Heat conduction from a sphere to a stagnant fluid. A heated sphere of radius R is suspended in a large, motionless body of fluid. It is desired to study the heat conduction in the fluid surrounding the sphere in the absence of convection. Set up the differential equation describing the temperature T in the surrounding fluid as a function of r, the distance from the center of the sphere. The thermal conductivity k of the fluid is considered constant.
Integrate the differential equation and use these boundary conditions to determine the integration constants: at r = R, T = T_K; and as r rightarrow infinity, T rightarrow T_infinity.
From the temperature profile, obtain an expression for the heal flux at the surface. Equate this result to the heat flux given by "Newton's law of cooling" and show that a dimensionless heat-transfer coefficient (known as the Nusselt number) is given by
Nu = hD/k = 2
in which D is the sphere diameter. This well-known result provides the limiting value of Nu for heat transfer from spheres at low Reynolds and Grashof numbers (see section 14.4).
In what respect are the Biot number and the Nusselt number different?