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Resource Conservation: Heat: Water Heaters: How We Made The Calculations




There are 5 Calculators in the Water Heaters section. It is fair to give
you full discloser as to how we make the calculations that appear in
those calculators. The following are the assumptions and formulas. 


The map that is used comes from the EPA. You would have to go there to
find out how they make that map. 

The heating elements are the ones we found installed on new water
heaters. There could be other standard heating elements.
1 Therm is defined as 100,000 BTU's. Most gas company websites we found
had a correction from 100 cubic feet to therm and the conversion is
about 3%. It is a little different for Propane.




Water Heat Cost Calculator:


The water heat cost calculator pulls from the section "What is
the energy required?". This is were we derive the amount of
energy required to raise the temperature of a gallon of water.
We do this in BTU's and KWHR. When you enter the input and
output temperature the calculator determines the difference in
temperature (T out  T in). Then it simply multiplies that by
the 0.00245 KWHR or 0.0000837 BTU to get the energy required.
Then, it also uses the efficiency factor to come up with the actual
energy used.
After that, the calculator will use the cost imputed to come up
with a cost to heat a gallon of water to the output temperature.


The 4 outputs:
Dishwasher: This is simply 100% of the calculation
of the cost to heat the water. This can also be used for the
"hot" cycle of your clothes washer.
Faucet: This is just 50% of the cost calculated to heat
the water to the output temperature. This is 50/50 mix of cold and hot water.
"warm cycle": for clothes washer for modern washing
machines is 40% hot and 60% cold water. So the number is 40% of the cost to heat
the water.
Shower water: Instead of figuring out % hot and % cold
required to get to 100 deg F we simply calculate the energy required to
raise the inlet water to a temp of 100 deg. F. This is simpler than doing the
% hot and cold. We do that later because it is required for other calculators.



Average Hot Water Recovery Rate:


Here we look at how fast the water recovers back to the set
temperature. Besides putting in your inlet and set temperatures
you put in the recovery efficiency. This is not the same as the
efficiency factor (EF). That is because the efficiency factor is
the overall efficiency of the unit over time (including standby
power loss). Recovery efficiency is the efficiency is the short
term conversion of the elements power to heating the water. That
is why recovery efficiency is higher than efficiency factor.
The recovery rate is a calculation to determine how many gallons an
hour you can heat water at the inlet temperature to the set
temperature. 

The industry specifies recovery rate for their hot water units as the number
of gallons their hot water heater can raise by 90 deg F in an hour.
This calculator calculates your recovery rate for your inlet
temperature and your set temperature (probably not a 90 deg
F range).
First, we subtract the cold temp from the hot and call
it our range (r). We next multiply r by 2.45 WHR (the energy
required to heat a gallon of water by that temperature
range), call it (E1). Next we divide E1 by the recovery
efficiency call it E2. Then we simply divide the wattage of
the element by E2. We do the same thing for gas except the
energy required is 0.0000837 Therm. This gives an answer in
gallons per hour. 


How Many Showers Can We Take Before They Are Cold?


Here we start by assuming that the water heater is
at full temperature. Then we assume that 70% is usable
at full temperature when the showers begin.
Therefore, if the tank is 50 gallons we assume 35 gallons are
usable at the set temperature. Next, we calculate the recovery
rate of the water heater (See "Average Hot Water Recovery Rate"
section above). We divide this by 60 to get a gallon/min rate.
Next we establish a "hot water use rate". We get this
by first calculating how many gallons of hot water are used by
each shower. That is done by calculating the fraction of hot
water used to make a 100 deg F shower stream. This takes on the
form:
fraction hot = 1  1/(1+(abs(Tc  100)/abs(Th100))
Where Tc is the inlet water temperature, Th is the set water
temperate,
abs is absolute value. Then based on your showers flow rate (gpm) and
shower time we get the total gallons of hot used for the shower (Gh). Next,
we add shower time to wait time (Tt). We get:
Hot Water use rate = Gh/Tt
In gallons/min.


Now, it is simply a "recovery rate" vs. "hot water use rate" problem.
We subtract the hot water use rate from the recovery rate. If
that number is positive you can take an infinite number of
showers under the conditions in the calculator. If that number
is negative you will run out of hot water eventually.
However, if the calculator reads a negative number you
have infinite showers and a positive number you can take that
many showers.
The reason for that, to make the calculation easier, we
divide the total available gallons at the start (70% of tank
capacity) by the difference of the two rates (hot water use rate
 recovery rate). That way, when you get a positive number you
know you have that many showers.



Standby Power Cost:


We have seen a number of web sites try to help you calculate the stand by power loss
of your water heater. None can do it because they would require you to take several
measurements that most are ill equipped to take (even us). Therefore, these web sites
have to make some assumptions and simplifications. We are no different.
What we use is the theoretical
heat loss from Physics for a cylinder that is insolated. We take that theoretical
heat loss and double it. We then assume that your heat loss is
between the theoretical and 2 times the theoretical. Your heat loss may actually be
a little more than
that.
The first thing we do in the calculator is to calculate the
surface area of the cylinder in square feet. We take your
diameter and height measurement and divide by 12. Then we cut
the diameter in half to get the radius (r).


surface area (SA) = 2*pi*r^{2} + 2*pi*height*r
Where pi is 3.1416.
Next, we subtract room temperature from the set
temperature of the water heater (Td).
We use:
Heat Loss = (SA*Td)/Rvalue
which is in the units of BTU/HR.
We either convert that to Therms/Hour by dividing by 100,000 or
KWHr by dividing by 3412.
Next we simply multiply by 8760 hours to get the heat loss per
year.
To get the cost we multiply the heat loss rate by the rates
inputted by the user of the calculator to get a dollar amount of
standby power loss for the year.



Water Heater Size:


This may be a bit confusing to people so we would like to clear it up here.
We give a 3 column output to show you where you might run out of hot water while doing
the tasks that require hot water. We show it this way so that you can be aware that you
may be able to all your tasks with a smaller tank. A smaller tank can save money on
standby power usage. However, you may need to wait a few minutes extra between tasks
to meet your hot water needs.
We ask you, in this calculator, to calculate your 2
hour peek hot water use. We also build in wait times in the
calculator for your typical use which you do not have to
use. These wait times will show just how much waiting you
need for many different tank sizes.
This calculator is basically 4 calculators in one. Let
us take a look at each one.
Faucet: It is hard for most people to put in faucet use at
exact times, so we ask you to put in a total amount of time and a flow rate
for the 2 hour period. The calculator assumes that this will be a
5050 mix of hot and cold water. The calculator then adds up all of the hot
water used and establishes a faucet flow rate in gallon/min of hot water
used during the entire 2 hour time. This seems like the simplest way to
account for hot water use through the faucet.
Faucet rate = gallons hot water/120 minutes.





Other Assumptiions:


Glossary Of Terms:


References:





