There are many variables involved with performing this calculation. The method I will give you is not exact by any means, but will give you a rough idea.
To determine a mixed water temp, take the temp of the warmer water and multiply it by its volume. For example: 3 gallons of 100˚F water=3x100=300. Then take the temp of the colder water an multiply it by its volume: 2 gallons of 35˚F water=2x35=70. Now add the two totals together and divide that sum by the total volume. 300+70=370÷5=74˚F.
Here is another example. 3 gallons of wort cooled in an ice bath to 90˚F. 90x3=270. Pour the wort into your fermenter containing 2 gallons of refrigerated water at 35˚F. 35x2=70. Now add 270 and 70 to get a total of 340. Now divide 340 by the total volume of 5 gallons. 340÷5=68˚F
This is way over simplified because some of the variables, for example, the specific heat of the two substances, are not the same. The specific heat of wort is different from that of plain water. The ambient temp at which the mixing occurs will vary. The thermal mass of the two containers, heat loss due to convection, conduction etc, just to name a few of the variables... Well, you get the picture.
John
