Algorithm (and/or heuristics) for low-flow showerhead cost savings calculator

low-flow-showerhead-cost-savings-algorithm.txt

  A quick discussion of a generic algorithm for a low-flow showerhead
water and energy savings calculator:

  0. First thoughts.

The easiest way to think about these costs, for a current or
prospective customer, are likely to be in:

* Savings per month.  (We think of mortgage or rent payments; utility,
car, cell phone contract payments, etc., in monthly terms.)

* Cost per 100 gallons saved.  That increment fits household usage
patterns quite nicely; one person taking one ten minute shower per day
saves about 300 gallons per month (2.5 gpm v. 1.5 gpm.)  (Cost per
gallon saved is often so low that you have to get into fractional
cents - and that's too many decimal places for most of us to be
comfortable with.)

  1. Water savings

Water savings are simple: multiply the number of gallons saved by the
cost per gallon.  The latter typically comes from the variable costs -
per gallon for water, and any additional per-gallon charges for
elevation, wastewater/sewage treatment, etc. - from one's water
district, private water company, etc.

The main complexity is likely to be a unit conversion to gallons, if
water and other variable costs aren't priced directly in gallons.  For
instance our own water district lists its prices in 'units,' with each
unit equivalent to 748 gallons (100 cubic feet).

Also, in some areas of the USA, water rates are tiered, so that
heavy water users are charged more as their usage goes up. If that's
the case, your savings will reflect the marginal rate of your highest
typical usage tier, as they'll be directly applied against that tier.
That can have the effect of increasing your potential cost savings.

  2. Energy savings

This turns out to be FAR less simple than I'd originally thought.  Two
helpful guides, each from state energy offices (Illinois and Nebraska,
respectively), as well as a unit conversion guide from an Ag extension
office at Iowa State, are the basis for the following calculation:

http://smartenergy.illinois.edu/pdf/newsletter6_6.pdf

http://www.neo.ne.gov/neq_online/july2003/july2003.02.htm
calculating energy / heating savings from low-flow showerheads

http://www.extension.iastate.edu/agdm/wholefarm/pdf/c6-86.pdf

  The calculation is based on the following factors:

a. How much thermal (heat) energy is needed to heat a gallon of water?

This is reflected by a static formula. (Most other calculations, below,
are variable and depend on the site's context.)

It takes 1 BTU to heat one pound of water by 1 degree F.  A gallon
weighs 8.3 pounds; hence 8.3 BTUs to heat a gallon by 1 degree F, or
830 BTUs to heat 100 gallons by 1 degree F.

b. How many degrees F do you need to heat your water?

Water coming in typically is around 55 degrees F (perhaps lower in
some cooler climes, warmer in others).  Most household water heaters
might be typically set to 120 to 140 degrees, with perhaps most at
120.  (For typical discussion of this range, see
http://www.structuretech1.com/2012/04/water-heater-temperature/)

So that's roughly 120 - 55 = 65 degrees F that the water needs to be
heated. 830 BTUs to heat 100 gallons by 1 degree * 65 degrees = 53,950
BTUs to heat 100 gallons by 65 degrees F.

c. Convert BTUs to therms (for natural gas) or kilowatts (for electricity).

This is also reflected by a static formula.

If your water is heated by natural gas, convert the BTU figure from b.,
above, into therms. Similarly, if your water is heated by electricity,
convert the BTU figure from b., above, to kilowatts.

1 therm = 100,000 BTUs
1 kilowatt hour of electricity = 3,412 BTU

Thus: 53,950 BTUs - needed to heat 100 gallons by 65 degrees F, as
in the example above - are equivalent to .5395 therms (natural gas)
or 15.81 kilowatt hours (electricity).

d. How efficient is your water heater?

Various water heater types, both gas and electric, turn various
proportions of their energy inputs into the heat energy directly used
to heat your water. If your hypothetical water heater is 75% efficient
(wastes only 25% of its inputs), that means you'll need to multiply
the therms or kilowatts subtotal from c. by 133%.

e. How hot do you set your shower water to?

Per that Nebraska State Energy Office page, "Typically, unmixed hot
water from the tap measures 110°F, which is quite warm (Water at 120°F
will produce a burn in 10 minutes). Mixing cold water lowers the
temperature. Water at 102 degrees seems just mildly warm. Most people
probably shower with water at about 105 degrees F."

If your water heater is heating your household water to 120 degrees F,
but want it at 105 F for showering, that means you will have adjusted
your shower handle(s) so that 87.5% of your water coming out of your
showerhead will be hot; the rest will be cold.  So you could multiply
the therms or kilowatts subtotal from d. by 0.875.

This should give you a reasonable intermediate estimate of energy
use in therms or kilowatts to heat a gallon of water. You can
then multiple that by your cost per therm or kilowatt - at the
margin, if you have tiered energy rates - to estimate your energy
cost savings. You can stop here if you like; everything below is
somewhat esoteric, if all you need is a 'ballpark' estimate.

(Any complexities here might - at most - involve tiered rates and
any distributed energy issues - such as a solar hot water heater,
or a PV panel that results in electricity credits.)

(One more complexity: if you figure that the 120 degree F hot water
from your water heater would might be, say, only 110 degrees F when
it reaches the shower arm, then the reduction value you apply here
might be a bit more modest than 0.875 - perhaps as high as 0.95. See g.,
below, for a discussion.)

f. How much does electricity or natural gas cost?

The U.S. Energy Information Administration (EIA) tracks the residential
cost of electricity, per kWh, both nationally and by state:
https://www.eia.gov/electricity/monthly/epm_table_grapher.cfm?t=epmt_5_6_a

They EIA also tracks the national residential cost of natural gas here:
https://www.eia.gov/dnav/ng/ng_pri_sum_dcu_nus_m.htm

But they list this in Mcf, "the volume of 1,000 cubic feet (cf)
of natural gas," so there's a conversion factor involved, per
http://www.eia.gov/tools/faqs/faq.cfm?id=45&t=7:
"In 2015, the average heat content of natural gas for the residential,
commercial, and industrial sectors was about 1,032 Btu per cf; one
Ccf = 103,200 Btu or 1.032 therms; [thus] one Mcf = 1.032 MMBtu or 10.32 therms."

So, if the average U.S. residential price for a thousand cubic feet
was $9.21, as it was in March 2016, that would be equivalent to
$9.21 (per Mcf) divided by 10.32 (per therm) = $0.892 per therm.
(That's consistent with the 'answer' result from a mid-June 2016
Google search, https://www.google.com/search?q=dollars+per+therm:
"The price paid by Washington area consumers for utility (piped) gas,
commonly referred to as natural gas, was $0.958 per therm in April,
9.0 percent above the national average of $0.879 per therm.")

g. How much heat loss is there between the water heater and
the shower arm?

One additional factor is that water loses heat when traveling in pipes
between the water heater and shower arm. I've seen casual estimates
asserting that heat loss in a range of 5-10%, so that means that to
achieve a comfortable 110 degree F temperature at the shower arm, you
might need to increase the percentage of hot water in your water stream
proportionally. (In a simple calculation, this factor could well be
ignored, but for accuracy, it should likely be accounted for as well.)

Clearly, distance between water heater and shower arm, insulation
of pipes, and wall temperatures (e.g. for walls exposed to outside
air) are all factors in this heat loss, but a simple temperature
measurement at exit via the shower arm should suffice.