how much 40 degree water will it take to cool 100gallons of 120 degree water to 100degrees

Question

I want to find the ratio of cold water to hot water I need to make 100 degree water for a nursing home I have a 100 gallon tank and I need to cool it from 120 to 100 degree and the water i'm adding to it is around 40 degrees

Answer 1

You could use Pearson's Square to determine the ratio. Here is a link, although the example deals with percentages rather than degrees:  http://www.ext.colostate.edu/pubs/livestk/01618.html

 

Your goal is water at 100 degrees. Subtract diagonally, larger number minus smaller. The temperature values carry across horizontally.

 

 

120°                           (100 - 40) = 60 parts 120° water

 

 

                  100°

 

 

40°                             (120 - 100) = 20 parts 40° water

 

 

So your ratio is 60:20, which simplifies to 3:1.

Mixing 3 parts 120° water with 1 part 40° water yields 100° water.

Answer 2

John,

 

So you have 100 gallons at 120 degrees and add x gallons at 40 degrees and you end up with 100+x gallons at 100 degrees when you are finished.  The easiest way is to treat this as a weighted average.  You have (100 gallons) (120 degrees of heat) + (x gallons)(40 degrees of heat) = (100 + x gallons)(100 degrees of heat)

 

Solve and distribute

12000 + 40 x = 10000 + 100x

-10000 - 40x = -10000 - 40x

2000 = 60 x

x = 2000/60 = 200/6 = 100/3 or 33 and 1/3 gallons of 40 degree water for every 100 gallons of 120 degree water, or reduced to least terms the ratio is 1 gallon (cold) / 3 gallons (hot)

 

Note this answer requires putting 133 1/3 gallons together.  But if you want to make the water in batches of 100 gallons at 100 degrees, this ratio will still work.  If you fill the tank 3/4 full with 120 degree water (i.e 75 gallons) and fill it 1/4 full with 40 degree water (ie 25 gallons), you will get 100 gallons at 100 degrees as

75 x 120 + 25 x 40 = 100 x 100

9000 + 1000 = 10000