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Home > Resource Conservation > Heat > Water Heaters > How We Make The Calculations  

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

water heating
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.

 

construction

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(Th-100))

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*r2 + 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 50-50 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:

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Glossary Of Terms:
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