Mass and momentum
Displayed element ΔP integrates (f/D + K/L)ρv²/2 along the branch. It excludes hydrostatic and acceleration contributions. Pump pressure rise is reported separately; a pump-only element has zero pipe-friction ΔP. Junction-node losses are reported separately in the port table. Gas equivalent-nozzle restrictions do not resolve an irreversible pressure-loss split and display unavailable.
Signed mass flow is positive from node A to B. Each unconstrained node satisfies Σṁout − Σṁin + demand = 0. Pressure boundaries supply or absorb the remaining balance.
dP/dx + G² d(1/ρ)/dx = −(fD/D + ΣK/L) G²/(2ρ) − ρg dz/dx
G = ṁ/A. Darcy friction uses 64/Re in laminar flow, iterated Colebrook above Re = 4000, and a linear blend between Re = 2300 and 4000. Fittings are specified as explicit K values, distributed along the equivalent branch. Individual bend, tee and area-change nodes apply losses at outgoing ports using their selected reference velocity. Pipe fitting inventories remain separate. Avoid entering the same loss twice.
1/√fD = −2 log₁₀[ε/(3.7D) + 2.51/(Re√fD)]
Energy and heat transfer
d(h + v²/2 + gz)/dx = q′/ṁ
q′ = U′(Tambient − T)
1/U′ = 1/(hᵢ2πrᵢ) + ln(rₒ/rᵢ)/(2πkwall) + ln(rins/rₒ)/(2πkins) + 1/(hₒ2πrins)
Adiabatic: q′ = 0. Isothermal: specified pipe temperature, with required heat calculated from the endpoint energy balance. Heat loss: cylindrical wall and insulation resistances; user-specified film coefficients (external radiation may be included in effective hₒ). A simple exchanger uses fixed duty or UA against a fixed utility temperature.
Mixing uses incoming component mass rates and mass-weighted enthalpy. Junction mixing transfers incoming enthalpy and kinetic energy to the node reference port. Port static pressure is P_ref + α ρv_ref²/2 − α ρv_port²/2 − Δp_loss. The reference is the largest-flow incoming port, or largest-flow outgoing port at a supply. Junction transfer uses a low-Mach approximation and is flagged above M = 0.3; no high-speed tee or transition choking model is implied. Liquid Cp is a common user input, including mixed-liquid cases.
Properties and equipment
Gas: ideal gas or Peng–Robinson with kᵢⱼ = 0, ideal-gas Cp polynomials from ThermoMix, and PR enthalpy departure. Density and enthalpy derivatives are numerical. Liquids: additive specific volumes, logarithmic viscosity mixing, or measured constant density/viscosity. No excess enthalpy or activity-coefficient model.
Pump is a lumped two-port machine with no pipe section, fittings or pipe heat transfer. Its elevation sets both connection-node elevations; connected pipes carry the elevation changes. Legacy pump pipe properties are ignored. Pump curve: actual suction volume flow versus head. Fan curve: suction volume flow versus pressure rise at reference density. Compressor curve: suction volume flow versus pressure ratio, with ideal-gas isentropic work/efficiency. Speed scaling uses affinity approximations. Curve extrapolation is flagged and invalidates design acceptance.
Valve Cv uses liquid-equivalent resistance, with density updated along the branch. Gas valves and orifices use a compressible equivalent-nozzle law, with upstream total pressure/temperature estimated from the adjoining pipe. Orifices use Cd × bore area; valves use effective area from Cv. Gas k and Z are frozen at the restriction inlet. Vendor xT, pressure recovery and IEC sizing are not included. Regulators throttle excess available pressure; fully open behaviour applies when the setpoint cannot be reached.
Choked-flow solver and scope
Each passive gas branch is solved as a pressure-to-mass-flow characteristic. Pipe capacity is found by bracketed integration to the critical momentum/energy condition. Adiabatic pipes approach sonic flow; isothermal pipes reach their isothermal critical condition (approximately M = 1/√γ for an ideal gas). Heat-loss pipes use the coupled property derivatives. The network solves continuity with the resulting capped flow.
r* = [2/(γ+1)]γ/(γ−1)
ṁ* = CdA P₀ / √(Z RsT₀) · √γ · [2/(γ+1)](γ+1)/(2(γ−1))
For restrictions above r*, ṁ = CdA P₀ √{2γ/[Z RsT₀(γ−1)] × [r2/γ − r(γ+1)/γ]}; below r*, use the capacity plateau. PR cases use a frozen-Z restriction approximation; this is not a rigorous real-gas critical-nozzle calculation. Pipe profiles stop at the critical exit state. A downstream node represents a mixed receiver; jet kinetic energy is included in its incoming total enthalpy. Critical pressure and downstream receiver pressure are reported separately. An infeasible fixed demand is rejected by the mass/flow residuals, not silently clipped.
Visual output: choked items override all colour maps in red and carry a CHOKED note, limiting mass flow and critical pressure. If the network has not converged, these remain provisional capacity indications.
Liquid reservoirs
Fixed surface pressure and elevation define a steady-state liquid reservoir. Connected pipes can share a take-off elevation or use individual depths/elevations and entry/exit losses. Hydrostatic pressure at rest is P_surface + ρg(z_surface − z_takeoff). For discharge into a pipe, subtract (1 + K_entry)ρv²/2. For inflow from a pipe, add (K_exit − 1)ρv²/2. Density comes from the selected liquid properties. Pipe elevation changes start at the take-off, avoiding double-counting reservoir depth. The surface is treated as negligible velocity. No level depletion or gas-space compression is simulated.
Junction geometry and losses
Bends use a specified K, or user K90 scaled by angle/90. Radius/diameter is recorded for traceability; it does not generate an automatic correlation. Tees use explicit outgoing-path K values and assigned A/B/C ports; geometry angle is recorded and does not alter manual K. Conical area changes use Crane-style angle/area-ratio loss coefficients; abrupt selects the 180° limit. These are not vendor-specific or detailed AFT tee correlations.
For β = small/large bore and s = sin(θ/2), expansion K (upstream basis) = (1−β²)² × (2.6s for θ ≤ 45°, otherwise 1). Contraction K = (1−β²)/β⁴ × (0.8s for θ ≤ 45°, otherwise 0.5√s). Manual K can use incoming or outgoing velocity.
AFT documentation: Crane conical area-change loss model. Abrupt contraction here uses the Crane 180° limit, not AFT’s alternative Idelchik abrupt correlation.
Drawing symbols and orientation
General liquid pump: circle with filled delivery triangle (2301). General compressor: circle with open delivery triangle (2302). Symbol basis: ISO 10628-2:2012 process diagrams and its ISO 14617 references. Fan, valves, heat exchanger and strainer use schematic vector outlines. Pressure/flow boundaries are model annotations. This is a process schematic, not a certified fabrication drawing.
Equipment symbols follow the local pipe tangent in the defined A-to-B direction. Flow arrows follow the calculated mass-flow direction, independently of static-pressure rise or fall. Connected bends and tees align their arms with the isometric pipe routes; assigned tee ports A/B form the run. Moving nodes updates the routes and symbols. Disconnected junctions use a default isometric pose until connected. Visual routing does not change hydraulic lengths, elevations or entered bend angles.
ISO 10628-2 reference
Scope and references
Steady-state, Newtonian, single-phase networks with subcritical and choked pipe/restriction branches. Downstream supersonic jets and shocks are not resolved. Fans and compressors retain their curve models and are not sonic-capacity maps. Pressure range defaults to 0.001 bara to 20 barg. Exact vacuum is excluded. No water hammer, tank inventory dynamics, two-phase hydraulics, solids, non-Newtonian flow, surge or rotating-machine stall maps.
Phase screening retains the supplied approximate vapour-pressure flash. The embedded package is not IAPWS steam tables; near-saturation steam and nonideal liquid mixtures require independent property validation. Red or amber flags must be resolved before using a result for design.
Built-in verification
The page's verification cases run as the engine's tests with every release, not on request: laminar Darcy f at Re = 1000; Hagen–Poiseuille pressure drop in one pipe (within 0.01 Pa) and its flow; the water loop's and the compressed-air network's mass balances (within 10⁻⁶ kg/s) and convergence; the choked orifice converging, flagged and holding its flow from atmospheric to 30 kPa back pressure; the critical ratio for γ = 1.4; and the °F and psig conversions. These checks establish numerical and unit-conversion behaviour, not full validation of every property correlation or item of equipment.
