I’ve noticed that my last few posts have been veering towards the metaphysical so I thought today I would talk about some kitchen science, literally. The question is what is the most efficient way to boil water. Should one turn the heat on the stove to the maximum or is there some mid-level that should be used? I didn’t know what the answer was so I tried to calculate it. The answer turned out to be more subtle than I anticipated.
So consider a pot of water heated by a stove from below. The stove can be gas or electric. Water boils because heat is transferred from the stove to the water. I will assume we are using a pot with high heat conductivity (e.g. stainless steel or copper) so the temperature at the bottom of the pot is the same as the burner. I will also assume that the main source of heat transfer from the bottom of the pot to the bulk of the water is through convection. The rate of convective heat transfer is proportional to the difference in the temperature between the bottom surface and the interior water temperature . The goal is to raise the water temperature to the boiling temperature degrees Celsius. At the same time the water is losing heat to the air and we can use Newton’s law of cooling, which is that the rate of heat transfer is proportional to the difference between the temperature of the air and . Now if we assume that the temperature of the burner and the air remain constant then we can model the change in the thermal energy of the pot as
where and are constants that are known or can be measured. Now in equilibrium the thermal energy is proportional to the temperature. So we will assume that for another measurable constant . My final assumption is that the temperature of the burner is proportional to the energy rate delivered to the burner (e.g. electrical power or gas flow). The time integral of the energy rate over the time it takes to boil the water is the total amount of energy used. I will assume that the energy rate is constant. By rescaling and lumping together parameters we arrive at the simple first order differential equation for the rescaled water temperature
where is the input power, is the rescaled room temperature, and the boiling temperature is a rescaled and . The energy used to boil water is , where is the time to boil.
Assume that at time the temperature is and we want to compute the time it takes to reach . The solution of (*) is
which can be easily obtained by solving (*) with the integrating factor or inferred intuitively because at time zero the temperature is and at time infinity it is and the approach to the asymptotic temperature is exponential. Setting gives the condition
which is a monotonically decreasing function of that is infinity at and zero as goes to infinity. So if the temperature is below boiling you will never boil water and if it infinite then it will take no time to boil.
The total energy used is thus
The goal is to find the minimum of (**). Given that this is a product of a monotonically increasing function with a monotonically decreasing function then it is possible that there could be a minimum between and infinity. We thus need to do a bit more of analysis.
The variables in equation (**) are expressed in scaled temperature units. If we assume that the initial water temperature is approximately the room air temperature then we can set , which simplifies the expression. We can then show that the energy is monotonically decreasing (since the derivative is always negative) so the most efficient way to boil water is to set the burner to the maximum. This would also hold true if the initial water temperature is below the room temperature. However, if the initial water temperature is higher than room temperature then there could be a setting less than maximum that is more efficient. I generally like to use cold water for cooking (since hot water has been sitting in the water heater) so I crank the dial to the maximum when I boil.
Addendum added 2009-9-12
Since and are not scaled identically, doesn’t imply that the room temperature is the same as the initial water temperature. In fact, is scaled by a smaller factor so it is less clear how high can be for energy to decrease with power. Additionally, I only estimated the energy for constant applied power. I think that there is probably a more efficient way if you let the power be time dependent. For example, if you slowly decreased power as the temperature of the water increased, that may be more efficient. It would also be a good exercise for using the variational principle, if anyone is interested.