Let's break down the problem step by step: **Given:** - Width of duct, \( W = 1.0 \) m - Height of duct, \( H = 0.3 \) m - Air gap thickness, \( t = 0.06 \) m - Duct wall temperature, \( T_d = 40^\circ C \) - Shield temperature, \( T_{sh} = 20^\circ C \) We're to **determine the rate of convection heat loss per unit length from the duct, in W/m.** --- ### **Step 1: Find Area for Heat Transfer per Unit Length** For a rectangular duct, the total area exposed to air per unit length (per meter length) is the perimeter times length (which is 1 m): \[ \text{Perimeter} = 2(W + H) = 2(1.0 + 0.3) = 2.6\, \text{m} \] So, for 1 m length, \[ A = 2.6 \times 1 = 2.6\, \text{m}^2 \] --- ### **Step 2: Find the Appropriate Heat Transfer Coefficient (h)** For air in a narrow gap, we use the natural convection correlation for air between two plates. For a conservative estimate and unless otherwise specified, we can use an average value for air: \[ h \approx 10~\text{W/m}^2\cdot\text{K} \] Alternatively, more accurate calculation would use Nusselt number correlation, but **unless given more details about the orientation and properties**, this is a reasonable approximation. --- ### **Step 3: Calculate the Rate of Convection Heat Loss per Unit Length** The convection heat loss is given by: \[ q_{\text{conv}} = h A (T_d - T_{sh}) \] Plug in the values: - \( h = 10~\text{W/m}^2\cdot\text{K} \) - \( A = 2.6~\text{m}^2 \) (per meter length) - \( T_d - T_{sh} = 40 - 20 = 20^\circ C \) \[ q_{\text{conv}} = 10 \times 2.6 \times 20 = 520~\text{W/m} \] --- ## **Final Answer** \[ \boxed{520~\text{W/m}} \] --- ### **Summary of Steps** 1. Find the area for heat exchange per unit length (\(A = 2.6~\text{m}^2\)). 2. Use an estimated convection coefficient for air (\(h = 10~\text{W/m}^2\cdot\text{K}\)). 3. Plug into the convection heat transfer formula to get the answer.