RU2551836C1 - Method for determining non-stationary heat flow - Google Patents

Abstract

FIELD: heating.

SUBSTANCE: method consists in measurement of difference and rate of variation of average temperatures of receiving and inverse surfaces of a heat meter. Novelty of the method consists in the fact that there additionally measured are rates of variation in an average temperature as to a surface area in cross-sections of the heat meter and temperature at points of its side surface.

EFFECT: improving accurate determination of a non-stationary heat flow.

3 dwg

Description

The present invention relates to the field of thermal measurements and can be used in the study of heat transfer and process control in metallurgy, energy and other sectors of the economy.

From the existing level, there is a known method for measuring heat fluxes, the essence of which is to measure two temperature drops on a sensing element with two hyperthermocouples, one after the other, calculate the rate of change of temperature difference measured by the second hyperthermocouple, and determine the unsteady heat flux using the formula (see a USSR N 1045011, G01K 17/08, 1982).

The disadvantage of this method is the low accuracy of the measurement of unsteady heat flow.

Of the known methods, the closest in technical essence is the method of measuring unsteady heat flux (see AS USSR N 1024751, G01K 17/08, 1981), which consists in measuring the temperature difference on the sensitive element, the rate of change of temperature at the receiving and the return surface of the sensing element and determining the magnitude of the unsteady flow according to the formula

, $$q=\frac{\lambda }{b}(t_0-t_b)+\frac{c\rho b}{6}(2t_0^{\text{'}\text{'}}+t_b^{\text{'}\text{'}})$$

,

where λ, s, ρ, b - thermal conductivity, heat capacity, specific gravity and thickness of the sensing element;

t ₀ , t _b - temperature of the receiving and return surfaces of the sensing element;

и $$t_b^{\text{'}}$$

- скорости изменения температур приемной и обратной поверхностей чувствительного элемента. $$t_0^{\text{'}\text{'}}$$

and $$t_b^{\text{'}\text{'}}$$

- the rate of change in temperature of the receiving and return surfaces of the sensitive element.

The disadvantage of this method is the low accuracy of measuring unsteady heat flow due to the large dynamic error and error due to the influence of convective heat transfer on the side surface of the heat meter.

The basis of the invention is the task of developing a method that improves the accuracy of determining unsteady heat flux.

To solve the problem in the known method of measuring unsteady heat flux, including measuring the difference and the rate of change of average temperatures of the receiving and return surfaces of the heat meter, the rate of change of the average temperature over the area and the temperature at the points of its lateral surface are additionally measured in sections of the heat meter, and the heat flux is determined by the formula

, $$q(\tau )=\frac{\lambda }{H}\left[{\overline{t}}_0^{(S)}(\tau )-{\overline{t}}_H^{(S)}(\tau )\right]+\frac{C}{H}{\displaystyle \sum _{k=0}^m{p}_k}\frac{\partial {\overline{t}}_k^{(S)}(\tau )}{\partial \tau }+{\overline{q}}_{\alpha }^{(H)}(\tau )$$

,

where λ, C - thermal conductivity and volumetric heat capacity of the material of the heat meter;

H, S, τ - thickness, cross-sectional area of the heat meter and time;

p _k - weight coefficient at an average rate of temperature change;

m + 1 is the number of sections of the heat meter in which the rate of change of the average temperature is measured;

- поправка на конвективный теплообмен боковой поверхности тепломера, определяемая по измеренной температуре в точках на боковой поверхности, его геометрическим размерам и коэффициенту теплообмена; $${\overline{q}}_{\alpha }^{(H)}(\tau )$$

- correction for convective heat transfer on the side surface of the heat meter, determined by the measured temperature at points on the side surface, its geometric dimensions and heat transfer coefficient;

, $${\overline{t}}_H^{(S)}(\tau )$$

, $${\overline{t}}_k^{(S)}(\tau )$$

- средняя по площади тепломера температура соответственно, на его приемной, обратной поверхностях и в сечениях, перпендикулярных распространению измеряемого теплового потока. $${\overline{t}}_0^{(S)}(\tau )$$

, $${\overline{t}}_H^{(S)}(\tau )$$

, $${\overline{t}}_k^{(S)}(\tau )$$

- the average temperature over the area of the heat meter, respectively, on its receiving, return surfaces and in sections perpendicular to the distribution of the measured heat flux.

An additional measurement in the sections of the heat meter of the rate of change of the area-average temperature and temperature at the points of its lateral surface allows one to reduce the dynamic error of the heat flux measurement and take into account the effect of convective heat transfer on the side surface of the heat meter.

To justify the increase in accuracy, it is necessary to consider the main provisions of the theory on which the claimed method is based. It is based on the mathematical description of heat transfer in the object of study in the form of an integral form of the heat equation. For a heat meter having the shape of a cylinder with a base radius R and thickness H, which perceives the measured unsteady heat flux with one base, and a convective heat flux acts on the side surface, the integral form of the heat equation has the form

, $$q(\tau )={\overline{q}}_{\lambda }^{(H)}(\tau )+{\overline{q}}_C^{(H)}(\tau )+{\overline{q}}_{\alpha }^{(H)}(\tau )$$

,

; $${\overline{q}}_C^{(L)}(\tau )=\frac{C}{H}\frac{\partial }{\partial \tau }{\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}{\overline{t}}^{(S)}}(x,\tau )dx$$

; $${\overline{q}}_{\alpha }^{(H)}(\tau )=\frac{2\alpha }{RH}{\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}t}(x,R,\tau )dx$$

; $${\overline{t}}^{(S)}(x,\tau )$$

- распределение по толщине тепломера средней на площади его поперечного сечения температуры; λ, С, α, τ - теплопроводность, объемная теплоемкость материала тепломера, коэффициент теплопередачи и время.Where $${\overline{q}}_{\lambda }^{(H)}(\tau )=\frac{\lambda }{H}\left[{\overline{t}}^{(S)}(0\tau )-{\overline{t}}^{(S)}(H,\tau )\right]$$

; $${\overline{q}}_C^{(L)}(\tau )=\frac{C}{H}\frac{\partial }{\partial \tau }{\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}{\overline{t}}^{(S)}}(x,\tau )dx$$

; $${\overline{q}}_{\alpha }^{(H)}(\tau )=\frac{2\alpha }{RH}{\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}t}(x,R,\tau )dx$$

; $${\overline{t}}^{(S)}(x,\tau )$$

- distribution over the thickness of the heat meter average over the area of its cross-section of temperature; λ, С, α, τ - thermal conductivity, volumetric heat capacity of the heat meter material, heat transfer coefficient and time.

. Данная модель является точным математическим описание теплопередачи в тепломере, не привязанным к решениям уравнения теплопроводности и условиям его выполнения. Она представляет собой баланс средних по толщине тепломера тепловых потоков, действующих на граничных поверхностях q(τ), $${\overline{q}}_{\alpha }^{(H)}(\tau )$$

проходящего через тепломер $${\overline{q}}_{\lambda }^{(H)}(\tau )$$

вследствие теплопроводности материала и аккумулированного $${\overline{q}}_C^{(H)}(\tau )$$

в нем из-за его теплоемкости. Справедливость данной формы представления уравнения теплопроводности была проверена на различных аналитических и дискретных моделях, а также экспериментально. Для одномерного варианта данная интегральная форма может быть получена путем двукратного интегрирования дифференциального уравнения теплопроводности с пределами от 0 до x и от 0 до H. Повторные интегралы вида $${\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}{\overline{t}}^{(S)}}(x,\tau )dx$$

, $${\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}t}(x,R,\tau )dx$$

определяются аналогично однократному интегралу $${\displaystyle \underset{0}{\overset{H}{\int }}{\overline{t}}^{(S)}}(x,\tau )dx$$

, $${\displaystyle \underset{0}{\overset{H}{\int }}t}(x,\tau )dx$$

, по приближенной формулеFor a parallelepiped-type heat meter, the integral form differs in the form of the expression for determining the thickness average convective heat flux $${\overline{q}}_{\alpha }^{(H)}(\tau )$$

. This model is an accurate mathematical description of heat transfer in a heat meter, not tied to the solutions of the heat equation and the conditions for its implementation. It represents the balance of heat flux averages over the thickness of the heat meter acting on the boundary surfaces q (τ), $${\overline{q}}_{\alpha }^{(H)}(\tau )$$

passing through a heat meter $${\overline{q}}_{\lambda }^{(H)}(\tau )$$

due to the thermal conductivity of the material and the accumulated $${\overline{q}}_C^{(H)}(\tau )$$

in it because of its heat capacity. The validity of this form of representation of the heat equation was tested on various analytical and discrete models, as well as experimentally. For the one-dimensional variant, this integral form can be obtained by double integration of the differential heat equation with limits from 0 to x and from 0 to H. Repeated integrals of the form $${\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}{\overline{t}}^{(S)}}(x,\tau )dx$$

, $${\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}t}(x,R,\tau )dx$$

are defined similarly to the single integral $${\displaystyle \underset{0}{\overset{H}{\int }}{\overline{t}}^{(S)}}(x,\tau )dx$$

, $${\displaystyle \underset{0}{\overset{H}{\int }}t}(x,\tau )dx$$

according to the approximate formula

, $${\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}{\overline{t}}^{(S)}}(x,\tau )dx={\displaystyle \sum _{k=0}^m{p}_k{\overline{t}}^{(S)}(x_k,\tau )}$$

, $${\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}t}(x,\tau )dx={\displaystyle \sum _{k=0}^m{p}_{k}t(x_k,\tau )}$$

, $${\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}{\overline{t}}^{(S)}}(x,\tau )dx={\displaystyle \sum _{k=0}^m{p}_k{\overline{t}}^{(S)}(x_k,\tau )}$$

,

; $$(k=\overline{\mathrm{0,}m})$$

, (i≠k) - весовой коэффициент; t(x_k, τ) - температура в точке x_k; m+1 - количество точек на интервале [0, H], данная формула основана на многочлене Лагранжа; она является точной для многочлена степени m.Where $$p_k={\displaystyle \underset{0}{\overset{H}{\int }}{\displaystyle \underset{0}{\overset{x}{\int }}{\displaystyle \prod _{i=0}^m(x-x_i)}}}/{\displaystyle \prod _{i=0}^m(x_k-x_i)dxdx}$$

; $$(k=\overline{0m})$$

, (i ≠ k) is the weight coefficient; t (x _k , τ) is the temperature at the point x _k ; m + 1 is the number of points on the interval [0, H], this formula is based on the Lagrange polynomial; it is exact for a polynomial of degree m.

In the simplest case, when the temperature distribution over the thickness of the heat meter is one-dimensional, i.e. there is no heat transfer from the side surface, and close to linear, the formula for determining the repeated integral takes the form

. $${\displaystyle \underset{0}{\overset{H}{\int }}dx}{\displaystyle \underset{0}{\overset{x}{\int }}t}(x,\tau )dx=\frac{H^2}{6}[2t(0\tau )+t(H,\tau )]$$

.

Then from the integral form of the heat equation, you can get the calculation formula of the prototype:

. $$q(\tau )=\frac{\lambda }{H}[(t(0\tau )-t(H,\tau )]+\frac{c\rho H}{6}(2t\text{'}\text{'}(0\tau )+t\text{'}\text{'}(H,\tau )]$$

.

The larger the number of points on a given thickness of the heat meter, the higher the degree of the polynomial that describes the temperature distribution, which corresponds to a decrease in the error of the determination of the repeated integral and, therefore, to an increase in the accuracy of determination of the unsteady heat flow. The graphs shown in Fig. 1 and Fig. 2 illustrate the effect additional measurement of temperature and its rate of change at the points of the heat meter.

); по четырем точкам с координатами - х=0r=0,75R, х=0,333H, r=0,75R, х=0,666H, r=0,75R, х=H, r=0,75R (

); по двум точкам с координатами - x₀=0, r=0,75R; х=0,5H r=0,75R (

); и рассчитанные по формулам аналога (

) и прототипа (

) (R=2,5 мм, H=1,2 мм). Погрешность по аналогу и прототипу находится в пределах от -40% до -60% и от -40% до 20%. Погрешность по заявленному способу, в зависимости от числа используемых точек, изменяется от ±0,5%, до ±2%. Повышение точности определения теплового потока достигнута за счет уменьшения динамической погрешности и учета влияния конвективного теплообмена боковой поверхности. Сущность изобретения поясняется чертежами, на которых изображено:Figure 1 shows an example of a change in time of the heat flux density acting on the surface of the heat meter (solid thin line), the values calculated by the formula of the proposed method (symbols) and by the prototype (bold line). Figure 2 shows the errors in determining the heat flux for a given change in heat flux, calculated by the formula of the claimed method: at six points (

); at four points with coordinates - x = 0r = 0.75R, x = 0.333H, r = 0.75R, x = 0.666H, r = 0.75R, x = H, r = 0.75R (

); at two points with coordinates - x ₀ = 0, r = 0.75R; x = 0.5H r = 0.75R (

); and calculated by analogue formulas (

) and prototype (

) (R = 2.5 mm, H = 1.2 mm). The error by analogue and prototype is in the range from -40% to -60% and from -40% to 20%. The error according to the claimed method, depending on the number of points used, varies from ± 0.5% to ± 2%. An increase in the accuracy of determining the heat flux was achieved by reducing the dynamic error and taking into account the influence of convective heat transfer on the side surface. The invention is illustrated by drawings, which depict:

Figure 1 is a graph of the time variation of the density of the heat flux acting on the surface of the heat meter (solid thin line), and the values calculated by the formula of the proposed method (symbols), and by the prototype (bold line).

) и прототипа (

).Figure 2 - error in determining the heat flux calculated by the formula of the claimed method with a different number of points, by the formulas of the analogue (

) and prototype (

)

Figure 3 is an example implementation of the inventive method.

, и сумматор 8, выходное напряжение которого связано линейной зависимостью U(τ)=Kq(τ) с измеряемым тепловым потоком, где K - масштабирующий коэффициент. В устройстве реализована расчетная формула заявленного способа с учетом выбранной приближенной формулы определения повторного интеграла и определения поправки на теплообмен боковой поверхности по сигналу термобатареи.A device that implements the claimed method consists of a heat meter 1 in the form of a cylinder with a radius of the base R and a device for processing analog signals. It uses an approximate formula for determining the repeated integral over two points, one of which is in the middle section, and which is accurate for the parabolic distribution of temperature over the thickness of the heat meter. The heat meter contains: differential thermocouple 2 and thermopile 3, the working junctions of which have the coordinates: x = 0 r = 0.75R, x = 0.5H r = 0.75R, x = H r = 0.75R, measuring average temperatures the base area of the cylinder in the corresponding sections of the heat meter along the x coordinate. The device contains: thermo-emf amplifiers. 4 and 5 for signals proportional respectively to the temperature difference and the weighted sum of the temperatures, a differentiator 6, a scaling voltage divider 7 to obtain a signal proportional to the magnitude of the correction for heat transfer of the side surface $${\overline{q}}_{\alpha }^{(H)}(\tau )$$

, and adder 8, the output voltage of which is connected by a linear dependence U (τ) = Kq (τ) with the measured heat flux, where K is the scaling factor. The device implements the calculation formula of the claimed method, taking into account the selected approximate formula for determining the repeated integral and determining the correction for heat transfer of the side surface by the signal of the thermal battery.

This method has undergone experimental studies in a laboratory setup and theoretical studies on various one-and two-dimensional models of heat meters by simulation method. To calculate the error in determining the heat flux, we used the results of simulation modeling of its measurement on a two-dimensional model with discreteness in the coordinates h = 0.1 mm of a cylindrical heat meter with a diameter of D = 5 mm and a thickness of H = 1.2 mm, having a thermal conductivity λ = 0.3 W / (m · K) and thermal diffusivity a = 1.5 m ² / s. On the lateral surface and the opposite receiving, boundary conditions of the third kind were adopted with the heat transfer coefficient α = 50 W / (m ² · K) and α = 100 W / (m ² · K), respectively.

Claims (1)

A method for determining an unsteady heat flow, which consists in measuring the difference and the rate of change of average temperatures of the receiving and return surfaces of the heat meter, characterized in that they additionally measure the rate of change of the average temperature over the area and the temperature at the points of its lateral surface in sections of the heat meter, and the heat flux is determined by the formula
$$q(\tau )=\frac{\lambda }{H}\left[{\overline{t}}_0^{(S)}(\tau )-{\overline{t}}_H^{(S)}(\tau )\right]+\frac{C}{H}{\displaystyle \sum _{k=0}^m{p}_k}\frac{\partial {\overline{t}}_k^{(S)}(\tau )}{\partial \tau }+{\overline{q}}_{\alpha }^{(H)}(\tau )$$

,
where λ, C - thermal conductivity and volumetric heat capacity of the material of the heat meter;
H, S, τ - thickness, cross-sectional area of the heat meter and time;
p _k - weight coefficient at an average rate of temperature change;
m + 1 - the number of sections of the heat meter, in which the rate of change of average temperature is measured:
$${\overline{q}}_{\alpha }^{(H)}(\tau )$$

- correction for convective heat transfer of the side surface of the heat meter;
$${\overline{t}}_0^{(S)}(\tau )$$

, $${\overline{t}}_H^{(S)}(\tau )$$

, $${\overline{t}}_k^{(S)}(\tau )$$

- the average temperature over the area of the heat meter, respectively, on its receiving, return surfaces and in sections perpendicular to the distribution of the measured heat flux.

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