I have a boiler (oil) with integrated water heater (3.5 gallon). I don't have problems with running out of hot water but I do wonder if having a tempering tank before the boiler would save money on heating the water. (I'm in the Boston area so the cold supply is really cold in winter.) Are there any rules of thumb, etc. for figuring out if this is a worthwhile thing to do?
Plumbing – How to calculate potential energy savings from using a tempering tank
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The volume of water that is being heated by the on-demand heater is very small, so the smallest commonly available expansion tank will work just fine. Just make sure that you are getting an expansion tank for potable water, not the one made for hydronic heating systems.
Model the water heater as a continuously stirred tank reactor (CSTR), so it is always at a uniform temperature. Assume the recovery time is not dependent on temperature and completely accounts for insulation losses and the like. Neglect losses in pipes and assume the operator controls the shower temperature to 105°F perfectly. Taking the stopping criterion from the question, the shower is over when the water in the tank becomes 105°F.
The recovery rate raises 140 gallons of water by 81.3°F in one hour. We'll say this is a constant heat input of 9118 W.
The key is that a constant-temperature shower removes a constant rate of heat from the tank, that which is associated with raising 2.5 gpm of water from 58.7°F to 105°F. This is 16986.5 W.
The difference is 7868.5 W. With constant heat, you don't need any complicated integrals.
The time for the tank to drop from 140°F to 105°F is 1568 s or 26 minutes.
The (maybe counterintuitive) fact that the variable flow rate from the water heater does not influence the rate of heat removal from the water heater comes from the fact that the incoming cold water is the same temperature at the shower and at the water heater.
Note that, because 
does not appear in the expression for 
, is now a constant.
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Best Answer
You're basically thinking of a tank near/above the boiler that pre-warms the cold feed from waste heat (the warmth of the boiler room) - If installation cost was no object, hand-wavingly it will help. So if you can do it at a fairly low cost, it should save a bit.
Getting to hard numbers is fairly difficult, or very easy if you knew some unknowable things.
A gallon of #2 fuel oil is roughly 138,000 BTUs - if we give your boiler the benefit of the doubt and say it's 85% efficient (a few might be more, a fair number will be 80 ) that means you'll get 117,000 BTUs from each gallon burned.
1 BTU heats one pound of water one degree Farenheit. 117,000 will heat 1800 lb of water from a "cold but not frozen" 34F to "might be a warm shower" 99F - that's roughly 225 gallons. If you prefer a "very hot" 120F shower, 170 gallons.
That's the easy part. The hard part (or mass of made-up fudge-factors) is "and what will the tempering tank do for the input water temperature" - to the extent that it directly sucks heat, it will make no difference (you're still burning fuel to heat your house and water, you're just shifting it around a bit) but to the extent that it "scavenges" heat that would otherwise be wasted, there is a savings. If the input water temperature was raised 21 degrees to 55, you'd get 225 gallons of very hot shower from your gallon of oil. If we make the ballpark assumption that you use 17 gallons per shower, you'd get 3 "free" showers (13 for the price of 10) Which is 25% savings looked at one direction, 30 the other. But "garbage in, garbage out" rules here - while I'm trying to speculate reasonably, it's just speculating. what the actual effect on your water heating will be is essentially unknown until you try it, though you can possibly make more educated guesses by considering things like the average temperature in the place you'd be putting the tank, your actual shower times and shower flow-rate, actual water temperatures in and as you shower, etc...
If you can fit the appropriate drain pipe below your shower, there is a neat widget that plumbs the cold-water to the shower around a copper drain pipe from the shower, pre-warming the cold water to the shower as you shower, so you can use less hot for the same shower temperature.