Question

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.

Short answer

Answer: The approximation \(U = h_i\) is considered reasonable in a double-pipe heat exchanger when the convection heat transfer coefficient inside the tube (\(h_i\)) is much larger than the convection heat transfer coefficient outside the tube (\(h_o\)) and the conduction resistance through the tube wall is small. This is likely to occur when the fluid inside the tube has a much higher heat transfer coefficient than the fluid outside, and the tube's wall is thin.

Step-by-step solution

Understanding Heat Transfer Resistances

In a double-pipe heat exchanger, the overall heat transfer coefficient, U, is determined by three resistances: 1. Convection inside the tube: \(h_i\) 2. Conduction through the tube wall: \(\frac{ln(r_o/r_i)}{2\pi\ k L}\) 3. Convection outside the tube: \(h_o\) These heat transfer resistances are connected in series, so their overall impact can be represented by their sum: \(\frac{1}{U} = \frac{1}{h_i} + \frac{ln(r_o/r_i)}{2\pi\ k L} + \frac{1}{h_o}\)

Evaluating the Importance of Other Resistances

The approximation \(U \approx h_i\) is reasonable when the resistances due to the conduction through the tube wall and convection outside the tube are significantly smaller than the resistance due to the convection inside the tube. In other words, when \(\frac{1}{h_i} >> \frac{ln(r_o/r_i)}{2\pi\ kL} + \frac{1}{h_o}\)

Providing a Condition for the Approximation

From the inequality obtained in step 2, we can provide a condition for the approximation to be reasonable: - If the convection heat transfer coefficient inside the tube (\(h_i\)) is much larger than the convection heat transfer coefficient outside the tube (\(h_o\)) and the conduction resistance through the tube wall is small, then the approximation \(U \approx h_i\) can be considered reasonable. In practical terms, this condition is likely to be met when the fluid inside the tube has a much higher heat transfer coefficient than the fluid outside, and the tube's wall is thin.