Heat Transfer - Computational Project

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ME 3345 Heat Transfer – Fall 2017 – Computational Project

Due date: September 28, 2017

Questions/clarifications: Contact Sampath Kommandur ([email protected])

Rules: This assignment may be completed individually or in groups of up to 3 people. By

committing to working in a group, a student assumes full responsibility for the grade assigned to

his/her group. No internal conflicts and/or problems with distribution of efforts within a group will

be considered in assigning the project grade (i.e., the same grade will be given to each member of

the group based on the evaluation of the submitted project report).

Problem description:

ASTM D5470-12 is a standardized test method for measurement of thermal resistance, and

calculation of thermal conductivity of materials. To overcome measurement uncertainties

associated with this technique, a modified measurement technique is proposed. (See figure on page

3.) The material of interest is sandwiched between two reference bars. The upper end of the upper

reference bar (URB) and the lower end of the lower reference bar (LRB) are maintained at fixed

temperatures, T1 and T2 (<T1), respectively. By placing thermocouples at appropriate locations in

the reference bars, temperature gradients and heat flow rates can be calculated. These data are fit

to thermal models to determine thermal conductivity.

The reference bars are made of Al 2024, whose thermal conductivity can be assumed to be kAl =

138 W/m-K, and are insulated on the sides. The LRB is typically maintained at near room

temperature and the insulation can be approximated to be perfect. Since the URB is maintained at

higher temperatures, there will be heat loss through its insulation. For a certain experimental set-

up, the heat leakage to the ambient through the insulation can be approximated with an effective

heat transfer coefficient of hins = 3 W/m2K. Convective heat transfer coefficient to the ambient

from uninsulated surfaces can be taken to be hair = 10 W/m2K. While a 3D analysis is required to

rigorously solve the heat transfer problem, it is reasonable to use a 2D approximation. The ambient

temperature is Tamb = 22 oC.

Part 1: Consider a sample material with ksample = 20 W/m-K placed between the reference

bars. The dimensions of the test material are given in Figure 1. Estimate the temperatures at

the locations of the 12 thermocouples, and at the interface between the sample and the reference

bars along the center (TH and TL). Also calculate the heat lost to the ambient due to convection

from the material. During the experiment, the ends of the reference bars are maintained at T1 =

100 oC and T2 = 15 oC, respectively.

Part 2: During the experiment, the readings of thermocouples TC 1-12 are the known

parameters. A user determined the thermal conductivity of the sample using an approximated

analysis that considers 1D heat flow along y-direction, and neglects other modes of heat

transfer.

1. Readings from TC 1-4 is fitted to a straight line, the slope of which gives the temperature

gradient. This gradient, coupled with kAl gives the amount of heat rate into the sample from

the URB.

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2. Readings from TC 4-8 is fitted to a second order polynomial. This polynomial fit is

extrapolated to obtain TH, the temperature at the interface between URB and the sample.

3. Readings from TC 9-12 is fit to a straight line, the slope of which can be used to obtain the

heat rate from the sample to the LRB (calculation similar to step 1).

4. The first order curve from step 3 (linear fit) is extrapolated to obtain TL. (Note: This is

different to the analysis for URB.)

5. The temperature gradient in the sample is assumed to be linear, and an effective thermal

conductivity of the sample is determined using Fourier’s law:

 

sample

meas sample

H L

L k q

T T 



sample q is the heat flux through the sample, which can be taken as the heat flux into the

sample (from step 1), heat flux out of the sample (from step 3), or the average value of the

two. Lsample is the length of the sample along the direction of heat flow.

Using the readings of TC 1-12 from the 2D model in Part 1, obtain kmeas using the analysis

described above. Calculate the error associated with the calculation using 1D model (Compare

kmeas with ksample). Comment on the change in error when using different values of sampleq

mentioned in step 5.

Part 3: In Parts 1 and 2, the sample thermal conductivity was significantly different from the

Al 2024, which will introduce relatively larger errors due to heat-shunting through the

insulation. To address this, a similar set-up was design with the reference bars made of carbon

steel (ksteel = 35 W/m-K). Repeat the calculations in part 1 and 2, and comment on whether

this modification is recommended.

Deliverables:

A maximum 5-page report must be written addressing the following:

1. Problem formulation

2. Any assumptions and idealizations

3. Governing differential equations with appropriate boundary conditions

4. Discretization of the domains, finite different equations, and brief description of the

solution method

5. Results to parts 1,2 and 3 including the following:

a. Contour plot of 2-D temperature field in the system

b. Variation of the temperature along the center line

6. Discussion and conclusions

7. Electronic copy of the executable file that was used to produce the results, emailed to

[email protected]. Subject line for the email and the name of the file should be:

“ME3345 Project LAST NAME and LAST NAME(S) OF GROUP MEMBERS”.

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TH

TL

𝑥

𝑦

Line of symmetry