Heat and Mass Transfer 2(2+0)

Lesson 10. Special cases: Fin insulated at the end, fin sufficiently long Variation of Heat Loss from Fins with Length

In order to determine the proper length of the fin, heat transfer from a fin of finite length is compared with that of infinite length.

Effectiveness of fin  

The purpose of use of fins is to enhance heat transfer from a surface. Performance of a fin is characterized by fin effectiveness, ε_f  and is defined as ratio of heat transfer from a finned surface to that of without a fin.     

Consider the case of a rectangular fin that is losing heat at the tip only. Its effectiveness is given expressed as       

                              

Divide numerator and denominator by    

Equation (9) gives the expression of effectiveness of a fin in terms of Biot Number, . If , there is no use of putting the fin as heat transfer rate will remain the same. If , heat transfer rate decreases with the addition of fin as it acts as an insulator. If , then this is desirable because heat transfer rate increases with the use of a fin.

Efficiency of fin

It is generally assumed that bond between a fin and surface, to which fin has been attached, is perfect and it does not offer any thermal resistance. Therefore, temperature at the base of the fin is maximum and equal to the surface temperature. Heat transfer from the base of fin to surrounding fluid will be maximum as temperature difference is maximum at the base of fin and is expressed as

Q_max = hA_{fin ­}(T_o – T_a) = h PL (T_o – T_a)                                                               (10)

However, due to conductive resistance of fin material, temperature decreases along length of fin, hence, heat transfer rate decreases due to decrease in temperature difference. To account for this decrease in heat transfer, fin efficiency is defined as

If fin tip is insulated, efficiency is expressed as

For a straight rectangular fin, A_c = w x 2δ and P= 2(w+2δ) as 2δ<< w, P=2w . Equation (11) becomes

Substituting the value of m from equation (13) in equation (12), we get

Efficiency of most of the fins generally used is above 90 percent.

Problems on fins

6.     Estimate the energy input required to solder together two very long pieces of bare copper wire 0.1625 cm in diameter with a solder that melts at 195°C. the wires are positioned vertically in air at 24°C and the heat transfer coefficient on the wire surface is 17 W/m²-deg. For the wire alloy, take the thermal conductivity 340 W/m-deg.

Solution        The physical situation approximates as two infinite fins with a base temperature                                         

                       of 195°C in an environment at 24°C with the given value of surface coefficient.

Example 5.4   A rod of 10mm square section and 160mm length with thermal conductivity of 50 W/m-deg protrudes from a furnace wall at 200°C, and is exposed to air at 30°C with convection coefficient comment on the result. Adopt a long fin model for the arrangement.

Solution        Heat dissipation for an infinity long fin is

Heat conducted upto any length is worked out by taking the difference of total heat and heat conducted at that section.

Heat convected upto 0.08 m length

Comments: Most of heat is dissipated in a short length of the fin. Accordingly it is   uneconomical to extend the fin beyond a certain value.

Example 5.6   Which of the following arrangement of pin fins will give higher heat transfer rate from a hot surface?

6 fins of 10 cm length

12 fins of 5 cm length

The base temperature of the fin is maintained at 200°C and the fin is exposed to a convection environment at 15°C with convection coefficient 25 W/m²-deg. Each fin has cross-sectional area 2.5 cm², perimeter 5 cm and is made of a material having thermal conductivity 250 W/m-deg.

Neglect the heat loss from the tip of fin.

Solution        The heat loss from n-fins is given by

Example 5.7   An array of 10 fins of anodized aluminium (k=180 W/m-deg) is used to cool a transistor operating at a location where the ambient conditions correspond to temperature 35°C and convective coefficient 12 W/m²-deg. Each fin measures 3 mm wide 0.4 mm thick 5 cm length and has its base at 60°C. Determine the power dissipated by the fin array.

Solution        Refer Fig. 5.7 for the fin array. The length of the fin is represented by projection perpendicular to the plane of the pipe.

                        For a fin of rectangular cross-section,

Example 5.8   A heating unit is made in the form of a vertical tube of 50 mm outside diameter and 1.2 m height. The tube is fitted with 20 steel fins of rectangular section with height 40 mm and thickness 2.5 mm. The temperature at the base of fin is 75° C, the surrounding air temperature is 20°C and the heat transfer coefficient between the fin as well as the tube surface and the surrounding air is 9.5 W/m²K. if thermal conductivity of the fin material is 55W/mK, make calculations for the amount of heat transferred from the tube with and without fin.

Solution        Heat flow rate from the tube surface without fin

Example 5.20 A horizontal steel shaft, 30 mm diameter and 600 mm long, has its first bearing located 100 mm from the end connected to the impeller of a centrifugal pump. If the impeller is immersed in a hot liquid metal at 500° C , work out the temperature at the bearing under the conditions: (a) the shaft is very long  (b)  the heat flow through the end of the shaft is negligible and   (c)  the heat is transferred to the surroundings from the end.

The temperature and convection coefficient associated with the fluid adjoining the shaft are 35°C and 68 kj/m²-hr-deg. For steel shaft, thermal conductivity k= 72 kj/m-hr-deg.

Solution        For the circular shaft

Example 5.23 A cylinder 5 cm diameter and 1 m long is provided with 10 longitudinal straight fins of material having thermal conductivity 120 W/m-deg. The fins are 0.75 mm thick and protrude 12.5 mm from the cylinder surface. The system is paced in an atmosphere at 40°C and the heat transfer coefficient from the cylinder and fins to the ambient air is 20 W/m²-deg. If the surface temperature of the cylinder is 150°C, calculate the rate of heat transfer and the temperature at the end of fins.

Consider the fin to be of finite length

Solution        For a fin of rectangular cross-section

Example 5.26 A steel rod (k= 30 W/m-deg) 1 cm in diameter and 5 cm long protrudes from a wall which is maintained at 100°C. The rod is insulated at its tip and is exposed to an environment with h =50   W/m²-deg and t_a=30°C. Calculate the fin efficiency, temperature at the tip of fin and the rate of heat dissipation.

Solution        The fin efficiency is given by