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I get asked this question with almost every house I design.  I lifted this article from Fine Homebuilding and have modified it in places.

Domestic water heating is one of "biggies" of energy consumption and there are many options.

The vast majority of homes have storage-type water heaters. There’s an insulated tank and either a gas burner or electric heating element (often two elements) to heat the water. An advantage of storage water heaters is that you don’t need a very large gas burner or a really high electric current flow to heat the water. The gas burner or element can chug along for hours, gradually warming up water in the tank. The water remains “thermally stratified” so that water drawn off from the top is always the hottest and even after 90% of the hot water is used up, the delivered water is still at full temperature. Storage electric water heaters also allow “off-peak” electricity to be used—more about this in a future column.

While storage water heaters are the most common, there’s a lot of interest in tankless models—sometimes referred to as “on-demand” or “instantaneous” water heaters. The advantage of these is that you don’t have water sitting all the time, losing energy through the tank walls. (Even with insulation, heat loss occurs.)

As with storage water heaters, tankless models can be either gas-fired or electric. For very small loads, such as with a remote lavatory that has only a sink (with a low-flow aerator), an electric tankless water heater can make a lot of sense since it obviates the need for running a gas line. But for whole-house needs—where a central water heater serves one or more bathrooms with showers, the kitchen sink, dishwasher, and clothes washer—a gas-fired tankless water heater is almost always a better choice than electric.

Providing enough electric current to instantaneously heat 4-5 gallons per minute (gpm), boosting the temperature more than 60 degrees F (as might be necessary if two showers are being used at the same time, or if a dishwasher or clothes washer is being used while someone is showering) would take a huge amount of electric current—on the order of 40 to 60 amps. Providing so much electricity would require special wiring and special circuit breakers, which are expensive. And from a big-picture standpoint, if a lot of people used these tankless electric water heaters, utility companies would have to build more power plants to have adequate electricity available during periods of time with high use of hot water, such as during the morning shower period. Utility companies love storage water heaters, because they spread out the demand.

If you decide to go with a whole-house, gas-fired (natural gas or propane) tankless water heater, be aware that because the burners on tankless water heaters are so large—150,000 to 200,000 Btu/hour, vs. 40,000 Btu/hour for a typical gas-fired storage water heater—larger-diameter gas lines are required (usually 3/4-inch instead of 5/8-inch). And to burn that much gas, a lot of air flow is required, which necessitates a large flue, and there is potential for significant air leakage if not properly installed.

Another issue with tankless gas water heaters is pilot vs. pilotless ignition. Tankless water heaters used to all have pilot lights, which burned gas all the time. The wasted energy from these pilot lights (about 5,000 Btu/hour) was about the same as the heat loss through the insulated walls of a storage water heater, so you didn’t end up with much energy savings.

Most new tankless water heaters have electronic ignition, and if you’re thinking of a tankless water heater I’d go with this option. The gas burner is ignited using an electrical spark. With pilotless ignition, today’s gas-fired tankless water heaters offer the highest efficiency of any water heater, except heat-pump models, which I’ll cover in a future column.

A few gas-fired tankless water heaters made by the Korean companies Takagi and Navien have “condensing technology” with an Energy Factor of up to 0.98 (Energy Factor is a measure of efficiency). Non-condensing, pilotless tankless water heaters have Energy Factors of .82 to .87, while conventional storage water heaters have energy factors of .58 to about .67 (up to .80 for condensing models).

So which is better: a storage or a tankless water heater? Despite the potential for higher efficiency with tankless technology, I’m partial to storage water heaters for most situations. They are less expensive, less prone to mechanical problems, and, with electric models, offer the potential for using off-peak electricity—which is significantly less expensive than propane. Storage water heaters also encourage thrift when showering, since there’s a finite amount of hot water.

Interestingly enough, my personal experience is as follows.  I took my heating contractor's advice when I decided to change over to propane heat in our country house.  He recommended a high efficiency propane water heater as well.   It cost over $1,000.00 compared to the $200 electric model I had.   After about ten years of service it started to leak but not before there were some repairs of about $200.  I found out that these gas fired heaters are subject to a level of stress much higher than electric water heaters which generally last 20+ years with maybe an inexpensive element replacement along the way.  I was surprised to learn that my electrician decided to go back to an electric hot water tank and I did the same.  There is just the two of us in the home now and our hot water needs are modest.  Dollar for dollar, this is the best decision for us.   We always have the option as does everyone, of pre-heating the water somehow, either by circuits on the roof in summer or around a wood fired stove in winter.

"Wood is good" says Patrick Moore, one of the founders of Greenpeace.  Greenpeace originated at UBC around the time I was there studying architecture.  That is when an earnest  concern for the environment began.  David Suzuki was teaching genetics but also preaching the fears of global cooling which changed to global warming, now simply climate change.  To be fair David, along with other environmentalists  felt that the melting of the ice caps would cool the oceans to a critical threshold which would enact a rapid cooling resulting in a new ice age.  We know that the last ice age about 100,00 years ago happened very quickly.  Large mammals we associate with much warmer climes were frozen in ice. Pretty scary stuff.  We could live with warm beaches in northern Canada but a mile of ice above our heads is a bit hard to adapt to.   Regardless, there was the continuing sense that mankind was having an effect in burning fossel fuels.  To what extent our activities  affect or can affect the weather remains a question but we know we do in terms of pollution.  Everyone agrees on that.  So what can we do?  In terms of architecture, one thing we can do is use more wood.  By the way, ice ages happen on average every 100,000 years, so we are due.

Moore believes wood is the answer to the future of human civilization  ("Confessions of a Greenpeace Dropout").  He outlines the history of wood harvesting where we practically decimated all the forests close to the cities, then we learned to farm trees the same way we grow crops.  It might surprise you to learn that China has added more forests than any other country because they realize its value for the environment and in construction.  Its growing middle class needs it.   And this is the irony.  The more harvesting, the more replanting and management is done.  This helps tremendously in carbon capture.  Untended forests create potential fire hazards.  Burning soil during forest fires releases massive amounts of carbon and other pollutents.

Trees are tremendous solar collectors. Wood kept dry and protected from sunlight will last for hundreds of years.    Properly used it is safer than metal construction.  Firemen tell me that metal floor joists are called "widow makers" as they collapse early in a fire situation.  Wood beams and joists stay integral much longer under similar conditions.  But like much in our society popular fears trump common sense and proper science.

You hear  a lot about "green" or sustainable construction methods.  When you analyze all the factors from harvesting to milling to the end result of the building,  including energy consumption, local participation, etc., the most sustainable material is "wood".  I've had many clients who want to be responsible home owners in terms of the environment but inevitably they want materials that, while they may be long lasting and need little maintenance, these materials are not "green" at  all.  In the three houses I have built for myself, I have used as little metal and plastics as I can.  Cedar siding, wood soffits and facias, hardwood floors, wood windows, etc.  Staining the siding once every 10 years and sometimes not all of it, is hardly high maintenance.  Trees such as cedar replace themselves often with no effort on our part.  In terms of energy conservation, more wood rots in the forest than could be used to heat every house in the country several times over.    It is unsurpassed as a building material offering a natural expression in design and construction.   Because of our proximity to a vast supply, we should be using much more wood in our buildings.