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Most energy-efficient method to cook brown rice?
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The energy efficiency of the appliance will only be a small dent in the energy consumption of cooking the rice. Most methods rely on boiling the water. Heating water up doesn't take very much energy - about 4 kJ/kg/K. This means that heating 1 liter of water from 20 C to 100 C takes about 320,000 Joules (1 kWh = 3,600,000 Joules).
Once water is heated up to 100 C, the next step is boiling where you phase change from liquid to vapor. The specific enthalpy of vaporization for water at atmospheric pressure is about 2000 kJ/kg, so the one Liter of water would take 2,000,000 Joules to completely vaporize. You could weigh your pot before and after cooking to see how much water you vaporized.
Let's assume you have 1 kg (about 1 liter) of water and 1/10 of it vaporizes. The minimum energy for this is about 500,000 Joules assuming 100% efficiency. Divide by efficiency, so if the appliance is 90% efficient it would be around 550,000 Joules, while 80% would be about 625,000 Joules.
So, to answer your question, the most energy efficient method will minimize vaporization. Soaking the raw rice for a few hours will help reduce boiling time. Then either a standard issue rice cooker, which should stop the boiling process as soon as the rice is finished.