Validation of a numerical method for interface-resolving simulation of multicomponent gas-liquid mass transfer and evaluation of multicomponent diffusion models

Abstract

The multicomponent model and the effective diffusivity model are well established diffusion models for numerical simulation of single-phase flows consisting of several components but are seldom used for two-phase flows so far. In this paper, a specific numerical model for interfacial mass transfer by means of a continuous single-field concentration formulation is combined with the multicomponent model and effective diffusivity model and is validated for multicomponent mass transfer. For this purpose, several test cases for one-dimensional physical or reactive mass transfer of ternary mixtures are considered. The numerical results are compared with analytical or numerical solutions of the Maxell-Stefan equations and/or experimental data. The composition-dependent elements of the diffusivity matrix of the multicomponent and effective diffusivity model are found to substantially differ for non-dilute conditions. The species mole fraction or concentration profiles computed with both diffusion models are, however, for all test cases very similar and in good agreement with the analytical/numerical solutions or measurements. For practical computations, the effective diffusivity model is recommended due to its simplicity and lower computational costs.

Appendices

Appendix 1. Numerical solution of Maxwell-Stefan eqs

A classical model for describing diffusion in multicomponent systems is the Maxwell-Stefan diffusion, see e.g. [26]. At constant temperature and pressure, the total concentration and the binary diffusion coefficients are constant, and the driving forces of the Maxwell-Stefan diffusion are the gradients of mole fractions. For the one-dimensional problems considered here, the Maxwell-Stefan equations for ideal mixtures have the form

(29)

To obtain the species distribution from the molar fluxes, the first order spatial derivative is discretized by a forward finite difference so that Eq. (29) is discretized at position k as

(30)

At the left wall (z = 0), fixed mole fractions are specified as boundary conditions and 40 nodes are used for the Stefan-tube example of Section 3.1.1.

Appendix 2. Numerical solution of non-equilibrium stage model (NESM)

Kenig et al. [70] suggested an analytical model for two-phase mass transfer in a steady state, which is based on the non-equilibrium stage model [26]. In the NESM, the reactor is divided into several stages. Between the stages, different kinds of fluxes (due to feeding, production and consumption) are transferred. Kenig et al. [70] provided the stage equation for the two-phase mass transfer with a heterogeneous model for each phase. The equations for species i in stage j for gas and liquid phases are

$$ \left(1+{r}_{\mathrm{G}}\right){\dot{V}}_{\mathrm{G},i}^j-{\dot{V}}_{\mathrm{G},i}^{j+1}-{\dot{V}}_{\mathrm{FG},i}^j-{J}_{\mathrm{G},i}^{V,j}{a}_{\mathrm{int}}^j=0 $$

(31)

$$ \left(1+{r}_{\mathrm{L}}\right){\dot{V}}_{\mathrm{L},i}^j-{\dot{V}}_{\mathrm{L},i}^{j+1}-{\dot{V}}_{\mathrm{FL},i}^j-{J}_{\mathrm{L},i}^{V,j}{a}_{\mathrm{int}}^j-{\dot{S}}_i^j{a}_{\mathrm{r}}^j=0 $$

(32)

Here, ${V}_{\mathrm{G},i}^j$ and ${V}_{L,i}^j$ denotes the molar flow rate to stage j with side stream, and r _G and r _L are the ratio of side stream to inter-stage flow. The second terms in Eq. (31) and Eq. (32) denote molar flow rates to stage j + 1. The third terms represent the molar flow rates by additional feed, and the fourth terms represent the mass transfer across the gas-liquid interface (with interfacial area ${a}_{\mathrm{int}}^j$). The fifth term, which appears only in the liquid phase equation, is the rate of heterogeneous reaction at the surface (with surface area ${a}_{\mathrm{r}}^j$).

For the test cases presented in section 3.2, side streams and additional feeds to the stages are neglected so that r _G = r _L = 0 and ${V}_{\mathrm{FG},i}^j={V}_{\mathrm{FL},i}^j=0$. Furthermore, the one-dimensional domain is divided into a number of neighboring stages so that Eq. (31) and Eq. (32) simplify to three types of equations. For stages containing one phase only it is, depending on the phase, either

$$ {\dot{V}}_{\mathrm{G},i}^j={\dot{V}}_{\mathrm{G},i}^{j+1}\mathrm{or}\kern0.5em {\dot{V}}_{\mathrm{L},i}^j={\dot{V}}_{\mathrm{L},i}^{j+1} $$

(33)

For the stage with the gas-liquid interface it is

$$ {\dot{V}}_{\mathrm{G},i}^j={J}_{\mathrm{G},i}^{V,j}{a}_{\mathrm{int}}^j\kern1em \mathrm{and}\kern1em {\dot{V}}_{\mathrm{L},i}^j={J}_{\mathrm{L},i}^{V,j}{a}_{\mathrm{int}}^j, $$

(34)

while for the stage with the reactive surface it is

$$ {\dot{V}}_{\mathrm{L},i}^j={\dot{S}}_i^j{a}_{\mathrm{r}}^j $$

(35)

For the present one-dimensional problems, the interfacial area (${a}_{\mathrm{int}}^j$) and surface area of reaction (${a}_{\mathrm{r}}^j$) are assumed to be identical so that the latter equations simplify to

$$ {J}_{\mathrm{G},i}^{V,j}={J}_{\mathrm{G},i}^{V,j+1},{J}_{\mathrm{L},i}^{V,j}={J}_{\mathrm{L},i}^{V,j+1} $$

(36)

and for the stage at the right wall

$$ {J}_{\mathrm{L},i}^{V,j}={\dot{S}}_i^j $$

(37)

These equations turn out to be the same as the flux balance equation. The solution procedure is, therefore, similar as for the flux balance equation by finite difference method. The solution of this elliptic problem is obtained by an iterative method with 21 stages including one stage at the middle of the domain (Z = 0.5) with the gas-liquid interface, and one stage at the right reactive wall (Z = 1).