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**Conduction: One Dimensional, Heat Transfer**

General heat conduction relation in cylindrical coordinate system (fig. 2.12) is derived (briefly) below.

**Fig.2.12. Cylindrical coordinate system (a) and an element of the cylinder**

The energy conservation for the system is written as,

Ó€ + Ó€Ó€ = Ó€Ó€Ó€ + Ó€V (2.21)

where,

I : Rate of heat energy conducted in

II : Rate of heat energy generated within the volume element

III : Rate of heat energy conducted out

IV : Rate of energy accumulated (Ó€V)

and the above terms are defines as,

Thus,

On putting the values in equation 2.21,

Thus the Laplacian operator is,

(a) (b)

**Fig.2.13. Spherical coordinate system (a) and an element of the sphere**

In a similar way the general expression for the conduction heat transfer in spherical body with heat source can also be found out as per the previous discussion. The Laplacian operator for the spherical coordinate system (fig.2.13) is given below and the students are encouraged to derive the expression themselves.

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