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How much does it cost for running Navien water heater for a year?

Energy Efficiency = Savings
Navien America pride ourselves on manufacturing the most cost-efficient Water Heater available. Their built-in hot water recirculation system recirculates hot water that would otherwise be lost, thus, creating a method that lowers the cost of customer's water bills each month. Navien's state-of-the-art technology means their product has a longer life span and lower operating costs than other water heaters, compounding customer savings. This, coupled with Navien's space-saving wall-mount design, which frees up your home's real estate space, only increases the money customers will save. And, Navien's 15-year warranty is the best in the industry—think of it as an added bonus!
On top of that, tankless water heater are preferred by customers most popularly due to their ability to save on energy and money. Unlike tank water heaters, the operating cost for tankless water heaters are comparatively low. Tankless hot water heaters are almost 23% more efficient when compared to the traditional water heaters. This means that you will be spending less energy when using these systems and hence as a result, saving money. What happens with tank water heaters is that they have to spend energy constantly to keep the hot water in the tank, HOT. Yes, your tank water heater uses a good amount of money/energy to keep heating the unused water so as to keep it ready for anyone who may have a use for it. On top of that, you lose good amount of heat. Although the tanks are insulated, no system can prevent heat loss, and hence as a result, you will lose heat and require constant reheating. Tankless water heaters provide hot water on demand, instantaneously when you need it, so you don't have to waste energy heating unused water. You're also not losing heat that's idly sitting in a tank.
To be able to know what goes behind the cost of running on demand water heaters, you will need to understand the two things that go behind it, fuel souece and EF (energy efficiency) rating. The EF rating defines how much your cost will be. Below is an explanation on the cost calculation of an on demand hot water heater.

For gas and oil water heaters:
You need to know the unit cost of fuel by Btu (British thermal unit) or therm. (1 therm = 100,000 Btu)
365 X 41045 ÷ EF X Fuel Cost (Btu) = estimated annual cost of operation OR 365 X 0.4105 ÷ EF X Fuel Cost (therm) = estimated annual cost of operation
Example: A natural gas water heater with an EF of .57 and a fuel cost of $0.00000619/Btu
365 X 41045/.57 X $0.00000619 = $163