Heat transfer problem solution BSL : Heat transfer from a radial circular fin

Heat Transfer Problem :
Heat transfer from a radial circular fin

Problem.

A pipe of radius R₀ has a circular fin of radius R₁ and thickness 2B on it (as shown in the figure below). The outside wall temperature of the pipe is T_w and the ambient air temperature is T_a. Neglect the heat loss from the edge of the fin (of thickness 2B). Assume heat is transferred to the ambient air by surface convection with a constant heat transfer coefficient h.

Figure. Radial circular fin on heated pipe.

a) Starting with a shell thermal energy balance, derive the differential equation that describes the radial temperature distribution in the fin.

b) Obtain the radial temperature distribution in the circular fin.

c) Develop an expression for the total heat loss from the fin.

Solution.

A circular fin is also called a radial fin or circumferential fin.

From a thermal energy balance over a thin cylindrical ring of width r in the circular fin, we get

Rate of Heat In - Out + Generation = Accumulation

The accumulation term (at steady-state) and the generation term will be zero. So,

(1)

where h is the (constant) heat transfer coefficient for surface convection to the ambient air and q_r is the heat flux for conduction in the radial direction.

Dividing by 4B r and taking the limit as r tends to zero,

(2)

(3)

If the thermal conductivity k of the fin material is considered constant, on substituting Fourier's law () we get

(4)

Let the dimensionless excess temperature be denoted by = (T - T_a)/(T_w - T_a). Then,

(5)

Using the chain rule on the left-hand side of the above equation and dividing throughout by r,

(6)

Alternatively, starting with the general expression for the cooling fin, we have

(7)

For the radial fin, the cross-sectional area (for conduction) is A = 2r 2B and the perimeter (for surface convection) is P = 4r. Also, r = R₀ + z, where z is the coordinate measuring distance from the outside wall of the pipe. Substituting these expressions for A and P with dr = dz, we get

(8)

The above equation on simplifying is identical to equation (5).

where c² = h/(kB). Note that I₀ and K₀ are modified Bessel functions (of order zero) of first and second kind, respectively.

The integration constants C₁ and C₂ are determined using the boundary conditions:

The second boundary condition suggests no heat loss through the edge of the circular fin (of thickness 2B), and requires the evaluation of derivatives of Bessel functions as given below:

On substituting the integration constants, the dimensionless temperature profile is

On multiplying the heat flux by the cross-sectional area for heat conduction, the total heat loss from the fin is obtained as

Heat Transfer Problem : Heat transfer from a radial circular fin

Heat Transfer Problem :
Heat transfer from a radial circular fin