General energy equation for control volume

Derivation

Consider a control volume (CV) with inlet i and exit e, over a time dt. Mass δm_i enters and δm_e leaves; heat δQ is received and shaft work δW_s is done by the CV.

   m_i, P_i, v_i  ->[   CV   ]-> m_e, P_e, v_e
                     ^Q    |W_s

Step 1: Apply the first law to a fixed mass (closed system). At time t this mass is the CV contents plus the mass δm_i about to enter; at time t + dt the same mass is the CV contents plus the mass δm_e that has just left.

Energy at time t: E_cv(t) + δm_i e_i
Energy at time t + dt: E_cv(t+dt) + δm_e e_e

where e = u + V²/2 + gz. The first law for this closed system:

δQ − δW_total = [E_cv(t+dt) + δm_e e_e] − [E_cv(t) + δm_i e_i]

Step 2: Work terms. The total work includes the shaft work and the flow work needed to push mass in and out:

δW_total = δW_s + δm_e P_e v_e − δm_i P_i v_i

Step 3: Substitute and collect terms:

δQ − δW_s = dE_cv + δm_e (e_e + P_e v_e) − δm_i (e_i + P_i v_i)

Since u + Pv = h:

δQ − δW_s = dE_cv + δm_e (h_e + V_e²/2 + g z_e) − δm_i (h_i + V_i²/2 + g z_i)

Step 4: Divide by dt (and sum over all inlets and exits):

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

This is the general energy equation. For steady flow dE_cv/dt = 0, and with one inlet and one exit (ṁ constant):

q − w = (h₂ − h₁) + (V₂² − V₁²)/2 + g (z₂ − z₁)