| Symbol | Description | Units |
|---|---|---|
| \(p\) | Pressure (gauge or absolute) | Pa |
| \(\rho\) | Density | kg/m³ |
| \(u\) | Velocity | m/s |
| \(U\) | Volume flow rate (\(uA\)), derived | m³/s |
| \(m\) | Mass (per cell) | kg |
| \(p_{\text{mom}}\) | Momentum in cell (\(\rho u A \Delta x\)) | kg·m/s |
| \(E_{\text{tot}}\) | Total energy in cell (\(\rho E A \Delta x\)) | J |
| \(U_{\text{int}}\) | Internal energy (0‑D volume) | J |
| \(e\) | Specific internal energy | J/kg |
| \(E\) | Specific total energy (\(e+u^2/2\)) | J/kg |
| \(T\) | Temperature | K |
| \(c\) | Speed of sound (\(\sqrt{\gamma R_g T}\)) | m/s |
| \(c_0\) | Ambient speed of sound | m/s |
| \(\rho_0\) | Ambient density | kg/m³ |
| \(A_{\text{cell}}\) | Cell cross‑sectional area | m² |
| \(A_{\text{face}}\) | Face cross‑sectional area | m² |
| \(A\) | Cross‑sectional area (general) | m² |
| \(u_{\text{face}}\) | Velocity at face (for boundary ports) | m/s |
| \(\gamma\) | Ratio of specific heats (\(c_p/c_v\)) | – |
| \(R_g\) | Specific gas constant | J/(kg·K) |
| \(c_p, c_v\) | Specific heats at constant pressure/volume | J/(kg·K) |
| \(\dot{m}\) | Mass flow rate | kg/s |
| \(h\) | Specific enthalpy (\(c_p T\)) | J/kg |
| \(\lambda\) | Lagrange multiplier (junction constraint) | – |
| \(\mathbf{J}\) | Skew‑symmetric interconnection matrix | – |
| \(\mathbf{R}\) | Dissipation matrix | – |
| \(R_g\) | Specific gas constant (note: distinct from \(\mathbf{R}\)) | J/(kg·K) |
| \(\mu\) | Dynamic viscosity | Pa·s |
| \(k_f\) | Fluid thermal conductivity | W/(m·K) |
| \(D_h\) | Hydraulic diameter | m |
| \(\epsilon\) | Pipe wall roughness | m |
| \(\text{Re}\) | Reynolds number | – |
| \(\text{Nu}\) | Nusselt number | – |
Note: The state variables \(m, p_{\text{mom}}, E_{\text{tot}}\) are the primary conserved quantities in a pipe cell. Density, pressure, velocity, and temperature are derived from these. The symbol \(R_g\) is used for the gas constant to avoid confusion with the dissipation matrix \(\mathbf{R}\).