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Old 12-12-10, 09:18 PM   #24
Lurking Renovator
Join Date: Jul 2010
Location: Mass,Merrimack Valley
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You may find this interesting:

(I copied and pasted as I could not post a link.)

Bottom Line: Yes, it saves energy, but not a huge amount.

Suppose you go on vacation one Saturday morning and return on Sunday evening a week later, which means you are away for a total of 8.5 days. Should you turn the water heater off or not? In reasoning this through I will use an electric water heater, but the same results apply to a water heater that runs on natural gas or propane. By the way, "turning off" an electric water heater generally means throwing the circuit breaker.

The easiest way to reason through this problem is to look at the cumulative heat lost in both cases. All the heat that was lost must be replaced by the electric resistance heater inside the tank.

Leave the Water Heater On:

If you leave the water heater on then the heating element cycles on and off occasionally to replace heat lost through the insulation.

The below graphic shows the rate of heat loss in btu/hr over 8.5 days for our 80 gallon R-16 hot water tank assuming the tank is left on. Because the thermostat in the water heater is keeping the water temperature roughly at a constant, then the rate of heat loss is constant over the entire 8.5 days. In the Hot Water Tank Heat Loss web page I computed this loss to be about 141 btu/hr.

Water Heater Loss When On

The "area under the curve" of this graph gives the total heat lost in btus over the 8.5 days. The calculation for total btu loss is:

BTU Loss on vacation = 141 btu/hr x 8.5 days x 24hr/day
= 28,764 btu

The only way to replace lost heat is by turning on the electric resistance heating element, so all of these btus are supplied by electricity. Divide the btus by 3,412 btu/kwh in order to convert the btu loss to kwh supplied by the electric company:

kwh supplied = 28,764 btu / (3,412 btu/kwh)
= 8.43 kwh

Multiply by $0.10/kwh to find the cost of replacing the lost heat:

Cost of kwh supplied = 8.43 kwh x $0.10/kwh = $0.84

So the cost of the electricity needed to keep the tank hot over this 8.5 day vacation is a little less than a dollar for our system.

Turn Off the Water Heater:

If you turn off the water heater then its temperature slowly coasts down according to Newton's Law of Cooling. No electricity is consumed while you are away, but once you return and throw the circuit breaker on again the heating element turns on and stays on until the tank is back up to temperature.

The below graphic shows the rate of heat loss in btu/hr over 8.5 days for our 80 gallon R-16 hot water tank assuming the tank is shut off. As the temperature slowly declines during those 8.5 days the rate of heat loss also declines.

Water Heater Loss When Off

The equation for this theoretical heat loss curve can be constructed as follows.

Start with the equation for the temperature over time:

T(t) = 60 + 60e-0.00351 t

(This equation is derived in the Sample Calculation at the bottom of the Newton's Law of Cooling web page.)

Next consider the equation for heat loss through the insulation:

H(t) = A(T(t)-TA)/R

H = Heat loss in btu/hr
A = Area of tank walls = 37.5 ft2 for our tank
T(t) = Temp of the hot water as a function of time
TA = Air temperature surrounding tank = 60 F
R = 16 ft2hrF/btu

(This equation is described in more detail in the Hot Water Tank Heat Loss web page.)

Substituting T(t) into the H(t) equation and simplifying:

H(t) = 141e-0.00351 t

This equation states mathematically what the above graph shows visually.

The "area under the curve" of the graph gives the total heat lost in btus over the 8.5 days. All of this lost heat is replaced at the end of the vacation when the electric resistance heating element turns on after your throw the breaker on.

To compute the area under this curve we take the integral of the H(t) equation from time = 0 to 204 hours (8.5 days).

Cumulative heat loss = ∫0204141e-0.00351 tdt
= 141*(e-0.00351*204-1)/(-.00351)
= 20,540 btu

Divide by 3,412 btu/kwh in order to convert the btu loss to kwh supplied by the electric company when we turn the breaker back on:

kwh supplied = 20,540 btu / (3,412 btu/kwh)
= 6.02 kwh

Multiply by $0.10/kwh to find the cost of replacing the lost heat:

Cost of kwh supplied = 6.02 kwh x $0.10/kwh = $0.60


So if we leave our hot water heater on it costs about $.84 of electricity to keep the water hot while we are away on an 8.5 day vacation, and if we switch the hot water heater off it costs about $.60 of electricity to heat the water back up to temperature when we return. The net energy savings is about 2.4 kwh, which translates into a cost savings of about $.24.

How might these results vary for other households? If you have a poorly insulated water tank, or lots of heat loss from the pressure relief valve and the hot water pipe, then you might save two or three times as much from turning off the breaker.

Does this result depend on the length of the vacation? The longer the vacation, the more you save. The only case in which turning off the water heater would not save energy and money would be a vacation so short (a few hours?) that the thermostat would not have turned on to top off the heat anyway.

So is it worth shutting off the breaker, then shutting it on again for a savings of $0.24? My immediate reaction was "No way". But wait. I would certainly stoop down to pick up a quarter that I saw on the sidewalk. Just how much effort would I be willing to expend to retrieve a quarter? If I saw a quarter at the bottom of a floor heating vent I would be willing to spend 30 seconds to remove the vent, pick out the quarter, then replace the vent. Thirty seconds is about how much time it takes me to turn off the breaker, then turn it on again when I get back from vacation, so logically I should be willing to do this.

What is the effective pay rate for shutting the breaker off and on again? To make the calculations easier, assume a savings of $.25 for 30 seconds of effort. That's $.50 per minute, or $30 per hour. But that is an after-tax rate of pay, which might translate to $40 per hour before tax, or a taxable salary of about $80,000/year. So for that brief 30 seconds I'm earning an implied salary of $80,000/year. Isn't math wonderful?

Possible Downside of Turning Off the Water Heater During Vacation?

Could the additional thermal cycling due to turning off your water heater tank during vacation cause it to fail sooner? If so, then the cost and energy savings of shutting off the tank may be entirely swamped by early tank failure. Early tank failure means that the purchase and installation cost of the tank is spread over fewer years, so you pay more per year for the tank. Building a water heater tank takes energy, so early tank failure means the energy embedded in making the tank is spread over fewer years, so the embedded energy use per year is higher.

Of course, if the tank were poorly insulated, then early tank failure might be a net positive.

The analysis of tank life will require some further research...

This site is still under construction…to be continued…

You can e-mail me at support(@ sign goes here)

He basically says that shutting off an electric water heater will have negligible savings and does the math to prove it.

I think you have the same setup I have, tank-less coil on the oil fired furnace.
One of the least efficient ways to make DHW.

I the summer with no heating requirement, I was using about 1 gal/day for DHW
about $90/mo. I switched to an 80 gal electric which should only cost about $25/ mo to run. It is less because I have a solar and wood stove pre heater.
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