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    General Mass Transfer Governing Equation

    The governing mass transfer equation is derived from a mass balance applied to a differential control volume and Fick's Law. It includes all possible mechanisms that can contribute to mass transfer in a system, and encapsulates the combined effects of mass storage, diffusion, convection, and generation or disappearance of a species, and allows the prediction of the concentration profile within the system. It combines the mass balance over a system with Fick's Law.

    It can be used to describe the concentration of a species in any kind of mass transfer situation. The result is a function that is valid at all points in a control volume, at all times (e.g., \( c_A(x,y,z,t) \)). While mass transfer can occur in three dimensions, it is simplified here to be solved with mass moving only in one direction (x,r). The governing mass transfer equation can be written in Cartesian or radial coordinates.
    Governing Mass Transfer Equation (Cartesian)
    $$ \frac{\partial c_A}{\partial t} + u\frac{\partial c_A}{\partial x} = D_{AB}\frac{\partial^2 c_A}{\partial x^2} + r_A $$
    where
    • \( c_A \) = concentration of species A \( [\frac{kg}{m^3}] \)
    • \( x \) = position (one-dimensional flow) \( [m] \)
    • \( t \) = time \( [s] \)
    • \( u \) = fluid velocity \( [\frac{m}{s}] \)
    • \( D_{AB} \) = diffusivity of species A in species B \( [\frac{m^2}{s}] \)
    • \( r_A \) = rate of generation or disappearance of species A per volume \( [\frac{kg}{m^3s}] \)
    Governing Mass Transfer Equation (Cylindrical)
    $$ D_{AB}\frac{1}{r} \frac{\partial}{\partial r} (r\frac{\partial c_A}{\partial r}) + r_A = \frac{\partial c_A}{\partial t} $$
    Governing Mass Transfer Equation (Spherical)
    $$ D_{AB}\frac{1}{r^2} \frac{\partial}{\partial r} (r^2\frac{\partial c_A}{\partial r}) + r_A = \frac{\partial c_A}{\partial t} $$

    In practice, many problems involve simplifying assumptions (e.g., system at steady state, no bulk flow in the system, or no generation or disappearance of species A), which allow certain terms to be removed. The meaning of each term and the conditions under which it should be retained or dropped are summarized below.

    TermWhat is it?When do I keep it?When can I drop it?
    \( \frac{\partial c_A}{\partial t} \) Stored mass Unsteady state, \( c_A \) changing with time (transient)Steady state (constant \( c_A \))
    \( u\frac{\partial c_A}{\partial x} \) Bulk flow (Convection) Fluid flow or bulk movement through the systemNo flow, typically in a solid
    \( D_{AB} \frac{\partial ^2 c_A}{\partial x^2} \) Diffusion Almost always in diffusion mass transfer problems Slow diffusion in relation to generation or convection
    \( r_A \) Generation Chemical reaction or conversion from/to species ANo chemical reaction or mass source

    Table 1: Governing equation terms and simplifications.

    Boundary and Initial Conditions

    To obtain a particular solution, we must solve the general governing equation utilizing known quantities called boundary conditions (BCs, which describe mass transfer at particular surfaces and interfaces) as well as initial conditions (ICs, for unsteady state or time-dependent problems).

    Surface Concentration Specified (BC)

    Concentration at the surface can be specified as a constant or a function of time.

    • For example: \( c_A(x=0)=c_{A,s} \).
    • Example of BC in mass transfer: At the surface of a material, the concentration of species A may be known because the surface is exposed to a surrounding phase or because equilibrium conditions determine the concentration at the interface. For mass transfer at a solid-fluid interface, the concentration needed depends on the domain being modeled. If the problem is set up for the solid, the surface concentration in the solid is needed. If the problem is set up for the fluid, the concentration on the fluid side is used.

    Surface Mass Flux Specified (BC)

    Mass flux at the surface is specified as a constant or a function of time. There are two special cases for this boundary condition:

    • Impermeable Condition: the surface is impermeable to the species, so the flux at the surface can be estimated as zero. For example: \( \frac{\partial c_A}{\partial x}(x=0)=0 \).
    • Symmetric Condition: the system has symmetric geometry and boundary conditions, making the flux at the centerline equal to zero. For example: \( \frac{\partial c_A}{\partial x}(x=L)=0 \).
    • Example of BC in mass transfer: If a boundary does not allow species A to pass through it, the surface can be treated as impermeable. If the geometry and concentration profile are symmetric about a centerline, the concentration gradient at that centerline is zero.

    Convection at the Surface (BC)

    When there is a fluid at the surface, the flow of mass due to diffusion can be set equal to the flow due to convection. There is one special case for this boundary condition:

    • When hm is large, the system looks like the boundary condition where the surface concentration can be specified.

    At the surface, the amount of mass diffused through the solid or liquid is equal to the amount of mass convected away by the fluid:

    Convection at the Surface
    $$ -D_{AB}\frac{\partial c_A}{\partial x}\bigg|_{x=0} = h_m\left(c_A^{fluid}\bigg|_{x=0} - c_{A,\infty}^{fluid}\right) $$

    The left side represents diffusion in the solid or liquid, while the right side represents convection in the fluid.

    Initial Condition

    An initial condition provides information about what is happening within the system (e.g., the concentration) at the start of our period of interest. This is necessary to solve a time-varying (i.e., unsteady state) problem.

    • For example: \( c_A(t=0)=c_{A,i} \) or \( c_A(t=0)=100 \frac{kg}{m^3} \).