Depth Required for 5.3% Carnot Efficiency in Ocean Power Plant

Calculating Required Ocean Depth for 5.3% Carnot Efficiency

Question:

If the power plant uses a Carnot cycle and the desired theoretical efficiency is 5.3%, from what depth must cold water be brought?

A. 100 m
B. 400 m
C. 800 m
D. Deeper than 1000 m

Answer:

Step 1: Understand the Carnot Efficiency Formula

The maximum theoretical efficiency of a heat engine operating on a Carnot cycle is given by:

η = 1 − (Tc / Th)

Where:

η = efficiency (in decimal form)
Th = temperature of hot reservoir (in Kelvin)
Tc = temperature of cold reservoir (in Kelvin)

Step 2: Insert Given Efficiency and Surface Temperature

Given Carnot efficiency: η = 5.3% = 0.053

Assume surface water temperature (warm reservoir): Th ≈ 20°C = 293 K

Now, solve for Tc:

η = 1 − (Tc / Th) →
Tc / Th = 1 − 0.053 = 0.947 →
Tc = 0.947 × 293 K ≈ 277.37 K

Convert Tc to °C: Tc ≈ 277.37 − 273.15 ≈ 4.2°C

Step 3: Determine the Required Ocean Depth

Ocean temperature decreases with depth, and in tropical oceans, water reaches temperatures around 4–5°C at depths greater than 1000 m.

Therefore, to access water at approximately 4.2°C (the Tc needed for 5.3% Carnot efficiency), cold water must be drawn from:

✅ Deeper than 1000 m

Conclusion:

To achieve a theoretical Carnot efficiency of 5.3% with warm surface seawater at around 20°C (293 K), cold seawater must be pumped from depths deeper than 1000 meters to reach temperatures around 4–5°C.

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