Validity of Methods for Analytically Solving the Governing Equation of Smoke Filling in Enclosures with Floor Leaks and Growing Fires

Abstract

The main objective of this paper is to evaluate the validity of the existing methods for analytically solving the governing equation of smoke filling in enclosures with floor leaks and growing fires. Since the complete form of the governing equation is inseparable, it can hardly be exactly solved analytically. Up to now, there are mainly five analytical solutions available in the literature. The first is derived by Mowrer (Fire Saf J 33: 93–114, 1999), the second is derived by Yamana and Tanaka (Fire Sci Technol 5: 31–40, 1985; Fire Sci Technol 5: 41–45, 1985), and the other three are derived by Delichatsios (Fire Saf J 38: 97–101, 2003; Fire Saf 39: 643–662, 2004). Among these solutions, the first has been incorporated into the SFPE handbook (SFPE handbook of fire protection engineering, 5th edn. Springer, New York, 2016), and one of the Delichatsios' solutions has been incorporated into the ISO standard (Fire safety engineering—Requirements governing algebraic equations—Smoke layers, 2006). These analytical solutions provide convenient ways for estimating the smoke-interface height. However, it should be noted that all of these solutions are approximate for the case of "floor leak & growing fire", so they must be used with caution. By comparing with the "exact" numerical solutions of the governing equation, errors of the five analytical solutions are calculated systematically. An absolute error contour plot and a relative error contour plot are made for each solution, which can be directly used to judge the validity of the solution. It is found that all the five analytical solutions sometimes produce considerable errors, and the error distribution characteristics are quite different from one solution to another. The error sources are analyzed. The analyses provide not only explanation for the error distribution characteristics but also deep insights into the analytical solutions. Based on the error contour plots made for each solution, the prediction of the corresponding solution can be corrected to a more accurate value. An example is presented to illustrate how to do this.

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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)

\(\dot{m}_{{\text{e}}}\) :

Expansion term (kg/s)

\(\dot{m}_{{\text{p}}}\) :

Plume term (kg/s)

n :

Exponent in power law fire model; while n = 2, the fire is t-squared fire

S :

Enclosure cross-sectional area (suppose to be constant) (m2)

T a :

Ambient temperature (K)

T g :

Smoke-layer temperature (K)

V g :

Volume of smoke-layer (m3)

t :

Time (s)

t c :

Characteristic time (s)

y :

Non-dimensional smoke-interface height

z :

Smoke-interface height (m)

α :

Fire growth factor (kW/sn)

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)

ρ g :

Density of smoke-layer (kg/m3)

σ :

Non-dimensional heat release rate

τ :

Non-dimensional time

kρc :

Factor related to thermal inertia of building material (W2s/m4K2)

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Acknowledgements

Financial supports from the National Key R&D Program of China (Grant Nos. 2018YFC0807900, 2018YFC0807903) and the Natural Science Foundation of Jiangsu Normal University (No. 16XLR051) are sincerely acknowledged.

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Correspondence to Yan Zhou.

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Zhou, Y., Meng, Q. Validity of Methods for Analytically Solving the Governing Equation of Smoke Filling in Enclosures with Floor Leaks and Growing Fires. Fire Technol (2021). https://doi.org/10.1007/s10694-021-01102-4

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Keywords

  • Enclosure
  • Smoke filling
  • Floor leak
  • Growing fire
  • Analytical solution
  • Validity
  • Error
  • Correction