Problem I:

Water $($density $=1000k\frac{g}{m^3}$ and viscosity $={10}^{-3}Pa*s)$ is pumped from an open reservoir through a cast iron pipe $($unknown length $L_p,D=4cm)$ through a bed of sand particles $($crosssectional area $1\mathrm{.}0m^2)$ for filtration. The water overflow is open to the atmosphere and at the same height as the reservoir level. For the filtration to work, fluidization must be prevented, because that could lead to contamination of the water downstream with sand particles. The sand particles can be considered to be spherical, have diameter $D_p=0\mathrm{.}6mm$ and their void fraction at minimum fluidization is known to be $0\mathrm{.}45.$ The total mass of particles in the bed is $1250kg.$ Without fluid flow, the height of the packed bed is $0\mathrm{.}85m.$ When the fluid flow is increased slowly, fluidization is observed when the bed height is equal to $1m.$

a$)$ Calculate the density of the particles.

b$)$ Calculate the maximum superficial velocity and volumetric flow rate at which water can be pumped through the bed.

c$)$ It is found that the maximum flow rate $Q_{\text{m}\text{a}\text{x}}$ is achieved when $600W$ is supplied to the pump, which has a $75\%$ efficiency. Calculate the total head loss of the piping network $($pipe$,$ valves, contractions, etc.$)$ that connects the reservoir to the pump and the pump to the bed. The water velocity in the bed overflow can be neglected.

d$)$ If we assume that all frictional losses in the piping network are due to the pipe itself, calculate the length $L_p$ of the pipe.

CAN YOU PLEASE DO ONLY $($C$)$ and $($D$)?$