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Heat transfer problem solution BSL : Heat transfer from a radial circular fin
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Home &gt; Engineering &gt; Transport Phenomena - Heat Transfer &gt;
Heat Transfer Problem :
Heat transfer from a radial circular fin
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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 Tw and the ambient air temperature is Ta. 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.
a)
Step. Differential equation from thermal energy balance
From a thermal energy balance over a thin cylindrical ring of width Dr 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 qr is the heat flux for
Dividing by 4p B Dr and taking the limit as Dr tends to zero,
(2)
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Heat transfer problem solution BSL : Heat transfer from a radial circular fin
(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 q = (T - Ta)/(Tw - Ta). 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 = 2p r 2B and the perimeter (for surface convection) is P =
4p r. 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).
b)
Step. Radial temperature profile by solving differential equation
Equation (6) is a modified Bessel equation of order zero. Its solution is
(9)
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:
(10)
(11)
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:
(12)
Equations (10) and (11) may be solved to yield C1 and C2. Thus,
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Heat transfer problem solution BSL : Heat transfer from a radial circular fin
(13)
On substituting the integration constants, the dimensionless temperature profile is
(14)
c)
Step. Total heat loss from the fin
The heat flux at the base of the fin (r = R0) is given by
(15)
On multiplying the heat flux by the cross-sectional area for heat conduction, the total heat loss from the fin is obtained as
(16)
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