# Why can a smaller hot water tank have a higher first-hour delivery rating?

I'm looking at hot water heaters, and have discovered a 40 gallon and a 50 gallon tank where the smaller one can deliver more hot water in the first hour than the larger one can, with a lower-BTU heater. I got nerd-sniped and am experimenting with the math on this, but I can't nail it down. Specifically:

• A 40 gallon tank with a 40,000 BTU/hr burner can provide 86 gallons of hot water in the first hour
• A 50 gallon tan with a 42,000 BTU/hr burner can provide 78 gallons of hot water in the first hour

Background: an important measurement of a hot water tank is how much hot water it can deliver in an hour starting from a fully-heated state. The most precise definition I've found of this is here - the amount of water raised 100 degrees F above the temperature of the inflowing cold water that the tank can deliver in that first hour. I'm looking for a more official definition still.

So far I've figured, based on 1 BTU raising 1 pound of water by 1 degree F and 8.34 pounds per gallon of water:

• The 40 gallon tank can raise its overall temperature by 119.9 degrees F per hour
• The 50 gallon tank can raise its overall temperature by 100.7 degrees F per hour
• If we think of a gallon of hot water flowing out of the tank and being replaced by cold water as that same gallon having its temperature lowered by the temperature delta (assume 100 degrees):
• 1 gallon flowing out of the tank lowers the total BTU of the tank by 834 BTU
• -834 BTU for the 40 gallon tank lowers the overall temperature by 2.5 degrees
• -834 BTU for the 50 gallon tank lowers the overall temperature by 2 degrees

But it's been too long since college calculus, and I'm not sure how to model the continuous change of temperature of the inflow/outflow and the burner. I'm also not accounting for heat loss out of the tank due to imperfect insulation from its surrounding environment.

How should this be modeled? What's the math behind how this works?

For example, given input water temperature $$t_{\rm input}$$ of 40 degrees F, hot water temperature $$t_{\rm output}$$ of 140 degrees F, and some constant flow rate $$r$$ of hot water out and cold water in, how does the tank temperature vary over time for both tanks? For simplicity, assume the BTUs from the burner are applied uniformly over the whole tank, and the lost heat energy from the flow is applied the same way. Is there some range of flow rates where the smaller tank stays hotter in the first hour than the larger tank?