Not wanting to sound like a butt hole but I donít see any reason to have an appliance to heat hot water. If the water is already hot why would one want to heat it even more?
It is a water heater or if an adjective is used to describe the appliance it would be a cold water heater. The appliance heats cold water not hot water.
Now just what does the entire above have to do with the question? Nothing, I was just bored and wanted to practice my typing skills.
There are several devices on the market today that can be used to measure the kilowatt per hour usage of just about any appliance. They can be programed to the amount you are charged per KWH and show just what that appliance cost a month. It takes about a week of use to get an accurate amount of cost.
I have a friend that several years back installed a coil on their wood stove to preheat the water before it entered the water heater. The installation cost around a thousand dollars and it was inspected to ensure its safety. According to him the utility bill did not reflect any difference for a period of one year and if anything the bill was slightly higher.
In order for this type of installation to work there will need to be a circulating pump installed as well as all sorts of relief valves to keep the water from getting above the 125 degrees or the scald point of the human body. The amount of power usage saved on the water heater will be offset by the pump or pumps depending on the type of system installed.
In my personal opinion you would be better served by insulating the water heater and maybe installing a time clock to control its usage.