How Mathematical Modeling Can Save Lives in High‑Rise Fires: Lessons from Hong Kong’s 31‑Story Tragedy

1. Fire Spread Mathematical Model

1.1 Chimney Effect and Vertical Propagation

The fire spread along the external wall scaffolding is driven by the chimney effect: hot air rises through narrow gaps, accelerating vertical flame propagation. A simplified speed model can be expressed as v = k \times \Delta T \times H / T_a, where v is the vertical spread velocity, k a baseline speed (≈0.5‑1 m/s), \Delta T the temperature difference between fire source and environment, H the height of the chimney channel, and T_a the absolute ambient temperature.

Baseline speed (≈0.5‑1 m/s)

Temperature difference between fire source and environment

Chimney channel height

Ambient absolute temperature

In the Hong Kong case, the 31‑storey building (~100 m tall) had external scaffolding that created narrow channels; under dry windy conditions (relative humidity 40‑50 %) the fire rose from the lower floors to the roof within minutes.

1.2 Time Function of Fire Development

Fire growth is commonly divided into ignition, growth, fully developed, and decay phases. In a closed space the heat release rate follows a t² model: HRR(t) = k_f \times t², where k_f is a fire growth coefficient determined by fuel type.

Slow type (wood)

Medium type (furniture)

Fast type (flammable finishes)

Ultra‑fast type (sprayed foam, etc.)

The building stored spray‑foam and other ultra‑fast combustible materials, explaining why the fire became uncontrollable in a very short time.

2. Evacuation Dynamics Model

2.1 Basic Evacuation Time Equation

The core safety question is whether the Available Safe Egress Time (ASET) exceeds the Required Safe Egress Time (RSET). RSET can be decomposed as:

Detection time (automatic alarm ≈30‑60 s)

Alarm confirmation and notification (≈60‑120 s)

Pre‑evacuation preparation (resident reaction, gathering belongings ≈60‑300 s)

Actual travel time (depends on floor height and crowd density)

2.2 Stairway Crowd Flow Model

For high‑rise residential buildings the stairwell is the primary escape route. If the stair width is w and crowd density is \rho (persons/m²), the flow rate is Q = w \times \rho. Walking speed v decreases with density and can be described by the Greenshields model:

Free walking speed ≈0.8 m/s (downward stair descent)

Jam density ≈5‑6 persons/m²

When density reaches jam density, congestion occurs, dramatically reducing evacuation efficiency—a dangerous situation in high‑rise fires.

2.3 Evacuation Time from Floor n

Assuming each floor is 3 m high and the effective stair length is about 6 m per floor, the ideal travel time from floor n to ground is: t_ideal = (3 m × n) / v_free Without congestion, descending from the 30th floor at 0.8 m/s takes roughly 112 seconds. In the Hong Kong incident, nearly 2,000 households attempted to evacuate simultaneously, causing congestion that extended the actual time by 3‑5 times, i.e., 15‑25 minutes—far exceeding the fire‑spread window.

3. Rescue Resource Allocation Optimization

3.1 Firefighter Response Time Model

Response time consists of travel time from the fire station, which depends on distance and traffic conditions. In this case, the fire was reported at 14:51 and the first unit arrived at 14:56, a 5‑minute response, illustrating Hong Kong fire services’ efficiency.

3.2 Rescue Capacity Constraints

Typical aerial ladder trucks operate up to 50‑70 m (≈15‑20 floors). The 31‑storey, ~100 m building exceeds this range, requiring internal stair evacuation or helicopter rescue. If N rescuers each can assist c persons per unit time, the maximum rescue flow is F_max = N × c. With hundreds of trapped occupants, this capacity quickly becomes a bottleneck.

4. Escape Decision Game‑Theoretic Analysis

4.1 To Evacuate or Wait for Rescue?

Residents face a binary decision: immediate evacuation (survival probability P_e) or staying to await rescue (survival probability P_w). The optimal choice depends on fire location, stair smoke, room sealing, rescue arrival time, and available oxygen.

If fire is below and stairs are filled with smoke, staying sealed may be preferable.

If escape routes are clear, early evacuation is optimal.

In the Hong Kong fire, dense smoke sealed doors and windows, leaving many residents unable to either escape or be rescued, creating a tragic dilemma.

5. Practical Escape Recommendations

Golden evacuation window : The first 3‑5 minutes after fire detection are critical; act immediately without gathering belongings.

Assess exit : Before opening a door, feel the handle; if hot or smoke is visible, stay inside and seek an alternative route.

Avoid elevators : Elevators may become smoke shafts; always use stairs and keep to the right to leave space for firefighters.

Stay low : Smoke rises, so crawl close to the floor; a wet cloth over mouth/nose can filter particles but does not protect against carbon monoxide.

If trapped : Move to a room far from the fire, close the door, seal gaps with wet cloth, open a window for rescue signals (do not break glass), and call 119 with precise location.

Daily preparedness : Keep smoke masks, flashlights, rescue ropes; know building evacuation routes; regularly participate in fire drills.

Monitor construction : Ensure scaffolding and repair materials meet fire‑resistance standards; promptly report any hazards to management and fire authorities.

The Hong Kong tragedy demonstrates that high‑rise fire safety is a race between fire spread, evacuation capacity, and rescue resources. Understanding the underlying dynamics and preparing accordingly can dramatically improve survival odds.