Pseudo-Transient Fire Simulation

Pressure Network Behaviour Under a Growing Fire

Abstract

A single steady-state pressure solve (see Nodal Pressure Network Analysis) answers the question "what are the pressures right now, given these conditions?" It cannot, by itself, show how a fire develops from ignition to full involvement, or at what moment the ventilation system is overwhelmed. This paper describes SootSafe's pseudo-transient approach: a sequence of 600 steady-state solves, one per simulated second, driven by a growing thermal source. The result is a time-resolved picture of pressure, flow, and temperature throughout the building as the fire evolves — including the precise second at which supply air reverses and the moment at which any pressure limit is breached.

1. The Need for Time Resolution

Fire-safety ventilation systems are designed to hold protected spaces above a pressure threshold — typically 10–50 Pa gauge above the fire room — long enough for evacuation and firefighting. Whether a system achieves this depends critically on the rate at which the fire grows. A system that is adequate at peak fire intensity may already have failed at the 90-second mark when the fire was still modest but the fans had not yet established the required pressure gradient.

Two questions that require time-resolved analysis:

When does a supply duct first reverse (backflow), indicating that the fire room pressure has overcome the supply fan?
For how long does the fire room exceed the allowable pressure limit on the door or partition?

Neither question can be answered by a single steady-state solve.

2. The Quasi-Static Assumption

True transient simulation of compressible flow in a building network is computationally expensive and unnecessary for the phenomena of interest. Fire-driven pressure changes in ventilated buildings evolve over tens of seconds — far slower than the acoustic speed at which pressure perturbations propagate through the ductwork. The network therefore reaches local pressure equilibrium within each second of fire growth.

This justifies the quasi-static (pseudo-transient) approach: at each simulated second, the network is treated as being in steady-state, with updated thermal boundary conditions. The coupling is one-way and explicit: the fire model drives the pressure model, not vice versa.

The simulation runs from $t = 0$ to $t = 600\,\text{s}$ in one-second steps, calling the full Newton–Raphson pressure solver at each step.

3. Fire Growth Models

3.1 The α·t² Power Law

For most occupancy types, fire heat release rate (HRR) is expected to follow a power-law growth phase:

$\dot{Q}(t) = \min\!\left(\alpha \cdot t^2,\; \dot{Q}_{\max}\right) \qquad \text{[kW]}$

where $\alpha$ [kW/s²] is the fire growth coefficient and $\dot{Q}_{\max}$ is the peak HRR at which the fire saturates (fuel-limited). The growth phase ends when $\alpha t^2$ reaches $\dot{Q}_{\max}$ ; thereafter the HRR is constant.

SootSafe provides four standard presets aligned with the Swedish/Nordic market and EN 1991-1-2:

Preset	α (kW/s²)	Typical occupancy
Slow	0.003	Storage of non-combustibles
Medium	0.012	Standard office, residential
Fast	0.047	Retail, theatre stages, upholstery
Ultra-Fast	0.188	Chemicals, high-rack storage

3.2 The ISO 834 Standard Fire

For structural fire resistance testing, ISO 834 prescribes a specific gas temperature–time curve rather than an HRR:

$T(t) = T_{\text{amb}} + 345 \cdot \log_{10}\!\left(8\, t_{\min} + 1\right) \qquad \text{[K]}$

where $t_{\min} = t / 60$ is time in minutes. This curve rises steeply in the first minutes and grows logarithmically thereafter. In this mode, the room temperature is a prescribed input to the pressure solver rather than a derived quantity; $dT/dt$ is estimated by finite difference between consecutive seconds.

3.3 Natural Fire Scenario — Three-Phase HRR Model

Neither the α·t² curve nor ISO 834 models the decay phase that occurs in a real fire as fuel is consumed. The full natural fire scenario ("naturligt brandförlopp" per Swedish practice, formalised in EN 1991-1-2 Annex A) comprises three phases: growth, fully developed, and decay. SootSafe implements this as FireGrowthCurve::NaturalFire, using a HRR-based three-phase model.

Growth phase — identical to α·t²:

$\dot{Q}(t) = \min\!\left(\alpha t^2,\; \dot{Q}_{\max}\right) \qquad [\text{kW}]$

The peak HRR is reached at $t_{\text{peak}} = \sqrt{\dot{Q}_{\max} / \alpha}$ .

Plateau phase — HRR is constant at $\dot{Q}_{\max}$ for duration $t_{\text{plateau}}$ .

Decay phase — HRR falls linearly from $\dot{Q}_{\max}$ to zero over $t_{\text{decay}} = t_{\text{plateau}}$ (symmetric).

The plateau and decay durations are derived from the total available fuel energy so the profile integrates exactly to the specified fuel load:

$E_{\text{total}} = q_f \cdot A_f \cdot 1000 \qquad [\text{kJ}]$

$E_{\text{growth}} = \frac{\alpha \cdot t_{\text{peak}}^3}{3} \qquad [\text{kJ}]$

$t_{\text{plateau}} = \frac{2\,(E_{\text{total}} - E_{\text{growth}})}{3\,\dot{Q}_{\max} \cdot 1000} \qquad [\text{s}]$

Additional inputs required:

Field	Description	Typical values (Swedish BBR / EN 1991-1-2 §E.4)
`fuel_load_mj_m2`	Fuel load density $q_f$ (MJ/m²)	Office: 400–600; Residential: 300–500; Retail: 600–900
`floor_area_m2`	Room floor area $A_f$ (m²)	From building geometry

The decay phase is important for smoke-control analysis: once HRR falls, the expansion-driven pressure gradient relaxes. A system that barely holds the required pressure differential during peak fire may recover later in the burn, or a system that manages peak conditions may still fail if intermediate states are unfavourable. Modelling only growth and plateau (the AlphaT2 variant) is conservative with respect to peak pressure but cannot answer how long the system is stressed.

This variant reuses the same adiabatic well-stirred dT/dt formula as AlphaT2 (§4.1), with HRR(t) now time-varying through all three phases. No new thermal model is introduced; the Wickström wall heat-loss correction (§4.1) applies equally.

Note: The EN 1991-1-2 Annex A temperature-prescribed parametric fire (requiring the opening factor $O = A_v \sqrt{h_v} / A_t$ ) is not yet implemented. It is planned as a future variant once the opening factor is added to the network model.

4. The Adiabatic Well-Stirred Room Model

4.1 Thermal State

For the α·t² curve, the fire room temperature evolves by integrating the energy balance of a well-stirred, adiabatic control volume:

$\frac{dT}{dt} = \frac{\dot{Q}(t) \cdot 1000}{\rho_{\text{amb}} \cdot V \cdot c_p}$

Symbol	Value / Source
$\rho_{\text{amb}}$	1.204 kg/m³ (air at 20 °C, ISO 10294-3)
$V$	room volume (m³, user input)
$c_p$	1 005 J/(kg·K) (specific heat of air at constant pressure, ISO 10294-3)

The denominator $\rho_{\text{amb}} \cdot V$ is the initial mass of air in the room. Using ambient density (rather than the instantaneous density, which decreases as the room heats) is consistent with the adiabatic, well-stirred approximation: the model tracks the bulk temperature of the gas layer, not its composition.

Temperature is advanced by forward Euler integration:

$T(t + 1) = T(t) + \frac{dT}{dt}(t) \cdot \Delta t, \qquad \Delta t = 1\,\text{s}$

This is first-order accurate in time. At one-second resolution, the truncation error is negligible relative to the uncertainty in $\alpha$ and $\dot{Q}_{\max}$ .

Future extension: The FireNodeParams struct reserves surface_area_m2 and wall_material fields for a future Wickström wall heat-loss correction:

$\dot{Q}_{\text{loss}}(t) = b \cdot A \cdot \frac{T(t) - T_{\text{amb}}}{\sqrt{t}}$

where $b = \sqrt{\rho c \lambda}$ is the thermal inertia of the wall material. Subtracting this from $\dot{Q}(t)$ before computing $dT/dt$ would give a more conservative (lower) temperature rise. The current adiabatic model is a conservative upper bound on expansion flow.

4.2 Expansion Flow — Coupling Fire to Pressure Network

A growing fire heats the air in the room, causing it to expand. In a sealed room this would raise the pressure; in a connected building this expansion drives air outward through all available paths. The expansion appears in the pressure-network model as a forced mass source at the fire node.

Differentiating the ideal gas law at constant pressure gives the expansion volumetric flow, which multiplied by ambient density yields the expansion mass rate (ISO 10294-3):

$\dot{m}_{\text{exp}}(t) = \frac{V \cdot \rho_{\text{amb}}}{T_{\text{amb}}} \cdot \frac{dT}{dt}(t) \qquad \text{[kg/s]}$

This is injected as a negative target_mass_rate at the fire node — a source term in the mass balance (see Nodal Pressure Network Analysis, §2.1). Every other node and branch then responds through the normal pressure-flow physics.

This formulation is physically equivalent to placing a virtual fan at the fire node whose output equals the thermal expansion rate, without introducing any artificial component into the network.

5. The Simulation Loop

At each second $t \in \{0, 1, \ldots, 600\}$ :

Compute fire state: evaluate $\dot{Q}(t)$ , $dT/dt$ , $T(t)$ , and $\dot{m}_{\text{exp}}(t)$ from the growth model.
Update network: set the fire node's temperature to $T(t)$ and its target mass rate to $-\dot{m}_{\text{exp}}(t)$ .
Solve: call the Newton–Raphson pressure solver (see Nodal Pressure Network Analysis, §4) to find the equilibrium pressure field at this instant.
Record events: test for backflow on supply ducts and pressure limit exceedances (§6).
Update probes: advance any virtual detector probes attached to the room (see Detector Response Prediction).
Store snapshot (if $t$ falls on the configured sample interval).
Advance temperature: apply forward Euler to obtain $T(t+1)$ .

Non-convergence at an individual timestep is tolerated — the snapshot for that second is omitted and the loop continues. This prevents a single ill-conditioned instant (e.g. at ignition when $\dot{m}_{\text{exp}} \approx 0$ ) from aborting the entire simulation.

6. Event Detection

6.1 Backflow on Supply Ducts

Supply air ducts are explicitly tagged in the network model (via a SupplyDuct component marker). At each timestep, the solved volumetric flow on each tagged edge is checked. The first second on which flow is negative (reversed against the supply direction) is recorded as a backflow event. Each edge produces at most one event regardless of how long backflow persists — the event marks the onset, not the duration.

Backflow on a supply duct indicates that the fire room pressure has exceeded the supply fan's static pressure capability: the system can no longer deliver clean air to the affected zone.

6.2 Pressure Limit Exceedance

At each timestep, the fire room gauge pressure is compared against a user-specified door pressure limit (typically 25–50 Pa, representing the force required to open a door against differential pressure). Every second on which the fire room pressure exceeds this limit is recorded as a pressure event, at full one-second resolution.

Sustained exceedance of the door limit implies that occupants may be unable to open doors in the fire compartment, potentially trapping them.

7. Output

7.1 Snapshots

The full pressure map and flow field at every sampled timestep, together with fire state variables:

Field	Description
`t_s`	Simulation time (s)
`temperature_k`	Fire room temperature (K)
`hrr_kw`	Heat release rate (kW); zero for ISO 834
`m_exp_kg_s`	Expansion mass flow injected (kg/s)
`node_pressures_pa`	Gauge pressure at every node (Pa)
`edge_flows_m3s`	Volumetric flow on every edge (m³/s)

Snapshots are downsampled to a configurable interval (default: every 5 seconds, giving 121 snapshots over 600 seconds). Events are always recorded at full 1-second resolution regardless of the snapshot interval.

7.2 Events

backflow_events: one entry per supply duct edge, recording the first second of reversal and the edge label.
pressure_events: one entry per second the fire room pressure exceeds the door limit.
activation_events: one entry per virtual detector probe that activates (see Detector Response Prediction).

8. Assumptions and Limitations

Assumption	Consequence	Direction
Quasi-static pressure	Acoustic transients and pressure wave propagation are not modelled; valid when fire growth time scale >> network acoustic time scale	Neutral — acoustic equilibration occurs in ~0.1 s; fire-driven pressure changes evolve over tens of seconds
Well-stirred room	Uniform temperature throughout the fire compartment; no vertical stratification or two-zone layering	Neutral for expansion flow — total expansion depends on HRR, not temperature distribution; mildly non-conservative for ceiling jet temperature estimates (the real hot layer is hotter than the room average)
Adiabatic	No heat loss through walls, floor, or ceiling in the current model; the adiabatic approximation gives a conservative (higher) temperature and expansion flow	Conservative — omitting wall heat loss over-predicts room temperature and expansion-driven pressure rise; this is the dominant conservative margin in the overall analysis chain
Ambient initial density	$\rho_{\text{amb}} \cdot V$ is computed once at $t = 0$ ; the reduction in air mass as the room heats is not tracked	Mildly non-conservative — as hot air is expelled the remaining air mass decreases, so the true temperature rise rate slightly exceeds the modelled rate; the discrepancy grows late in the simulation
Fixed network topology	Doors, dampers, and openings remain in their initial state throughout the simulation; the effect of door opening or fire-barrier failure is not modelled	Context-dependent — conservative for fire room over-pressurisation analysis (real doors open and relieve pressure); potentially non-conservative for protected-space pressurisation analysis (assumes fans and barriers remain operational throughout)
Single fire room	One active fire source per simulation run	Non-conservative if simultaneous ignitions in multiple rooms are a credible scenario for the occupancy under review
Forward Euler	First-order time integration; adequate at 1-second resolution for the time scales of interest	Mildly non-conservative in principle — forward Euler underestimates a monotonically growing rate; negligible at 1-second resolution for the time scales of interest
Natural fire decay phase	Only α·t² growth and a constant plateau are modelled; the cooling phase (HRR falling as fuel is consumed) is absent — see §3.3	Non-conservative for scenarios where the fire burns out before evacuation is complete — the model sustains peak expansion pressure indefinitely rather than allowing it to decay
Natural fire decay phase	Implemented via `FireGrowthCurve::NaturalFire` (§3.3) — three-phase profile (growth, plateau, linear decay) with durations derived from fuel load and floor area. The `AlphaT2` variant still omits decay for cases where a perpetually growing fire is the design assumption	Handled for `NaturalFire`; `AlphaT2` remains conservative (no decay) by design

9. Relationship to Manual Practice

Manual transient analysis in traditional tools requires re-entering network boundary conditions at each time point of interest — a process that discourages fine time resolution and leads engineers to check only a few selected instants. The pseudo-transient loop automates this entirely: 600 pressure solves run in a fraction of a second, and the results are indexed so the engineer can scrub through the full 10-minute scenario at any granularity.

10. Standards Referenced

Standard	Scope
ISO 10294-3	Expansion flow formula; air physical constants
EN 1991-1-2	α·t² fire growth curves and occupancy classification (§3.1); Annex A three-phase HRR natural fire model (§3.3); Annex A temperature-prescribed parametric fire (planned)
ISO 834	Standard temperature–time curve for fire resistance testing
Boverket (BBR)	Swedish building code — door pressure limits and evacuation requirements

11. Summary

The pseudo-transient simulation extends the steady-state pressure solver into the time domain by coupling a fire thermal model to the pressure network at one-second resolution. The key elements are:

α·t², ISO 834, or three-phase natural fire curves prescribe how the fire room temperature evolves.
The adiabatic well-stirred model converts HRR to a temperature rise rate.
The expansion flow formula (ISO 10294-3) translates that temperature rise into a mass source that drives the pressure network.
600 sequential pressure solves produce a complete time history of pressures and flows across the building.
Backflow and pressure events are detected automatically at one-second resolution, pinpointing exactly when and where the ventilation system is first overwhelmed.