$(50)$ Finding the Flow Rate through a Water Bio$-$Treatment System

Contaminated water at room temperature is to be treated in a large packed$-$

bed bioreactor as shown on the next page to remove sulphate and metal ions.

The pressure rise across the pump is $50$kPa.

a$)(10)$ Calculate the specific work $($in $\frac{J}{k}g)$ of the pump.

b$)(25)$ Calculate the volumetric flowrate $($in $\frac{L}{s})$ through the system.

c$)(10)$ Calculate the gauge pressure $($in kPa $)$ going into the pump.

d$)(5)$ Calculate the power requirement for the pump $($in $kW)$ assuming $60\%$

efficiency.

Data:

All pipe: $,$ Welded steel, $4\mathrm{.}0$cmID

All elbows: $,90$ standard radius

Bed packing: $,$ spherical biofilm support, $3mm$ diameter, $35\%$ porosity

Water data: $,=997k\frac{g}{m^3},=8\mathrm{.}57{10}^{-4}k\frac{g}{m}\frac{?}{s}$

Hints:

Ignore expansion and contraction at the ends of the packed bed, and

assume fully turbulent flow in the pipes.

Rearrange Ergun's equation into a friction loss $(E_f)$ term for this problem

Use the equation of continuity to relate the superficial velocity in the

packed bed to the average velocity in the pipes.

The best way to solve this problem is to write Bernoulli's equation for the

entire system $($including the packed bed$)$ and solve it for average velocity in

the pipes. You may find the quadratic equation useful:

$x=\frac{-b+-\sqrt[\phantom{2}]{b^2-4ac}}{2a}$ when $a{x}^2+bx+c=0$