Is there a easy formula for figuring the cost of running a natural gas furnace if you know the cost per cu.ft; the BTU of the furnace and the run time?
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Simpler if you have the cost per BTU, which the gas company should be able to give you. (It varies somewhat depending on the makeup of the gas, but most companies price by "therm" (100K BTU) rather than cu ft because of this.)
Then, to a first approximation, the cost is just the cost per BTU times the BTU/HOUR times the number of hours run. (Note for BTU/HOUR you should use the input or "bonnet" number, not the heat output.)
The other issue is that burner time and run time are no longer the same thing. In the old days, the blower came on when the burner came on and it went off when the burner went off. Now, in order to improve efficiency, the blower kicks in 10-20 seconds before the burner comes on (due to no standing pilot light) and the blower runs several minutes after the burner stops to pull the residual heat out of the heat exchanger. Depending on the conditions, I have even had our burner cycle on and off a couple of times with the blower never shutting down.
>>In the old days, the blower came on when the burner came on and it went off when the burner went off.
Acctually in the old days the blower came on after the burner was on for several seconds. This was done so that the heat could build up in the bonnet before the blower came on and you wouldn't get a blast of cold air when you expected warm air to come out of you registers. On the other end the blower also ran a few seconds after the burner shut off to cool down the bonnet.
Hi-eff furnaces have a start fan that introduces the combustion air to the sealed chamber. That is what you hear before ignition, not the blower fan. There is also a safety built in for this feature. If the combustion air switch does not prove the gas valve and igniter will not work.
You are correct about the blower running after the unit stops burning. It will wring more heat out of the still hot heat exchanger.
BTUin x hours x $ per C.F/1000 is the simple answer.
"Run time", as logged by a thermostat is the amount of time the stat is calling for heat. Depending on the length of the startup sequence and the cycle time, it should be reasonably accurate, but a little longer than the time the gas valve is open.