Whenever you turn on a gas, propane or electric tankless water heater the hot water starts heating up as soon as the minimum flow rate is met for the hot water demand. It takes a few seconds for the water to get heated up to the requested temperature. But, before the hot water starts to flow out of your shower or sink faucet the cold water that is in the pipe has to be pushed through your faucet. This is often referred to as pushing a "cold water slug".
Customers frequently ask us how much water is contained in this cold water slug and how long it will take to push this slug through their faucet. This can be easily calculated by using the formula for the volume of a cylinder.
The formula for the volume of a cylinder is V = Pi x r x r x h
where Pi = 3.1416; r = the radius of the water line; and h = height (or in this case the length) of the line
If your water line is 1/2" then r = .25" and r x r = .0625" and Pi x r x r = 0.19635 square inches
If your water line is 3/4" then r = .375" and r x r = .1406" and Pi x r x r = 0.4417875 square inches
Now to get all units in inches calculate h (the length of the water line) by taking feet x 12". As an example let's assume we have a 40' water line of 1/2" pipe. 40' = 480" so we calculate the Volume as:
V = 0.19635 square inches x 480" = 94.248 cubic inches
Now there are 1728 cubic inches in a cubic foot and there are 7.48 gallons in a cubit foot. Therefore there are 0.0043 gallons in a cubic inch.
So, 94.248 cubic inches * 0.0043 gallons = 0.4 gallons of water.
If you are running this cold water slug through a 1 GPM sink faucet then it will take .4 x 60 seconds or 24 seconds to push the cold water slug through the 1 GPM faucet. Add a few more seconds for the tankless water heater to bring the water up to temperature and remember that there are about 15' of internal pipe in a gas or propane tankless water heater that also starts out with cold water. In this example it will take just more that 1/2 minute to get the hot water flowing out of the faucet.