Calculating wort expansion along with pre and post boil boil volumes
 

First off I must confess I am math challenged. With this in mind I will describe my last brew day I am using ½ barrel Sanke kegs and taking volume measurements by marking my mash paddle from the center of the domed bottom of the keg. I first measure the water going into HLT for my mash water I add 11 gallons and mark this level on my mash paddle the water is 60 F. For my mash I heat 7 gallons of water to 165 in my mash tun for a 10 gallon batch of beer (21 lbs of grain) mash at 154. I sparge with 170 f water to collect about 11 gallons of wort at 132 F the gravity is 1.040 (adjusted to 1.053 at 60 F) it’s about 1 inch over the 11 gallon mark I made on my paddle earlier (I collected more for my estimate of thermal expansion) I boil the wort for 2 hours (2 hours because I'm using Pilsner malt normally I boil for 1 hour). I end up with 8 gallons of 1.070 wort 2 gallons short and .010 over my target OG. Does anyone have a source for thermal expansion and boil off rates, like I said I am using Sanke kegs as are a lot of other folks out there. I figured I would end up with 10 gallons by starting with 11 but I ended up with only 8 I left only 1 quart behind in the boil kettle. So either my guess on thermal expansion is off or boil off was estimated too low. I am using Beer smith software, don’t know enough about the software maybe I can plug numbers in there?

First off, I don't know how you managed to get 11 gallons into a 1/2 gallon Sanke :D. Second, no, I don't know the formulas for water expansion, but I can't imagine a google search wouldn't have them. I'm lazy and use BTP. But I would say that for a 2hr boil, going from 11gallons to 8 gallons seems like a high, but quite reasonable boil off rate. That's probably what my system would do (not a Sanke). Looking at BTP real quick, it looks like "11gal" water at 60F is "11.46gal" at boiling.

Coefficient of Thermal Expansion for water is 207*10^-6. Volumetric change = alphav*°C*Volume.

In this case, 207*10^-6*84.5*11= 0.19 gallons. hrm, funny...less than Ketchepillar's number, but maybe that's cuz I used a simpler formula.

Anyway, losing 0.2 gallons to thermal expansion and losing 2.8 gallons to evaporation doesn't sound bad. I lost about 1.25 gallons per hour, if not a bit more.

Wort expansion and water expansion are two totally different things.

Try to use a hydrometer correction table to figure this out.

The quick and easy calculation is:
At or near boiling: Measured Volume X 0.96 = Volume @ 68 F
At or near sparge temps (~170-ish): Measured Volume x 0.97 = Volume @ 68 F

Don't measure your gravity at high temps and adjust...it seems to almost never work and you'll usually be way off. Cool it to at most, room temp, and then measure your gravity (then adjust if you're at room temp).