Temperature at the fin base

Consider steady one-dimensional heat conduction in a pin fin of constant diajneter D with constant thermal conductivity. The fm is losing heal by conveclion to the ambient air at T with a heat transfer coefficient of A. The nodal network of the fm consists of nodes 0 (at Ihe base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of Ax. Using the energy balance approach, obtain the finite difference formulation of (his problem to determine T, and T2 for Ihe case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in C. 

The efficiency of a square sheet fin which sits on a tube of radius ro is to be determined approximately using a difference method. The fin has side length s = 4ro, a constant thickness Sf and thermal conductivity Af. The heat transfer coefficient ctf is constant over the surface area of the fin. The temperature at the fin base is do, the ambient temperature is ds. The difference method for the calculation of the dimensionless temperature field d+ = (d — ds)/(do — ds) will be based on the square grid with Ax = 0.40ro, illustrated in Fig. 2.63. Due to symmetry it is sufficient to determine the 12 temperatures 4) to 4. ... 

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 temperature of the plate to which the fins are attached is normally known in advance. Therefore, at the fin base we have a specified temperature boundary condition, expressed as... 

The condition at the fin base remains the same as expressed in Eq. 3 -59. The application of these two conditions on the general solution (Eq. 3-58) yields, after some manipulations, this relation for the temperature distribution ... 

L that is attached to the surface with a perfect contact (Fig. 3-40). Hiis time heat is transfered from the surface to the fin by conduction and from the fin to the surrounding medium by convection with the same heat transfer coefficient h. The temperature of the fin is at the fin base and gradually decreases toward the fin tip. Convection from the fin. surface causes the temperature at any cross section to drop somewhat from the midsection toward the outer surfaces. However, the cross-sectional area of the fins is usually very small, and thus the temperature at any cross section can be considered to be uniform. Also, the fin lip can be assumed for convenience and. simplicity to be adiabatic by using the corrected length for the fin instead of the actual length. 

In the limiting case of zero thermal resistance or infinite thermal conductivity (k — < ), the temperature of the fin is uniform at the base value of 7. The heat transfer from the fin is maximum in this case and can be expressed as... 

IOS Obtain a relation for the fin efficiency for u fin of constant cross-sectional area/lc, perimeterp, length L, and thermal conductivity k exposed to convection to a medium at T with a heat transfer coefficient h. Assume the this are sufficiently long so that the temperature of the fm at the tip is nearly T. Take the temperature of the fin at the base to be Ti, and neglect heat transfer front the fin tips. Simplify the relation for (n) a circular fin of diameter D and (h) rectangular fins of thickness t. 

As already shown in section 1.2.3, heat transfer between two fluids can be improved if the surface area available for heat transfer, on the side with the fluid which has the lower heat transfer coefficient, is increased by the addition of fins or pins. However this enlargement of the area is only partly effective, due to the existence of a temperature gradient in the fins without which heat could not be conducted from the hn base. Therefore the average overtemperature decisive for the heat transfer to the fluid is smaller than the overtemperature at the hn base. In order to describe this effect quantitatively, the fin efficiency was introduced in section 1.2.3. Its calculation is only possible if the temperature distribution in the hn is known, which we will cover in the following. Results for the hn efficiencies for different hn and pin shapes are given in the next section. 

FIN EFFICIENCY. The outside area of finned tube consists of two parts, the area of the fins and the area of the bare tube not covered by the bases of the fins. A unit area of fiin surface is not so efficient as a unit area of bare tube surface because of the added resistance to the heat flow by conduction through the fin to the tube. Thus, consider a single longitudinal fin attached to a tube, as shown in Fig. 15.15, and assume that the heat is flowing to the tube from the fluid surrounding the fin. Let the temperature of the fluid be T and that of the bare portion of the tube T . The temperature at the base of the fin will also be T . The temperature drop available for heat transfer to the bare tube is T — T , or AT . Consider the heat transferred to the fin at the tip, the point farthest away from the tube wall. To reach the wall of the tube, this heat must flow by eonduction through the entire... 

In Example 3, T represents the temperature of the fin at the location denoted by the position X which varies from zero to L. Note that is measured firom the base of the fin. The boundary conditions for this problem tell us that the temperature of the fin at its base is base> and the temperature of the tip of the fin will equal the dr tempetatute, provided that the fin is very long. The solution then shows how the temperature of the fin varies along the length of the fin. 

IO3. A wedge-shaped fin is used to cool machine-gun barrels. The fin has a triangular cross section and is L meters high (from tip to base) and W meters wide at the base. This longitudinal fin is i meters long. It loses heat through a constant heat transfer coefficient h to ambient air at temperature 7. The flat base of the fin sustains a temperature Th- Show that the temperature variation obeys... 

The most common plafing bath uses fluoride to complex the fin. A typical solution contains 45 g/L staimous chloride, 300 g/L nickel chloride hexahydrate, and 55 g/L ammonium bifluofide. It is operated at pH 2.0—2.5 usiag ammonium hydroxide temperature is 65—75°C and current about 200 A/m. The bath has excellent throwing power. Air agitation is avoided. The deposit is bright without additives. Anodes are cast nickel, and the fin is replenished by additions of staimous chloride. AHoy anodes of 72% fin have been used to a much lesser extent. Tia-nickel deposits are covered by ASTM (136) and ISO (137) specifications. One other bath based on pyrophosphate has appeared ia the Hterature, but does not seem to be ia commercial use. 

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)]. 

A longitudinal tin on the outside of a circular pipe is 75 mm deep and 3 mm thick. If tire pipe surface is at 400 K. calculate the heat dissipated per metre length from the fin to the atmosphere at 290 K if the coefficient of heat transfer from its surface may be assumed constant at 5 W/m2 K, The thermal conductivity of the material of the fin is 50 W/m K and the heat loss from the extreme edge of the fin may be neglected. It should be assumed that the temperature is uniformly 400 K at the base of the fin. 

Thermal conductivity of the solid is constant Fluid is at temperature that is uniform and constant Heat transfer coefficient between fin and fluid is constant The temperature of the base of the fin is constant. 

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. 

We shall defer part of our analysis of conduction-convection systems to Chap. 10 on heat exchangers. For the present we wish to examine some simple extended-surface problems. Consider the one-dimensional fin exposed to a surrounding fluid at a temperature T as shown in Fig. 2-9. The temperature of the base of the fin is T0. We approach the problem by making an energy balance on an element of the fin of thickness dx as shown in the figure. Thus... 

All of the heat lost by the fin must be conducted into the base at x = 0. Using the equations for the temperature distribution, we can compute the heat loss from... 

From Fig. 2-12 ly = 82 percent. The heat which would be transferred if the entire fin were at the base temperature is (both sides of fin exchanging heat)... 

The total efficiency for a finned surface may be defined as the ratio of the total heat transfer of the combined area of the surface and fins to the heat which would be transferred if this total area were maintained at the base temperature T0. Show that this efficiency can be calculated from... 

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