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

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.

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)

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.

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.