Approximate closed-form solution for effectiveness factors of catalyst pellets with arbitrary shapes

This study presents a universal closed-form analytical solution for predicting the effectiveness factor ( E ) of porous catalyst pellets with arbitrary geometries governed by first-order reaction kinetics. By introducing a generalized shape factor ( F p = L min A p / V p ) that maps any pellet geometry onto a continuously varying F p ∈ [1, 3], we have applied the diffusion-reaction equation in terms of F p , which is exact for the three basic shapes of catalysts, to catalyst pellets of any non-basic shapes. We then obtained the following expression of the effectiveness factor in terms of the modified Bessel function of the first kind: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mi>E</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:msub> <m:mi>I</m:mi> <m:mfrac> <m:msub> <m:mi>F</m:mi> <m:mi>p</m:mi> </m:msub> <m:mn>2</m:mn> </m:mfrac> </m:msub> <m:mrow> <m:mfenced close=")" open="(" separators="|"> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mi>p</m:mi> </m:msub> <m:msub> <m:mi>λ</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:mfenced> </m:mrow> </m:mrow> <m:mrow> <m:msub> <m:mi>λ</m:mi> <m:mi>p</m:mi> </m:msub> <m:msub> <m:mi>I</m:mi> <m:mfrac> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:msub> <m:mrow> <m:mfenced close=")" open="(" separators="|"> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mi>p</m:mi> </m:msub> <m:msub> <m:mi>λ</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:mfenced> </m:mrow> </m:mrow> </m:mfrac> </m:mrow> </m:math> E=I_Fₚ2(Fₚλ ₚ)λ ₚI_Fₚ-22(Fₚλ ₚ) where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:msub> <m:mi>λ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>=</m:mo> <m:mfrac> <m:msub> <m:mi>V</m:mi> <m:mi>p</m:mi> </m:msub> <m:msub> <m:mi>A</m:mi> <m:mi>p</m:mi> </m:msub> </m:mfrac> <m:msqrt> <m:mrow> <m:mi>k</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>D</m:mi> <m:mi>e</m:mi> </m:msub> </m:mrow> </m:msqrt> </m:mrow> </m:math> λ ₚ=VₚAₚ√k/Dₑ is the generalized Thiele modulus. Rigorous validation of this expression of effectiveness factor confirmed that it converts to the known analytical expressions for basic shapes, correct asymptotic limits, E →1 as λ p →0, and E ˜ 1/ λ p as λ p → ∞ for all values of F p . Furthermore, this expression of the effectiveness factor yields exceptional agreement (<5 % error) with exact numerical simulations for complex shapes (finite/hollow cylinders, cones, parallelepipeds). The solution eliminates complex computational simulations or shape-specific approximations, enabling the rapid optimization of industrial catalyst pellets.