homebrew mashout: calculation of how much water to add to mash to raise temperature to 168 F (75 C) to stop enzymatic activity

 I worked out the approximate math to figure out how much boiling water to add to a completed mash to for the mashout - to raise the temperature of the mash to 170 F to stop the enzymatic activity.

Using conservation of energy:  (heat content of individual components before) = (heat content of combined mixture)

$$ H_{mws} + H_{gs} + H_{aws} = H_{wf} + H_{gf} $$

starting individual terms of heat content are:

mash water at start:  $H_{mws} = C_w M_{mw} T_s$

* $C_w$ heat capacity of water

* $M_{mw}$ mass of "mash water"

* $T_s$ starting temperature

grain in mash at start:  $H_{gs} = C_g M_g T_s$

* $C_g$ heat capacity of grain

* $M_g$ mass of grain

add water at start:  $H_{aws} = C_w M_{aw} T_{aw}$

* $M_{aw}$ mass of add water

* $T_{aw}$ temperature of add water

final individual terms of heat content are:

water, final:  $H_{wf} = C_w (M_{mw} + M_{aw}) T_f$

* $T_f$ final temperature

grain, final:  $H_{gf} = C_g M_g T_f$

heat before = heat after

$$ H_{mws} + H_{gs} + H_{aws} = H_{wf} + H_{gf} $$

$$ C_w M_{mw} T_s + C_g M_g T_s + C_w M_{aw} T_{aw} = C_w (M_{mw} + M_{aw}) T_f + C_g M_g T_f $$

Solve for $M_{aw}$:

$$ C_w M_{aw} T_{aw} - C_w M_{aw} T_f = C_w M_{mw} T_f + C_g M_g T_f -C_w M_{mw} T_s - C_g M_g T_s $$

$$ M_{aw} C_w (T_{aw} - T_f) = C_w M_{mw} T_f + C_g M_g T_f -C_w M_{mw} T_s - C_g M_g T_s $$

$$ M_{aw} = \frac{C_w M_{mw} T_f + C_g M_g T_f - C_w M_{mw} T_s - C_g M_g T_s}{C_w (T_{aw} - T_f)} $$

$$ M_{aw} = \frac{C_w M_{mw} (T_f - T_s) + C_g M_g (T_f - T_s)}{C_w (T_{aw} - T_f)} $$

$$ M_{aw} = \frac{(T_f - T_s) (C_w M_{mw}  + C_g M_g)}{C_w (T_{aw} - T_f)} $$

The numerator is for the materials in the starting mash mixture (water + grain) - their change in temperature times their heat capacity - this is their energy change that will happen as a result of mixing in the add water.

The denominator is for the add water - it is the heat capacity times the change in temperature for the add water, and thus the energy change of the add water.

A lot of terms in here but all on the right side are known!

$T_f = 75.5 \degree C$

* desired final temperature

* this is just an example from one recipe may vary

$T_s = 65.5 \degree C$

* measure this value to use in the calculation

* this is just an example

$C_w = 1 \frac{cal}{g \degree C}$

* specific heat of water

$M_mw = 10 kg$

* mass of the water used to mash

* this is just an example from one recipe may vary

$C_g = 0.4 \frac{cal}{g \degree C}$

* specific heat of grain 

* there is wide variation here see below for references

$M_g = 3.4 kg$

* mass of grain

* this is just an example from one recipe may vary

$T_{aw} = 100 \degree C$

* temperature of add water

* typically it is boiling water but can adjust as needed

specific heat of grain references:

https://www.homebrewtalk.com/threads/specific-heat-of-grain.246356/

https://ojs.openagrar.de/index.php/JKA/article/view/375/1207