Effectiveness of a countercurrent heat exchanger is given by:
$\varepsilon = \frac{1 - \exp\left[-\text{NTU}\left(1 - \frac{C_{\min}}{C_{\max}}\right)\right]}{1 - \left(\frac{C_{\min}}{C_{\max}}\right)\exp\left[-\text{NTU}\left(1 - \frac{C_{\min}}{C_{\max}}\right)\right]}$
If the same liquid at the same mass flow rate is used as heating and cooling media ($C_{\min} = C_{\max}$), then the effectiveness is given by:
Options:
(A) \frac{\text{NTU}}{1 - \text{NTU}}
(B) \frac{\text{NTU}}{1 + \text{NTU}}
(C) \frac{1}{1 + \text{NTU}}
(D) \frac{1 - \text{NTU}}{1 + \text{NTU}}
Official Answer Key:
B
Step-by-Step Solution & Explanation
Same fluid, same flow rate on both sides ⟹ C_min = C_max ⟹ C_r = C_min/C_max = 1. The given ε formula becomes an indeterminate 0/0 as C_r→1; taking the limit (L'Hôpital, or the well-known balanced-counterflow result) gives ε = NTU/(NTU+1), the standard limiting form for a balanced counter-flow exchanger.
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