In a thin-walled double-pipe heat exchanger, when is the approximation \(U=h_{i}\) a reasonable one? Here \(U\) is the overall heat transfer coefficient and \(h_{i}\) is the convection heat transfer coefficient inside the tube.

In a double-pipe heat exchanger, heat is transferred between two fluids through the walls of the tubes. This process involves three heat transfer mechanisms: convection inside the tube, conduction through the tube wall, and convection outside the tube. The overall heat transfer coefficient (\(U\)) accounts for all three mechanisms, while the convection heat transfer coefficient inside the tube (\(h_i\)) accounts for only the convection inside the tube.

The overall heat transfer coefficient (\(U\)) combines the effects of convection (\(h_i\) and \(h_o\)) and conduction (\(k_w\)) in a series of resistances. By accounting for the length of the exchanger tube (\(L\)) and the thickness of the tube wall (\(x_w\)), we can relate \(U\) to the three mechanisms using the following equation: \(1/U = (1/h_i) + (x_w/k_w) + (1/h_o)\) Here, \(h_o\) is the convection heat transfer coefficient outside the tube, and \(k_w\) is the thermal conductivity of the tube wall material.

The approximation \(U = h_i\) is reasonable when the resistance from both the conduction through the tube wall and the convection outside the tube are negligible compared to the resistance from the convection inside the tube. This condition can be expressed mathematically as follows: \(1/U \approx 1/h_i \Rightarrow (1/h_i) >> (x_w/k_w) + (1/h_o)\) In practice, this approximation is reasonable when the convective resistance inside the tube (\(1/h_i\)) is much larger than the sum of the convective resistance outside the tube (\(1/h_o\)) and the conductive resistance through the tube wall (\(x_w/k_w\)). This typically occurs when the inner fluid has a very low thermal conductivity or when there is strong turbulence inside the tube, resulting in a high convective resistance.

In a thin-walled double-pipe heat exchanger, the approximation \(U=h_{i}\) is reasonable when the resistance from convection inside the tube dominates over the other two resistances (conduction through the tube wall and convection outside the tube). This may occur when the inner fluid has a low thermal conductivity or under conditions of strong turbulence within the tube.