# someone who knows about thermodynamics!

it's a slightly odd question, but I'm a plumber who has a problem at a dairy!
I need to heat 1500 litres of milk from 33 to 38℃ using a tank of water stored at around 75 which will pass through a water jacket around a vat containing the milk....... but what size of tank do I need to store the water! I'm hoping a brilliant student can come to my aid.
thanks in advance to whoever can help.
This is one of the most random questions I've ever heard, but I'll give it a go

Some preliminary research:
1ml of milk weighs 1.04g
1ml of water weighs 1.00g
Specific Heat Capacity of milk: 3.97 J/g °C (skimmed), 3.89 J/g °C (whole).... I'll use 3.93 J/g °C as a mean of the two! -> 3930J/kg °C
Specific Heat Capacity of water: 4200 J/kg °C

This entire calculation will be done assuming that no energy is lost to the surroundings (which is really unrealistic, but isn't calculable as there is not that much additional information);
Using the equation E=mcΔ𝜃 we can calculate that for 1500l (1560kg) of milk to raise by 5°C;
E=(1560)(3930)(5)
E=30654000J

This will need to be acquired from the water tank stored at 75°C
Assuming we wish to find the minimum amount of water for the two tanks of liquid to reach equilibrium at 38°C we can once again use E=mcΔ𝜃
as: 30654000=m(4200)(75-38)
Rearranged to give m: 30654000/(4200*37)
Which equals 197.258687259kg
Using our conversion rate we can assume this is 197.26l (2dp)
Once again, this is unrealistic as energy is bound to be lost to the surroundings but it should give a good idea

Quite fun to do but so random
- Apologies if any of this is wrong I did it while eating
38C is yoghurt making temperature isn't it?

I don't have any inside dairy knowledge but presumably time is money and waiting for the water and milk to reach equilibrium at 38C is going to take a long time because as the temperature of the milk and water get closer together the rate at which the temperature of the milk increases goes down.

What I'd naively expect is for the hot water tank to be heated (by immersion heater or something similar) and as the heat is transferred into the the milk, more heat is being continuously added to the hot water by the immersion heater until the milk reaches 38C when the circulating pump and immersion heater are switched off. if it's required to keep the temperature at 38C for a long period of time, such as would be needed to make yoghurt, the pump and immersion heater might need to be switched several times... or maybe it'd need cooling - large scale commercial beer fermentation actually needs to be cooled down once it's started because the process of fermentation produces heat, but increased temperature slows down the process of fermentation, it's the opposite of what you'd expect if you homebrew small batches of beer of wine where keeping it warm enough is the usual problem.

But if I wanted to know for sure I'd ask a friendly dairy farmer.