Advanced Materials Research Vol. 445 (2012) pp 893-898
© (2012) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMR.445.893
A Method of Analysis to Estimate Thermal Down-shock Stress Profiles
in Hollow Cylinders When Subjected to Transient Heat/Cooling Cycle
1
2
M.A. Ali , S.T.Hasan , and D.P. Myriounis
1,2,3:
3
Faculty of Arts, Computing, Engineering and Sciences, Sheffield Hallam University, Sheffield,
United Kingdom.
Mohammed.A.Nasser@student.shu.ac.uk, S.Hasan@shu.ac.uk, D.Myriounis@shu.ac.uk
Keywords: Thermal shock, down-shock, thermal stresses, cylindrical shell.
Abstract. An empirical solution for the thermal shock stresses in cylindrical shell presented when
cylinder is subjected to heating or re-heating case and down-shock cooling by forced air case.
Linear equations are developed to describe the severity of thermal shock loading. When thermal
gradient and time period are in consideration, it is shown the equations displays good
approximation for major characteristics of the thermal shock stress profiles.
Introduction
Many industries deal with components which are subjected to higher loads at elevated
temperatures due to the increasing performance requirements. The difficult prediction of a material
to be well chosen for its performance under transient thermal cycling become a reason of primary
damage under a large changes in temperature over a period of time. Specially, the most interesting
cases are the heating or re-heating and cooling cycles during start-up and shut-down operations
which cause thermal stresses superimposed with mechanical loadings. For example in land-based
turbines, pressure vessel and heat exchanger in conventional and nuclear power plants, thermal
fatigue may occur as a result of plasticity at the crack tip and associated stresses arising from the
thermal gradients. Research effort has been concentrating on the stresses under transient cooling
condition; H. F. Nied, has observed that the transient thermal stresses of a hollow cylinder under
rapid cooling are dependent on the time, inner-to-outer radius and the coefficient of diffusivity of
the material,[1]. G. A. Kardomateas, has presented the transient thermal stress distribution with time
through the wall thickness of the hollow cylinder subjected to a constant temperature at one side
and convection heat transfer at different temperature in the other side, [2]. For dynamic wave
propagation studies, H. Cho, has studied the elastodynamic solution for thermal shock stress in
thick cylinder shell subjected to rapid thermal loading showing the wave formation of the elastic
stresses and the important value of the thermal shock distribution throughout the shell wall similar
works due to heat transfer into a medium, [3]. H. Cho, has presented the dynamic fluctuation of the
thermal shock stresses employing practical inertia quantity and material properties through the
orthotropic thick cylinder shell made of glass/epoxy, [4]. G. A. Kardomateas, has observed the
fluctuation of the transient thermal stresses with time through the thickness of composite hollow
cylinder affected by a constant temperature at one side and heat transfer into a medium under
various temperature conditions at the other side, [5]. This paper produces the estimation of the
transient temperature and thermal stresses in hollow cylinder which takes place as in start-up and
shutdown conditions in nuclear power plant by classifying them to heating or re-heating and downshock cooling cases.
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Materials and Manufacturing Technologies XIV
Transient Temperature Formulation
To analyse the hollow cylinder subjected to transient thermal loading in heating and down shock
cooling cases, a temperature distribution is estimated to evaluate the thermal stresses induced due to
the thermal loading. During the transient thermal loading the temperature gradients are introduced
across the hollow cylinder wall thickness in which results vary with position on the shell wall
thickness and time.
The transient temperature distributions for a hollow cylinder geometric model are given by
employing the solution of the general conduction heat equation given by:
,
, 1 ,
= + , < < ,
> 01
Where D is the coefficient of diffusion (i.e., = . ; , , are being the thermal conduction
coefficient of the cylinder material, the mass density, and the specific heat. Carslaw and Jaeger, has
modelled a hollow cylinder using the application of Laplace transformation in a cylindrical region
and the Bessel's function of the first and second kinds employing the boundary conditions of each
case studied, [6].
I- Heating Case
In this case the outer surface of the hollow cylinder is subjected to heating flux and the inner
surface will be kept at a constant temperature and the initial temperature assumed to be at a selected
temperature.
The initial and general boundary conditions are:
, = = 200! ° , = 02
, = # = 400! ° , = , > 03
, &
= , = , > 04
The distribution of temperature , through the shell wall in
this case is :
,
= '( + ' ln + + ',- ./ 0- . 1 0- − 1 0- . / 0- 35
∞
-5(
Where ,# are the initial and inner surface temperatures, F is the heat flux supplied to the
outer surface of the hollow cylinder, a, b are inner and outer radii and '( , ' , ',- are constant
shown in appendix A. Also, 0- are the positive roots of the equation:
./ 0- . 1( 0- − 1 0- . /( 0- 3 = 06
II- Down-shock Cooling Case
The outer surface of the hollow cylinder will be considered at constant temperature and the inner
surface will be down shock cooled by forced air and the initial temperature assumed to be at room
temperature.
Advanced Materials Research Vol. 445
895
The initial and general boundary conditions are:
, = = 25! ° , = 07
89:,;
8
<
= − # − #= , = , > 08
, = ? = 600! ° , = , > 09
The distribution of temperature , through the shell wall in this
case is:
,
= A( + A ln + + A,- ./ 0- . 1( 0- − 1 0- . /( 0- 310
∞
-5(
Where ,# ,#= ,? are the initial, inner surface, forced air and outer surface temperatures, H
is the coefficient of surface heat transfer when the inner surface is exposed to cooling by forced air
depending on the air velocity and the diameter of the pipe and A(, A, A,- are constant shown in
appendix A. Also, 0- are the positive roots of the equation as follow:
./( 0- . 1 0- − 1( 0- . / 0- 3 = 011
Thermal Stress Formulation
An elastic relation for resultant thermal stresses have been derived by Timoshinko and Goodier,
induced in rectangular plate of uniform thickness and hollow cylinder due to non-symmetrical
temperature distribution, [7].
The general thermal redial stresses formulas for a hollow cylinder is given by:
σBCDEFBGHI
B
αE 1 r − a Q
=
N Trdr − N Trdr12
1 − ν r b − a H
H
Where E, α, and ν are the young's modulus, linear coefficient of thermal expansion and poisons'
ratio respectively also, a, b are inner and outer radii. From the substitutions of the temperature
distribution in the general formulas in the cases studied the thermal stresses estimation is given:
I- Thermal Stress of Heating Case
The final radial stress for the heating case is given by:
R = R( + R + R, 13
Where R( , R STR, are the solutions of Eq.5.for this case.
II- Thermal Stress of Down-shock Cooling Stress Case
The final radial stress for the heating case is given by:
R = R( + R + R, 14
Where R( , R STR, are the solutions of Eq.10 for this case.
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Materials and Manufacturing Technologies XIV
Results and Discussions
The thermal shock loading are estimated in this research, in a hollow cylinder shell subjected to
heating and down shock cooling cases. The Laplace transformation and Bessel's functions are used
for solving the transient temperature under boundary conditions of each case and the integral
transformation have been applied for solving the thermal stresses in cylindrical coordinates. A
numerical program is designed to carry out the estimate of the transient temperature distribution and
the thermal stresses for a hollow cylinder geometry made of AISI 410 Martensitic stainless steel
employed in the heating and down shock cooling cases. In the heating case, the temperature
distribution passing through the shell wall as the boundary conditions shown in Fig.1. Transient
temperature appears as a maximum when the time period reaches close to zero which is mostly
clear in Fig.2. From the results the temperature gradients are affected by a period of time and then
relax later with increasing time.
The radial stress quantities are maximum when the temperature distribution is minimum through
the shell wall and also effected by time as illustrated in Fig.3. Thermal loads are affected by the
time periods, the diffusivity property of the material, supplied heat flux and the ratio of the inner
and outer radii. In the cooling case, the transient temperatures calculated are of relevance with a
period of time as shown in Fig.4.The roles of the convection coefficient of the forced air with the
diffusivity property of the material, in addition to the inner and outer radii have a very important
implicit effect in the thermal loading results As well as, the effects of the thermal expansion factor
and the stiffness property of the material on the thermal stresses. The maximum radial stress is
mostly clear due to the effect of sudden cooling from inner side and heating from the outer and it
will experience the opposite with the transient temperature gradient becoming highest when the
time reaches close to zero as shown in Fig.5. When the cooling load increased by increasing the
convection coefficient of the forced air the curves tends toward the outer surface in transient
temperature distribution and thermal stresses through the shell wall and vice versa when the
temperature of the outer surface increased. The description of Figs. 6 and 7, indicate the important
effect of the forced air on the temperature distribution which is increased due to the sudden downshock cooling when the air temperature decreased. The thermal loading results appear very close
because of the small thickness of the hollow cylinder and the thermal property of the material
selected. In the two cases the values of the stresses which occurs at (t=0), has the expected values of
VWX
600,6
800
600,5
700
600,4
600,3
600,2
t = 0.5 sec
600,1
t = 0.7 sec
600
599,9
0,004
t = 1 sec
0,0045
0,005
r - Radius ( m)
T - Temperature ( ºC )
T - Temperature ( º C )
U (CZY[ positive for radial thermal stress.
600
500
400
r=0.004 (m)
300
200
r=0.0045
(m)
100
0
0
1
2
3
t- Time ( sec )
Figure 1. The temperature distribution with time
through the shell wall in heating case.
Figure 2. The temperature distribution with time
in heating case.
5,00E+11
700
0,00E+00
600
-5,00E+11
0
2
T- Temperature ( º C )
σr - Radial stresses ( N/ m2)
Advanced Materials Research Vol. 445
4
-1,00E+12
r=0.004
(m)
r=0.0045
(m)
-1,50E+12
-2,00E+12
897
500
400
r = 0.004
m
r = .0045
m
r = 0.005
m
300
200
100
-2,50E+12
0
0
-3,00E+12
1
2
3
4
t- Time (sec )
t -Time ( sec )
Figure 4. The temperature distribution with
time in down shock cooling case.
Figure 3. The radial stresses distribution with time
in heating case.
2,00E+09
599,9
2
4
r = 0.004 m
-4,00E+09
r = .0045 m
-6,00E+09
r = 0.005
m
-8,00E+09
-1,00E+10
Temperature - ( º C )
0
-2,00E+09
599,8
599,7
599,6
Ta= 0 ºC
599,5
Ta = 50
ºC
Ta= 100
ºC
599,4
599,3
0,004 0,0042 0,0044 0,0046 0,0048 0,005
t- Time (sec )
Figure 5. The radial stresses distribution with
time in down shock cooling case.
Figure 6. The effect
of the forced
r - Radius
(m) air on the
temperature distribution through the shell wall at
t=0.5 sec.
2,00E+06
σr - Radial stresses ( N/m2 )
σr - Radial stress ( N/m2)
0,00E+00
0,00E+00
0,004
-2,00E+06
0,0042
0,0044
0,0046
0,0048
0,005
-4,00E+06
-6,00E+06
-8,00E+06
-1,00E+07
Ta = 0 ºC
Ta = 50 ºC
-1,20E+07
-1,40E+07
r - Radius (m)
Figure 7. The effect of the forced air on the radial stresses
distribution through the shell wall at t= 0.5 sec.
Ta= 100 ºC
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Materials and Manufacturing Technologies XIV
Conclusion
The thermal shock loading are obtained from this research in a hollow cylinder shell subjected to
heating and down shock cooling cases. The combined effects of the time, material properties and
the ratio of the inner and outer radii of the hollow cylinder on the transient temperature gradient are
clearly domenstrated.Hence; their functions on the thermal stresses must be taken in consideration.
Also, the thermo-elastic stresses are mostly influenced by the thermal expansion factor and stiffness
properties of the material. In the heating case, the temperature gradient solutions affected by a
period of time and its maximum appearance when the time reaches close to zero, on the contrary,
the radial thermal stress minimum. In the cooling case, the down shock is clear depends on the
convection coefficient effect and the temperature of the forced air which is appeared very close
because of the small thickness of the shell wall and the material property selected.
Appendix A
'( = + # ,' = ',- = −]^ C_`a .
b
?\
b
&. / 0- d
0- / 0- − /( 0- /( 0- . c0- # /( 0- − #<
# − #= ,e
l
jY ?`a .k 9n C9no: .jY ?`a C`a 9p jq #`a r
A( = + ?, A = −
A,- = ]^ C_`a .
m
= 5.5 × 10Cg . h.i . T C. `a jq b #`a CjY b ?`a References
[1]
H.F. Nied and F. Erdogan, in: Transient Thermal Stress Problem for a Circumferentially
Cracked Hollow Cylinder, submitted to Journal of Thermal Stresses, Vol. 6 (1983), p. 1-14.
[2]
G.A. Kardomateas, in: The Initial Phase of Transient Thermal Stresses due to General
Boundary Thermal Loads in Orthotropic Hollow Cylinder, submitted to Journal of Applied
Mechanics, Vol. 57(1990), p. 719.
[3] H. Cho, G. A. Kardomateas and C. S. Valle, in: Elastodynamic Solution for the Thermal Shock
Stresses in an Orthotropic Thick Cylinder Shell, submitted to Journal of Applied Mechanics,
Vol. 65(1998), p. 184.
[4]
H. Cho and G. A. Kardomateas, in: Thermal Shock Stresses due to Heat Convection at a
Bounding Surface in a Thick Orthotropic Cylindrical Shell, submitted to International Journal
of Solids and Structures, Vol. 38 (2001), p. 2769-2788.
[5]
G. A. Kardomateas, in: Transient Thermal Stresses in Cylindrically Orthotropic Composite
Tubes, submitted to Journal of Applied Mechanics, Vol. 56 (1989), p. 411.
[6]
H. S. Carslaw and J. C. Jaeger, in: Conduction of Heat in Solids, edited by Oxford Science
Publications, Oxford (1959).
[7]
S. P. Timoshenko and J. N. Goodier, in: Theory of Elasticity, edited by McGraw-Hill,
Kogakusha, Ltd (1970).