Q$3.($Learning Outcome $3)$ The following equations are used to calculate the thermal response of a detector or sprinkler located at or near a ceiling whose area is large enough to neglect the effects of smoke layer development. When the detector or link temperature reaches its activation temperature, then the detector will be activated.

The rate of temperature rise of the detector response is modelled by

$\frac{T_{D,t+t}-T_{D,t}}{t}=(T_{\text{j}\text{e}\text{t}\text{,}t+t}-T_{D,t})(1-e^{-\frac{1}{}})+(T_{\text{j}\text{e}\text{t}\text{,}t+t}-T_{\text{j}\text{e}\text{t}\text{,}t})(e^{-\frac{1}{}}+\frac{1}{}-1)$

Where

$=\frac{\text{R}\text{T}\text{I}}{\sqrt[\phantom{2}]{v_{\text{j}\text{e}\text{t}t}t}}$

$v_{\text{j}\text{e}\text{t}\text{,}t}=\{\begin{array}{ccc}0\mathrm{.}95(\frac{Q_t}{z}{)}^{\frac{1}{3}},& \text{f}\text{o}\text{r}& \frac{r}{z}0\mathrm{.}15\end{array}$

$0\mathrm{.}2\frac{Q_t^{\frac{1}{3}}{z}^{\frac{1}{2}}}{r^{\frac{5}{6}}},$ for $\frac{r}{z}>0\mathrm{.}15$

$T_{\text{j}\text{e}\text{t}\text{,}t}=\{\begin{array}{ccc}{T}_{}+\frac{16\mathrm{.}9e^{}{?}^{\frac{2}{3}}}{z^{\frac{5}{3}}},& \text{f}\text{o}\text{r}& \frac{r}{z}0\mathrm{.}18\end{array}$

$T_{}+\frac{5\mathrm{.}38}{z}(\frac{Q_t^{}}{r}{)}^{\frac{2}{3}},$ for $\frac{r}{z}>0\mathrm{.}18$

$Q_t^{},$ Total theoretical fire heat release rate at time $t(kW)$

$r,$ Radial distance of the detector$/$sprinkler from the vertical axis of the fire $($m$)$

RTI Response Time Index of detector$/$sprinkler

$T_{\text{j}\text{e}\text{t}\text{,}t+t},$ Temperature of the jet at the next time step, $t+t(C)$

$T_{\text{j}\text{e}\text{t}\text{,}t},$ Temperature of the jet at the previous time step, $t(C)$

$T_{},$ Ambient space and initial detector$/$sprinkler temperature $(C)$

$T_{D,t},$ Detector or sprinkler temperature at time, $t(C)$

$v_{\text{j}\text{e}\text{t}\text{,}t},$ Velocity of the ceiling jet gases at the time step, $t(\frac{m}{s})$

z Vertical entrainment distance; the difference between the height of the ceiling and the base of the flames $($m$)$

$Q_{\text{p}\text{e}\text{a}\text{k}}^{}=*T^2$

Assume the fire will develop as a t$-$square growth fire. Four fire classes named as $"$U$"-$ ultra$-$fast, $"$F$"-$ fast, $"$M$"$ medium and $"$S$"-$ slow will be considered. The fire growth rate table is summarized at the right table.

$(1)$ Plot the t$-$square growth fire curve till the activation of the detector$/$sprinkler;

$(2)$ Plot the detector$/$sprinkler temperature $(T_{D,t}),$ ceiling jet temperature $(T_{\text{j}\text{e}\text{t}\text{,}t})$ varying with time till the activation of the detector$/$sprinkler; and

$(3)$ Save the iteration results for all the variables including $Q_t^{},v_{\text{j}\text{e}\text{t}\text{,}t},,T_{\text{j}\text{e}\text{t}\text{,}t}$ and $T_{D,t}$ to a text file named as "output.txt$"$ at the end of the simulation.