This page discusses the physics related to heat transfers and losses in brewing vessels used in home brewing. We are interested in this topic because it allows us to predict what the mash temperature will be after adding a known amount of strike water to a vessel. It also allows us to predict what the mash end temperature will be, which is lower than the start temperature due to heat loss through the vessel walls.
I have prepared a mash calculator which will help plan the mash schedule. Read on to see how it works.
This topic has been discussed by other authors, but either the effect of the brewing vessel on the temperature is ignored, or the approach to it is too complicated and uses unnecessary information. I will show that we do not need to know anything at all about the vessel beforehand, such as its shape, dimensions, materials, or weight. The temperature response of a calibrated strike water infusion is going to tell us all we need to know about the vessel, namely its heat capacity and heat loss coefficient.
A common theme in brewing is adding hot water to an unheated brewing vessel, for example mash lauter tun (MLT) or hot liquor tank (HLT). When we do this there are two processes at work determining the temperature in the vessel:
The five minute division between the two phases is arbitrary and is based on practical experience. It is also used by brewing software such as Beer Tools Pro.
In the equalization phase, heat transfers occur between objects of different temperature so that they end up at the same temperature. We assume that there is only little heat loss to the environment in this phase, and that little loss there is can be included into the calculation of the vessel's heat capacity. This is a fair assumption because at first there really is no heat loss to the environment, since the vessel walls are at ambient temperature. As the interior of the vessel heats up, its exterior will also heat up and the heat loss will gradually build to reach its peak at the end of this phase.
Before the equalization phase, each object has its own energy based on its heat capacity and temperature. At the end of the equalization phase, heat transfer has occurred so that all objects have acquired the same temperature. Based on the principle of conservation of energy, the sum of the energies going into the process is the same as the final energy of the equalized system.
This concept is easy to understand for the strike water and the grist because they have a uniform temperature throughout themselves before and after the equalization phase. It is a little less obvious what is going on with the vessel because at the end of this phase it is not at a uniform temperature throughout its body. We use the concept of an equivalent thermal mass for the vessel. This equivalent thermal mass is assumed to be heated to the interior temperature.
Conservation of energy gives the following equation:
...where c is heat capacity and T is temperature. Subscript w is for strike water, g for grist, v for vessel, and eq for equalized (the mash temperature).
We compute each ingredient's heat capacity by multiplying its mass in kg by its specific heat. The specific heat of water is 4.186 KJ/kg/K. The specific heat of malt is about 0.44 times the specific heat of water. You can add other ingredients to the equation if you wish. The heat capacity of the vessel is found by calibration, see below.
Once we know all the other quantities involved, we can solve this equation for strike water temperature, Tw, or any other quantity.
In the heat loss phase, the rate of heat loss is proportional to the temperature differential between the inside and outside of the vessel. When the temperatures are equal there is no heat loss. When the temperatures are unequal the loss is characterized by a constant heat loss coefficient. Each brewing vessel has its own heat loss coefficient, and it is found by calibration, see below. The better the insulation of the vessel the smaller the heat loss coefficient.
The following equation describes the heat loss.
...where P is the heat loss in Watt, hlc is the heat loss coefficient, Tmash is the interior temperature, and TA is the ambient temperature, both in °C or K.
The previous equation can be used to calculate how much power is needed to keep a vessel at a constant temperature.
The heat loss equation gives rise to an exponential temperature decay described by the following equation.
...where T(t) is the function of temperature versus time, TA is the ambient temperature, T5 is the temperature at 5 minutes after adding the strike water (or the mash start temperature), τ (Greek tau) is the temperature time constant.
The time constant τ is computed by the following equation, using previously defined symbols:
Using the temperature function T(t) we can compute the temperature at any point during the heat loss phase, for example T(60 min) to find the mash end temperature.
We do calibration to find the two vessel calibration constants, cv and hlc.
Calibration is performed by adding a carefully measured quantity of hot water to the vessel and measuring its temperature at three specific times.
I feel that it is more accurate to weigh the water than to use a volume measure. I put a one gallon pitcher on my electronic kitchen scale, set it to grams, and zero it. Add water from the tap until it is just over one gallon. Dip a ladle in the pitcher and draw up a small amount of water. Drizzle the water back until the scale reads 3782 gram (the mass of 1 US gallon of water at 60°F is 3.7816 kg, equivalent to 133.393 oz). I used three gallons measured in this way to calibrate my 7.5 gallon picnic cooler HLT and five gallon water cooler MLT.
Before you start, let the vessel acquire room temperature for a few hours with the lid off. Measure the temperature inside the vessel and record as Tv.
The temperature of the water should be in the range that you expect to use in the vessel. It is not important to heat it to an exact temperature, but it is important that you measure the temperature accurately just before adding it to the vessel. Record this as the T0 temperature. Put the lid on the vessel and let it sit.
At the five minute mark, record the T5 temperature. At 65 minutes record the T65 temperature.
The vessel's heat capacity is calculated by the following equation:
As before, cw is the mass of the water in kg multiplied by 4.186 KJ/kg/K. One US gallon of water at 60°F weighs 3.782 kg. The temperatures can be in any unit you like, it doesn't matter.
The vessel's heat loss coefficient is calculated by the following equation:
... where τcal is the calibration time constant described by the following equation:
... where ln is the natural logarithm and TA is the ambient temperature. Here again you can use any temperature unit as long as you do so consistently.
You can see and modify the detailed calculations I used to develop this method in this MathCAD 14.0 worksheet (right-click, Save Target As). If you do not have MathCAD installed on your computer you can view this static webpage which shows the worksheet with all its equations. A MathCAD worksheet is easy to read and is rather self-explanatory, but I put together this primer which explains the basics of how it works.