If an instant, on demand water heater is to expensive or for some reason not an option for you, look into solar.
One of the most energy efficient water heating methods is the heat pump water heater. This method is proven to be two times more efficient over the standard electric water heater.
The most efficient is a geo-thermal heat pump, if properly installed.
over a blue flame
A tankless water heater is much more efficient than a traditional water heater because it only heats the water that is being used instead of heating and reheating water stored in a tank. Rinnai, Rheem, and Bosch all make versions of tankless water heaters that are well-reviewed.
Buderus Heating systems supply Boiler and heating systems which are listed on their website as designed to be the most efficient Boiler and heating systems within the area of Maine.
I assume you mean hot water tank. The hot water tank was supposed to be the most efficient way to heat your water because heat rises. Since heat rises most of the heat energy would be absorbed by the water in the tank and very little of that energy would escape through the top of the tank. However; building scientists are now saying a direct hot water heating system is more efficient than a hot water tank because in a direct system you are not heating water until you need it, so you are not heating water you won't be using. If you frequently use hot water a hot water tank is the way to go. If you rarely use hot water a direct hot water (tankless) heating system is the way to go.
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Gas-fired, storage-based water pumps are the most energy efficient type.
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Energy efficient home heating systems can usually be installed by whatever company or store you purchase them from.For the most part,these systems are installed just like regular heating systems but cost less to maintain due to their lower energy usage.
Geothermal heating and cooling systems are the most cost effective. Although they may be slightly more expensive to install, the cost of use long term is greatly decreased from traditional sources of heating and cooling.
I have no experience in this, however A. O Smith claims that there brand of gas water heaters are the most efficient. They also offer a hybrid water heater.