Steady/unsteady system; mass conservation for CV

Steady and unsteady state

• Steady state: the properties (P, T, v, u, h) and flow rates at any point of the system or control volume do not change with time. The mass and energy in the control volume are constant: dm_cv/dt = 0, dE_cv/dt = 0. Example: a turbine running at constant load.
• Unsteady (transient) state: properties or flow rates at a point change with time, so the mass or energy inside the control volume changes. Example: filling or emptying a gas cylinder.

Conservation of mass for a control volume

Mass entering − mass leaving = rate of change of mass inside:

dm_cv/dt = Σṁ_i − Σṁ_e

with ṁ = ρ A V = A V/v. For an unsteady process over a time interval:
m₂ − m₁ = Σm_i − Σm_e

Conservation of energy for a control volume

Energy enters with heat, with mass (specific energy plus flow work P v), and leaves as work and with mass:

dE_cv/dt = Q̇ − Ẇ_shaft + Σṁ_i (h_i + V_i²/2 + g z_i) − Σṁ_e (h_e + V_e²/2 + g z_e)

where E_cv = m u + m V²/2 + m g z of the contents. Here h = u + Pv includes the flow work.

For the unsteady process (uniform state, uniform flow) integrated over time:

Q − W = (m₂e₂ − m₁e₁) + Σm_e (h_e + V_e²/2 + g z_e) − Σm_i (h_i + V_i²/2 + g z_i)

Neglecting KE and PE: Q − W = m₂u₂ − m₁u₁ + Σm_e h_e − Σm_i h_i.

For steady state the right-hand storage terms vanish: Σṁ_i = Σṁ_e and Q̇ − Ẇ = Σṁ_e(h + V²/2 + gz)_e − Σṁ_i(h + V²/2 + gz)_i.