# Conservation of mass species equation

$\frac{\partial }{{\partial t}}\int_V {{\rho _i}dV + \int_A {{\rho _i}({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} } = - \int_A {{{\mathbf{J}}_i} \cdot {\mathbf{n}}dA} + \int_V {{{\dot m'''}_i}dV}$

The surface integrals in the above equation may be converted to volume integrals by applying

$\int_A {{\mathbf{\Omega }} \cdot {\mathbf{n}}dA} = \int_V {\nabla \cdot {\mathbf{\Omega }}dV}$

as follows:

$\int_A {{\rho _i}({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA = \int_V {\nabla \cdot {\rho _i}{{\mathbf{V}}_{rel}}dV} } \qquad \qquad(1)$
$\int_A {{{\mathbf{J}}_i} \cdot {\mathbf{n}}dA} = \int_V {\nabla \cdot {{\mathbf{J}}_i}dV} \qquad \qquad(2)$

Substituting eqs. (1) and (2) into the integral conservation of mass species equation and considering $\frac{d}{{dt}}\int_V {\rho \phi } dV = \int_V {\frac{{\partial (\rho \phi )}}{{\partial t}}} dV$, the entire left-hand side of the integral conservation of mass species equation is included in a single volume integral, i.e.,

$\int_V {\left( {\frac{{\partial {\rho _i}}}{{\partial t}} + \nabla \cdot {\rho _i}{{\mathbf{V}}_{rel}} + \nabla \cdot {{\mathbf{J}}_i} - {{\dot m'''}_i}} \right)dV = 0} \qquad \qquad(3)$

The only condition that makes eq. (3) true regardless of the shape and size of the control volume is that the integrand must equal zero, i.e.,

$\frac{{\partial {\rho _i}}}{{\partial t}} + \nabla \cdot {\rho _i}{{\mathbf{V}}_{rel}} = - \nabla \cdot {{\mathbf{J}}_i} + {\dot m'''_i},{\rm{ }}i = 1,2, \cdots ,N \qquad \qquad(4)$

The first term on the left-hand side is the rate of increase of mass of the species i per unit volume, and the second term is net rate of additions of mass of the ith species per unit volume by convection. The terms on the right-hand side are the net rate of mass of ith species per unit volume by diffusion, and rate of production of species i by chemical reaction.

Equation (4) is the equation of conservation of mass for species. If the total number of species is N, a total of N − 1 independent equations for conservation of species mass can be obtained.

After defining the mass fraction of species i as

${\omega _i} = \frac{{{\rho _i}}}{\rho } \qquad \qquad(5)$

eq. (4) can be rewritten as

$\left[ {\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \rho {{\mathbf{V}}_{rel}}} \right]{\omega _i} + \rho \left[ {\frac{{\partial {\omega _i}}}{{\partial t}} + \nabla \cdot {\omega _i}{{\mathbf{V}}_{rel}}} \right] = - \nabla \cdot {{\mathbf{J}}_i} + {\dot m'''_i} \qquad \qquad(6)$

According to the continuity equation, $\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \rho {{\mathbf{V}}_{rel}} = 0$, the first bracket on the left-hand side of eq. (6) is zero. The second bracket on the left-hand side is the substantial derivative of the mass fraction. Therefore, the conservation of species mass in terms of mass fraction becomes

$\rho \frac{{D{\omega _i}}}{{Dt}} = - \nabla \cdot {{\mathbf{J}}_i} + {\dot m'''_i} \qquad \qquad(7)$

Assuming binary system of A and B, one can use Fick’s law in eq. (7) to yield

$\rho \frac{{D{\omega _A}}}{{Dt}} = - \rho \nabla \cdot ({D_{AB}}\nabla {\omega _A}) + {\dot m'''_A} \qquad \qquad(8)$

which is useful in determining the diffusion in dilute liquid solution at constant temperature and pressure. Equation (8), with ${\dot m'''_A} = 0$, is similar to energy equation with no internal heat source and viscous dissipation:

$\rho {c_p}\frac{{DT}}{{Dt}} = \nabla \cdot (k\nabla T) + q''' + \nabla {{\mathbf{V}}_{rel}}:{{\mathbf{\tau }}_{rel}}$

and therefore it is used for analogy between heat and mass transfer analysis.

In the preceding discussion to develop eq. (4), the mass fraction and mass flux were used. The species equation can also be developed in term of molar concentration (or molar fraction) and molar flux. By following a similar procedure, the species equation is

$\frac{{\partial {c_i}}}{{\partial t}} = - \nabla \cdot {{\mathbf{\dot n''}}_i} + {\dot n'''_i} \qquad \qquad(9)$

where the molar flux relative to the stationary coordinate axes can be obtained from (see Introduction to Mass transfer):

${{\mathbf{\dot m''}}_i} = {\rho _i}{\mathbf{\tilde V}} + {{\mathbf{J}}_i} = {\omega _i}\sum\limits_{j = 1}^N {{{{\mathbf{\dot m''}}}_j}} + {{\mathbf{J}}_i}$

i.e.,

${{\mathbf{\dot n''}}_i} = {c_i}{{\mathbf{\tilde V}}^*} + {\mathbf{J}}_i^* \qquad \qquad(10)$

Substituting eq. (10) into eq. (9), we have

$\frac{{\partial {c_i}}}{{\partial t}} + \nabla \cdot ({c_i}{{\mathbf{\tilde V}}^*}) = - \nabla \cdot {\mathbf{J}}_i^* + {\dot n'''_i} \qquad \qquad(11)$

where the first term on the left-hand side is rate of increase of mole of the species i per unit volume, and the second term is net rate of additions of mole of the ith species per unit volume by convection. The terms on the right-hand side are net rate of mole of ith species per unit volume by diffusion, and the molar rate of production of species i by chemical reaction.

For a binary system of components A and B with constant pressure, eq. (11) reduces to

$\frac{{\partial {c_A}}}{{\partial t}} + \nabla \cdot ({c_A}{{\mathbf{\tilde V}}^*}) = - c\nabla \cdot ({D_{AB}}\nabla {x_A}) + {\dot n'''_A} \qquad \qquad(12)$

where c is the mixture molar concentration and ${{\mathbf{\tilde V}}^*}$ is molar-averaged velocity defined in Chapter 1. Equation (12) can be applied to low density gases with constant temperature and pressure.

The production rate of the ith species, ${\dot m'''_i}$ (or ${\dot n'''_i}$), is still unknown at this point, but it can be obtained by analyzing the chemical reaction. If the number of chemical reactions taking place in the system is Nc, the mass production rate is (Kleijn, 1991; Mahajan, 1996)

${\dot m'''_i} = \sum\limits_{j = 1}^{{N_c}} {{a_{ij}}{M_i}{\Re _j}} \qquad \qquad(13)$

where aij is the stoichiometric coefficient that describes the proportions of the mole numbers of reactants disappearing and mole numbers of products appearing as a result of the reaction process (see Section 2.3.3). The net reaction rate of the jth chemical reaction ${\Re _j}$ is the difference between the forward and backward reactions, i.e.,

${\Re _j} = {\Re _{{j^ + }}} - {\Re _{{j^ - }}} \qquad \qquad(14)$

If chemical reactions take place in gas mixture, the forward and backward reaction rates are

${\Re _{{j^ + }}} = {r_{{j^ + }}}(p,T)\prod\limits_{i = 1}^{{N_r}} {{{\left( {\frac{p}{{{R_u}T}}{x_i}} \right)}^{\left| {{a_{ij}}} \right|}}} \qquad \qquad(15)$
${\Re _{{j^ - }}} = {r_{{j^ - }}}(p,T)\prod\limits_{i = 1}^{{N_p}} {{{\left( {\frac{p}{{{R_u}T}}{x_i}} \right)}^{\left| {{a_{ij}}} \right|}}} \qquad \qquad(16)$

where Nr and Np are number of reactants and products, respectively. The two reaction rate constants, ${r_{{j^ + }}}$ and ${r_{{j^ - }}}$, depend on the specific chemical reaction under consideration. These constants are related by

$\frac{{{r_{{j^ - }}}(p,T)}}{{{r_{{j^ + }}}(p,T)}} = \frac{1}{{{K_j}(T)}}{\left( {\frac{{{R_u}T}}{{{p^0}}}} \right)^{\sum\limits_{i = 1}^N {{a_{ij}}} }} \qquad \qquad(17)$

where p0 is the standard pressure and Kj(T) is the thermodynamic equilibrium constant of the jth chemical reaction:

${K_j}(T) = \exp \left[ {\frac{{ - \Delta \bar G_j^0(T)}}{{{R_u}T}}} \right] \qquad \qquad(18)$

where $\Delta \bar G_j^0(T)$ is the standard Gibbs energy for the jth chemical reaction; its value depends on the specific chemical reaction considered.

For the case in which only one chemical reaction (Nc = 1) leads to production of the ith component, the molar production rate can be obtained by a simplified from

${\dot n'''_i} = k'''c_i^n \qquad \qquad(19)$

where the index n represents the order of the reaction and k'''n is a rate constant (1/sn) depends on the temperature.

As before, the mass flux, Ji, in eqs. (4) and (7) includes mass fluxes due to ordinary diffusion driven by the concentration gradient, pressure diffusion, body force diffusion, and thermal (Soret) diffusion [see eq. (1)]. For a binary mixture, the mass flux can be calculated by eq. (16).

Equation (7) is valid for the case that chemical reaction occurs in the entire volume – referred to as homogeneous reacting system. For the case where the chemical reaction takes place on a surface – referred to as heterogeneous reaction – the source term in eq. (7) will not appear, and the rate of production will be accounted as a boundary condition. More discussion about modeling of the heterogeneous reacting system can be found in Faghri and Zhang (2006).

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Kleijn, C.R., 1991, “A Mathematical Model of the Hydrodynamics and Gas Phase Reaction in Silicon LPCVD in a Single Wafer Reactor,” Journal of Electrochemical Society, Vol. 138, pp. 2190-2200.

Mahajan, R.L., 1996, “Transport Phenomena in Chemical Vapor-Deposition Systems,” Advances in Heat Transfer, Academic Press, San Diego, CA.