Continuity equations (often known better as mass, energy or mole 'balances') form the basis of all dynamic models. A continuity equation is simply a balance of some conserved quantity (mass, component mass, moles , component moles, energy or momentum) over some capacity capable of storing that quantity. The general form for a continuity equation is:
Rate of Accumulation of quantity in capacity = rate of quantity into capacity - rate out of capacity + rate of generation of quantity within capacity.
Most mistakes in forming continuity equations occur in the rate of accumulation term - these are usually caused by people trying to jump to the final answer in one go. I find the best way to sort out the accumulation term is to write it as
Rate of accumulation = d( Amount of 'stuff' held up in the capacity)/dt
where the 'stuff' is the quantity the balance is being carried out on.
You can only form continuity equations based on the conserved quantities listed above. You can't create volume balances, temperature balances, pressure balances, etc. You may be able to derive equations that tell you how quickly the liquid volume, or pressure, or temperature, changes in a capacity, but these are always derived from one of the conserved quantities.
Continuity equations always have units of 'unit of conserved quantity'/'unit of time'. If you are carrying out a mass balance all the terms in your equations should have units like kg/hr, or g/s, or lbs/day, etc. MAKE SURE THAT YOUR UNITS ARE CONSISTENT THROUGHOUT THE EQUATION. If you use 'kg' in one term you must use this unit in all the other terms. The same applies to the time unit you use: if 'min' are used in one term it must be used in all of the terms (mixing time units in an equation is a common mistake). Also, if your model involves multiple continuity equations, make sure that the units are the same across ALL the equations.