Steam

Steam distribution network in ring, star or mixed layout: nodal solver with valves, scenarios and automatic DN

The calculator solves a whole steam network — ring, star or mixed — at steady state: mass balance at every tee, momentum in every section by an IAPWS-IF97 march with coupled heat loss, and enthalpy mixing at the nodes, with each consumer valve and the line valves solved by IEC 60534-2-1, in every scenario at once.

When to use

Use it when the question does not fit a single line: a ring header feeding several consumers from both sides, a star with a manifold and branches, or a mixed network with a pressure-reducing valve ahead of a group of consumers. In a ring the flow split depends on the diameters and the consumption, so the pressure at each consumer only comes out of the network solved as a whole. In every scenario the calculator checks whether each consumer valve can hold its set point, the velocity of every section, model validity through Mach, low-flow sections, dead legs and sections that reverse direction between scenarios. In Size mode it picks the smallest DN of the series that meets velocity and pressure in the worst operating scenario.

What the calculator solves

An isolated steam line answers a simple question: with this flow and this diameter, how much pressure reaches the end. In a network the question changes, because the flow of each section is not an input. In a ring, each consumer receives steam from both sides, and the split comes out of the pressure balance: it depends on the diameters, lengths, fittings and where the consumption is in that scenario. The calculator solves this with a steady-state Newton nodal solver, with mass balance at the tees and momentum in every section.

Physics of each section

Every section — supply, header, lines and branches — is calculated by the same coupled march as the steam line: heat loss through the insulation (ISO 12241:1998), energy balance, friction, fittings per piece, elevation and acceleration, with the state updated by IAPWS-IF97 at every segment. Pressure drop and heat loss are coupled, so the steam quality drops along the header and the condensate formed in each section shows in the result. At the tees, the enthalpy is the mix of the incoming streams, propagated in the flow direction.

Valves, separator and network parts

The consumer valve sits next to it, at the branch end (D170); the line valve sits at the end of a section outside any ring, ahead of the branches it feeds (D175), and splits the network into parts, each with a single pressure source. Both follow IEC 60534-2-1. Ahead of every valve LOGOS assumes a separator with a steam trap (D169): the valve receives dry saturated steam and the upstream line carries the consumption plus the drained condensate. The reduction is isenthalpic, so the steam leaves the line valve slightly superheated.

Scenarios, start-up and automatic DN

Maximum operation is always computed; minimum operation is created automatically when some consumer has a minimum flow. Start-up scenarios put cold consumers with the valve 100 % open and solve the network until the flows close. Between the operating scenarios the calculator points out the sections that reverse direction — they go through zero flow in some intermediate condition — and, in every scenario, the low-flow zones and dead legs. In Size mode, sections set to Auto get the smallest DN of the ASME B36.10M series that meets velocity and pressure in the worst operating scenario, with a final check by the complete model.

Declared limits

The calculation is steady-state: it points to where condensate accumulates, but does not compute water hammer. Two-phase steam in the line is treated as a homogeneous mixture, valid for quality from about 0.95. Mach states the validity of the model, not a design limit. A scenario that does not converge, or a network part whose fixed point does not close, is reported as does not meet (D226), never as an approximation. The range goes up to 100 bar(a).

Formulas and fundamentals

Mass balance at the tee (node) Σ ṁ_section (tee) = ṁ_consumption

At every tee without a fixed pressure, the sum of the flows of the sections arriving and leaving (sign = direction) closes with the consumption connected to it. ṁ [kg/h]. It is one of the two families of equations of the Newton solver; the supply has a fixed pressure and the flow injected there is a result.

Momentum in the section (coupled march) P_out = P_in − Δp_friction − Δp_fittings − Δp_elevation − Δp_acceleration

Each section is marched in segments: heat loss through the insulation, energy balance, friction, fittings and elevation, new IAPWS-IF97 state and acceleration term. Heun's predictor-corrector (second order) couples pressure drop and heat loss. The segment count is refined until doubling it changes the outlet pressure by less than 0.0001 bar (D149).

Local loss per fitting Δp_fittings = Σ n_i · K_i · ρ·v²/2

n_i = number of each piece in the section; K_i = local loss coefficient from the same table used by the hydraulics calculators (Crane TP-410 / Azevedo Netto); ρ = steam density [kg/m³]; v = velocity [m/s]. Steam pieces outside the list (separator, Y-strainer) go in as 'Other K'.

Enthalpy mixing at the tee h_mix (tee) = Σ ṁ_i·h_i / Σ ṁ_i

The enthalpy of each node is the flow-weighted average of the incoming streams, propagated in the topological order of the flow direction — not by pressure, which breaks when the flow reverses. h [kJ/kg]. This is how the steam quality degrades along the ring.

Valve flow — IEC 60534-2-1, Eq. N6 W = 31.6·F_P·Kv·Y·√(x·P₁·ρ₁), x ≤ F_γ·x_T, Y = 1 − x/(3·F_γ·x_T) ≥ 0.667

W = mass flow [kg/h]; Kv = flow coefficient (Cv = 1.1561·Kv); F_P = piping geometry factor (without a valve DN, F_P = 1); Y = expansion factor; x = pressure drop ratio; x_T = pressure differential ratio factor; P₁ and ρ₁ = pressure and density at the valve inlet. It is the same physics as the LOGOS steam control valve calculator, applied to each consumer valve and to the line valves.

Valve regulating and separator ahead of it (D169) P₂ = P_set (valve regulating); ṁ_line = ṁ_consumption / x₁

With the valve regulating, its outlet pressure is the set point. Assumption D169: there is a separator with a steam trap right ahead of every valve; the moisture reaching it is drained and the valve receives dry saturated steam. That is why the upstream line carries the consumption divided by the quality x₁ at the valve inlet — consumption plus the drained condensate.

Standards & methods

IAPWS-IF97 (IAPWS R7-97(2012)) — thermodynamic properties of water and steam, including the supplementary backward equations
IAPWS R12-08 (viscosity) and IAPWS R15-11 (thermal conductivity), industrial form
IEC 60534-2-1:2011 — flow capacity of control valves (Eq. N6, Kv form)
ISO 12241:1998 — heat loss through the insulation (LOGOS implements the 1998 edition)
ASME B36.10M — pipe series for the automatic DN (Sch 40, Sch 80 or Sch 80 up to 2 in)
Crane TP-410 / Azevedo Netto — local loss coefficients K per fitting
NR-13 (Brazil) — documentation reference and interface with the legally qualified professional; LOGOS does not perform legal classification nor replace the construction code

Typical reference values

Quantity	Typical range	Note
Velocity limit per section	40 m/s saturated or wet; 70 m/s superheated	LOGOS default criterion (D124), adjustable. Checked independently of pressure. Downstream of a line valve the steam leaves slightly superheated, so the 70 m/s criterion applies.
Mach (validity of the marching model)	≤ 0.3 in range; 0.3–0.5 warning; > 0.5 refined calculation required	LOGOS criterion (D127), not a pipe design limit. In operation, Mach above 0.5 is reported as does not meet; at start-up it is a critical alert (D165). At two-phase points it uses the homogeneous equilibrium speed of sound, HEM (D231b).
Minimum valve opening	10 %	Below it the calculator gives attention: control becomes imprecise. Consider a smaller valve or two in parallel (split range).
Pipe roughness (default)	0.045 mm	Commercial steel condition adopted by LOGOS; for an existing line, set it from the pipe condition or calibrate it with field data.
Discretization tolerance per section	0.0001 bar between N and 2N segments	Adaptive refinement (D149). If a section reaches the segment limit without closing, there is a warning — usually very high velocity.
Fixed-point closure of each network part	1 kg/h or 0.05 %, up to 40 rounds	A network part that does not close within this limit does not support 'meets': the scenario is reported as does not meet (D226).
Calculator range	up to 100 bar(a); up to 40 tees, 60 sections, 40 branches and 10 line valves	Pressure above 100 bar(a) is rejected (D220). Two-phase steam in the line is treated as a homogeneous mixture, valid for quality from about 0.95.

Worked example

Ring header with four tees, two consumers on the ring and a pressure-reducing valve for two heat exchangers

Inputs

Supply (2 boilers in parallel, saturated steam): 12 bar(g)
Ring header: 4 sections of ID 102.26 mm: 540 m in total
Dryer / Evaporator on the ring (max. · min.): 3,000 · 800 / 2,000 · 500 kg/h
Set point / rated Cv — Dryer and Evaporator: 9 / 40 and 8 / 25 bar(g) / —
PRV-01 on the 80 m line: set point / Cv: 5 / 25 bar(g) / —
Exchanger A / Exchanger B after PRV-01 (max.): 800 / 1,200 kg/h
Insulation (λ 0.05 W/(m·K), indoors, 20 °C) · design margin: 50 mm · 0.2 — · bar
Start-up scenario: Dryer + Evaporator cold: 0 bar(g) at the consumer

Results

Supply flow (maximum operation): 7,087 kg/h
Ring split from TE-01 (H-01 / H-04): 3,790 / 3,298 kg/h
Dryer: valve inlet P · req. Cv · opening: 11.45 · 31.0 · 93 % bar(g) · — · —
Evaporator: inlet P · quality · drained condensate: 11.31 · 0.976 · 50.0 bar(g) · — · kg/h
PRV-01: inlet P · req. Cv · outlet T: 11.42 · 15.7 · 170.9 bar(g) · — · °C
Exchanger B: valve inlet P · margin: 4.80 · 0.80 bar(g) · bar
Start-up: Dryer flow · branch velocity: 4,902 (163 %) · 47.7 kg/h · m/s
Verdict: maximum / minimum operation / start-up: meets / meets / attention —

At maximum operation the steam leaves TE-01 along both sides of the ring: 3,790 kg/h through H-01 and 3,298 kg/h through H-04, and the lowest-pressure tee of the header (TE-03) sits at 11.65 bar(g). The supply carries 7,087 kg/h for 7,000 kg/h of consumption; the difference is the condensate drained at the separators ahead of the Dryer, Evaporator and PRV-01 valves (15.8 + 50.0 + 21.4 kg/h). Every valve regulates with the required Cv below the rated one (Dryer 31.0 of 40; Evaporator 18.6 of 25; PRV-01 15.7 of 25), so every consumer meets with room above the 0.2 bar margin. The Evaporator receives steam at quality 0.976, and the separator ahead of its valve drains 50.0 kg/h (D169); at minimum operation its inlet quality drops to 0.905 and the drain rises to 52.6 kg/h with only 500 kg/h of consumption. After PRV-01 the isenthalpic reduction delivers steam at 170.9 °C, slightly superheated. No section reverses between the operating scenarios. At start-up, with Dryer and Evaporator cold and their valves 100 % open, the Dryer draws 4,902 kg/h (163 % of maximum operation) and the Evaporator 3,010 kg/h (150 %); the branches reach 47.7 and 46.0 m/s, above the 40 m/s criterion, and the scenario is reported as attention (D165), with a maximum Mach of 0.10 — within the model range.

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Common mistakes

Checking the ring only at maximum operation. It is between scenarios that sections stop, carry little flow or reverse direction — and that is where condensate accumulates. Enter the consumers' minimum flow (the minimum operation scenario is created automatically) and add the peak scenarios.
Placing the pressure-reducing valve on a ring section. The line valve can only sit on a section outside any ring, at its end, next to the arrival tee and ahead of the branches it feeds; steam passes it only in the from → to direction.
Ignoring start-up with a cold consumer. With the valve 100 % open discharging into a consumer at 0 bar(g), the warm-up flow exceeds the operating flow: in the example below, 163 % at the Dryer and 150 % at the Evaporator, with both branches above the velocity criterion. Opening the valve on a ramp is recommended.
Forgetting that the line ahead of the valve carries the consumption plus the condensate. Under assumption D169 the moisture reaching the valve is drained at the separator; that flow adds to the station trap load and makes the upstream line carry more than the consumption.
Leaving a stopped consumer branch without drainage. A branch without consumption stays pressurized without flow (dead leg): steam condenses and condensate accumulates. Provide a steam trap at the low point, next to the consumer valve.
Reading Mach as a design limit. It states the validity of the marching model; the pipe design criterion is velocity. And a scenario that did not converge is not an acceptable approximation: the calculator marks it as does not meet.
Entering a required pressure on a branch that has a valve. With a valve at the consumer the criterion is the set point at its outlet; the required pressure applies only to a branch without a valve, compared at the branch end with the design margin.

Frequently asked questions

Why are ring and star in the same calculator?

Because the engine is the same. You describe tees, sections between them and branches to the consumers; connecting the last tee to the first closes a ring. The engine finds the rings by itself and solves any topology — ring, star or mixed — with the same nodal solver.

How does the solver solve the network?

By Newton at steady state. The unknowns are the flow of every section (the sign gives the direction) and the pressure of every node without a fixed pressure; the equations are the mass balance at the nodes and the momentum in every section, whose outlet pressure comes from the IAPWS-IF97 march. The Jacobian is by finite differences, with a line search on the residual. The node enthalpy is the mix of the incoming streams, propagated in the flow direction.

Where is the consumer valve and what is the set point?

The valve sits next to the consumer (D170): the branch runs from the tee to it, and the set point is the pressure at its outlet. The consumer meets when the valve regulates — inlet pressure above the set point and required Cv not above the rated one. With the required Cv above the rated one, the valve is limited and the consumer gets the pressure the fully open valve delivers: does not meet. The design margin also applies to consumers with a valve: if P₁ − set point falls below it, the verdict is meets without margin (D232).

What does the calculator do in a start-up scenario?

The chosen consumers are cold at the start-up pressure (default 0 bar(g), explicit assumption) with the valve 100 % open at the rated Cv. The flow depends on the valve inlet pressure, which depends on the network, so the network is solved again until the flows close (D171). Velocity above the operating criterion is attention; Mach between 0.3 and 0.5 is a compressibility warning; above 0.5 it is a critical alert, with no favourable start-up verdict (D165).

How does the automatic DN work?

Sections marked Auto get the smallest DN of the chosen series that meets, in the worst operating scenario, the velocity limit and the pressure at every consumer and line valve (set point plus margin). The flows come from the solved network; with them the network is marched in the flow direction and sections step up for velocity or for the highest loss per metre on the path to the consumer that does not meet, then are pruned. In a ring the solve → size cycle repeats until the DNs stop changing; if they alternate, the larger one is kept. The final result is verified by the complete model, with the D149 refinement. Start-up scenarios do not size.

Does the calculator compute water hammer?

No. The calculation is steady-state. It points to the conditions that favour condensate build-up and water hammer: low-flow zone in the header (for saturated steam, steam flow below k times the condensate formed in the section; for superheated steam, flow below the minimum for the steam to leave still superheated), dead legs and sections that reverse between scenarios (D138). In those sections, ensure drainage.

What happens if the calculation does not converge?

The scenario is not accepted as a result. If the solver finds no solution, the scenario is reported as not converged; if the fixed point between the network and the valve flows reaches 40 rounds without closing, the result shown is the last iteration and the scenario is reported as does not meet (D226). If the march of a section fails (pressure exhausted near sonic choking or outside the IF97 range), velocity, ΔP and heat of that section are not evaluated and the affected consumers are reported as does not meet.

Glossary

Ring header: Distribution header closed in a loop. Each consumer receives steam from both sides, and the flow split changes with the consumption of each scenario.
Tee (node): Network point where sections and branches meet. It is where the mass balance closes and where the streams mix in enthalpy.
Branch: Section leaving a tee and running to a consumer valve. The pressure at the branch end is the pressure at that valve inlet (P₁).
Line valve: Pressure-reducing valve at the end of a section outside any ring, ahead of the branches it feeds. It splits the network into parts, each with a single pressure source; the set point is the pressure at its outlet.
Set point: Pressure at the valve outlet that the valve throttles to hold. At start-up it is replaced by the cold consumer pressure.
Dead leg: Branch with the consumer stopped: it stays pressurized without flow, steam condenses and condensate accumulates. It calls for a steam trap at the low point.
Low-flow zone: Header section with flow so low that the condensate formed in it dominates. Flagged by D138 together with the sections that reverse direction between scenarios.
Design margin: Pressure reserve required beyond the minimum: over the required pressure, without a valve; as P₁ − set point at the valve inlet, with a valve (D137/D232).