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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 TR and the temperature far away from the sphere is Ta.

 

figure : heated sphere in infinite stagnant fluid 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

 

 

 

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)

 

 

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:

 


 

Related Problems in Transport Phenomena :

 

Transport Phenomena - Heat Transfer Problem : Forced convection heat transfer for plug flow in circular tube
- Problem of determining Nusselt number for forced convection in cylindrical coordinates

 


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