Abstract
For the smoke filling in enclosures with floor leaks and growing fires, a comprehensive governing equation has been set up by Copper. However, this governing equation can hardly be exactly solved analytically because it is inseparable. Up to now, the existing analytical solutions for it are all approximate, including the so-called SFPE solution, the Deli-2 solution, the Deli-3 solution, etc. Among these approximate solutions, the SFPE solution almost always overestimates the smoke-interface height, while the Deli-2 solution almost always underestimates the interface height. In general, the Deli-3 solution is the most accurate but it can generate considerable error when the smoke interface is near the floor level. In this note, a new approximate solution is derived, which is actually a correction of the SFPE solution. The basic idea of the new solution is to resume the effect of the expansion term that is neglected in the deduction of the SFPE solution. It is found that, while the smoke interface in not very close to the floor level, the accuracy of the new solution is generally of the same order of magnitude as that of the Deli-3 solution; while the smoke interface is near the floor level, the accuracy of the new solution can be one order of magnitude higher than that of the Deli-3 solution. Moreover, it is shown that the Deli-2 solution is the limit of the new solution as the non-dimensional heat release rate approaching to infinity.
Abbreviations
- c p :
-
Specific heat of air at constant pressure, kJ/(kg K)
- e :
-
Absolute error of an analytical solution
- E :
-
Relative error of an analytical solution
- g :
-
Acceleration due to gravity, m/s2
- H :
-
Height of enclosure from floor to ceiling, m
- k :
-
Factor used to reduce the expansion term
- \(\dot{m}_{{\text{e}}}\) :
-
Expansion term, kg/s
- \(\dot{m}_{{\text{p}}}\) :
-
Plume term, kg/s
- r :
-
Coefficient in the compensation term of the new solution
- S :
-
Enclosure cross-sectional area (suppose to be constant), m2
- T a :
-
Ambient temperature, K
- t :
-
Time, s
- t c :
-
Characteristic time, s
- y :
-
Non-dimensional smoke-interface height
- y s :
-
SFPE solution of y
- z :
-
Smoke-interface height, m
- α :
-
Fire growth factor, kW/s2
- λ c :
-
Fraction of heat release rate lost to enclosure boundaries
- λ r :
-
Fraction of heat release rate lost due to radiation from fire plume
- ρ a :
-
Density of ambient air, kg/m3
- σ :
-
Non-dimensional heat release rate
- τ :
-
Non-dimensional time
- τ d :
-
Non-dimensional time when the smoke interface reaches the floor level
References
Zukoski EE (1978) Development of a stratified ceiling layer in the early stages of a closed-room fire. Fire Mater 2:54–62
Zhou Y, Meng Q (2021) Validity of methods for analytically solving the governing equation of smoke filling in enclosures with floor leaks and growing fires. Fire Technol. https://doi.org/10.1007/s10694-021-01102-4
Karlsson B, Quintiere JG (2000) Enclosure fire dynamics. CRC Press, Boca Raton
Cooper LY (1982) A mathematical model for estimating available safe egress time in fires. Fire Mater 6:135–144
Cooper LY (1983) The development of hazardous conditions in enclosures with growing fires. Combust Sci Technol 33:279–297
Mowrer FW (1999) Enclosure smoke filling revisited. Fire Saf J 33:93–114
Mowrer FW (2016) Enclosure smoke filling and fire-generated environmental conditions. SFPE handbook of fire protection engineering, 5th edn. Springer, New York, pp 1066–1101
Delichatsios M (2003) Closed form approximate solutions for smoke filling in enclosures including the volume expansion term. Fire Saf J 38:97–101
Delichatsios M (2004) Tenability conditions and filling times for fires in large spaces. Fire Saf J 39:643–662
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A. Data Table
Appendix A. Data Table
See Table 1.
Rights and permissions
About this article
Cite this article
Zhou, ZY., Zhou, Y. A Correction of the SFPE Solution for Smoke Filling in Enclosures with Floor Leaks and Growing Fires. Fire Technol 58, 29–39 (2022). https://doi.org/10.1007/s10694-021-01156-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10694-021-01156-4