Module 1, lesson 2 · 35 minutes

Density, speed, flow

Three quantities and one multiplication: how many people a corridor, gate or ramp can move each minute.

You’ll be able to

define density, speed, specific flow and flow, and convert between their units;
compute the flow through a given width;
explain why flow, not a head count, is the quantity you plan movement with.

The final whistle goes and a concourse fills. Someone in the control room asks how many people are out there. It is a reasonable question with an unhelpful answer, because the number that decides whether the concourse empties in eight minutes or twenty is not how many people there are. It is how many can pass through each exit every minute.

That rate is called flow. It comes from three things you can see from a CCTV camera: how tightly people are packed, how fast they are walking, and how wide the opening is. Multiply them together and you have most of what an engineer means by “capacity”. This lesson builds that multiplication carefully, with its units, because every later module leans on it.

Three quantities and their product

Density, written ρ (rho), is the number of people per square metre of floor, in p/m². Keith Still recommends always expressing it this way round, as people per square metre rather than square metres per person, . A bigger number then always means a more crowded space. Lesson 1 builds a feel for what the numbers look like.

Walking speed, v, is how fast people move, in metres per second. In a sparse crowd it is whatever people choose; in a dense one it is whatever the crowd allows.

Put the two together and you get the quantity that matters most. Specific flow, Jₛ, is the number of people crossing a line per metre of its width per second: density times speed. Total flow, J, is the specific flow times the width of the line, in people per second (or per minute).

$J_s = \rho\, v \qquad J = J_s\, w = \rho\, v\, w$

Jₛ = ρ · v J = Jₛ · w = ρ · v · w

ρ: density, p/m²
v: speed, m/s
Jₛ: specific flow, p/m/s
w: width of the opening, m
J: flow, p/s
60: multiply p/s by 60 for p/min

Why does the product work? Stand at a line across a corridor and ask who will cross it in the next second. Everyone within v metres upstream will, because that is how far they walk in a second. That strip is v metres long and w metres wide, so it holds ρ × v × w people. Figure 1 shows it.

The people who cross a line in the next second are the ones already within v × 1 s of it. At 1.5 p/m² and 1.2 m/s across 3 m, that strip holds about 5 people, so the flow is about 5 p/s.

The four quantitiesUse these units throughout the course.
Quantity	Symbol	Unit	What it tells you
Density	ρ	p/m²	How tightly packed the crowd is, at a place and a moment
Speed	v	m/s	How fast people are walking
Specific flow	Jₛ = ρv	p/m/s or p/m/min	Throughput per metre of width: comparable between openings
Flow	J = Jₛw	p/s or p/min	Throughput of the whole opening: what empties a space

The two traditions of the field use different clocks. Fruin and the UK Green Guide quote flows per minute (p/m/min); the Jülich experimenters and Helbing quote them per second (p/m/s). Converting is a factor of 60: 1 p/m/s = 60 p/m/min. Check which one a source uses before you compare numbers.

How fast people walk

Ulrich Weidmann’s 1993 report for ETH Zürich’s transport institute collected 52 published values of free walking speed. Their mean is 1.34 m/s (4.83 km/h); they range from 0.97 to 1.65 m/s, and 28 of the 52 lie between 1.25 and 1.45 m/s. When nothing is in the way, that is what a crowd does.

Free speed is only the starting point. As density rises, people slow down to keep from treading on the person in front, and the next lesson is about exactly how. For now, treat speed as something you measure, not something you assume.

Weidmann’s Transporttechnik der Fussgänger (1993) is still the most quoted source for pedestrian speeds and the speed–density curve. It is a literature synthesis: its strength is breadth, and its numbers are averages over very different studies.

Flow through a width: a worked example

A corridor is 4 m wide. The crowd in it is at 1 p/m² and walking at 1.0 m/s.

Specific flow: Jₛ = 1 p/m² × 1.0 m/s = 1 p/m/s.
Flow: J = 1 p/m/s × 4 m = 4 p/s.
Per minute: 4 × 60 = 240 p/min.

Now compare the Green Guide. For sizing exits it uses a rate of passage of 82 p/m/min on level surfaces and 66 p/m/min on stepped ones. At the level rate the same 4 m corridor is credited with 4 × 82 = 328 p/min, so our example crowd is moving below the rate the Guide plans with.

A gate does not care how many people are queuing behind it. It passes a flow, and the queue is what is left over. Exit widths, egress times and capacities are all built from flow per metre of width, multiplied by width and by time.

Leave the defaults (1 p/m², 1.0 m/s, 4 m) and press Play. Within about 10 seconds the count at the line settles near the computed 240 p/min.
Halve the width. Then put it back and halve the speed instead. Why do both give the same flow?
Tick “Tie speed to density” and drag density slowly from 0.5 to 5 p/m². Watch the flow rise, peak and fall.

Flow is a product, so halving any one factor halves it. But speed is not free to choose: once it depends on density, packing more people in eventually lowers the flow. That is the fundamental diagram, the subject of the next lesson.

With the box unticked you can set speed and density independently, which no real crowd allows. That is deliberate: it isolates the arithmetic. The tied mode uses Weidmann’s speed–density relation.

The Green Guide’s 4th edition (1997) used 109 p/m/min on level routes and 73 on stairs. The 5th edition (2008) lowered these to 82 and 66, following the standard BS EN 13200-1. In Still’s words, the 109 figure “was adjusted DOWN to 82 people per metre per minute to bring the Green Guide – 5th Edition – in line with the rest of the world”. The “73 for stairs” you may still see quoted is out of date.

A stand has two exits, 1.2 m and 2.4 m wide. The crowd approaching each is at the same density and walking at the same speed. What is the ratio of their flows?

1 : 2. Density and speed set the specific flow, which is the same for both; total flow is specific flow times width, so doubling the width doubles the flow.

Three regimes, in words

Because speed falls as density rises, the product ρv does not grow for ever. It behaves in three ways as a crowd packs in:

Free flow. Density is low and people walk at their own speed. Add people and flow rises almost in proportion.
Congested. People slow down for each other, but the extra density more than makes up for it: flow still rises, more slowly, to a maximum. On Weidmann’s curve that maximum is about 1.23 p/m/s at 1.75 p/m², where people walk at 0.70 m/s.
Jammed. Past the maximum, speed falls faster than density rises. Flow falls, reaching zero at jam density, 5.4 p/m² in Weidmann’s relation.

The same flow can therefore come from two very different crowds: a fairly sparse one walking briskly, or a dense one shuffling. The second is the dangerous one, and a flow counter alone cannot tell them apart. You need density too. The curve behind all three regimes is the subject of the next lesson, the fundamental diagram.

Standing is not moving

Densities quoted in guidance are not all the same kind of number. The Green Guide allows standing areas up to 47 people per 10 m², 4.7 p/m², for spectators who are watching, not walking. A moving crowd at that density would be close to a standstill.

For moving crowds Still puts the critical range lower, and his own pages do not quite agree on where: one gives 2–3 p/m², another 2–4 p/m². Heuristics drift like this, which is why the course always tells you where a threshold comes from and what it was derived for.

4.7 p/m² is a holding figure for a terrace. Using it to size a concourse, a ramp or a queue that has to keep moving puts the design on the wrong side of the flow maximum.

A corridor 3 m wide carries 180 people per minute. What is the specific flow, in p/m/min and in p/m/s?

180 ÷ 3 = 60 p/m/min, which is 60 ÷ 60 = 1 p/m/s. On its own that does not tell you whether the corridor is comfortable: it could be a sparse crowd walking fast or a dense one walking slowly.

“More people in the corridor means more people through it.” Only up to the flow maximum. Beyond it, adding people slows everyone more than it adds bodies, and flow falls.
“4.7 p/m² is a safe density for a moving crowd.” It is a standing figure from the Green Guide. Moving crowds reach their critical range well below it.
“Flow is a property of the crowd.” Flow is measured at a line, across a width. The same crowd produces different flows at a 1.2 m door and a 2.4 m door.