You are tasked with scale$-$up of a CSTR$,$ from pilot$-$scale to demonstration$-$scale. The

required settings should be optimal for an aerobic mammalian cell culture process. While the

pilot$-$scale reactor volume was $200L,$ the demonstration$-$scale bioreactor is to have a liquid

volume $V_L$ of $3000L.$ It will require $2$ marine$-$type impellers for mixing. The bioreactor is

designed with the following geometric ratios $($assume that the tank is a cylinder$)$:

$\frac{D_{\text{I}}}{D_T}=0\mathrm{.}35,[$diameter of impeller $/$ diameter of tank$]$

$\frac{x}{D_T}=0\mathrm{.}4,[$distance of impeller from tank bottom $/$ diameter of tank$]$

The culture has a density ${}_L$ of $1050kg{m}^{-3}$ and a viscosity $$ of $9\mathrm{.}5{10}^{-4}kg{m}^{-1}{s}^{-1}.$ The power

number of a marine$-$type impeller is $0\mathrm{.}35$ when the mixing regime is turbulent.

a$)$ Calculate $D_T,H_L,D_{\text{I}},x(m)$ and the cross$-$sectional area $A$ of the pilot$-$scale and

demonstration$-$scale tanks $(m^2).$ State your assumed $\frac{H_L}{D_T}$ ratio for the bioreactors.

b$)$ If the impeller speed is $N=150$ revolutions per minute $(rpm$ at the pilot$-$scale,

calculate the impeller speed at demonstration$-$scale assuming a constant $\frac{P}{V}$ ratio.

Also calculate the Reynolds number $N_{Re}($dimensionless$)$ and the power

consumption $P(W)$ of the agitation system $($ungassed$)$ at demonstration$-$scale.