Question

By making a mass balance over a volume element (Δr)(rΔθ)(Δz) derive the equation of continuity in cylindrical coordinates.

Answer 1

Let us consider the elementary control volume with respect to (r, 8, and z) coordinates system. Fig. 4.7 is self explanatory.

In r direction

Mass flux in= p [v, r dθ dz]

Mass flux out =

Mass flux in- Mass flux out

Neglecting term containing (d r)² i.e., higher order.

In θ direction

Mass flux in= p [v _θ d r dz]

Mass flux out =

Mass flux in – Mass flux out

In Z direction              Mass flux in= p [v _θ d r dz]

Mass flux out =

Mass flux in- Mass flux out

Total mass flux accumulation in r, θ and z directions

But we have the mass flux accumulation = (p r d θ dz)

Equating both we get,

Dividing both sides by r d r dθ dz we get,

Equation (4.23) is 3-D continuity equation in r, θ, z coordinates for compressible and unsteady flow.

Different forms of 2-D continuity equation under different flow condition (r, e coordinates)

(1) 2-D Compressible unsteady