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2020 | OriginalPaper | Chapter

6. Unsteady or Transient Heat Conduction

Author : Rajendra Karwa

Published in: Heat and Mass Transfer

Publisher: Springer Singapore

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Abstract

This chapter is devoted to the transient state of heat conduction, i.e. the heating or cooling where the temperature of the solid body varies with time as well as in the space. For bodies with a very high thermal conductivity combined with a low value of the convective heat transfer coefficient, lumped heat capacity analysis has been presented. For determination of temperature variation with time and spatial position in plates (whose thickness is small compared to the other dimensions), cylinders (whose diameter is small compared to its length) and spheres, solution based on Heisler charts has been given. Numerical method of solving transient conduction problems has been presented with a number of illustrative examples.

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previous chapter Steady-State Two-Dimensional Heat Conduction
next chapter Convective Heat Transfer
Appendix
Available only for authorised users
Footnotes
1
For two infinite plates, the differential equations are
$$\frac{{\partial^{2} t_{1} }}{{\partial x^{2} }} = \frac{1}{\alpha }\frac{{\partial t_{1} }}{\partial \tau }$$
(i)
$$\frac{{\partial^{2} t_{2} }}{{\partial y^{2} }} = \frac{1}{\alpha }\frac{{\partial t_{2} }}{\partial \tau }$$
(ii)
and their temperature distributions are
$$t_{1} = t_{1} (x,\tau )$$
(iii)
$$t_{2} = t_{2} (y,\tau )$$
(iv)
Let the solution to Eq. (6.29) is a simple product solution of above functions, i.e.
$$t_{{}} = t_{1} (x,\tau )t_{2} (y,\tau )$$
(v)
Then the derivations are
$$\frac{{\partial^{2} t}}{{\partial x^{2} }} = t_{2} \frac{{\partial^{2} t_{1} }}{{\partial x^{2} }}$$
(vi)
$$\frac{{\partial^{2} t}}{{\partial y^{2} }} = t_{1} \frac{{\partial^{2} t_{2} }}{{\partial y^{2} }}$$
(vii)
$$\frac{\partial t}{\partial \tau } = t_{2} \frac{{\partial t_{1} }}{\partial \tau } + t_{1} \frac{{\partial t_{2} }}{\partial \tau }$$
(viii)
Using Eqs. (i) and (ii), Eq. (viii) transforms to Substitution of the value of ∂t/∂τ satisfies Eq. (6.29):
$$\frac{\partial t}{\partial \tau } = \alpha t_{2} \frac{{\partial^{2} t_{1} }}{{\partial x^{2} }} + \alpha t_{1} \frac{{\partial^{2} t_{2} }}{{\partial y^{2} }}$$
$$t_{2} \frac{{\partial^{2} t_{1} }}{{\partial x^{2} }} + t_{1} \frac{{\partial^{2} t_{2} }}{{\partial y^{2} }} = \frac{1}{\alpha }\left( {\alpha t_{2} \frac{{\partial^{2} t_{1} }}{{\partial x^{2} }} + \alpha t_{1} \frac{{\partial^{2} t_{2} }}{{\partial y^{2} }}} \right)$$
Therefore, the assumed product solution, Eq. (v), is correct and the dimensionless temperature distribution for the rectangular bar can be given by Eq. (6.31).
 
Literature
Grober H, Erk S, Grigull U (1961) Fundamentals of heat transfer. McGraw-Hill Book Co, New York
Heisler MP (1947) Temperature charts for induction and constant temperature heating. Trans ASME 69:227–236
Holman JP (1992) Adapted for SI units by White PRS, heat transfer. McGraw-Hill Book Co, New York
Metadata
Title
Unsteady or Transient Heat Conduction
Author
Rajendra Karwa
Copyright Year
2020
Publisher
Springer Singapore
Book
Heat and Mass Transfer

Print ISBN: 978-981-15-3987-9

Electronic ISBN: 978-981-15-3988-6

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-981-15-3988-6_6

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