Question

Two finned surfaces with long fins are identical, except that the convection heat transfer coefficient for the first finned surface is twice that of the second one. What statement below is accurate for the efficiency and effectiveness of the first finned surface relative to the second one? (a) Higher efficiency and higher effectiveness (b) Higher efficiency but lower effectiveness (c) Lower efficiency but higher effectiveness (d) Lower efficiency and lower effectiveness (e) Equal efficiency and equal effectiveness

Short answer

The first finned surface has higher efficiency and higher effectiveness compared to the second finned surface.

Step-by-step solution

1. Identify relevant parameters

Let's denote the convection heat transfer coefficients for the first and second finned surfaces as h_1 and h_2, respectively. Given that h_1 = 2 * h_2. Now we need to relate fin efficiency and fin effectiveness to the convection heat transfer coefficient h. To do this, let's use the standard fin efficiency formula for a straight rectangular fin: Fin Efficiency (η) = tanh(m * L) / (m * L), where m = sqrt(h * P / (k * A_c)), L is the fin length, P is the perimeter, k is the thermal conductivity, and A_c is the cross-sectional area for the fin.

2. Efficiency Comparison

Now, we can directly compare the efficiency of the first and second finned surfaces. For example, let the efficiency of the first fin be η_1 and that of the second fin be η_2. Since h_1 = 2 * h_2, the value of m for the first fin will be sqrt(2) times the value of m for the second fin, as m = sqrt(h * P / (k * A_c)) Thus, m_1 = sqrt(2) * m_2. Fin Efficiency comparison: η_1 = tanh(m_1 * L) / (m_1 * L) η_2 = tanh(m_2 * L) / (m_2 * L)

3. Determine Effectiveness Comparison

For comparing effectiveness, we can use the definition of fin effectiveness: Fin Effectiveness (ψ) = 1 + η(m * L) Therefore, ψ_1 = 1 + η_1 * m_1 * L ψ_2 = 1 + η_2 * m_2 * L Now we can compare ψ_1 and ψ_2.

4. Conclusion

Based on the comparison of efficiency and effectiveness, we can revisit the statement options: (a) Higher efficiency and higher effectiveness: Since η_1 > η_2 due to higher convection heat transfer coefficient, and the effectiveness formula states ψ = 1 + η(m * L), it's possible that ψ_1 > ψ_2. (b) Higher efficiency but lower effectiveness: This is not possible due to the way the effectiveness formula is formulated (ψ = 1 + η(m * L)). (c) Lower efficiency but higher effectiveness: This is not possible since higher h results in higher fin efficiency. (d) Lower efficiency and lower effectiveness: This option is not possible due to higher convection heat transfer coefficient. (e) Equal efficiency and equal effectiveness: This is not possible since η_1 > η_2 and ψ_1 > ψ_2. Therefore, the correct answer is: (a) Higher efficiency and higher effectiveness.

Key concepts

Fin Efficiency

Fin efficiency is a measurement of how well a fin can transfer heat from the base to the tip. It compares the fin's actual heat transfer with the maximum possible transfer under ideal conditions. The formula used to calculate fin efficiency often involves parameters such as the perimeter of the fin, thermal conductivity, and convection heat transfer coefficient. The fin efficiency (\(\eta\)) can be represented mathematically as:
\[ \eta = \frac{\tanh(m \cdot L)}{m \cdot L} \]where:

• \(m\) is a term that depends on the convection heat transfer coefficient (\(h\)), perimeter (\(P\)), thermal conductivity (\(k\)), and cross-sectional area of the fin (\(A_c\)).
• \(L\) is the length of the fin.

The higher the efficiency, the better the fin is at transferring heat over its entire length.

Convection Heat Transfer Coefficient

The convection heat transfer coefficient (\(h\)) is a critical parameter in heat transfer that quantifies how easily heat is transferred from a solid surface to a fluid or vice versa. It is essential in determining how effective a finned surface will be at cooling or heating applications.
A higher convection heat transfer coefficient means that the fin can more efficiently transfer heat to the surrounding environment. In the given exercise, the first surface has a convection heat transfer coefficient twice that of the second, meaning it has a greater potential for heat transfer. This increase in \(h\) also affects other calculations related to fin efficiency and effectiveness.

Thermal Conductivity

Thermal conductivity (\(k\)) is a material property that describes the ability of a material to conduct heat. It appears in the formula for calculating the efficiency and effectiveness of fins, particularly in terms of the variable \(m\):
\[m = \sqrt{\frac{h \cdot P}{k \cdot A_c}}\]Thermal conductivity directly influences the performance of the fin in dissipating heat. A higher thermal conductivity means heat is more easily conducted through the fin, improving its overall efficiency.
In practical applications, materials with high thermal conductivity are often chosen for fins to maximize their heat dissipation capacity.

Fin Effectiveness

Fin effectiveness (\(\psi\)) is another important measure, indicating how effective a fin is at enhancing heat transfer relative to a planar surface without fins. The formula for fin effectiveness takes the form:
\[ \psi = 1 + \eta(m \cdot L) \]This formula shows how fin efficiency impacts fin effectiveness.

• Increased fin efficiency results in greater fin effectiveness, indicating that fins are not only optimizing heat transfer locally but are also significantly enhancing overall heat transfer capabilities.
• The value \(m \cdot L\) accounts for the relationship between the convection heat transfer coefficient and fin geometry.

In the exercise, due to a higher convection heat transfer coefficient, fin effectiveness for the first surface is greater, aligning with the given solution conclusion.