# Air Conditioning Math

## Introduction

I get some strange phone calls. I recently received one from a customer who wanted to know how many "tons of air conditioning" he needed to cool some equipment he had purchased from my company. After I assisted this customer with his problem, he asked me if I knew where this strange unit came from. Here is the story I told him.

## Background

This unit of air conditioning is a real fossil. It goes back to the first days of the air conditioning/refrigeration industry. The ton came into use by the refrigeration industry because early refrigerators were used to make ice. The ton represents the amount of cooling capacity needed to make 1 ton of ice per day. In the US, home air conditioners are usually rated in BTUs/hour, and commercial refrigeration units are usually rated in terms of tons of ice frozen per day.

The temperature of liquid water reduces by roughly 1 °C for every calorie per gram. When liquid water is at 0 °C and you continue to extract heat, water begins to undergo a phase transition from liquid to ice. The energy required to make this transition is referred to as heat of fusion (symbol ?H°m). The heat of fusion for water is 79.72 cal/g.

We can compute the energy required for one ton of water to transition from liquid to ice as shown in Equation 1.

 Eq. 1 $E=\left( \frac{1\text{ ton}}{\text{day}} \right)\cdot \left( \frac{2000\text{ lb}}{1\text{ ton}} \right)\cdot \left( \frac{1000\text{ g}}{2.20\text{ lb}} \right)\cdot \left( \frac{79.72\text{ cal}}{\text{g}} \right)=3.03\text{E8 }\frac{\text{J}}{\text{day}}\text{ = 287000 }\frac{\text{BTU}}{\text{day}}$

We usually say that 12000 BTUs per hour equals on 1 ton of refrigeration per day. Equation 2 illustrates this calculation.

 Eq. 2 $\frac{1\text{ ton}}{\text{day}}\text{=287000}\frac{\text{BTU}}{\text{day}}\cdot \frac{1\text{ day}}{24\text{ hour}}=11958\frac{\text{BTU}}{\text{hr}}$

## Conclusion

I must admit that I find the units of measure used in the US confusing. I wish things were different, but I am a realist. All I can do is try to shed some light on the subject. If you're more interested in just cooling your room down rather than running calculations, check out the Coolest Gadgets review of various portable air conditioners that will do the job.

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### One Response to Air Conditioning Math

1. Matthew Egan says:

I think it's so important to optimize your home with the best unit that is going to save you money and save on the cost of energy. I really enjoyed this post and feel it shed light on a subject that is really important right now.