I've just installed a 40 gallon electric hot water tank into a residence. When you run the water it starts at a nice hot temp and shifts to a cooler temp. This happens back and forth from hot to warm then back to hot and so on... This happens at the basin and the tub/ shower which both have separate take offs from the main lines below. Any suggestions? They are on a well system and it seems to go cooler as the pump kicks on, and warms back up as the system looses pressure. Help please. Greg.

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I've just installed a 40 gallon electric hot water tank

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Does that mean that you did the installation? If so, did you at any time heat the cold water input pipe close to the water heater?

I did the installation, and I used a 3/4" Female adapter with a union which I sweated together prior to tightening it on the cold inlet. There was no heat added to the inlet on the tank which would have probably compromised the dip tube. I should add that I removed the drain at the bottom of the tank and installed a tee to accommodate a copper heat exchanger which will be hooked up to a wood fired boiler. One side of the tee is now the drain, and the other goes up to the cold inlet which is also teed' off. One side of that tee accommodates the heat exchanger and the other side goes into the cold inlet of the tank. The exchanger is valved off, and not in use. Im wondering if I need a check valve between the cold isolation valve and the tee which branches to the exchanger and inlet tube? Of course I would also add a expansion tank if I were to do so. Let me know your take on this scenario. Thanks

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Im wondering if I need a check valve between the cold isolation valve and the tee

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I would say so. As water make up water in the cold water input pipe flows past the tee it is at a lower pressure than the water in the pipe from the drain (see the Bernoulli principle). That would cause the water from the bottom of the water heater to be recirculated back to the top of the tank.
Not sure how that would cause the hot water out to alternate between hot and cold unless so much turbulence in the tank that the in coming cold water is moving to the top of the tank.
Easy enough to install check valve and find out.