During the incubation process, an egg needs to lose sufficient water to create the air cell for the embryo to start lung respiration. The optimal moisture loss (which equals weight loss) is approximately 12-14% until pipping, or on average 0,6% per day. The moisture loss of the egg is not necessarily constant, as especially in single stage incubation we often try to decrease the moisture loss in the first period and increase it in the final stages of incubation, to limit the activity of the sprayer in the machines and with that get a more uniform temperature distribution.
As the moisture is a liquid inside the egg but leaves the egg as a gas, it needs to be evaporated. Evaporation costs energy, which means that the egg will be cooled by the loss of moisture. When we know some physical properties of eggs and water, we can estimate how much that cooling effect will be.
First of all we need to know the egg size and the moisture loss at a given moment. Lets assume that the egg weight is 60g and the moisture loss is 0,6% per day. This means that the egg will lose 60 x 0,6% = 0,36 g of water per 24 hours, or 0,015 g per hour.
To evaporate a gram of water, energy is needed. This so called latent heat for evaporation is 2,257 kJ per gram of water. This means that if 1 gram of water is evaporated from the egg, the egg will lose 2,257 kJ or 2257 J of energy. As the egg is losing 0,015 g of water per hour, the loss in energy in that hour will be 2257 x 0,015 = 34 J per hour or 0,009 J per second. As 1 J/s is equal to 1 Watt it means that the heat loss through evaporation is 0,009 Watt.
To calculate the cooling effect of this moisture loss, we need to know how much the temperature of an egg changes with the loss of a specific amount of energy. This is called the specific heat, which is for eggs 3,18 J per gram per oC. This means that it costs 3,18 x 60 = 190,8 J to change the temperature of a 60 gram egg with 1 oC. In our calculation we remove every second 0,009 J and every hour 34 J. As it takes 190,8 J to change the temperarture of the egg with 1 oC, the egg will lose every hour 34 / 190,8 = 0,18 oC.If no heat would be added, the egg would continously lose a bit of energy and as a result get colder and colder.
However, an egg that will get colder because of its evaporation will start to be warmed again by the surrounding air, as a temperature difference between the egg and the surrounding air will bring energy towards the egg. That means that a new equibrillium will be formed, in which the heat loss (evaporation) and the heat uptake (heat exchange) will be in balance again. When the embryo starts to produce heat, this will have an influence on the equillibrium as well.
We can calculate the heat loss through evaporation (34 J/h for a 60 g egg), but the heat uptake at a given temperature difference between egg and air is less easy to calculate. This heat uptake depends on factors like size of the egg (surface/weight ratio) but especially on air velocity, which can be very variable in a machine. Also the temperature inside the egg will not be totally uniform, as the constant heat loss and heat uptake will create a small gradient inside the egg.
However, if we assume that the air velocity over the egg is 0,5 m/s, we can calculate the resulting temperature if we know the moisture loss of the egg. It goes too far into detail to describe the way it can be calculated here, but a reference to a more detailed article is given at the bottom of this article.
With this calculation we can estimate that a 60 gram egg with a moisture loss of 0,6% per day and an air velocity of 0,5 m/s will have a continuous temperature which is less than 0,1oC (approximately 0,15-0,16oF) lower than the temperature of the air. A smaller egg, a lower moisture loss or a higher air velocity will decrease the temperature difference even more.
When we have a fertile egg and the embryo is growing, there will be a moment when the cooling effect of the moisture loss will be compensated by the heat production of the growing embryo. This means that in our example, if the heat production of the embryo is 0,009 Watt, the temperature inside the egg will be equal to the temperature of the air. If the heat production is less (smaller embryo) the egg will be colder than the air, and consequently if the heat production is more, the egg will be warmer.
Although the heat production of a very young embryo is not that easy to measure, based on literature we can estimate that the heat production of an embryo around day 7-8 of incubation is approximately 0,009 W, so after a week of incubation the temperature of the egg will be more or less equal to the temperature of the air.
Especially in single stage incubation we will often see that this balance between egg and air temperature is already reached at an earlier age. This is caused by the fact that in single stage incubation we tend to have a higher relative humidity and therefore a more limited moisture loss in the first week of incubation, as we tend to keep the ventilation very low to even no ventilation at all. If the moisture loss in this period is not an average of 0,6% per day but only 0,3% per day, the cooling effect will also be only 50% of the 0,009 Watt that we calculated. In this situation, the egg temperature will already be equal to air temperature after 4-5 days of incubation. A higher air velocity will contribute to this effect.
A more detailed explanation of the calculation can be found in: Meijerhof and van Beek; Mathematical modelling of temperature and moisture loss in hatching eggs, 1993. Journal of theoretical biology 165, 21-47.