Discussion
Water (Cp = 4.18 kJ/kg.K) at 80°C enters a counterflow heat exchanger with a mass flow rate of 0.5 kg/s. Air (Cp = 1 kJ/kg.K) enters at 30°C with a mass flow rate of 2.09 kg/s. If the effectiveness of the heat exchanger is 0.8, the LMTD (in °C) is
Answer: C
The LMTD is 10 °C.
The heat capacity rates are equal:
\[ \begin{aligned} C_{w} &= 0.5 \times 4.18 = 2.09\ \text{kW/K} \\ C_{a} &= 2.09 \times 1 = 2.09\ \text{kW/K} \end{aligned} \]
With effectiveness 0.8 and a maximum temperature difference of 80 - 30 = 50 °C, each stream changes by 0.8 × 50 = 40 °C: water leaves at 40 °C and air at 70 °C. In a counterflow exchanger with equal capacity rates, the temperature difference is the same at both ends, 80 - 70 = 40 - 30 = 10 °C, so the LMTD equals that constant difference.
40 °C is the temperature change of each stream, 20 °C and 5 °C have no basis here. A check: Q = 2.09 × 40 = 83.6 kW, and UA = Q/LMTD = 8.36 kW/K, so NTU = 4 and \(\varepsilon = 4/5 = 0.8\).
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