[NaDoume]

Greetings everybody.
 
In the quest of completing my reactor design I've gone over multiple books dealing with heterogeneous kinetics. One thing that has struck me is  that authors consistently neglect the variation in effectiveness factor and concentration profiles caused by change in reaction order above first order(they only consider 0th and 1st orders). Their corresponding equations are related to shape and type of operation(more commonly whether or not isothermal) only.

For those unfamiliar with the terms , not that I am proficient on the matter, the effectiveness factor is essentially the ratio of actual rate of reaction over the rate of reaction that’d occur if the surface concentration of reactant  were the same everywhere within the pellet. So it gives insight on the rate of mass transfer within the pellet.

The concentration profile is the fraction of the diffusing reactant's concentration over its concentration on the catalyst surface , at varying distances within the catalyst pellet.

The equations are derived from material balances as indicated in the attachments for a first order isothermal reaction. The attachments are taken from  -->  (Peter Harriot, Chemical Reactor Design,2002 page 151). The first attachment is for concentration profile and the second is for the effectiveness factor..  All books looked at for this particular matter seemed to only make concentration profiles and provide effectiveness factors for first order reactions. Generally they are depicted as if they are of standard form for a given shape irrespective of other factors.. So as all sources consistently don't  look at higher than 1st orders ive come up with two reasonings to justify this: either solutions are too simple/obvious and for convenience/simplicity demonstration is given for 1st order(so indeed just a coincidence that first orders were only seen) or they are too complicated and not worthwhile. A particular source provided a justification for this apparent consistency by claiming that a "reasonable" approximation of the effectiveness and concentration profile can be obtained using the equations derived (from the material balance on the catalyst pellet) for the an isothermal 1st order reaction, without however giving any evidence of validity.

Specifically I am looking at the concentration profile and isothermal effectiveness factor for spherical catalyst pellets hosting a 2nd order heterogeneous reaction. I've plotted the concentration profile , as if the reaction order were first, but doing this really doesn't feel right.
 Doing the material balance is straightforward, correct me if I'm wrong but the only thing that will change are the units of K and the concentration term will be raised to the power of 2. Subsequently like in all derivations for this matter, i've neglected the (dR)²⁺ term. But I have trouble deriving the equations as I don't understand the differentials involved. I understand that we are differentiating with respect to radius, but i don't understand what happens to the second order differential?should't at the end of the differentiation be a first order differential left?  Are there any intermediate steps not presented or do I have to de-rust my mathematics ? In the use of the equation I do NOT need  proof, just a valid relationship so to demonstrate the evolution of  my component within pellet.

Thank you all for your time

(If I may say so, I'd like to express special gratitude to Billnotgatez for pinpointing me to the right direction and respecting my content by not sending it to oblivion)

(....hopefully this time I will not get raided by the forum police , but by their plentiful engineering knowledge, trimmed and designed to perfection so to enlighten amateurs like me to the path of knowledge )