THERMOELASTIC PROBLEM OF AN ISOTROPIC CIRCULAR ANNULAR FIN

HIRANWAR P.C.; KHOBRAGADE N.W.

THERMOELASTIC PROBLEM OF AN ISOTROPIC CIRCULAR ANNULAR FIN

HIRANWAR P.C.¹, KHOBRAGADE N.W.²*
¹Department of Mathematics, RTM Nagpur University, Nagpur-440033, MS, India
²Department of Mathematics, RTM Nagpur University, Nagpur-440033, MS, India
* Corresponding Author : khobragadenw@gmail.com

Received : 18-06-2012 Accepted : 25-06-2012 Published : 27-06-2012
Volume : 3 Issue : 2 Pages : 103 - 108
J Stat Math 3.2 (2012):103-108

Cite - MLA : HIRANWAR P.C. and KHOBRAGADE N.W. "THERMOELASTIC PROBLEM OF AN ISOTROPIC CIRCULAR ANNULAR FIN." Journal of Statistics and Mathematics 3.2 (2012):103-108.

Cite - APA : HIRANWAR P.C. , KHOBRAGADE N.W. (2012). THERMOELASTIC PROBLEM OF AN ISOTROPIC CIRCULAR ANNULAR FIN. Journal of Statistics and Mathematics, 3 (2), 103-108.

Cite - Chicago : HIRANWAR P.C. and KHOBRAGADE N.W. "THERMOELASTIC PROBLEM OF AN ISOTROPIC CIRCULAR ANNULAR FIN." Journal of Statistics and Mathematics 3, no. 2 (2012):103-108.

Copyright : © 2012, HIRANWAR P.C. and KHOBRAGADE N.W., Published by Bioinfo Publications. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

Abstract

In this paper, an attempt has been made to determine the temperature distribution, displacement, and thermal stresses of a thin annular fin when the boundary conditions are known. Integral transform techniques have been utilized to obtain the solution of the problem. The results are obtained in the form of infinite series in terms of Besselâ€™s function.

Keywords

Integral transform, Transient problem Thermal stresses, Circular annular fin.

Introduction

The transient thermal stresses in an annular fin have been investigated by Wu (1997) and the solution has been obtained with the help of exponential-like solution. Deshmukh (2003) analyzed the transient thermal stresses applying Hankel transform and Fourier transform techniques, where temperature transfer condition was prescribed on the surface of an annular fin. The boundary value problem described by Wu et al. and Deshmukh et al. occurs in design application.
The present paper attempts to generalize the problem considered by Shang-Sheng Wu and obtain the exact solution of the problem. This paper further investigates the transient thermal stresses by the use of finite Marchi- Zgrablich transform and Laplace transform techniques. The results are obtained in the form of infinite series in terms of Besselâ€™s function.

Statement of the Problem

Consider an isotropic circular annular fin occupying the space $D=\left \{ (x,y,z)\in R^{3}:a\leq (x^{2}+y^{2})^{1/2}\leq b,\: 0\leq z\leq l \right \}$ . The material of the fin is isotropic, homogenous and all properties are assumed to be constant. We assume that the fin is of a small thickness and its boundary surfaces remain traction free (as shown in [Fig-1] ).
The governing equations and boundary conditions for the stress field [5,8] consist of:
a non-zero stress strain-displacement equation [7]

$\varepsilon _{r}=\frac{\partial u}{\partial r},\: \varepsilon _{\phi }=\frac{u}{r}\; \; \; \; \; (1)$

a single equilibrium equation

$\frac{d\sigma _{r}}{dr}+\frac{\sigma _{r}-\sigma _{\phi }}{r}=0\; \; \; \; \; (2)$

two equations of stress-strain-temperature relations [9]

$\sigma _{r}=\frac{E}{1-\nu ^{2}}\left [ \varepsilon _{r} +\nu \varepsilon _{\phi}-(1+\nu )aT \right ]\; \; \; \; \; (3)$

$\sigma _{\phi }=\frac{E}{1-\nu ^{2}}\left [ \varepsilon _{\phi }+\nu \varepsilon _{r}-(1+\nu )aT \right ]\; \; \; \; \; (4)$

and, two boundary conditions

$\sigma _{r}=0$ at $r=a\; \; \; \; \; (5)$

$\sigma _{r}=0$ at $r=b\; \; \; \; \; (6)$

Combining equations (1)-(4), integrating twice with respect to r, and applying the boundary conditions (5,6), one obtains the stress-displacement relations as

$\sigma _{r}=\frac{aE}{r^{2}}\int_{a}^{r}(T-T_{\infty })\eta d\eta +\frac{aE}{b^{2}-a^{2}}\left ( 1-\frac{a^{2}}{b^{2}} \right )\int_{a}^{b}(T-T_{\infty })\eta d\eta\; \; \; \; \; (7)$

$\small \sigma _{\phi }=-aE(T-T_{\infty })+\frac{aE}{r^{2}}\int_{a}^{r}(T-T_{\infty })\eta d\eta +\frac{aE}{b^{2}-a^{2}}\left ( 1-\frac{a^{2}}{b^{2}} \right )\int_{a}^{b}(T-T_{\infty })\eta d\eta\; \; \; \; \; (8)$

Substituting these expressions, for the radial & tangential stresses into the stress equilibrium equation (2), leads to the following governing equation for the thermoelastic equilibrium of the circular annular fin in qualitative agreement with equations found earlier [3 and 6] :

$\small k\left ( \frac{\partial ^{2}T}{\partial r^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{\partial ^{2}T}{\partial z^{2}} \right )-\frac{2h}{l}(T-T_{\infty })=\rho c\frac{\partial T}{\partial t}\; \; a\leq r\leq b,\: 0\leq z\leq l,\: t> 0\; \; (9)$

Introducing the following dimensionless parameters as defined in the nomenclature (Appendix A):

$\small \theta =\frac{k(T-T_{\infty })}{q_{b}a},\; \xi =\frac{r}{a},\; \zeta =\frac{z}{a},\; L=\frac{l}{a},\; \tau =\frac{(kt)}{(\rho ca^{2})},\; R=\frac{b}{a},\; N^{2}=\frac{2ha^{2}}{kl}$

The equation (9) can be written in the dimensionless form as:

$\small \frac{\partial ^{2}\theta }{\partial \xi^{2}}+\frac{1}{\xi}\frac{\partial \theta }{\partial \xi }+\frac{\partial ^{2}\theta}{\partial \zeta ^{2}}-N^{2}\theta =\frac{\partial \theta}{\partial \tau },\; 1\leq \xi \leq R,\; 0\leq \zeta \leq L,\; \tau > 0\; \; \; \; \; (10)$

The dimensionless radial and tangential stresses Sr and Sf in terms of the dimensionless displacement function are,

$\small S_{r}=-\left ( \frac{1}{\xi ^{2}} \right )\int_{1}^{\xi }\theta \xi d\xi +\left ( \frac{1}{\xi ^{2}} \frac{\xi ^{2}-1}{R^{2}-1}\right )\int_{1}^{R}\theta \xi d\xi \; \; \; \; \; (11)$

$\small S_{\phi }=-\theta +\left ( \frac{1}{\xi ^{2}} \right )\int_{1}^{\xi }\theta \xi d\xi +\left ( \frac{1}{\xi ^{2}} \frac{\xi ^{2}+1}{R^{2}-1}\right )\int_{1}^{R}\theta \xi d\xi \; \; \; \; \; (12)$

The various dimensionless boundary conditions are defined to determine the influence of the thermal boundary conditions on the thermal stresses as:
The initial condition

$\small \theta (\xi \zeta \tau )=0$ , for all $\small 1\leq \xi \leq R$ , $\small 0\leq \zeta \leq L$ and $\small \tau =0\; \; \; \; \; (13)$

the boundary conditions

$\small \left [ \theta (\xi ,\zeta ,\tau )+k_{1} \frac{\partial \theta (\xi ,\zeta ,\tau )}{\partial \xi }\right ]_{\xi =1}=F_{1}(\zeta ,\tau )$ , for all $\small 0\leq \zeta \leq L$ and $\small \tau > 0\; \; \; \; \; (14)$

$\small \left [ \theta (\xi ,\zeta ,\tau )+k_{2} \frac{\partial \theta (\xi ,\zeta ,\tau )}{\partial \xi }\right ]_{\xi =R}=F_{R}(\zeta ,\tau )$ , for all $\small 0\leq \zeta \leq L$ and $\small \tau > 0\; \; \; \; \; (15)$

where k₁ and k₂ are the radiation constants on the two annular fin surfaces

$\small \left [ \theta (\xi ,\zeta ,\tau )+c \frac{\partial \theta (\xi ,\zeta ,\tau )}{\partial \zeta }\right ]_{\zeta =0}=f(\xi ,\tau )$ , for all $\small 1\leq \xi \leq R$ and $\small \tau > 0\; \; \; \; \; (16)$

$\small \left [ \theta (\xi ,\zeta ,\tau )+c \frac{\partial \theta (\xi ,\zeta ,\tau )}{\partial \zeta }\right ]_{\zeta =L}=g(\xi ,\tau )$ , for all $\small 1\leq \xi \leq R$ and $\small \tau > 0\; \; \; \; \; (17)$

where $\small F_{1}(\zeta ,\tau )$ and $\small F_{R}(\zeta ,\tau )$ are known constants and here it is set to be zero, as this assumption, commonly made in the literatures [6] and [8] leads to considerable mathematical simplification and the function $\small f(\xi ,\tau )$ and $\small g(\xi ,\tau )$ are assumed to be known.
The equations (10) to (17) constitute the mathematical formulation of the problem under consideration.

Solution of the Problem

Convergence of the Series Solution

Special Case and Numerical Results

Set

$f(\xi ,\tau )=(1-e^{-t})e^{\xi }(1+c),\; g(\xi ,\tau)=(1-e^{-t})e^{\xi }e^{L}(1+c),\; a=0.5,$

$b=1,\; c=1,\; R=2,\; k=0.375,\; k_{1}=0.25,\; k_{2}=0.25,\; \tau =1\, sec\: and$

$L=2,\; I=1,\; \eta =1$ in the equation (24) to obtain

$\theta (\xi ,\tau )=\sum_{n=1}^{\infty }\frac{1}{c_{n}}\left \{ \frac{2\pi ka^{2}}{\eta ^{2}} \sum_{m=1}^{\infty }m(-1)^{m+1}\left [ \left ( \frac{m\pi a}{\eta } \right ) cos\left ( \frac{m\pi aL}{\eta }\right )-sin\left ( \frac{m\pi aL}{\eta }\right )\right ]\right.$

$\left. \times \int_{0}^{\tau }\left ( 1-e^{-t}\right )e^{\xi }(1+c),e^{-\left [ \left ( \frac{m\pi a}{\eta } \right )^{2}+\mu _{n}^{2}+N^{2} \right ](\tau -t)}dt\right \}$

$\times S_{0}(k_{1},k_{2},\mu_{n}\xi )r_{0}\; S_{0}(k_{1},k_{2},\mu_{n}r_{0})+\frac{2\pi ka^{2}}{\eta ^{2}}\sum_{m=1}^{\infty }(2m-1)(-1)^{m+1/2}$

$\times \left [ sin \left ( \frac{(m-1/2)\pi aL}{\eta }\right)-\left ( \frac{(m-1/2)\pi a}{\eta } \right )cos \left ( \frac{(m-1/2)\pi aL}{\eta } \right )\right ]$

$\left. \times \int_{0}^{\tau }\left ( 1-e^{-t}\right )e^{\xi }e^{L}(1+c)e^{-\left [ \left ( \frac{(m-1/2)\pi a}{\eta } \right )^{2}+\mu _{n}^{2}+N^{2} \right ](\tau -t)}dt\right \}$

$\times S_{0}(k_{1},k_{2},\mu_{n}\xi )\; r_{0}\; S_{0}(k_{1},k_{2},\mu_{n}r_{0})\; \; \; \; \; (28)$

Conclusion

The temperature, displacements and thermal stresses of a thin annular fin have been obtained, using finite Integral transform techniques, when the boundary conditions are known. The results are obtained in terms of Besselâ€™s function in the form of infinite series.
The series of solutions converge, provided we take sufficient number of terms in the series. Since the thickness of annular fin is very small, the series of solution given here will be definitely convergent. Any particular case of special interest can be derived by assigning suitable values to the parameters and functions in the series of expressions. The temperature, displacement and thermal stresses that are obtained can be applied to the design of useful structures or machines in engineering applications.
[Fig-2] and [Fig-3] shows that as time and radius increases, temperature and radial stresses goes on increasing respectively, but in [Fig-4] as time and radius increases, the Tangential stresses goes on decreasing. [Fig-5] and [Fig-6] shows that as time and radius increases, temperature and radial stresses goes on decreases respectively, but in [Fig-7] as time and radius increases, the Tangential stresses goes on increasing.

Appendix A

[Table-1]

Acknowledgement

The authors are thankful to University Grant Commission, New Delhi for providing the partial financial assistance under major research project scheme.

References

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Images

	Fig. 1- Cross-sectional view of an isotropic circular annular fin.
	Fig. 2-
	Fig. 3-
	Fig. 4-
	Fig. 5-
	Fig. 6-
	Fig. 7-
	Table 1-

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