CC BY
14
DOI: 10.17277/amt.2018.04.pp.069-081
Optimization of the Heat Pulse Duration, Basic Construction Size of the Device and Conditions of Experimental Data Processing when Measuring Thermophysical Properties of Heat Insulating Materials by Pulse Plane Heat Source Method to Improve the Testing Laboratory
Management System
S.V. Ponomarev*, E.V. Bulanov, V.O. Bulanova, A.G. Divin, G.V. Mozgova, S.S.S. Al-Busaidi
Tambov State Technical University, Sovetskaya St., 116, Tambov, 392000, Russia; * Corresponding author. Tel.: +7 (4752) 63 08 70. E-mail: [email protected]
Abstract
Using the developed mathematical models of errors in measuring the heat conductivity and the thermal diffusivity coefficient of heat insulating materials, the authors propose a method for selecting the optimal: (1) value of the heat pulse duration; (2) basic construction size of the device, and 3) conditions for implementing the algorithm of the experimental data processing.
Keywords
Heat conductivity; thermal diffusivity coefficient; measurement; minimization of errors; optimal data processing conditions; rational value of construction size; heat pulse duration.
© S.V. Ponomarev, E.V. Bulanov, V.O. Bulanova, A.G. Divin, G.V. Mozgova, S.S.S. Al-Busaidi, 2018
Introduction
The need for information on thermophysical properties of heat insulating materials arises while designing new technological processes, controlling product quality in conditions of real-life production processes, as well as while doing mathematical modeling and solving optimization problems of modernized industries [1 - 5]. The main approach to obtaining knowledge of thermophysical properties for new substances and materials remains their experimental measurement [1 - 8].
Traditional methodology of implementing methods of "instantaneous" heat sources did not pay due attention to the choice of [1 - 4]:
1) optimal conditions for measuring and processing primary information;
2) the rational constructionsizes of the used measuring devices, 3) the actual heat pulse duration
Tpulse .
Only recent publications [5 - 10] have discussed the optimization of the mode parameters in the measurement process and rational values of construction sizes of measuring devices. However the
choice of the optimal heat pulse duration has not been considered.
The aim of the research results presented in the article is to increase the accuracy of measuring thermophysical properties of heat insulating materials by the pulse plane heat source method by selecting the optimal conditions for both the measurement process of thermophysical properties and the algorithm for processing the experimental data.
To achieve this goal, the following tasks have been set and solved:
1) a mathematical formulation of the problem of choosing the optimal conditions for carrying out the experiment and subsequent processing of the obtained experimental data was made for the pulse plane heat source method;
2) the problem of choosing the optimal: a) heat pulse duration; b) basic construction size of the measuring device; c) parameters of the algorithm for processing experimental data, has been solved;
3) recommendations on the implementation of the pulse plane heat source method when measuring thermophysical properties of heat insulating materials have been formulated.
A significant drawback of the known methods of a plane "instantaneous" heat source is that the mathematical model of the temperature field T(x, t) in the sample is represented in the form [1 - 10]:
cp
dT (x, t) = ^d 2T (x, t)
ct
dx
W (x, t),
t> 0, - <x> < x < +<x>;
T(x, 0) = To = 0;
T (-«, t) = T (+®, t) = T0 = 0,
where the internal heat source W(x, t) is given as a plane instantaneous pulse W (x, t) = Qn5(x )5(t) , but in fact the heat is supplied to the heater for a period of time 0 <X<XpulSe-
The designations used above: x, t is the spatial coordinate of the sample and time; cp, X is volumetric heat capacity and heat conductivity of the studied material; T0 is the initial temperature of the material (at time t = 0) taken as the start of the temperature scale in each experiment, i.e. T0 = 0; Qn is the amount of heat released per unit surface of the flat heater at x = 0 and at time t = 0; S(x), S(t) - the symbolic Dirac delta functions [1, 5, 11, 12], xpulse is the duration of
the real (not instantaneous) heat pulse applied to the heater.
The physical model of the measuring device is a
cell (Fig. 1) where a sample which consists of three plates: the lower plate 2, the middle plate 1 and the top plate 3 is placed. The highest requirements are imposed on the accuracy of manufacturing the middle plate 1 of a given thickness x, the upper and lower edges of which must be made strictly parallel to each other. The low-inertia flat heater 4 is usually placed between the lower plate 2 and the middle plate 1, and the primary temperature measuring transducer, e.g. a thermocouple 5, is installed between the middle and the top plates.
The measuring cell shown in Fig. 1 includes the following main elements.
I. The sample of the studied material made in the form of three plate elements 1, 2, 3. Note that the thickness x of the plate element 1 along the x axis is selected within the range of 2 to 10 mm depending on the thermophysical properties of the studied heat insulating material. The heights L2 and L3 along the x axis of elements 2 and 3 of the studied heat insulating material should be about 60 mm.
II. The electric heater 4 placed between the sample elements 1 and 2. Dimensions Hy = 100 mm
/
/ / x / z
Fig. 1. A scheme illustrating the structure of a measuring cell and the mutual arrangement of its constituent parts
and Hz = 100 mm of all three elements 1, 2, 3 (of the studied sample) along the y and z axes are chosen according to the size of the electric heater 4 made of a permalloy sheet fixed in electrical insulating (dielectric) holders 6. To obtain the desired electrical resistance ^heater=1,57 Ohm of the heater 4, the permalloy sheet is provided with slots 7, and the outer elements of the slotted heater are provided with electrical contacts 8 to which wires intended for supplying electric voltage power are connected.
III. Primary temperature measuring transducer made in the form of a thermocouple 5 butt welded (from Chrome and Copper wires) and placed between the elements 1 and 3 of the studied sample.
IV. Easily removable thermal insulation shown in Fig. 1 in the form of dashed lines 9. This easily removable insulation is made of foam plastic in the form of three constituent parts, the internal dimensions of which are 2-3 mm larger than the outer overall dimensions (along the y, z axes) of the elements 1, 2 and 3 of the studied sample.
V. To reduce the thermal resistances that occur at the points of contact:
1) of the sample elements 1 and 2 with the heater 4;
2) of the sample elements 1 and 3 with each other and with the thermocouple 5, the structure of the measuring cell involves the use of the constant mass creating the force F, shown in Fig. 1 with an arrow, and ensuring mutual pressing of the elements 1, 2, 3 to each other, to the heater 4 and the thermocouple 5 with a constant force, which allows to stabilize the value of thermal resistances and minimize the effect of their changes on the results of measuring thermophysical properties, i.e. thermal diffusivity a, volume heat capacity cp and heat conductivity X = acp.
Constructions of similar measuring devices and their connection schemes to information-measuring and control systems are also considered in [1 - 5].
The mathematical model of the temperature field T (x, t) in a flat sample (in the case of using a pulsed plane heat source) on the basis of the classical theory of heat conductivity [1 - 4, 11, 12] can be written as:
dT (X'T) = a üfel, t > 0,0 < x <®; (1)
dx
dx 2
. dT(0, x) ( )
- ^-= ^(T,Tpulse) =
dx
T (x, 0) = To = 0; (2)
rh(T)- h(T --)];(3) T K x) = T, = 0, (4)
X
where a = — is the thermal diffusivity coefficient; cP
q(x, Tpulse) is pulse plane heat source with duration xpulse; qc is the heat flux supplied to the sample through the surface x = 0 during the time interval 0 < t < Tpulse h(T), h(T - Tpulse) are the individual asymmetric step functions given by the ratio [12]:
0 at t < 0,
1 at t > 0;
0 at t < t
pulse'
1 at t > t
(5)
pulse
Tpulse is the heat pulse duration of q(x, Tpulse).
The ratio used in the mathematical model is graphically illustrated in Fig. 2.
pulse
Wm/m
2
T pulse ) = 4ch(T) - qch(T - Tpulse )
Y[max - T0 ]
T = T0
- qch(T-Tpulse )
. t . [[ X)-T0 ],°c
T max T0i
a)
[T (x, T )- T0 ] = f (xpulse , a )
----i
tpulse
b)
Fig. 2. The change of physical quantities in time t:
a - the heat pulse q(x, Tpulse) = qch(x) - qch(x - Tpulse) representing the algebraic sum of step functions qch(x)u - qch(x - Tpulse);
b - the temperature difference [[(x, x) - T0 ] at a distance x from the pulse plane heat source
T
X
max
The solution of the boundary value problem (1) - (4) with the simpler boundary condition (3) in the form
dT ( 0, t)
-X-
dx
= qch (T)5
according to the monograph [4] can be written as follows
t(x, t) - to =
2qc I
a t ierfc
2y[cñ
(6)
Using the superposition principle [11, 12] and the known ratio (6), the solution of the boundary value problem (1) - (4) stated above takes the following form:
T (x5 T)- T0 = ^ f ( T,Tpulse5 a)5
(6a)
where
f ( X5 T Tpulse5 a):
Vax ierfc
f
^föTj at 0 < T S TP"l'e;
Vax ierfc
f
xierfc
f
lyfax „
a
T T pulse ) X
2\/O (T _ Tpulse )
at t > t
(7)
pulse
is a
œ
i* 1 2 ierfc(u)=l erfc(w)dw = —,= e-u -uerfc(u)
u Vn
special function [11] which is an integral of the
2 w 2
function erfc (W)= 1 - erf (w); erf (w)=^ f e-W dW [T(x, t)-T0 ] in the time t;
vn 0 ______•........ ...
U (T - Tpulse )
^O (T - Tpulse )
X - X
pulse
= U (t )
t - T
pulse
The dependency graph (8) is shown in Fig. 2b. Fig. 2b shows that the change in the temperature difference [T(x, t)- T0 ] calculated by formula (8) at
the instant of time t = Tmax reaches the maximum value [Tmax — To ] = [T (x, t max ) - To ], and this moment
of time t = Tmax corresponds to the definite value of the dimensionless function
Um = U (W ) =
2y[aXi]
The traditional approach to carrying out the experiment and subsequent processing of the obtained data in the measurement of thermophysical properties by the plane "instantaneous" heat source method consists of the following stages [1 - 4]:
1) a sample of the studied material is made in the form of three plates, a flat heater and a thermocouple are placed between these plates (see Fig. 1), and then the uniform distribution of the temperature field T (x, t) = T0 = const inside the sample of the studied material is expected to be achieved;
2) for a determined period of time 0 < t < Tpulse a
constant electric power P is applied to a flat electric heater with area S and a change in the temperature difference is recorded from the thermocouple signal
is a Gauss error function [11, 12].
For the instants of time t > tpulse the solution (6a)
with consideration of (7), takes the following form
qcx
T (x5 t )-To = ierfc [u (t)] ierfc [U (x - xpulse )
U (x )
where U (t) = —; U ( 2v ax
U (X - Xpulse )
(8)
t - T
pulse
2}ja (T - Tpulse )
is dimensionless functions depending on x, t, Tpulse, a, and
3) the maximum value of the temperature
difference [Tmax - To ]=[ttmax ) — To ] and the value
of the time moment t = Tmax corresponding to this maximum value [Tmax - T0 ] are determined according to the obtained experimental data;
4) according to the obtained values of Tmax, Tmax - T0, taking into account the known values of the
distance x and the total amount of heat Qn supplied to the sample, the required values of the thermal diffusivity coefficient a and heat conductivity 1 of the studied material are calculated from the known formulas
a =
2t]
I =
Qn
Tmax X^/2Пë
T
x
x
2
x
The traditional procedure for carrying out the experiment and processing the obtained data has the following typical drawbacks:
1) high relative error in determining the time moment t = Tmax (about 15 - 20 %),
2) lack of recommendations for the choice of:
- the optimal conditions for processing experimental data;
- the optimal thickness x of the middle sample plate;
- the optimal value of the heat pulse duration
Tpulse.
The developed technique for carrying out the experiment and processing experimental data
Using the technique for carrying out the experiment and then processing the obtained data (when measuring thermophysical properties by the pulse plane heat source method) developed and described in this paper, a sample of the studied material is made in the form of three plates between which a flat heater and a thermocouple are placed. After the uniform distribution of the temperature field T(x, t) = T0 = const is achieved in the sample, a constant power pulse P is applied to the flat electric heater with an area S for a determined period of time 0 < x < xpulse, and an array of changes in the time of
the temperature difference [[(x, x) - T0 ] is recorded in the thermocouple. According to the obtained array, the maximum temperature difference [pmax - T0 ] in the sample and the corresponding value of the time moment Tmax are determined, using which (taking into account the known x and qc = P/(2S)) the thermal
diffusivity coefficient a and the heat conductivity X are calculated for the studied material according to the formulas (10) and (11) below from the experimental data corresponding to the optimal values of the dimensionless parameter y.
The methodology proposed by the authors of this article introduce a dimensionless parameter
y = (T(x, t) - To )/(Pmax - To ) ,
(9)
representing the ratio of the current temperature difference value [[(x, t)- T0 ] to the maximum value
of this difference [[max - T0 ] = [T(x, Tmax )- T0 ] at the
time moment t = Tmax.
Fig. 2b shows that each temperature difference value y (Tmax - T0 ) = T (x, t') - T0, and consequently,
each value y, in the interval 0 < t < Tmax corresponds to a particular time moment t' and the value of the
dimensionless function U ( t') = xj ( 2>/ôr').
In order to determine the value of the
U ( t') = x/( 2^07)
dimensionless
function
corresponding to the given value of the dimensionless parameter y from the data obtained during the experimental measurement of thermophysical properties (in the form of temperature differences array [T (x, t)- T0 ] corresponding to the known time moments t), the following approach is used. If we substitute t = Tmax into the formula (8), we obtain
Tmax - T0 = T (x, Tmax )- T0 =
ierfc [U (xmaX )] ierfc [u (
qcx X
max Tpulse j
U (tmax )
Tmax Tpulse j
(8a)
and after dividing (8) by (8a) the ratio (9) takes the following form
Y
T ( x t) - To
T - T
* max -1 0
ierfc [U (t)] ierfc [U tpulse)
U (t )
U (t)
ierfc [U (TmaX)^ i
ierfc
U (max tpulse
(9a)
U (max )
U (max tpulse )
During the experiment in the numerical modeling of the measurement process, an array of temperature differences [T(x, x)- T0 ] was determined, and then, using the search method, the maximum value of this difference Tmax - T0 was found, the analytical
expression for which is obtained from (8) at t = Tmax in the form (8a). With the known Tmax - T0 and given y the root of the equation (9a) was found as the value of the dimensionless function U (t') . Considering the
known value of the dimensionless function U ( t') and
the known corresponding values of x and t' , on the
basis of the ratio U ( t') = xj (2yjaz''), the sought value
of the thermal diffusivity coefficient has been calculated by the formula
a = x
'( 4t'( (T'))2 ).
(10)
5a,
marg
Aa a
Taking into account (8) and (9), the formula for calculating the heat conductivity takes the following form
Ax , At' , \ AU (t') * = _; 5t' = _ ; 5U w =
1 = O where
U (t'),
t t
pulse'
qcxj [T (x, t')
- T
(11)
O
- ierfc
U (t')
t t
pulse
: ierfc [U (t')]/u (t')
expresses relative errors in the determination of the corresponding physical quantities a, x, t', U(t') .
After the transition (according to recommendations [1, 14, 15]) from the marginal estimation (5a)marg to the root-mean-square (RMS)
estimation (5a)RMS of the error in determining the thermal diffusivity coefficient, we obtain
U(t'^t/(t'-Tpulse)]/(U(t'Ut/(t'-Tpulse)). (5a)RMs = ^4(Sr)2 + (5t')2 + 4[5U(t')_2. (13)
The choice of the optimal values of the dimensionless parameter y and the heat pulse duration Tpulse
In accordance with the theory of errors [1, 14, 15] and the available experience of solving similar problems [5 - 10, 13, 16 - 22], after making the logarithm of the dependence (10) and the subsequent definition of the differential from the left and right parts we get the following:
ln a = 2ln x - ln4 - ln t' - 2lnU ( t' ); d lna = 2d ln x - d ln4 - d lnT - 2d ln U (t');
da 2 dx d 4 dT ^ dU (t') a x 4 t' U (t') .
(12)
According to the theory of errors, it has been carried out [1, 14, 15]:
1) the replacement of differentials da « Aa,
dx « Ax, dT « At', dU (t')«AU (t') by absolute errors Aa, Ax, At', AU (t') ;
2) taking into account that the differential of the constant is d 4 = 0;
3) the replacement of the signs "-"by signs "+" in (12), and the formula for calculating the so-called marginal estimation of the relative error in measuring the thermal diffusivity has been obtained
i
Aa v a ,
marg
Ax At' AU (t')
= 2— +-+ 2--V"
x t' U (t')
or
where
(5a )marg = 25x + 5t' + 25U (t'),
Let us consider the procedure for determining the errors included in the last expression (13) in a more detailed way. Bearing in mind that the value of the time moment depends on the dimensionless parameter Y, i.e. t'= t'(y),, we get
5 [U ( t' ( y ))] = 5U (y )«
dU ( y ) = 1 dU' 1 dU'
U '(y ) U ' dY U ' dY
Ay .
To determine the absolute error Ay we can transform the formula (9) (by analogy with the above)
(5y)
RMS
AT 2 AT
T(x,t')-T, _ T — T max 0 _
= 5(max -T0
or
AY = Y 5(Tmax - T0 = 5 (Tmax - T)))^
(14)
where AT is an absolute error in measuring tempera-
tare difference; 5(rmaX - T0 )=At/(TmaX - T0 ) is a
relative error in measuring the maximum value of the
temperature difference (tmax - T0) ; ay, (5y)rms is
absolute and root-mean-square relative errors in determining the dimensionless parameter y from the experimentally measured values of the temperature
differences [T (X, t')- T0 ] and [Tmax - T0 ] .
The relative error 5t' in the determination of the time moment t' in (13) is also related to the errors in measuring the temperature differences [t(x, t') -T0 j .
d\T ( x, t')- T01
From the ratio —-
дт
AT At
we get
at
T (x, t')- T0
ST'
(15)
where At', 5t' is absolute and relative errors in determining the time moment corresponding to a given value of the dimensionless parameter y.
Substituting (14), (15) into the formula (13), we obtain the following ratio
(5a)
RMS
4(5x)2
AT
which is used in further calculations to get (for the case of measuring the thermal diffUsivity coefficient) the optimal values:
1) the dimensionless parameter y;
2) the basic construction size x of the middle plate of the studied sample (see Fig. 1).
Then the work was done to obtain a ratio for calculating the root-mean-square estimation of the relative error (5X)RMS in measuring the heat
conductivity X. Taking into account that for each known heat pulse duration the values:
1) the time moment т' = т'( у);
2) the dimensionless function U (x'(y)) = U'(у)
depend on the dimensionless parameter y, the previously obtained formula (11) for calculating the heat conductivity has been transformed to the form
X =
4cx
T
( x,T (Y))
- T
J n
f (y )
(17)
where F(y)s O[u(x'(y)), U(x'(yH^)
Then, according to the recommendations of the theory of errors [1, 14, 15], after making the logarithm of the dependence (17) and the subsequent determination of the differentials from the left and right parts, we obtain:
ln X = ln qc + ln x + ln F[ y] - ln[T (x, t) - T0]; d ln X = d ln qc + d ln x + d ln F[y] - d ln[T (x, t) - T0 ];
a T (x T)- To
T dT
- — 5 (Tmax - To )
U ay
(16)
dX dqc dx dF[y] d[T(x, t) - T0] X qc x F [y] [T (x, t) - T,]
Having replaced the differentials dX«AX,
dx « Ax, dqc = Aqc , dF[y] « AF[y],
d[T(x,t')-T0] s a[T(x,t')-T0] d~ A with the corresponding absolute errors AX, Ax, Ay, AF[y], Aqc, A[T (x, t')- T0 ] as adopted in the theory of errors [1, 14, 15], we obtain
AX Aqc Ax AF[y] A[T(x, t) - T0]
X
x F[y] [T(x, t) - To]
Having changed signs "-" with signs "+", we obtain an expression for calculating the so-called marginal estimation [1, 14, 15] of the relative error in measuring heat conductivity
fAXl =Aqc + Ax + AF [y] + A[T (x, t) - TQ] I X yl qc x F[y] [T(x, t) - T„]
or
(5X) where
marg
= 5qc + 5x + 5F[y] + 5 T(x, t) - T0 , (18)
AX Ax Aqc
5Xmarg , 5x ^ —, 5qc =-ZL, 5F[y]: X x qc
x
x
2
5 [ T (x, t')- T0 J are relative errors in determining the
corresponding values X, x, qc, F(y), [T (x, t') - T0 ] .
By analogy with the above procedure for determining errors [1, 14, 15], we obtain:
1 <5F[y]
is an absolute error in measuring temperature differences [ T (x, t') - T0 ] .
Taking into account the above calculations, the expression (18) takes the following form
SF [y] =
F [ Y] dY
Ay ;
(SX) = Sqc +Sx+
V /marg ±c
1 dF ( y) F(y) dy
AY = Y5 (max - T0 = 5 (Tmax - T0 ))+7
A T7
^ [T ( x, T) - T0 ] = ■
^Vl + T2 S (Tmax - T0 ) +
AT
t(x, t) - t,
where
T(x, t) - T, AT = A[ T (x, t')- T, ]
After the transition (according to recommendations [1, 5, 6, 14, 15]) from the marginal estimation (5X)marg to the root-mean-square estimation
(5X)RMS of the relative error in the determination of heat conductivity, we obtain
(SX)
RMS
(S, )2 + (Sx)2 + j -f-S (Tmax - T, )
[ F (y) dy J
2 f +
V
AT
T(x, t) - T, J
(19)
To reduce the errors in the measurement of heat conductivity (5X)RMS during the measuring process, it
is desirable to provide such modes of conducting experiments in which the maximum temperature difference in the sample would not change very much, ie. [Tmax - T0 ]« const. For that purpose, in each
experiment a constant amount of heat Qconst = 2qc t pulse = const should be brought about to
the unit of the sample surface (1 m2). Here we note that the value of Qconst has dimensions J/m2. Then the power consumed by the heater, ensuring the implementation of the formulated condition Qconst« const, can be calculated using the formula P = QconstSITpulse . In this case, the heat flux (from the
heater with power P and area S) supplied to the lower side of the middle plate (indicated by the position 1 in Fig. 1) for a period of time 0 < t < tpulse can be
calculated by the formula
P
plate 2.5 - 5.0 mm, the thermal diffusivity coefficient
—7 2
a = (1.0 - 1.5)-10 m /s and heat conductivity 1 = 0.05 - 0.2 W/(m-K)) in order to obtain the temperature difference in the range of 3 - 7 °C the amount of heat supplied to a unit surface should be in the range of values Qconst « 50 - 80 kJ/m2.
With the values of Qconst = 55 kJ/m2 and the heater
area S = 0.01 m , the power consumed by the heater can be calculated by the formula
550
-. (21)
P = -
pulse
Let us consider the calculation of the error component included in the formula (19).
After making the logarithm of the expression (20), defining the differentials of its left and right parts, and implementing the recommendations of the theory of errors [1, 14, 15], we get:
,c =
2S
(20)
Numerical calculations and experimental studies have shown the following. When using samples of heat insulation materials (with the thickness of the middle
5qc =V(5P)2 + (5S)2 =^AP/P(tpulse))2 +(AS/S)2 .
(22)
When performing calculations, the values P (Tpulse)
have been calculated by the formula (21). After substitution (22) in (19) we get:
(SX)
RMS
AP
P(xpulse)
(SS )2 + (Sx)2
■(y2 + O^Fl^ (max - T0 )) +(AT/(T ^ T)-T0 ^
Table 1
, , , , (^a)p^iQ (SX)r'M'S
The dependence of errors (oa)RMS, (oX)RMS,
5 p = v Rivis ^ .rms from the value of the heat pulse duration t pulse with the thickness of the middle plate in the sample x = 3,55 mm
Root-mean-square error 5 10 15 Xpulse? 20 s 25 30 35 40
(5X)rms 5.1105 5.1272 5.1448 5.1933 5.2423 5.3014 5.3704 5.4489
(Öö)rms 6.6725 6.4696 6.3493 6.2802 6.2418 6.2223 6.2149 6.2147
£ mean 5.8915 5.7984 5.7521 5.7367 5.742 5.7619 5.7925 5.8318
When calculating according to (23), it was assumed that AP = 0.5 W and the relative error in measuring the area S of the heater is SS = 0,5 %.
The results of calculating the dependences of
errors (Sa)RMS, (SX,)
RMS
and their arithmetic mean
values Sm
duration
(Sa)RMS + (SX)
RMS
2
from the heat pulse
pulse
are presented in Table 1.
Table 1 shows that with increasing xpulse the error (5X)rms monotonously increases, the error (5a)RMS decreases, and their arithmetic mean value Smean takes the minimum values in the interval 15 < Tpulse < 25 s.
Results of numerical simulation of the root-mean-square relative errors in the measurement of thermal diffusivity coefficient a and heat conductivity X
Using the obtained formulas (16) and (23), the dependences of the root-mean-square relative errors (5a)RMS, (SX)RMS at the heat pulse duration Tpulse = 21 s have been calculated and shown in Fig. 3.
The following initial data have been used: SS = 0.5 %,
gconst = 55 kJ/m2, 5 < Tpulse < 40 s, a = 1.06 • 10-7 m2/s,
X = 0.194 W/(mK), AP = 0.5 W, x = 2-8 mm, Ax = 0.1 mm, AT = 0.05 K.
During the research it has become obvious that the minimum values of the relative errors (Sa)RMS,
(SX)RMS depend not only on the dimensionless
parameter y, but also on the distance x from the heater placement plane to the thermocouple installation plane,
which measures the temperature difference [ T (x, t') - T0 ] . In this regard, it has been decided to
construct lines of equal error levels on the plane with coordinates y and x. The results of this work are presented in Fig. 4.
The results of calculations presented in Fig. 4 show that (using the initial data in the calculations) acceptable values of the root-mean-square relative errors (Sa)RMS of measuring the thermal diffusivity coefficient a occur at the values of a dimensionless parameter in the range 0.35 < ya < 0.59 and at the values of the basic construction size of the measuring device within the limits 4.0< x < 5.0 mm. The minimum values of the error (Sa)RMS are achieved at Yapt« 0.465 and xapt « 4.5 mm.
At the same time, acceptable values of the root-mean-square relative errors (SX)RMS in measuring the
heat conductivity X occur in the range of values 0.90 < y < 1.0 and 2.4 < x < 3.6. At the same time,
the minimum values of the errors (SX)RMS are
X
X
achieved at Yopt = 1 and xopt« 3.0 mm.
Thus, in order to achieve the minimum values of error (Sa)RMS and (SX)RMS when measuring the
thermal diffusivity coefficient a and heat conductivity X of the studied material, a measuring transducer should be used with the distance between the temperature meter and the heater in the range 3.0 < x < 4.5 mm, in which connection the following formula can be accepted
a + X xopt + xopt _
xopt =—2— =35 mm.
12 11 10
9 8 7 6 5
0.25
(5a) 12
11
10
9 8 7 6 5
0.05
0.35
0.45
0.55
RMS
0.65
a)
0.75
0.85
0.95 у
- x = 2.2
x = 2.5
- x = 2.9
— x = 3.5
x = 4.0
x = 5.0
-x = 3.5 -x = 4.0 x = 4.2 x = 5.0 -x = 6.0
0.25
0.45
0.65
0.85
Y
b)
Fig. 3. Dependencies of the root-mean-square relative errors (8X)rmS and (8a )RMS (on the dimensionless parameter у for different values of the distance x from the location of the plane pulse heat source to the plane in which the thermocouple measuring the temperature difference (x, т') - T0 J is located) during measurements of:
a - heat conductivity X; b - thermal diffusivity coefficient a
x, mm
(Sa)RMs = 6.0
(5a)RMS = 5.8587; x = 4.5 mm
'(Sa)RMs = 6.4
(SX)RMS = 6.0 (5X)rms = 5.4 \
2
^ (SX)rms = 5.2
1 i (SX)RMS = 5.0165; x = 3.0 mm
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 J Fig. 4. Lines of equal levels in root-mean-square relative errors (5a )RMS and (SX)RMS constructed with an optimal heat pulse duration Tpulse of 21 s
7
6
5
4
3
AM&T —
To clarify the optimal value of the heat pulse duration Tpulse ensuring the achievement of the
minimum values of relative errors (5a)RMS, (51)RMS and the arithmetic mean values of the errors
_ [(5aVRMS +(51 X
5
/RMS
in
2
measuring
thermophysical properties a and X, calculations have been performed using formulas (16) and (23) (for the already determined optimal values), the results of which are shown in Fig. 5.
Fig. 5 shows that when changing the heat pulse duration tpulse the arithmetic mean value of root-
mean-square estimations of relative errors takes on minimal values at Tpu^se« 21 s in the range of
18 < tpulse < 24 s.
The above dependence presented in Fig. 5 may leave a wrong impression that when taking into account the influence of the heat pulse duration t pulse,
the measurement errors decrease only by 0.025 -0.063 %. In fact, the application of the measurement and data processing methods proposed in this paper allows the arithmetic mean value of the root-mean-square estimations of relative errors
[(5a )RMS +(5X )i
5
RMS
2
to be reduced by
5.78
5.76
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Fig. 5. Dependencies of arithmetic mean values
[( Sa )
RMS
+ ( 5X )
RMS.
Tpulse, s
5,
2
of root-mean-square
estimations of relative errors (5«)rms , (5X)rms for measuring the thermal diffusivity coefficient a and heat conductivity X on the heat pulse duration Tp„ise when processing data using the pulse plane heat source method considered in this article
5 - 8 % compared with the traditional plane "instantaneous" heat source method [1 - 4].
To illustrate this fact we performed calculations of the thermal diffusivity coefficient a and heat conductivity X values for various heat pulse duration
T pulse values usin§:
- the calculated ratios (10) and (11) proposed in this article;
- the calculated ratios [1 - 4]
ßconst x0
«inst =
Xinst 2л/2Л
2Tmax 2V2ne T max^max
(24)
used in the implementation of the traditional plane "instantaneous" heat source method. In these calculations the exact values aexact = 1.06 • 10-7 m2/s, Xexact = 0.194 W/(mK), xopt = 3.75 mm, have been
used and the power value has been calculated by the formula (21).
After calculating the values of a and X by the formulas (10) and (11), as well as ainst and Xinst according to the formulas (24), the errors
5a = a - aexact • 100%, 5X = X - Xexact-100%,
Xe
a.
exact
exact
5ainst = «inst «exact -100%, aa
5Xinst =
exact
X — X
inst exact
X
-100 %,
exact
have been calculated and then the arithmetic mean
, — [5a + 5X] - [5ainst + 5Xinst ] values 5 = -—-—- and 5inst = inst ^—have
been found.
As a result, the graphs 5 = /1(Tpulse) and
5inst = />(Tpulse) presented in Fig. 6 have been built.
The graphs presented in Fig. 6 show that the following results have been obtained when numerically simulating the measurement of thermophysical properties:
1) when using the plane pulse heat source method proposed in the article, the arithmetic mean values of
the data processing errors 5 = /1(Tpulse) do not exceed 1 %;
2) when processing data with calculation ratios (24), which are the basis of the traditional plane "instantaneous" heat source method [1 - 4], the arithmetic mean values of data processing errors
5
5.82
5.80
5.74
5.72
5.70
8™t, %
the relative measurement errors (ôa)RMS
5.9-6.7 %
Sinst /2(Tpulse)
8 -/1(Tpulse)
11
16
21
26 Tpulse?s
Fig. 6. Comparison of experimental data processing errors Sinst = f2 (Tpulse ) in the case of using the traditional plane "instantaneous" heat source method and experimental data processing errors S = /¿(Tp^) using the plane pulse heat source method considered in this article
^inst
[5ajnst + 5XjnJ 2
= /2(Tpulse) reach 5-8 % with
and (5X)RMS« 5.0-6.0 %.
Note that measuring the thermophysical properties of the studied material, the thermal diffusivity coefficient a and heat conductivity X of which differ from those used in this article (in the source data of the above calculations), should be done as follows:
1) by conducting preliminary measurements it is necessary to experimentally determine the indicative values of the thermal diffusivity coefficient aindic and heat conductivity Xindic of the studied material;
2) by analogy with the method described in the article above it is necessary to carry out calculations (with aindic and Xindic values found) in order to determine (clarify) the optimal values of the parameter уопт and construction sizes xapt and xXpt of the middle
plate used to measure the thermal diffusivity coefficient a and heat conductivity X;
3) by making samples with two middle plates
the heat pulse duration in the range 17 < тpulse < 30 s. with thickness xopt and xopt, or with °ne middle plate
Conclusion
Thus:
1) using the approach for choosing the optimal value of the dimensionless parameter y, the rational construction size x of the middle plate in the sample of the studied material and the heat pulse duration tpulse
proposed in the article reduces the relative measurement errors of thermal diffusivity coefficient and heat conductivity from 10-12 to 4-5 %.
2) to achieve the minimum values of error (5a)RMS in measuring the thermal diffusivity
coefficient a, a sample of the studied material with a plate thickness 4.0 < x < 5.0 mm should be used and the experimental data should be processed at 0.45 < y < 0.48;
3) to ensure the minimum error values (5X)RMS
when measuring heat conductivity X, it is required to use a sample with a thickness of the middle plate 2.8 < x < 3.2 mm, and to process the experimental data at 0.95 <y < 1.0;
4) if it is necessary to simultaneously measure the thermal diffusivity coefficient a and heat conductivity X in one experiment, then the thickness x of the middle plate of the sample from the studied material should be chosen from the range 3.5 < x < 4.0 mm, that ensures
with thickness Xopt = ( x^ + x^ )/2 ;
4) by conducting a series of experiments (with fabricated samples) to carry out measurements and subsequent processing of the obtained data, and as a result to obtain the values of the desired thermal diffusivity coefficient a and heat conductivity X of the studied material.
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