Network model

A network model is a computer model that represents a building as a network of spaces, called nodes, connected by flow paths such as doors, windows, cracks, shafts and ducts. Each node has a single pressure and temperature, and air flows through each path from high to low pressure. By solving the whole network at once, the model gives the pressures in all spaces and the airflows through all paths, driven by wind, fans and temperature differences between inside and outside. Network models are mainly used to study smoke movement through a building and to design smoke control systems that work by pressure differences, such as stair pressurisation.

The best-known example is CONTAM, a public-domain program of the United States National Institute of Standards and Technology (NIST). It was developed for indoor air quality but is widely used for smoke control. NIST used it to evaluate smoke management options for the World Trade Center towers. Network models are less suited to the space around the fire, because each space is treated as fully mixed and the buoyant flow of hot smoke is not captured.

Principle

Emmons describes the airflow through a building as flow through a complex system of pipes and orifices. A network model puts this idea into practice. Every room, corridor or shaft becomes a node, and every leakage path between two nodes becomes a link. Stairwells and other shafts are divided into a vertical series of nodes, one for each floor. The flow through each link follows from the pressure difference across it, using the same orifice relation as for vents and cracks: $$\dot{V}=CA\sqrt{\frac{2\mathrm{\Delta }P}{\rho }}$$

Klote notes that the flow through a path is, in general, proportional to the pressure difference raised to a power between 0.5 and 1. For all paths except extremely narrow cracks, a power of 0.5 is reasonable, which gives the square-root law above. The flow coefficient $C$ depends on the shape of the path, turbulence and friction, and is generally between 0.6 and 0.7. At each node, the mass flowing in must equal the mass flowing out. The HVAC network solver in the Fire Dynamics Simulator, for example, writes this nodal balance as: $$\sum _j{\rho }_j{u}_j{A}_j=0$$

Symbol	Description	Unit
$\dot{V}$	volume flow through a flow path	m³/s
$C$	flow coefficient	–
$A$	flow area (leakage area) of the path	m²
$\mathrm{\Delta }P$	pressure difference across the path	Pa
$\rho$	density of the air entering the path	kg/m³
${\rho }_j$, $u_j$, $A_j$	density, velocity and area of the flow in path $j$ at a node	kg/m³, m/s, m²

Together, the flow relations and the nodal balances form a set of equations with the node pressures as unknowns. Solving this set gives the pressures throughout the building and the flow rates through all paths. Klote compares leakage paths with electrical resistances: paths in parallel add up, and paths in series combine so that the smallest opening controls the flow. Van Rossum stresses that every model is a simplified imitation of reality, and that the modeller must justify what has been left out.

Driving forces

A network model needs the forces that set the air in motion. Klote names stack effect, buoyancy, expansion, wind and the heating, ventilating and air conditioning (HVAC) system as the main driving forces of smoke movement. In a fire they usually act in combination. Van Rossum gives a similar list and notes that ordinary ventilation systems can also strongly influence how smoke spreads.

The stack effect arises when the air in a shaft is warmer or colder than the outside air. Klote gives the example of a building 60 m tall with its neutral plane at mid-height. With −18 °C outside and 21 °C inside, the pressure difference at the top and bottom of a shaft is about 55 Pa. In the 110-storey World Trade Center towers, the stack effect at the winter design temperature reached about 280 Pa at the top and bottom. This is an order of magnitude more than the design pressure difference for unsprinklered buildings.

Wind is harder to describe. Wind pressures vary across each face of a building and fluctuate in time, so the input requires many assumptions. Without wind tunnel data, the pressure coefficients used are at best rough approximations, and this uncertainty outweighs any uncertainty in the model itself. Fans are entered as supply or exhaust flows, so that the model can simulate pressurised stairwells and zoned smoke control, in which the fire zone is exhausted.

Worked example: a pressurised stairwell

Klote gives a hand method for a pressurised stairwell in a building with equal floors and no vertical leakage between floors. The only driving forces are the pressurisation fan and the temperature difference between inside and outside. In network terms, each floor has two paths in series: from the stairwell into the building, and from the building to the outside. Because of the stack effect, the pressure difference across the stairwell wall grows with height: $$\mathrm{\Delta }{P}_{SB}=\mathrm{\Delta }{P}_{SBb}+\frac{by}{1+{(A_{SB}/A_{BO})}^2},b=\frac{gP}{R}(\frac{1}{T_O}-\frac{1}{T_S})$$

The flow of pressurisation air then follows from integrating the flow through all stairwell doors and walls: $$\dot{V}=\frac{2}{3}NC{A}_{SB}\sqrt{\frac{2}{\rho }}\frac{\mathrm{\Delta }{P}_{SBt}^{3/2}-\mathrm{\Delta }{P}_{SBb}^{3/2}}{\mathrm{\Delta }{P}_{SBt}-\mathrm{\Delta }{P}_{SBb}}$$

Symbol	Description	Unit
$\mathrm{\Delta }{P}_{SB}$	pressure difference between stairwell and building at height $y$	Pa
$\mathrm{\Delta }{P}_{SBb}$, $\mathrm{\Delta }{P}_{SBt}$	pressure difference at the bottom and at the top of the stairwell	Pa
$y$	height above the bottom of the stairwell	m
$A_{SB}$	flow area between stairwell and building, per floor	m²
$A_{BO}$	flow area between building and outside, per floor	m²
$b$	temperature factor	Pa/m
$g$, $P$, $R$	acceleration due to gravity, atmospheric pressure, gas constant of air	m/s², Pa, J/(kg·K)
$T_O$, $T_S$	absolute temperature outside and in the stairwell	K
$N$	number of floors	–

Klote's example is a stairwell of 20 storeys of 3.3 m each, with a single-leaf door on every floor. The gap around each door has an area of 0.024 m², and the exterior and stairwell walls are of average tightness. With 560 m² of exterior wall per floor, the leakage to the outside is 0.118 m² per floor. With 52 m² of stairwell wall per floor, the wall leakage is 0.006 m², so the total flow area from the stairwell is 0.030 m² per floor. For an outside temperature of −10 °C and a stairwell at 21 °C, the temperature factor is 1.39 Pa/m.

The designer chooses 20 Pa at the bottom of the stairwell, above a minimum of 13 Pa. At the top, 66.1 m higher, the method then gives 106 Pa, which stays below the maximum of 137 Pa. The required supply of pressurisation air is about 3.9 m³/s.

A calculation in Python reproduces these values: 106.2 Pa and 3.88 m³/s with a flow coefficient of 0.65. Because the building leaks much more easily than the stairwell, about 94 % of the stack pressure acts across the stairwell wall. If the flow area per floor were 0.050 m² instead of 0.030 m², the required supply would rise to about 6.5 m³/s. Klote stresses that these leakage areas are difficult to evaluate and even more difficult to control. For buildings with vertical leakage or open stairwell doors, Klote recommends computer analysis instead. The programs he describes for this purpose are network models.

Such calculations support code requirements for pressurised escape routes. In Australia, AS 1668.1, referenced by the National Construction Code, requires an airflow of at least 1 m/s through each open doorway to the fire-affected compartment. This applies with all doors to that compartment and the main discharge doors of the exits open. The resulting large airflows require pressure relief, because otherwise the pressures could cause excessive door opening forces or stop doors from closing automatically.

Software and applications

In 1982 the National Bureau of Standards, the predecessor of NIST, developed a computer program specifically for smoke control analysis. In 1993 a research project of the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) compared algorithms for network flow. The best one became the core routine of the public-domain program CONTAM96. Some programs of this kind calculate only the steady airflows and pressures, while others also follow the smoke concentrations throughout a building. The NIST investigation of the World Trade Center notes that ASHRAE handbooks recognise CONTAM as an appropriate tool for pressurisation-based smoke management. So does NFPA 92A, a standard of the National Fire Protection Association.

In CONTAM, the building is drawn as zones connected by airflow paths on a sketch that only needs the correct connections. The shape and size of the zones on the sketch do not matter. Results are the same for flow into a long corridor or a square room, as long as the zone volume, temperature and openings stay the same. The program models wind and stack effects and contains simplified and detailed models of HVAC ductwork. Door positions and fan capacities can change over time through schedules.

Network solvers are also built into other fire models. The Fire Dynamics Simulator (FDS), a computational fluid dynamics model, contains an HVAC network solver in which ducts and junctions form a network of nodes and ducts. Its solver is based on MELCOR, a program for nuclear power plant containments, and draws on the Fire and Smoke Simulator (FSSIM), which NIST's reference guide calls a network fire model. FDS also uses such ducts to represent leakage through walls between closed compartments.

Case study: the World Trade Center towers

Network models played a large part in the smoke management studies of the World Trade Center. Between 1976 and 1979, a prototype stair pressurisation system was tested against Local Law No. 5 of New York City. The criteria were 12.5 Pa across closed stair doors with three doors open, 25 Pa with all doors closed and a door opening force of at most 156 N. The system failed to meet them: with three doors open the pressure differences were low, and with all doors closed the opening forces approached the limit. The bombing of 1993 showed how open doors change the network. Many stair doors were held open by evacuating occupants, which increased the airflow up the stairs due to the stack effect, and smoke spread through the stairways was substantial.

After this event, a consultant modelled the smoke movement with SMOKESIM, a precursor of CONTAM, and found general agreement with the reported smoke spread. The study found that exhausting the stairs was not effective, because it increased the stack effect and drew more smoke into the stairs on the lower floors. Venting the elevator shafts improved the overall conditions. Stair pressurisation was feasible but needed pressure relief at the top of each stair segment to keep door opening forces acceptable. Positive pressure could not be maintained with all stair doors open, as during a mass evacuation.

For its investigation after 11 September 2001, NIST rebuilt the model of one tower with 116 levels, more than 3,200 zones and more than 14,000 flow paths. The temperature in the impact zone was varied between 400 and 1,000 °C to bound the results. Doubling or halving the assumed holes between floors showed that the pressure differences were not overly sensitive to their size.

Measurements of airflows and pressure differences made in 1996 were used for calibration. However, no single building configuration could match all the data, because there were too many independent variables, such as leakage, weather, fan settings and changes to floor plans. NIST therefore carried three configurations through the analysis. After 180 model runs, NIST advised against treating any single result as an absolute answer and reviewed the full set of results as a whole.

Limitations

The main simplification is that each zone is fully mixed, with a single temperature, pressure and contaminant concentration at any moment, although the pressure still varies with height within a zone. A hot smoke layer, as found near a fire, is therefore not represented, and the buoyant flow of smoke close to the fire is not captured. CONTAM agrees well with measured airflows at or near ambient temperature, but it has not been validated for buoyant flows. It is therefore best suited to sprinklered or strongly compartmented buildings and to the far field, away from the fire, where the smoke has cooled.

The model does not calculate the temperature of the smoke as it moves through the building. The user must prescribe the temperature of each zone. Near-field conditions must come from other tools, such as a zone model that solves the energy balance of the fire room. NIST notes that feeding the results of one model into another compounds the uncertainty, and that more research on such coupling is needed. Leakage areas are a further source of uncertainty, because they depend mainly on workmanship rather than on construction materials.

Van Rossum warns that a model is only a tool in the engineer's toolkit and does not replace the engineer's own thinking. The user must check whether the scenario lies within the field of application of the model and must understand how the model works. Network models are one of several types of fire modelling tools, which range from tests and simple hand calculations to complex computer simulations.

Network models in evacuation analysis

The term network model is also used in evacuation modelling, with a different meaning. There, the building is a network through which occupants move, and the model looks for the optimal flow of people. According to the International Fire Engineering Guidelines, such models assume that everyone starts to evacuate at the same moment, knows the building very well and always takes the shortest route. They help to find bottlenecks in large buildings, but they omit human behaviour and do not give realistic evacuation times.

See also

• CFAST
• Fire Dynamics Simulator
• Computational fluid dynamics
• Verification and validation
• Sensitivity analysis

References

• Australian Building Codes Board. (2020a). Data sheets: Fire Safety Verification Method (Handbook annex, Print version 1.1).
• Emmons, H. W. (2002). Vent flows. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 2-32–2-41). National Fire Protection Association; Society of Fire Protection Engineers.
• Ferreira, M. J., & Strege, S. M. (2005). Smoke management systems (NIST NCSTAR 1-4D). National Institute of Standards and Technology.
• Klote, J. H. (2002). Smoke control. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 4-274–4-291). National Fire Protection Association; Society of Fire Protection Engineers.
• McGrattan, K., Hostikka, S., Floyd, J., McDermott, R., Vanella, M., Mueller, E., & Paul, C. (2026). Fire Dynamics Simulator technical reference guide: Vol. 1. Mathematical model (6th ed., NIST Special Publication 1018-1). National Institute of Standards and Technology. https://doi.org/10.6028/NIST.SP.1018
• National Research Council of Canada, International Code Council, Department of Building and Housing, & Australian Building Codes Board. (2005). International fire engineering guidelines (2005 ed.). Australian Building Codes Board.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.