CHE $447$: Heat and Mass Transfer in Chemical Engineering Processes

Homework Problem Set #$1$

Due Thursday January $18,2024$ before $11$:$59$ PM on Blackboard.

Consider a stagnant, conical pipe containing "species $A"$ with the concentration profile

and $L=10cm.$ At $z=0,$ the radius of the pipe $1$

$cm$ and at $z=10cm,$ the radius of the pipe is $3cm.$ Assume that the concentration profile

is radially uniform for a given value of $z.$ Calculate the total moles of species A within the

pipe.

Consider an open$-$top, well$-$mixed tank. At time $=0s,$ the contents are drained by gravity

through a circular hole at the bottom with area $A_1=0\mathrm{.}1m^2.$ Also at time $=0s,$ a stream

containing species $A$ at a concentration $C_{A0}=10mo\frac{l}{m^3}$ is pumped into the top of the tank

at a constant flow rate $F_{in}=0\mathrm{.}8\frac{m^3}{s}.$ The tank has a cross$-$sectional area $A_2=10m^2$ and

is initially filled with water at a height $h=5m.$ The initial concentration of species $A$ in the

tank is $0mo\frac{l}{m^3}.$

a$.$ Perform an overall mass balance on the liquid contents of the tank to derive an

equation for $d\frac{h}{d}t,$ the rate of change of the liquid level, as a function of Fin, A$2,$

$A1,h,$ and $g($the gravitational constant$).$ Assume that the velocity of the liquid level

in the tank $(v_2)$ is significantly less than the exit velocity of the fluid through the

drainage hole $(v_1).$ Assume that the density of the liquid is constant and unaffected

by the mixing of species $A.$

b$.$ Solve the differential equation in part a and plot $h$ as a function of time over the

interval $0t600$ seconds.

c$.$ Perform a mass balance on species $A$ to derive an equation for dCAdt as a

function of $F_{in},A_2,A_1,h,g,C_A$ and $C_{A0}.$ Solve this differential equation and plot $C_A$

as a function of time over the interval $0t600$ seconds. Also plot the total moles

in the system or $\{$:$A_2^{**}{h}^{**}CA)$ in a separate plot, using the same time interval.

d$.$ Is the assumption $v_1{v}_2$ valid? Justify your answer.