Mass and energy equations for CV; gas filling process

A control volume (CV) is a fixed region of space across whose boundary mass, heat and work may cross. Mass and energy are conserved for it.

General mass conservation (continuity)

Rate of change of mass inside CV = total mass inflow rate − total mass outflow rate

dm_cv/dt = Σ ṁ_i − Σ ṁ_e

General energy conservation (first law for CV)

dE_cv/dt = Q̇ − Ẇ + Σ ṁ_i (h_i + V_i²/2 + g z_i) − Σ ṁ_e (h_e + V_e²/2 + g z_e)

Here Q̇ is heat added to the CV, Ẇ is work done by the CV (shaft work, excluding flow work, which is already inside enthalpy h = u + Pv), and E_cv = m(u + V²/2 + gz).

Gas filling process (unsteady, uniform-state uniform-flow)

A rigid cylinder (tank) of volume V is filled from a large supply line at constant state (P_i, T_i, h_i). The valve is open from t = 0 to t. No gas leaves (ṁ_e = 0).

Assumptions: the state inside the tank is uniform at any instant; the line state h_i is constant; changes in kinetic and potential energy are negligible; the tank is rigid, so W = 0.

Mass equation. Integrate dm_cv/dt = ṁ_i over the filling time:

m₂ − m₁ = m_i

where m₁, m₂ are the initial and final masses in the cylinder and m_i is the mass entering.

Energy equation. Integrate dE_cv/dt = Q̇ + ṁ_i h_i:

m₂u₂ − m₁u₁ = Q + m_i h_i

Substituting m_i = m₂ − m₁:

Q = m₂u₂ − m₁u₁ − (m₂ − m₁) h_i

Special cases

• Adiabatic filling (Q = 0): m₂u₂ − m₁u₁ = (m₂ − m₁) h_i.
• Initially evacuated cylinder (m₁ = 0): u₂ = h_i. For an ideal gas, cv T₂ = cp T_i, so T₂ = k T_i. The final gas is hotter than the line gas, because the flow work of the incoming gas is converted to internal energy.
• Final mass from the gas law: m₂ = P₂V/(RT₂).