Question

Consider two double-pipe counterflow heat exchangers that are identical except that one is twice as long as the other one. Which heat exchanger is more likely to have a higher effectiveness?

Short answer

Answer: The double-pipe counterflow heat exchanger that is twice as long as the other one is more likely to have a higher effectiveness.

Step-by-step solution

Factors affecting heat exchanger effectiveness

In a double-pipe counterflow heat exchanger, the effectiveness (ε) depends on several factors like the heat capacity rates of the fluids involved, the heat transfer coefficient, and the surface area for heat transfer. The formula for the effectiveness of a counterflow heat exchanger is given by: ε = \frac{1 - e^{-NTU(1-C_r)}}{1 - C_r e^{-NTU(1-C_r)}} Here, NTU (Number of Transfer Units) measures the size of the heat exchanger relative to the capacity rates of the fluids and is defined as: NTU = \frac{UA}{C_{min}} where U is the overall heat transfer coefficient, A is the heat transfer surface area, and C_{min} is the minimum heat capacity rate of the fluids (C_{min} = min(C_h, C_c), where C_h and C_c are the hot fluid and cold fluid heat capacity rates, respectively). C_r is the capacity rate ratio, defined as C_r = \frac{C_{min}}{C_{max}}. From the given information, the heat exchangers are identical except for their length. We can assume that their heat capacity rates, heat transfer coefficients, and fluid properties are the same for both heat exchangers. Now, let's consider the impact of length on the surface area and effectiveness.

Impact of length on the surface area and NTU

The surface area (A) for heat transfer in a double-pipe heat exchanger is directly proportional to the length (L) of the exchanger: A = k*L where k is a constant factor that depends on the geometry and dimensions of the heat exchanger. The NTU of the longer heat exchanger (L2) will be twice that of the shorter heat exchanger (L1) as L2 = 2*L1: NTU_2 = \frac{UA_2}{C_{min}} = \frac{2kL_1U}{C_{min}} = 2 * NTU_1

Impact of increased NTU on the effectiveness

Since the heat capacity rates (C_h and C_c) and their ratio (C_r) are the same for both heat exchangers, the only difference in their effectiveness will be due to the difference in their NTUs, as shown in the formula: ε = \frac{1 - e^{-NTU(1-C_r)}}{1 - C_r e^{-NTU(1-C_r)}} As NTU_2 = 2 * NTU_1, the effectiveness of the longer heat exchanger will be higher than that of the shorter heat exchanger as the term e^{-NTU(1-C_r)} will decrease with increasing NTU, resulting in a higher numerator and lower denominator in the effectiveness formula.

Conclusion

Based on the analysis, we can conclude that the double-pipe counterflow heat exchanger that is twice as long as the other one is more likely to have a higher effectiveness. This is because the longer heat exchanger has a larger surface area for heat transfer, resulting in a higher NTU and, consequently, a higher effectiveness.