Solid component, molar flux

Different combinations of spatial finite difference formulas were tried to determine the best set for our system of equations. The two point upwind formula was found to be best for the solids component molar fluxes. The low order formula was used because most of the gasifier reactions turn off abruptly when a component disappears and this creates sharp discontinuities. Higher order formulas tend to flatten out discontinuities, and in some cases, this causes material balances to be lost which then leads to numerical instability problems. Maintaining component material balance is an important aid to preserving numerical stability in the calculations. The low order formulas minimized these difficulties. 

The concentration of component A at the surface between the solid and the liquid film is given by cA0, whilst the concentration of A in the bulk flow is represented by cA5. We will presume c = N/V = const. As the material only moves in the y-direction, there is no need to note the mass transfer direction in terms of vectors. The molar flux from the solid into the liquid, according to (1.158), is... 

The total molar flux of mobile component A with respect to a stationary reference frame Na is evaluated at the solid-liquid interface when 0 > 0. It is necessary to consider the component of this flux in the normal coordinate (i.e., radial)... 

Answer The mass transfer calculation is based on the normal component of the total molar flux of species A, evaluated at the solid-liquid interface. Convection and diffusion contribute to the total molar flux of species A. For thermal energy transfer in a pure fluid, one must consider contributions from convection, conduction, a reversible pressure work term, and an irreversible viscous work term. Complete expressions for the total flux of speeies mass and energy are provided in Table 19.2-2 of Bird et al. (2002, p. 588). When the normal component of these fluxes is evaluated at the solid-liquid interface, where the normal component of the mass-averaged velocity vector vanishes, the mass and heat transfer problems require evaluations of Pick s law and Fourier s law, respectively. The coefficients of proportionality between flux and gradient in these molecular transport laws represent molecular transport properties (i.e., a, mix and kxc). In terms of the mass transfer problem, one focuses on the solid-liquid interface for x > 0 ... 

SOLUTION. The steady-state molar density profile of reactant A, given by (13-14) in the presence of a first-order irreversible chemical reaction, is employed to calculate the r component of the molar flux of A at the solid-liquid interface [i.e., r = (t)]. Then, one constructs an unsteady-state macroscopic mass balance... 

Estimate the external resistance to mass transfer by invoking continuity of the normal component of intrapellet fluxes at the gas/porous-solid interface. Then use interphase mass transfer coefficients within the gas-phase boundary layer surrounding the pellets to evaluate interfacial molar fluxes. 

The molar flux of a component i diffusing in a single pore of a porous solid, for example, a solid catalyst or a solid reactant like coke, coal, or an ore, is given by ... 

A particle with a porous product layer is divided into three zones the gas film around the product layer, the porous product layer, and the unreacted solid material. The structure of the particle is shown in Figure 8.2. The gas-phase component A diffuses through the gas film and product layer to the interface, where chemical reactions occur. The gas-phase product, P, has the opposite transport route. The molar flux of A is denoted as Na, and the positive transport direction is given in Figure 8.2. 

In equation 6.85, ([J (x, r)] nL) is the total liquid-solid particles interface, averaged, molar diffusive flux of component f. Note that Ag - = Asr +Asjf The second term of the right-hand side of equation 6.85 is 0 because at the catalyst interface convective fluxes are 0. On the other hand, from equation 6.13 the molar diffusive flux through the boundary layer at the liquid-solid interface must be equal to the reaction rate at the liquid-solid interface ... 

Calculations involving diffusion processes in inhomogeneous multicomponent ionic systems have been recently performed by Kirkaldy [30] and Cooper [38]. They worked with the same assumptions that have been made in this section in which quasi-binary systems have been discussed constant molar volume of the solid solution, and independent fluxes of ions, which are coupled only by the electrical diffusion potential. The latter can be eliminated by the condition zJi 0 which means that local electroneutrality prevails. With these assumptions, and with a knowledge of the thermodynamics of the multicomponent system (which is a knowledge of the activity of the electroneutral components as a function of composition), the individual ionic fluxes can be calculated explicitly with the help of the ionic mobilities and the activity coefficients of the components. 

---