These calculations allow you to estimate the amount of heat provided by a volume of hot water so you can predict how much that heat will change the temperature of the mash. This method makes a few simplifications, one of which is the assumption that no heat will be lost to the surroundings, but we can minimize this error by pre-heating the tun.
Most of the thermodynamic constants used in the following equations have been rounded to single digits to make the math easier. The difference in the results is at most a cup of hot water and less than 1¡F. Experience has shown the equation to be fairly reliable and consistent batch-to-batch.
When mixing hot water with dry grain for the initial infusion, the equation is algebraically simplified so that the amount of grain does not matter, only your initial grain temperature, the target mash temperature, and the ratio (r) of water to grain in quarts per pound.
NOTE: These equations also work for degrees Celsius, liters and kilograms. The only difference is that the thermodynamic constant of .2 changes to .41.
Initial Infusion Equation:
Strike Water Temperature Tw = (.2/r)(T2 - T1) + T2
r = The ratio of water to grain in quarts per pound.
Wa = The amount of boiling water added (in quarts).
Wm = The total amount of water in the mash (in quarts).
T1 = The initial temperature (¡F) of the mash.
T2 = The target temperature (¡F) of the mash.
Tw = The actual temperature (¡F) of the infusion water.
G = The amount of grain in the mash (in pounds).
The infusion water does not have to be boiling, a common choice is to use the sparge water at 170¡F. Then TB becomes 170 ¡F and more water (Wa) will be needed to make up the additional quantity of heat.
This example will push the envelope with three rests. We are going to mash 8 lbs. of grain through a 104 ¡F, 140 ¡F, and 158 ¡F (40, 60, and 70 ¡C) multi-rest mash schedule. For the purposes of this example, we will assume that the temperature of the dry grain is 70 ¡F (21 ¡C). The first infusion will need to take the temperature of the mash from 70 ¡F to 104 ¡F. We will start with an initial water ratio of 1 qt/lb. Using the initial infusion equation, the strike water temperature is:
For the second infusion, to bring the temperature to 140 ¡F, we need to use the mash infusion equation. At 1 qt/lb, Wm is 8 qt. We will assume that our boiling water for the infusions has cooled somewhat to 210 ¡F.
Wa = (T2 - T1) X (0.2G + Wm) / (Tw - T2)
Wa = (140 - 104) X (1.6 + 8) / (210 - 140)
Wa = 36 X 9.6 / 70 = 4.9 qt
For the third infusion, the total water volume is now 8 + 4.9 = 12.9 qt.
Wa = (158 - 140) X (1.6 + 12.9) / (210 - 158) Wa = 18 X 15.1 / 52 = 5.2 qt
The total volume of water required to perform this schedule is 8 + 4.9 + 5.2 = 18.1 qt, or 4.5 gallons). The final water-to-grain ratio has increased to 2.26 qt/lb (18.1 / 8).