Brewing a chocolate ... whoops, carob ... porter. Crude heat capacity calculations...

I started a batch of homebrew today, aiming for a chocolate carob* porter.  My overall goal is to make a peanut butter porter, and possibly even a chocolate-peanut butter porter, but since I haven't brewed in a while I thought I'd start with an established recipe, and if that works try the peanut butter.

* turns out I used carob instead of chocolate...

I'm following the recipe "Goat Scrotum Ale (formerly Tumultous Porter)" from the book "The New Complete Joy of Home Brewing" by Charlie Papazian, pages 199-200.  It was the first porter listed and included chocolate in the laundry list of possible extras, so it sounded like a good starting point.  I did a half-batch, because I wanted to not have too much around so I could move on to the peanut butter version.  Here's what I used:

• 2.5 lbs. CBW special dark malt extract (Briess)
• 0.5 lb crystal malt
• 0.125 lb black patent
• 0.125 lb roasted barley
• 0.75 oz. crystal hops (boiling)
• 0.125 oz. galena hops (finishing)
• 0.5 c brown sugar
• 0.5 c blackstrap molasses
• 1 tsp. gypsum (pH fun!)
• 0.5 lb corn sugar
• 3 oz. unsweetened Baker's chocolate 3 oz. unsweetened Carob
• yeast (Muntons)

The day before, I boiled 1.75 gallons of water on the highest setting on our electric stove, vigorous boil for 10 minutes to try to drive out any gases in the water.  I washed out my water-cooler bottle / carboy with soap and hot water, and then transferred the boiled water into it.  I stoppered it (with a just-washed stopper).  After a while the water cooled enough so that the pressure inside dropped, causing the carboy to start to collapse, and the noise reminded me that I needed to loosely stopper it, so I did that.

The next section is about specific heat and is math heavy; click here to skip ahead

Specific heat

The pre-boiled water from above will be combined with the wort, and I wanted to know what temperature it should be at so that when I combine them the mixture will be 70 F (20 C).  I used the following logic:  I want the total heat of the combined mixture to yield 20 C, the wort is 0.75 gallons at 100 C, the pre-boiled is 1.75 gallons at temperature X.  The basic equation we start with is:

heat content = C * delta_T * M     (1)

C is the heat capacity.  delta_T is the change in temperature, M is the mass of water.  What this equation says is that when the sample changes temperature by delta_T, the heat content goes up (positive delta_T) or down (negative delta_T) an amount proportional to the amount of material (more material, more heat involved) and the ability of the material to store heat (heat capacity).

If we have a sample at temperature T, how do we estimate the heat content?  One way to do it is to imagine our sample started at 0 K and then was heated to its current temperature.  Plugging into the above we have:

heat content = C * (T - 0) * M = C * T * M

(T is in Kelvin).  We are interested in water between 0 and 100 C, which is 273 to 373 K, and the immediate, glaring fault with the above is that C is not constant from 0 to 373 K.  We know there is a big difference between the heat capacity of ice and the heat capacity of water (water is almost twice that of ice).  We can rewrite the above taking this into account:

heat content = Cw*(T - 273 K) * M +  Ci * (273 K) * M

Cw is the heat capacity of water (liquid), Ci is the heat capacity of ice.  Again it is not realistic to assume these two heat capacities are constant in their respective temperature ranges, but for Cw it will be close enough for our crude purposes ("what's a factor of 2 among friends?"). For ice, we are not really interested in different temperatures of ice, so we could choose a fake Ci such that Ci * (273 K) is the correct heat content of ice at 273 K (i.e. mean value theorum).  (hint:  it's going to cancel out anyway).

From conservation of energy, we know that the heat content of the individual samples will equal the heat content of the combined mixture:

(heat content wort) + (heat content of pre-boiled) = (heat content of mixture)        (2)

These heat contents are:

heat content wort = Cw * (373 K - 273 K) * Mw) + Ci * (273 K) * Mw

heat content of pre-boiled =  Cw * (X - 273 K) * Mp + Ci * (273 K) * Mp

heat content of mixture = Cw * (293 K - 273 K) * Mm + Ci * (273 K) * Mm

Mw is the mass of the wort, Mp is the mass of the pre-boiled, Mm is the mass of the mixuture:

Mm = Mw + Mp      (3)

Subbing into equation (2) yields:

Cw * (373 K - 273 K) * Mw + Ci * (273 K) * Mw + Cw * (X - 273 K) * Mp + Ci * (273 K) * Mp = Cw * (293 K - 273 K) * Mm + Ci * (273 K) * Mm

Before anyone panics, let's do a quick rearrangement to put all the ice related terms together:

 Cw * (373 K - 273 K) * Mw + Cw * (X - 273 K) * Mp = Cw * (293 K - 273 K) * Mm + [Ci * (273 K) * Mm - Ci * (273 K) * Mw - Ci * (273 K) * Mp]

 (the ice terms are in square brackets)

Cw * (373 K - 273 K) * Mw + Cw * (X - 273 K) * Mp = Cw * (293 K - 273 K) * Mm +  Ci * (273 K) * [Mm - Mw - Mp]

Mm - Mw - Mp = 0, from equation (3)

Cw * (373 K - 273 K) * Mw + Cw * (X - 273 K) * Mp = Cw * (293 K - 273 K) * Mm

Cw cancels out and we can do the subtraction (could have done that earlier):

(100 K) * Mw + (X - 273 K) * Mp = (20 K) * Mm

Solve for X!

(X - 273 K) * Mp = (20 K) * Mm - (100 K) * Mw

X - 273 K =  [(20 K) * Mm - (100 K) * Mw] / Mp

We can get the mass of the water from the density:

Mi = d * Vi

Our next horrible assumption is that density is the same for all of these samples.  We know this is wrong - the density of boiling water (wort) is lower than room temperature water (mixture).  But it's close enough.

X - 273 K =  [(20 K) * d * Vm - (100 K) * d * Vw] / (d * Vp)

The density cancels out:

 X - 273 K =  [(20 K) * Vm - (100 K) * Vw] / Vp

 We can plug in values:  Vm = 2.5 g, Vw = 0.75 g, Vp = 1.75 g:

X - 273 K = [(20 K) * (2.5 g) - (100 K) * (0.75 g)] / (1.75 g)

X - 273 K = -14 K

X is then -14 Celsius.  Since it was around 0 C out today, I could safely put the mixture outside in a snow bank, knowing it would not get too cold.

If you want to learn more about specific heat, talk to your friendly neighborhood physical chemist of chemical engineer!

Finishing the beer

I actually bought ingredients for the full recipe (double the above list).  For the grains, the total weight I bought was 1.6 lbs., I mixed them all thoroughly in a bowl, then weighed out the 0.8 lbs. that is half the total.  I ground them up using the grinder attachment to our Kitchen aid.  I initially had it on the finest setting, which produced a powder!  I dialed it back to the coarsest setting, which was still probably too fine.  The shells were definitely more than just cracked.  

I put the grains in a 4 gallon pot with 0.75 gallons of cold water, and set it on the highest heat possible.  While it came to a boil I measured out all the other ingredients - dry ingredients in one bowl, malt extract in one measuring cup, blackstrap in another.  Once it boiled, I scooped out the grains using a metal strainer.  I then poured in the blackstrap, then rinsed the pyrex measuring cup in the wort to dissolve the last of the syrup.  I repeated that procedure for the malt extract.  I then dumped in all the dry ingredients, and stirred for awhile to make sure nothing was getting burnt on the bottom of the pot.  I boiled for a total of 47 minutes, during the boil I worked on cleaning and sterilizing the carboy and transfer equipment.  For the last 2 minutes, I added the galena hops.

At the end it looked like I had a pretty thick mixture, so I decided to pour some of the pre-boiled water into the wort to make it easier to transfer.  I then poured some of the pre-boiled water into the sterilized carboy (5 g capacity) so that there was ~ 3 in. of water.  I then used an auto-siphon to transfer the wort into the carboy.

A problem I've had with past home brew is that 2 weeks after bottling the beer doesn't taste that good, but then after 2 more weeks it is much better, and it continues to improve for approximately 6 months.  My brother-in-law suggested I try to aerate the wort more.  In that spirit, during the transfer I pressed the hose against the inside of the neck of the carboy, causing the wort to fan out and cover about 1/3 the circumference of the carboy.  I rotated the hose around so that wort was exposed to different sides of the carboy.

Towards the end, the siphon stopped, even though there appeared to be a good amount of liquid.  I attempted to tilt the pot with the wort and restart the siphon, but the auto-siphon stopped working as well.  I'm guessing that it had gotten clogged, I got a little more over by swirling the auto-siphon in an attempt to unclog it, then restarting the siphon.

I then added the rest of the pre-boiled water to the carboy, I would have stopped at the half-way mark on the carboy, but I ran out just as that was reached.  I stirred it up with a sterilized long handle spoon, measured the temperature (sterilized thermometer dangling from a string to reach the wort) - it read 78 F, so I poured in the yeast, then stirred violently to get it dissolved and aerate some more.

In a further spirit of aeration, I violently sloshed the wort in the carboy several times and said "Aerate you #&*^#(#*!!!!".  I put the carboy in a corner of the dinning room against an inner wall of the house (outer wall has baseboard heating and might get colder when the heat is off), in a cardboard box (to protect from light).  I then cursed repeatedly as I tried to get my overflow hose onto the stopper with the airlock, eventually I got it.

3 hours later, no bubbling yet - the damn overflow hose fell off though, of course...

Edit:  it fell off again.  I switched to a newer hose that is newer and more flexible.  Soon after that, we started to notice bubbling!  I'm thinking I will epoxy the hose in place.