Abstract
In a fire, “every second counts”. What this paper shows is that when a fire grows as the square of time, the weight of one and the same minute’s delay changes greatly with how the fire grows and when it was noticed, and the earlier it was noticed, the heavier the delay that follows. No new theorem or law is claimed.
First, the growth coefficients follow from the standard classes——for the four fires reaching $1055$ kW in $600$, $300$, $150$ and $75$ seconds, $\alpha=\mathbf{0.002931}$, $\mathbf{0.011722}$, $\mathbf{0.046889}$ and $\mathbf{0.187556}$ kW/s$^2$ (Section 2).
Second, the same $60$ seconds weigh from $\mathbf{1.7551}$ to $\mathbf{12.9489}$ times, by how the fire grows——in the heat released so far, from $\mathbf{2.3252}$ to $\mathbf{46.5962}$ times (Section 2).
Third, and this is the core. The weight of a delay is $(1+\Delta t/t_d)^2$ and depends on the time of noticing $t_d$——a $60$-second delay after noticing at $50$ seconds gives $\mathbf{4.840000}$ times, after noticing at $400$ seconds $\mathbf{1.322500}$ times (Section 3).
Fourth, a fire noticed at $120$ seconds and delayed $60$ has exactly $\mathbf{2.250000}$ times the heat release and $\mathbf{3.375000}$ times the heat released——$(3/2)^2$ and $(3/2)^3$ (Section 3).
Fifth, the type of growth changes the nature of a delay——noticed at $100$ seconds and delayed $60$, linear growth gives $\mathbf{1.600000}$ times, squared growth $\mathbf{2.560000}$, and exponential growth that multiplies by $e$ every $60$ seconds $\mathbf{2.718282}$, the last alone independent of when the fire was noticed (Section 4).
Sixth, the separator is whether growth follows a power or an exponential——for a power the weight of a delay is set by when the fire was noticed; for an exponential only by how long the delay was (Section 5).
Keywords: t-squared fire, fire growth, detection delay, heat release rate, sprinkler.
Scope (what is not claimed)
No new theorem or law is claimed——the model of a fire whose heat release grows as the square of time and its classes of growth are standard. Only the growth phase is considered——limits to spread, the ventilation-controlled plateau and burnout are not included. Noticing is placed at $100$ kW——the heat release at which a detector responds varies with ceiling height and detector type; $100$ kW is a representative value for comparison. No fire protection is designed——the choice and placement of detectors and sprinklers and the quantity of water discharged are not treated, and no “within so many seconds” is stated. Heat taken by water is estimated up to the evaporation of water at $20$ degrees——the water actually needed to suppress a fire depends strongly on how much of it reaches the fire. Relation to earlier papers: Paper 558 showed that the same “ten per cent” becomes $1.4641$ or $0.7513$ depending on which dimension——this paper shares the shape of one operation giving different results, but what changes is not the dimension but where on the growth curve the fire was noticed. Paper 283 showed that the same “double it” gives results differing by orders of magnitude according to mechanism——here the same delay changes in nature with the type of growth: dependent on the time of noticing for a power, independent for an exponential. Paper 300 showed that sharing a root is decidable——“the faster the fire, the heavier the delay” and “the earlier noticed, the heavier the delay” share one root, the ratio $\Delta t/t_d$. What is added is deriving the coefficients of four fires from the standard classes and setting out the weight of the same $60$-second delay, showing that the weight of a delay is $(1+\Delta t/t_d)^2$ and depends on the time of noticing, comparing the nature of a delay under linear, squared and exponential growth, and placing the separator on power against exponential.
1. Notation and the Question
Setting: consider a fire whose heat release grows as $Q(t)=\alpha t^2$. If it is noticed at a heat release $Q_d$, the time of noticing is $t_d=\sqrt{Q_d/\alpha}$, and a further delay $\Delta t$ gives
$$\frac{Q(t_d+\Delta t)}{Q(t_d)}=\left(1+\frac{\Delta t}{t_d}\right)^{2},\qquad \frac{E(t_d+\Delta t)}{E(t_d)}=\left(1+\frac{\Delta t}{t_d}\right)^{3},$$where $E=\alpha t^3/3$ is the heat released so far.
Observation 1.1 in a fire, “every second counts”. How much a minute weighs is set by how the fire grows——a minute is the same length every time, but how much the fire grows in it is not.
Question: does the same $60$-second delay weigh the same for every fire?
2. Growth Coefficients and the Same Delay
Computation 2.1: $\alpha$ from the standard classes, which sort fires by the time taken to reach $1055$ kW, and the effect of a $60$-second delay after noticing at $100$ kW.
| growth | $\alpha$ (kW/s$^2$) | time to $100$ kW | heat release | heat released |
|---|---|---|---|---|
| slow ($600$ s) | $\mathbf{0.002931}$ | $\mathbf{184.7247}$ s | $\mathbf{1.7551}$ times | $\mathbf{2.3252}$ times |
| medium ($300$ s) | $\mathbf{0.011722}$ | $\mathbf{92.3624}$ s | $\mathbf{2.7212}$ times | $\mathbf{4.4890}$ times |
| fast ($150$ s) | $\mathbf{0.046889}$ | $\mathbf{46.1812}$ s | $\mathbf{5.2865}$ times | $\mathbf{12.1548}$ times |
| ultra-fast ($75$ s) | $\mathbf{0.187556}$ | $\mathbf{23.0906}$ s | $\mathbf{12.9489}$ times | $\mathbf{46.5962}$ times |
Observation 2.2 the same $60$ seconds weigh from $1.7551$ to $12.9489$ times. In every row $\alpha t_d^2$ returns exactly to $100$ kW (to $10^{-9}$)——an ultra-fast fire reaches $100$ kW in $23.0906$ seconds, so $60$ seconds is two and a half times its whole growth so far; for a slow fire it is only a third.
3. When It Was Noticed
Computation 3.1: for the same squared fire, only the time of noticing $t_d$ was varied, followed by a $60$-second delay.
| time of noticing | heat release after a $60$-second delay |
|---|---|
| $50$ s | $\mathbf{4.840000}$ times |
| $100$ s | $\mathbf{2.560000}$ times |
| $200$ s | $\mathbf{1.690000}$ times |
| $400$ s | $\mathbf{1.322500}$ times |
Observation 3.2 this is the core. With squared growth the weight of the same delay depends on when the fire was noticed, and the earlier, the heavier——“noticed early, so safe” does not follow: having noticed early, the next minute is the heaviest of all. The weight is set by the delay divided by the time the fire has already been growing.
Computation 3.3: a fire noticed at $120$ seconds and delayed $60$ has $(180/120)^2=\mathbf{2.250000}$ times the heat release and $(3/2)^3=\mathbf{3.375000}$ times the heat released (to $10^{-12}$).
4. Types of Growth
Computation 4.1: noticed at $100$ seconds and delayed $60$, under three kinds of growth.
| growth | heat release after a $60$-second delay |
|---|---|
| linear ($Q\propto t$) | $\mathbf{1.600000}$ times |
| squared ($Q\propto t^2$) | $\mathbf{2.560000}$ times |
| exponential ($e$ times in $60$ s) | $\mathbf{2.718282}$ times |
Observation 4.2 with exponential growth the weight of a delay does not depend on when the fire was noticed. $e^{\Delta t/\tau}$ contains no $t_d$——with power growth the same delay weighs more the earlier it comes. The weight of a delay in a squared fire depends on the time because the growth is a power.
Computation 4.3: the heat taken by water was also estimated. $1$ kg of water at $20$ degrees takes $\mathbf{2.591880}$ MJ up to evaporation, so taking $1$ MW needs $\mathbf{0.385820}$ kg of water per second.
Observation 4.4 the water needed is also proportional to the heat release. If a $60$-second delay makes the heat release $2.25$ times, the water needed to take it away is $2.25$ times too——a delay accumulates as water to be paid later.
5. The Separator
Observation 5.1 the separator is whether growth follows a power or an exponential. For a power the weight of a delay is $(1+\Delta t/t_d)^n$, set by when the fire was noticed——for an exponential it is $e^{\Delta t/\tau}$, set only by how long the delay was.
Observation 5.2 the shape of Paper 558, with something else changing. There the same “ten per cent” became $1.4641$ or $0.7513$ depending on the dimension——here the same “minute” becomes $1.322500$ or $4.840000$ depending on where on the growth curve the fire was noticed.
Observation 5.3 the question is posed as in Paper 283. There the same “double it” gave results differing by orders of magnitude according to mechanism——here the same delay changes in nature with the type of growth: dependent on the time for a power, independent for an exponential.
Observation 5.4 one root, under Paper 300. “The faster the fire, the heavier the delay” and “the earlier noticed, the heavier the delay” both come from the one ratio $\Delta t/t_d$——a fast fire has a short $t_d$, and so does a fire noticed early.
Observation 5.5 what is not said. No fire protection is designed, and no “within so many seconds” is stated——what is shown is that the weight of a minute is not one number but is set by how the fire grows and the ratio to when it was noticed.
6. Roll Call and Closing
- $\alpha$ for four fires (Computation 2.1): from $\mathbf{0.002931}$ to $\mathbf{0.187556}$ kW/s$^2$.
- the same $60$ seconds weigh $\mathbf{1.7551}$ to $\mathbf{12.9489}$ times (Computation 2.1): by how the fire grows.
- heat released $\mathbf{2.3252}$ to $\mathbf{46.5962}$ times (Computation 2.1): it goes as the cube.
- noticed at $50$ seconds $\mathbf{4.840000}$ times, at $400$ seconds $\mathbf{1.322500}$ (Computation 3.1): the core.
- noticed at $120$ seconds, delayed $60$: $\mathbf{2.250000}$ times (Computation 3.3): heat released $\mathbf{3.375000}$ times.
- linear $\mathbf{1.600000}$, squared $\mathbf{2.560000}$, exponential $\mathbf{2.718282}$ (Computation 4.1): the type changes the nature.
- only the exponential ignores when the fire was noticed (Observation 4.2): no $t_d$ in the formula.
- $\mathbf{0.385820}$ kg of water per second per MW (Computation 4.3): the water needed is proportional.
- the separator is power or exponential (Observation 5.1): when it was noticed, or how long the delay.
- the shape of Paper 558 (Observation 5.2): what changes is the place on the curve, not the dimension.
Closing: In a fire, “every second counts”. Yet the weight of one and the same minute changes greatly with how the fire grows and when it was noticed——sorting fires whose heat release grows as $\alpha t^2$ by the time taken to reach $1055$ kW gives $\alpha$ from $\mathbf{0.002931}$ to $\mathbf{0.187556}$ kW/s$^2$. A $60$-second delay after noticing at $100$ kW multiplies the heat release by $\mathbf{1.7551}$ for a slow fire and $\mathbf{12.9489}$ for an ultra-fast one, and the heat released so far by $\mathbf{2.3252}$ to $\mathbf{46.5962}$. The weight of a delay is $(1+\Delta t/t_d)^2$, and the earlier a fire is noticed, the heavier the delay that follows——a $60$-second delay weighs $\mathbf{4.840000}$ times after noticing at $50$ seconds and $\mathbf{1.322500}$ after noticing at $400$. For a fire noticed at $120$ seconds it is exactly $\mathbf{2.250000}$ times in heat release and $\mathbf{3.375000}$ in heat released, and the water needed rises by the same factor, $\mathbf{0.385820}$ kg per second for each MW. Linear growth gives $\mathbf{1.600000}$, squared $\mathbf{2.560000}$, exponential $\mathbf{2.718282}$, and only the exponential ignores when the fire was noticed. The separator is whether growth follows a power or an exponential——for a power the weight of a delay is set by when the fire was noticed, for an exponential only by how long the delay was. Under Paper 300, “faster fire, heavier delay” and “noticed earlier, heavier delay” share the one ratio $\Delta t/t_d$. Placed among the earlier papers——as the same ten per cent of Paper 558 became different factors by dimension, the same minute becomes different factors by its place on the curve. As the same doubling of Paper 283 differed by mechanism, the same delay changes in nature with the type of growth. To be honest——only the growth phase was considered, without limits to spread, the ventilation plateau or burnout. Noticing at $100$ kW is a representative value, and no detector or sprinkler was designed. The water estimate is the heat up to evaporation, without the share that reaches the fire. And no “within so many seconds” is stated: what is shown is that the weight of a minute is not one number.
References
- Drysdale, D. (2011). An Introduction to Fire Dynamics (3rd ed.). Wiley.
- Hurley, M. J. (Ed.) (2016). SFPE Handbook of Fire Protection Engineering (5th ed.). Springer.
- Karlsson, B., & Quintiere, J. G. (2000). Enclosure Fire Dynamics. CRC Press.
- Yamagishi, Y. (2026). One and the Same “Double the $Q$” Becomes $5.69\times10^{18}$, $32$, and $8$ [Paper 283]. Zenodo. 10.5281/zenodo.22171127
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Same “Ten Per Cent” Becomes 1.4641 or 0.7513, Depending on Which Dimension [Paper 558]. Zenodo. 10.5281/zenodo.23014230
Reference
Yamagishi, Y. (2026). Fire Grows as the Square of Time, So the Weight of the Same Minute’s Delay Changes [Paper 577]. YOI: https://www.yamagishiyuuki.com/yoi/577
DOI: 10.5281/zenodo.23014319
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 577. This paper is a preprint and has not been peer reviewed.