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 100 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, = 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):

or

( )

The rate of heat removed from the sample bar by the calorimeter is given by the first law

of thermodynamics (Cengel & Boles, 2006):

( out in)

where is the mass flow rate of water measured by the flow meter, and in and 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):

p

(J K

-1 mol

-1)

where is in degrees Kelvin. The numerical coefficients are as follows: ,

, , ,

.

The valid range for this equation is K at near-atmospheric pressure. The cross-sectional area of the rod is

s

where s is the diameter of the sample rod as measured by using digital Vernier calipers. The axial temperature gradient in the rod is given by

where and are the sample temperatures measured by the thermocouples installed

in the sides of the rods. For the sample, T 03 and T 0 . 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

( ) ( out in)

s ( )

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:

[(

)

(

p p)

(

out out)

(

in in)

(

)

(

s s)

(

H H)

(

)

]

[ ( p( out in)

s ( ))

( p( out in)

s ( ))

( p out

s ( ))

( p in

s ( ))

( p( out in)

s ( )

)

( p( out in) s

s ( )

)

( p( out in) H

s ( ) )

( p( out in)

s ( ) )

]

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

Tin

Tout

Water-Cooled

Calorimeter

Electric

Heater

Filter

Ball

Valve

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

Tout

Copper 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)

(°C)

TC03 Block Temperature (°C)

TC03 Calibration Prediction (°C)

(°C)

(°C)

48.71971 0.0024658 49.03569 48.55204 0.16766 0.17443

71.90097 0.0022709 71.59958 71.84586 0.055103 0.061674

96.9267 0.0073763 95.92662 96.95989 0.033194 0.044870

122.20842 0.0078219 120.48850 122.31634 0.10792 0.12004

147.43681 0.0073761 144.98130 147.60148 0.16467 0.17635

172.57366 0.0062257 169.27022 172.67615 0.10249 0.11301

197.89573 0.0046092 193.73850 197.93598 0.040250 0.049159

222.60953 0.0045024 217.57960 222.54834 0.061189 0.069992

247.26067 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)

(°C)

TC04 Block Temperature (°C)

TC04 Calibration Prediction (°C)

(°C)

(°C)

48.71971 0.0024658 48.89668 48.54351 0.17619 0.18295

71.90097 0.0022709 71.47360 71.84470 0.056260 0.062831

96.9267 0.0073763 95.80870 96.96047 0.033777 0.045453

122.20842 0.0078219 120.37820 122.31816 0.10974 0.12186

147.43681 0.0073761 144.88528 147.61142 0.17461 0.18629

172.57366 0.0062257 169.18238 172.68798 0.11432 0.12484

197.89573 0.0046092 193.64406 197.93439 0.038661 0.047570

222.60953 0.0045024 217.48726 222.54248 0.067048 0.075851

247.26067 0.0047109 241.27156 247.08978 0.17088 0.17989

Table 3: Summary of calibration uncertainties.

Device Calibration Uncertainty

TC03 0.176°C

TC04 0.186°C

Table 4: Length measurements and uncertainties.

Measurement

(mm) 50.880 ± 0.0254

(mm) 44.958 ± 0.0254

Labview virtual instrument for taking temperature and voltage data:

http://www.cs.wright.edu/people/faculty/sthomas/reader06.vi