## 825

The analysis of fins is based on a simple energy balance between one-dimensional conduction down the length of the fin and the heat convected from the exposed surface to the surrounding fluid. The basic equation that applies to most fins is d2B 1dA dB +

Surface effectiveness K is defined as the actual heat transfer from a finned surface to that which would occur if the surface were isothermal at the base temperature. Taking advantage of fin efficiency, we can write

when 0 is (T - Tx), the temperature difference between fin and fluid at any point; A is the cross-sectional area of the fin; S is the exposed area; and x is the distance along the fin. Chapman2 gives an excellent discussion of the development of this equation.

The application of equation I.5 to the myriad of possible fin shapes could consume a volume in itself. Several shapes are relatively easy to analyze; for example, fins of uniform cross section and annular fins can be treated so that the temperature distribution in the fin and the heat rate from the fin can be written. Of more utility, especially for fin arrays, are the concepts of fin efficiency and fin surface effectiveness (see Holman3).

Fin efficiency n/ is defined as the ratio of actual heat loss from the fin to the ideal heat loss that would occur if the fin were isothermal at the base temperature. Using this concept, we could write

Qfm = aL( Tb - Tjv f n/ is the factor that is required for each case. Figure I.6 shows the fin efficiency for several cases.

Equation I.6 reduces to

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