A bag of catalyst has recently been delivered to ACE Pte. Ltd$..$ The Certificate of Analysis of the catalyst provides the particle size distribution by dynamic light distribution, as laid out in Table $1.$ Fv is the cumulative volume distribution $($normalized$)$ of the particles, and x is the diameter in micrometer.

Part $1$

As the catalyst will be used in fluidized bed reactors, it is necessary to determine its surface volume mean for further calculations.

Using the cumulative volume distribution given, determine the

$($a$)$ the harmonic mean of volume distribution,

$($b$)$ the arithmetic mean of surface distribution, and

$($c$)$ the surface volume mean.

All values should be calculated using appropriate conversions from the cumulative volume distribution values given. An Excel spreadsheet should be used for the calculations.

Hint: Make use of the general equation and table from the textbook, given in the next page.

Table $1.$ PSD for the catalyst particles

$\backslash$table$[[x($micron$),$Cumulative normalized Fv$],[30,0],[50,0\mathrm{.}005],[60,0\mathrm{.}01],[90,0\mathrm{.}016],[110,0\mathrm{.}026],[130,0\mathrm{.}057],[150,0\mathrm{.}087],[190,0\mathrm{.}143],[250,0\mathrm{.}222],[300,0\mathrm{.}338],[390,0\mathrm{.}72],[420,0\mathrm{.}909],[450,0\mathrm{.}993],[500,1]]$