On the relationship between Aris and Sherwood numbers and friction and effectiveness factors

We present a general solution of the diffusion–reaction problem for linear kinetics and an expression for the effectiveness factor (ηη) in terms of the shape normalized Thiele modulus (ΦΦ) for a catalyst particle of arbitrary shape and with an arbitrary activity profile. We also show that the Sherwood number (ShΩShΩ) or the dimensionless mass transfer coefficient between the interior of a particle or region (denoted by ΩΩ) and its boundary (∂Ω∂Ω) is related to the effectiveness factor and Thiele modulus by η=1/(1+Φ2/ShΩ)η=1/(1+Φ2/ShΩ). Further, the coefficients in the expansion of ηη in terms of ΦΦ (the Aris numbers, AriAri) are related to the asymptotic Sherwood number (ShΩ∞ShΩ∞) obtained in the limit of slow reaction. We show that the curve ShΩShΩ versus ΦΦ is universal for most common particle or channel geometric shapes and for the case of uniform activity is described by the two asymptotes ShΩ=ShΩ∞=1/Ar1ShΩ=ShΩ∞=1/Ar1 for Φ⪡1Φ⪡1 and ShΩ≈ΦShΩ≈Φ for Φ⪢1Φ⪢1. For two-dimensional ducts we show that the friction factor times Reynolds number (fRe  ) is equal to 8ShΩ∞8ShΩ∞ and provide a physical interpretation of this result. In the second part of this work, we derive low-dimensional models for solving multicomponent nonlinear diffusion–reaction problems using the concept of an internal mass transfer coefficient. We also present low-dimensional models for catalytic reactors using external and internal mass transfer coefficients. Finally, the Aris/Sherwood numbers are presented for some commonly used catalyst particles in packed-bed reactors and washcoat shapes in catalytic monoliths.