RU2161301C2 - Method of non-destructive determination of thermal physical properties of materials - Google Patents

Abstract

FIELD: technical physics. SUBSTANCE: heat insulated surface of object of test is exposed to action of thermal pulses of constant power and repetition period. Difference of temperatures between two points on surface of object of test differently spaced from line of action of source is measured under thermal action. Period Δτ of feed of thermal pulses is assigned in correspondence with relation specified in advance. Thermal action by pulses is conducted, after each i-th pulse temperature difference between two points on surface of object of test is measured. Value of dynamic parameter computed beforehand is found across each i-th measurement step. Then value of dynamic parameter across i-th measurement step is compared with value of this parameter across i-1 measurement step. Test is terminated when value of dynamic parameter across i-th measurement step turns lesser than value of dynamic parameter across i-1 measurement step, then thermal physical properties are calculated with use of certain formulas. EFFECT: increased speed and accuracy of determination of sought-for thermal physical properties. 6 dwg, 1 tbl

Description

 The present invention relates to the field of technical physics, and in particular to the field of determining the thermophysical properties of materials.

 A known method for the comprehensive determination of the thermophysical characteristics of materials (ed. St. USSR N 1381379, G 01 N 25/18, 1988), consisting in the thermal effect on the surface of a semi-infinite thermally investigated body from a linear heat source by pulses with a given duty cycle and fixing the number of applied on the surface of the pulses from the onset of heat exposure to the point in time when the temperature at two points on the surface, different from the line of action of the source, reaches a steady-state constant value, and oschnost heat pulses applied to the surface of the test body is changed from the start of exposure until they reach a steady temperature value control points according to the relationship given in the description, and then define the required thermal characteristics of the formulas given in the description. The disadvantages of this method are: the need to control the amplitude of the thermal pulses, the lack of control over the course of thermostating, insufficient speed and the lack of accounting for the fact that the real test objects have finite dimensions.

 Closest to the proposed technical essence is a method of non-destructive testing of the thermophysical characteristics of materials (ed. St. USSR N 1402892, G 01 N 25/18, 1988), which consists in thermal action in the contact plane of the investigated semi-limited body and heat insulator from a linear heat source by pulses constant power, measuring the steady-state excess temperature at different points of the contact plane of the body under investigation and the heat insulator located at fixed distances from the action line of the heat source a, determining the time intervals from the moment of supply of one heat pulse to the times when the temperature at the controlled points becomes equal to its initial value, then the minimum frequency of thermal pulses is set and the frequency of thermal pulses begins to increase in accordance with the dependence given in the description, determine such a frequency of thermal pulses at which the steady-state value of the excess temperature at the first control point located at a short distance from the line of action of the heat source, will become equal to the predetermined value in advance, then continue to increase the frequency of repetition of thermal pulses according to the initially indicated dependence until the steady-state value of the excess temperature at the second control point becomes equal to the predetermined value, and determine this frequency of repetition of thermal pulses, and the desired thermophysical characteristics are determined by the formulas given in the description. This method provides the possibility of non-destructive testing of thermophysical characteristics - thermal conductivity and thermal diffusivity. However, the speed and accuracy of determining the thermophysical properties are insufficient due to the need for complex preparatory operations, and also due to the lack of accounting for the fact that real test objects have finite dimensions.

 The technical task of the invention is to increase the speed and accuracy of determination of the desired thermophysical properties.

где

Fо₁ и Fо₂ определяются из уравнения

где δ - допустимая погрешность определения теплофизических свойств, a_max - максимальное значение из диапазона определения температуропроводности, k - целое положительное число большее 5, r₁ - расстояние от линии действия источника до ближней точки контроля поверхности объекта измерения,

Fо - число Фурье, осуществляют тепловое воздействие импульсами, измеряют после каждого 1-го импульса разность температур между двумя точками поверхности объекта испытания, определяют величины b_1i и b_0i согласно формулам

где T_j ^* - разность температур после каждого j-го импульса, Δτ - период подачи импульсов, k - целое положительное число большее 5, сравнивают величину b_1i с b_1i-1, испытания заканчивают при выполнении условия b_1i < b_1i-1 и рассчитывают теплофизические свойства по формулам

(1)
где α,β - постоянные, определяемые конструктивными и режимными параметрами, применяемого устройства; b_0i-1, b_1i-1 - коэффициенты, непосредственно определяемые из снятой зависимости разности температур от времени на (i-1)-ом шаге измерения.This is achieved by the fact that in the non-destructive method of determining the thermophysical properties of materials, the thermally insulated surface of the test object is exposed to a line with constant-power heat pulses and a period of repetition, and before the thermal effect, the temperature difference is measured between two points of the surface of the test object that are different from the source action line at distances r ₁ and r ₂ , r ₂ > r ₁ , until this temperature difference becomes less than the predetermined value, while the second point and is located at a distance from the line of action of the source r ₂ not greater than the thickness of the test object, designate the period of supply of thermal pulses based on the ratio

Where

Fo ₁ and Fo _{2 are} determined from the equation

where δ is the permissible error in the determination of thermophysical properties, a _max is the maximum value from the range of thermal diffusivity, k is a positive integer greater than 5, r ₁ is the distance from the source line to the near point of control of the surface of the measurement object,

Fо is the Fourier number, the thermal action is performed by pulses, after each 1st pulse the temperature difference between two points of the surface of the test object is measured, the quantities b _1i and b _{0i are} determined according to the formulas

where T _j ^* is the temperature difference after each j-th pulse, Δτ is the pulse supply period, k is a positive integer greater than 5, compare the value of b _1i with b _1i-1 , the tests are completed under the condition b _1i <b _1i-1 and calculate the thermophysical properties by the formulas

(1)
where α, β are constants determined by the structural and operational parameters of the device used; b _0i-1 , b _1i-1 - coefficients directly determined from the measured dependence of the temperature difference on time at the (i-1) -th measurement step.

 The essence of the proposed method is illustrated by the following theoretical justification.

(2)
где q_о - мощность одного импульса на единицу длины нагревателя, τ₀ - длительность одного импульса, Δτ - период следования импульсов.In the case of a linear pulsed heat source, the power on the heater is a periodic function of time (Fig. 1) and can be written as

(2)
where q _о is the power of one pulse per unit length of the heater, τ ₀ is the duration of one pulse, Δτ is the pulse repetition period.

Подставив (2) в (3), предварительно разложив (2) в ряд Фурье, имеем следующее выражение для описания температурного поля системы с линейным нагревателем, действующим на поверхности полуограниченного тела:

где

- экспоненциальный интеграл.According to the source method, the formula that determines the temperature field in half space from a linear heat source with an arbitrary law of power supply to the heater has the form

Substituting (2) in (3), having previously expanded (2) in the Fourier series, we have the following expression for describing the temperature field of a system with a linear heater acting on the surface of a semi-bounded body:

Where

is the exponential integral.

стоящая в правой части уравнения (4), представляет собой ограниченную и периодическую функцию времени и при числе импульсов n > 5-7 ею можно пренебречь, отнеся к случайной погрешности измерения температуры. То есть для точки поверхности полуограниченного тела будет справедливо следующее выражение:

Так как реальные объекты измерения имеют конечные размеры, то формула (5) будет справедлива только некоторое ограниченное время испытания. С целью фиксирования времени, которое исследуемый объект можно считать полуограниченным, в предлагаемом способе измеряется разность температур между двумя точками его поверхности, причем дальняя точка находится на расстоянии от нагревателя, не большем толщины объекта испытания. Тогда можно записать выражение для этой разности температур

где T^* - разность температур между двумя точками поверхности объекта испытания, τ - время, r₁ - расстояние от линии действия источника тепла до ближней точки контроля,

- отношение расстояний между нагревателем и дальней точкой контроля температуры r₂ и нагревателем и ближней точкой контроля температуры r₁.Amount

standing on the right side of equation (4) is a limited and periodic function of time, and for the number of pulses n> 5-7 it can be neglected, referring to the random error of temperature measurement. That is, for the surface point of a semi-bounded body, the following expression will be valid:

Since real measurement objects have finite dimensions, formula (5) will only hold for a limited limited test time. In order to fix the time that the test object can be considered semi-limited, the proposed method measures the temperature difference between two points of its surface, and the far point is at a distance from the heater, not more than the thickness of the test object. Then we can write the expression for this temperature difference

where T ^* is the temperature difference between two points of the surface of the test object, τ is time, r ₁ is the distance from the action line of the heat source to the nearest control point,

- the ratio of the distances between the heater and the far point of temperature control r ₂ and the heater and the near point of temperature control r ₁ .

где

- число Фурье (безразмерное время)

Характерной особенностью функции Θ^*(s,Fo) является то, что для любого заданного значения s > 1 на графике _Θ*_{= Θ}*_(s,Fo) будет наблюдаться точка перегиба (фиг. 2), соответствующая значению Fо^*:

которое получено из решения уравнения

Касательная к точке Θ^*(s,Fo^*) кривой Θ^*(s,Fo) имеет вид (фиг.2)
_Θ*_{(s,Fo) = p(s)(ln[Fo]+h(s)), (10)}
где

На фиг. 3 представлена зависимость p=p(s), а на фиг. 4 - зависимость h= b(s). Точка перегиба будет соответствовать максимуму кривой

и будет соответствовать значению

Так как дальняя точка поверхности, относительно которой измеряется температура ближней точки поверхности объекта испытания, связана с его толщиной, то для точек экспериментальной кривой, расположенных правее Fо^* (Fо > Fо^*) начнут сказываться граничные условия у поверхности объекта испытания противоположенной поверхности, на которую оказывается тепловое воздействие, и в уравнение (7) необходимо вводить дополнительный член, который будет тем больше, чем правее относительно Fо^* находится точка экспериментальной кривой (чем больше величина (Fо - Fо^*)). Так как способ предполагает неразрушающее определение теплофизических свойств, то учет этих граничных условий в явном виде затруднителен.Expression (6) can be rewritten in dimensionless form

Where

- Fourier number (dimensionless time)

A characteristic feature of the function Θ ^* (s, Fo) is that for any given value s> 1 on the graph _Θ * _{= Θ} * _{(s, Fo) there} will be an inflection point (Fig. 2) corresponding to the value of Fо ^* :

which is obtained from the solution of the equation

The tangent to the point Θ ^* (s, Fo ^* ) of the curve Θ ^* (s, Fo) has the form (Fig. 2)
_Θ * _{(s, Fo) = p (s) (ln [Fo] + h (s)), (10)}
Where

In FIG. 3 shows the dependence p = p (s), and in FIG. 4 - dependence h = b (s). The inflection point will correspond to the maximum of the curve

and will match the value

Since the far point of the surface, relative to which the temperature of the near point of the surface of the test object is measured, is related to its thickness, then for the points of the experimental curve located to the right of Fо ^* (Fо> Fо ^* ) the boundary conditions at the surface of the test object of the opposite surface, thermal effect occurs, and it is necessary to introduce an additional term in Eq. (7), which will be the greater, the more the point of the experimental curve is located to the right of Fo ^* (the larger the value ( Fo - Fo ^* )). Since the method involves a non-destructive determination of thermophysical properties, it is difficult to explicitly take into account these boundary conditions.

Данное уравнение можно записать в двух формах

и

где z = ln[τ] - новая переменная, α и β - постоянные, определяемые режимными и конструктивными особенностями устройства, реализующего способ:

β = h(s)-2ln[r₁], (16)
b₁ и b₀ - величины, непосредственно определяемые из экспериментально снятых термограмм:

λ,a - искомые теплофизические свойства.Taking into account the foregoing, and also taking into account the fact that the temperature difference is measured with a certain random error, which also includes the discarded sum of a series of expression (4), on the experimentally recorded thermogram in the coordinates T ^* = T ^* (ln [τ]) select the rectilinear (working) section corresponding to the equation

This equation can be written in two forms

and

where z = ln [τ] is a new variable, α and β are constants determined by the regime and design features of a device that implements the method:

β = h (s) -2ln [r ₁ ], (16)
b ₁ and b ₀ - values directly determined from experimentally captured thermograms:

λ, a are the desired thermophysical properties.

Для определения рабочих участков экспериментально снимаемых термограмм в предлагаемом способе использовали следующее:
1. свойства функции (7), согласно которым на термограммах в координатах T^*(ln[τ]) рабочим участкам соответствуют прямолинейные отрезки;
2. качественную информацию, полученную при анализе выражения (7), на основе которого было получено расчетное соотношение (12). Рабочему участку термограммы будет соответствовать вершина кривой

3. необходимость статистической обработки результатов эксперимента, так как разность температур в эксперименте снимается в дискретных точках с определенной случайной погрешностью.Then the expression for determining the thermophysical properties of the studied materials will have the form

To determine the working areas of experimentally removed thermograms in the proposed method used the following:
1. properties of function (7), according to which on the thermograms in the coordinates T ^* (ln [τ]) the straight sections correspond to the working sections;
2. qualitative information obtained in the analysis of expression (7), on the basis of which the calculated relation (12) was obtained. The top of the curve will correspond to the working portion of the thermogram

3. the need for statistical processing of the experimental results, since the temperature difference in the experiment is removed at discrete points with a certain random error.

i-k, k+1,..., n (21)
на основе следующих формул:

где T_j ^* - значение разности температур, полученное в результате эксперимента в точке с номером j для i-го отрезка, b_0i, b_1i - оценка коэффициентов α_0i, α_1i уравнения (21),

- температура, рассчитываемая по уравнению (21), z = ln[τ].
На графике зависимости b_1i от

будет наблюдаться вершина, соответствующая рабочему участку термограммы. С целью уменьшения времени эксперимента в предлагаемом методе на каждом шаге измерения вычисляют b_1i которое сравнивается с b_1i-1. Испытания заканчивают, когда b_1i< b_1i-1, то есть тогда, когда b_1i начнет уменьшаться. Для того, чтобы гарантировать определение теплофизических свойств с заданной точностью, в предлагаемом способе назначают период подачи импульсов исходя из следующего. Для определения теплопроводности используется коэффициент b₁, непосредственно определяемый из термограммы, который представляет собой величину, прямо пропорциональную значению p(s), которое в свою очередь есть максимум кривой

соответствующее значению Fо^*. Поэтому для точек, лежащих правее и левее Fо^*, погрешность будет иметь вид

Решив это уравнение относительно Fо, получим два значения - Fо₁, лежащее левее Fо^*, и Fо₂, лежащее правее Fо^*. На основе Fо₁ и Fо₂ назначается частота подачи импульсов

где δ- погрешность определения теплофизических свойств, a_max - максимальное значение из диапазона определения температуропроводности, k - целое положительное число большее 5, r₁ - расстояние от линии действия источника до ближней точки контроля поверхности объекта измерения.Assuming that at least k points belong to the working section of the thermogram, and in total there are n points on the thermogram, we consider sequentially segments of thermograms with the numbers of points 1 ... k, 2 ... k + 1, ..., nk ... n. Denote each of the segments by the index i (i = k ... n). For each of these segments we construct the equations of linear dependencies

ik, k + 1, ..., n (21)
based on the following formulas:

where T _j ^* is the value of the temperature difference obtained as a result of the experiment at the point with number j for the ith segment, b _0i , b _1i is the estimate of the coefficients α _0i, α _{1i of} equation (21),

is the temperature calculated by equation (21), z = ln [τ].
In the graph of the dependence of b _1i on

a vertex corresponding to the working portion of the thermogram will be observed. In order to reduce the experiment time in the proposed method, at each measurement step, b _1i is calculated which is compared with b _1i-1 . Tests end when b _1i <b _1i-1 , that is, when b _1i begins to decrease. In order to guarantee the determination of thermophysical properties with a given accuracy, in the proposed method, a pulse supply period is assigned based on the following. To determine the thermal conductivity, we use the coefficient b ₁ directly determined from the thermogram, which is a value directly proportional to the value of p (s), which in turn is the maximum of the curve

corresponding to the value of Fo ^* . Therefore, for points to the right and left of Fo ^* , the error will have the form

Solving this equation with respect to Fo, we obtain two values - Fo ₁ lying to the left of Fo ^* , and Fo ₂ lying to the right of Fo ^* . Based on Fo ₁ and Fo _{2, the} pulse frequency is assigned

where δ is the error in determining the thermophysical properties, a _max is the maximum value from the range of thermal diffusivity, k is a positive integer greater than 5, r ₁ is the distance from the source line to the near point of control of the surface of the measurement object.

Since the points lying between τ ₁ and τ ₂ are used to determine the coefficient b _{1 , /} the real error in determining the thermophysical properties will be less than the value δ on the right side of equation (26).

The implementation of the method is illustrated in the diagram shown in FIG. 6. When implementing the method using the test sample 1, which in real conditions can be a finished product. A linear heater 2 and a sensor 3 are placed on the free surface of the sample, which measures the temperature difference between two points of the surface of the sample located at a distance r ₁ and r ₂ from the heater, so that the following condition is met:
H _n <r ₂ ,
where r ₂ is the distance from the heater to the furthest point of control, H _n is the thickness of the sample (product).

In preparation for testing, a thermal contact is created between the heater and the sample, as well as between the sensor and the sample. The temperature control process is monitored using a measuring and computing device (IVU) 4. When the temperature difference T ^* becomes less than the predetermined value determined by the accuracy of temperature measurement, the IVU supplies an electric current with pulses of constant power and duration to the heater using a stabilized power source 5 with a frequency Δτ determined from expression (27). Simultaneously with the supply of electric current after each pulse, a temperature difference T ^{* is} measured.

At each i-th step, the value of the coefficient b _{1i is} determined according to formula (22) and compared with the value of b _1i-1 at the (i-1) -th measurement step. The tests are completed at the i-th step under the condition b _1i <b _1i-1 . Thermophysical properties are determined by formulas (1) in accordance with the above methodology.

 In order to verify the method, measurements were carried out on samples of polymethyl methacrylate (PMMA) and ripor with known thermophysical properties, as well as samples of fluoroplast-4 and caprolon-B, the thermal conductivity of which was determined on an IT-3 thermophysical instrument by the stationary method. The table shows the results of determining the thermophysical properties of the above materials. Tests showed the consistency of the obtained values of thermal conductivity and thermal diffusivity with the known and determined stationary method.

The proposed method allows to reduce the systematic error of measuring the thermophysical properties of materials by monitoring the progress of temperature control by the temperature difference between two points on the surface of the test sample (product). Since the temperature difference between two points of the surface of the sample is fixed during the entire time of the test, the influence of the systematic error due to the fact that not all the time of the test the sample can be considered semi-limited decreases. Monitoring the value of parameter b ₁ during the test reduces the time of the experiment. Since the proposed method uses not individual points of the thermogram, but its portion, the random component of the error in determining the thermophysical properties of the material under study decreases. In the case of studying the thermophysical properties of dispersed materials according to the proposed method, their average integral values are determined, which also increases the accuracy of determining the thermophysical properties.

Claims (1)

A non-destructive method for determining the thermophysical properties of materials, which consists in the fact that the heat-insulated surface of the test object is affected by a line with constant-power heat pulses and a repetition period, characterized in that the temperature difference between two points of the test object’s surface that are different from the action line is measured first source at distances r ₁ and r ₂ , r ₂ > r ₁ , until this temperature difference becomes less than a predetermined value, at the second point is located at a distance from the line of action of the source r ₂ not greater than the thickness of the test object, the period of supply of thermal pulses is assigned based on the ratio

Where

and Fo ₁ and Fo _{2 are} determined from the equation

where δ is the permissible error in determining the thermophysical properties, and _max is the maximum value from the range of thermal diffusivity, k is a positive integer greater than 5, r ₁ is the distance from the source line to the near point of control of the surface of the measurement object,

Fo is the Fourier number, a thermal effect is performed by pulses, after each i-th pulse, the temperature difference between two points of the surface of the test object is measured, the quantities b _1i and b _{0i are} determined according to the formulas

where T * _j is the temperature difference after each j pulse;
Δτ is the pulse supply period;
k is a positive integer
compare the value of b _1i with b _1i-1 , the tests are completed when the condition b _1i <b _{1i-1 is} fulfilled and the thermophysical properties are calculated by the formulas

where α, β are constants determined by the structural and operational parameters of the device used;
b _{0 i-1} , b _{1 i-1} - coefficients directly determined from the measured dependence of the temperature difference on time at the i-1 measurement step.

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