Problem $3(35$ points$)$: FRB $6\mathrm{.}59$

Here we are doing a variant of Problem $6\mathrm{.}59$ from Felder, Rousseau, and Bullard. The original problem states: "Nitrogen

is bubbled through a liquid mixture that initially contains equimolar amounts of benzene and toluene. The system pressure

is $3$atm and the temperature is $80C.$ The nitrogen flow rate is $10\mathrm{.}0$ standard liters per minute. The gas leaving the bubbler

is saturated with benzene and toluene vapors."

The original problem in the text then asked you to estimate the initial flow of benzene and toluene. Now you are in

position to calculate all of the variables as they change over time. Assume the tank begins with $1$mol of total mixture.

$($a$)(1)$ Find the total molar flow rate of the output stream, $n_v^{},$ in terms of the molar flow rate of nitrogen, $n_{N2}^{},$ and the

vapor mole fractions of benzene and toluene, $y_B$ and $y_T,$ respectively.

$($b$)(1)$ Using Raoult's law, write the relationships between $x_B$ and $y_B$ and between $x_T$ and $y_T.$

$($c$)(8)$ Write the transient material balance for the total moles in the tank.

$($d$)(8)$ Write the transient material balance for the moles of benzene in the tank, $n_B.$

$($e$)(7)$ The number of moles of benzene is the product of the liquid mole fraction, $x_B,$ and the total moles, $n_{\text{T}\text{o}\text{t}}.$

Manipulate the balance on the moles of benzene to have $d\frac{x_B}{d}t$ and $d\frac{n_{\text{T}\text{o}\text{t}}}{d}t$ rather than $d\frac{n_B}{d}t.$

$($f$)(5)$ Use Matlab and ode$45$ to solve the differential equations for $d\frac{x_B}{d}t$ and $d\frac{n_{\text{T}\text{o}\text{t}}}{d}t$ on the time interval tin$[0,10]$

minutes.

$($g$)(3)$ What happens right before $10$ minutes elapse?

$($h$)(2)$ Speculate on what is wrong with the equations $($i$.$e$.,$ what is wrong with the assumptions that went into your

equations$)$ that would allow this event in part $($g$)$ to happen. In other words, where does the physics of the problem

break down?