Question

Cylindrical pin fins are used to increase the heat transfer from a flat plate of 100 mm × 100 mm. Four hundred fins are added to the plate. Each fin is 10-mm long, has a diameter of 2 mm, a thermal conductivity of 180 W/m∙K, and a fin efficiency of 90%. The plate is at Tb = 85°C and the ambient temperature is T∞ = 20°C. Convection occurs at the fin tips, and the convection heat transfer coefficient for the fin array is 50 W/m2 ∙K. Determine (a) The total heat transfer rate from the fin array. (b) The temperature at the fin tip.

Answer 1

The data given in the question are,

Area of the plate = 100100 = 10000mm2 = 1000010-6m2

Length of fin, L = 10mm = 1010-3m

Diameter of each fin, d = 2mm = 210-3m

Thermal conductivity of the fin, k = 180W/m.K

Convection heat transfer coefficient, h = 50W/m2.K

Fin efficiency, = 90%

Base temperature of the plate, Tb = 85°C

Ambient temperature, Tinf = 20°C

Total number of fins = 400

(a)

Here to find the heat dissipated through the fins, let us take a single fin from the array for analysis.

For ideal fin, the temperature of the fin is constant throughout. Let Qmax be the heat dissipated in case of an ideal fin. Then,

where, P is the fin perimeter.

The actual heat dissipated through a single fin will be,

The total heat dissipated through the entire fin = 4000.1838 = 73.513W

Area on the flat plate without fins is,

Heat transfer through this area,

Now the total heat dissipated = 73.513+28.416 = 101.929W

(b)

For a fin with convective heat loss form the tip, the temperature, T, at any point 'x' distance from the plate is given as,

where,

At the tip of the fin, x=L, hence,

Therefore we will obtain the temperature at the tip of the fin as, Ttip = 83.063°C