Asian Journal of Current Engineering and Maths 2: 6 Nov – Dec (2013) 349 - 352.
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ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS
Journal homepage: http://www.innovativejournal.in/index.php/ajcem
TRANSIENT THERMAL STRESSES DUE TO INSTANTANEOUS INTERNAL HEAT
GENERATION IN A THIN ANNULAR DISC
C. M. Bhongade1*. M. H. Durge2
2
1Department of Mathematics, Shri. Shivaji College, Rajura, Maharashtra, India
Department of Mathematics, Anand Niketan College, Warora, Maharashtra, India
ARTICLE INFO
ABSTRACT
Corresponding Author
C. M. Bhongade
Department of Mathematics, Shri.
Shivaji College, Rajura,
Maharashtra, India
The present paper deals with the determination of temperature change,
displacement and thermal stresses in a thin annular disc defined by
with internal heat generation. Heat dissipates by convection from inner circular
surface (
and outer circular surface (
. Also, initially the annular
disc is at arbitrary temperature
Here we modify Kulkarni et al. (2008) for
homogeneous heat convection along radial direction. The governing heat
conduction equation has been solved by the method of integral transform
technique. The results are obtained in a series form in terms of Bessel’s
functions. The results for temperature change, displacement and stresses have
been computed numerically and illustrated graphically.
©2013, AJCEM, All Right Reserved.
Key Words: Transient thermal
stresses, instantaneous internal
heat generation, annular disc.
INTRODUCTION
During the second half of the twentieth century,
nonisothermal problems of the theory of elasticity became
increasingly important. This is due to their wide
application in diverse fields. The high velocities of modern
aircraft give rise to aerodynamic heating, which produces
intense thermal stresses that reduce the strength of the
aircraft structure. The steady state thermal stresses in
circular disk subjected to an axisymmetric temperature
distribution on the upper face with zero temperature on
the lower face and the circular edge has been considered by
Nowacki (1957). Gogulwar and Deshmukh (2005)
determined thermal stresses in a thin circular disc with
heat sources. Roy Choudhary (1972) discussed the quasistatic thermal stresses in a thin circular disk subjected to
transient temperature along the circumference of a circle
over the upper face with lower face at zero temperature
and fixed circular edge thermally insulated. Recently
Bhongade and Durge (2013) considered thick annular disc
and discuss thermal stresses due to arbitrary heat is
applied on the upper surface and heat dissipates by
convection from the lower boundary surface into the
surrounding at the zero temperature and the circular edges
are thermally insulated. The present paper deals with the
determination of displacement and thermal stresses in a
thin annular disc defined by
with internal heat
generation. Heat dissipates by convection from inner
circular surface (
and outer circular surface (
.
Also initially the annular disc is at arbitrary temperature
Here we modify Kulkarni et al. (2008) for
homogeneous heat convection along radial direction. The
governing heat conduction equation has been solved by the
method of integral transform technique. The results are
obtained in a series form in terms of Bessel’s functions. The
results for temperature change, displacement and stresses
have been computed numerically and illustrated
graphically. A mathematical model has been constructed of
a thin annular disc with the help of numerical illustration
by considering copper (pure) annular disc. No one
previously studied such type of problem. This is new
contribution to the field. The direct problem is very
important in view of its relevance to various industrial
mechanics subjected to heating such as the main shaft of
lathe, turbines and the role of rolling mill, base of furnace of
boiler of a thermal power plant, gas power plant and the
measurement of aerodynamic heating.
FORMULATION OF THE PROBLEM
Consider a thin annular disc occupying space D
defined by
Initially the annular disc is at
arbitrary temperature
Heat dissipates by convection
from the inner circular surface (
and outer circular
surface (
. For time
heat is generated within the
annular disc at the rate
Under these prescribed
conditions, the displacement and thermal stresses in a thin
annular disc due to internal heat generation are required to
be determined.
Following Roy Choudhuri (1972), we assume that
a annular disc of small thickness h is in a plane state of
stress. In fact, “the smaller the thickness of the annular disc
compared to its diameter, the nearer to a plane state of
stress is the actual state”. The displacement equations of
thermoelasticity have the form
(1)
(2)
where
- displacement component
e - dilatation
T – temperature
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Bhongade et.al/Transient thermal stresses due to instantaneous internal heat generation in a thin annular disc
and and are respectively, the Poisson’s ratio and the
linear coefficient of thermal expansion of the thin annular
disc material. Introducing
On applying the finite Hankel transform defined in the Eq.
(16) and its inverse transform defined in Eq. (17), one
obtains the expression for temperature as
we have
(3)
where
is the Lame constant and
symbol.
In the axially-symmetric case
(4)
is the Kronecker
(21)
DISPLACEMENT POTENTIAL AND THERMAL STRESSES
Using Eq. (22) in Eq. (5), one obtains
and the differential equation governing the displacement
potential function
is given as
(22)
Solving Eq. (23), one obtains
(5)
at
The stress function
at
and
(8)
(6)
(7)
are given by
(9)
In the plane state of stress within the annular disc
(10)
The temperature of the annular disc satisfies the heat
conduction equation,
(23)
Using Eq. (24) in Eqs. (8) and (9), one obtains expression
for thermal stresses as
(11)
with the boundary conditions,
(12)
(24)
(13)
and the initial condition
at
(14)
where k is the thermal conductivity of the material of the
disc, is the thermal diffusivity of the material of the disc,
q(r, t) is internal heat generation and
and
are the
relative heat transfer coefficients on the inner and outer
surface of the thin annular disc.
Equations (1) to (14) constitute the mathematical
formulation of the problem.
SOLUTION OF THE HEAT CONDUCTION EQUATION
To obtain the expression for temperature T(r, t), we
introduce the finite Hankel transform over the variable r
and its inverse transform defined in Ozisik (1968) are
(15)
(16)
where
(17)
and
,
(18)
The normality constant
(19)
and
,
,
…… are the positive
transcendental equation
(2)
where r is the radius measured in meter,
is well
known diract delta function of argument r.
The internal heat source
is an instantaneous line
heat source of strength situated at the centre of the thin
annular disc along the radial direction and releases its heat
instantaneously at the time
of the
(20)
where
and
(25)
SPECIAL CASE AND NUMERICAL CALCULATIONS
Setting
(1)
is Bessel function of the first kind of order n
is Bessel function of the second kind of order n.
and
Material Properties
The numerical calculation has been carried out for a copper
(pure) annular disc with the material properties defined as,
Thermal diffusivity
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Bhongade et.al/Transient thermal stresses due to instantaneous internal heat generation in a thin annular disc
Specific heat
Thermal conductivity
Shear modulus
Youngs modulus
Poisson ratio
Coefficient of linear thermal expansion
Lame constant
Roots of Transcendental Equation
The
are the roots of transcendental
equation
.
The numerical calculation and the graph has been carried
out with the help of mathematical software Matlab.
DISCUSSION
In this paper a thin annular disc is considered and
determined the expressions for temperature, displacement
and stresses due to internal heat generation within it. As a
special case mathematical model is constructed for
considering copper (pure) thin annular with the material
properties specified above.
From figure 1, it is observed that temperature increases
for
,
and
along radial
direction.
Temperature
decreases
for
and
. Overall behavior of temperature
along radial direction is decreasing from the inner circular
surface to outer circular surface of a thin annular disc.
From figure 2, it is observed that the displacement
function decreases for
,
and
along radial direction. It increases for
and
along radial direction.
Overall behavior of displacement function along radial
direction is decreasing from the inner circular surface to
outer circular surface of a thin annular disc.
From figure 3, it is observed that the radial stress
decreases from for
,
and
along radial direction. It increases for
and
along radial direction.
Overall behavior of radial stress
is tensile towards outer
circular surface along radial direction.
From figure 4, it is observed that the angular stress
function
for
,
and
along radial direction. It increases for
and
along radial direction.
Overall behavior of angular stress function
is
compressive towards outer circular surface along radial
direction.
Figure 1 Temperature function
Figure 2 Displacement function
Figure 3 Radial stress function
.
Figure 4 Angular stress function
.
CONCLUSION
We can conclude that due to instantaneous
internal heat generation, temperature and displacement
decreases from the inner circular surface to a outer circular
surface along radial direction of thin annular disc, whereas
the radial stress
is tensile in nature and the angular
stress function
is compressive in nature along radial
direction of thin annular disc. The results obtained here are
useful in engineering problems particularly in the
determination of state of stress in a thin annular disc, base
of furnace of boiler of a thermal power plant, gas power
plant and the measurement of aerodynamic heating.
REFERENCES
1. Bhongade C. M. and Durge M. H. (2013) Some study of
thermoelastic
steady
state
behavior
of
thick annular disc with internal heat generation. IOSR
Journal of Mathematics7, 47-52.
2. Gogulwar V. S. and Deshmukh K.C. (2005) Thermal
stresses in a thin circular disc with heat sources. J. Ind.
Acad. Math27, 129-141.
3. Kulkarni V. S., Deshmukh K.C. and Warbhe S. D.(2008)
Quasi- static thermal stresses due to heat generation in
a thin hollow circular disk. Journal of thermal stresses31,
698-705.
351
Bhongade et.al/Transient thermal stresses due to instantaneous internal heat generation in a thin annular disc
4. Naotake Noda, Richard B Hetnarski and Yoshinobu
Tanigawa (2003) Thermal Stresses, 2 nd edn., Taylor
and Francis, New York, 259-261.
5. Nowacki W. (1957) The state of stresses in a thick
circular disk due to temperature field. Bull. Acad. Polon.
Sci., Scr. Scl. Tech. 5, 227.
6. Ozisik M. N. (1968) Boundary Value Problems of Heat
Conduction, International Text Book Company, Scranton,
Pennsylvania.
7. Roy Choudhary S. K. (1972) A note of quasi static stress
in a thin circular disk due to transient temperature
applied along the circumference of a circle over the
upper face. Bull Acad. Polon. Sci., Ser. Scl. Techn.20, 21.
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