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

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.