In the growth phase, before a fire runs out of fuel or air, its heat release rate is well described by \( \dot{Q} = \alpha t^{2} \), where \( t \) is time since effective ignition and \( \alpha \) is a growth coefficient set by the fuel and its arrangement. Fire protection engineering sorts fuels into four standard categories by how long they take to reach one megawatt: ultrafast about 75 seconds, fast about 150, medium about 300 and slow about 600. Stacked plastics and foam furniture are fast or ultrafast; densely packed paper records are slow. The quadratic form matters because it means growth is not steady: a fire that took two minutes to reach the first megawatt reaches the fourth in another two. The confusion this clears up is imagining that detecting a fire twice as late costs twice as much fire; because the curve is quadratic, a detection delay costs far more than proportionally. Engineers call a chosen curve a design fire: it is an input you assume, not a result the model produces. After this Concept you can estimate a fire's size at a chosen time from its growth category, and see what a detection delay costs.
In fire dynamics, the t-squared (t²) fire growth model describes heat release rate as a function of time using the equation HRR = α(t − t_kn)², where α is a fire growth coefficient corresponding to a…