Wednesday, March 18, 2009

thermal equilibrium of hot wort and cold water

Regarding my wonderment about the calculation for final temperature of a mixture of hot wort and cold water, I dug out my old Halliday and Resnick Fundamentals of Physics (second edition!) and turned to chapter 20, Heat and the First Law of Thermodynamics. It's a matter of solving for the equivalence between the heat lost by the wort mass and the heat gained by the water mass. The specific heat of wort and water can probably be assumed to be equal and therefore be factored out. If I'm thinking straight that means that

mwo(Tf - Two) = mwa(Twa - Tf)

where m is mass, T is temperature, wo is wort, wa is water, and f is final.

Really, when solving for final temperature, this just works out to an average of the temperatures weighted by mass:

mwoTf - mwoTwo = mwaTwa - mwaTf

mwoTf + mwaTf = mwaTwa + mwoTwo

Tf(mwo+ mwa) = mwaTwa + mwoTwo

Tf = (mwaTwa + mwoTwo) / (mwo+ mwa)

As it's the proportions that matter it's fair to use volume instead of mass.

So, if the wort boils down to 2.5 gallons and is at 212 degrees, adding it directly to 2.5 gallons of 38 degree water from the refrigerator would result in 125 degree wort in the fermenter. That is still going to take a long time to cool to yeast-pitching temperature relying solely on heat conduction through a plastic bucket into room-temperature air. That's all time during which oxidation damage can occur and off-flavors can be created.

So I think the technique espoused by The Cellar's witbier instructions is suboptimal. At a 1:1 ratio of wort to water, the wort should be cooled to about 120 degrees before being added to the refrigerator-temperature water in the fermenter in order to hit an 80 degree target.

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