Mr Kilowatt
How many kilowatts needed to heat water in water heater
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Kilowatt hours of electricity needed to heat water ... the math formula:
1.) Basic formula > it takes .0002931 Kwh to raise 1 pound of water 1°F

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.)
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 65°F 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

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.
4.) BTU formula > 1 BTU = 2.931 x 10–4 Kwh (kilowatt hours).
     BTU = 2.931 x .0001 = .0002931 Kwh
     Therefore it takes .000293 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 .0002931 = 6.84 Kwh

Calculation using 30 Gallon water heater:
> Same problem as above
> Water in tank weighs 220.2 lbs
> Multiply 250.2 x 70 x .0002931 = 5.13 Kwh
Formula for heating water inside tank:
The number of gallons of water that are heated Per Hour (for Electric heater) = (Kw rating of element x 3413) divided by (temperature rise x 8.34)
8.34 is weight of 1 gallon water. Resource: Water heater formulas and terminology .pdf

Calculation #1
Let's say you have 5500 watt elements which are 55 Kw.
Let's say incoming water temperature is 65°F and you want to heat tank to 105°, or 40° rise in temperature. (40 temp rise not shown on the AO Smith chart)

5500 watt or 5.5Kw elements x 3413 = 18771.5
40 degree temperature rise x 8.25 = 333.6
So 18771.5 divided by 330 =
So in this specific situation, the element would heat 56.26 gallons per hour.
Calculation #2
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° (65 temp rise is not shown on chart)
4.5 KW x 3413 = 15358.5
65 degree temperature rise x 8.34 = 542.1
15358.5 divided by 536 = 28.
So one element would heat 28.33 gallons per hour.
Note: the chart shows 30.8 for 60° rise, so you would expect 65 rise in temperature would result in lower output than 60. This says that reasonable estimates can be made using the AO Smith chart.

Measuring water temperatures can be accomplished using a PID sensor taped to pipe and covered with insulation, or by installing temperature gauge on pipe, or bleed water out of TP valve and use cooking thermometer.
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Considerations
Calculations do NOT consider heat loss from tank, ambient room temperature, thickness of insulation, low voltage, approximate calibration of thermostats, and inefficiencies such as elements coated with sediment etc. There are too many variables to make any calculation exact, so all number for heating water are approximations which help determine actual performance vs potential problems with heater.
The number of gallons per hour is per EACH element, but 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, unless heater is wired for simultaneous operation.
Resource:
How to wire simultaneous water heater thermostats
0.0002931 Kwh to raise 1 pound of water 1°F
0.0024444 Kwh to raise 1 gallon of water 1°F
0.146666  Kwh to raise 1 gallon of water 60°F (60-120°F)
5.866689  Kwh to raise 40 gallons of water 60°F (60-120°F)
1 Kw = 3413 BTU
2.42 watt required to raise 1 gallon water 1 degree
1 Kwh will raise 6.8 gallon water 60 degrees F
8.33 BTU will raise 1 gallon of water 1 degree at 100% efficiency
11 BTU required to raise 1 gallon of water 1 degree at 75% efficiency (gas)
1 cubic foot natural gas = 1000 BTU
1 therm 100,000 BTU or 100 cubic feet
1 cubic foor propane = 2500 BTU
1 gallon propane = 91,250 BTU
More formulas

Water heater formulas and terminology/ pdf
Electrical formulas /pdf
Cost for 1 bath
15 gallons hot for bath x 8.34 llb per gallon x 60 degree rise in temperature x .14 cets per kwh = 1050
 3413 BTU content of fuel x .95 efficiency rating of heater = 3139
1050 divided by 3193
= 33¢ cost of hot water per bath inside standard 15' x 30" x 60" Princeton bathtub.

9 gallon shower cost 19.8¢
Temperature of water
Cost if you use 40 Gallon per day Cost if you use 30 Gallon per day
Raise from 32° to 120° 8.60 Kwh x 14¢ = $1.20 6.45 Kwh x 14¢ = 90¢
Raise from 50° to 120° 6.84 Kwh x 14¢ = 96¢ 5.13 Kwh x 14¢ = 72¢
Raise from 70° to 120°> 4.88 Kwh x 14¢ = 68¢ 3.66 Kwh x 14¢ = 51¢
Raise from 85° to 120° 3.42 Kwh x 14¢ = 48¢ 2.56 Kwh x 14¢ = 36¢

See Family chart for setting water heater timer
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Ultra low watt density
Ultra low watt density
What does watt rating mean
Wattage is just a rating that does not measure heating power
Wattage over time, or watt-hours or Kilowatt-hours (Kwh) is measure of heating power.
If you have 1250 watts for 1/2 hour, then you have 625 watt-hours or .625 Kwh.
If you have 1250 watts for 2 hours of the day, and 750 watts for 3 hours, then you have 4.75 Kwh... which should raise 40 gallons of water from 75° to 120
°

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Select and understand water heater elements

3-phase water heaters
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Temperature for killing bacteria

Same 4500 watt element @ 240 volts and @ 120 volts
Wattage drops when voltage is reduced
See water heater label showing difference between 208V - 240V

Resistive loads such as light bulbs and elements burn less hot when voltage is reduced.
Examples: ordinary household dimmer for incandescent bulbs, or power brown-out when household voltage is reduced and light bulbs dim.
Same is true for water heater elements.
Lower wattage heats less water per hour See chart
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Select and understand water heater elements

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water consumption bathtub
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Regular bath uses 13- 15 gallons hot water (20 gal water total)
Set water heater timer for 45-60 minutes for hot bath

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What size water heater for oversize bathtub?

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Recommend 80 gallon first hour delivery
Electric: minimum 60-to-80 gallon size just for that one tub
Gas: minimum 50 gallon size just for that one tub

Electric heat pump water heaters generally cannot produce enough hot water.

Tankless promise unlimited hot water, but temperature of water is 120 F and water in tub may be cool by time tub fills. Cost of maintenance and repairs never pay back. All costs are energy.

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