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Journal of Power Technologies XX (X) (2011) X–X
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A Mathematical Analysis of Two Dimensional Steady State
Heat Conduction in the Coil of an Induction Heater Using
Finite Element Method
Debabrata Roy *, Ashok Kumar Naskar**,Pradip Kumar Sadhu***
*Electrical Engineering Department,Batanagar Institute of Engineering, Management & Science A unit of Techno India™ Group
B7-360/New, Ward No.30, Putkhali, Maheshtala Kolkata - 700141, West Bengal, India
** Department of Electrical Engineering, Seacom Engineering College, Howrah-711302, IndiaAffiliation, Address, Ciry ZIP Code,
Country
** *Electrical Engineering Department, Indian School of Mines (under MHRD, Govt. of India), Dhanbad - 826004, India
E-Mail: [email protected], [email protected], [email protected]
Abstract
In developing heaters typically and induction heater in specific temperature limits can be a key issue
disturbing the efficiency of the overall policy. Since typical loading of induction heater is commonly
costly. The estimation of temperature rise by tools of mathematical modelling becomes a lot of and a lot
of necessary. Excepting for providing an additional correct illustration of the matter. The projected model
might in addition cut back computing prices. The paper develops a 2-Dimentional (2-D) steady state
thermal model in polar co-ordinates by means of finite element formulation and arch shaped components.
A temperature time methodology is utilized to calculate the distribution of loss in various elements of the
induction heater. Overwhelming these loss distributions as input for finite element analysis. Additional
precise temperature distributions are obtained. The projected model is applied to predict the temperature
rise within the coil of the induction heater 3200 W totally encircled fan-cooled induction heater. The
temperature distribution has been determined considering convection from the outer air gap surface and
circular finish surface for each entirely encircled and semi encircled structures.
Keywords: Induction Heater, Coil, Design Performance, FEM.
1. Introduction
In electrical engineering design problems,
heating is the most vital burden that limits the
designing to a certain extend. More and more
heating reduces the life span of a product
considerably. Different types of losses, in
induction heater equipment are the main sources
of heat. After prediction of that heat generation,
proper cooling method has to be established to
limit this heating effect. For this, the exact point
source of the heat is to be determined. Being
specific, in case of an induction coil, in the
normal operating condition it is not easy to
determine the exact region where the heat is
being generated as it is laid below the surface
and also for its high power handling is very
dangerous for any measuring device. But in a
faulty condition when abrupt rise in the
temperature occurs, it is to be predicted before
2
Author name / Journal of Power Technologies 00 (2011) 000–000
the temperature-build up for safety of the
insulation. There the problem arises when we
need to determine the exact point where the heat
is originated as the heat generation may be
unequal for a induction heating coil when it
delivers heat. Now, to solve this problem,
differential equations have got some advantages.
But, due to the gradual growth of computers and
microprocessors, various types of numerical
methods are upheld nowadays. Finite element
method, the method that has many advantages
over any other type of solution method, is
extensively used to solve heat conduction
problems in 2-Dimentional (2-D) and in 3Dimentional (3-D) also. After considering some
geometrical restrictions, we get the solution
almost equal to the original values and hence
this method is advantageous, hence is used here
from now.
The paper deals with a basic problem related
to heat flow through the induction coil in
common steady state working conditions. The
body of induction coil consisting of copper,
lead, insulations, coolant etc may be considered
as a composite. Because of volumetric, line or
point source, heat may produce in the induction
coil. Maximum heat is produced or generated
due to eddy current loss in the induction coil. In
usual operating condition of the induction heater
heat produces continuously in the induction coil
[1-3]. In case of a faulty condition of the
induction heater heat can discrete in the
induction coil. Now problem is solved such a
manner that it does not affect. The 2Dimentional (2-D) steady state finite-element
procedure for the thermal analysis of induction
coil provides the chance for in-depth studies of
coil heating issues. By virtue of the new,
expressly derived arch part, along with associate
economical information measure and Gauss
routine, extraordinarily massive issues will be
effectively resolved. A novel two-dimensional
finite element procedure in cylindrical polar coordinates, with expressly derived answer
matrices, has been applied to the answer of the
heat conduction equation throughout steady state
condition. Though the results are approximate,
the strategy is quick, cheap and leads itself to
immediate visual photos of the temperature
pattern in an exceedingly 2-Dimentional (2-D)
slice of core conductor (single)bounded by two
90o inclined planes divided into arc shaped
elements of the coil.
For this end the heat of the conductor is
limited to a definite limit [5-8]. For full solution
of the heat [16,17]distribution of a 2Dimentional (2-D) i.e. 2-D cross-sectional area
of the single-core induction coil is calculated to
solve the problem [9-11]. The thermal
conduction of copper within the core are taken
along with for simplifying the calculation. In
this numeric analysis, the 2-Dimentional (2-D)
planes divided into arch shaped elements for
symmetry and these are divided into finite
elements as shown in Fig.2.Throughout the
solution region arch shaped elements are used
[12-14]. This is shown in Fig.2, taken from the
Fig.1 [15-16].
Fig. 1: Cross-sectional area of the coil
3
Fig. 2: Slice of core conductor delimited by two 90o
inclined planes divided into arc formed components.
convection and normal convection for the
turbulent and steady state flow in the cylindrical
core of the conductor. These depend upon
Reynolds’s number and Prandlt number.
2. Thermal Constants
For the steady state problem in two dimensions,
the following properties are required for each
different
element
materials.
Thermal
conductivity in radial direction
. i.e
2.007 for Copper and Insulation Thermal
conductivity
in
circumferential
.Direction
. i.e.
1.062 for Copper and Insulation. A typical set of
thermal constants are mentioned above for
copper wires of induction coil.
3. Calculation Of Eddy Current Loss
Force Convection
(2)
Where,
k = fluid thermal conductivity
= 70 watt/m
P = fluid density = 1.022kg/m3
Now calculate the eddy current loss in the
induction coil in following:
V = fluid velocity = 17.5m/s
µ = fluid viscosity = 2.06 kg/m-s
(1)
Re = Reynolds’s number
Table I. Parameter for find out Eddy Current Loss
Constant Dependent
upon the Material
(
Maximum Flux
Density
)
Frequency
Thickness
Area
0.1m
Where, Cp = fluid specific heat
=1008.34 J/kg
4. Convective Heat Transfer Coefficient
We will consider two convective heat
transfer coefficients which are the forced
Pr = Prandlt number
4
Author name / Journal of Power Technologies 00 (2011) 000–000
Table II. The heat transfer co-efficient for the cylindrical
curved surface of the conductor.
Item
Fluid
thermal
conductivity
Hydraulic
Diameter
Fluid density
Fluid velocity
Fluid
viscosity
Reynolds’s
number
Fluid specific
heat
Prandlt
number
Heat transfer
co-efficient
(3)
Natural Convection
(4)
GrashhoffNumber
So
(5)
To compute the temperateness transfer
coefficient for the rod-shaped curve surface of
the conductor the Reynolds number
and
therefore the Prandtl number calculated on the
base of fluid thermal conductivity, hydraulic
diameter, fluid density, fluid s velocity, fluid
viscosity and fluid heat energy are measure
and also given in Table-II.
Symbol
K
Value
70 watt/m
D
0.5m
P
V
µ
1.022kg/m3
17.5m/s
2.06 kg/m-s
Re
Cp
Pr
H
1008.34 J/kg
5
5. Steps Of Solving Finite Element Problem
Regardless of approach used to find the
element properties, the solution of a continuum
problem by the finite element method always
follows an orderly step-by-step process.
Step-1: Discretize the continuum: The
continuum on solution region is divided into
elements.
Step-2: Select Interpolation Function: Each
element is assigned nodes, and the interpolation
function is chosen to represent the variation of
the field variable over the element. A
polynomial is often selected as on interpolation
function as it is easy to integrate and
differentiate, the degree of the polynomial
depending on the number of nodes assigned to
the element.
Step-3: Find the Element properties: The
matrix equations expressing the properties of
individual elements are determined. For this, we
may use one of the four approaches mentioned
earlier.
Step-4: Assemble the Element properties to
obtain the system equations: The matrix
equations are combined to obtain the behavior
of the entire solution region.
Step-5: Solving the System equation: The
assembly process in step-4 needs a set of
simultaneously equation which can be solved to
obtain the unknown nodal values of the field
variable, after assigning the boundary
conditions.
Step-6: Make additional computation if
desired: Sometimes it may require using the
solution of the system to calculate other
important parameters.
the temperature distribution is expressed in
general forms as,
(5)
(6)
Here, T is the potential function
(Temperature)
.
V is the medium permeability (Thermal
conductivity) watt /m oC.
q is the flux (heat flux) watt / mm2.
Q is the forcing function (Heat source).
Fig .3: 2-Dimentional (2-D) heat conduction domain
6. 2-D Steady State Heat Conduction In
Cylindrical Polar Co-Ordinates
The problem concerns the steady state
temperature distribution and heat conduction in
a 2-Dimentional (2-D) domain as defined in Fig
below. The governing differential equation for
Combining equations (5) and (6), one obtains
the general partial differential equation
describing the 3-Dimentional (3-D) heat
conduction problems.
(7)
In cylindrical polar co-ordinates, the equation
above can be expressed as,
6
Author name / Journal of Power Technologies 00 (2011) 000–000
(8)
Vr, and Vθ are thermal conductivities in the
radial
and
circumferential
directions
respectively.
6.1. Finite Element Equations (Galerkin’s
Method):
The solution of equation (8) can be obtained by
assuming the general functional behaviour of
the dependent field variable in some way so as
to approximately satisfy the given differential
equation and boundary conditions. Substitution
of this approximation into the original
differential equation and boundary conditions
then results in some error called a residual. This
residual is required to vanish in some average
sense over the entire solution domain.
The approximate behaviour of the potential
function within each element is prescribed in
terms of their nodal values and some weighted
functions N1, N2 … such that,
(9)
Equation (10) expresses the desired averaging to
the error or residual within the element
boundaries. But it does not admit the influence
of the boundary. Since we have made no
attempt to choose the Ni so as to satisfy the
boundary conditions, we must use integration
by parts to introduce the influence of the natural
boundary condition.
6.2. Arch Element Interpolation Function
Consider the arch shaped element of Fig.4
below, formed by circle arch radii a, b. Radii
inclined at an angle of 2α.
The interpolation function can now be defined
in terms of set of non-dimensional co-ordinate
by non-dimensional zing the cylindrical polar
co-ordinates r,
using,
(11)
The arch element with non-dimensional coordinates is shown in Fig.4 and Fig.5
The weighting functions are strictly functions of
the geometry and are termed interpolation
functions. The interpolation functions determine
the order of the approximating polynomials for
the heat conduction problem.
The methods of weighted residuals determine
the m unknowns Ti in such a way that the error
over the entire solution domain is small. This is
accomplished by forming a weighted average of
the error and specifying that this weighed
average vanishes over the solution domain.
The required equations governing the behaviour
of an element is given by the expression,
(10)
Fig. 4: 2-D Arch Shaped Element
7
The temperature at any point within the element
is given in terms of its nodal temperature by
T=TANA+TBNB+TCNC+TDND (12)
Here the N’s are the interpolation functions
chosen as follows.
;
2.
The value of potential varies linearly
between any two adjacent nodes on the element
edges.
3.
The value of the potential function in
each element is determined by the order of the
finite element. The order of the element is the
order of polynomial of the spatial co-ordinates
which describes the potential within the
element. The potential varies as a quadratic
function of the spatial co-ordinates on the faces
and within the element.
6.3. Boundary Condition:
;
We are considering the portion of conductor
bounded by a plane passing through the center
of a single core and another plane which is 90o
displaced clockwise from the previous plane.
The temperature distribution is assumed
symmetrical across two planes, with the heat
flux normal to the surface being zero. From the
other two boundary surfaces heat is transferred
by convection to the surface. The boundary
conditions may be written in terms of δ T / δ n ,
the temperature gradient normal to the surface.
(13)
Mid core horizontal surface
(14)
Mid core vertical surface
(15)
Conductor surface
Fig. 5: Non-Dimensional Arc element
It is seen that the interpolation functions satisfy
the following conditions.
1.
At any given vertex ‘A’, the
corresponding interpolation function NA has a
value of unity and other shape functions NB,
NC have a zero value at this vertex thus at node
j, Nj=1 but Ni=0; where i=j.
(16)
Where, T = Surface temperature and TC =
conductor surface temperature
8
Author name / Journal of Power Technologies 00 (2011) 000–000
Table III. Nodal temperatures in induction coil
7. Statistics Analysis & Performance Of
Solution
The steady state response of the coil of
induction heater has been scientifically
determined with its surface temperature
mounted at a potential temperature of 200.56℃
.Now heat is transferred from the surface by
natural convection to the encircling space and
therefore the resolution is compared. The
temperature distribution in additional realistic
within the case of convection than within the
case of assumed mounted extreme temperature
on the boundary. The temperature distribution
of 2-Dimentional (2-D) core conductor of coil at
steady state, in each cases, has been numerically
obtained then calculative the heat transfer coefficient at the mean of the temperatures as
tabulated below in Table- II. In both cases the
temperature. Fig. 6 shows that temperature
variations with completely different nodal
points on the coil each in temperatures Force
convection and natural convection once same
current is passing through the coil. The system
of global equations, as determined by equation,
should be resolved to work out the nodal
temperatures. The answer of this set of linear
equation is set by the Gauss methodology that
takes the advantage of the banded nature of the
matrices. To avoid computer memory, the
bilaterally symmetrical matrix is half
information measure, with efficiency keep by
Gauss routine, by that extraordinarily massive
issues are often effectively resolved.
NODE
NUMB
ERS
FREE
CONVECT
ION
TEMP
FORCEDCONVECTIO
NTEMP
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
195.64
174.98
149.22
125.13
104.06
86.03
70.21
195.46
182.28
161.79
138.79
115.33
93.22
74.01
194.94
199.96
200.56
192.54
175.10
148.21
112.23
194.62
210.32
226.69
232.26
223.52
195.92
146.60
155.78
136.73
114.06
93.71
76.64
62.46
50.06
155.63
143.26
124.42
103.68
83.19
64.80
50.17
155.18
159.39
158.43
148.52
129.06
100.23
62.54
154.89
168.97
182.20
183.33
168.33
133.28
74.78
9
8. Conclusion
Fig. 6: Node vs Temperature
The 2-Dimentional (2-D) steady state finiteelement procedure for the thermal analysis of
induction coil provides the chance for in-depth
studies of coil heating issues. By virtue of the
new, expressly derived arch part, along with
associate economical information measure and
Gauss routine, extraordinarily massive issues
will be effectively resolved. A novel twodimensional finite element procedure in
cylindrical polar co-ordinates, with expressly
derived answer matrices, has been applied to the
answer of the heat conduction equation
throughout steady state condition. Though the
results are approximate, the strategy is quick,
cheap and leads itself to immediate visual
photos of the temperature pattern in an
exceedingly 2-Dimentional (2-D) slice of core
conductor (single)bounded by two 90o inclined
planes divided into arc shaped elements of the
coil.
References
[1] Armor, A. F.. "1981 Power Engineering
SocietyPrize Paper Transient, Three-Dimensional,
Finite-Element Analysis of Heat Flow in TurbineGenerator Rotors", IEEE Power Engineering
Review, 1981.
[2] Contuzzi, N.; Campanelli, S. L.; Ludovico, A. D.
(2011). 3D finite element analysis in theselective
laser melting process, International Journal of
Simulation Modelling, Vol. 10, No. 3,113-121,
doi:10.2507/IJSIMM10(3)1.169.
[3] Ternik, P.; Rudolf, R. (2012). Heat transfer
enhancement for natural convection flow of
waterbased nanofluids in a square enclosure,
International Journal of Simulation Modelling, Vol.
11,No. 1, 29-39, doi:10.2507/IJSIMM11(1)3.198.
[4] Ternik, P.; Rudolf, R. (2014). Laminar forced
convection heat transfer characteristics from
aheated cylinder in water based nanofluids,
International Journal of Simulation Modelling,
10
Author name / Journal of Power Technologies 00 (2011) 000–000
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
Vol.13,
No.
3,
312-322,
doi:10.2507/IJSIMM13(3)5.271
Sarkar, D., and A.K. Naskar. (2013). Computation
of thermal condition in an induction motor during
reactor starting, International Journal of Electrical
Power & Energy Systems, 2013.
Naskar, A. K., and D. Sarkar.(2015). New approach
for computational analysis of temperature rise
phenomena in the rotor of an induction motor,
Energy Systems.
Huai, Y.. "Computational analysis of temperature
rise phenomena in electric induction motors",
Applied Thermal Engineering
Law, A. M. (2015). Simulation Modeling and
Analysis, 5th edition, McGraw-Hill, New York
Blecha, P.; Prostrednik, D. (2011). Influence on the
failure probability, Proceedings of the 22nd
International DAAAM Symposium, 11-12
Holden, J .(2003). An inductive charger with a
large air-gap, The Fifth International Conference
on Power Electronics and Drive Systems 2003
PEDS
Kr.
Naskar,
Ashok,
Nirmal
Kr.
Bhattacharya,Sourav Saha, and S. N. Kundu.(2013)
Thermalanalysis of underground power cables
usingtwo dimensional finite element method,
2013IEEE 1st International Conference on
Condition Assessment Techniques in Electrical
Systems(CATCON).
Naskar, A.K., and D. Sarkar.(2012) Approximate
analysis Of 2-dimensional heat conduction in the
rotor of an induction motor during reactor starting,
2012 IEEE 5th India International Conference on
Power Electronics (IICPE), 2012.
C. Carretero, J. Acero, R. Alonso, J.M. Burdio,
Interference emission estimation of domestic
induction cookers based on finite element
simulation, Spanish MICINN under Project
TEC2010-19207,Project CSD2009-00046, and
Project IPT-2011-1158-920000, by the DGA-FSE,
and by the Bosch and Siemens Home Appliances
Group, 2011.
D. Istardi, A. Triwinarko, Induction Heating
Process Design Using COMSOL®Multiphysics
Software, Telkomnika, 9, 2, pp. 327–334 (2011).
Y. Boadi, T. Tsuchida, Todaka, M. Enokizono,
(2005). Designing ofsuitable construction of high-
frequency induction heating coil by using finiteelement method, IEEE Trans. Magn., 41, 10, pp.
4048–4050
[16] T. A. Jankowski, D. P. Johnson, J. D. Jurney, J. E.
Freer, L. M. Dougherty, S. A. Stout, (2009).
Experimental
Observation
and
Numerical
Prediction of Induction Heating Graphite Test
Article, In The Proceeding of the COMSOL
conference 2009, Boston
[17] Jarosław Milewski, Wojciech Bujalski, Marcin
Wołowicz, Kamil Futyma, Jan Kucowski (2015)
Off-design operation of an 900 MWclass power plant with utilization of low
temperature heat of flue gases Journal of
Power Technologies 95 (3) (2015) 221–
227
[18]
Kamil
Lubikowski,Stanisław
Radkowskia,
Krzysztof Szczurowskia, Michał Wikarya (2015)
Seebeck
phenomenon,
calculation
method
comparisonJournalofPowerTechnologies95(Polish
EnergyMix) 63–67