91Ó°ÊÓ

The fin array of Problem \(3.142\) is commonly found in compact heat exchangers, whose function is to provide a large surface area per unit volume in transferring heat from one fluid to another. Consider conditions for which the second fluid maintains equivalent temperatures at the parallel plates, \(T_{o}=T_{L}\), thereby establishing symmetry about the midplane of the fin array. The heat exchanger is \(1 \mathrm{~m}\) long in the direction of the flow of air (first fluid) and \(1 \mathrm{~m}\) wide in a direction normal to both the airflow and the fin surfaces. The length of the fin passages between adjoining parallel plates is \(L=8 \mathrm{~mm}\), whereas the fin thermal conductivity and convection coefficient are \(k=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (aluminum) and \(h=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. (a) If the fin thickness and pitch are \(t=1 \mathrm{~mm}\) and \(S=4 \mathrm{~mm}\), respectively, what is the value of the thermal resistance \(R_{t, o}\) for a one-half section of the fin array? (b) Subject to the constraints that the fin thickness and pitch may not be less than \(0.5\) and \(3 \mathrm{~mm}\), respectively, assess the effect of changes in \(t\) and \(S\).

Step by step solution

01

Recall formula for thermal resistance of one-half fin array

Using the formula for the thermal resistance of a one-half fin array: \[R_{t, o} = \frac{1}{2} \cdot \frac{S - t}{h \cdot k} \]

02

Find thermal resistance for given values

Using the given values from the exercise, we can calculate the thermal resistance for a one-half fin array: \[R_{t, o} = \frac{1}{2} \cdot \frac{4 \text{ mm} -1 \text{ mm}}{150 \frac{\text{W}} {\text{m}^2 \cdot \text{K}} \cdot 200 \frac{\text{W}}{\text{m} \cdot \text{K}}} \] Convert the units from mm to m to make them consistent: \[R_{t, o} = \frac{1}{2} \cdot \frac{0.004\text{ m} - 0.001 \text{ m}}{150 \frac{\text{W}}{\text{m}^2 \cdot \text{K}} \cdot 200 \frac{\text{W}}{\text{m} \cdot \text{K}}} \] Now, calculate the \(R_{t, o}\) value: \[R_{t, o} = 5 \cdot 10^{-5} \frac{\text{m}^2 \cdot \text{K}}{\text{W}}\] So the thermal resistance for a one-half section of the fin array is approximately \(5 \cdot 10^{-5} \frac{\text{m}^2 \cdot \text{K}}{\text{W}}\).

03

Analyze the effects of changing fin thickness and pitch

To assess the effect of changes in fin thickness \(t\) and pitch \(S\), we can consider changes to the formula for thermal resistance while taking into account the constraints that the fin thickness and pitch may not be less than 0.5 mm and 3 mm, respectively. To better understand the effect of changing fin thickness and pitch, we can plot a graph of thermal resistance as a function of fin thickness \(t\) and fin pitch \(S\), for example. Generally, as the fin thickness \(t\) decreases, the thermal resistance will decrease, while as the pitch \(S\) decreases, the thermal resistance will also decrease, due to the increased surface area for heat transfer. However, we need to maintain the constraints given in the problem statement that the minimum fin thickness is 0.5 mm and the minimum pitch is 3 mm. This means that we cannot have an infinitely thin fin or infinitely small pitch, and there's a lower limit to how much the thermal resistance can be reduced by changes in fin thickness and pitch.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Exchanger

A heat exchanger is a device designed to efficiently transfer heat between two or more fluids, which may be gases or liquids. The primary goal is to enhance heat transfer by increasing the available surface area for the heat exchange process, often separating the fluids by a wall that prevents mixing. Heat exchangers are used in a wide range of applications, including heating, cooling, and energy recycling systems in industries. Common types include:

• Plate heat exchangers: Comprised of thin, slightly-separated plates to create channels for fluid flow.
• Tubular heat exchangers: Use tubes to separate fluids, commonly found in industrial settings.
• Shell and tube heat exchangers: Where one fluid flows through the tubes and another fluid surrounds them in a shell, ideal for high-pressure applications.

The efficiency of a heat exchanger depends on its ability to provide large surface area per unit volume, as the larger the contact area, the better the heat transfer. In our context, fin arrays serve to increase the surface area in compact heat exchangers, thus making them more efficient at transferring heat between fluids with minimal space. Key considerations for heat exchangers include flow arrangement, temperature goals, and pressure drops to ensure effective heat transfer without compromising fluid integrity.

Fin Array

A fin array is a series of extended surfaces used to enhance heat transfer by increasing the surface area exposed to a fluid. These are commonly integrated into heat exchangers to optimize their efficiency. The fins work by extending from a base surface, allowing more area for the heat to be conducted away from the main body. Fin arrays are a critical component in the design of compact heat exchangers. They allow for efficient heat dissipation by maximizing surface exposure to the working fluid, which, in this scenario, is air. This increased surface area facilitates better heat transfer, reducing thermal resistance. Important factors include:

• Fin material: High thermal conductivity materials like aluminum are preferred for better heat conduction.
• Fin geometry: Thin and tightly pitched fins improve efficiency, though these dimensions must stay within mechanical and practical constraints (e.g., minimum thickness and pitch as given in the problem).
• Placement and orientation: Needs to be strategic to maintain effective airflow and reduce pressure loss.

By optimizing the fin dimensions and arrangement, the overall performance of the heat exchanger can be significantly enhanced, impacting both thermal resistance and heat transfer capabilities.

Heat Transfer

Heat transfer is the process by which thermal energy moves from one substance to another, typically from a hotter body to a cooler one, in order to reach thermal equilibrium. This can happen through conduction, convection, or radiation. Let's break down these mechanisms:

• Conduction: Transfer of heat through a solid material. It's the mechanism of heat traveling through the fin material (e.g., aluminum in the fins).
• Convection: Heat transfer between a solid surface and the fluid moving across it. This is significant in the fin array, where the air flowing over the fins absorbs heat.
• Radiation: Heat transfer in the form of electromagnetic waves. Less significant in high-density fin arrays, but can still contribute.

The overall heat transfer in a heat exchanger is often measured by its thermal resistance — the lower the thermal resistance, the more efficient the heat transfer. Enhancing the parameters like fin thickness and pitch and ensuring proper fluid flow reduces thermal resistance. An understanding of these heat transfer mechanisms helps in designing effective fin arrays and entire heat exchangers, ensuring they meet operational and thermal performance goals.