Problem $1.$ Scented oil $(=0\mathrm{.}88\frac{g}{c}{m}^3)$ and surfactant filled water $(\{$:$=1\mathrm{.}1\frac{g}{c}{m}^3)$ enter a mixing

device shown below with circular inlet pipes of radius $R_{in}=0\mathrm{.}05m$ and a circular outlet pipe of

radius $R_{\text{o}\text{u}\text{t}}=0\mathrm{.}2m.$ The liquids mix homogenously to form the active ingredient used in air

fresheners. The oil enters at $v=30\frac{m}{s}$ and the water enters at $v=22\frac{m}{s}.$ If the exit velocity is

$v=3\mathrm{.}5\frac{m}{s},$ what is the density of the resulting mixture?

a$)(2$ pt$.)$ Choose a fixed control volume, identify the important control surfaces, and draw

respective velocity vectors and normal vectors at each influx and efflux point on the

control volume $($you may redraw the diagram if you like$).$

b$)(3$ pt$.)$ Write out the conservation of mass equation and simplify the integral relationship

by evaluating each relevant control surface. Report you simplified conservation of mass

equation in terms of the relevant variables like $v,,$ and $A.$

c$)(5$ pt$.)$ Solve your expression to find the density of the mixture.Scented oil $($

r $=0\mathrm{.}88$ g$/$cm$3)$ and surfactant filled water $($

r $=1\mathrm{.}1$ g$/$cm$3)$ enter a mixing

device shown below with circular inlet pipes of radius Rin $=0\mathrm{.}05$ m and a circular outlet pipe of

radius Rout $=0\mathrm{.}2$ m$.$ The liquids mix homogenously to form the active ingredient used in air

fresheners. The oil enters at v $=30$ m$/$s and the water enters at v $=22$ m$/$s$.$ If the exit velocity is

v $=3\mathrm{.}5$ m$/$s$,$ what is the density of the resulting mixture?