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QB_DFE_EXP_001
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NAT • 2 Marks
Section 7: Dairy and Food Engineering
Heat and Mass Transfer
Subtopic: Working principles of heat exchangers
Question Statement:
A counter-current double pipe heat exchanger cools milk from $70.0^\circ\text{C}$ to $25.0^\circ\text{C}$ using chilled water that enters at $10.0^\circ\text{C}$ and leaves at $30.0^\circ\text{C}$. The Logarithmic Mean Temperature Difference ($LMTD$) in $^\circ\text{C}$ is ________ (round off to 2 decimal places).
Step-by-Step Solution & Explanation
1. For counter-flow heat exchanger:
• Hot fluid: $T_{h,in} = 70.0^\circ\text{C}, T_{h,out} = 25.0^\circ\text{C}$
• Cold fluid: $T_{c,in} = 10.0^\circ\text{C}, T_{c,out} = 30.0^\circ\text{C}$
2. Temperature differences at ends:
$\Delta T_1 = T_{h,in} - T_{c,out} = 70.0 - 30.0 = 40.0^\circ\text{C}$
$\Delta T_2 = T_{h,out} - T_{c,in} = 25.0 - 10.0 = 15.0^\circ\text{C}$
3. Log Mean Temperature Difference:
$LMTD = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)} = \frac{40.0 - 15.0}{\ln(40.0 / 15.0)} = \frac{25.0}{\ln(2.6667)}$
$\ln(2.6667) \approx 0.98083$
$LMTD = \frac{25.0}{0.98083} \approx 25.042^\circ\text{C} \approx 25.04^\circ\text{C}$
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