Conservation of mass

Conservation of mass
Consider flow through the pipe-work shown in Figure 1, in which the fluid occupies the whole cross section of the pipe. A mass balance can be written for the fixed section between planes 1 and 2, which are normal to the axis of the pipe. The mass flow rate across plane 1 into the section is equal to ?l Ql and the mass flow rate across plane 2 out of the section is equal to ?2 Q2, where ? denotes the density of the fluid and Q the volumetric flow rate.
Thus, a mass balance can be written as
Mass flow rate in = mass flow rate out + rate of accumulation within section
Where Vis the constant volume of the section between planes 1 and 2, and ?av is the density of the fluid averaged over the volume V. This equation represents the conservation of mass of the flowing fluid: it is frequently called the ‘continuity equation’ and the concept of ‘continuity’ is synonymous with the principle of conservation of mass.
In the case of unsteady compressible flow, the density of the fluid in the section will change and consequently the accumulation term will be non-zero. However, for steady compressible flow the time derivative must be zero by definition. In the case of incompressible flow, the density is constant so the time derivative is zero even if the flow is unsteady.
Thus, for incompressible flow or steady compressible flow, there is no accumulation within the section and consequently equation 1.4 reduces to