 Open Access
 Total Downloads : 225
 Authors : Mayank Mangal
 Paper ID : IJERTV5IS100171
 Volume & Issue : Volume 05, Issue 10 (October 2016)
 DOI : http://dx.doi.org/10.17577/IJERTV5IS100171
 Published (First Online): 13102016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Analysis of Steady State Heat Conduction through Tapered Section
Mayank Mangal1
UG Student,
Department of Mechanical Engineering, SRM University Chennai, India
Abstract It is very difficult to analyze and calculate the thermal behavior of tapered section material. The thermal behavior involves the study of heat transfer rate and heat flux. The tapered section has importance as support or flanges in boilers, chimneys at the lower end. In most of the industries their involvement in the heat loss is considered to be negligible, but it is not so as it appeared to be, so through this paper we analyze the thermal behavior of different material of tapered section according to their thermal conductivity to withstand on the particular heat production without any failure. For finding heat flow rate through the section, we are using Fourier law and comparing the result with the Solid work (simulation) 2011 Software and Finally, the variation of heat flow rate with respect to other factors.
Keywords Fourier law, Thermal Conductivity, Tapered Section, Heat Flux, Steady State

INTRODUCTION
1.1 Heat Transfer
It is a form of energy which occurs due to temperature difference. It is exchange of thermal energy between physical systems. These objects could be two solids, a solid and a liquid or gas, or even within a liquid or gas. The
Mathematically it can be expressed as,
T=f (x, y, z)
Under unsteady state, temperature varies with time. Unsteady state conditions are precursor to steady state conditions.

Conduction Heat Transerf
It is one of the most fundamental modes of heat transfer which occurs due to microscopic collision of particles and movement of electrons with in a body also known to be as internal energy. It is also called as diffusion as shown in Fig1. The rate at which energy is transferred is directly dependent on the temperature difference. Conduction is mediated by the combination of vibrations and collisions of molecules, of propagation and collisions of phonons, and of diffusion and collisions of free electrons.

Conduction Rate Equation
As we just now see that conduction is the heat transfer from one to other end. For one dimensional, heat conduction through an area A can be expressed as,
direction of heat transfer is from a region of high temperature to another region of lower temperature, and is governed by
Where,
Q= KA
——————— (3)
the Second Law of Thermodynamics. Heat transfer changes the internal energy of the systems from which and to which the energy is transferred. Heat transfer will occur in a direction that increases the entropy of the collection of
Q Is the heat transfer rate in J/s
K Is the thermal conductivity unit
(.)
A Is the area of cross section in m2
systems. The unit of heat transfer rate is J/s or watt.
The amount of internal energy changed U during t time
Is the temperature gradient.
interval and let Q amount of heat decreases.
0
0
U= ————– (1)
List of commonly used materials and their thermal conductivity
Materials
Thermal conductivity
1.Plain carbon steel
43
2.cast carbon steel
37
3.Cast alloy steel
38
4.Grey Cast iron
45
Materials
Thermal conductivity
1.Plain carbon steel
43
2.cast carbon steel
37
3.Cast alloy steel
38
4.Grey Cast iron
45
Then heat flux is defined as rate of heat transfer per unit area.
q=

Steady and Unsteady Heat Transfer
————— (2)
There are two modes of heat transfer steady and unsteady state. In case of steady state heat transfer, the temperature within the system does not change with time and temperature, it is a function of space coordinates only, but it is independent of time.
Consider a plane at a distance X from side of width b. Now we are interested to calculate the value of y as shown in the figure. From the similarity of triangles in PQR and POT, we can say that
=
() 2
=
= ().()
2
———– (4)
Figure 1


RELATED WORK
So from equation (4) we can find the whole length z as
= + 2
In order to check this we can use the boundary conditions as

X=0 we get Z=b

X=L we get Z=a
Now, we are using the Fourier law in our case,
Variety of research is being conducted on, To study the heat transfer through materials. The research papers dealing with
=
the thermal analysis of materials in industries have been studied. Some of the research paper reviews are given below
H. Baig and M. A. Antar explains the study on Natural
(.) =
Where m is the thickness
convection analysis of heat transfer across multilayer building blocks. Their study is mostly is used in construction of thermal power plants and in chimneys to prevent from the material failure.


MATHEMATICAL ANALYSIS

Deriving the result.
As we have discussed above the Fourier law of heat conduction in equation (3) now we apply this law in case of tapered section.
Consider a tapered section as shown in the figure 2
Figure 2
.[+[()()]] = — (5)
Through the relation (5) we can find heat transfer rate at any point with in the section as shown in the figure 2.

Assumptions

The flow of heat to the ground and sided faces due to conduction are considered to be negligible to make the calculation easier.

The radiation happens is to be negligible as the value of Stefan Boltzmann constant is 5.688 x 108 .If we multiply this factor with temperature difference which is in the order of hundreds gives a very less value.

The convective heat transfer is not considered here. In order to involve it we just need to add the term in equation (5).

The trapezium section is considered to be symmetrical about central axis.


EXPERIMENTAL AND SOLID WORK SIMULATION RESULT

Temperature distribution
Fig.3 PLAIN CARBON STEEL
The above graph shows variation of heat flux in x direction as the distance increases Heat flux decreases. This variation is for the material of plain carbon steel.
Fig.4 CAST CARBON STEEL
Fig.3 and Fig.4 clearly shows the temperature distribution at 473 K in the furnace and 293K outside that is surrounding temperature. The extreme red part shows the maximum temperature (473K) and blue part shows minimum temperature (293K).

Graphical Result
The variations of Heat flux at different nodes are shown below and its variation with respect to different axis and thickness can be explained. Here we can easily understand that our assumption the flow of heat through furnace to the ground Y and sides of the support Z is veryless in comparison to the direction of X.
Node
X (mm)
Y (mm)
Z (mm)
HEAT FLUX
(Cal/(scm^2))
2789
67.5555
20.3729
9.21875
1.14
328
68.663
20.0688
8.4375
1.10
362
68.663
20.0688
5.9375
1.10
2927
68.663
20.0688
4.21875
1.10
2694
69.7704
19.7648
3.75
1.08
2371
68.663
20.0688
4.21875
1.10
169
68.663
20.0688
8.4375
1.12
151
68.663
20.0688
2.5
1.18
201
68.663
20.0688
5.9375
–
Heat Flux Vs Distance
Heat Flux Vs Distance
70
69
68
67
X
70
69
68
67
X
66
65
Distance
66
65
Distance
Heat Flux
Heat Flux
Figure 5


CONCLUSION
Now from the equation derived equation number 5, it can be easily predicted that
( )
That means Q decreases with increase of X which matches with graph shown above. Hence the relation, predicted is correct.

FUTURE WORK
To analyze the heat flow through the pipes or boilers of different material in ANSYS and concluding some result from it.

REFERENCES

Analysis of steady state heat conduction in different composite wall, International Journal of Innovative research in Science, Engineering and technology volume4, Issue7, July2015.

Holman J.P Heat and Mass Transfer Tata McGrawHill 2008.

Heat and Mass transfer data book by New age international Publishers.

H. Baig and M. A. Antar, Conduction / Natural convection analysis of heat transfer across multilayer building blocks, 5th European Thermal Sciences Conference, the Netherlands, 2008.

Ali, Y.M. and Zhang, L.C., 2005, "Relativistic heat conduction," International Journal of Heat and Mass Transfer 48, 23972406. doi: 10.1016/j.ijheatmasstransfer.2005.02.003