Specific flow

Specific flow is the number of people passing a point on an escape route per unit time and per unit effective route width. It combines the density of the moving population with its speed. Nelson and Mowrer distinguish this width-normalised quantity from the total number passing the same point each second. Its usual SI unit is persons/(s·m), while density is expressed in persons/m². Multiplication by effective width converts specific flow into total flow.

Specific flow provides a connection between walking speed, crowd density and exit capacity. It is central to the hydraulic evacuation model, which represents occupants as moving streams. The relationship is conditional on the population, route and movement assumptions. A maximum tabulated specific flow therefore does not mean that an exit continuously achieves that rate during a real evacuation.

Density, speed and width

Density describes how many people occupy a unit area of a route. Speed describes their progress along its direction of travel. Specific flow combines these two quantities at a particular location. A faster stream can carry more people at a given density, while a denser stream carries more people only if its speed remains sufficient.

The occupant load is a population count associated with a building or space. It is not automatically the density at a doorway or along a corridor. Occupants may reach that location at different times. Nelson and Mowrer consequently describe updating route density from the people entering and leaving during successive time intervals.

The basic relationship is simple, but its units matter. Multiplying metres per second by persons per square metre gives persons per second per metre. One length dimension cancels; the remaining length dimension represents route width. Multiplying once more by effective width gives a count rate in persons per second.

F_s = SD

F_c = F_sW_e

Symbol	Description	SI unit
Fs	Specific flow	persons/(s·m)
S	Movement speed along the route	m/s
D	Density of people on the route	persons/m²
Fc	Total calculated flow past the chosen point	persons/s
We	Effective width of the route	m

The cited documents contain inconsistent unit labels in their explanatory text. Nelson and Mowrer's definition and table identify flow per unit width, although one sentence prints an area denominator. The ABCB definition also specifies unit effective width, but its symbol list omits that dimension. The units above follow the definitions and the dimensional result of multiplying speed by density. The distinction matters when comparing a quoted specific flow with a measured door count, because the latter also depends on opening width.

The relationship with crowd density

Nelson and Mowrer present a model in which speed decreases approximately linearly across an intermediate density range. For level corridors, aisles, ramps and doorways, their coefficient is 1.4 m/s. The density coefficient is 0.266 m²/person in SI units. These coefficients describe the cited empirical model; they are not properties shared by every possible population.

The chapter applies the linear relationship between approximately 0.54 and 3.8 persons/m². Below the lower value, individuals can move more independently at their own pace. Above the upper value, its simplified treatment assumes that movement cannot resume until congestion decreases. Extending the equation beyond those conditions would misrepresent the stated model.

S = k(1 − aD)

F_s = kD(1 − aD)

Symbol	Description	SI unit
k	Speed coefficient for the specified route element	m/s
a	Density coefficient, 0.266 for the cited SI formulation	m²/person
D	Route density within the model's applicable range	persons/m²
S	Movement speed	m/s
Fs	Specific flow	persons/(s·m)

Specific flow first rises as density increases, because more people pass within each metre of width. It reaches a maximum when the density gain is offset by the speed reduction. At still greater density, the reduction in speed dominates. Adding more people to an already crowded route can therefore reduce its throughput in this model.

The ABCB's 2020 data sheets reproduce the same general speed relationship and density limits. They identify a typical unimpeded adult speed near 1.2 m/s and require consideration of slower occupants. This does not justify applying the linear formula at zero density, where its intercept is larger than that typical unimpeded speed. The free movement and congested regimes require consistent treatment.

Maximum specific flow and route type

For the stated quadratic expression, the maximum occurs at a density of one divided by twice the density coefficient. Substituting 0.266 gives approximately 1.88 persons/m². Nelson and Mowrer round the corresponding location to 1.9 persons/m². Their tabulated maximum for level routes is approximately 1.3 persons/(s·m).

Specific flow rises and then falls as increasing density reduces speed. Based on Nelson and Mowrer (2002), pp. 3-370–3-371.

The maximum is a feature of the particular speed–density curve. It is not the highest density that the route can contain. Nor does it mean that everybody moves fastest at that density. People move faster at lower densities, but fewer people are available to pass each metre of width.

Stairs have different coefficients and maximum flows in the cited tables. Their values depend on the riser and tread configuration represented by the source data. The chapter gives lower maximum specific flows for its listed stairs than for level routes. Substituting the level-route coefficient into a stair calculation would therefore change the model being applied.

Speed on stairs is measured along the line of movement. Vertical storey height alone is not the distance travelled, and travel across landings must also be included. The same distinction applies when translating a stair flow into a travel-time estimate. A consistent specific-flow calculation does not remove the need to define the route geometry correctly.

The same curve can associate a particular submaximum flow with two different densities. One state has fewer people moving faster; the other has more people moving slowly. A flow count alone therefore does not identify the crowd's density or speed. Additional information is needed to distinguish the two possible states.

The rounded density limits should also be interpreted with the equation. With the displayed coefficient, its algebraic zero occurs at approximately 3.76 persons/m². This corresponds to the chapter's rounded upper limit near 3.8 persons/m². The negative values obtained by continuing the expression beyond that point have no physical meaning in this movement model.

Effective width and total throughput

The effective width is the usable flow width after allowing for boundary clearances. Nelson and Mowrer relate these clearances to lateral body movement and balance near walls or fixed obstacles. Their treatment distinguishes the clear physical opening from the portion used in the hydraulic calculation. The model's flow coefficient and its width convention belong together.

For a doorway, the cited tables deduct 0.15 m on each side. A corridor wall instead has a listed boundary allowance of 0.20 m on each side. Handrails and obstacles have their own treatment. The applicable deductions depend on the route element, so a single width allowance cannot be transferred indiscriminately between geometries.

Effective width is the clear opening less the boundary allowances used by the flow model. Based on Nelson and Mowrer (2002), pp. 3-369–3-371, and ABCB (2020), p. 280.

At an unchanged specific flow, total calculated throughput is proportional to effective width. This proportionality concerns the modelled stream at that point. A wider opening cannot discharge more people than arrive, and its downstream route may impose another restriction. The width relationship is therefore one part of a connected route calculation.

A narrow transition can require a higher outgoing specific flow to pass the same incoming total flow. If that requirement exceeds the downstream maximum, the flow cannot remain continuous at the incoming rate. A queue forms upstream. Its growth is the difference between arriving and departing numbers per unit time.

The boundary deduction represents a modelling convention, rather than a claim that the edge zones are physically inaccessible. Its purpose is to relate the selected width to the empirical flow relationship. Using a coefficient defined per effective width with the full clear width changes the predicted capacity. The calculation must retain the width definition associated with its source.

Worked comparison at three densities

Consider an illustrative level doorway with a clear width of 1.20 m. Applying the cited 0.15 m allowance on each side gives an effective width of 0.90 m. Assume the model's level-route coefficients and compare densities of 1.0, 1.9 and 3.0 persons/m². These selected inputs demonstrate the relationship and do not describe an observed evacuation.

At 1.0 person/m², the equation gives a speed of approximately 1.028 m/s. Specific flow is also approximately 1.028 persons/(s·m), because density is numerically one. Multiplying by 0.90 m gives a total flow of approximately 0.925 persons/s. The stream moves relatively quickly but remains less densely occupied than at the curve's peak.

The intermediate-density example has the greatest throughput despite a lower walking speed than the least crowded case. Based on Nelson and Mowrer (2002), pp. 3-370–3-371.

At 1.9 persons/m², speed falls to approximately 0.692 m/s. Specific flow increases to approximately 1.316 persons/(s·m), close to the mathematical maximum. The doorway's resulting total flow is approximately 1.184 persons/s. The higher density more than compensates for the lower speed in this part of the curve.

At 3.0 persons/m², speed decreases further to approximately 0.283 m/s. Specific flow is then approximately 0.848 persons/(s·m), and total doorway flow is approximately 0.764 persons/s. This denser stream discharges fewer people per second than either previous case. The result illustrates why density and capacity cannot be treated as interchangeable quantities.

Suppose 90 people pass the doorway at each constant calculated rate. Division of population by total flow gives approximately 97, 76 and 118 seconds respectively. These are passage durations at the selected point under sustained-flow assumptions. They exclude detection, response, approach travel and any later movement to a place of safety.

Transitions, queues and changing conditions

A route commonly includes several elements in series. People may pass from a room into a corridor, through a door and onto a stair. The density and speed can change at each transition even when the same total flow continues. Consequently, assigning one specific-flow value to the entire route can conceal the controlling restriction.

At a merge, incoming total flows are added before comparison with downstream capacity. Specific flows cannot simply be added without accounting for their respective widths. Each incoming specific flow must first be multiplied by its effective width. The resulting total is then assessed against the downstream maximum total flow.

Conservation does not determine how people from competing routes share access. Nelson and Mowrer describe outcomes ranging from shared entry to dominance by one incoming stream. They propose examining different dominance assumptions where the sequence matters. Equal division at a merge is therefore an additional assumption, rather than an inevitable consequence of the flow equations.

When arrivals vary, passage time cannot always be found by dividing the entire population by one maximum rate. The exit may initially be underused, later operate near capacity, and finally empty after the queue clears. Tracking arrivals and departures preserves this sequence. It also distinguishes a capacity limit from an average flow over the whole evacuation.

A local reduction in width also changes the space available upstream as a queue develops. The arriving rate can then change because upstream density and speed change. Maintaining the original approach flow indefinitely would ignore that feedback. Nelson and Mowrer's treatment of transitions therefore connects the calculation of each route segment with the neighbouring segments.

Interpretation and limitations

The hydraulic relationships summarise observations with empirical scatter. Nelson and Mowrer caution that a calculated movement time is a model result, not a guaranteed actual time. Behaviour, route choice and interruptions can affect the realised flow. Such effects remain relevant when specific-flow equations are embedded in more extensive evacuation modelling.

Population characteristics also matter. The ABCB data sheets explicitly state that their doorway-flow equation is not suitable for people with mobility impairment. Their speed discussion requires further reductions for older people and those with restricted mobility. A general-population maximum cannot be used as evidence that an assisted evacuation will achieve the same throughput.

Much of the movement evidence discussed by Nelson and Mowrer concerns smoke-free conditions. Smoke can reduce visibility and affect the eyes, changing movement performance. Their discussion distinguishes these influences from crowd density. Applying an unobstructed flow curve does not establish either movement capability or acceptable exposure in a smoke-filled route.

A numerical maximum describes movement under the model's assumptions; it is not a complete performance criterion. The same throughput can be associated with different queue locations and different waiting times for particular groups. Merge dominance can especially affect which occupants leave first. Total discharge alone cannot describe those differences in individual experience.

A useful interpretation therefore preserves the complete chain of assumptions. The population, density range, route type, effective width and arrival pattern must match the calculation. Its result describes flow under those stated conditions. Claims about total evacuation performance require the remaining time components and the conditions along the route to be assessed as well.

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.