## Transport Phenomena - Heat Transfer Problem :

Heat conduction from a sphere to a stagnant fluid

Problem.

A heated sphere of diameter *D* is placed in a large amount of stagnant fluid. Consider the heat conduction in the fluid surrounding the sphere in the absence of convection. The thermal conductivity *k* of the fluid may be considered constant. The temperature at the sphere surface is *T*_{R} and the temperature far away from the sphere is *T*_{a}.

**Figure.**Heated sphere in a large amount of stagnant fluid.

a) Establish an expression for the temperature *T* in the surrounding fluid as a function of *r*, the distance from the center of the sphere.

b) If *h* is the heat transfer coefficient, then show that the Nusselt number (dimensionless heat transfer coefficient) is given by

__Hint__: Equate the heat flux at the sphere surface to the heat flux given by Newton's law of cooling.

Solution.

a)

From a heat balance over a thin spherical shell in the surrounding fluid,

where *S* is the rate of generation of heat per unit volume. In this case, *S* = 0 in the fluid.

Since the thermal conductivity

*k*for the fluid is constant, on substituting Fourier's law we get