_i\left(\overline{x},Fo\right)$$

will be self-similar with respect to the coordinate Fo. The solution to problem (1) - (6) has the following form

, $${\Theta }_i\left(\overline{x},Fo\right)=AFo+F_i\left(\overline{x}\right),i=\overline{1.3}$$

,

- функции, имеющие видwhere A is a constant coefficient; $$F_i\left(\overline{x}\right)$$

- functions having the form

$$F_i\left(\overline{x}\right)=\{\begin{array}{c}A\frac{a_2}{a_{\mathrm{one}}}\frac{{\overline{x}}^2}{2}-\frac{{\lambda }_2}{{\lambda }_{\mathrm{one}}}\overline{x}+C_{\mathrm{one}},i=\mathrm{one,}\\ \left(A-{\overline{W}}_2\right)\frac{{\overline{x}}^2}{2}+C_{21}\overline{x}+C_{22},i=2\\ A\frac{a_2}{a_3}\frac{{\overline{x}}^2}{2}-\left(-\frac{q_2}{q_{\mathrm{one}}}\frac{{\lambda }_2}{{\lambda }_3}+A\frac{a_2}{a_3}\right)\overline{x}+C_3,i=3.\end{array}$$

The constants A, C ₁ , C ₂₁ , C ₂₂ , C _{3 are} determined from the boundary conditions (3) - (6), as well as from the heat balance equation recorded for the system of layers 1-3 in figure 1. In particular, for A an expression of the form

$$A=\frac{\mathrm{one}+{\overline{W}}_2\left(l_2/l_3-l_{\mathrm{one}}/l_3\right)+\frac{q_2}{q_{\mathrm{one}}}}{\frac{c_{\mathrm{one}}{\rho }_{\mathrm{one}}}{c_2{\rho }_2}\left(l_{\mathrm{one}}/l_3\right)-l_{\mathrm{one}}/l_3-\frac{c_3{\rho }_3}{c_2{\rho }_2}\left(l_2/l_3\right)+\frac{c_3{\rho }_3}{c_2{\rho }_2}+l_2/l_3},\left(7\right)$$

from which, taking into account ₁ ρ ₁ = c ₃ ρ ₃ = сρ, we can obtain an expression for calculating the volumetric heat capacity of the samples

$$c\rho =c_2{\rho }_2\frac{\frac{\mathrm{one}+\frac{W_2}{q_{\mathrm{one}}}\left(l_2-l_{\mathrm{one}}\right)+\frac{q_2}{q_{\mathrm{one}}}}{A}+l_{\mathrm{one}}/l_3-l_2/l_3}{\left(\mathrm{one}+l_{\mathrm{one}}/l_3-l_2/l_3\right)}.\left(8\right)$$

In the study of porous, fibrous or bulk materials, their volumetric heat capacity will consist of two components - volumetric heat capacity of the solid phase and volumetric heat capacity of the gas phase filling the pores, i.e.

cF = (cF) _Tp + (cF) _gf.

The last expression is provided (cF) _tf >> (cF) _gf takes the form

cρ≈ (cρ) _tf , or cρ≈c _tf m / V _tf = c _tf / ν _tf .

Thus, the specific heat of the solid phase will be determined from the expression

c _{tf tf} = cρν,

and taking into account (8) we get

$$c_{tf}=c_2{\rho }_2\frac{\frac{\mathrm{one}+\frac{W_2}{q_{\mathrm{one}}}\left(l_2-l_{\mathrm{one}}\right)+\frac{q_2}{q_{\mathrm{one}}}}{A}+l_{\mathrm{one}}/l_3-l_2/l_3}{\left(\mathrm{one}+l_{\mathrm{one}}/l_3-l_2/l_3\right)}{\nu }_{tf}.\left(9\right)$$

, W²=U²/(RSh_н), с₂ρ₂=c_нρ_н. Тогда (9) примет видDenote h ₀ = l ₁ is the thickness of the sample, h _n = l ₂ -l ₁ is the thickness of the heat source (heater), 2h ₀ + h _n = l ₃ , G ₁ = 1 + h ₀ / (2h ₀ + h _n ) - (h ₀ + h _n ) / (2h ₀ + h _n ), $$Q=\frac{\mathrm{one}+\frac{U^2/\left(RS{h}_n\right)}{q_{\mathrm{one}}}{h}_n+\frac{q_2}{q_{\mathrm{one}}}}{A}$$

, W ² = U ² / (RSh _n ), with ₂ ρ ₂ = c _n ρ _n . Then (9) takes the form

$$c_{tf}=c_n{\rho }_n\frac{\left(Q+G_{\mathrm{one}}-\mathrm{one}\right)}{G_{\mathrm{one}}}\cdot {\nu }_{tf}.\left(10\right)$$

Thus, in comparison with the prototype, an additional measurement of the specific volume of the solid phase during the experiment allows us to determine the specific heat capacity of the solid phase of the investigated material.

The circuit of the measuring cell is shown in figure 2. The studied samples (or bulk layer) are placed in the chamber 4 between the gas-permeable sheath 5 with a heating element and a temperature meter deposited on it, which are made of manganin and copper wires, respectively. The external surfaces of the test samples are brought into thermal contact with thin copper plates 6, on the external surface of which are placed flat heaters 7, which, in turn, are brought into thermal contact with heat flux sensors 8. The described system is thermally insulated from the environment by insulation 9. In the design the cell provides a cylinder 10 with a piston 11, performing a reciprocating motion. The cavity of the cylinder 10 is connected to the chamber 12 and the pressure meter (not shown in Fig. 2).

The use of heat flux sensors in the design of the measuring cell makes it possible to measure heat fluxes q _1u and q _2u from the surfaces of the heaters 7. This allows the known power allocated to the heaters and determined by the expression U ² / (RS), where U, R, S is the voltage, supplied to the heater, its resistance and area, determine heat fluxes through the first and second samples according to the formulas

q ₁ = U ² / (RS) -q _1u q ₂ = U ² / (RS) -q _2u .

нагревательного элемента 5. Вычисляют безразмерную среднеинтегральную температуру $${\overline{\Theta }}_2\left(Fo\right)=\frac{{\overline{T}}_2\left(\tau \right)-T_0}{q_1\left(2h_0+h_{н}\right)/{\lambda }_{н}}$$

и число Фурье $$Fo=\frac{a_{н}\tau }{{\left(2h_0+h_{н}\right)}^2}$$

. При достижении регулярного теплового режима второго рода регистрируют $${\overline{\Theta }}_2\left(Fo\right)$$

и в заданном интервале безразмерного времени [Fo∗,Fo∗∗] вычисляют А по формуле $$A=\left[{\overline{\Theta }}_2\left(F{o}^{\ast \ast }\right)-{\overline{\Theta }}_2\left(F{o}^{\ast }\right)\right]/\left[F{o}^{\ast \ast }-F{o}^{\ast }\right]$$

.The method for determining the specific heat of materials is implemented as follows. Before placing the analyzed material in the chamber, its mass m and atmospheric pressure P _{atm are} determined. The test bulk material is poured into the chamber 4 (FIG. 2) and the chamber is sealed. A constant voltage U is applied to the heaters 8 and 5 of the measuring cell. At each time step τ, the average integral temperature is measured $${\overline{T}}_2\left(\tau \right)$$

heating element 5. Calculate the dimensionless average integral temperature $${\overline{\Theta }}_2\left(Fo\right)=\frac{{\overline{T}}_2\left(\tau \right)-T_0}{q_{\mathrm{one}}\left(2h_0+h_n\right)/{\lambda }_n}$$

and Fourier number $$Fo=\frac{a_n\tau }{{\left(2h_0+h_n\right)}^2}$$

. Upon reaching the regular thermal regime of the second kind $${\overline{\Theta }}_2\left(Fo\right)$$

and in a given interval of dimensionless time [Fo ∗, Fo ∗∗] calculate A by the formula $$A=\left[{\overline{\Theta }}_2\left(F{o}^{\ast \ast }\right)-{\overline{\Theta }}_2\left(F{o}^{\ast }\right)\right]/\left[F{o}^{\ast \ast }-F{o}^{\ast }\right]$$

.

During the experiment, the total volume of chambers 4 and 10 is reduced by ΔV = k · m, where k is the coefficient of proportionality. The change in pressure ΔP is measured. The specific volume of the solid phase is determined from the equation

, $${\nu }_{tf}=\frac{V}{m}-k\frac{P_{\mathrm{but}tm}}{\Delta P}={\nu }_m-k\frac{P_{\mathrm{but}tm}}{\Delta P}$$

,

where V is the total volume of the chamber 4, in which the test material and chambers 10 are placed; m is the mass of the test material; P _atm is atmospheric pressure; ΔP - pressure change in the chamber with the studied material; ν _m is the specific volume of the investigated material; k is the coefficient of proportionality.

The determination of the specific volume of the solid phase of the analyzed material is carried out discretely with a given step.

The desired specific heat is calculated by the formula (10).

Claims (1)

A method for determining the specific heat capacity of materials, namely, that the first and second identical samples are brought into thermal contact in a plane with a heat source, the external surfaces of the samples are brought into thermal contact with reference samples, the external surfaces of the reference samples are brought into thermal contact with flat heat sources, heat is brought to the test samples, the specific power of the heat sources is recorded, the temperature is measured with a constant step in time throughout the experiment, I calculate t heat fluxes through the studied samples, determine the desired thermophysical characteristics by the formulas, characterized in that they additionally isolate the volume with the formed samples of known mass from bulk or porous material, change the volume of their gas space by a constant value, measure the pressure change, determine the specific volume of the solid phase samples, as well as heat fluxes from those surfaces of planar heat sources that are not brought into thermal contact with reference samples, calculate heat sweat ki first through q ₁ = U ² / (RS) -q _1u and second q ₂ = U ² / (RS) -q _2u test samples, specific heat calculated by the formula

Where

- specific volume of the solid phase of the samples; V is the volume of the chamber filled with the test material, m is the mass of the test material; P _atm is atmospheric pressure; ΔP - pressure change in the chamber with the studied material; k is the coefficient of proportionality; with _n ρ _n - volumetric heat capacity of the material from which the heat source is made; G ₁ = 1 + h ₀ / (2h ₀ + h _n ) - (h ₀ + h _n ) / (2h ₀ + h _n ) is the coefficient of thickness of the sample h ₀ and heat source h _n , U is the voltage supplied to the sources heat, R, S - resistance and surface area of the heat source, q _1u , q _2u - measured heat fluxes from the surfaces of the first and second plane heat sources,

, A is the slope of the straight section of the dependence

;

- temperature measured in the plane of contact of the studied samples; T ₀ is the initial temperature.

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