For a double-pipe heat exchanger, the inside and outside heat transfer coefficients are 100 and 200 W m\(^{-2}\) K\(^{-1}\), respectively. The thickness and thermal conductivity of the thin-walled inner pipe are 1 cm and 10 W m\(^{-1}\) K\(^{-1}\), respectively. The value of the overall heat transfer coefficient is _________ W m\(^{-2}\) K\(^{-1}\).
Show Hint
For heat exchangers, the overall heat transfer coefficient depends on the thermal resistances due to convection on both sides of the pipe and conduction through the pipe material.
The overall heat transfer coefficient \( U \) for a double-pipe heat exchanger can be calculated using the following equation:
\[
\frac{1}{U} = \frac{1}{h_i} + \frac{r_i \ln(r_o / r_i)}{k} + \frac{1}{h_o}
\]
where:
- \( h_i \) is the inside heat transfer coefficient,
- \( h_o \) is the outside heat transfer coefficient,
- \( r_i \) and \( r_o \) are the inner and outer radii of the pipe,
- \( k \) is the thermal conductivity of the pipe material.
Given:
- \( h_i = 100 \, \text{W m}^{-2} \text{K}^{-1} \),
- \( h_o = 200 \, \text{W m}^{-2} \text{K}^{-1} \),
- The thickness of the pipe is 1 cm, so \( r_o = r_i + 0.01 \) m, and the thermal conductivity \( k = 10 \, \text{W m}^{-1} \text{K}^{-1} \).
Substituting these values into the equation, we find that the overall heat transfer coefficient \( U \) is approximately \( 62.5 \, \text{W m}^{-2} \text{K}^{-1} \). Therefore, the correct answer is (C).
