Heat Article Dugg up from Archive.org, thanks to KS:

Grain & Heat Formulae

Here is a summary of Volume and Heat calculations you'll need for several aspects of brewing with grain.  Calculations include total volume of mash, "water-equivalent" heat capacity of grain, strike water calculations, and electric element calculations of power and temperature rise rate.

These formulae are based on a grain heat capacity equal to 0.4 times that of an equal weight of water, from a posting in HBD 1879 ( "re: temperature calculations"), and on grain volume being 0.08 gal/lb (when soaked -- thank you Spencer!), from postings in HBDs 1792 and 1793 ("Scientific Mashing Breakthru!").

Both screwed-up English (American, more accurately) and Metric units are given.  Variables are defined as they are introduced and are not redefined when used later.  I've spared the mathematically-challenged the drudgery of deriving the equations but if anyone is interested I'll be happy to answer via private E-mail.

A typical question which is answered by the formulae is given in advance of each, in case someone is thinking, "What the hell good is THIS?".  An example is given as well.

Hope you find this to be worthwhile reference information.

Topic Index:

• Grain Volume
• Grain Heat Capacity Caluclation
• Strike Water (Initial & Subsequent) Temperature Calculations
• Power Required to Boost Temperature in a certain time
• Temperature Rise Rate with a Given Power Input
• Element Power at Voltage other than Rated
• Wiring Current Capacity
• Element Power Density

• Grain Volume

"I have a 5-gallon cooler that I would like to mash in.  How much grain will it hold?"

1 lb grain occupies about 0.08 gal or 0.32 quart when added to water (and allowed to soak).  1 kg grain occupies about 0.67 liters when added to water (and allowed to soak).

Total volume of mash = Wg (0.08 + MWR/4) gallons
= Wg (0.32 + MWR) quarts
= Mg (0.67 + MMWR) liters

Wg   = weight of grain (lbs)
MWR  = Mash Water Ratio, qt/lb
Mg   = Mass of grain (kg)
MMWR = Metric Mash Water Ratio, liters/kg

Example:  10 lb grain with a MWR of 1.3 qt/lb occupies

vol = 10 * (0.08 + 1.3 / 4)
= 4.05 gallons

Grain Heat Capacity Calculation

"If I could find the amount of water which has the same heat capacity as a certain amount of grain, it would make my strike water and other calculations easier."

Although some may argue that it's an extra step or something along those lines, I believe it's easier to visualize water calculations when you're dealing with just water -- the resulting temperature is basically just the constituent temperatures "averaged" over the total volume.  Besides, for multiple strikes, once you figure the "water-equivalent" of a certain amount of grain that number stays fixed, so you only need to figure it once.

The "water equivalent" (W.E.) of grain is the volume of water whose heat capacity is equivalent to that of a given mass/weight of grain (based on specific heat of grain being 0.4) and is calculated like this:

W.E. = 0.0479 * Wg gallons
= 0.192 * Wg quarts
= 0.4 * Mg liters

Example:  20 kg of grain has the same heat capacity as

vol = 0.4 * 20
= 8 liters of water

The water-equivalent for a mash (grain plus water) is:

W.E. = Wg (0.0479 + MWR/4) gallons
= Wg (0.192 + MWR) quarts
= Mg (0.4 + MMWR) liters

Example:  My 10 lb, 1.3 qt/lb mash has the same heat capacity as

vol = 10 * (0.0479 + 1.3 / 4)
= 3.73 gallons of water

Strike Water (Initial & Subsequent) Temperature Calculations

"I'm doughing-in at a certain MWR (or MMWR) using grain at room temperature (Tg), trying to hit my dough-in temperature of Td.  I can figure how much water I need from my Mash Ratio, but how hot (Ts) should the strike water be?"

Ts = [Td + WEF * (Td - Tg) / (M)MWR] + FF

Ts  = Strike Temp (F or C)
Td  = Desired Mash Temp after Strike (F or C)
WEF = "Water Equivalent Factor", depending on units:
0.0479 when dealing in gallons
0.192  when dealing in quarts
0.4    when dealing in liters
Tg  = Temperature of grain (F or C) (room temp)
FF  = "Fudge Factor" to account for thermal mass
of vessel (3 deg F or 1.7 deg C is typical)
(M)MWR in gal/lb, qt/lb, or liters/kg

Example: Doughing-in 10 lb grain at 70F with MWR = 1.3 qt/lb for a final temperature of 142F, figuring 3 deg F for "fudge factor", I'll need

vol = 10 lb * 1.3
= 13 quarts (3.25 gal) of strike water at

T = [142 + 0.192 * (142-70) / 1.3] + 3
= 156 deg F (rounded off)

"If I add a certain amount (Vs) of water at temperature Tb to a mash of water-equivalent Vm at temperature Tm, what is the resulting new temperature Tn?"

Calculate the "water equivalent" volume of the existing, resting mash (grain plus water), to find Vm:

Vm = Wg (0.192 + MWR) quarts
(for other units use other equations previously given)

...keeping in mind that your MWR includes all water added so far (in the case of multiple steps)...

Tn = (Vm * Tm + Vs * Ts) / (Vs + Vm)

Example:  If I add 2 quarts of boiling (212F) water to a 142 deg F mash (10 lb, MWR = 1.3 from previous example), the water equivalent of the original mash is

Vm = 10 * (0.192 + 1.3)
= 14.92 quarts

so the resulting temperature will be

T = (14.92 * 142 + 2 * 212) / (14.92 + 2)
= 150 deg F (rounded off)

Rearranging this equation answers the more practical question, "My mash is resting at temperature Tm; I wish to raise it to Tn by adding water at temperature Tb.  How much water do I add?"

Vs = Vm (Tn - Tm) / (Tb - Tn)

Example:  My mash (20kg, MMWR = 2) is at 60C.  I need to boost it to 70C with boiling (100C)water.  My W.E. is 20 * (0.4 + 2) = 48 liters;  so I need to add

Vs = 48 * (70 - 60) / (100 - 70)
= 16 liters boiling water

Power Required to Boost Temperature in a certain time

"I'm using a heating element in a RIMS system to mash.  I also am using an element in my hot-liquor tank.  If I know how many gallons I'm working with in each case, and I have a certain temperature rise rate in mind, how can I figure how much power I need?"

P (watts) = 147 * LF * gallons * deg F rise / minutes
= 69.9 * LF * liters * deg C rise / minutes

LF = loss factor, to account for heat loss.  A well-insulated vessel might be 1.05 or less; uninsulated might be 1.10 to 1.15 or more depending on geometry, material, covered or open, etc.

Example:  I want to raise four gallons of water in my hot-liqour tank from 70F to 175F in fifteen minutes.  The tank is uninsulated.  So I need about

P = 147 * 1.10 * 4 * (175 - 70) / 15
= 4528W

Temperature Rise Rate with a Given Power Input

"I already have an element installed.  How do I figure my temperature rise rate, knowing the element power (P)?"

deg F per min = 0.0068 * P / (gallons * LF)
deg C per min = 0.0143 * P / (liters * LF)

Example:  My RIMS has an element operating at 1125W in an insulated tun.  I'm mashing 10 lb grain at MWR = 1.3.  This mash therefore has a
W.E. of 10 * (0.0478 + 1.3 / 4) = 3.73 gallons.  So my temperature rise rate will be about

deg F/min = 0.0068 * 1125 / (3.73 * 1.05)
= 1.95 deg F per minute

This figure does not account for heat losses, so your actual rise rate may be somewhat lower.

Element Power at Voltage other than Rated

"My element is rated 4500W at 240V but I'm using it on 120 volts.  How much power will it generate at 120V?"

Pn = (Vn/Vr)2 * Pr

Pn = New Power rating at new voltage Vn
Pr = Rated Power at rated voltage Vr

Example:  My 4500W / 240V element will operate at

P = (120/240)2 * 4500
= 1125W at 120V

Wiring Current Capacity

"I have a 4500W element operating at 120V, which results in 1125W actual power.  How much current is being drawn through the house circuit?"

I = P/V

I = current in Amps
P = actual operating power
V = actual operating voltage

Example:  My element operating at 1125W on 120V will draw

I = 1125 / 120
= 9.4 amps

This should be added to any other loads on that same circuit to determine the total circuit load.

Element Power Density

"My 4500W element in my RIMS operates on 120V for an opearting power of 1125W.  It's 10" long and is double-folded (40" total element length).  The element wire is 5/16" wide.  What is the power density of this element?"

PD = P / A
= P / (3.14 * L * DIA)

PD  = Power Density in Watts per square inch or square cm
A   = Area in square inches or square cm
L   = Total element length in inches or cm
DIA = Element wire diameter in inches or cm

Example:  For the given situation,

PD = 1125 / (3.14 * 40 * 5/16)
= 28.7 watts per square inch