Question $1$

A fixed bed reactor used for ammonia production is loaded with monodisperse spherical particles

having a diameter of $3mm$ and a density of $6\mathrm{.}8gc{m}^{-3}.$ The reactor is fed with an inlet nitrogen

pressure of $1$ bar and operated at $325C.$ The conversion is low at those conditions, so the

concentration of nitrogen is assumed to be constant outside the particles. The effective diffusivity

of nitrogen in the particles is $3\mathrm{.}6c{m}^2{h}^{-1}.$ The turnover frequency of nitrogen on the metal sites of

the catalyst particles is $0\mathrm{.}04$ molecules. site ${?}^{-1}{s}^{-1}$ at the operating temperature. The metal surface area

in catalyst particles is determined to be $40m^2{g}^{-1}$ and the surface area of each metal surface atom

is $10{}^2.$ Assume that the inlet molar fractions of nitrogen and hydrogen are stoichiometric, and

the reaction is first order with respect to nitrogen.

$($a$)$ Estimate the Thiele modulus ${}_S.$

$($b$)$ Estimate the effectiveness factor $.$

$($c$)$ If the packing density of catalyst particles is $0\mathrm{.}7,$ find the reaction rate of nitrogen per unit

volume of the reactor $(mol\frac{N_2}{m^3}(s)).$

$($d$)$ Ammonia is synthesized using a microreactor with a diameter of $30mm$ and loaded with

$15g$ of catalyst particles. Find the range of total inlet flowrates $(mo\frac{l}{s})$ at $4$ bar total

pressure and $325C$ for a stoichiometric mixture of reactants, if given that the nitrogen

conversion at the outlet should not exceed $0\mathrm{.}03.$ For such low conversion, the rate can be

assumed uniform $($constant$)$ through the packed bed and pressure drop can be neglected.

$($e$)$ Create the Arrhenius plot $($ln R vs $1/$T$)$ at a temperature range of $450-750K$ and find the

range of the apparent activation energies that can be extracted $($show your plot and

calculation$).$ The activation energy for the rate constant at this temperature range is $65$

$k\frac{J}{m}ol$ and the effective diffusivity is assumed to follow the correlation:

$D_c=D_{c,}{T}^{1\mathrm{.}5}$

$T$ is the temperature in $K$ and the parameter $D_{c,}$ can be obtained using the given value of

$D_c$ at $325C.$