An Asymptotic Solution for Washcoat Pore Diffusion in Catalytic Monoliths: Reformulation and Extension to Small Concentrations

Abstract

A recent publication (Bissett in Emission Control Sci. Technol. 1(1), 3–16, 2015) proposed an alternative to the so-called 1 + 1D modeling of aftertreatment reactors with nontrivial washcoat pore diffusion. Rather than numerically solve the 1D reaction-diffusion problem within the washcoat(s), asymptotic results based on small diffusion resistance give the concentration profiles within the washcoat analytically, and these are integrated within the overall solution for transient reactor performance. The description of the asymptotic solution in the former publication is suitable for the formal derivation and demonstration that all special properties of this solution follow from small diffusion resistance alone, but experience has shown that alternative descriptions and further extensions to accommodate small washcoat concentrations are desirable and perhaps necessary for practical application. In this paper and in a less formal style, we provide the new alternatives and analysis necessary for small concentrations.

Appendix

This appendix contains the details we have found necessary for the robust and routine solution of the nonlinear equation for the implicit solution of H in region 3. A possible alternative is to derive a fifth-order polynomial for either x_m,0 or 1 − x_m,1 by eliminating the other two variables in Eq. (14) and then solving for the root in [0,1]. We chose not to pursue this option and so cannot speak to its pitfalls.

For any fixed, positive $\left({\omega}_g,{\overline{\omega}}^{(1)},{\overline{\omega}}^{(2)}\right),$ we must solve the nonlinear, scalar equation for $H{\overline{\omega}}^{(1)}$ from Eq. (14) in the form

$$ G\left(H{\overline{\omega}}^{(1)}\right)={\left(H{\overline{\omega}}^{(1)}\right)}^2\left[{\left(\frac{x_{m,0}}{H{\overline{\omega}}^{(1)}}\right)}^3+{\left(\frac{1-{x}_{m,1}}{H{\overline{\omega}}^{(1)}}\right)}^3\right]-1=0 $$

(18)

where, for purposes of evaluating G, we obtain the following two functions of $H{\overline{\omega}}^{(1)}$ from Eq. (14):

$$ \frac{x_{m,0}}{H{\overline{\omega}}^{(1)}}=\frac{\frac{1}{3}{D}_{inv}^{(1)}K\frac{\omega_g}{{\overline{\omega}}^{(1)}}}{1+\sqrt{1+\frac{1}{3}{\left({D}_{inv}^{(1)}K\right)}^2\frac{\omega_g}{{\overline{\omega}}^{(1)}}H{\overline{\omega}}^{(1)}}} $$

(19)

$$ \frac{1-{x}_{m,1}}{H{\overline{\omega}}^{(1)}}=\frac{\frac{D_{inv}^{(1)}}{D_{inv}^{(2)}}\frac{{\overline{\omega}}^{(2)}}{{\overline{\omega}}^{(1)}}}{1+\sqrt{1+3{\left(\frac{D_{inv}^{(1)}}{D_{inv}^{(2)}}\right)}^2\frac{{\overline{\omega}}^{(2)}}{{\overline{\omega}}^{(1)}}H{\overline{\omega}}^{(1)}}}. $$

(20)

The square roots in Eqs. (19) and (20) come from treating the last two equalities in Eq. (14) as quadratics for x_m,0 and (1 − x_m,1), respectively. The scalings implicit in these equations are to take advantage of

$$ H{\overline{\omega}}^{(1)}={x}_{m,0}^3+{\left(1-{x}_{m,1}\right)}^3 $$

$\Rightarrow \kern0.4em 0\le H{\overline{\omega}}^{(1)}\le 1$ (with x_m,0 ≤ x_m,1). Moreover, both $\frac{x_{m,0}}{H{\overline{\omega}}^{(1)}}$ and $\frac{1-{x}_{m,1}}{H{\overline{\omega}}^{(1)}}$ are finite as H → 0, and $G\left(H{\overline{\omega}}^{(1)}\right)$ has a unique root in $\left[0\le H{\overline{\omega}}^{(1)}\le 1\right]$. The root is unique because $G\left(H{\overline{\omega}}^{(1)}\right)$ is a monotonic function, which can be inferred from

$$ \frac{dG}{d\left(H{\overline{\omega}}^{(1)}\right)}=2\frac{x_{m,0}^3}{{\left(H{\overline{\omega}}^{(1)}\right)}^2}\frac{1+\frac{1}{4}{D}_{inv}^{(1)}\kern0.10em K{x}_{m,0}}{1+{D}_{inv}^{(1)}\kern0.10em K{x}_{m,0}}+2\frac{{\left(1-{x}_{m,1}\right)}^3}{{\left(H{\overline{\omega}}^{(1)}\right)}^2}\frac{D_{inv}^{(2)}+\frac{3}{4}{D}_{inv}^{(1)}\left(1-{x}_{m,1}\right)}{D_{inv}^{(2)}+3{D}_{inv}^{(1)}\left(1-{x}_{m,1}\right)}. $$

This derivative is also used in Newton’s method for the solution of Eq. (18).

If desired, in particular asymptotic limits, H can be explicitly derived without need for the nonlinear solve. These limits also can be the most problematic if left for the nonlinear solve.

$$ {x}_{m,0}\ll \left(1-{x}_{m,1}\right) $$

(A.1)

When this asymptotic limit holds, x_m,0 drops out of the equality between the first and third equality of Eq. (14), which gives a quadratic for (1 − x_m,1), whose explicit solution is

$$ 1-{x}_{m,1}=\frac{3}{2{\overline{\omega}}^{(2)}}\left[{\overline{\omega}}^{(1)}+\sqrt{{\overline{\omega}}^{(1)}\left({\overline{\omega}}^{(1)}+\frac{8{D}_{inv}^{(2)}}{9{D}_{inv}^{(1)}}{\overline{\omega}}^{(2)}\right)}\right] $$

Then H is explicitly given by the last equality in Eq.(14), and j⁽²⁾ is given by the second equality in Eq. (16). Then j⁽¹⁾ is explicit after using Eq. (19) to eliminate $\left(\frac{x_{m,0}}{H{\overline{\omega}}^{(1)}}\right)$ from the first equality of Eq. (15):

$$ {j}^{(1)}=\frac{-2K{\omega}_g}{1+\sqrt{1+\frac{1}{3}{\left({D}_{inv}^{(1)}K\right)}^2H{\omega}_g}}. $$

Finally, x_m,0 is given by solving for it in the first equation of Eq. (15).

$$ {x}_{m,0}=-\frac{1}{6}{D}_{inv}^{(1)}{j}^{(1)}H $$

Since the goal is to determine the fluxes from $\left({\omega}_g,{\overline{\omega}}^{(1)},{\overline{\omega}}^{(2)}\right)$, it is preferable to have a test for this asymptotic region from the concentrations directly. To this end, we note

$$ \frac{x_{m,0}}{1-{x}_{m,1}}=\frac{\frac{1}{6}{D}_{inv}^{(1)}K{\omega}_g}{{\overline{\omega}}^{(2)}}\frac{\left[2\frac{D_{inv}^{(2)}}{D_{inv}^{(1)}}+3\left(1-{x}_{m,1}\right)\right]}{1+\frac{1}{2}{D}_{inv}^{(1)}K{x}_{m,0}}\le \frac{\frac{1}{6}{D}_{inv}^{(1)}K{\omega}_g}{{\overline{\omega}}^{(2)}}\frac{\left(2\frac{D_{inv}^{(2)}}{D_{inv}^{(1)}}+3\right)}{1}={B}_{22}\frac{\omega_g}{{\overline{\omega}}^{(2)}} $$

(21)

so the right-hand-side of this inequality can be used to test for this limiting condition.

$$ \left(1-{x}_{m,1}\right)\ll {x}_{m,0} $$

(A.2)

When this asymptotic limit holds, x_m,1 drops out of the equality between the first and second equality of Eq. (14), which gives a quadratic for x_m,0, whose explicit solution is

$$ {x}_{m,0}=\frac{2}{D_{inv}^{(1)}K}\left(\gamma +\sqrt{2\gamma +{\gamma}^2}\right), $$

where here $\gamma \equiv \frac{3}{4}{D}_{inv}^{(1)}K\frac{{\overline{\omega}}^{(1)}}{\omega_g}$. Then H is explicitly given by the second equality in Eq. (14), and j⁽¹⁾ is given by the second equality in Eq. (15). Then j⁽²⁾ is explicit after using Eq. (20) to eliminate $\left(\frac{1-{x}_{m,1}}{H{\overline{\omega}}^{(1)}}\right)$ from the first equality of Eq. (16):

$$ {j}^{(2)}=\frac{6{\overline{\omega}}^{(2)}}{D_{inv}^{(2)}+\sqrt{{\left({D}_{inv}^{(2)}\right)}^2+3{\left({D}_{inv}^{(1)}\right)}^2H{\overline{\omega}}^{(2)}}} $$

Finally, 1 − x_m,1 is given by solving for it in the first equation of Eq. (16).

$$ 1-{x}_{m,1}=\frac{1}{6}{D}_{inv}^{(1)}{j}^{(2)}H $$

To test for this asymptotic region using the concentrations, a similar estimate to that used in Eq. (21) shows that $\frac{1-{x}_{m,1}}{x_{m,0}}\le \frac{3{B}_{12}}{D_{inv}^{(2)}K}\frac{{\overline{\omega}}^{(2)}}{\omega_g}.$

$$ {x}_{m,0},\left(1-{x}_{m,1}\right)\ll 1 $$

(A.3)

If both

$$ \frac{1}{2}{D}_{inv}^{(1)}K\kern0.1em {x}_{m,0}\ll 1\kern2.5em \frac{3}{2}\frac{D_{inv}^{(1)}}{D_{inv}^{(2)}}\left(1-{x}_{m,1}\right)\ll 1 $$

(22)

are assumed in Eq. (14), then those equations simplify to an explicitly solvable set of 3 equations for 3 unknowns, H, x_m,0, and 1 − x_m,1.

$$ H=\sqrt{\frac{{\overline{\omega}}^{(1)}}{\left[{\left(\frac{1}{6}{D}_{inv}^{(1)}K\kern0.1em {\omega}_g\right)}^3+{\left(\frac{D_{inv}^{(1)}}{2{D}_{inv}^{(2)}}{\overline{\omega}}^{(2)}\right)}^3\right]}} $$

$$ {x}_{m,0}=\frac{1}{6}{D}_{inv}^{(1)}K\kern0.1em H{\omega}_g $$

(23)

$$ 1-{x}_{m,1}=\frac{D_{inv}^{(1)}}{2{D}_{inv}^{(2)}}H{\overline{\omega}}^{(2)}. $$

(24)

Finally, by substituting x_m,0/H from Eq. (23) into the first equation from Eq. (15),

$$ {j}^{(1)}=-K{\omega}_g. $$

This argument for j⁽¹⁾ is better than using the second equation of Eq. (15) and small x_m,0 because it avoids dropping any additional terms beyond those in Eq. (14). Similarly, substituting (1 − x_m,1)/H from Eq. (24) into the first equation from Eq. (16),

$$ {j}^{(2)}=\frac{3{\overline{\omega}}^{(2)}}{D_{inv}^{(2)}}. $$

This allows both of the conditions in Eq. (22) to be satisfied in terms of the concentrations if

$$ {\overline{\omega}}^{(1)}\max \left[{\left(\frac{D_{inv}^{(1)}K}{6}\right)}^4{\omega}_g^2,{\left(\frac{D_{inv}^{(1)}}{2{D}_{inv}^{(2)}}\right)}^4{\left({\overline{\omega}}^{(2)}\right)}^2\right]\ll {\left(\frac{D_{inv}^{(1)}K}{6}{\omega}_g\right)}^3+{\left(\frac{D_{inv}^{(1)}}{2{D}_{inv}^{(2)}}{\overline{\omega}}^{(2)}\right)}^3. $$

This test is complicated by needing to satisfy two criteria, but it roughly amounts to ${\overline{\omega}}^{(1)}\ll \left({\omega}_g,{\overline{\omega}}^{(2)}\right),$ where the usual asymptotic notation is generalized slightly to be even-handed when $O\left({\omega}_g\right)=O\left({\overline{\omega}}^{(2)}\right).$ The quantitative implementation of this asymptotic condition will be more restrictive than the previous two asymptotic limits because the neglected terms here in Eq. (14) are $O\left(\sqrt{{\overline{\omega}}^{(1)}/{\omega}_g}\right)$ or $O\left(\sqrt{{\overline{\omega}}^{(1)}/{\overline{\omega}}^{(2)}}\right),$ rather than $O{\left({\omega}_g/{\overline{\omega}}^{(2)}\right)}^3$ or $O{\left({\overline{\omega}}^{(2)}/{\omega}_g\right)}^3$ in the previous asymptotic limits.