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How many kilowatts needed to heat water:  Mr Kilowatt
Formula for heating water inside tank:
Gallon Per Hour (Electric) = (KW x 3413) divided by (temp. rise x 8.25) or (KW x 414) divided by (temp rise.)
Let's say you have 5500 watt elements.
let's say incoming water temperature is 65F and you want to heat tank to 105, or 40 rise in temperature.

5500 watt or 5.5Kw elements x 3413 = 18771.5
40 degree temperature rise x 8.25 = 330
So in this specific situation, the element would heat 56.88 gallons per hour

If you have 4500 watt elements and incoming water temperature is 55, and you want to heat to 120, then the temperature rise is 65
4.5 KW x 3413 = 15358.5
65 degree temperature rise x 8.25 = 536
So one element would heat 28 gallons per hour

This number is for EACH element, but does NOT consider heat loss from tank, room temperature, or thickness of insulation etc.
Remember, with residential simultaneous tank, only ONE element is ON at one time.
Upper element is ON, or lower element is ON, or BOTH elements are off. At no time are both elements on.

Kilowatt hours of electricity needed to heat water ... the math formula:

1.) Basic formula > it takes .0002931 Kwh to raise 1 pound of water 1F

2.)
Basic number >A gallon of water weighs 8.34 lbs (pounds)

3.) "A BTU is amount of heat needed to raise 1 
pound of liquid water by 1 from 60 to 61 F at constant pressure of one atmosphere. Other definitions of BTU exist which are based on different water temperatures and cause the results to vary by .5%." (For analysis of water heaters, .5% is ignored. Variation in atmospheric pressure is also ignored. Water heater 'science' is an approximation since calcium carbonate in tank will vary electrical efficiency and tank capacity.)

4.)
BTU formula > 1 BTU = 2.931 x 10–4 Kwh (kilowatt hours).
     BTU = 2.931 x .0001 = .0002931 Kwh
     Therefore it takes .000239 Kwh to raise 1 pound of water 1
F
 
5.) 30 Gallons of water x 8.34 = 250.2 lbs
     40 Gallons of water x 8.34 = 333.6 lbs

Calculation using 40 Gallon water heater:
> Ordinary 40 gallon water heater in attic. Temperature in attic = 50
F. Temperature of water in tank = 50 F.
> Water in tank weighs 333.6 lbs.
> How much electricity is needed to raise temperature of full tank to 120
F.
> 120
 minus 50  = 70 > so the temperature needs to go up 70
> Multiply 333.6 x 70 x .0002391 = 5.58 Kwh

Calculation using 30 Gallon water heater:

> Same problem as above
> Water in tank weighs 220.2 lbs
> Multiply 250.2 x 70 x .0002391 = 4.18 Kwh
Water heater Cost estimate
Temperature of water
Cost if you use 40 Gallon per day Cost if you use 30 Gallon per day
Raise from 32 to 120 7.01 Kwh x 14 = $1.20 5.26 Kwh x 14 = 73
Raise from 50 to 120 5.58 Kwh x 14 = 78 4.18 Kwh x 14 = 58
Raise from 70 to 120 3.99 Kwh x 14 = 56 2.99 Kwh x 14 = 42
Raise from 85 to 120 2.79 Kwh x 14 = 39 2.09 Kwh x 14 = 29

See Family chart for setting  water heater timer
Electric water heater efficiency
Pre-2015: 93%
Beginning April 2015: 95%
Efficiency of electric water heater Energy factor is NOT efficiency rating. There is a subtraction factor based on size of water heater.

To find efficiency for 50 gallon heater, you take EF .96 and subtract (.0003 x tank volume measured in gallons). The result equals .945 .... so the efficiency for 50 gallon tank, manufactured after April 2015 is .945 or rounded up to .95 as seen on bottom row of chart.

The 50 gallon gas water heater has EF .96, but has .945 efficiency.
The 30 gallon has EF .96, but efficiency of .951
So 30 gallon has minimally higher efficiency than 50 gallon. Source for chart AO Smith
How much does it cost to run gas water heater

True efficiency of heater has numerous variables.
Efficiency declines over time with all appliance. Thermostat calibration is approximate . Other variables include elevation
Other variables include elevation above sea level, ambient room temperature, condition of heater, water hardness and quality, scale build up, etc.
First hour delivery is reduced by new efficiency standard
The older 2012 efficiency Standard was based on thermostat set at 135F
The new 2015 efficiency Standard is based on 125F to reduce standby cost of keeping tank hot throughout day.

A lower thermostat setting means you have less hot water in the tank. It reduces first hour delivery, so the water heater might run longer to keep up with demand. This factor might eliminate the expected savings of new water heater vs older heater.

Energy label shows comparative first hour delivery for each model, but the exact number varies with incoming water temperature, condition of heater and thermostat calibration.  Calculate first hour  Ways to increase amount of hot

Electric water heater efficiency can be restored

Burned out lower element means less hot water. Upper element keeps top of tank warm. Opening is 1-1/8" so several pipe and hose sizes fit inside tank. How to replace element and clean sediment Full efficiency is restored to electric water heater after sediment is cleaned out and new lower element installed
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