The expansion vessel is designed to take care of the excess water produced when you heat water. In normal central heating this expands by 4%. Obviously it needs somewhere to go, rather than blow the boiler (and anyone near it) up. Its also an important safety device, in the normal use of your system it gets used only if other things are not working properly.
The water to water heat exchange occurs in your hot water cylinder. Your boiler heats an amount of water, this circulates through a coil pipe in your cylinder, in doing so it heats the water in your cylinder.
Understanding the functions of these items is only part of a bigger issue. There are many potential mistakes to be made when fitting hot water /central heating systems.
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