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A Correction of the SFPE Solution for Smoke Filling in Enclosures with Floor Leaks and Growing Fires

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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.

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Figure 1
Figure 2

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

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Authors and Affiliations

  1. Department of Physics, Hebei Normal University, Hebei, 050024, China

    Zhi-Yu Zhou

  2. Jiangsu Key Laboratory of Fire Safety in Urban Underground Space, China University of Mining & Technology, Xuzhou, 221116, Jiangsu, China

    Yan Zhou

Authors
  1. Zhi-Yu Zhou
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  2. Yan Zhou
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Corresponding author

Correspondence to Yan Zhou.

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Appendix A. Data Table

Appendix A. Data Table

See Table 1.

Table 1 Values of τd and r/σ for Different Values of σ from 10–14 to 10
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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

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  • DOI: https://doi.org/10.1007/s10694-021-01156-4

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