Q.54 Fermentation medium is cooled from 121 °C to 30 °C in a double pipe heat
exchanger. If cold water is flowing in the counter-current direction and is heated
from 10 °C to 70 °C, then the Log-Mean Temperature Difference (LMTD) is
________ °C (rounded off to the nearest integer).
Answer: LMTD = 53°C
LMTD Calculation
For a counter-current heat exchanger, first identify temperature differences at both ends:
- Hot fluid (fermentation medium) enters at 121°C and exits at 30°C.
- Cold water enters at 10°C and exits at 70°C.
Temperature differences at each end:
ΔT₁ = 121°C - 70°C = 51°CΔT₂ = 30°C - 10°C = 20°C
Apply the LMTD formula for counter-current flow:
LMTD = (ΔT₁ - ΔT₂) / ln(ΔT₁ / ΔT₂)
= (51 - 20) / ln(51 / 20)
= 31 / ln(2.55)
≈ 31 / 0.936
≈ 52.6°C
≈ 53°C (rounded)
Why Counter-Current Matters
Counter-current flow maximizes the LMTD compared to parallel flow because it maintains larger average temperature gradients throughout the exchanger. In parallel flow, the temperature difference decreases quickly along the length, lowering efficiency.
For example, parallel flow would use:
ΔT₁ = 121°C - 10°C = 111°CΔT₂ = 30°C - 70°C = -40°C (invalid, needs adjustment)
This shows that counter-current design is more efficient for cooling fermentation mediums in biotech applications.
