Finite Control Volume - Conservation of Mass (continued)

Differing forms for Conservation of Mass

There are differing versions of this integral form for conservation of mass depending on the conditions of the flow and of the fluid.

If the flow is steady (no change with time), then ∂/∂t ∫ ρ dV is zero.

cv

Steady flow exists when the flow is established and doesn’t change with time.

Conservation of mass for steady flow then gives the following relation (in integral form)

∫ ρ V . n dS = 0 or ∫ ρ V . n dS  +   ∫ ρ V . n dS = 0

cs cs in cs out

1. Steady, Incompressible Flow ρ = constant (integral form)

ρ ∫ V . n dS = ρ ∫ V . n dS

cs in cs out

Note: This integral form holds for a “nonuniform” velocity profile.

2. Steady, compressible, Uniform Flow (uniform velocity profile across cs)

ρ_in A_in V_in = ρ_out A_out V_out

3. Steady, Incompressible, Uniform Flow (uniform velocity profile across cs)

A_in V_in = A_out V_out

More complicated cases occur in the application of control surfaces and control volumes.

Two of these are:

1. Moving, nondeformable control volume
2. Deformable control volume

Click here for examples.

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