$(1)$

A test is being carried out to determine the specific surface of alumina powder. In the test, a packed bed of the powder is placed in a $0\mathrm{.}1m$ diameter tube and the pressure drop of a flow rate of air passing through the bed is measured.

Test results are as follows:

Mass of alumina charged to bed: $2kg$

Depth of alumina: $0\mathrm{.}1m$

Density of alumina: $4000k\frac{g}{m^3}$

Flow rate of air: $3\mathrm{.}9{10}^{-5}{m}^3{s}^{-1}$

Pressure drop across the bed: $1090N{m}^{-2}$

Under the conditions of the test, the following data applies for the air stream:

Density: $1\mathrm{.}15k\frac{g}{m^3}$

Viscosity: $1\mathrm{.}8{10}^{-5}kg{m}^{-1}{s}^{-1}$

Using the Carman$-$Kozeny relationship:

Calculate the specific surface of the material.

Calculate the mean diameter of the alumina particles if they may be assumed to be spherical.