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# 物理代写|传热学代写Heat Transfer代考|ME422 CONDUCTION THROUGH FINS WITH UNIFORM CROSS-SECTIONAL AREA

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## 物理代写|传热学代写Heat Transfer代考|CONDUCTION THROUGH FINS WITH UNIFORM CROSS-SECTIONAL AREA

From Newton’s Law of Cooling, the heat transfer rate can be increased by either increasing the temperature difference between the surface and fluid, the heat transfer coefficient, or the surface area. For a given problem, the temperature difference between the surface and fluid may be fixed, and increasing the heat transfer coefficient may result in more pumping power. One popular way to increase the heat transfer rate is to increase the surface area by adding fins to the heated surface. This is particularly true when the heat transfer coefficient is relatively low such as the air-side of heat exchangers (e.g., the car radiators) and air-cooled electronic components. The heat transfer rate can increase dramatically by increasing the surface area many times with the addition of numerous fins. Therefore, heat is conducted from the base surface into the fins and dissipated into the cooling fluid. However, as heat is conducted through the fins, the surface temperature decreases due to a finite thermal conductivity of the fins and the convective heat loss to the cooling fluid. This means the fin temperature is not the same as the base surface temperature and the temperature difference between the fin surface and the cooling fluid reduces along the length of the fins. It is our job to determine the fin temperature in order to calculate the heat loss from the fins to the cooling fluid.
In general, the heat transfer rate will increase with the number of fins. However, there is a limitation on the number of fins. The heat transfer coefficient will reduce if the fins are packed too close. In addition, the heat transfer rate will increase with thin fins that have a high thermal conductivity. Again, there is limitation on the thickness of thin fins due to manufacturing concerns. At this point, we are not interested in optimizing the fin dimensions but in determining the local fin temperature for a given geometry and working conditions. We assume that heat conduction through the fin is 1-D steady state because the fin is thin. The temperature gradient in the other two dimensions is neglected. We will begin with the constant cross-sectional area fins and then consider variable cross-sectional area fins. The following is the energy balance of a small control volume of the fin with heat conduction through the fin and heat dissipation into cooling fluid, as shown in Figure 2.7. The resulting temperature distributions through fins of different materials can be seen from Figure $2.8$.

## 物理代写|传热学代写Heat Transfer代考|Fin Performance

Most often, before adding the fins, we would like to know whether it is worthwhile to add fins to the smooth, heated surface. In this case, we define the fin effectiveness. The fin effectiveness is defined as the ratio of the heat transfer rate through the fin surface to that without the fin (i.e., convection from the fin base area).

$$\eta_{\varepsilon}=\frac{q_{\text {with fin }}}{q_{\text {without fin }}}$$
The fin effectiveness must be greater than unity in order to justify using the fins. Normally, it should be greater than 2 in order to include the material and manufacturing costs. In general, the fin effectiveness is greater than 5 for most of the effective fin applications. For example, for the long fins (case 4 fin tip boundary conditions), the fin effectiveness is
$$\eta_{\varepsilon}=\frac{q_{\text {with fin }}}{q_{\text {without fin }}}=\frac{\theta_b \sqrt{h P k A_c}}{h \theta_b A_c}=\sqrt{\frac{k P}{h A_c}}>1-5$$

## 物理代写|传热学代写Heat Transfer代考|Fin Performance

$$\eta_{\varepsilon}=\frac{q_{\text {with fin }}}{q_{\text {without fin }}}$$

$$\eta_{\varepsilon}=\frac{q_{\text {with fin }}}{q_{\text {without fin }}}=\frac{\theta_b \sqrt{h P k A_c}}{h \theta_b A_c}=\sqrt{\frac{k P}{h A_c}}>1-5$$

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