No, a central air conditioner does not need a water supply.
While some cooling systems, like swamp coolers, actively use water, a central air conditioner operates differently. It isn't connected to a water supply line or a water tank. According to the reference, if you see water around systems like water heaters, boilers, or even swamp coolers, it's expected because those systems use water. However, this is not the case for central air conditioners.
