## Abstract

To model the flame tilt angle toward wall surface, experimental and theoretical studies were carried out. Cubical-shaped flexible polyurethane foam having 0.5 m edge length was burned at a position close to a wall made of ceramic fiber board. The separation distance between wall and specimen surfaces was in the range of 0.01 m to 0.2 m. Flame tilt angle increased with separation distance, but ceased at more than certain separation distance. To develop a mathematical model for flame tilt angles, the ratio of buoyancy force to inertia of entrainment air was considered. A virtual cylinder with a radius of *n*-times of fire source diameter was considered as an entry of entrained air flow. The effect of blocking by wall surfaces was considered by the angle between wall ends seen from fire source. In case of flames collided with wall surface, the flame tilt angle was calculated assuming that flame bends along with wall surface while keeping the same total flame length as in open configuration. Flame tilt angle increased with separation distance in case of flame collide on wall surface. The prediction results were compared with experimental data and existing models. The proposed model could predict the changes of flame tilt according to the separation distance fairly well. As an overall comparison, the proposed model is in better agreement with experimental data than existing models.

## Keywords

Flame tilt Separation distance Near-wall burning Buoyancy Inertia force Blocking effect## List of symbols

- \( A_{in} \)
Area that receives the inertia force \( F_{in} \) (m

^{2})- \( A_{out} \)
Area that receives the inertia force \( F_{out} \) (m

^{2})- \( c \)
Entrainment coefficient (–)

- \( D \)
Representative diameter of the fire source (m)

- \( D_{1} \)
Diameter of the burning area on perpendicular axis to the wall (m)

- \( D_{2} \)
Diameter of the burning area on parallel axis to the wall (m)

- \( F_{B} \)
Buoyancy of the plume (N)

- \( F_{in} \)
Inertia force to push the flame away from the wall (N)

- \( F_{out} \)
Inertia force to the flame towards the wall surface (N)

- \( H_{m} \)
Collision height of the flame (m)

- \( \bar{H}_{m} \)
Collision height of the imaginary flame induced by inertial force (m)

- \( L_{m} \)
Flame length (m)

- \( \bar{L}_{m} \)
Average flame length in open configuration (m)

- \( L_{1} \)
\( \left( {d + \frac{{D_{1} }}{2}} \right)/\left( {\sin \bar{\theta }} \right) \) (m)

- \( L_{2} \)
\( \bar{L}_{m} - L_{1} \) (m)

- \( N \)
Frame number contacting to the wall surface (–)

- \( N_{0} \)
Total number of frames analyzed at one second, \( N_{0} = 30 \)

- \( Q \)
Heat release rate (kW)

- \( Q^{*} \)
Non-dimensional heat release rate (–)

- \( T_{0} \)
Temperature of air (K)

- \( W \)
Width of the wall (m)

- \( b_{c} \)
Characteristic distance (m)

- \( c \)
Entrainment coefficient (–)

- \( c_{p} \)
Specific heat of air [kJ/(kg K)]

*d*The shortest separation distance from the tip of burning area to the wall surface (m)

- \( d_{0} \)
Initial separation distance (m)

- \( d_{c} \)
Separation distance between center of the fire source and the wall surface (m)

- \( d^{\prime} \)
Burnt-out distance (m)

- \( {\text{g}} \)
Acceleration of gravity (m/s

^{2})- \( k \)
Constant, k = 1.9 m/(kW

^{1/5}s)- \( n \)
Constant determined by experiment (–)

- \( u \)
The velocity of the entrainment flow into the gap between flame and the wall (m/s)

- \( u_{0} \)
Horizontal velocity of entrained air into the plume (m/s)

- \( u^{\prime} \)
The incoming velocity at the center of the gap between wall and fire source (m/s)

- \( w \)
Upward velocity in a zone of intermittent flame (m/s)

- \( \beta \)
Angle between

*y*-axis and straight line passing through origin point and the edge of the wall- \( \theta \)
Flame tilt angle (rad)

- \( \bar{\theta } \)
Calculated flame tilt angle by inertia forces and buoyancy (rad)

- \( \rho_{0} \)
Density of air (kg/m

^{3})- \( \rho_{f} \)
Density of plume (kg/m

^{3})- \( \upomega \)
Horizontal angle measured from x-axis (rad)

- ∆
Difference

## Notes

### Acknowledgements

Funding was provided by Japan Society for Promotion of Science (Grant No. 26289204).

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