Nondimensionalization

The dimensionless equations are derived for the special case of constant physical properties. Let r = r'/R, c = c'/c₀, T = T'/T₀, and R₁ = R'/c₀ⁿ. Thus,

and

at the origin

while at the outer boundary

The dimensionless groups are

The corresponding equations for cylindrical and planar geometry are obtained by replacing r² by r and 1, respectively, in the differential equations.

The dimensionless groups have physical meaning. The most important parameter is the Thiele modulus phi. The group 1/(k₀ c₀^n-1) is a characteristic time for reaction, while R²/D_e is a characteristic time for diffusion. The Thiele modulus squared is thus the ratio of two characteristic times, diffusion to reaction. If the reaction is very fast, its characteristic time is small and the Thiele modulus is large; the concentration in the center of the pellet may be small because the reaction is so fast that the material can't diffuse to the center before it is reacted. Likewise, if the diffusion is very fast its characteristic time is small and the Thiele modulus is small; now the concentration may not vary very much within the pellet because the diffusion is fast enough to keep the pellet filled with reactant. The Thiele modulus thus measures the relative importance of the diffusion and reaction phenomena.

The Biot number for mass transfer, Bi_m, is the ratio of two characteristic times: R²/D_e for diffusion of mass transfer across the inside of the pellet and R/k_g for mass transfer across the boundary layer outside the pellet. For large Bi_m the characteristic time for internal diffusion is large compared with the external diffusion, and internal diffusion dominates. The Biot number for heat transfer, Bi, is likewise the ratio of the characteristic time R² r C_p T₀ / k_e for diffusion of heat internal to the pellet to the characteristic time R r C_p T₀ / h_p for heat transport across the thermal boundary layer external to the catalyst pellet. Typically the Biot number for mass is large (say 100 or larger), making internal diffusion important. The Biot number for heat is smaller (about five) making external heat transport important. The effect of temperature is also related to the dimensionless heat of reaction beta and the dimensionless activation energy

where E is the activation energy and is the gas constant. Clearly, if b or g is small the temperature effect is less.

Continue on to: Combination of Variables, Effectiveness Factor, Linear Problem, Nonlinear Problem.

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