A Closed-Form Solution of the Smoke Filling Time and Descent History in Enclosure Growing Fires with Floor Leaks

Abstract

For the smoke filling time or smoke descent history in enclosure fires with floor leaks, the existing close-formed solutions are all based on the hypothesis that the expansion term is negligible. However, when the smoke interface is near to the floor level, the expansion term is more important than the plume entrainment term and the existing solutions give unrealistic predictions. For those scenarios where the expansion term cannot be neglected, there is not yet a closed-form solution for the smoke filling time or the smoke descent history. Based on asymptotic analysis, a closed-form solution is derived in this work when the smoke fills the room fully, namely, when the smoke interface reaches the floor level. In addition, a closed-form solution is derived for the smoke filling history, which describes when the smoke interface reaches any given height. In the two solutions, both the expansion term and the plume entrainment term are considered. By comparing with the results from the numerical solution for a wide range of scenarios, it is shown that the relative error of the suggested solution for smoke descent history is in the range from − 2.8% to 1.7%. To be conservative, we can reduce 1.7% from the prediction value of the suggested solution, which causes no more than 4.5% underestimation of the smoke filling time for any given height. An illustrative example is presented to demonstrate the calculation procedure and practical usefulness of the proposed method.

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Abbreviations

A f :

Floor area covered by the growing fire, m2

c p :

Specific heat of air at constant pressure, kJ/(kg K)

D :

Diameter of the fire source, m

E :

Relative error

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

p :

Positive number introduced in Equation (29), being set as 1.63 by regression

q :

Positive number introduced in Equation (37), being set as 3.4 by regression

$$\dot{q}^{\prime\prime}$$ :

Heat release rate per unit area (HRRPUA) of floor covered by fire, kW/m2

$$\dot{Q}$$ :

Heat release rate, kW

r :

Positive number introduced in Equation (35), being set as 0.368 by regression

S :

Enclosure cross-sectional area (supposed to be constant), m2

T a :

Ambient temperature, K

t :

Time, s

t c :

Characteristic time defined in Equation (13), s

X :

Non-dimensional variable defined in Equation (4)

X 0 :

Value of X when the smoke interface reaches the floor level

X 0L :

Asymptotic value of X0 as Λ approaches to 0

X 0s :

Asymptotic value of X0 as Λ approaches to infinity

y :

Non-dimensional smoke-interface height defined in Equation (10)

y s :

Non-dimensional smoke-interface height predicted by the SFPE solution [8, 12]

Y :

Non-dimensional variable defined in Equation (3)

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

Λ :

Non-dimensional variable defined in Equation (5)

ρ a :

Density of ambient air, kg/m3

σ :

Non-dimensional heat release rate at the characteristic time, defined in Equation (12)

τ :

Non-dimensional time defined in Equation (11)

τ 0 :

Value of τ when smoke interface reaches the floor level

τ 0L :

Asymptotic value of τ0 as σ approaches to 0

τ 0s :

Asymptotic value of τ0 as σ approaches to infinity

τ n :

Value of τ calculated by numerical solution

τ p :

Asymptotic limiting solution of τ while the expansion term is neglected

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Acknowledgements

Michael Delichatsios acknowledges the partial support of Leverhulme Emeritus Scholarship EM-2020-033.

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Correspondence to Yan Zhou.

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Zhou, Y., Delichatsios, M., Deligiannis, M. et al. A Closed-Form Solution of the Smoke Filling Time and Descent History in Enclosure Growing Fires with Floor Leaks. Fire Technol 59, 271–287 (2023). https://doi.org/10.1007/s10694-023-01372-0

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• DOI: https://doi.org/10.1007/s10694-023-01372-0

Keywords

• Enclosure
• Smoke filling time
• Floor leak
• Growing fire
• Closed-form solution
• Asymptotic analysis