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 oxygen-acetylene 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 tin-antimony 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):

Q=kA dTdx

or k= QA (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):

Q=mC p (T out−T in )

where m is the mass flow rate of water measured by the flow meter, and T in and T out 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+ A2T+ A3T 2+A4 T 3+ A5T 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=π4

Ds2

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

dTdx

=T H−T L

LCC

where T H and T L are the sample temperatures measured by the thermocouples installed in the sides of the rods. For the sample, T H=TC03 and T L=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

k= QA ( dT /dx )

=4 mC p (T out−T in ) LCC

π Ds2 ( T H−T L )

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:

∆ k=[( ∂ k∂m ∙ ∆ m)

2

+( ∂ k∂ Cp

∙ ∆ Cp)2

+( ∂k∂T out

∙ ∆ Tout )2

( ∂ k∂T in

∙ ∆ T in)2

+( ∂k∂ LCC

∙∆ LCC)2

+( ∂ k∂ Ds

∙∆ Ds)2

+( ∂k∂T H

∙∆ T H)2

+( ∂ k∂ T L

∙ ∆ TL)2]

1 /2

¿ [( 4 ∆ mCp (T out−T in ) LCC

π Ds2 (T H−T L) )

2

+(4 m∆ Cp (Tout−T in ) LCC

π Ds2 (T H −T L) )

2

+( 4 mCp ∆ T out LCC

π Ds2 (T H−T L) )

2

+( 4 mCp ∆T in LCC

π Ds2 ( T H−T L ) )

2

+ (4 m C p (T out−T in ) ∆ LCC

π D s2 (T H−T L) )

2

+( 4 mCp ( Tout−T in ) LCC ∆ D s

π Ds3 (T H−T L ) )

2

+( 4 mCp (T out−T in ) LCC ∆T H

π Ds2 (T H−T L )2 )

2

+( 4 mCp (T out−T in ) LCC∆ T L

π Ds2 (T H−T L)2 )

2]1 /2

Specific Heat of WaterThe 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 HeadPressure Tank

Tin

Tout

Water-Cooled Calorimeter

Electric Heater

Filter

BallValve

Turbine Flow Meter

TL

TH

Variable AC Transformer

Copper Rod Sample

+ −

Digital Voltmeter

Figure 1: Schematic diagram of the experimental setup.

Ceramic Wool Insulation

Tin

ToutCopper Tubing Calorimeter

Copper Heat Spreader Plate

Ceramic Wool Insulation

Layers

Aluminum Foil Layers

Electric Heater

Copper Heat Spreader Plate

Coolant Water In

LCC

TL

TH

Coolant Water Out

Copper Rod Sample

Sample Thermocouples

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)

∆ T WELL95 (°C)

TC03 Block Temperature (°C)

TC03 Calibration Prediction (°C)

∆ T BF (°C)

∆ T CAL (°C)

48.71971 0.0024658 49.03569 48.55204 0.16766 0.1744371.90097 0.0022709 71.59958 71.84586 0.055103 0.06167496.9267 0.0073763 95.92662 96.95989 0.033194 0.044870

122.20842 0.0078219 120.48850 122.31634 0.10792 0.12004147.43681 0.0073761 144.98130 147.60148 0.16467 0.17635172.57366 0.0062257 169.27022 172.67615 0.10249 0.11301197.89573 0.0046092 193.73850 197.93598 0.040250 0.049159222.60953 0.0045024 217.57960 222.54834 0.061189 0.069992247.26067 0.0047109 241.35802 247.09599 0.16467 0.17368

Table 2: Values used to determine the calibration uncertainty of sample thermocouple TC04.

PRTD Block Temperature (°C)

∆ T WELL95 (°C)

TC04 Block Temperature (°C)

TC04 Calibration Prediction (°C)

∆ T BF (°C)

∆ T CAL (°C)

48.71971 0.0024658 48.89668 48.54351 0.17619 0.1829571.90097 0.0022709 71.47360 71.84470 0.056260 0.06283196.9267 0.0073763 95.80870 96.96047 0.033777 0.045453

122.20842 0.0078219 120.37820 122.31816 0.10974 0.12186147.43681 0.0073761 144.88528 147.61142 0.17461 0.18629172.57366 0.0062257 169.18238 172.68798 0.11432 0.12484197.89573 0.0046092 193.64406 197.93439 0.038661 0.047570222.60953 0.0045024 217.48726 222.54248 0.067048 0.075851247.26067 0.0047109 241.27156 247.08978 0.17088 0.17989

Table 3: Summary of calibration uncertainties.

Device Calibration Uncertainty

TC03 0.176°CTC04 0.186°C

Table 4: Length measurements and uncertainties.

Measurement

Ds (mm) 50.880 ± 0.0254LCC (mm) 44.958 ± 0.0254

Labview virtual instrument for taking temperature and voltage data: http://www.cs.wright.edu/people/faculty/sthomas/reader06.vi