Derive equation of continuity

a steady flow process - flow where the flow rate do not change over time - through a control volume where the stored mass in the control volume do not change - implements that inflow equals outflow This statement is called the Equation of Continuity. Common application where the Equation of Continuity are used are pipes, tubes and ducts with flowing fluids or gases, rivers, overall processes as power plants, diaries, logistics in general, roads, computer networks and semiconductor technology and more. The Equation of Continuity and can be expressed as: m = ρi1 vi1 Ai1 + ρi2 vi2 Ai2 +..+ ρin vin Aim = ρo1 vo1 Ao1 + ρo2 vo2 Ao2 +..+ ρom vom Aom (1) where m = mass flow rate (kg/s) ρ = density (kg/m3) v = speed (m/s) A = area (m2) With uniform density equation (1) can be modified to q = vi1 Ai1 + vi2 Ai2 +..+ vin Aim = vo1 Ao1 + vo2 Ao2 +..+ vom Aom (2) where q = flow rate (m3/s) ρi1 = ρi2 = . . = ρin = ρo1 = ρo2 = . .= ρom.