I am in the process of replacing some old wiring for a friend and when mapping the electric I found that the water heater is only hooked up to 110 volts. I traced the 10/2 wire from the unit back to the service box and found that the black wire from the heater goes to a 20 amp single breaker. The red wire (at the heater) is connected to the white 10/2 line and is grounded at the service box. So this unit has been running on 110v for years. It heats water fine, but has a slow recovery.
At any rate, I am going to replace the heater and run a new 220v. Line to it. But I really want to know how this has been possible for years, and what that would do to the electric bill.

Yes,it does work but has slow recovery. Why run another line? Use the 10/2 with a hot black and a hot white, mark it with black tape. Use a double breaker. PS, it's 240 volts

Could be the water heater is either purchased as 120 volt , see nameplate on unit, or someone installed 120 heating elements.

Could be the unit has original 240 volt heating elements. If this is the case, the elements would get warm, I doubt there would be enough to heat the water to an acceptable temperature.

To calculate the power in watts a standard 240 volt 4500 watt heating element would produce at 120 volts is:

R=E Squared/P
R= 240 x 240/ 4500
R= 12.8 ohms

P=E Squared/R
P= 120 x 120/ 12.8 ohms
P= 1125 Watts

Without getting into the actual BTU calculations, I doubt if 240 volt elements powered by 120 volts will heat the water.

Need to look more closely at the water heater elements to confirm the voltage rating.

If the unit is rated 240 volts, and are 4500 watt elements, and are powered by 120 volts, and is re-wired to be powered by 240 volts, the wattage consumed will rise up the difference between 4500 watts and the 1125 watts, net increase of 3375 watts, and would increase the electric bill.

I did this myself. I had a new 40 gal 220v water heater, but only 110v wiring. The heater did need a switch of its own, but it heated the water to full heat, and gave me good long showers, although the water did cool quicker than if it was on a 220v switch. Yes, it took longer to heat, but there was no-one else waiting for the shower when I was done, so really, that was not any inconvenience at all. I'm sure my electric bill was most likely the same to heat the water as it would have been if it was hooked to a 220v switch. The volts were half of 220, so it took twice as long to heat the water on 110v, using half the amount of volts to complete it's task for twice the time.

Of course the water heated to its full heat... o_O It won't turn the elements off till the temperature gauge reaches the heat you have set it for, even if it takes twice the time to heat it.

I did this myself. I had a new 40 gal 220v water heater, but only 110v wiring. The heater did need a switch of its own, but it heated the water to full heat, and gave me good long showers, although the water did cool quicker than if it was on a 220v switch. Yes, it took longer to heat, but there was no-one else waiting for the shower when I was done, so really, that was not any inconvenience at all. I'm sure my electric bill was most likely the same to heat the water as it would have been if it was hooked to a 220v switch. The volts were half of 220, so it took twice as long to heat the water on 110v, using half the amount of volts to complete it's task for twice the time.

Of course the water heated to its full heat....o_O It won't turn the elements off till the temperature gauge reaches the heat you have set it for, even if it takes twice the time to heat it.There is no such thing as "110v wiring". Wiring has a maximum voltage, how you use it is up to you. You, same as the OP two months ago, could have simply changed the terminations to make it a 240v feed. This assumes it is a dedicated circuit.

If the wiring to a water heater was connected to 120v hot and a neutral, it could have very easily been changed to the proper 240v and a two-pole breaker. Why yours was done this way is anybody's guess.

Also, electric water heaters are resistive heat. The resistance of the element is the constant. As the voltage varies so does the wattage and amperage. It is also not a linear scale, it is a sliding scale.
If you take a resistive heat element and half the voltage going to it, such as this 240v, 4500 watt water heater element, and feed it with 120v, you are halving the voltage, but you will only get 1/4 the wattage, or 1125 watts, out of it.
This is Ohm's Law at work.

At any rate, I am going to replace the heater and run a new 220v. Line to it. But I really want to know how this has been possible for years, and what that would do to the electric bill.

The bottom line answer is, theoretically, there should be no major change in your electric bill using 120 or 240 volt service. 120v would probably take 4 times longer to heat with 1/4 the power, but eventually heats to the same temperature. To prove this, you have to take some measurements which wouldn't be worth the time. The reason you are switching to 240v is to heat faster and maintain heat longer.

So, tkrussel, he's using 4 times more watts but only for maybe 1/4 as long. It's a wash, no?

110/115/120 are essentially the same and not relevant to this problem. Same with 220 vs. 230 vs. 240.

Stanfortyman: What BigIvyGD means is 110v wiring is usually grounded 10/2, in other words, white and black 10 gauge wire with an additional bare or green ground wire. 10/3 wire has the additional red wire for 220/230/240 volt applications or 3-way circuits.

Would either of you electricians use 10/2 with the white as an additional hot for 220/240? Don't do that in my house, I might have to sell it sometime.

Very common to have a 10/2 used on straight 240VAC. White is retasked as a conductor and marked with black / red tape or black / red marker at the tank and panel. Electric H20 and AC at my house, 2 yrs old, are wired that way. 10/3 is more often used for split 120/240VAC applications such as an electric dryer. Would have to be a very special application to see 10/3 used on a 3 way circuit but you won't do it with a regular switch.