A lift with a volume $V_L$ is ventilated by the small vertical gap left between its

closed doors. The total volumetric flow$-$rate at which the atmosphere of the lift

can pass to the outside is $R.$ An open$-$top canister of liquid nitrogen $(N_2)$ is being

transported in the lift, starting at time $t=0,$ and is supplying a volumetric flow$-$rate

$Q_{in}$ of cold $N_2$ gas into the lift's atmosphere. The cold gas sinks to the floor of the

lift, displacing the air, which is why no one is allowed to travel in the lift with the

canister.

$($a$)$ If the volume of the cold $N_2$ gas in the lift at time $t$ is $Q_{\text{o}\text{u}\text{t}}{N}_2{Q}_{\text{o}\text{u}\text{t}}=\frac{V(t)}{V_L}R{N}_{2}V(t)V(t)V(t)Q_{in}=R{Q}_{in}$ what $is$ the long$-$time steady state volume $of$ cold $N_2$

gas $in$ the lift?

$[2$ marks$]$

$(c)$ Sketch a graph showing how $V(t)$ varies over time and include scales $on$

your axes.

$[4$ marks$]$

$(d)$ Perform a material balance and derive a first$-$order differential equation for

the time dependence $ofV(t).$

$[4$ marks$]$

$(e)$ Solve the equation you obtained $in$ part $(d)$ for $V(t).$

$[6$ marks$]$

$(f)$ How does your solution $to$ the above problem change $if{Q}_{in}=R$

$[2$ marks$]V(t),$ show that

the volumetric flow rate $Q_{\text{o}\text{u}\text{t}}at$ which cold $N_2$ gas leaves the lift $is$ given

$by$

$Q_{\text{o}\text{u}\text{t}}=\frac{V(t)}{V_L}R$

$[2$ marks$]$

$(b)$ Assuming $Q_{in}$ what $is$ the long$-$time steady state volume $of$ cold $N_2$

gas $in$ the lift?

$[2$ marks$]$

$(c)$ Sketch a graph showing how $V(t)$ varies over time and include scales $on$

your axes.

$[4$ marks$]$

$(d)$ Perform a material balance and derive a first$-$order differential equation for

the time dependence $ofV(t).$

$[4$ marks$]$

$(e)$ Solve the equation you obtained $in$ part $(d)$ for $V(t).$

$[6$ marks$]$

$(f)$ How does your solution $to$ the above problem change $if{Q}_{in}=R$

$[2$ marks$]$