CC BY
10
DOI: 10.17277/amt.2019.03.pp.066-079
Optimization of the Linear Pulsed Heat Source Method and Basic Structural Dimensions of the Device for Measuring Thermophysical Properties of Solid Materials to Improve the Laboratory Management System
S.V. Ponomarev*, V.O. Bulanova, E.V. Bulanov, A.G. Divin
Tambov State Technical University, 106, Sovetskaya, Tambov, 392000, Russia * Corresponding author. Tel.: +7 4752 63 08 70. E-mail: [email protected]
Abstract
Using the methods of metrology and the theory of thermal conductivity, the mathematical models of relative errors in measuring the volumetric heat capacity and thermal diffusivity of solid materials were developed using the linear pulsed heat source method, which created a method for choosing the optimal conditions for processing the experimental data, the main structural dimensions of the measuring device, and optimal duration of the heat pulse.
Keywords
Thermal diffusivity; volumetric heat capacity; measurement; relative errors; minimization; heat pulse; data processing; structural dimensions; optimization.
© S.V. Ponomarev, V.O. Bulanova, E.V. Bulanov, A.G. Divin, 2019
Introduction
In the conditions of rapid development of technologies and creation of new materials, it is important to study their thermophysical properties (TPPs). In the last decade, research into the development and modernization of new methods for implementing the so-called "instantaneous" sources of heat (moisture) has been quite actively conducted [1-16].
Traditional methods of implementing methods of "instantaneous" heat sources did not pay enough attention to the choice of (1) optimal conditions for measuring and processing primary information; (2) the rational structural dimensions of the measuring devices used; (3) the actual duration of the heat pulse Tp [1-4]. Only recently, publications [5-10] addressed the issues of optimizing the operating parameters of the measurement process and the rational values of the structural dimensions of measuring devices, but the question of choosing the optimal duration of a thermal pulse was not discussed.
The purpose of this article is to come up with recommendations on choosing the best (optimal from the point of view of minimizing the errors in measuring TPPs): (1) conditions for processing the data obtained
during the experiment; (2) distance between the heater and the temperature meter; (3) duration of the heat pulse supplied to the linear heater.
To achieve this goal the following problems were set and solved:
(1) mathematical formulation of the problem of choosing the optimal conditions for the experiment and subsequent processing of the experimental data for the linear pulsed heat source method were proposed;
(2) the problem of choosing (a) the optimal duration of the heat pulse; (b) the main structural dimensions of the measuring device; (c) parameters of the experimental data processing algorithm was solved;
(3) recommendations on the implementation of the linear pulsed heat source method when measuring TPPs of solid materials were given.
In the mathematical model of the temperature field T(r, t) in radial coordinate system:
dT (r, t) 1 d
cp-= X--
dT r dr
dT (r, t)
dr
w(r, t);
t> 0; 0 < r <<x>; T (r,0) = T = 0;
(1) (2)
dT (0, t)
= 0;
dr
T (o, t) = To = 0
(3)
(4)
in which the internal heat source W(r, t) was previously set as a linear instantaneous pulse
W (r, t) = Qlin 5(r )8(t),
to achieve the goal formulated above, a source of heat in the form of a heat pulse with a duration of Tp is set
W(r, t) = qHn 5(r - 0)[(t - 0) - h( - Tp )], (1a)
providing for the supply of heat to the linear heater for a period of time 0 < t < Tp .
The above designations are as follows: r, t is the spatial coordinate of the sample and time; cp, 1 are volumetric heat capacity and thermal conductivity of the material under study; T0 is initial material temperature (at time t = 0), assuming that the start of temperature scale in each experiment i.e. T0 = 0; Qlin is the total amount of heat released in a unit length of a linear heater if r = 0 at time t = 0; S(r), S(t) are Dirac's symbolic delta functions [3, 11, 12], is Tp the
duration of the real (not instantaneous) heat pulse supplied to the heater; qlin = Qlin/Tp is the amount of
heat released by the unit of length of the linear source of heat per unit of time; H(t- o), h(T-Tp )are single step
functions [11, 12].
The physical model of the measuring device
The physical model of the measuring device is a cell (Fig. 1) into which a sample consisting of two plates is placed - the bottom one and the top one. A linear electric heater (e.g. made in the form of thin metal wire made of nichrome, manganin or constantan) with a length L is placed between the upper face of the lower plate and the lower face of the upper plate, while a temperature meter (in the form of a resistance thermometer made from copper or tungsten wire or in the form of a thermocouple) is placed at the distance r from the heater in the same plane. Diagrams and designs of similar measuring devices were considered in [1-6, 13, 19, 20, 22].
Presented in Fig. 1 the measuring cell includes the following main elements:
- a sample of the material made in the form of two plates 1, 2. Heights L1 and L2 along the axis x of the plates 1 and 2 the heat-insulating material under investigation have to be about 60 mm;
Fig. 1. Measuring cell of the pulsed linear heat source method with the mutual arrangement of its components
- the measuring plate 3, consisting of a frame on which two wire elements are stretched (heater and temperature meter), the design of which will be described below;
- easily removable thermal insulation conditionally shown in Fig. 1 as dashed lines 4. This easily removable insulation is made of foam plastic in the form of three component parts, the internal dimensions of which are 2-3 mm higher than the external dimensions (along the y, z axes) of plates 1 and 2 of the sample of the material under study;
- to reduce the thermal resistances that occur at the points of contact of the plates 1 and 2 of the sample between themselves and with the wire elements of the measuring plate 3, the design of the measuring cell involves the use of a constant-weight load, which creates the force F shown in Fig. 1 by an arrow, and ensuring mutual pressing of plates 1 and 2 to each other with a constant force, which allows stabilizing the value of thermal resistances and minimizing the effect of their changes on the measurement results of the desired thermophysical properties, namely, thermal diffusivity a, volumetric heat capacity cp and thermal conductivity 1 = a cp.
The design of the measuring plate (a holder of the linear heater and temperature meter)
The method used is based on the use of two thin wires 1 and 2 (a linear heater and a temperature meter), stretched over a strong, but rather thin frame 3 (Fig. 2).
x
y
5 7 \ ^
10, oc
7
5
v
V
6 Y>7
^4
100,00
0,00
Fig. 2. Drawing of the measuring plate for the linear pulsed heat source method
The frame 3 of the measuring plate of the linear pulsed heat source method consists of a dielectric material (textolite) measuring 120 by 140 mm and a thickness of 2-3 mm. Inside it is cut a square hole 100 by 100 mm in size, in the middle of which two wire elements 1 and 2 are stretched. The linear heater 1, madein the form of thin metal wire made of nichrome, manganin or constantan, heats the sample surface when a pulse tp, is applied to it, and a temperature meter 2 is placed at a distance r from the heater (in the form of a resistance thermometer (made of copper or tungsten wire) or thermocouple). The distance r between the wire elements 1 and 2 is set (and adjusted) by the braces 6. The heater 1 and the temperature meter 2 are fixed on the frame 3 by means of springs 4. The wires 1 and 2 are soldered at both ends to insulated copper wires 5 of large cross section. Solder joints are marked with 7.
The mathematical model of the temperature field
The mathematical model of the temperature field T(r, t) in the material under study (in the case of a pulsed linear heat source) can be written in the form (1) - (4) with a modified function of the source of heat W(r, t), given by the formula (1a).
The solution of the boundary value problem (1) -(4) with a continuously operating constant source of heat W (r, t) = qlin s(r - 0) h(t - 0), has the form [2]
[T (r, t)-T ] = - ^ Ei 4nx
f 2^ r
4aT
(5)
where qlin = Qlin/tp is the amount of heat released by
the unit of length of the linear source of heat per unit of time; a = x/cp is temperature conductivity coefficient; H(t - 0) is single step function [16, 17], having the form:
, . f0 over t< 0; h(t-0) = l
[1 over t > 0.
If (in the case of a pulsed heat source considered in the article, the duration 0 < t < tp) wet W(r,t) = qlin8(r - 0)[h(t)- h(t -tp)], in the basis of the principle of superposition and the well-known solution (5), we obtain that if t > tp:
[T (r, t)-T) ] = -
qli
4nx
Ei
r
4at
2 N\ f - Ei
J
v
4a(T-Tp)j
(6)
Thus, the general solution of problem (1) - (4) taking into account (5) and (6) takes the form:
[T (r, t)- T0 ]
glin
4nx
glin
4nx
Ei (- U(t)) over 0 < t < tp; [Ei (- U(t))- Ei(- U(t-t
over t > t
p
(7)
M e-t
where Ei(u )= j— dt = -j— dt is integral
exponential function [2, 11, 12]; U (t) =-,
4at
r 2
U (t-tp ) = —;-r are dimensionless functions
4a(-Tp )
depending on r, t, tp, a, and
J2
u(t - tp ) -
4a(-Tp)
p> 4at
t-t
p
U (t)-
t - tr
v t j
The traditional approach to the experiment and the subsequent processing of the data
The traditional approach to the experiment and the subsequent processing of the data when measuring TPPs using the linear "instantaneous" heat source consists of several steps [1-4]:
3
2
r
—cc
2
2
r
t
(1) manufacture of two massive plates from the material under study (their thickness should be less than ten to twenty times the distance r between the electric heater and the temperature meter);
(2) placing the heater and temperature meter at a distance r from each other between the two plates, and then achieving a uniform distribution of the temperature field T(r, t) = T0 = const inside the sample of the material under study;
(3) supply of constant power P for a specified short period of time 0 < t < Tp to a linear electric heater of length L and recording the change in temperature difference over time [TT(r, t)- To ] by a signal from a temperature meter;
(4) determination from the obtained experimental data of the maximum value of the temperature difference [max - To ] = [(r, Tmax) - To ] and the value of the moment of time t = Tmax, corresponding to this maximum value [max - T0 ] ,as well as the amount of heat Qlin = qlinTp , released in a unit of length of the heater;
(5) calculation by the obtained values Tmax, [Tmax - T0 ] ,taking into account the known r, QUn, the desired values of thermal diffusivity a and thermal conductivity 1 of the material under study.
For the traditional order of the experiment it is typical to have:
1) high relative error in determining the point in time t = Tmax (about 15-20 %),
2) lack of recommendations on selecting:
- optimal conditions for processing experimental
data;
- optimal distance r between the linear heater and the temperature meter;
- the optimal value of the duration Tp of the
thermal pulse.
The method of processing experimental data
The method of processing experimental data proposed in this article is based on the use of a dimensionless parameter
t (r, t)- T0
T -T
max 0
(8)
which is a temperature difference relation [[(r, t)- To ] (at time t) to maximum temperature
difference [[( Tmax ) - T0 ] = [Tmax - T0 ], taking place
at time t = Tmax.
Moreover, each value of the temperature difference y [Tmax - T0 ] = [[(r, t) - T0 ], that is, each value of the dimensionless parameter y corresponds to a specific value of time t.
In the mathematical modeling of measuring TPPs first with a constant step At in time t by formula (7), the values of temperature differences [T(r, Ti )- T0 ] and relevant times Ti, i = 1, 2, ..., n, were calculated and recorded (in the form of arrays), and then by the data array [T(r, Ti) - T0 ], i = 1, 2, ., n, the maximum
value [Tmax - T0 ] of this difference was found
[T (r, Tmax )- T0 ] =
qii,
\Ei
- U(Tmax Y
t — t
lmax
- Ei[- U(Tmax)]
(9)
After this, the method of interpolation, the value of time t, corresponding to the value of the temperature difference [T (r, t) - T0 ] = y [Tmax - T0 ] was found.
Dividing the dependence (7) for t > Tp by (9), we obtain that
Y = ■
T (r, t)- to
T — T max 0
Ei
- U (t)-
t — t„
- Ei[- U (t)]
Ei
- U(Tmax )-
T — T
max p
- Ei[— U(Tmax )]
(10)
If the duration of Tp of heat pulse is known from
the experiment, temperature difference values [T(r, t; )- T0 ], and the corresponding values of the moments of time t;, i = 1, 2, ..., n, then by solving equation (10) we find the value of the dimensionless value
„2
U (t') = -
4aT'
(11)
corresponding to the specified value of the parameter y, moreover, the value of the moment of time t' = t'(y) is a function of the value of the parameter y.
It follows from (11) that the calculated ratio for calculating the coefficient of thermal diffusivity is
a = ■
4t'U (t')
(12)
T
2
r
After transforming relation (6), a formula was obtained for calculating the volumetric heat capacity
cp =
qiint
nr 2 [ (r, t)-To ]
: U(t) Ei
- u (t>-
t-t„
- Ei[- U(t)] . (13)
Having obtained the formulas (12) and (13), the question arises: "At what value of the dimensionless parameter y will the minimum errors of measurement of the desired a and cp of thermal diffusivity coefficient and volumetric heat capacity take place?".
The mathematical model of root-mean-square
(RMS) estimation of relative errors (5a )RMS
in the measurement of the thermal diffusivity
According to the theory of errors [3, 14, 15], after the logarithm of dependence (12) and the subsequent determination of the differential from the left and right parts (by analogy with the one stated in [3, 5, 6, 9, 10, 13-22]), we obtain:
lna = 2ln r - ln4 - ln t- ln U(t);
d ln a = 2d ln r - d ln 4 - d ln t - d ln U(t);
da dr d 4 dT dU (t) a r 4 t U (t)
According to the theory of errors [3, 14, 15] we substitute differentials da « Aa, dr « Ar, dT « At,
dU (t)»AU (t) with absolute errors
Aa, Ax, At, AU(t) and obtain
Aa 2 Ar At AU (t)
a r t U (t)
where it is taken into account that the differential constant d 4 = 0.
Substituting the signs "-"with the signs "+", we obtain [3, 14, 15] the expression for calculating the so-called marginal estimate of the relative error in measuring the coefficient of thermal diffusivity
f Aa ^
' marg
Ar At AU (t) >
= 2-----or (oa )
r t U (t)
marg
= 25r + St +
Aa
+ 5 U(t) , where 5amarg = — :
Ar At
5r = —, 5t = —,
5U (t) =
AU (t)
- relative errors in determining the
corresponding physical values a, r, t, U(t).
After the transition (by analogy with the recommendations [3, 5, 6, 9, 10, 13-22]) from the limit (5a )marg to the mean-square estimate (5a )RMS of the
relative error in determining the thermal diffusivity, we obtain
(5a)RMs =V4(5r)2 +(5t)2 +[5U(t)]2 . (14)
Let us consider in more detail the procedure for determining the errors included in the last expression (14). Taking into account that the value of the moment of time depends on the dimensionless parameter y, that is t = t(y) , we obtain
dU (y) 1 dU 1 dU 5[U(t(y))] = 5U(y) - —y =--dY «--ay.
U(y) U dY U dY
To determine the absolute error Ay we perform (by analogy with the above) the transformations of formula (8) and obtain
(50
RMS
1
AT
T (r, t>- To
+
AT
T - T
1 max 10.
= 5(max - T0 >
1
— +1
Y 2
or
AY = Y5(Tmax - To >
1 +1 = 5((max - To >+7 , (15)
where AT is absolute measurement error of temperature differences; 5(Tmax - T0) is relative error of measurement of the maximum value of the temperature difference (Tmax - T0); ay , (5y)rms are absolute and
root mean square relative error in determining the dimensionless parameter y from the experimentally measured values of temperature differences
[T(r, t)- T0 ] and [Tmax - T0 ].
Included in (14) the relative error 5t determining the value of the point in time t is related to the errors in measuring the temperature differences [T(r, t)- T0 ].
From the expression
d[T(r, t)-To ] AT
dr
At
, we obtain
r
2
2
Y
a
a
r
r
Ax AT determining the value of the point of time т,
St = = [d\T(r т) T 11 , corresponding to the given value of the dimensionless
т т J T , ^ 0 * I parameter у.
[ дт J Substituting (15), (16) into formula (14), we obtain
where At, ôt are absolute and relative errors of the relation
(Sa )
RMS
1 du (t(y)) гг-л -
——-VY + 1 S(Tmax - T0 )
u (t(y)) dY
(17)
used by us in further calculations in order to identify the optimal value of the dimensionless parameter у (when measuring thermal diffusivity a).
The mathematical model of RMS estimation of relative errors (Scp)RMS in measuring the volumetric heat capacity
Formula (13) can be represented as
qlinт
cp =
nr 2 [T (r, t)- To]
F \U (t)1,
(18)
where F\U (т)] = U (т)Ш
- U (t)-
т-т
- E/\- U (т)]}
r \T(r, t)-To1
According to the theory of errors [3, 14, 15] we substitute differentials dcp « Acp, dr « Ar , dT « At ,
dqlin « Aqlin, dF[U(t)]«AF[U(t)] , with absolute
errors Acp, Ar, At , Aqlin , AU(t) , AF[U(t)], and
obtain
Acp Aqlin at AF [U (t)] Ar A[ (r, t)- To ] -=-+ — +--2---
cp qlin
F [U (t)] r \T(r, t)-To]
where it is taken into account that the differential constant dn = 0 having substituted the signs "-" with the signs "+", we obtain an expression for calculating the so-called marginal estimate of the relative error [3, 14, 15] of measuring the volumetric heat capacity
^Acp^j Aqlin At AF [U (t)] Ar A[T (r, t)- To ] =-+ — +-+ 2— + -
Then, according to the theory of errors [3, 14, 15], after taking the logarithm function (18) and subsequent determination of the differentials of the left and right sides, we obtain:
ln cp = ln qlin + ln t + ln F [U (t)] -- ln n - 2ln r - ln[(r, t) - T0 ];
d ln cp = d ln qHn + d ln t + d ln F[U(t)] -- d ln n - 2d ln r - d ln[[(r, t) - T0 ];
dcp dqHn dT dF [U (t)] dn dr d [[ (r, t)- To ] -=-+ — +----2---
cp qlin T F [U (t)] n
cp
or
' marg
4lh
f\U(t)] r \\(r, t) - To ]
(Scp)marg = Sqlin + st + SF\U(т)] + 2Sr + S\\(r, т) - To ],
/marg
(19)
Acp Ar At Aqlin where Scpmarg «-, Sr « —, St « — , Sqlin «-,
cp r T qlin
SF[U(t)] are relative errors in determining the
corresponding physical values cp, r, t, U(t), qlin.
Acting by analogy with the above procedure for determining errors, we obtain:
1 dF [U (T(Y))]
SF \U (T(Y))] =
AY = Y S ((max - T0 )
F\U (T(Y))] dY i
■ay ;
+ 1 = S(Tmax - To ;
S\T (r, t)-to 1 =
AT
ST = -
T (r, t)-To AT
fd\T (r, t)-To дт
form
With this in mind, the expression (19) takes the
2
2
1
т
t
1
(5cP)marg =5glin +-
AT
dF [U (t(y))]
(d[T (r, t)-T0i
dT
F [U (t(y))] dy
5(?max -T0))i+7 + 25r +
AT
T (r, T)-T0.
After the transition (by analogy with the recommendations [3, 6, 7, 10, 14-20, 22]) from the limit estimate (5cp)marg to the mean-square estimate (5cp)RMS of the error in measuring the volumetric heat capacity, we obtain
(5cp)
RMS
(5glin )2
dF [U (t(y))]
[f[u(t(y))] dy
5(Tmax -T0)]i+7} + 4(5r)2 + {—-J }T (r, t
AT
(20)
Finding a formula for choosing the optimal value of the duration Tp
of heat pulse supplied to the linear heater
In the process of the research, it became apparent that the relative mean-square errors (5a)RMS
RMS
of of
measuring the thermal diffusivity a and (5cp)
measuring the volumetric heat capacity cp additionally depend on the duration Tp and the thermal impulse.
When making measurements, it is desirable to ensure that the requirement for supplying a linear heater of such a value of power P, at which the maximum temperature difference [T(r, Tmax) -- T0] = [Tmax - T0] achieved at the moment of time t = Tmax during each experiment distance r from the heater remains approximately the same and is within certain limits, which is necessary for the following reasons:
- if this maximum difference [Tmax - T0] is small, then the relative error of measuring the values of temperature differences [T(r, t) - T0] will be too large, which can lead to an increase in the relative errors (5a)RMS, (5cp)RMS in measurement of the desired
thermophysical properties (TPPs);
- if this maximum [Tmax - T0] is too big, the assumption that the heat transfer processes in the sample described by a linear mathematical model (1) - (4) is not satisfied, which in turn will lead to increased resulting errors (5a)RMS, (5cp)RMS in
measurement of the desired thermophysical properties due to nonlinearities that are not taken into account by the linear boundary value problem (1) - (4).
To fulfill the requirement (that [Tmax - T0] « « const), at each value of the duration Tp of heat pulse
linear heater must ensure the creation of power density P
qlin = —, at which a constant total amount of heat is L
emitted within the sample per unit length of the heater in each experiment
Qlin = glin Tp = const, (21)
where qlin is power density supplied by the heater power P and length L to the sample during the time period 0 < t < Tp .
The numerical calculations showed that in the study of samples of solid materials with a distance from the heater to the temperature meter 3 < r < 6 mm to obtain the temperature difference [Tmax - T0] = 3-7 °C,
total amount of heat QHn, released in a unit of length L of the electric heater should be maintained within P
Qlin =— Tp = 1000-2000 J/m. Therefore, as shown in L
Fig. 3a, b, the data and results of determining the optimal value Tppt, ensuring minimum errors (5a)RMS and (5cp)RMS of measurement a and cp, were obtained for Qlin = 1500 J/m.
Let us consider the calculation of the error 5glin , included in the formula (20) in more detail. From the above it follows
glin =-
Qlh
and glin =-
P
L '
(22)
1
T
2
2
1
T
p
5,6 5,6 5,4
5,2 ' 6 4,8 4,6 4,4 4,2 4
0,1
0,2
6,6 6,1 7,6 7,1 6,6 6,1 5,6 5,1
4,6 0,15
0,25
0,3
rms, % k tl 7
ft f
.0 t> = / r = 5.7 k / /
= 6. 0 / N7
' = 5.0 vs
tv
x x x-*
0,4
0,5
a)
0,6
0,7
0,3
0,9 Y
)RMS, % V\ ^r = 5.3 r = 4.0 II
i
r = 6.0 r = 6.0 f >
. r = 5.0 r
Jj}:
*—Si— )K X X
0,35
0.45
0,55 b)
0,65
0,75
0,35
0,95 Y
Fig. 3. Dependencies of root-mean-square relative errors (5a)RMS and (Scp)RMS on the dimensionless parameter y for
different values of the distance r from the linear pulsed source of heat to the temperature meter during the measurement:
a - coefficient of thermal diffusivity a; b - volumetric heat capacity cp
P Ota
i.e. — =-. At the same time electrical power P,
L ip
supplied to the flat heater should be chosen from the
ratio P =
Qlin L
ratio and the implementation of other recommendations of the theory of errors [3, 6, 7, 10, 14-20, 22], we obtain the formula
5qlm =V(5P )2 + (8L )2 =
and, for Qlin= 1500 J/m, L = 0.1 m it
x.
p
turns out that
i \ 150 p(Tp ) = —.
Tp
(23)
r 2
AP + AL
^ P(Tp )J _ L _
After logarithm (22), the definition of the
differentials from the left and right parts of the resulting formula
(24)
in which the value P(xp) was calculated by the formula (23).
After substituting (24) into (20), we obtain the
2
(5cp]RMS =
AP
PM
+
AL L
dF[U (t(y))]
[FtU(T(y))]
5((max -T0)]+7[ + 4(5r)2 +
AT
№ (r, t)-T0 ]
dx
AT
[T (r, x)-T0
(25)
The results of numerical modeling of the root-mean-square estimates of the relative errors in the measurement of the thermal diffusivity a and the volumetric heat capacity cp
Using the obtained formulas (17) and (25), we calculated the dependences on the dimensionless parameter y of the root-mean-square relative errors (5a )RMS, (5cp)RMS, for the duration of the heat pulse
xp = 21 s. The calculation results are presented in
Fig. 3. At the same time, the following initial data were
—7 2
used in the calculations: a = 1.06-10 m/s, cp = 1.83-106 J/(m3-K), AP = 0.1 W, r = 2-8 mm, Ar = 0,1 mm, AT = 0.05 K, AL/L = 5L = 0.5 %.
Fig. 3 shows that the minimum values of relative errors (5a)RMS, (5cp)RMS depend not only on the
parameter y, but also on the value of the distance r between the linear pulse heater and the temperature meter. In this regard, it was decided to build lines of equal error levels on the plane with coordinates (y, r) for the duration of the heat pulse xp= 21 s. The results
of this work are presented in Fig. 4.
The results of calculations presented in Fig. 4 show that (with the initial data used in the calculations) the minimum values of the root-mean-square relative errors (5a)RMS of measuring the thermal diffusivity a
are achieved with the values of the dimensionless parameter ya in the range 0.41 < ya < 0.55 and with the values of the main structural dimension of the measuring device within 5.4 < r < 6.0 mm, and
y0apt - 0.48, r0apt - 5.7 mm.
At the same time, the minimum values of the mean square relative errors (5cp)RMS of measuring the
volumetric heat capacity cp occur at 0.78 < ycp < 0.84
and 4.5 < r < 5.1 mm, and yopt - 0.81, rocppt - 4.8 mm.
Thus, to achieve the minimum values of error (5a)RMS and (5cp)RMS in measuring the thermal
diffusivity a and volumetric heat capacity cp of the material under study, a measuring transducer should be used with the distance between the temperature meter and the heater in the range 5.1 < r < 5.4 mm, and, you can take
ra + rcp opt opt
r t =-= 5.25 mm.
2
To determine the optimal value of the duration xp
of heat pulse, ensuring the achievement of the minimum values of relative errors (5a )RMS, (5cp)RMS
and arithmetic mean values of errors
5 (5a ]RMS +(5cp]RMS , • ,
5 =- of measuring thermophysical
2
properties a and cp, calculations were made using formulas (17) and (25) with optimal values yapt = 0.48 ;
Y opt = 0.81; ropt = 5 25 mm, the results of which are
shown in Fig. 5a.
Fig. 5a shows that when the duration xp of the
heat pulse changes, the arithmetic mean value of the root mean square estimates of the relative errors takes
minimum values for xppt - 21 s, in the range
18 < xp< 24 s.
From dependence in Fig. 5a the reader might have the wrong impression that when the influence of the duration xp of the thermal pulse is taken into account, the measurement errors decrease by only 0.20-0.25 %. In fact, the application of the measurement method and the data processing technique proposed in the article allows reducing the arithmetic mean value of the mean square estimates of the relative errors
TT (5a ]RMS + (5cp]
5 =
RMS
2
by 10-20 % in comparison
with the traditional method of linear "instantaneous"
heat source [1-5, 7, 22].
2
2
2
1
T
r. 6.5
6.0
5.5 --
5.0 --
4.5 --
4.0 --
3.5
3.0.
mm
MrMS = 4.25 (5a >RMS = 43
J.
(5a >
RMS
(5a)RMS = 4.2069, r = 5.7 mm
(5cp)rms = 5.21
(0cp)RMS = 4.2069, r = 4.8 mm
h-1-1-1-1-1-h
0.4 Y apt = 0.48 0.5
(Mrms = 5.23
(5cp)rms = 5.2
-I-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-4—i-1-1-1-1
0.3
0.6
0.7
Y oPt
0.9 Y
Fig. 4. Lines of equal levels of root-mean-square relative errors (5a)RMS = fa (y, r) and (5cp)RMS = f (y, r), constructed for optimal duration xp = 21 s of the heat pulse
To illustrate this fact, we performed calculations of the coefficient of thermal diffusivity a and volumetric heat capacity cp for various values of the duration T p of the heat pulse using:
- the calculated ratios proposed in this article (12) and (13);
- calculated ratios [1-5, 7, 22]
ainst ='
4t
cpinst =
Qiin
n er 2 [Tmax - T0 ]
(26)
were used in these calculations aexact = 1.06 -10
-7 2
m /s,
6 3
cpexact = 1.83 -10 J/(m -K), ropt = 5.25 mm, and the
power value was calculated by the formula (23).
After calculating the values of a and cp by formulas (12) and (13), as well as ainst and cpinst by
formula
(26),
errors
5a = -
a - a,
exact
100%,
exact
5cp = CP CPexact100%;
Cpexact
5ainst = '
-100%,
5cPinst =
Cpinst Cpexact
100 %, and the arithmetic mean
Cpexact
— [5a + 5cp] values 5 =-
and 5inst =
[5ainst +5cPinst]
were
used in the implementation of the traditional method of linear "instantaneous" heat source. The exact values
22 calculated.
As a result, dependency graphs 5 = f1(xp ) h 5inst = f2 (xp) presented in Fig. 5b were obtained.
As can be seen from the graphs shown in Fig. 5b, using the numerical modeling of measuring thermophysical properties, the following results were obtained:
1) when using the linear pulsed heat source method proposed in the article, the arithmetic average
of the data processing errors 5 = f (xp) do not exceed
1 %;
2) when processing data using the calculated relations (26), which underlie the traditional method of a linear "instantaneous" heat source [1-5, 7, 22], the arithmetic mean values of the data processing errors
[5ainst +5cpinst]
5inst = '
2
= f2 (tp ) reach 14-25 % with a
exact
heat pulse duration in the range of 16 < t p< 30 s.
2
r
8, % 4.65
4.60
4.55
4.50
4.45
0
8inst,8, %
25 20 15 10 5 0
10
15
20
a)
25
30
35 40 Tp, s
11
16
21
26
Tp, s
b)
Fig. 5. Dependences of arithmetic mean values 8 = ■
(8a )RMS + (8cP)
RMS
2
of mean square estimates
of relative errors (5a)RMS, (5cp)RMS of measuring thermal diffusivity a and volumetric heat capacity cp
on the duration Tp of the heat pulse when processing data:
a - by the linear pulsed heat source method considered in this article; b - traditional method of linear "instant" heat source [1-5, 7, 22]
5
1
6
The block diagram of the measuring installation is shown in Fig. 6. The thermoelectric converter 8, in the form of a thermocouple, is connected to the normalizer 2, amplifying and linearizing the signal of the thermocouple. Using the data acquisition board 3 of type E-14-140M manufactured by L-CARD (Russia), the thermal converter signal (emf) is measured, processed in a personal computer program 4 implemented in the LabView programming environment. The program also through a discrete output of the data acquisition board 3 controls the switch 7, connecting the heater 9 to the power supply 6 for a given time. The obtained temperature response is processed in the program according to the method described above and the coefficients of thermal diffusivity and volumetric heat capacity of the studied material are calculated.
Experiments with the internal plant tissue of potatoes (Table 1) showed that the effective thermal conductivity of healthy tissue of the Udacha potato variety, as well as its volumetric heat capacity, are slightly different from the defective one, in particular, from tissue affected by dry rot, late blight, and alternariosis. The difference in the thermophysical properties of tissues can be explained by the different water content in them, as well as by a change in their structure.
The first block of Table 1 shows the arithmetic mean values of the coefficients of thermal conductivity and thermal diffusivity of the five measurement results for healthy plant tissue of potatoes. The boundaries of the confidence interval, taking into account the Student's coefficient, are ± 0.01 W/(m-K) and
-7 2
± 0,15-10 m/s, respectively, for the thermal
Fig. 6. Block diagram of the measuring installation:
1 - test material; 2 - normalizer; 3 - data acquisition board E14-140M; 4 - computer processing unit; 5 - visual display unit; 6 - power supply; 7 - switch; 8 - thermoelectric converter; 9 - wire linear heater
Potato tubers with internal plant tissue of different quality were used as the test material (Fig. 7).
Fig. 7. Placing a thermocouple and heater on the potato tuber plant tissue
Table 1
Thermophysical properties of healthy and defective internal plant tissue of potato
Test specimen # measurement À,, W/(m-K) a, (x107), m2/s 3 3 cp (x 103), kJ/(m-K)
Healthy potato plant tissue 1 0.515 1.412 3.647
2 0.510 1.383 3.684
3 0.491 1.441 3.436
4 0.505 1.421 3.554
5 0.514 1.43 3.594
Mean value 0.507 1.417 3.560
rms deviation 0.010 0.022 0.011
Plant tissue of late blight potato 1 0.554 1.422 3.892
2 0.542 1.462 3.707
3 0.541 1.460 3.705
4 0.561 1.360 4.125
5 0.539 1.380 3.905
Mean value 0.550 1.420 3.800
rms deviation 0.009 0.046 0.017
conductivity and thermal diffusivity of the plant tissue, taking into account the Student's coefficient. The measured values in the sample were obtained for the same potato tuber.
The second block of Table 1 shows the arithmetic mean values of the coefficients of thermal conductivity and thermal diffusivity of the five measurement results for plant tissue of potato affected by late blight. The boundaries of the confidence interval, taking into account the Student's coefficient, are ± 0.025 W/(m-K)
_7 2
and ± 0.15-10 m/s, respectively, for the thermal
conductivity and thermal diffusivity of the plant tissue, taking into account the Student's coefficient. The measured values in the sample were obtained for the same potato tuber, but at different probe positions.
The time of the active stage of measuring the thermal conductivity and thermal diffusivity coefficients, as well as the specific volumetric heat capacity for one test sample does not exceed 1 min, and the product is heated less than 15 K, which fully meets the requirements for preserving the original properties of the test sample.
Recommendations for the implementation of the linear pulsed heat source method proposed in the article
It is noteworthy that when measuring the thermophysical properties of the test material, the thermal diffusivity a and volumetric heat capacity cp of which differ from the set values (in the initial data of the above calculations), one should do as follows:
1) by conducting preliminary measurements, it is necessary to determine the approximate values of the thermal diffusivity coefficient aapp and volumetric heat
capacity cpapp of the test material;
2) acting similar to the above mentioned in this article, it is necessary to:
а) carry out calculations (for the found values of aapp and cpapp) in order to determine (refine) the
two optimal values of the parameter y£pt, yopt, as well
as the structural dimensions ropt and rocpp of the
distance between the temperature meter and the linear heater;
б) take the distance between the temperature meter and the linear heater (the main structural size of the
ra + rcp
measuring device) equal to r t = ———, and
2
calculate the value of the duration xppt of the thermal
pulse in order to achieve 5 = min;
3) set the distance between the temperature meter and the linear heater in the measuring transducer
ropt
ra + rcP
opt opt
2
4) by carrying out a series of experiments (with manufactured samples), carry out measurements and subsequent processing of the obtained data (at found
values of yapt, yopt and xp = xopt) and, as a result,
obtain the values of the desired thermal diffusivity coefficient a and volumetric heat capacity cp of the test material.
Conclusion
Using the approach proposed in the article to the choice of two optimal values of the dimensionless parameter y and the rational structural size r, which determines the relative position of the temperature meter and heater in the sample of the test material, provides a significant increase in the accuracy of measurements of the sought values of the thermal
diffusivity a and volumetric heat capacity cp. Using the optimal value of the duration of the thermal pulse supplied to the linear heater, allows to further reduce the measurement error of the desired thermal physical properties.
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