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Question: How do I figure out how much of my mash to drain for heating to raise the temperature to achieve starch conversion, or to move from starch conversion to mash out? And, is there a formula?
Answer: Yes, there is a formula. Basically the amount of heat drop in the run-off has to equal the heat gain needed by the remaining mash to get a certain temperature rise. One critical part of the formula is to know that one pound of water takes 1.0 BTU (British Thermal Unit) to raise its temperature by 1°F. In contrast, grain takes only 0.5 BTU to raise its temperature by 1°F. One pound of water has a volume of 1 pint, so start out by converting your mash water volume into pints.
Then you need to select a mash concentration and whether you want to add more water as you go along or not. I prefer the step-infusion method, where you add water for the protein rest and then again for starch conversion. Then I draw some off and heat up for mash-out. From the phrasing of your question, I assumed you want to add all the water at the beginning to start the protein rest. Then you want to pull off some wort and heat for a starch conversion and later once again for a mash-out. The way the math works out, this approach risks destruction of the amylase starch conversion enzymes too early, so I don't recommend it. For the purposes of illustration, lets use the following 5 gallon recipe:
We'll assume the bucket of grain is starting at a room temperature of 70°F. To get the grain to 120°F, it needs 10 lb x 0.5 BTU/lb x (120-70) = 250 BTU. The water then has to lose 250 BTU to get the whole mixture to temperature. For the water heat balance, 20 lb x 1.0 BTU/lb x (T-120) = 250 where T is the water temperature needed. A bit of algebra shows that T = 250/(20 x 1.0) + 120 = 132.5°F. Note that this assumes no heat loss. In reality, you'll want to add a few degrees to this.
Now we repeat the process to figure out how much heat we need to get to starch conversion at 153°F. The mash has two components now, so the math is a bit uglier:
(10 lb x 0.5 BTU/lb + 20 lb x 1.0 BTU/lb) x (153-120) = 825 BTU. So the step water has to be: T = 825/(10 x 1.0) + 153 = 235°F. But wait! The water can't get above 212°F! If I dump in boiling water, the mash will only be about 146°F. There are a couple ways around this. You could just add more water for the step, or run the protein rest a bit warmer. I usually drain off some liquid and heat that up and recycle it back in. So, back through the math again. I use this opportunity to check for heat loss, and make small adjustments up or down depending on the recipe. Also, I make sure the mash is well stirred so the temperature is uniform throughout. In this case, I'll drain off about 4 pints and figure out how much to heat it to raise the mash from 146°F to 153°F. Note that the mash now has 26 pounds of water and 10 pounds of grain. So we need (10 lb x 0.5 BTU/lb + 26 x 1.0 BTU/lb) x (153-146) = 217 BTU. So the 4 pints has to be heated to T = 217/(4 x 1.0) + 153 = 207°F. Again, make sure to stir the mash well to distribute the heat. Now with the mash at 153°F, I can cover it up and have a homebrew. Note that if the temperature drops too much during starch conversion, you can repeat the process.
Now one final time with the math to do mash-out. Drain out 16 pints for example. The mash now has 14 pounds of water and 10 pounds of grain, and needs to get from 153°F to 170°F. So it needs (10 lb x 0.5 BTU/lb + 14 x 1.0 BTU/lb) x (170-153) = 323 BTU. So heat the wort to T = 323/(16 x 1.0) + 170 = 190°F. Note that you need to take out at least 10 pints of wort to keep the temperature under boiling.
One final word about mash-out. Unless you're making a recipe that has some extra residual sweetness, you probably don't need a mash-out. I'll usually let the grain bed set as I draw off the initial runnings. I'll heat it to about 180°F or so and pour it back in without stirring. I'll switch over to sparge water at 170°F as soon as the runoff is clear.