GATE ME 2015 Set 3 — Question 22

In a condenser, one fluid undergoes a phase change (vapor to liquid) at a constant temperature. This means its heat capacity is effectively infinite. Therefore, this fluid has the maximum heat capacity, $C_{max} = \infty$.The heat capacity ratio is defined as $C_r = \frac{C_{min}}{C_{max}}$.Substituting $C_{max} = \infty$, we get:
$C_r = \frac{C_{min}}{\infty} = 0$.The effectiveness of a heat exchanger where one fluid's temperature is constant (i.e., for boiling or condensation, where $C_r = 0$) is given by the formula:$$\epsilon = 1 - \exp(-NTU)$$This relationship is independent of the flow arrangement (parallel, counter, cross-flow, etc.) when $C_r = 0$.Let's check the options:
(A) $\frac{1-\exp [-NTU(1+C_r)]}{1+C_r}$ is the effectiveness for a parallel flow heat exchanger. For $C_r=0$, it simplifies to $1-\exp(-NTU)$.
(B) $\frac{1-\exp [-NTU(1-C_r)]}{1-C_r \exp [-NTU(1-C_r)]}$ is the effectiveness for a counter flow heat exchanger. For $C_r=0$, it simplifies to $1-\exp(-NTU)$.
(C) $\frac{NTU}{1+NTU}$ is the effectiveness for a counter flow heat exchanger when $C_r=1$.
(D) $1-\exp(-NTU)$ is the general formula for the case where $C_r=0$.Since the question asks for the effectiveness of the condenser, which is a special case where $C_r=0$, the correct expression is given in option (D).