Due to stiffness (rapid pressure changes), explicit solvers require very small ( \Delta\theta ). Implicit methods or adaptive step size are recommended. Typical run time for one operating point: 0.5–2 seconds on a modern CPU for a chamber model; full 3D CFD models may take hours.
For many gases (especially refrigerants like R134a or hydrocarbons), ideal gas law fails. A real gas equation like Peng-Robinson or NIST REFPROP correlations is used:
[ p = \fracRTv - b - \fraca(T)v(v+b) + b(v-b) ]
Where ( v ) is specific volume, ( a(T) ) and ( b ) are fluid-specific parameters. Due to stiffness (rapid pressure changes), explicit solvers
Indicated work per cycle: $$ W_ind = \int_V_max^V_min P_in-chamber , dV $$
Indicated power: $$ \dotWind = \fracn \cdot z_160 \cdot Wind $$
Choose model depending on speed, heat transfer, oil-flooding, and desired accuracy. For many gases (especially refrigerants like R134a or
Modern profiles (e.g., SRM “N” profile) minimize blow-hole area and leakage using conjugate curve generation. Mathematical description: $$ x(t) = R \cos t + a \cos(kt) \quad \text(example epicyclic) $$
Friction losses (bearings, oil shear, rotor meshing) are modelled as torque losses ( T_loss ):
[ P_shaft = P_ind + T_loss \cdot \omega ] Friction losses (bearings, oil shear, rotor meshing) are
Mechanical efficiency:
[ \eta_m = \fracP_indP_shaft ]
$$ \eta_v = \frac\dotmdelivered\rhosuction \cdot \dotV_theor $$
Where $\dotVtheor = \fracz_1 \cdot n \cdot Vmax60$ for male rotor speed $n$ (RPM).
Accounting for leakage: $$ \eta_v = 1 - \frac\sum \dotmleak\rhosuction \cdot \dotV_theor $$