Wright State University, Department of Mechanical and Materials Engineering
ME 495: THERMAL-FLUID SCIENCES LABORATORY
Determination of the Thermal Conductivity of a Metallic Rod
Objective: Utilize Fourier’s law of heat conduction to determine the thermal
conductivity of a metallic rod with a round cross section for heat flow.
Experimental Procedure: Based on scoping runs made by the students, an experimental
procedure will be determined.
Report: The objective of the written report is to show that the students are capable of
collecting the appropriate data to determine the thermal conductivity of the rod. The
report must include an experimental procedure, hand calculations, and a discussion of the
results including appropriate plots and conclusions.
CAUTION: The heater voltage must not be greater than 110 V. Do not let any
temperature in the system go above 200oC!
Experimental Setup
The objective of the experiment was to measure the thermal conductivity of two sample
metallic rods using Fourier’s law of heat conduction. Heat was transferred to the rod
using an electric heater at one end of the rod, and heat was extracted at the other end
using a water-cooled calorimeter, as shown in Figure 1. The cooling water was supplied
by a constant head pressure tank which maintained a constant flow rate. The coolant
water flow was filtered and controlled using a ball valve. The mass flow rate of coolant
was measured using an electronic turbine flow meter. The temperature increase in the
cooling water was measured using thermocouple probes inserted into the coolant stream.
Power was supplied to the electrical heater using a variable AC transformer and
measured using a digital voltmeter. The axial temperature gradient within the sample rod
was measured using thermocouple probes mounted in the sample in small-diameter holes,
as shown in Figure 2. Layers of ceramic wool insulation and aluminum foil provided
convective and radiative insulation along the sides of the samples and around the heater
and calorimeter. The thermal conductivities of the samples were determined by using the
heat removed by the calorimeter due to heat losses from the heater to the ambient.
However, the electrical power input to the heater was calculated to provide information
on the validity of the measured heat removed by the calorimeter.
In order to supply a sufficient amount of heat to the sample, a 6.35-mm-thick copper heat
spreader plate was silver-soldered (Alloy number 20233, Ag56/Cu22/Zn17/Sn5, solidus
temperature 620°C, liquidus temperature 650°C) to each sample rod using an oxygenacetylene torch as shown in Figure 3 and Figure 4. An electric heater (Marathon Heater,
Model ST060-060B, 𝑅𝐻 = 30.0 Ω) was directly mounted to the copper heat spreader plate
for heat input. A 12.5-mm-thick piece of ceramic fiber insulation (FiberFrax
Durablanket) was held next to the electric heater with a steel backer plate to provide even
pressure against the heater. Electrical power was supplied to the heater by a variable AC
transformer (Powerstat, Model BP57515). The voltage across the heater was monitored
using a digital multi-meter (National Instruments, Model USB-4065). Calorimeters were
constructed using 6.35-mm-thick copper plate and copper tubing as shown in Figure 5.
Grooves were machined in the plates using a 6.35-mm ball end mill to a depth of 1.6 mm.
Copper pipe (6.35-mm-outside diameter) was soldered onto the copper plate using tinantimony solder (Alloy number 2011, Sn95/Sb5, solidus temperature 235°C, liquidus
temperature 240°C) in a temperature-controlled furnace. The calorimeters were
successfully pressure-checked to 1.3 atm. The calorimeters were then soldered to the
copper rod samples using tin-lead solder (Alloy number 2030, Sn62/Pb38, eutectic
temperature 183°C) in the temperature-controlled furnace, as shown in Figure 6. Brass
fittings were used to place the thermocouple probes (Omega, Model EMQSS-062G-12)
in the coolant stream of the calorimeter, as shown in Figure 7. The 1.016-mm-diameter
(0.040 inch) sample thermocouple probes (Omega, Model EMQSS-040G-12) were
mounted in the sides of the rods to a depth of 19.1 mm. The mounting holes were drilled
with a precision micro drill press (Dayton, Model 2LKU8) using a 1.041 mm (0.041
inch) solid carbide drill bit, as shown in Figure 8. The sample thermocouples were held in
place using aircraft wire, as shown in Figure 9. The temperatures sensed by the
thermocouples were monitored and recorded by using a data acquisition system (DAS),
which consisted of a data acquisition board (National Instruments, Model SCC-68), four
thermocouple modules (National Instruments, Model SCC-TC01) mounted to the DAQ
Board, and a data acquisition card (National Instruments, Model PCI-6221) installed in a
PC. The assembled system is shown in Figure 10, where the samples were uninsulated.
Figure 11 shows one of the fully insulated samples, where four layers of ceramic fiber
insulation and four layers of aluminum foil were installed using aircraft wire. Both
samples were insulated in the same manner to provide a meaningful comparison.
Thermal Conductivity Calculation
The thermal conductivities of the copper rod samples were calculated using Fourier’s law
of heat conduction (Incropera & DeWitt, 1990):
𝑄̇ = 𝑘𝐴
𝑑𝑇
𝑑𝑥
or
𝑘=
𝑄̇
𝐴(𝑑𝑇/𝑑𝑥)
The rate of heat removed from the sample bar by the calorimeter is given by the first law
of thermodynamics (Cengel & Boles, 2006):
𝑄̇ = 𝑚̇𝐶𝑝 (𝑇out − 𝑇in )
where 𝑚̇ is the mass flow rate of water measured by the flow meter, and 𝑇in and 𝑇out are
the inlet and outlet temperatures of the water measured by the calorimeter thermocouples,
respectively. The specific heat at constant pressure is evaluated at the average of the inlet
and outlet temperatures, and is given by (Lide & Kehiaian, 1994):
𝐶p = 𝐴1 + 𝐴2 𝑇 + 𝐴3 𝑇 2 + 𝐴4 𝑇 3 + 𝐴5 𝑇 4 (J K-1 mol-1)
where 𝑇 is in degrees Kelvin. The numerical coefficients are as follows: 𝐴1 = 917.5,
𝐴2 = −10.1016, 𝐴3 = 0.0454134, 𝐴4 = −9.07517 × 10−5, 𝐴5 = 6.80700 × 10−8 .
The valid range for this equation is 273 ≤ 𝑇 ≤ 373 K at near-atmospheric pressure.
The cross-sectional area of the rod is
𝐴=
𝜋 2
𝐷
4 s
where 𝐷s is the diameter of the sample rod as measured by using digital Vernier calipers.
The axial temperature gradient in the rod is given by
𝑑𝑇 𝑇𝐻 − 𝑇𝐿
=
𝑑𝑥
𝐿CC
where 𝑇𝐻 and 𝑇𝐿 are the sample temperatures measured by the thermocouples installed
in the sides of the rods. For the sample, 𝑇𝐻 = TC03 and 𝑇𝐿 = TC04. The center-to-center
distance between the sample thermocouples was measured as follows. Each sample was
placed on a Starrett Crystal Pink precision granite surface. A precision pin was placed
into the bottom hole and then the height to the top of the pin was zeroed using a Mitutoyo
height gage with an attached Interapid indicator. This distance was then replicated using a
precision slide gage block. The height to the top of the second hole was then measured
using the same pin and that height was replicated using precision gage blocks placed onto
the slide gage block. The height was then calculated based on the gage blocks used to
zero the height of the second hole.
In terms of the eight measured quantities, the thermal conductivity of the sample rods is
given by
4𝑚̇𝐶𝑝 (𝑇out − 𝑇in )𝐿CC
𝑄̇
𝑘=
=
𝐴(𝑑𝑇/𝑑𝑥)
𝜋𝐷s2 (𝑇𝐻 − 𝑇𝐿 )
Measurement Uncertainty Estimates
Thermal Conductivity
The root-sum-square uncertainty for the thermal conductivity is given in terms of the
eight measured quantities as follows:
2
2
2
2
𝜕𝑘
𝜕𝑘
𝜕𝑘
𝜕𝑘
𝜕𝑘
∆𝑘 = [(
∙ ∆𝑚̇) + (
∙ ∆𝐶p ) + (
∙ ∆𝑇out ) (
∙ ∆𝑇in ) + (
∙ ∆𝐿CC )
𝜕𝑚̇
𝜕𝐶p
𝜕𝑇out
𝜕𝑇in
𝜕𝐿CC
1/2
2
2
2
𝜕𝑘
𝜕𝑘
𝜕𝑘
+(
∙ ∆𝐷s ) + (
∙ ∆𝑇H ) + (
∙ ∆𝑇L ) ]
𝜕𝐷s
𝜕𝑇H
𝜕𝑇L
2
2
2
4 ∆𝑚̇ 𝐶p (𝑇out − 𝑇in )𝐿CC
4𝑚̇ ∆𝐶p (𝑇out − 𝑇in )𝐿CC
4𝑚̇𝐶p ∆𝑇out 𝐿CC 2
=[(
) +(
) +(
)
𝜋𝐷s2 (𝑇𝐻 − 𝑇𝐿 )
𝜋𝐷s2 (𝑇𝐻 − 𝑇𝐿 )
𝜋𝐷s2 (𝑇𝐻 − 𝑇𝐿 )
4𝑚̇𝐶p ∆𝑇in 𝐿CC 2
+( 2
)
𝜋𝐷s (𝑇𝐻 − 𝑇𝐿 )
2
2
4𝑚̇𝐶p (𝑇out − 𝑇in ) ∆𝐿CC
4𝑚̇𝐶p (𝑇out − 𝑇in )𝐿CC ∆𝐷s
+(
) +(
)
𝜋𝐷s2 (𝑇𝐻 − 𝑇𝐿 )
𝜋𝐷s3 (𝑇𝐻 − 𝑇𝐿 )
1/2
2
2
4𝑚̇𝐶p (𝑇out − 𝑇in )𝐿CC ∆𝑇H
4𝑚̇𝐶p (𝑇out − 𝑇in )𝐿CC ∆𝑇L
+(
) +(
) ]
𝜋𝐷s2 (𝑇𝐻 − 𝑇𝐿 )2
𝜋𝐷s2 (𝑇𝐻 − 𝑇𝐿 )2
Specific Heat of Water
The calculation of the uncertainty of the heat removed by the calorimeter requires an
estimate of the uncertainty of the value of the specific heat of the water coolant. Since
this information was not available in the archival reference (Lide & Kehiaian, 1994), a
conservative value of 1% of the reading was taken.
Bibliography
Cengel, Y., & Boles, M. (2006). Thermodynamics: An Engineering Approach. New
York: McGraw-Hill.
Incropera, F., & DeWitt, D. (1990). Fundamentals of Heat and Mass Transfer. New
York: Wiley.
Lide, D., & Kehiaian, H. (1994). CRC Handbook of Thermophysical and
Thermochemical Data. Boca Raton: CRC Press.
Constant Head
Pressure Tank
Ball
Valve
Filter
Water-Cooled
Calorimeter
Turbine
Flow Meter
Tin
TL
Tout
Copper Rod
Sample
TH
Electric
Heater
Digital
Voltmeter
+
−
Variable AC
Transformer
Figure 1: Schematic diagram of the experimental setup.
Tin
Coolant
Water In
Ceramic Wool
Insulation
Layers
Copper Tubing
Calorimeter
Copper Heat
Spreader Plate
Coolant
Water Out
Copper Rod
Sample
TL
Sample
Thermocouples
LCC
Aluminum Foil
Layers
Tout
TH
Copper Heat
Spreader Plate
Electric
Heater
Ceramic Wool
Insulation
Steel Backer Plate
Figure 2: Schematic diagram of the experimental setup, cont.
Figure 3: Cut-away view of the assembled heat spreader plates, calorimeter, and sample rod.
Figure 4: Copper heat spreader plate soldered to the sample rod using silver solder.
Figure 5: Copper calorimeter constructed using tin-antimony solder.
Figure 6: Calorimeter soldered to the sample rod using tin-lead solder.
Figure 7: Brass fittings used to place 1/16-inch-diameter thermocouple probes into the coolant
stream of the calorimeter.
Figure 8: Setup for drilling 0.041-inch-diameter holes for sample thermocouples using the
precision micro drill press.
Figure 9: Installed 0.040-inch-diameter sample thermocouple probes held in place with stainless
steel aircraft wire.
Figure 10: Uninsulated experimental setup.
Figure 11: Fully insulated sample.
Figure 12: Calibration equation for sample thermocouple TC03.
Figure 13: Calibration equation for sample thermocouple TC04.
Table 1: Values used to determine the calibration uncertainty of sample thermocouple TC03.
PRTD Block
Temperature (°C)
48.71971
71.90097
96.9267
122.20842
147.43681
172.57366
197.89573
222.60953
247.26067
∆𝑻WELL95
(°C)
0.0024658
0.0022709
0.0073763
0.0078219
0.0073761
0.0062257
0.0046092
0.0045024
0.0047109
TC03 Block
Temperature (°C)
49.03569
71.59958
95.92662
120.48850
144.98130
169.27022
193.73850
217.57960
241.35802
TC03 Calibration
Prediction (°C)
48.55204
71.84586
96.95989
122.31634
147.60148
172.67615
197.93598
222.54834
247.09599
∆𝑻BF (°C)
0.16766
0.055103
0.033194
0.10792
0.16467
0.10249
0.040250
0.061189
0.16467
∆𝑻CAL
(°C)
0.17443
0.061674
0.044870
0.12004
0.17635
0.11301
0.049159
0.069992
0.17368
Table 2: Values used to determine the calibration uncertainty of sample thermocouple TC04.
PRTD Block
Temperature (°C)
48.71971
71.90097
96.9267
122.20842
147.43681
172.57366
197.89573
222.60953
247.26067
∆𝑻WELL95
(°C)
0.0024658
0.0022709
0.0073763
0.0078219
0.0073761
0.0062257
0.0046092
0.0045024
0.0047109
TC04 Block
Temperature (°C)
48.89668
71.47360
95.80870
120.37820
144.88528
169.18238
193.64406
217.48726
241.27156
TC04 Calibration
Prediction (°C)
48.54351
71.84470
96.96047
122.31816
147.61142
172.68798
197.93439
222.54248
247.08978
Table 3: Summary of calibration uncertainties.
Device
TC03
TC04
Calibration
Uncertainty
0.176°C
0.186°C
Table 4: Length measurements and uncertainties.
Measurement
𝐷s (mm)
50.880 ± 0.0254
𝐿CC (mm)
44.958 ± 0.0254
Labview virtual instrument for taking temperature and voltage data:
http://www.cs.wright.edu/people/faculty/sthomas/reader06.vi
∆𝑻BF (°C)
0.17619
0.056260
0.033777
0.10974
0.17461
0.11432
0.038661
0.067048
0.17088
∆𝑻CAL
(°C)
0.18295
0.062831
0.045453
0.12186
0.18629
0.12484
0.047570
0.075851
0.17989