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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 TR and the temperature far away from the sphere is Ta.
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
a)
Step. Differential equation from heat balance
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
Step. Temperature profile by solving differential equation
On integrating,
The integration constants are determined using the boundary conditions:
where R is the radius of the sphere.
On substituting the integration constants, the temperature profile is
b)
Step. Nusselt number from heat flux
Using Fourier's law and differentiating the temperature profile, the heat flux is
Equating the heat flux at the sphere surface (r = R) to the heat flux as per Newton's law of cooling, we get
The Nusselt number (which is the dimensionless heat transfer coefficient) is
where D is the diameter of the sphere.
Note:
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