Chapter 4. Measurement of Insulation Thermal Conductivity with Irradiation
Thermosensor Heater plate
SUS bobbins 30 mm
Sample (SUS304)
Figure 4.6: Experimental setup for the reference measurement. An I-shaped SUS304 sample is bolted on the sample holder. The temperature difference between two temperature sensors is measured by adjusting the heater power.
Temperature [K]
5 10 15 20 25 30
]-1 K⋅-1 m⋅Thermal conductivity [W
0.5 1 1.5 2 2.5 3 3.5
Measurement Database
Figure 4.7: Comparison of the measured thermal conductivity of SUS304 with database. The green line and its yellow band indicate the data and its error provided from NIST material database [83]. The dot with error bar is the measured data with the estimated uncertainty.
(copyright: ELSEVIER [81])
3) Thermal conductivity is calculated by normalizing the inverse of thermal resistance with geometrical parameters.
One of the measurements at 6 K is demonstrated as follows. Fig. 4.8 shows the measured temperature of Tl, Tu, as input current to the sample heater is scanned. The lower side temperature Tl is controlled at 6 K with a closed loop control. The data of temperature difference, ∆T =Tu −Tl, is obtained in a stable time range at each heater power Q and collected in histogram as shown in Fig. 4.9, in which the mean of ∆T and the mean of Q are 216.8 mK and 1.923 mW, respectively.
Regarding the thermal resistance of the insulation layer as constant in a small temper-ature range, the data set of tempertemper-ature difference ∆T with small correction described in Sec. 4.4.2 is fitted with a linear function of heater power Qas shown in Fig. 4.10, in which the slope of the fit function is 106±3 K/W corresponding to the thermal resistanceR(T) at T = 6.2 K. The measurement error described in Sec. 4.4.3 is taken into account in the fitting process. Thereby, the thermal conductivity is calculated be 0.026 W·m−1·K−1 by normalizing the inverse of the thermal resistanceR(T)−1 with a thickness of the insulation layer of 0.28 mm and a cross sectional area of 100 mm2.
4.4.2 Data Correction
Since the temperature at contact surface of the insulation layer in this test sample cannot be measured directly, the temperature gradient in aluminum bar lying between
Chapter 4. Measurement of Insulation Thermal Conductivity with Irradiation
0 500 1000 1500 2000 2500 3000
Time [sec]
4.55 4.60 4.65 4.70 4.75 4.80
Temperature [K]
Upper Sensor Lower Sensor
0.0 0.2 0.4 0.6 0.8 1.0
Heater Power [mW]
Heater Power
0 500 1000 1500 2000 2500 3000 3500 4000 Time [sec]
5.95 6.00 6.05 6.10 6.15 6.20 6.25 6.30 6.35
Temperature [K]
Upper Sensor Lower Sensor
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Heater Power [mW]
Heater Power
Figure 4.8: The measured temperature of sample with the sample heater power scanned at 4.7 K (left) and 6.2 K (right). Red (Blue) line indicates the temperature at the upper (lower) side of sample, and black curve shows the measured heater power.
Temperature difference [mK]
195 200 205 210 215 220 225 230 235
Counts
0 50 100 150 200 250 300 350 400 450
Entries: 4509 Mean: 216.77 Std. Dev.: 4.51 Minimum: 198.0 Maximum: 232.0
Heater power [mW]
1.92305 1.9231 1.92315 1.9232 1.92325 1.9233
Counts
0 50 100 150 200 250 300 350
Entries: 4509 Mean: 1.923 Std. Dev.: 3.46e-05 Minimum: 1.9231 Maximum: 1.9233
Figure 4.9: Histograms of temperature difference (left) and heater power (right) distributions in a selected time range from 2680 sec until 3420 sec when the current of 4 mA is fed. (copyright:
ELSEVIER [81])
the temperature sensor and insulation layer is taken into account. Regarding the thermal conductivity in aluminum as constant at each side of the insulation layer, the temperature difference on the insulation layer ∆Tins can be calculated from the measured temperature difference ∆Tmeas at the sensors with the following approximation.
∆Tins = ∆Tmeas−Qd
A{ ∆Tu RTu
Tu−∆TukAl(T)dT + ∆Tl RTl+∆Tl
Tl kAl(T)dT} ' ∆Tmeas−Qd
A{ 1
kAl(Tu)+ 1
kAl(Tl)} (4.4)
where d, A, Q and kAl are the distance from the temperature sensor to the insulation layer, the cross sectional area of aluminum bar, the heat through the test sample and the thermal conductivity of aluminum bar, respectively.
The thermal conductivity in aluminum bar of the test sample is measured by using
Heater Power [mW]
0 0.5 1 1.5 2 2.5 3
Temperature Difference [mK]
0 50 100 150 200 250
300 FitMeasurement
Fitting function:
Q + b
⋅ T = a
∆ Entries: 8
2.503
± a: 106.39
4.129
± b: 1.51 T = 6.2 K
Figure 4.10: The temperature difference data as a function of heater power. Each dot indicates measured data with the error bars including both statistical and systematic uncertainties except for geometrical uncertainty of the sample. Red line is obtained by linear fit of the measured dataset, and the band represents the propagated error on the fitted line. (copyright:
ELSEVIER [81])
additional sensor and Tu with a distance of 30 mm on aluminum bar as shown in Fig. 4.11.
Since the temperature difference between the sensors below 10 K is too small to be measured due to the higher thermal conductivity of aluminum bar and limited length, the thermal conductivity was measured from 10 K until 22 K. The data is extrapolated down to 4 K with Hust’s semi-empirical equations [61]. The RRR for the aluminum A1070 is calculated to be around 15 by fit which is a reasonable value for 1000 series aluminum [66].
The correction is performed on data points of the temperature difference before the fit.
It contributes less than 5% on temperature (e.g. 15 mK correction for 300 mK difference in the measurement at 6 K).
4.4.3 Uncertainties
Due to the instability of cooling power and other unknown reasons, the temperature drift in the test sample was often observed. The drift could cause the fluctuation of temperature difference in test sample. To take into account the systematic effect as well as statistical fluctuation in the measurement, the standard deviation of measured temperature difference is regarded as measurement error for each data point in the fitting procedure. Also, the uncertainty of the sensor readout is linearly added to the error of each data point. The uncertainty in the sensor resistance measurement is calculated from the electronic accuracy of readout electronics [84], and converted into the error in temperature (±1.5 mK at 4 K, ±6.3 mK at 10 K, ±10 mK at 20 K). The fitting error
Chapter 4. Measurement of Insulation Thermal Conductivity with Irradiation
0 5 10 15 20 25 30 35
Temperature [K]
0 50 100 150 200 250 300 350 400
T he rm al C on du ct iv ity [W
·m
−1 ·K
−1]
RRRAl = 14.8 +/- 1.0 Fit
Measurement
Figure 4.11: Thermal conductivity of the aluminum in the test sample. Red dots with error bars indicate the measured thermal conductivity with the estimated uncertainties including the systematics. Black solid line is obtained by fitting the data with Hust’s equations [61] and green band is its fitting error. (copyright: ELSEVIER [81])
calculated by least mean square method is regraded as a propagated uncertainty for the measurement of thermal resistance.
The effect of the boundary resistance between the temperature sensor and aluminum bar is also checked by assuming the boundary is Cu / Apiezon-N / Cu, and using the data given in Ref. [85]. The estimation shows that the temperature difference between the sensor and the sample is negligibly small in the temperature range from 4 K to 20 K.
Dimension Measurement
The thickness of the insulation tape in the test sample is difficult to be measured directly since the tape is glued between the aluminum bars after the heat treatment. The length of aluminum bars without insulation tapes is measured before the heat treatment, and compared with the total sample length with the insulation tape after the heat treatment.
The measurement results using a vernier caliper are consistent with the specification of tape thickness, 0.28 mm, within the vernier resolution. Thus 0.03 mm is assigned as the uncertainty of the thickness measurement. The thickness uncertainty is not taken into account in the fitting since it is a correlated error in each sample.
Kapitza Resistance
Because the prepreg tape is sandwiched with two aluminum bars in this measurement, a Kapitza resistance between the aluminum bar and preperg tape could cause the uncertainty at the low temperature region. Here, a method in the measurement of Kapitza resistance
Temperature [K]
4 6 8 10 12 14 16 18 20
Resistance [K/W]
0 20 40 60 80 100 120 140 160
/ ndf
χ2 0.329 / 7
p0 1217 ± 4875 p1 0.00169 ± 0.0006007 p2 0.9883 ± 0.1339
/ ndf
χ2 0.329 / 7
p0 1217 ± 4875 p1 0.00169 ± 0.0006007 p2 0.9883 ± 0.1339
Sample A (before irradiation)
Measurement Kapitza Resistance Thermal Resistance Total Resistance
Temperature [K]
4 6 8 10 12 14 16 18 20
Resistance [K/W]
0 20 40 60 80 100 120 140 160
/ ndf
χ2 1.003 / 13
p0 4360 ± 3241 p1 0.002015 ± 0.0004113 p2 0.9049 ± 0.07369
/ ndf
χ2 1.003 / 13
p0 4360 ± 3241 p1 0.002015 ± 0.0004113 p2 0.9049 ± 0.07369
Sample A (1 MGy)
Measurement Kapitza Resistance Thermal Resistance Total Resistance
Temperature [K]
4 6 8 10 12 14 16 18 20
Resistance [K/W]
0 20 40 60 80 100 120 140 160
/ ndf
χ2 0.8516 / 8
p0 3117 ± 4780 p1 0.001865 ± 0.0005402 p2 0.9318 ± 0.1043
/ ndf
χ2 0.8516 / 8
p0 3117 ± 4780 p1 0.001865 ± 0.0005402 p2 0.9318 ± 0.1043
Sample B (before irradiation)
Measurement Kapitza Resistance Thermal Resistance Total Resistance
Temperature [K]
4 6 8 10 12 14 16 18 20
Resistance [K/W]
0 20 40 60 80 100 120 140 160
/ ndf
χ2 1.155 / 14
p0 4307 ± 3457 p1 0.00196 ± 0.0004329 p2 0.9053 ± 0.0806
/ ndf
χ2 1.155 / 14
p0 4307 ± 3457 p1 0.00196 ± 0.0004329 p2 0.9053 ± 0.0806
Sample B (5 MGy)
Measurement Kapitza Resistance Thermal Resistance Total Resistance
Temperature [K]
4 6 8 10 12 14 16 18 20
Resistance [K/W]
0 20 40 60 80 100 120 140 160
/ ndf
χ2 0.4226 / 10
p0 979.2 ± 3537 p1 0.001566 ± 0.0003868 p2 0.9728 ± 0.0937
/ ndf
χ2 0.4226 / 10
p0 979.2 ± 3537 p1 0.001566 ± 0.0003868 p2 0.9728 ± 0.0937
Sample C (before irradiation)
Measurement Kapitza Resistance Thermal Resistance Total Resistance
Temperature [K]
4 6 8 10 12 14 16 18 20
Resistance [K/W]
0 20 40 60 80 100 120 140 160
/ ndf
χ2 1.121 / 15
p0 3164 ± 3101 p1 0.001842 ± 0.0003743 p2 0.9091 ± 0.07502
/ ndf
χ2 1.121 / 15
p0 3164 ± 3101 p1 0.001842 ± 0.0003743 p2 0.9091 ± 0.07502
Sample C (0.2 MGy)
Measurement Kapitza Resistance Thermal Resistance Total Resistance
Figure 4.12: The possible Kapitza resistance (blue) and thermal resistance (green) of prepreg tape separated with a fit method for each sample before and after the irradiation. The red line is the total thermal resistance, the marker with error is the thermal resistance measured in this experiment, and the band is the error from the fit process.
between liquid helium and polyimide tape [86] is used to investigate the uncertainties of the Kapitza resistance between prepreg tape and aluminum bar. From the acoustic
Chapter 4. Measurement of Insulation Thermal Conductivity with Irradiation
mismatch model [87], the measured thermal resistance is given by R = RuK+Rc+RlK
= 1
αnTun−1 + l
Ak(T)+ 1
αnTln−1 (4.5)
where RuK (RKl ) is the Kapitza resistance between the upper (lower) aluminum bar and surface of prepreg tape, Rc is the thermal resistance of prepreg tape, α is the Kapitza coefficient, A is the cross-sectional area, and n is 4 given by the acoustic mismatch theory, respectively. Due to a small temperature difference across prepreg tape, the thermal resistance can be simplified to
R = 2
αnTln−1(1−1 2
∆T
Tl +O(∆T)2) + l Ak(T)
≈ 2
nαTl1−n+ l
Ak(T) (4.6)
= RK+Rc (4.7)
Assuming the parameter of n is 4 and the second term can be expressed as a power function of Rc=T−p2/p1 at low temperature, the thermal resistance can be rewritten as
R(T) = 1
2p0T−3+ 1
p1Tp2 (4.8)
where p0 is the Kapitza coefficient. Fitting the measurement data with this fit function, the thermal resistance and Kapitza resistance can be separated as given in Fig. 4.12. The Kapitza resistance between the prepreg tape and aluminum bar is small within the fit error enough with respect to the thermal resistance of prepreg tape in this temperature range, therefore, the uncertainty from the Kapitza resistance is ignored in this analysis.