Contents
A hydraulic evacuation model represents the movement of people along escape routes through relationships between crowd density, speed, flow and effective width. Nelson and Mowrer describe an approach that generally assumes simultaneous departure, uninterrupted movement decisions and occupants able to maintain the stream's pace.[1] It provides a reproducible estimate of movement under those assumptions. The result does not automatically include detection, interpretation of warnings or the time before people begin moving.[2]
The method connects walking speed with density and specific flow, then applies route width to calculate throughput. It can support hand calculations for simple routes or more extensive network calculations.[3] Its value lies in making capacities, bottlenecks and queue formation explicit. Interpretation still requires evidence about the occupants and their response, because actual evacuation can depart from the idealised flow assumptions.[1]
What the hydraulic representation includes
The hydraulic analogy treats a group as a stream passing through connected route elements. Corridors, doors and stairs have geometries that influence the stream's speed and flow. The approach focuses on quantities that can be expressed consistently across these elements. It does not imply that people behave as an actual liquid.[1][3]
Nelson and Mowrer identify three common assumptions: occupants begin together, decisions do not interrupt the flow, and most people can keep pace with the group. These assumptions simplify the movement problem. They also define important limits on the interpretation of its output. A population requiring assistance cannot automatically be represented by the same unrestricted stream.[1]
The calculated movement duration contributes to required safe egress time. Other components account for detection, notification, decisions and the interval before evacuation starts. A short flow calculation is therefore compatible with a much longer total response. Comparing its movement output directly with fire development would omit those additional periods.[1]
The source distinguishes modelled movement time from actual evacuation performance. It describes the calculation as a reproducible baseline for examining routes or changes to a system. Actual movement often takes longer because of interruptions and uneven use, but the source also recognises the opposite possibility. The hydraulic result is consequently not an automatic lower bound for every observed event.[2]
Geometry, population and effective width
The route is divided where its dimensions or character change. A room entrance, a narrowing corridor and a stair entrance can each form a transition. Merging streams require additional attention because several populations may compete for one downstream element. Treating the whole route as one uniform passage can conceal these differences.[4]
The initial population determines how many people must pass through the system. Local density instead describes the number occupying a particular area at a particular time. These quantities are related but cannot be substituted for one another. Where arrival times differ, density can be updated from the number entering and leaving each route element.[5]
Effective width is the clear route width after allowing for boundary clearances. Nelson and Mowrer associate these clearances with body movement and balance beside fixed surfaces. Their tables use different allowances for doors, corridor walls, stairs, handrails and obstacles. The selected width convention must match the empirical flow relationship.[6]
For example, both the SFPE chapter and the ABCB's 2020 data sheets assign 0.15 m on each side of a doorway. Their corridor-wall allowance is 0.20 m on each side. These are inputs to the cited calculation method. They are not statements that every person maintains an identical physical clearance throughout an evacuation.[6][7]
Stairs require a consistent distance convention. The chapter defines speed along the line of movement, and its stair treatment relates vertical rise to distance along that line. Travel across landings must then be added. Using only storey height would underestimate the distance associated with the stated speed.[8]
Speed, flow and passage time
The cited model represents speed as decreasing approximately linearly with density over an intermediate range. For level routes, its SI coefficients are 1.4 m/s and 0.266 m²/person. Different stair configurations have different speed coefficients. These are empirical relationships with observed variability, rather than exact laws governing every individual.[2][8]
The following equations connect movement speed, specific flow and total flow. Passage time is obtained by dividing population by a sustained total flow. That final step assumes that the chosen point remains supplied at the calculated rate. Intermittent arrivals or downstream restrictions require a more detailed account.[4][9]
| Symbol | Meaning | SI unit |
|---|---|---|
| S | Speed along the movement path | m/s |
| k | Speed coefficient for the route type | m/s |
| a | Density coefficient, 0.266 in this formulation | m²/person |
| D | Density of occupants in the route | persons/m² |
| Fs | Specific flow per effective width | persons/(s·m) |
| We | Effective route width | m |
| Fc | Total calculated flow | persons/s |
| P | Population passing the selected point | persons |
| tp | Passage duration at sustained flow | s |
Nelson and Mowrer apply the linear speed relationship between densities of approximately 0.54 and 3.8 persons/m². Below this range, occupants can move more independently at their own pace. At the upper end, the simplified model approaches stopped movement. It should not be extended to produce negative speeds above the stated range.[8]
Combining speed with density produces a curved flow relationship. Flow increases initially as more people occupy the stream, then decreases when crowding reduces speed sufficiently. The source places maximum flow near 1.9 persons/m². Its level-route maximum is approximately 1.3 persons/(s·m), with different values for the listed stairs.[4]
The ABCB data sheets reproduce these general relationships and distinguish travel time from doorway queuing. They also identify a typical upper unimpeded adult speed near 1.2 m/s. This requires the low-density regime to be treated consistently. The intercept of the congested linear expression is not automatically the chosen free walking speed.[10]
Transitions, capacity and queues
At a simple transition, incoming total flow is compared with the downstream route's capacity. If the downstream element can carry that flow, its specific flow adjusts to its effective width. A wider route can therefore carry the same total stream at a smaller specific flow. Width and total flow must remain distinct in the accounting.[9]
Exit capacity constrains what can leave a bottleneck. When arriving flow exceeds the maximum downstream flow, occupants accumulate upstream. Nelson and Mowrer express queue growth as the difference between those rates. The calculation must account for this accumulation instead of passing the entire demand through the restriction immediately.[9]
A time-step balance makes the distinction clear. The people waiting at the end of an interval equal those waiting initially, plus arrivals, minus departures. Departures are constrained by capacity and by the number available to leave. Maintaining maximum discharge after the queue and arriving population disappear would violate that accounting.[5][9]
Where two streams merge, their total flows are added. Their specific flows must first be multiplied by the respective effective widths. The combined demand is then compared with the downstream capacity. Adding specific-flow values directly would ignore the widths of the contributing routes.[9]
The equations do not determine which stream receives priority. Nelson and Mowrer allow possibilities ranging from shared entry to dominance by one route. They recommend examining dominance cases when different populations are of concern. Equal allocation is an additional modelling choice, and it can strongly affect the waiting time of a particular group.[9]
Worked doorway and queue example
Consider two illustrative streams arriving at a doorway at 0.8 and 0.7 persons/s. Their combined arrival rate is 1.5 persons/s. The clear opening is selected as 1.20 m, giving 0.90 m effective width after the cited doorway allowances. Assume the level-route coefficients and a density of 1.9 persons/m² at discharge.[8][7]
The speed equation gives approximately 0.692 m/s. Multiplying by density gives approximately 1.316 persons/(s·m). Multiplying again by effective width gives a discharge rate of approximately 1.184 persons/s. The example therefore has greater arrival demand than the doorway can discharge under its stated assumptions.[4]
The difference is approximately 0.316 persons/s. If both arrivals and discharge remain steady for 60 seconds, about 19 people accumulate upstream. This is a continuous-flow approximation to a count of people. It demonstrates queue growth and does not prescribe an acceptable queue or verify the safety of its location.[9]
For comparison, increase the illustrative clear opening to 1.50 m. Effective width then becomes 1.20 m, and the same specific flow gives approximately 1.579 persons/s. This exceeds the selected combined arrival rate. The alternative has spare modelled discharge capacity, although other route elements could still constrain the real system.[4][9]
Suppose the previously accumulated queue exists when the greater discharge becomes available. With continued arrivals of 1.5 persons/s, it shrinks at only approximately 0.079 persons/s. Removing the accumulated 19 people would take approximately 241 seconds. Queue clearance depends on spare capacity above ongoing arrivals, rather than on total discharge capacity alone.[9]
These calculations hold density, incoming flow and width constant for each comparison. They do not predict how people would reorganise during a physical alteration to an exit. The width change is a mathematical comparison between flow conditions. It isolates the effects of capacity and arrival demand within the cited hydraulic relationships.[2][4]
From movement estimates to evacuation performance
Hydraulic output can be used within broader evacuation modelling. Its movement estimates must then be combined with the response assumptions appropriate to the population and scenario. Detection and warning do not guarantee immediate departure. Occupants may investigate, consult others, collect information or undertake other actions before moving towards an exit.[1][11]
Route use can also differ from the optimum distribution assumed in a simplified calculation. Some exits may be heavily used while others remain underused. A calculation that divides population equally between exits embeds a behavioural assumption as well as a capacity calculation. The chosen allocation therefore needs its own justification.[2]
The source's worked building example explicitly separates a first approximation from more detailed tracking of route flows. The detailed treatment examines door restrictions, stair entry and merging from successive floors. It shows why a population divided by a final exit capacity is not always sufficient. Travel before discharge and the sequence of joining streams can affect completion times.[12]
Conditions must also remain suitable along the movement route. Smoke can affect visibility and irritation, and the cited movement evidence is largely drawn from smoke-free conditions. An unobstructed hydraulic speed cannot establish safe passage through smoke. Exposure and movement performance require attention to the conditions actually expected during the relevant period.[9]
Travel time for the first person and passage time for the whole group answer different questions. A route may already contain people while its final discharge is still developing. The source's building example adds travel considerations to its estimate of exit passage. A complete movement calculation must identify the chosen starting and finishing events.[12]
The spatial position of a queue also matters to the calculation. An upstream stream cannot continue filling a route without changing its density. Subsequent speed and arrival flow can therefore differ from their initial values. Updating occupancy in successive intervals connects queue storage with the movement relationships, instead of treating every segment as permanently operating at maximum flow.[5][9]
Applicability, uncertainty and reporting
The hydraulic method is most transparent when its inputs and assumptions are visible. Population, route geometry, width deductions, speed relations and arrival sequence define the result. Reporting only the final time conceals which element controlled the calculation. It also makes comparison with another calculation difficult when the width or population conventions differ.[2][5][4]
Different occupants may require different movement representations. The ABCB data sheets warn that their doorway-flow expression is unsuitable for people with mobility impairment. They also require slower movement to be considered for older people and those with restricted mobility. A general-population flow relation cannot demonstrate the adequacy of assistance arrangements.[10][7]
The empirical relationships themselves contain variability. Nelson and Mowrer show observed movement speeds scattered around a fitted relationship. Additional uncertainty arises from departure timing, route allocation and merger priority. Giving a numerical result to many decimal places does not remove those uncertainties in the scenario or its underlying evidence.[2][9]
A useful assessment can examine how conclusions change when justified input assumptions change. Different arrival sequences or merge priorities may expose delays hidden by a single optimised case. The source specifically recommends bounding dominance cases where multiple routes matter. Such comparisons keep the model focused on its purpose: explaining movement consequences under stated conditions.[9]
The final interpretation should preserve the distinction between capacity, passage time and complete evacuation time. Capacity describes potential throughput at an element. Passage time describes a selected population crossing a point under specified flow conditions. Complete evacuation performance also includes response, movement elsewhere and whether the intended destination remains suitable.[1][4]
Notes
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Nelson and Mowrer (2002), pp. 3-367–3-368.
- ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 Nelson and Mowrer (2002), p. 3-368.
- ↑ 3.0 3.1 Nelson and Mowrer (2002), pp. 3-370–3-372.
- ↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Nelson and Mowrer (2002), p. 3-371.
- ↑ 5.0 5.1 5.2 5.3 Nelson and Mowrer (2002), p. 3-369.
- ↑ 6.0 6.1 Nelson and Mowrer (2002), pp. 3-368–3-369, Table 3-14.1.
- ↑ 7.0 7.1 7.2 Australian Building Codes Board (2020), p. 280, Table 21 and §1.5.3.
- ↑ 8.0 8.1 8.2 8.3 Nelson and Mowrer (2002), p. 3-370, Tables 3-14.2–3-14.4.
- ↑ 9.00 9.01 9.02 9.03 9.04 9.05 9.06 9.07 9.08 9.09 9.10 9.11 9.12 Nelson and Mowrer (2002), p. 3-372.
- ↑ 10.0 10.1 Australian Building Codes Board (2020), pp. 278–280, §§1.5.2–1.5.3.
- ↑ Nelson and Mowrer (2002), pp. 3-375–3-376.
- ↑ 12.0 12.1 Nelson and Mowrer (2002), pp. 3-373–3-375.
References
- Australian Building Codes Board. (2020). Data sheets: Fire Safety Verification Method (Version 1.1).
- Nelson, H. E., & Mowrer, F. W. (2002). Emergency movement. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-367–3-380). National Fire Protection Association.