Calculating the Cost of Electricity

[JFletcher]

With residential loads, most times cost would be (watts x hours)/1000 x kw/hr electrical cost. However, I was trying to figure out the cost of heating water, say for a hot shower or bath. Would this be a matter of converting BTU to kw/hr? I've worked out the following:

Gallons of water x 8.34 lbs/gal = lbs of water
a BTU is raising one pound of water 1*F
Total lbs of water x temperature change (ΔT) = Total BTU
1kW-hr = 3412 BTU, or 1 BTU = 0.29W-hr


So, say you take a shower that uses 10 gallons of hot water @ 120*F. Water coming into the water heater is, say, 60*F. Assume a kW-hr of electricity costs $0.12.

10gal x 8.34lbs/gal x 60 = 500.4 BTU

500.4 BTU x 0.29 W-hr = 145 W-hrs = .145 kW-hrs x $0.12 kW-hrs = $0.02 per shower?

This seems very low. I know I'm not compensating for pressure nor heat loss through pipes, but still, where is my math off?

This would be the equivalent of sticking a 150W flood bulb into a thermally insulated 10gal pail of 60* water and that pail being 120* in an hour.

 

[iwire]

No expert here but don't see anything accounting for losses. The water is constantly dropping temp due to contact with the tank and pipes etc.

 

[mgookin]

With residential loads, most times cost would be (watts x hours)/1000 x kw/hr electrical cost. However, I was trying to figure out the cost of heating water, say for a hot shower or bath. Would this be a matter of converting BTU to kw/hr? I've worked out the following:

Gallons of water x 8.34 lbs/gal = lbs of water
a BTU is raising one pound of water 1*F
Total lbs of water x temperature change (ΔT) = Total BTU
1kW-hr = 3412 BTU, or 1 BTU = 0.29W-hr


So, say you take a shower that uses 10 gallons of hot water @ 120*F. Water coming into the water heater is, say, 60*F. Assume a kW-hr of electricity costs $0.12.

10gal x 8.34lbs/gal x 60 = 500.4 BTU

500.4 BTU x 0.29 W-hr = 145 W-hrs = .145 kW-hrs x $0.12 kW-hrs = $0.02 per shower?

This seems very low. I know I'm not compensating for pressure nor heat loss through pipes, but still, where is my math off?

This would be the equivalent of sticking a 150W flood bulb into a thermally insulated 10gal pail of 60* water and that pail being 120* in an hour.

84 lbs H₂O * 60 deg = 5040 BTU.

 

[JFletcher]

84 lbs H₂O * 60 deg = 5040 BTU.

That's it; off by a factor of 10. $0.20 sounds much more reasonable. Thanks for the catch.

iwire, heat losses via piping/tank were excluded from the calculations; plastic vs copper pipe, insulated/non-insulated, recirc system, distance from tank to tub, tank/tankless, etc. are too many (small) variables to consider. If I used just 10 gal of hot water a day, loss would be a much bigger factor than say a family of 5 getting baths, doing a load or two of laundry (with hot water), running a load of dishes through the dishwasher, and so on.

 

[iwire]

iwire, heat losses via piping/tank were excluded from the calculations; plastic vs copper pipe, insulated/non-insulated, recirc system, distance from tank to tub, tank/tankless, etc. are too many (small) variables to consider.

I disagree, heat loss when doing these calculations would not be so small not to consider. IMO to ignore them makes the result mealiness.

It would be like figuring the cost of running a motor using only its mechanical output as source for the calculations.

 

[mgookin]

That's it; off by a factor of 10. $0.20 sounds much more reasonable. Thanks for the catch.

iwire, heat losses via piping/tank were excluded from the calculations; plastic vs copper pipe, insulated/non-insulated, recirc system, distance from tank to tub, tank/tankless, etc. are too many (small) variables to consider. If I used just 10 gal of hot water a day, loss would be a much bigger factor than say a family of 5 getting baths, doing a load or two of laundry (with hot water), running a load of dishes through the dishwasher, and so on.

If all you're doing is taking that one shower daily and you have an uninsulated 50 year old tank, it might cost you 20x that or more.

If you have on-demand hot water and the unit is in the shower wall, you might be right at that number.

Assuming you have a 5 year old modern 240V electric water heater system and your water does not have to travel far to reach that shower, I'd double your number for good measure, at least.

Maybe put a logger on the heater and find out for sure.

 

[gar]

150109-1000 EST

JFletcher:

You lost a factor of 10.

Total lbs of water x temperature change (ΔT) = Total BTU

Using your numbers we get:
8.34 * 10 * 60 = 5004 BTU
kWh = 5004 / 3412 = 1.47 kWh for 10 gallons raised 60 F

At your cost of 0.12 per kWh the shower cost is 0.12 * 1.47 or about $ 0.18.

Using slightly different constants derived from the Handbook of Chemistry and Physics, 40th Edition, p 2107 and using the density of water at 75 F I calculated 1.459 kWh for your 10 gallons and 60 F rise. So your answer is correct except for the 10 factor.

Do you really take 10 gallon showers? Is your electric cost really 0.12 / kWh? You should calculate your electric cost on an average basis from the total bill for 1 year divided by the kWh used.

From a recent bill of mine:

510 kWh @ 0.06912 energy cost to me (includes actual cost and some overhead)
253 kWh @ 0.08257 this is a dumb idea of the Michigan Senate and House
to try to reduce energy consumption.

actual cost to DTE for the energy is about 0.03 to 0.04 per kWh but is going up because of government regulations.

763 kWh @ 0.05003 for distribution (power lines and company overhead)
783 kWh @ 0.00276 for energy optimization

So my cost might appear to be about 510*0.069 + 253*0.083 + 763*(0.050+0.003) = 96.63 for the 763 kWh used or 0.127 per kWh. But the electric bill was 105.24 or 0.138 per kWh. I happened to pick a low use month. Becasuse of the Michigan rule my costs per kWh on average over a whole year are greater than this one month. I have not yet calculated my total year average cost for 2015. From 2014 my average cost was about 0.155 per kWh.

When talking to people around our area they will say their electric cost is $ 0.12 per kWh, but it is actually higher if calculated on total billing. Taxes and other misc fees are really also part of the electric cost.

By comparison my natural gas cost is about 1/5 of my electric per unit of energy. I don't use electricity for water heating or clothes drying because gas is so much less costly. 50 years ago the ratio was about 1/3.

.

 

[K8MHZ]

I disagree, heat loss when doing these calculations would not be so small not to consider. IMO to ignore them makes the result meaningless.

I agree. If the loss was too small to consider, the heater would be able to hold a constant temp without the elements coming on for a couple of days.

It won't do that, of course, and the loss would equate to how much energy was used to bring the temp back to the set rating after not being heated for that amount of time.

 

[mgookin]

150109-1000 EST

JFletcher:

...

When talking to people around our area they will say their electric cost is $ 0.12 per kWh, but it is actually higher if calculated on total billing. Taxes and other misc fees are really also part of the electric cost.

...

.

That is a hard one for people to understand. They think the "rate" is what it costs them.
Many of the taxes (all?) are based on usage.

Divide money paid by kWh used and there's your true "rate".

 

[K8MHZ]

That is a hard one for people to understand. They think the "rate" is what it costs them.
Many of the taxes (all?) are based on usage.

Divide money paid by kWh used and there's your true "rate".

That's what I do. With taxes, etc., it works out to about 13 cents per kWh here.

 

[GoldDigger]

I disagree, heat loss when doing these calculations would not be so small not to consider. IMO to ignore them makes the result mealiness.

It would be like figuring the cost of running a motor using only its mechanical output as source for the calculations.

When comparing relative cost of resistance, heat pump, natural gas and propane the heat losses only affect the magnitude of the cost savings, not the cost ratio.
But when comparing the cost of storage versus tankless heating the sum total and nature (pipe versus tank) of the heat losses are critical.

 

[JFletcher]

Thanks for the replies. The amount of hot water, price per kw hour, water heater setpoint and groundwater temps were all approximated, just so I'd have some numbers to crunch into the equations.

Loss can be hugely variable; this house has 40' or so of uninsulated copper pipe between the water heater and the shower. All that pipe (and water) is near ambient temp. That said, turning on the hot water in the tub (flowrate: ~3 gpm) gets all the cold water out of the line and hot water to the tub in 20 seconds (1 gallon), so a 10 gallon shower would be 11 gallons. A recirc system would have a much higher loss, keeping the entire hot water loop ready to roll at 120*; 100' of uninsulated copper pipe would be steadily bleeding away heat. I agree with iWire there in that such losses could/would be substantial and couldnt be ignored in any kind of even ballpark calculations.

In theory, a water heater located in a room that was the same ambient temperature as the water, and no leaks, would not have cyclic losses.

The hotel where I used to work had about a 600' loop of large copper pipe (insulated and recirculated) for one wing. Its loss to ambient would also affect heating and cooling costs (reduced heat in the winter, more AC in the summer). That's beyond the scope of my calculations but certainly not to be disregarded to figure out accurate costs. Commercial boilers also run a much higher delta-T with the use of mixing valves to reduce the storage tank size.

 

[mgookin]

Thanks for the replies. The amount of hot water, price per kw hour, water heater setpoint and groundwater temps were all approximated, just so I'd have some numbers to crunch into the equations.

Loss can be hugely variable; this house has 40' or so of uninsulated copper pipe between the water heater and the shower. All that pipe (and water) is near ambient temp. That said, turning on the hot water in the tub (flowrate: ~3 gpm) gets all the cold water out of the line and hot water to the tub in 20 seconds (1 gallon), so a 10 gallon shower would be 11 gallons. A recirc system would have a much higher loss, keeping the entire hot water loop ready to roll at 120*; 100' of uninsulated copper pipe would be steadily bleeding away heat. I agree with iWire there in that such losses could/would be substantial and couldnt be ignored in any kind of even ballpark calculations.

In theory, a water heater located in a room that was the same ambient temperature as the water, and no leaks, would not have cyclic losses.

The hotel where I used to work had about a 600' loop of large copper pipe (insulated and recirculated) for one wing. Its loss to ambient would also affect heating and cooling costs (reduced heat in the winter, more AC in the summer). That's beyond the scope of my calculations but certainly not to be disregarded to figure out accurate costs. Commercial boilers also run a much higher delta-T with the use of mixing valves to reduce the storage tank size.

On demand heater as close as possible to point of use is most efficient.

 

[GoldDigger]

On demand heater as close as possible to point of use is most efficient.

Except for the fact that on demand is not an option with current heat pump technology!

 

[Besoeker]

With residential loads, most times cost would be (watts x hours)/1000 x kw/hr electrical cost. However, I was trying to figure out the cost of heating water, say for a hot shower or bath. Would this be a matter of converting BTU to kw/hr? I've worked out the following:

Gallons of water x 8.34 lbs/gal = lbs of water
a BTU is raising one pound of water 1*F
Total lbs of water x temperature change (ΔT) = Total BTU
1kW-hr = 3412 BTU, or 1 BTU = 0.29W-hr


So, say you take a shower that uses 10 gallons of hot water @ 120*F. Water coming into the water heater is, say, 60*F. Assume a kW-hr of electricity costs $0.12.

10gal x 8.34lbs/gal x 60 = 500.4 BTU

500.4 BTU x 0.29 W-hr = 145 W-hrs = .145 kW-hrs x $0.12 kW-hrs = $0.02 per shower?


This seems very low. I know I'm not compensating for pressure nor heat loss through pipes, but still, where is my math off?

This would be the equivalent of sticking a 150W flood bulb into a thermally insulated 10gal pail of 60* water and that pail being 120* in an hour.

If ever there was a case for SI units......................

 

[dfmischler]

If ever there was a case for SI units......................

Right. Note that this ignores the change in density of water as the temperature increases.

10 gallons ~= 37.8541 liters
60 degrees F ~= 33.33 degrees C
~4184 joules to raise 1 liter (kg) of water 1 degree C (i.e. 1 kilocalorie)
1 watt = 1 joule per second
1 hour = 3600 seconds
1 kWh = 3 600 000 joules (i.e. 3600 * 1000)

37.8541 * 33.33 * 4184 ~= 5 278 857 joules

5 278 857 / 3 600 000 ~= 1.47 kWh

1.47 kWh * $.12 ~= $.18

 

[Besoeker]

Right. Note that this ignores the change in density of water as the temperature increases.

10 gallons ~= 37.8541 liters
60 degrees F ~= 33.33 degrees C
~4184 joules to raise 1 liter (kg) of water 1 degree C (i.e. 1 kilocalorie)
1 watt = 1 joule per second
1 hour = 3600 seconds
1 kWh = 3 600 000 joules (i.e. 3600 * 1000)

37.8541 * 33.33 * 4184 ~= 5 278 857 joules

5 278 857 / 3 600 000 ~= 1.47 kWh

1.47 kWh * $.12 ~= $.18

Yes, but if you really went SI, you wouldn't have to do the gal/litre or F/C conversions in the first place. And no need to for the kCal.

 

[kwired]

The hotel where I used to work had about a 600' loop of large copper pipe (insulated and recirculated) for one wing. Its loss to ambient would also affect heating and cooling costs (reduced heat in the winter, more AC in the summer). That's beyond the scope of my calculations but certainly not to be disregarded to figure out accurate costs. Commercial boilers also run a much higher delta-T with the use of mixing valves to reduce the storage tank size.

Exactly, people also don't consider heat given up by lighting, computers, televisions, etc. They all add to summer cooling load but provide relief to winter heating load.

A HVAC buddy of mine once told me you are better off having a 80 gallon water heater tank then having two 40 gallon tanks because of the heat loss and it's impact on cooling load. He said they figure about a 1/4 ton of cooling is needed for each water heater tank.