Question

Two finned surfaces are identical, except that the convection heat transfer coefficient of one of them is twice that of the other. For which finned surface is the \((a)\) fin effectiveness and \((b)\) fin efficiency higher? Explain.

Short answer

In summary, for two identical finned surfaces with the only difference being their convection heat transfer coefficients, the surface with the higher convection heat transfer coefficient (surface 2) would have higher fin effectiveness. However, we cannot make a definitive statement about the fin efficiency of the two surfaces as it depends on multiple factors, including the fin geometry and material properties.

Step-by-step solution

Define fin effectiveness and fin efficiency

Fin effectiveness is the ratio of the heat transfer rate through the fin to the heat transfer rate if the entire surface were at the base temperature. Fin efficiency is the ratio of the actual heat transfer rate through the fin to the heat transfer rate if the entire fin were at the base temperature. Mathematically, fin effectiveness \(\epsilon_f\) and fin efficiency \(\eta_f\) are given by the following formulas: \[ \epsilon_f = \frac{Q_f}{Q_{f,max}} \] \[ \eta_f = \frac{Q_f}{Q_b} \] where \(Q_f\) is the heat transfer rate through the fin, \(Q_{f,max}\) is the maximum heat transfer rate if the surface were at the base temperature, and \(Q_b\) is the heat transfer rate through the base.

Derive relations between fin effectiveness, fin efficiency, and convection heat transfer coefficient

Firstly, we'll need to express \(Q_f\) in terms of fin geometry, material properties, convection heat transfer coefficient (\(h\)), fin temperature (\(T_f\)), and ambient temperature (\(T_\infty\)). The heat transfer rate \(Q_f\) can be found using the following formula: \[ Q_f = hA(T_f - T_\infty) \] where \(A\) is the surface area of the fin. Next, we'll express the maximum heat transfer rate, \(Q_{f,max}\), using the same variables: \[ Q_{f,max} = h_\text{max}A(T_f - T_\infty) \] where \(h_\text{max}\) denotes the maximum convection heat transfer coefficient. Now we can express fin effectiveness and fin efficiency in terms of these variables: \[ \epsilon_f = \frac{hA(T_f - T_\infty)}{h_\text{max}A(T_f - T_\infty)} = \frac{h}{h_\text{max}} \] \[ \eta_f = \frac{hA(T_f - T_\infty)}{h_bA_b(T_b - T_\infty)} \] where \(h_b\) is the convection heat transfer coefficient of the base, \(A_b\) is the base area, and \(T_b\) is the base temperature.

Compare fin effectiveness and fin efficiency of the two finned surfaces

According to the given information, one finned surface has twice the convection heat transfer coefficient of the other, i.e., \(h_2 = 2h_1\). Using the relations derived above, we can compare their fin effectiveness and fin efficiency: (a) Fin effectiveness of surface 1: \[ \epsilon_{f_1} = \frac{h_1}{h_\text{max_1}} \] Fin effectiveness of surface 2: \[ \epsilon_{f_2} = \frac{h_2}{h_\text{max_2}} = \frac{2h_1}{h_\text{max_2}} \] Considering that the two surfaces are identical except for the convection heat transfer coefficient, we can conclude that the fin effectiveness of the surface with a higher convection heat transfer coefficient (surface 2) is higher. (b) Fin efficiency can be compared using the following relations for the two finned surfaces: \[ \eta_{f_1} = \frac{h_1 A_1}{h_{b_1}} \] \[ \eta_{f_2} = \frac{h_2 A_2}{h_{b_2}} = \frac{2h_1 A_1}{h_{b_2}} \] Without directly comparing these two expressions, we are unable to make a definitive statement about the fin efficiency of the two surfaces. Fin efficiency depends not only on the convection heat transfer coefficient but also on the fin geometry, material properties, and other factors.

Key concepts

Heat Transfer Coefficient

In the study of thermodynamics and heat transfer, the heat transfer coefficient plays a vital role. It is a measure of the thermal conductivity between a solid surface and a fluid per unit area and temperature difference. Mathematically, it is commonly denoted by the symbol \(h\), and its unit is \(W/m^2K\) in the International System of Units (SI).

When we analyze the heat transfer coefficient within the context of fins, it determines how efficiently a fin can transfer heat to the surrounding fluid, which can be air or liquid. The higher the heat transfer coefficient, the better the fin will be at dissipating heat away from the base material. In the case of the two finned surfaces with different heat transfer coefficients, the surface with the higher \(h\) will be more effective at transferring heat, as it can move more thermal energy per unit time and area for a given temperature difference.

Finned Surfaces

Finned surfaces are an enhancement technique used to increase the surface area in contact with a fluid to improve heat transfer. They are often attached to objects that generate heat, like the components in a computer or an engine block in a car. Fins provide a way of spreading the heat over a larger area, which allows for more efficient cooling.

The design of a fin is crucial for its performance. Factors such as the fin's dimensions, the material it's made of, and the spacing between each fin play significant roles in fin efficiency and effectiveness. Fin efficiency is particularly dependent on these design aspects since they determine how well the fin conducts heat from the base to the tip. Two identical fins with different heat transfer coefficients will have differing capabilities in dissipating heat. The fin with the higher heat transfer coefficient will typically be more effective as it implies a better conduction of heat from the surface into the fluid.

Convection Heat Transfer

Convection heat transfer is essentially the movement of thermal energy through a fluid, which can be either a gas or liquid, by the combined effects of conduction and mass flow. When discussing convection heat transfer in relation to fins, we are generally referring to the process of transferring heat between the fin surfaces and the surrounding fluid.

There are two types of convection: natural and forced. Natural convection occurs as a result of differences in density due to temperature gradients within the fluid, while forced convection happens when a pump, fan, or another external force moves the fluid over a surface. In the problem with the two finned surfaces, enhancing the convection heat transfer coefficient could be interpreted as improving the means of forced convection by, for instance, increasing the speed of the fan to cool a computer's CPU. The performance of fins is closely tied to the efficiency of the convection process, and it's clear that the fin with a higher heat transfer coefficient, indicative of superior convection heat transfer attempts, would theoretically have greater capability to dissipate heat effectively.