Liquid partially filling a vessel

The term g(dz/dt) in equation (A 1.3) is disposed of very quickly for a vessel that is always completely filled with either liquid or gas assuming a homogeneous mixture, the centre of gravity will stay in the same position, and so g(dz/dt) = 0. Hence equation (A 1.3) reduces to 

When the liquid in a vessel has a free boundary, the centre of gravity of the liquid will move as the level changes. There will normally be a level controller to ensure that this movement is small, but in fact the term g(dz/dt) can be expected to be insignificant, even 

A formal differentiation of (A 1.12) with respect to time gives  

Now for a liquid, both the specific volume, v, and the specific energy, , are overwhelmingly dependent on temperature, with only a weak dependence on pressure. Hence we may use the approximations  

Substituting from equations (A1.14) and (A1.15) into equation (A 1.3), we have  

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