Question

The heat transfer surface area of a fin is equal to the sum of all surfaces of the fin exposed to the surrounding medium, including the surface area of the fin tip. Under what conditions can we neglect heat transfer from the fin tip?

Short answer

Answer: The heat transfer from the fin tip can be considered negligible under the following conditions: 1. If the heat transfer coefficient (h) is small, making convection from the fin tip insignificant. 2. If the surface area of the fin tip (A_tip) is very small compared to the overall surface area of the fin exposed to the surrounding medium. 3. If the temperature difference between the fin tip (T_tip) and the surrounding medium (T_infinity) is small, causing a minor contribution of the fin tip to the overall heat transfer.

Step-by-step solution

Understanding the heat transfer from a fin

A fin is a surface that extends from an object to increase the rate of heat transfer to or from the environment by increasing the surface area. The temperature of the object and the surrounding air may vary along the fin length, and this temperature difference causes heat to transfer by conduction along the fin and by convection from the fin's surfaces into the surrounding medium.

Calculating heat transfer from the fin tip

The heat transfer from the fin tip is a result of convection from the surface of the tip to the surrounding medium. This can be expressed mathematically using Newton's law of cooling, which states that the rate of heat transfer, Q, is proportional to the temperature difference between the surface and the surrounding medium and the surface area: Q_tip = h * A_tip * (T_tip - T_infinity), where h is the convection heat transfer coefficient, A_tip is the surface area of the fin tip, T_tip is the temperature of the fin tip, and T_infinity is the temperature of the surrounding medium.

Comparing heat transfer from the fin tip to the overall heat transfer

To know whether the heat transfer from the fin tip can be neglected, we need to compare it with the overall heat transfer from the entire fin. Generally, if the heat transfer from the fin tip is significantly smaller than the overall heat transfer from the fin, it can be neglected. To do this, we can use the following relationship: Q_total = Q_conductive + Q_convective, where Q_total is the total heat transfer, Q_conductive is the heat transfer through conduction along the fin, and Q_convective is the heat transfer due to convection from the fin's surfaces. If Q_tip << Q_total, then the heat transfer from the fin tip can be neglected.

Identifying the conditions for negligible heat transfer from the fin tip

There are several conditions where the heat transfer from the fin tip can be considered negligible: 1. If the heat transfer coefficient h is small, making convection from the fin tip insignificant. 2. If the surface area of the fin tip A_tip is very small compared to the overall surface area of the fin exposed to the surrounding medium. 3. If the temperature difference between the fin tip (T_tip) and the surrounding medium (T_infinity) is small, causing a minor contribution of the fin tip to the overall heat transfer. Under any of these conditions, the heat transfer from the fin tip can be neglected in the analysis.

Key concepts

Conduction and Convection

To fully understand how heat transfer works in objects like fins, we need to delve into the concepts of conduction and convection. Imagine a metal rod with one end heated up. The heat travels down the rod to the cooler end; this process is known as conduction. In conduction, heat is transferred through the material without the material itself moving. It occurs due to the energy exchange between adjacent atoms and molecules.

Now, picture that same rod in the air. The heat travels from the rod to the surrounding air. This movement of heat from the rod to the air is convection, which requires a fluid, such as air or water, to carry away the heat. Unlike conduction, convection involves the bulk movement of molecules within fluids (gases and liquids), taking heat away from the object.

In the context of a fin, conduction occurs along the length of the fin, transferring heat from the base where it's attached to an object (like an engine or a heat sink) to the tip. Once the heat reaches the surface of the fin, it's then transferred to the surrounding fluid through convection. The efficiency of a fin in transferring heat to the environment depends on this seamless interplay between conduction along the fin and convection away from its surfaces.

Newton's Law of Cooling

Newton's law of cooling describes the rate at which an object changes temperature through radiation or convection as a function of the difference in temperature between the object and its environment. This principle can be applied to understand the heat transfer from objects, like the tip of a fin, to their surroundings. The law states that the rate of heat loss of a body is directly proportional to the difference in temperatures between the body and its surroundings.

The mathematical representation of Newton's law of cooling is given by the equation: $Q=h\times A\times (T_{object}-T_{surroundings})$, where 'Q' is the rate of heat transfer, 'h' is the convection heat transfer coefficient, 'A' is the surface area, and $T_{object}-T_{surroundings}$ is the temperature difference. This law provides us with a fundamental understanding of how the fin tip transfers heat to the air around it. For those trying to solve heat transfer problems, Newton's law of cooling is a crucial formula to analyze how quickly a fin or any other object will cool down.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient, denoted as 'h', is a parameter that represents the convection heat transfer characteristics of a fluid in contact with a solid surface. It quantifies the efficiency with which the fluid (air, water, oil, etc.) can absorb heat from the solid surface.

The units of 'h' are typically in $W/m^2\times K$ (watts per square meter per kelvin), and this coefficient varies depending on the fluid's properties, the velocity of the fluid over the surface, and the temperature difference between the surface and the fluid. It plays a pivotal role in calculations involving Newton's law of cooling.

The higher the value of 'h', the more efficient the fluid is at absorbing heat from the surface. That's why in certain situations, when the convection heat transfer coefficient is low, the heat transfer from the fin tip to the surrounding environment might be so minimal that it can be neglected in thermal analysis. This is one of the scenarios where a designer may decide not to consider the heat transfer from the fin tip, simplifying the thermal management problem.