Galway-Mayo Institute of Technology
Department of Mechanical & Industrial Engineering
School of Engineering
41031 Heat Transfer
HEAT TRANSFER LAB 2 APPLICATION OF FINITE
DIFFERENCE NUMERICAL METHOD - EXERCISE
NO. 3 – RECTANGULAR FIN EXTENDED SURFACE
LECTURER: DR. J. LOHAN
Patrick Livingstone | ID Number: G00353888 | 05/ Dec /2019
Table of Contents: DECLARATION OF ORIGINALITY: ............................................................................................................ 1
INRODUCTION: ......................................................................................................................................... 2
Aim: .......................................................................................................................................................... 2
Finite difference Method: ........................................................................................................................ 2
METHOD ANALYSIS: .................................................................................................................................. 3
RESULTS: ..................................................................................................................................................... 4
Fibre Glass Board: .................................................................................................................................... 4
10 W/m2K:............................................................................................................................................. 4
100 W/m2K: .......................................................................................................................................... 4
1000 W/m2K: ........................................................................................................................................ 5
Stainless Steel: .......................................................................................................................................... 5
10 W/m2K:............................................................................................................................................. 5
100 W/m2K: .......................................................................................................................................... 6
1000 W/m2K: ........................................................................................................................................ 6
Aluminium: .............................................................................................................................................. 7
10 W/m2K:............................................................................................................................................. 7
100 W/m2K: .......................................................................................................................................... 7
1000 W/m2K: ........................................................................................................................................ 8
Isoflux Lines: ............................................................................................................................................ 8
AVERAGE FIN TEMPERATURE (TAverage): .................................................................................................. 9
Fibre Glass Board: .................................................................................................................................... 9
Stainless Steel: .......................................................................................................................................... 9
Aluminium: .............................................................................................................................................. 9
CONCLUSION: ........................................................................................................................................... 10
REFERENCE: ............................................................................................................................................... 10
PAGE | 1
DECLARATION OF ORIGINALITY:
I, Patrick Livingstone am the author of this work, using my own words (except where attributed to others). I
know that plagiarism is to use another’s work and pretend that it is one’s own, and that this is forbidden.
Signatures of Author,
______________________
Patrick John Livingstone.
PAGE | 2
INRODUCTION:
Figure 3 Provides the dimensions of a rectangular fin that is mounted on a base-plate that operates at 120⁰C.
The ambient air temperature is 18⁰C and the convective heat transfer varies from 10, 100 and 1000 W/m2K. The
fin materials being considered are:
• Glass Fibre Board (k = 0.04 W/mK),
• Stainless Steel (k = 18 W/mK)
• Aluminium (k = 180 W/mK)
AIM:
To use the finite difference numerical method to support the thermal design of one of the 4 prescribed problems
defined on the attached exercise sheet.
FINITE DIFFERENCE METHOD:
Consider the 2-D body shown on the right, that is sub divided into equal sized square segments. Note ∆X = ∆Y.
The corners in each segment is identified by ‘m’ & ‘n’ Coordinates in the ‘x’ & ‘y’ directions, and these also
define ‘Nodal Points’. The object of the analysis is to establish the temperature at these Nadal points using the
Equation below as a governing condition. ‘Finite Differences’ are used to approximate differential increments
in the temperature and place coordinates. The smaller we choose these finite increments the closer to the true
temperature distribution will be approximated.
(𝑇𝑚 + 1, 𝑛 + 𝑇𝑚, 𝑛 + 1 + 𝑇𝑚 − 1, 𝑛 + 𝑇𝑚, 𝑛 − 1)
4
This formula only applies to the interior squares (nodes).
PAGE | 3
METHOD ANALYSIS:
Using the data and variables given graphs were created to represent the heat transfer of each material. Here is
an example of a Calculation domain that was used within in Excel to help generate the data. This example was
used for Stainless Steel with a convective heat transfer coefficient of 10W/m2K. The variables were changed
for each material and tested through each variable of convective heat transfer coefficient (10 / 100 / 1000
W/m2K).
Formulae used for calculating fin temperatures:
Interior node =(B7+C6+D7+C8)/4
Right exterior =(B9+$B$17*$B$18+(C8+C10)/2)/(2+$B$17)
Top exterior =(C14+$B$17*$B$18+(B13+D13)/2)/(2+$B$17)
Bottom exterior =(C14+$B$17*$B$18+(B15+D15)/2)/(2+$B$17)
Exterior top right corner =($B$17*$B$18+(B20+C21)/2)/(1+$B$17)
Exterior bottom right corner =($B$17*$B$18+(C23+B24)/2)/(1+$B$17)
Convective Heat Transfer Coefficient, h (W/m2K) = 10
Characteristic Dimension, S (m) = 0.01
Thermal Conductivity, k (W/mK) 18
Biot Number (h.S)/k = 0.0055556
Surrounding Air Temp (Deg. C) = 18
PAGE | 4
RESULTS:
FIBRE GLASS BOARD:
10 W/m2K:
100 W/m2K:
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
Temp °C
Length (mm)
Temp °C
Length (mm)
PAGE | 5
1000 W/m2K:
STAINLESS STEEL:
10 W/m2K:
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
Temp °C
Length (mm)
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
Temp °C
Length (mm)
PAGE | 6
100 W/m2K:
1000 W/m2K:
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
Temp °C
Length (mm)
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
Temp °C
Length (mm)
PAGE | 7
ALUMINIUM:
10 W/m2K:
100 W/m2K:
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
Temp °C
Length (mm)
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
Temp °C
Length (mm)
PAGE | 8
1000 W/m2K:
ISOFLUX LINES:
0
20
40
60
80
100
120
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Convective Heat Transfer Coefficent: 3D Wireframe Temperature Distribution
0-20 20-40 40-60 60-80 80-100 100-120
Temp °C
Length (mm)
PAGE | 9
0
20
40
60
80
100
120
140
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
Centre Line Temperature Profiles @ 100 W/m2K
Glass Fibre Board Stainless Steel Aluminium
AVERAGE FIN TEMPERATURE (TAverage):
FIBRE GLASS BOARD:
• 10 W/m2K = 71.03177702
• 100 W/m2K = 69.37196835
• 1000 W/m2K = 69.06816349
STAINLESS STEEL:
• 10 W/m2K = 98.97884917
• 100 W/m2K = 89.11935939
• 1000 W/m2K = 74.13685709
ALUMINIUM:
• 10 W/m2K = 100
• 100 W/m2K = 97.808638
• 1000 W/m2K = 86.21351905
Length (mm)
Temp (°C)
PAGE | 10
CONCLUSION:
After conducting this research experiment, I have concluded that Aluminium would be the best suited material
of the three for the construction of the heat sink fin. Aluminium has the highest averages from all the materials,
this is possibly due to its higher thermal conductivity, but in using the Finite Difference Method, we have seen
that the temperature distribution is more evenly spread out over the Aluminium plane and the material reaches
a higher temperature further away from the 120oC base plate than the other materials. This would make
Aluminium the best suited material as with a higher average and a more evenly distributed temperature range
there is more contact with the surrounding 18oC, thus more chance of the fin transferring the heat to the
surrounding air.
The Infinite difference method is a very simple and effective way of analyzing/modelling the heat transfer
through a 2D object with strong degree of accuracy.
REFERENCE:
Information has been taken from: Heat transfer module notes and the excel file supplied for the experiment,
both supplied by John Lohan.
