A balanced counter-flow plate heat exchanger is used for chilling pasteurized skim milk. Since both fluid streams have equal heat capacity rates ($C_{\text{min}} = C_{\text{max}}$, so capacity ratio $C_r = 1.0$), the Number of Transfer Units is $NTU = 2.50$. Using the effectiveness-NTU relation $\varepsilon = \frac{NTU}{1 + NTU}$, the thermal effectiveness of the heat exchanger expressed in percentage is ________ % (round off to two decimal places).
Answer Key:
71.43
Step-by-Step Solution & Explanation
Multi-Approach Solution:
**Method 1: Direct Substitution into Balanced Counter-Flow Formula**
For a counter-flow heat exchanger with $C_r = \frac{C_{\text{min}}}{C_{\text{max}}} = 1.0$:
$\varepsilon = \frac{NTU}{1 + NTU}$
Given $NTU = 2.50$:
$\varepsilon = \frac{2.50}{1 + 2.50} = \frac{2.50}{3.50} = \frac{5}{7} \approx 0.7142857$
Expressed as a percentage:
$\varepsilon = 71.43\%$
**Method 2: Limit of General Counter-Flow Relation**
General formula: $\varepsilon = \frac{1 - \exp[-NTU(1 - C_r)]}{1 - C_r \exp[-NTU(1 - C_r)]}$. Applying L'Hôpital's rule as $C_r \to 1$ yields $\varepsilon = \frac{NTU}{1 + NTU} = \frac{2.5}{3.5} = 71.43\%$.