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I suspect that something has gone awry with the electric water heater.

We do get water that is plenty hot, but it doesn't seem to last as long as it used to. My suspicions are further aroused because one day the water heater breakers had tripped.

Is this indicative of one of the elements failing, or would that just give us lukewarm water?

update

2 days after asking this question, the breaker tripped again, and this time a reset did not get hot water flowing again.

I did the whole testing procedure on the elements and thermostats. Both elements were definitely bad, so I drained the tank (it helped to use the pump from my rain barrels), then went to the hardware store and found a full kit with 2 elements and thermostats for $32 USD, so I went ahead and replaced everything. I also bought the special wrench tool to remove the screw-in elements, as it was easier than finding a 1 1/2" socket and cheaper than buying a wrench that would fit.

All-in-all, a relatively fast and inexpensive repair.

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It sure looks like you are on the right track with you assessment. Most electric water heaters have two thermostats, one that controls the main tank element, and one that controls the quick recovery element. There are several very helpful test procedures available online or in the This Old House website. The basics of the test are very simple if you have knowledge of using a volt/ohm meter.(VOM) Be absolutely sure the breaker and power to the heater is off before attempting any OHM measurements on the elements. You will check for proper operation of the therostats. This is usually done with the power on (220VAC) , which can be dangerous if one is not competent in testing live circuits with a volt meter. Basically, you will first verify the 220 volt input at each thermostat. As you draw down the hot water in the tank,replacing it with cold water, the output relays(just a double pole switch for testing purposes) in the thermostats will close, putting 220VAC on the output terminals to the elements. You can also simply turn up the temp on the thermostats and check the outputs as you are now asking for hotter water. Assuming you verify proper operation of the thermostats, the next step will be checking the resistance of the elements with the ohm meter. BE SURE POWER IS OFF. Disconnect the output wires from the thermostats and measure the ohms of each element. Knowing the factory spec is very helpful, however a very high reading or a "0" reading will indicate a bad element. I will try to find a good link and post it in the comments section for you, but a good procedure is easy to find as stated above. http://homerepair.about.com/od/plumbingrepair/ss/element_replace.htm

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The measured resistance should be around 13 ohm for a 4500W/240V element.

    V * V   240V * 240V
R = ----- = ----------- = 12.8Ω
      P        4500W
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    Welcome to Home Improvement. Your derived resistance is correct, but your formula is wrong. Resistance is equal to voltage squared divided by wattage. (Edited.) – Daniel Griscom Oct 25 '17 at 1:02
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Water heater elements fail one of two ways. They fail open, in which case they just stop working. That is the usual mode. Or they fail shorted, meaning current is bypassing some or all of the resistor, flowing more current and usually tripping the breaker.

An open in all elements means no hot water.

An open in some elements means the water heater still reaches full temperature, but takes longer to recover.

In one case a guy had a problem where he could not supply 240V to his water heater. We advised him to connect 120V instead. Half voltage equals half current equals 1/4 useful power, so 4-5x the recover time, but it worked.

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