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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 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

equation : Nu = hD/k = 2

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

a)

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

equation : d/dr (r^2 q_r) = S r^2 (1)

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 (equation : q_r = -k dT/dr) we get

equation : d/dr (r^2 dT/dr) = 0 (2)

On integrating,

equation : r^2 dT/dr = C_1 or T = - C_1/r + C_2 (3)

The integration constants are determined using the boundary conditions:

equation : BC 1: r = infinity, T = T_a or C_2 = T_a (4)

equation : BC 2: r = R, T = T_R or -C_1 = (T_R - T_a) R (5)

where R is the radius of the sphere.

On substituting the integration constants, the temperature profile is

T = (T_R - T_a) R/r + T_a or (T - T_a)/(T_R - T_a) = R/r (6)

b)

Using Fourier's law and differentiating the temperature profile, the heat flux is

equation : q_r = -k dT/dr = k (T_R - T_a) R/r^2 (7)

Equating the heat flux at the sphere surface (r = R) to the heat flux as per Newton's law of cooling, we get

k (T_R - T_a)/R = h (T_R - T_a) or hR/k = 1 (8)

The Nusselt number (which is the dimensionless heat transfer coefficient) is

equation : Nu = hD/k = 2 (9)

where D is the diameter of the sphere.

Note:

Nu = hD/k_fluid or Nu = convection/(conduction in fluid) (10)

Bi = hD/k_solid or Bi = convection/((internal) conduction in solid) (11)

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