These considerations can be very important. If you use baseboard electric heat during the heating season and the heat-pump water heater is in your heated space, you’re “robbing from Peter to pay Paul,” and your overall energy use (heat plus hot water) may be noticably affected. If the heat pump is in the basement and there’s a lot of waste heat from an inefficient furnace, the impact on your heating costs may be insignificant. If you’re in a southern climate and use air conditioning much of the year, the heat-pump water heater can remove a significant amount of ambient heat, reducing your cooling energy use.
Both stand-alone and add-on heat-pump water heaters are available. The former look much like conventional storage-type water heaters (though are usually taller, with the heat-pump on top), while the latter are boxes that sit next to or on top of conventional water heaters, with insulated pipe connections between the two and a pump to circulate water.
Unfortunately, heat-pump water heaters are expensive, and most products on the market are relatively new, without a long track record of performance. Early heat-pump water heaters, 10 or 20 years ago, often had reliability problems. Because they use fans to draw in air from the room, they can be noisy—typically louder than a refrigerator. Ask about noise when selecting a heat-pump water heater (noise ratings range from about 55 to 65 dB), and think about an out-of-the-way spot to install it.
Natural gas and propane water heaters are very similar, varying only in the burner orifices, gas valves, and pilot lights. Both offer the potential for very high burner efficiencies. EFs are dependent on standby losses, which are higher compared to electric water heaters because of the uninsulated flue extending up the center of the tank. Most gas-fired water heaters have pilot lights, which reduce the EF somewhat; the highest-efficiency models have electronic burner ignition. Compared with electric water heaters, the EF ratings of gas-fired water heaters are lower: As with electric storage water heaters, the minimum EF is based on tank volume: about 0.52 for an 80-gallon model and about 0.59 for a 40-gallon model. However, natural gas prices are a lot lower than electricity prices (per unit of energy), so gas-fired water heaters usually cost less to operate than electric models. (Propane prices are typically a lot higher and vary significantly from place to place.)
The highest-efficiency gas water heaters use condensing technology. The heat exchangers in condensing water heaters extract so much heat that the water vapor in the flue gas cools and condenses. This condensation has an energy benefit (capturing that latent heat of vaporization of the water vapor), but it also means that condensing water heaters should not be vented into a masonry or metal chimney due to corrosion concerns. They are typically vented through a sidewall, with a drain for the condensate. Gas-fired condensing, storage-type water heaters have EFs as high as 0.86.
Oil-fired stand-alone water heaters are far less common than gas-fired, have lower efficiency, and tend to be more expensive. With so many better options available, these units do not make much economic sense.
The alternative to a storage water heater is to heat the water as it is used. That’s the principle of a tankless water heater (aka instantaneous or demand). They can be located at the point of use (a bathroom or kitchen) or centrally, with distribution handled as with storage water heaters. Tankless water heaters can be gas or electric, though central tankless water heaters are usually gas-fired. Because hot water is not stored, the heating capacity must be great enough to meet all of the hot water demand.
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One option not talked about (or at least I have never heard of it happening) is using a solar water heater controller and for a collector use a coil of 3/4" copper tubing mounted behind a wood burning stove. When the coil is warmer than the tank, then the pump would come on and heat the water in the water heater. When the stove is cold, it would not come on, but would save a lot of kilowatts while the stove is hot.
Too extend your mentioned option, it does not even need a pump, if the setup is such that it will thermosiphon to the hot water tank.
Also, there are heat exchangers that can be retrofit to the inside of the wood heater, obtaining even more hot water. Of course, the usual cautions are that it can get too hot and flash into steam if not properly designed; and also the exchanger might take enough heat out of the firebox to cool the gases enough to deposit creosote on the heater walls and chimney -- a potential fire hazard.