Finned surfaces equations

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. 

Very long, thin fins of thickness b, width W are attached to a black base that is maintained at a constant temperature Tb, as shown in the figure. There is a larger number of fins. The fin surface is diffuse-gray, and they are in a vacuum at temperature, Te = 0 K. Write the equation that describes the local fin temperature. 

Finally, the tennination step can be envisioned as a -elimination followed by desorption of the product ole fin from the catalyst surface (Equation (26)). 

The fin tips, in practice, are exposed to the surroundings, and thus the proper boundary condition for the fin tip is convection that also includes the effects of radiation. The fin equation can still be solved in this case using the convection at the fin tip as the second boundary condition, but the analysis becomes more involved, and it results in rather lengthy expressions for the temperature distribution and the heat transfer. Yet, in general, the fin tip area is a small fraction of the total fin surface area, and thus the complexities involved can hardly justify the improvement in accuracy. 

This differential equation covers all forms of extended surfaces, as long as the aforementioned assumptions are met. The different fin or pin shapes are expressed by the terms Aq = Aq(x) for the cross sectional area and Af = Af(x) for the fin surface area over which the heat is released. So for a straight fin of width b perpendicular to the drawing plane in Fig. 2.11, with a profile function y = y(x) we obtain the following for the two areas... 

Convection is the transfer of heat to or from, and within, flowing fluids. Section 6.3 of this chapter provides a more extensive treatment of convective heat transfer. For the analysis of heat-transfer fins, the rate of heat transfer to/from the fin surface from/to the ambient fluid is given by the equation... 

Equation (4.13-5) is only an approximation, since the temperature on the outside surface of the bare tube is not the same as that at the end of the fin because of the added resistance to heat flow by conduction from the fin tip to the base of the fin. Hence, a unit area of fin surface is not as efficient as a unit area of bare tube surface at the base of the fin. A fin efficiency r]y has been mathematically derived for various geometries of fins. 

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... 

Equation 2-5 gives a value for U based on the outside surface area of the tube, and therefore the area used in Equation 2-3 must also be the tube outside surface area. Note that Equation 2-5 is based on two fluids exchanging heat energy through a solid divider. If additional heat exchange steps are involved, such as for finned tubes or insulation, then additional terms must be added to the right side of Equation 2-5. Tables 2-1 and 2-2 have basic tube and coil properties for use in Equation 2-5 and Table 2-3 lists the conductivity of different metals. 

In the foregoing development we derived relations for the heat transfer from a rod or fin of uniform cross-sectional area protruding from a flat wall. In practical applications, fins may have varying cross-sectional areas and may be attached to circular surfaces. In either case the area must be considered as a variable in the derivation, and solution of the basic differential equation and the mathematical techniques become more tedious. We present only the results for these more complex situations. The reader is referred to Refs. 1 and 8 for details on the mathematical methods used to obtain the solutions. 

Here 6 is the thickness, Am the average thermal conductivity, and Am the average area of the wall without fins. Overall heat transfer for finned walls can be calculated using the same relationships as for an unfinned wall. The only change being that the fin area multiplied by the fin efficiency replaces the surface area of the fins in the equations. 

In which, a0 is the heat transfer coefficient at pressure p0 and reference heat flux q0 = 20 000 W/m2 calculated from the equation given in 2 for a flat tube. F(p+/y/ip) is obtained from the function F(p+) in 2, when in place of p+ = p/pCT the quantity p+/y/ p is used with the area ratio ip = Ai/A. Af is the surface area of the finned tube, A that of a flat tube of the same core diameter as the finned tube. It is... 

Fins such as those shown in Figures 6.5 and 6.7 usually have a constant cross-sectional area for conduction and are usually several times as high as they are thick. This allows the use of the one-dimensional conduction equation to calculate the temperature profile in fhe fin (Figure 6.8). It is also usually assumed that the film heat-transfer coefficient is uniform over the surface (nonconservative) and that the fin tip is adiabatic (i.e., no heat transfer, which is a slightly conservative assumption). 

Enhanced surfaces can often significandy increase the effective heat-transfer coefficient in condensation, especially if the condensing heat-transfer coefficient is the limiting factor in the overall heat-transfer-coefficient equation. Such enhancements include low fins on horizontal tubes, which increase the heat-transfer area, and fluting on vertical tubes and plane surfaces, which thins the condensate film over part of the surface by surface-tension effects. However, these improvements are limited by condensate retention between the fins and flooding of the drainage paths [7, 34],... 

The coefficient cannot be accurately found by the use of the equations normally used for calculating the heat-transfer coefficients for bare tubes. The fins change the flow characteristics of the fluid, and the coefficient for an extended surface differs from that for a smooth tube. Individual coefficients for extended surfaces must be determined experimentally and correlated for each type of surface, and such correlations are supplied by the manufacturer of the tubes. A typical correlation for longitudinal flnned tubes is shown in Fig. 15.17. The quantity T>e is the equivalent diameter, defined as usual as 4 times the hydraulic radius, which is, in turn, the cross section of the fin-side space divided by the total perimeter of fins and tube calculated as in Example 15.4. 

Rectangular Isothermal Fins on Vertical Surfaces. Vertical rectangular fins, such as shown in Fig. 4.23a, are often used as heat sinks. If WIS > 5, Aihara [1] has shown that the heat transfer coefficient is essentially the same as for the parallel-plate channel (see the section on parallel isothermal plates). Also, as WIS - 0, the heat transfer should approach that for a vertical flat plate. Van De Pol and Tierney [270] proposed the following modification to the Elenbaas equation [88, 89] to fit the data of Welling and Wooldridge [283] in the range 0.6 < Ra < 100, Pr = 0.71,0.33 < WIS < 4.0, and 42 < HIS < 10.6 ... 

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