This paper describes the numerical results of the
heat transfer from heat generating wire at different conditions
by finite element method. The parametric effects on heat transfer
were investigated. The varied parameters included ambient
conditions, as well as the shape of the cross-section. The numerical
results show that the type of the medium where the
heat generating wire immerges has strong effects on the heatdissipation
rate. As the size of the diameter the heat dissipation to
the ambient is decreased. The effects of free convection conditions
on the wire surface are also significant. However, the effect of the
shape of the cross-section is not very strong. The results presented
in this paper provide useful information for the application of
heat generating wire

Chavda, Bhargav M.

Parmar, Dipak J.

Nirma University Journal of Engineering and Technology (NUJET)

30 NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1, NO.1, JAN-JUN 2010
HEAT TRANSFER ANALYSIS OF HEAT
GENERATING WIRE USING FINITE ELEMENT
METHOD
Dipak J Parmar and Bhargav M. Chavada
Abstract—This paper describes the numerical results of the
heat transfer from heat generating wire at different conditions
by finite element method. The parametric effects on heat transfer
were investigated. The varied parameters included ambient
conditions, as well as the shape of the cross-section. The nu-
merical results show that the type of the medium where the
heat generating wire immerges has strong effects on the heat-
dissipation rate. As the size of the diameter the heat dissipation to
the ambient is decreased. The effects of free convection conditions
on the wire surface are also significant. However, the effect of the
shape of the cross-section is not very strong. The results presented
in this paper provide useful information for the application of
heat generating wire.
Index Terms—Heat transfer, finite element, heat dissipation
I. INTRODUCTION
HEAT conduction issue is one of the hot topics in theresearch area. It has wide applications in industry and
our daily life, such as electrical furnaces, electrical heating
appliances, and ventilation of computer chips. Among those
heat conduction problems, the heating elements are the core
topic of the heat conduction problems. Different heating ele-
ments demonstrate different characteristics in heat conduction.
Among many different structures of the heating elements, the
wire has been used in many areas because of its compact
structure and large thermal expansion. This study has analyzed
the heat transfer of the heat generating wire under different
conditions by using the finite element method. It provides
some useful information on designs and applications of heat
generating wire.
II. FINITE ELEMENT METHOD FOR HEATCONDUCTION
In general, heat conduction problems can be studied by
experimental, analytical, and numerical methods. In the ex-
perimental method, measurements may be very difficult in
some applications. In analytical method, it is impossible to get
accurate solutions of differential equations in most problems.
With the advancement of computational technologies, the
numerical simulation is now widely used to help resolving
the difficulty. Various numerical methods can be used to solve
heat conduction problems, such as finite differential method,
variation perturbation. Boundary element method and finite
element method. In this study, the finite element method is
Dipak Parmar is with Department of Mechanical Engineering Institute
of technology Nirma University Ahmedabad-382481,India and Bhargav M.
Chavada is with G.H.Patel College of Engineering,V.V.Nagar 38xxxx, India
dipakjparmar@nirmauni.ac.in
used. Heat conduction problems could be divided into two
categories: the steady state and the unsteady state. In a .three
dimensional heat conduction with internal heat generation
problem, the unsteady state temperature distribution T(x, y,
z,t) in a Cartesian coordinator system in the domain can be
solved by the following differential equation’”:
pc
∂T
∂τ
− ∂
dx
(
k
∂T
dx
)
− ∂
dy
(
k
∂T
dy
)
− ∂
dz
(
k
∂T
dz
)
−Q = 0;
(1)
The equation is a heat balance equation, where t is time, k is
heat conductivity, p is the density, c is the heat capacity, and
Q is the rate of the heat generated per volume from the heat
source of the material. The boundary conditions for solving
this equation should meet the following conditions in the Ω
domain:
Γ1 + Γ2 + Γ3 = Γ
where,Γ1 + Γ2 + Γ3 = Γ are 1st, 2nd and 3rd types of boundary
conditions, respectively. If T and Q are unchanged with the
time, i.e. ∂T∂τ -o , the Equation (1) becomes the steady state
heat conduction equation. This study considers the steady state
heat conduction problems with the heat generation. In the
finite element method, Galerkin’s approach can be used to
convert the heat conduction equation (1) to the finite element
equation”’. The corresponding finite element matrix of the
unsteady state heat conduction is given as follows:
CT +KT = F (2)
Where, C is the matrix of the heat capacity, K is the matrix of
the heat conduction, T is the matrix of temperature, and F is
the matrix of the temperature loading. In Equation (2) if the
first term becomes zero, the equation becomes the steady state
heat conduction finite element equation. For a 2-Dimentional
heat conduction problem, the elements of the matrix could be
expressed as
Kv =
∑
e
Kev +
∑
e
Hev (3)
Fi =
∑
e
F eQt +
∑
e
F eqt +
∑
e
F em (4)
Each term in Equations 3 and 4 can be expressed as
followings: The contribution of each cell to the matrix of heat
conduction:
Keij =
∫
Qe
(
k ∂Ni∂x
∂Ni
∂x + k
∂Ni
∂y
∂Ni
∂y
)
∂ Ω
NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1,NO.1, JAN-JUN 2010 31
The correction of heat transfer boundary condition of each cell
to the matrix of heat conduction:
Heij =
∫
Γe3
hNi Nj ∂ Γ
The temperature loading generated by heat souse in each cell:
F eQi =
∫
Ωe
ρ QNi Nj ∂ Ω
The temperature loading from constant surface heat flux:
F eqi =
∫
Γe2
qNi ∂ Γ
The temperature loading from the convection on the surface:
F eHi =
∫
Γe3
HTa Ni ∂ Ω
Where, Ni, Nj are the weighing functions. This approach has
converted the steady state heat conduction differential equation
in a space domain ? into the initial value problem of an
ordinary integration equation of temperature T at a number
of nodes. For a 3-Dimentional heat conduction problem, a
similar method can be used for deriving the corresponding
finite element equation of heat conduction.
III. RESULTS AND DISCUSSION
Based on the above heat conduction analysis using the finite
element method, the heat transfer from a wire at different
conditions was analyzed in this study. A wire is a typical
structure used for generating heat. Figure 1(a) shows the
schematic of a wire. The shape of the cross-section of the wire
is circular. The finite element model of the wire is shown in
figure 1(b).
Fig. 1. Model of solid wire
Table 1 lists a typical set of parameters conductivity, am-
bient temperature used for the baseline case in this study.
In theoretical analysis, an infinite wire is assumed. In the
numerical analysis the length of the wire is taken 5m and
adiabatic condition is assumed at the two ends of wire.
Fig. 2. FEM of wire
Diameter of wire 0.5m
Length of wire 5m
Heat generation rate 30 kW/m3
Convection heat transfer coefficient 60 W/m2K
Conductivity of Pure Copper 401 W/m K
Ambient Temperature 30oC
Table: - 1 Set of parameters for wire.
Based on the above heat conduction analysis using the finite
element method, the problem of heat generating wire is solved
by ANSYS 9 analysis software for above set of parameters.
The temperature distribution with in the heat generating wire
is shown in figure 2.
Fig. 3. Temperature distribution in Heat Generating Wire
A. The Effects of Medium Type
Heat transfer between a heat generator and its surroundings
is affected by the type of mediums. In this paper, calculations
are performed for a pure copper wire with the circular cross-
section in still water, air and a magnesia brick. Only heat
transfer by conduction in the medium is considered. Figure
3 shows the surface temperature versus the heat generation
at different types of mediums. At the same heat generation
32 NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1, NO.1, JAN-JUN 2010
Fig. 4. Temperature distribution in Heat Generating Wire
rate, the surface temperature is different at different medium.
The air and water has small heat conductivity, the corre-
sponding surface temperature is therefore higher. However,
the Mg brick’s heat conductivity is high, causing high heat
transfer rate between the wire and the Mg brick and thus
low surface temperature of the wire. In general the smaller
the heat conductivity of the medium, the less heat transfers
between the wire and the medium. It is also clear that the
surface temperature increases linearly with the heat generation.
However, the slope is different at different medium. The
smaller the heat conductivity of the medium, the larger the
slope, which indicates again less heat is transferred.
Fig. 5. Effect of Medium type
B. The Effects of Free Convection
When the free convection in the air is considered, the surface
temperature decreases significantly as shown in Figure 4. This
indicates significant impacts of the free convection to the heat
transfer between the wire and the air. In the calculation, the
free convection coefficient is specified based on experimental
data. When heat generation increases, the surface temperature
also increases, but with smaller increment because the free
convection causes lower heat resistance. The above results
indicate that the trend of the surface temperature versus the
heat generation is very similar for different medium. In the
following parametric studies, only air will be used as the
Fig. 6. Effect of Free Convection
medium, free convection is considered, and the constant heat
generation is considered.
C. The Effects of Materials
Different material has different heat conductivity. Figure 5
shows the effects of material of wires. It plots the relationship
between the surface temperature and the thermal conductivity
for 6 different materials. The surface temperature decrease
slightly with the increases of heat conductivity when the heat
conductivity is less than 50 W/m.K and then show little effects
afterwards.
D. The Effects of Diameter
Figure 6 shows the effect of the diameter of the wire on the
surface temperature. For a given heat generation rate, when the
diameter increases, initially the surface temperature decreases
very rapidly and then the surface temperature decreases grad-
ually.
Fig. 7. Effect of Materials
E. Effects of Heat Generation Rate
Figure 7 shows the effect of heat generation rate. As
expected, the higher the heat generation rate, the higher the
surface temperature.
NIRMA UNIVERSITTY JOURNAL OF ENGINEERING AND TECHNOLOGY, VOL.1,NO.1, JAN-JUN 2010 33
Fig. 8. The Effects of Diameter
F. Effects of Ambient Temperature
This paper has also studied the effects ambient temperature
on heat transfer. As shown in Figure 8 at low ambient
temperature, there is little change of the surface temperature
with the ambient temperature. At higher ambient temperature,
the surface temperature increases with the increase of ambient
temperature.
Fig. 9. Effects of Heat Generation Rate
IV. CONCLUSION
This paper has established the heat conduction model of
the finite element method based on the basic theory of heat
conduction. Numerical analysis was conducted for various heat
generating wire. A number of computations were performed to
investigate parametric effects on heat transfer. It is found that
the type of medium and the free convection have significant
impacts on heat transfer between the wire and the medium.
The effects of the area and the diameter of the wire are
relatively large. There is almost no effect of type of materials.
In general, the increase of heat generation rate and ambient
temperature causes higher surface temperature.
Fig. 10. Effects of Ambient Temperature
REFERENCES
[1] J.P. Holman, 1976. ”Heat Conduction”, McGraw-Hill. pp.5.
[2] T.M. Shih,1984, ”Numerical Heat Conduction”, Hemisphere Publishing
Corporation. pp.225
[3] F.P. Incropera, and D. P. Dewitt. 1981, ”Fundamentals of Heat Conduc-
tion”, John Wiley Sons, Inc.pp.519-527.
[4] Kun Xiangqian, 1986, ”the applications of the finite element method in
heat conduction”, science publisher, Beijing.
[5] Bian Jiantao, 2002 ”Numerical analysis of Heat transfer from heating
coil”
Dipak Parmar is Assistant professor in the Depart-
ment of Mechanical Engineering, Institute of Tech-
nology, Nirma University, he has obtained B.E. From
S.P. University in 2003 and M.E. in Jet propulsion
and gas Turbine in 2006 from the M.S. University.
He is life member of ISTE.
Bhargav Chavda is Asst. professor in the De-
partment of Mechanical Engineering, G. H. Patel
college of Engineering at Vallabh Vidyanagar, he
has obtained B.E. in mechanical engineering from
L.E.college of Engineering in 2003 and M.E. in
Thermal Science from M.S. University Baroda in
2006. His research interest are Heat transfer analysis,
Fluid flow analysis and Finite Element method.


HEAT TRANSFER ANALYSIS OF HEAT
GENERATING WIRE USING FINITE ELEMENT
METHOD

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HEAT TRANSFER ANALYSIS OF HEAT GENERATING WIRE USING FINITE ELEMENT METHOD

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