Additional Math Pages & Resources

Wednesday, December 14, 2011

How to get into hot water

A common idiom in English is "you're in hot water" which means you're in trouble. Why? Is the phrase derived from the old image of an explorer being cooked in a pot by a bunch of cannibals?

I think hot water is universally considered to be a very good thing, as any one who's bathed in luke warm or cold water can attest. Today in the math blog we'll investigate this subject - a moderate amount of elementary math is enough for the main issues related to hot water.

US Average Soil Temperatures
Most of our water comes from pipes buried well below ground level, where it stays a relatively constant 40-70° F or so, depending on location and weather. Since our body temperature is around 100° F, we prefer to wash (if we can) in water that is a bit warmer than the water coming out of the pipes - say about 104–120° F.

A water heater in our homes should be able to raise water temperature about 40-60° F and supply us with at least 5-6 US gallons per minute (gpm) or more. Why? Because sink faucets and appliances can draw up to 3 gpm (each).

You may know that plumbing codes restrict shower water flow to 2.5 gpm. But remember, this is for ONE shower head. If TWO or more showers might be going at the same time, you need a much greater supply of hot water, or there will be angry shouting in your house!

A BTU is the energy required to raise one pound of water by one degree F and a gallon of water weighs 8.3 pounds. Assuming 100% efficiency, raising 40 gallons of 50° F water up sixty degrees to 110° F requires a fair amount of energy, as you can see here:

(40 gal × 8.3 lbs) × (110 − 50) degrees
332 lbs x 60 degrees
19920 BTU
19920 / 100,000 
(natural gas is sold by the "therm" which is 100,000 BTU)
.20 therms

NOTE: For more on Natural Gas, check my first blog on the subjectHere's the second one, too.

We have a number of choices for heating our water. None are without compromises:
  • A 40,000 BTU/hr gas heater takes about a half hour to heat 40 gallons, at 100% efficiency. At $1 per therm, the cost of the gas would be about 20 cents. The tank will give us at least a 16-minute shower (40 gallons ÷ 2.5 gpm = 16 min) or longer since we usually mix some cold into the hot water flowing from the tap.
  • A typical electric water heater with a 4500 watt heating element takes a bit more than an hour to heat the same tank of water for roughly $ .50. 
  • A tankless water heater with 40,000 BTU/hr can only supply about 1.6 gpm. To supply 2 showers (5 gpm and 60° F temperature rise), tankless heaters need capacity of about 150,000 BTU per hour, which means larger gas lines or electrical wiring.   
  • Combination boilers in the UK and Europe heat water for both the shower and heating the house. They are compact and efficient and can generate 100,000 BTU or more. Since many are located inside the house, some fancy plumbing or venting may be required.
  • Solar heating is "free" but not fast. It requires a solar array and piping. The details vary so greatly with location, climate, and sun exposure that I can't do any reasonable math on solar water heating. While the heat is free, solar requires large holding tanks, because it works best when we are least likely to be taking hot showers (in the middle of a sunny day).
Lukewarm water is no treat. Neither is a cold shower.

Hot water is a luxury we all appreciate, enjoy and (most of us) happily pay for!