Fin efficiency

The heat flow Qt for a straight fin with a rectangular profile was calculated in the last section. Using (2.74) and j4q0 = bSt it follows from (2.76) that 

The efficiency of a straight rectangular fin only depends on the dimensionless group  

The efficiencies of other fin shapes can be found in the same manner from the temperature distribution in the fins. Fig. 2.14 shows rjt as a function of mh for a 

For the frequently used annular fins of constant thickness f, y(r) = 5f/2 has to be put into (2.70) for the profile function. The fin efficiency r/f is dependent on two dimensionless groups mh according to (2.78) and the radius ratio (r0+h)/r0 = 1+ h/r0, cf. Fig. 2.12. This yields a complicated expression containing modified Bessel functions. F. Brandt [2.13] found the rather accurate approximation equation 

For 7f 0.5 it deviates by no more than 1% from the exact values. 

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The subscripts / and o correspond to inner and outer surfaces of tube, respectively. In these equations, Pi is a reference area for which U is defined, and T[ is the total efficiency of a finned heat-transfer surface and is related to the fin efficiency, Tl by... 

Fin efficiency is defined as the ratio of the mean temperature difference from surface to fluid divided by the temperature difference from fin to fluid at the base or root of the fin. Graphs of fin efficiency for extended surfaces of various types are given by Gardner [Tmn.s. Am. Soc. Mech. Eng., 67,621 (1945)]. 

Fin efficiencies and fin dimensions are available from manufacturers. Ratios of finned to inside surface are usually available so that the terms Aj, A, and A may be obtained from these ratios rather than from the total surface areas of the heat exchangers. 

The fin efficiency is found from mathematically derived relations, in which the film heat-transfer coefficient is assumed to be constant over the entire fin and temperature gradients across the thickness of the fin have been neglected (see Kraus, Extended Suiface.s, Spartan Books, Baltimore, 1963). The efficiency cui ves for some common fin configurations are given in Figs. Il-3(k7 and 11-30 ,... 

The fin heat transfer is determined by using fin efficiency. The fin efficiency is calculated using a theoretical approach where the whole fin is considered to be at the same temperature as the fin base. The required parameters necessary to determine the fin efficiency are shown in Fig. 9.10. 

The fin efficiency is defined by the division of the actual by the theoretical heat transfer, i.e.. 

The actual heat flow is calculated by multiplying the fin outer surface area by the fin efficiency. The outer surface area is easy to determine hence, if the fin efficiency is known, the heat transfer from the fin is easily calculated. 

Figure 10-154. Finned transfer efficiency is never as great per unit area as the bare pipe therefore, fin efficiency must be calculated to arrive at correct h , shell-side heat transfer coefficient. (Used by permission Technical paper. Brown Fintube Co., A Koch Engineering Company, Houston, Texas.)...

In this case, the low value of mL indicates a fin efficiency of almost 1.0, though where mL tends to 1.0 the efficiency falls to about 0.8. 

The fin surface area will not be as effective as the bare tube surface, as the heat has to be conducted along the fin. This is allowed for in design by the use of a fin effectiveness, or fin efficiency, factor. The basic equations describing heat transfer from a fin are derived in Volume 1, Chapter 9 see also Kern (1950). The fin effectiveness is a function of the fin dimensions and the thermal conductivity of the fin material. Fins are therefore usually made from metals with a high thermal conductivity for copper and aluminium the effectiveness will typically be between 0.9 to 0.95. 

The combustion gases flow across the tube banks in the convection section and the correlations for cross-flow in tube banks can be used to estimate the heat transfer coefficient. The gas side coefficient will be low, and where extended surfaces are used an allowance must be made for the fin efficiency. Procedures are given in the tube vendors literature, and in handbooks, see Section 12.14, and Bergman (1978b). 

Estimation of fin efficiencies of regular tubes arrayed in circumferential fins (with D.-Y. Kuan and H.T. Davis). Int. J. Heat Mass Trans. 27,148-151 (1984). 

Extended surfaces are used to augment heat transfer from the base area. To compare and evaluate these extended surfaces, two performance factors are used fin efficiency and fin effectiveness. In the... 

Problem For a fin with a uniform cross section with x = L, find the fin efficiency. 

The fin efficiency is defined as Energy actually radiated by fin Energy radiated if entire fin at temp Tj... 

To indicate the effectiveness of a fin in transferring a given quantity of heat, a new parameter called fin efficiency is defined by... 

It is interesting to note that the fin efficiency reaches its maximum value for the trivial case of L = 0, or no fin at all. Therefore, we should not expect to be able to maximize fin performance with respect to fin length. It is possible, however, to maximize the efficiency with respect to the quantity of fin material (mass, volume, or cost), and such a maximization process has rather obvious economic significance. We have not discussed the subject of radiation heat transfer from fins. The radiant transfer is an important consideration in a number of applications, and the interested reader should consult Siegel and Howell IV1 for information on this subject. 

These values may be inserted into Eq. (2-33a) to calculate the temperatures at different x locations along the rod, and the results are shown in the accompanying figure. We notice that the glass behaves as a very long fin, and its behavior could be calculated from Eq. (2-32). The fin efficiencies are calculated from Eq. (2-38) by using the corrected length approximation of Eq. (2-40). We have... 

To compare the heat flows we could either calculate the values from Eq. (2-36) for a unit value of 0O or observe that the fin efficiency gives a relative heat-flow comparison because the maximum heat transfer is the same for all three cases i.e., we are dealing with the same fin size, shape, and value of h. We thus calculate the values of ry from Eq. (2-38) and the above values of mLc. 

For this example we can compute the heat transfer by using the fin-efficiency curves in Fig. 2-12. The parameters needed are... 

The actual heat transfer is then the product of the heat flow and the fin efficiency ... 

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