Integral conservation of mass species equation

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The continuity equation states that the total mass for a closed system is constant. For a system containing one phase but more than one component, the total mass of the system is composed of different species. If the concentrations of each of these species are not uniform, mass transfer occurs in a way that makes the concentrations more uniform. Therefore, it is necessary to track the individual components by applying the principle of conservation of species mass. For a system of multiple components, each component can have its own mass density and velocity. The conservation of mass for the ith species in a single phase system for a fixed-mass system is obtained by applying eq.

{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA} }

from transformation formula for the ith species with Φ = mi and φ = ρi / ρ (where ρi is the concentration of the ith component), i.e.,

{\left. {\frac{{d{m_{k,i}}}}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {{\rho _{k,i}}dV + \int_A {{\rho _{k,i}}({{\mathbf{V}}_{k,i,rel}} \cdot {{\mathbf{n}}_k})dA} }     \qquad \qquad(1)

If there is no chemical reaction, the total mass of the ith species for a closed system remains constant. Chemical reactions, on the other hand, will result in the production or consumption of the ith species, which can be modeled as a mass source or sink for the ith species, i.e.,

{\left. {\frac{{d{m_i}}}{{dt}}} \right|_{system}} = \int_V {{{\dot m'''}_i}dV}    \qquad \qquad(2)

where {m_i^{'''}} is the mass production rate for the ith species, which can be either positive (mass production) or negative (mass consumption). For the cases without chemical reaction, {m_i^{'''}}=0. The conservation of species mass for a control volume is therefore

\frac{\partial }{{\partial t}}\int_V {{\rho _i}dV + \int_A {{\rho _i}({{\mathbf{V}}_{i,rel}} \cdot {\mathbf{n}})dA} }  = \int_V {{{\dot m'''}_i}dV}     \qquad \qquad(3)

which is valid for each component in the control volume. If the total number of species is N, summation of the conservation of mass of all species results in

\frac{\partial }{{\partial t}}\int_V {\left[ {\sum\limits_{i = 1}^N {{\rho _i}} } \right]dV + \int_A {\left[ {\sum\limits_{i = 1}^N {({\rho _i}{{\mathbf{V}}_{i,rel}}) \cdot {\mathbf{n}}} } \right]dA} }  = \int_V {\left[ {\sum\limits_{i = 1}^N {{{\dot m'''}_i}} } \right]dV}     \qquad \qquad(4)

The summation of the densities of the individual species is equal to the bulk density of the multicomponent substance:

	\rho  = \sum\limits_{i = 1}^N {{\rho _i}}     \qquad \qquad(5)

The bulk velocity of the multicomponent substance is the mass-averaged velocity of the velocity of each individual species:

\rho {{\mathbf{V}}_{rel}} = \sum\limits_{i = 1}^N {({\rho _i}{{\mathbf{V}}_{i,rel}})}     \qquad \qquad(6)

The right-hand side of eq. (4) must be zero, because the total mass of species produced is equal to the total mass of species consumed, i.e.,

 \sum\limits_{i = 1}^N {{{\dot m'''}_i}}  = 0   \qquad \qquad(7)

Substituting eqs. (5) – (7) into eq. (4), the continuity equation \int_V {\left( {\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot \rho {{\mathbf{V}}_{rel}}} \right)dV = 0} is obtained. This means that we can write conservation of species mass for N species, but only N − 1 of these equations are independent. In practice, one can use these N − 1 equations for conservation of species mass in combination with the continuity equation, eq. \int_V {\left( {\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot \rho {{\mathbf{V}}_{rel}}} \right)dV = 0}, to describe multicomponent systems.

In each equation of the conservation of species mass, there are two new unknowns: mass density of the ith species ρi, and the species mass velocity {{\mathbf{V}}_{i,rel}}. This yields more unknowns than the number of equations. So in order to have a properly-posed equation set, it is necessary to reduce the number of unknowns in each equation from two to one. The second term on the left-hand side of eq. (3) represents the species i mass flow across the surface of the control volume, which results from convection by bulk flow and diffusion relative to the bulk convection.

\int_A {{\rho _i}({{\mathbf{V}}_{i,rel}} \cdot {\mathbf{n}})dA}  = \int_A {{\rho _i}({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA}  + \int_A {{{\mathbf{J}}_i} \cdot {\mathbf{n}}dA}     \qquad \qquad(8)

where {{\mathbf{J}}_i} is the diffusive mass flux vector of species i, which includes mass fluxes due to ordinary diffusion driven by the concentration gradient, and thermal (Soret) diffusion.

Substituting eq. (8) into eq. (3), one obtains an expression for the conservation of species mass that contains only one new additional variable, ρi:

\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}     \qquad \qquad(9)

The above analysis is based on the assumption that the control volume contains only one phase. For a control volume containing Π phases, the conservation of species mass is (Faghri and Zhang, 2006).

 \sum\limits_{k = 1}^\Pi  {\left[ {\frac{\partial }{{\partial t}}\int_{{V_k}(t)} {{\rho _{k,i}}dV}  + \int_{{A_k}(t)} {{\rho _{k,i}}({{\mathbf{V}}_{k,rel}} \cdot {{\mathbf{n}}_k})dA} } \right]}  \\ 
  = \sum\limits_{k = 1}^\Pi  {\left[ { - \int_{{A_k}(t)} {({{\mathbf{J}}_{k,i}} \cdot {{\mathbf{n}}_k})dA}  + \int_{{V_k}(t)} {{{\dot m'''}_{k,i}}dV} } \right]}  \\ 
 \end{array}   \qquad \qquad(10)


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.

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