Interfacial mass, momentum, energy, and species balances

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The conservation laws for transport phenomena can be reduced to local partial differential equations if they are considered at a point that does not belong to a surface of discontinuity, such as an interface. When considering a discontinuous point, appropriate jump conditions relating the values of the fundamental quantities on both sides of the interface should be considered. Jump conditions at an interface were discussed in Section 3.3.6, but the effects of surface tension and disjoining pressure associated with a nonflat liquid-vapor interface were not taken into account. It is the objective of this subsection to specify mass, momentum, and energy balance at a nonflat liquid-vapor interface, as well as species balance in solid-liquid-vapor interfaces.


Mass Balance

At a liquid-vapor interface, the mass balance is

{{{\dot{m}}''}_{\delta }}={{\rho }_{\ell }}({{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{I}})\cdot \mathbf{n}={{\rho }_{v}}({{\mathbf{V}}_{v}}-{{\mathbf{V}}_{I}})\cdot \mathbf{n}


where {{{\dot{m}}''}_{\delta }} is mass flux at the interface due to phase change and VI is the velocity of the interface. For a three-dimensional interface, there are three components of velocity: the normal direction, and two tangential directions, denoted by n, t1, and t2, respectively. Therefore, the velocity components should be defined according to these directions, as follows:

\mathbf{V}\cdot \mathbf{n}={{V}_{\mathbf{n}}}


\mathbf{V}\cdot {{\mathbf{t}}_{1}}={{V}_{{{\mathbf{t}}_{1}}}}


\mathbf{V}\cdot {{\mathbf{t}}_{2}}={{V}_{{{\mathbf{t}}_{2}}}}


The interfacial mass balance can be rewritten in these terms by:

{{{\dot{m}}''}_{\delta }}={{\rho }_{\ell }}\left( {{V}_{\ell ,\mathbf{n}}}-{{V}_{I,\mathbf{n}}} \right)={{\rho }_{v}}\left( {{V}_{v,\mathbf{n}}}-{{V}_{I,\mathbf{n}}} \right)


Momentum Balance

For a situation where the effects of surface tension and disjoining pressure are negligible, the momentum balance at the liquid-vapor interface can be described by eq. (3.173), i.e.,

\left( {{{\mathbf{{\tau }'}}}_{\ell }}-{{{\mathbf{{\tau }'}}}_{v}} \right)\cdot \mathbf{n}={{{\dot{m}}''}_{\delta }}({{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{v}})


where n is the normal direction of the interface and points toward the vapor. The left-hand side of eq. (6) represents the force per unit area acting on the interface, while the right-hand side represents the change of momentum across the interface. For applications involving thin film evaporation and condensation, the effects of surface tension and disjoining pressure will create additional forces on the interface; in these cases, the left-hand side of eq. (6) should be modified. The force per unit area created by the surface tension as indicated by eq. (51) is \sigma (T)(1/{{R}_{I}}+1/{{R}_{II}})\mathbf{n}. The force per unit area created by the disjoining pressure is -{{p}_{d}}\mathbf{n}. For the situations where the interfacial temperature is not a constant, the contribution of the Marangoni effect, (d\sigma /dT)\nabla {{T}_{\delta }}, should also be included. Therefore, the momentum balance at the interface becomes

  & \left( {{{\mathbf{{\tau }'}}}_{\ell }}-{{{\mathbf{{\tau }'}}}_{v}} \right)\cdot \mathbf{n}+\sigma (T)\left( \frac{1}{{{R}_{I}}}+\frac{1}{{{R}_{II}}} \right)\mathbf{n}-{{p}_{d}}\mathbf{n} \\ 
 & -\left( \frac{d\sigma }{dT} \right)(\nabla {{T}_{\delta }}\cdot \mathbf{t})\mathbf{t}={{{{\dot{m}}''}}_{\delta }}({{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{v}}) \\ 


On the left-hand side, the first term is the stress tensor, the second term is the capillary pressure, the third term is the disjoining pressure, and the fourth term is the Marangoni stress. The right-hand side is the momentum transfer due to inertia. In this equation, the tangential direction, t, can either be t1 or t2. The stress tensor is:

{\tau }'=-p\mathbf{I}+2\mu \mathbf{D}-\frac{2}{3}\mu \left( \nabla \cdot \mathbf{V} \right)\mathbf{I}


The deformation tensor can be written for a reference frame that is adjusted to the interface:

\mathbf{D}=\frac{1}{2}\left[ \nabla \mathbf{V}+{{\left( \nabla \mathbf{V} \right)}^{T}} \right]=\left[ \begin{matrix}
   \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{\mathbf{n}}}} & \frac{1}{2}\left( \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{1}}}}}{\partial {{x}_{\mathbf{n}}}} \right) & \frac{1}{2}\left( \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{2}}}}}{\partial {{x}_{\mathbf{n}}}} \right)  \\
   \frac{1}{2}\left( \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{1}}}}}{\partial {{x}_{\mathbf{n}}}} \right) & \frac{\partial {{V}_{{{\mathbf{t}}_{1}}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}} & \frac{1}{2}\left( \frac{\partial {{V}_{{{\mathbf{t}}_{1}}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{2}}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}} \right)  \\
   \frac{1}{2}\left( \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{2}}}}}{\partial {{x}_{\mathbf{n}}}} \right) & \frac{1}{2}\left( \frac{\partial {{V}_{{{\mathbf{t}}_{1}}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{2}}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}} \right) & \frac{\partial {{V}_{{{\mathbf{t}}_{2}}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}  \\
\end{matrix} \right]


The normal direction of the interface is [1 0 0], the first tangential direction is [0 1 0] and the second tangential direction is [0 0 1]. Therefore,

  & {\tau }'\cdot \mathbf{n}=\left[ -\begin{matrix}
   p & 0 & 0  \\
\end{matrix} \right]+ \\ 
 & 2\mu \left[ \begin{matrix}
   \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{\mathbf{n}}}} & \frac{1}{2}\left( \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{1}}}}}{\partial {{x}_{\mathbf{n}}}} \right) & \frac{1}{2}\left( \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{2}}}}}{\partial {{x}_{\mathbf{n}}}} \right)  \\
\end{matrix} \right]-\frac{2}{3}\mu \left[ \begin{matrix}
   \nabla \cdot \mathbf{V} & 0 & 0  \\
\end{matrix} \right] \\ 


This can be reduced to the three components to obtain:

{\tau }'\cdot \mathbf{n}\cdot \mathbf{n}=-p+2\mu \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{\mathbf{n}}}}-\frac{2}{3}\mu \nabla \cdot \mathbf{V}=-p+\frac{4}{3}\mu \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{\mathbf{n}}}}-\frac{2}{3}\mu \left( \frac{\partial {{V}_{{{\mathbf{t}}_{1}}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{2}}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}} \right)


{\tau }'\cdot \mathbf{n}\cdot {{\mathbf{t}}_{1}}=\mu \left( \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{1}}}}}{\partial {{x}_{\mathbf{n}}}} \right)


{\tau }'\cdot \mathbf{n}\cdot {{\mathbf{t}}_{2}}=\mu \left( \frac{\partial {{V}_{\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{{{\mathbf{t}}_{2}}}}}{\partial {{x}_{\mathbf{n}}}} \right)


The momentum equation balance at the interface is then broken into its three components, as follows:

Normal Direction

  & -{{p}_{\ell }}+{{p}_{v}}+\frac{4}{3}\left( {{\mu }_{\ell }}\frac{\partial {{V}_{\ell ,\mathbf{n}}}}{\partial {{x}_{\ell ,\mathbf{n}}}}-{{\mu }_{v}}\frac{\partial {{V}_{v,\mathbf{n}}}}{\partial {{x}_{v,\mathbf{n}}}} \right) \\ 
 & -\frac{2}{3}\left[ {{\mu }_{\ell }}\left( \frac{\partial {{V}_{\ell ,{{\mathbf{t}}_{1}}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{\ell ,{{\mathbf{t}}_{2}}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}} \right)-{{\mu }_{v}}\left( \frac{\partial {{V}_{v,{{\mathbf{t}}_{1}}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{v,{{\mathbf{t}}_{2}}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}} \right) \right] \\ 
 & +\sigma \left( \frac{1}{{{R}_{I}}}+\frac{1}{{{R}_{II}}} \right)-{{p}_{d}}={{{{\dot{m}}''}}_{\delta }}\left( {{V}_{\ell ,\mathbf{n}}}-{{V}_{v,\mathbf{n}}} \right) \\ 


Tangential 1

{{\mu }_{\ell }}\left( \frac{\partial {{V}_{\ell ,\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{\ell ,{{\mathbf{t}}_{1}}}}}{\partial {{x}_{\mathbf{n}}}} \right)-{{\mu }_{v}}\left( \frac{\partial {{V}_{v,\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{v,{{\mathbf{t}}_{1}}}}}{\partial {{x}_{\mathbf{n}}}} \right)-\left( \frac{d\sigma }{dT} \right)\left( \frac{\partial {{T}_{\delta }}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}} \right)={{{\dot{m}}''}_{\delta }}\left( {{V}_{\ell ,{{\mathbf{t}}_{1}}}}-{{V}_{v,{{\mathbf{t}}_{1}}}} \right)


Tangential 2

{{\mu }_{\ell }}\left( \frac{\partial {{V}_{\ell ,\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{\ell ,{{\mathbf{t}}_{2}}}}}{\partial {{x}_{\mathbf{n}}}} \right)-{{\mu }_{v}}\left( \frac{\partial {{V}_{v,\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{v,{{\mathbf{t}}_{2}}}}}{\partial {{x}_{\mathbf{n}}}} \right)-\left( \frac{d\sigma }{dT} \right)\left( \frac{\partial {{T}_{\delta }}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}} \right)={{{\dot{m}}''}_{\delta }}\left( {{V}_{\ell ,{{\mathbf{t}}_{2}}}}-{{V}_{v,{{\mathbf{t}}_{2}}}} \right)


The non-slip condition at the liquid-vapor interface requires that {{V}_{\ell ,{{\mathbf{t}}_{1}}}}={{V}_{v,{{\mathbf{t}}_{1}}}} and {{V}_{\ell ,{{\mathbf{t}}_{2}}}}={{V}_{v,{{\mathbf{t}}_{2}}}}. The momentum balance at the tangential directions becomes

{{\mu }_{\ell }}\left( \frac{\partial {{V}_{\ell ,\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{\ell ,{{\mathbf{t}}_{1}}}}}{\partial {{x}_{\mathbf{n}}}} \right)={{\mu }_{v}}\left( \frac{\partial {{V}_{v,\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{v,{{\mathbf{t}}_{1}}}}}{\partial {{x}_{\mathbf{n}}}} \right)+\left( \frac{d\sigma }{dT} \right)\left( \frac{\partial {{T}_{\delta }}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}} \right)


{{\mu }_{\ell }}\left( \frac{\partial {{V}_{\ell ,\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{\ell ,{{\mathbf{t}}_{2}}}}}{\partial {{x}_{\mathbf{n}}}} \right)={{\mu }_{v}}\left( \frac{\partial {{V}_{v,\mathbf{n}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}}+\frac{\partial {{V}_{v,{{\mathbf{t}}_{2}}}}}{\partial {{x}_{\mathbf{n}}}} \right)+\left( \frac{d\sigma }{dT} \right)\left( \frac{\partial {{T}_{\delta }}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}} \right)


For most applications, the evaporation or condensation rate – {{{\dot{m}}''}_{\delta }} – is not very high; therefore, it can be assumed that {{{\dot{m}}''}_{\delta }}({{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{v}})\doteq 0. If the liquid and vapor phases are further assumed to be inviscid ({{\mathbf{\tau }}_{\ell }}={{\mathbf{\tau }}_{v}}\doteq 0), the momentum equation at the interface can be reduced to

{{p}_{v}}-{{p}_{\ell }}=\sigma (T)\left( \frac{1}{{{R}_{I}}}+\frac{1}{{{R}_{II}}} \right)-{{p}_{d}}



The energy balance at the interface can be obtained from eq. (3.180), i.e.,

\left( {{{\mathbf{{q}''}}}_{\ell }}-{{{\mathbf{{q}''}}}_{v}} \right)\cdot \mathbf{n}-(\mathbf{n}\cdot {{\mathbf{{\tau }'}}_{\ell }})\cdot {{\mathbf{V}}_{\ell ,rel}}+(\mathbf{n}\cdot {{\mathbf{{\tau }'}}_{v}})\cdot {{\mathbf{V}}_{v,rel}}={{{\dot{m}}''}_{\delta }}\left[ \left( {{e}_{v}}+\frac{\mathbf{V}_{v,rel}^{2}}{2} \right)-\left( {{e}_{\ell }}+\frac{\mathbf{V}_{\ell ,rel}^{2}}{2} \right) \right]


where the work done by surface tension and disjoining pressure is neglected. If the velocity of the reference frame is taken as the interfacial velocity VI, eq. (20) can be rewritten as

  & \left( {{k}_{v}}\nabla {{T}_{v}}-{{k}_{\ell }}\nabla {{T}_{\ell }} \right)\cdot \mathbf{n}-(\mathbf{n}\cdot {{{\mathbf{{\tau }'}}}_{\ell }})\cdot ({{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{I}})+(\mathbf{n}\cdot {{{\mathbf{{\tau }'}}}_{v}})\cdot ({{\mathbf{V}}_{v}}-{{\mathbf{V}}_{I}}) \\ 
 & ={{{{\dot{m}}''}}_{\delta }}\left\{ \left[ {{e}_{v}}+\frac{{{({{\mathbf{V}}_{v}}-{{\mathbf{V}}_{I}})}^{2}}}{2} \right]-\left[ {{e}_{\ell }}+\frac{{{({{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{I}})}^{2}}}{2} \right] \right\} \\ 


where the heat fluxes in the liquid and vapor phases have been determined by Fourier’s law of conduction.

Equation (21) can be rewritten in terms of enthalpy (h = e + p / ρ), i.e.,

  & \left( {{k}_{v}}\nabla {{T}_{v}}-{{k}_{\ell }}\nabla {{T}_{\ell }} \right)\cdot \mathbf{n}-(\mathbf{n}\cdot {{{\mathbf{{\tau }'}}}_{\ell }})\cdot ({{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{I}})+(\mathbf{n}\cdot {{{\mathbf{{\tau }'}}}_{v}})\cdot ({{\mathbf{V}}_{v}}-{{\mathbf{V}}_{I}}) \\ 
 & ={{{{\dot{m}}''}}_{\delta }}\left[ {{h}_{\ell v}}-\left( \frac{{{p}_{v}}}{{{\rho }_{v}}}-\frac{{{p}_{\ell }}}{{{\rho }_{\ell }}} \right)+\frac{1}{2}\mathbf{V}_{v}^{2}-{{\mathbf{V}}_{v}}\cdot {{\mathbf{V}}_{I}}-\frac{1}{2}\mathbf{V}_{\ell }^{2}+{{\mathbf{V}}_{\ell }}\cdot {{\mathbf{V}}_{I}} \right] \\ 


where {{h}_{\ell v}} is the difference between the enthalpy of vapor and liquid at saturation, i.e., the latent heat of vaporization. The stress tensor in eq. (22) can be expressed as

{\tau }'=-p\mathbf{I}+2\mu \mathbf{D}-\frac{2}{3}\mu \left( \nabla \cdot \mathbf{V} \right)\mathbf{I}=-p\mathbf{I}+\tau


Since the relative velocities at the interface satisfy and , we have

{{p}_{\ell }}\left( {{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{I}} \right)\cdot \mathbf{n}-{{p}_{v}}\left( {{\mathbf{V}}_{v}}-{{\mathbf{V}}_{I}} \right)\cdot \mathbf{n}={{{\dot{m}}''}_{\delta }}\left[ -\left( \frac{{{p}_{v}}}{{{\rho }_{v}}}-\frac{{{p}_{\ell }}}{{{\rho }_{\ell }}} \right) \right]


Substituting eqs. (23) and (24) into eq. (22), the energy balance can be written as

  & \left( {{k}_{v}}\nabla {{T}_{v}}-{{k}_{\ell }}\nabla {{T}_{\ell }} \right)\cdot \mathbf{n}-\left( \mathbf{n}\cdot {{\tau }_{\ell }} \right)\left( {{\mathbf{V}}_{\ell }}-{{\mathbf{V}}_{I}} \right)+\left( \mathbf{n}\cdot {{\tau }_{v}} \right)\left( {{\mathbf{V}}_{v}}-{{\mathbf{V}}_{I}} \right) \\ 
 & ={{{{\dot{m}}''}}_{\delta }}\left[ {{h}_{\ell v}}+\frac{1}{2}\mathbf{V}_{v}^{2}-{{\mathbf{V}}_{v}}\cdot {{\mathbf{V}}_{I}}-\frac{1}{2}\mathbf{V}_{\ell }^{2}+{{\mathbf{V}}_{\ell }}\cdot {{\mathbf{V}}_{I}} \right] \\ 


To simplify the energy equation, the kinetic energy terms are considered negligible and no-slip conditions are assumed at the interface,

{{V}_{\ell ,\mathbf{t}}}={{V}_{v,\mathbf{t}}}={{V}_{I,\mathbf{t}}}


Therefore, the energy equation can be rewritten as

  & \left( {{k}_{v}}\frac{\partial {{T}_{v}}}{\partial {{x}_{\mathbf{n}}}}-{{k}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial {{x}_{\mathbf{n}}}} \right) \\ 
 & ={{{{\dot{m}}''}}_{\delta }}\left[ {{h}_{\ell v}}+\frac{4}{3}{{\nu }_{\ell }}\frac{\partial {{V}_{\ell ,\mathbf{n}}}}{\partial {{x}_{\mathbf{n}}}}-\frac{2}{3}{{\nu }_{\ell }}\left( \frac{\partial {{V}_{\ell ,{{\mathbf{t}}_{1}}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{\ell ,{{\mathbf{t}}_{2}}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}} \right) \right. \\ 
 & \left. \text{    }-\frac{4}{3}{{\nu }_{v}}\frac{\partial {{V}_{v,\mathbf{n}}}}{\partial {{x}_{\mathbf{n}}}}+\frac{2}{3}{{\nu }_{v}}\left( \frac{\partial {{V}_{v,{{\mathbf{t}}_{1}}}}}{\partial {{x}_{{{\mathbf{t}}_{1}}}}}+\frac{\partial {{V}_{v,{{\mathbf{t}}_{2}}}}}{\partial {{x}_{{{\mathbf{t}}_{2}}}}} \right) \right] \\ 


where are kinematic viscosities of liquid and vapor phases, respectively. The energy balance at the interface can be simplified by assuming that the change in the kinetic energy across the interface is negligible, i.e.,

\left( {{k}_{v}}\nabla {{T}_{v}}-{{k}_{\ell }}\nabla {{T}_{\ell }} \right)\cdot \mathbf{n}={{{\dot{m}}''}_{\delta }}{{h}_{\ell v}}


Equations (1), (19), and (28) are widely used in the analysis of evaporation and condensation.

Species Balance

In general, to obtain a detailed solution one needs to solve the continuity, species, momentum, and energy equations in each phase (presented in Chapters 3 and 4) and use the above interfacial balances to couple the conditions in each phase. This provides information for dependent variables, such as pressure, density, mass fraction, velocity, and temperature, as functions of time and space. Such an approach requires detailed numerical simulation in all phases. One may often be interested in a restricted solution for making an order of magnitude analysis or doing an analytical solution for a limiting case, in contrast to a detailed numerical solution. In some applications, dependent variables are also coupled, preventing the solution of one dependent variable independent of the others. In practice, one often must make assumptions to simplify the interface conditions or complexity of the multiphase problem. In other cases one can neglect, based on physical significance, the resistance to mass transfer and/or heat transfer in one of the phases; this permits simplification of the solution techniques and the solution of conservation equations in only one phase.

Some dependent variables, such as temperature, are continuous across the phases; other variables, such as concentration, are discontinuous. For a general interface between phases k and j in a multi-component system, a local balance in mass flux of species i must be upheld. The total species mass flux, {{{\dot{m}}''}_{i}}, at an interface is:

{{{\dot{m}}''}_{i}}={{\rho }_{k,i}}\left( {{\mathbf{V}}_{k,i}}-{{\mathbf{V}}_{I}} \right)\cdot \mathbf{n}={{\rho }_{j,i}}\left( {{\mathbf{V}}_{j,i}}-{{\mathbf{V}}_{I}} \right)\cdot \mathbf{n}


The velocity of species i in phase k and phase j is {{\mathbf{V}}_{k,i}} and {{\mathbf{V}}_{j,i}}, respectively. These velocities are defined as:

{{\rho }_{k,i}}{{\mathbf{V}}_{k,i}}={{\mathbf{J}}_{k,i}}+{{\omega }_{k,i}}{{\rho }_{k}}{{\mathbf{V}}_{k}}


{{\rho }_{j,i}}{{\mathbf{V}}_{j,i}}={{\mathbf{J}}_{j,i}}+{{\omega }_{j,i}}{{\rho }_{j}}{{\mathbf{V}}_{j}}


Using the interfacial species mass balance, and substituting the definition of the species velocity, the interfacial species mass flux is:

{{{\dot{m}}''}_{i}}={{\mathbf{J}}_{k,i}}\cdot \mathbf{n}+{{\omega }_{k,i}}{{\rho }_{k}}\left( {{\mathbf{V}}_{k}}-{{\mathbf{V}}_{I}} \right)={{\mathbf{J}}_{j,i}}\cdot \mathbf{n}+{{\omega }_{j,i}}{{\rho }_{j}}\left( {{\mathbf{V}}_{jk}}-{{\mathbf{V}}_{I}} \right)


Remembering the overall mass conservation at the interface,

{\dot{m}}''={{\rho }_{k}}\left( {{\mathbf{V}}_{k}}-{{\mathbf{V}}_{I}} \right)\cdot n={{\rho }_{j}}\left( {{\mathbf{V}}_{j}}-{{\mathbf{V}}_{I}} \right)\cdot n


The interfacial species mass flux is:

{{{\dot{m}}''}_{i}}={{\mathbf{J}}_{k,i}}\cdot \mathbf{n}+{{\omega }_{k,i}}{\dot{m}}''={{\mathbf{J}}_{j,i}}\cdot \mathbf{n}+{{\omega }_{j,i}}{\dot{m}}''


In some problems, the species mass flux will be specified, and the total mass flux is simply a sum of all the species mass fluxes.



If the species mass flux is not specified, the total mass flux at an interface can be calculated from the interfacial species mass flux equation, eq. (34):

{\dot{m}}''=\frac{\left( {{\mathbf{J}}_{j,i}}-{{\mathbf{J}}_{k,i}} \right)\cdot n}{{{\omega }_{k,i}}-{{\omega }_{j,i}}}


Since the species equation is second order in space, and there are two phases, there needs to be a secondary condition relating the species mass fraction in phases k and j. This can be done by expressing the mass fraction of species i in phase k as a function of the species mass fraction i in phase j, or vice versa.

{{\omega }_{k,i}}={{\omega }_{k,i}}\left( {{\omega }_{j,i}} \right)


{{\omega }_{j,i}}={{\omega }_{j,i}}\left( {{\omega }_{k,i}} \right)


Note that, in general, the species mass fraction is not a continuous function, and there is almost always a jump condition at the interface when two or more species are present. To make this point clear, the simplest case of a binary mixture is considered. For a binary mixture, the species diffusion flux, J, can be calculated by Fick’s law.

{{\mathbf{J}}_{k,1}}\cdot n=-{{\rho }_{k}}{{D}_{k,12}}\nabla {{\omega }_{k,1}}\cdot \mathbf{n}


{{\mathbf{J}}_{j,1}}\cdot n=-{{\rho }_{j}}{{D}_{j,12}}\nabla {{\omega }_{j,1}}\cdot \mathbf{n}


There are several examples of different phenomena of a binary mixture in the presence of a liquid/vapor, solid/vapor, or liquid/solid interface. In other examples, the species flux in one of the phases is unimportant; therefore, only one phase must be considered.

A classical example of species flux is the condition of zero mass flux due to an impermeable surface where species A does not diffuse to the stationary media \left( \nabla {{x}_{A}}\cdot n=0 \right) or \left( \partial {{x}_{A}}/\partial y=0 \right). The case of constant surface concentration is more typical; however, all types of mass transfer boundary conditions are demonstrated in three categories below.

a. The mass transfer from solid or liquid to a gas stream Evaporation and sublimation are typical examples of mass transfer from liquid or solid to a gas mixture, as shown in Fig. 1, where {{x}_{A,\ell }}, xA,s, xA,g are molar fractions in liquid, solid, and gas, respectively, and {{{\dot{m}}''}_{A}} is mass flux for species A. There are two cases shown in Fig. 1. In case I, a pure liquid/solid is evaporating/sublimating into a gaseous mixture and the evaporation /sublimation rate is controlled purely by the species gradient in the gas phase. In case II, the liquid/solid is not pure, therefore concentration gradients also exist in the liquid/solid phases. The mass transfer rate is limited by both phases. Also note that the concentration gradient in the solid [Fig. 1(b)] is steeper and does not penetrate as deeply when compared to the liquid concentration gradient [Fig. 1(a)]. Also, the liquid concentration gradient is steeper and does not penetrate as far compared to the gas. These trends are due to the ratio of the mass diffusivities in each of the phases.

A simplified relationship for the mass concentration condition at the interface can be obtained assuming the gas mixture is approximated by the ideal gas and the solid or liquid has a high concentration of A. The condition within the gas stream is of interest, because the main resistance to mass transfer is within that region. With the preceding assumptions, the partial vapor pressure of A in the gas mixture at the interface can be approximated from Raoult’s Law:

Figure 1 Species concentration and mass transfer from solid and or liquid to a gas mixture.
Figure 1 Species concentration and mass transfer from solid and or liquid to a gas mixture.

{{\left. {{p}_{A}} \right|}_{y={{0}^{-}}}}={{\left. {{x}_{A,\ell }} \right|}_{y={{0}^{+}}}}\left( {{p}_{A,sat}} \right)


{{\left. {{p}_{A}} \right|}_{y={{0}^{-}}}}={{\left. {{x}_{A,s}} \right|}_{y={{0}^{+}}}}\left( {{p}_{A,sat}} \right)


where pA is the partial vapor pressure of A in the gas mixture, {{\left. {{x}_{A,\ell }} \right|}_{y={{0}^{+}}}} is the mole fraction of species A in liquid, {{\left. {{x}_{A,s}} \right|}_{y={{0}^{+}}}} is the mole fraction of species in solid, and pA,sat is the normal saturation pressure of species A at the surface interface. Clearly, if the solid or liquid is made of pure species A, then {{\left. {{x}_{A,\ell }} \right|}_{y=0}}{{\left. ={{x}_{A,s}} \right|}_{y=0}}=1 and eqs. (41) and (42) reduce to

{{\left. {{p}_{A}} \right|}_{y={{0}^{-}}}}={{p}_{A,sat}}


This means that the partial vapor pressure of species A in the gas mixture at the interface is equal to the normal saturation pressure for species A, which is a function of interfacial temperature and can be obtained from thermodynamic tables in Appendix B. At the interface, species A and B in the gas phase must be in equilibrium with species A and B in the liquid or solid phase, except in extreme circumstances. Knowing {{\left. {{p}_{A}} \right|}_{y={{0}^{-}}}} and total pressure p, one can easily calculate the mole fraction xA,g and mass fraction ωA,g of species A at the interface on the gas side by the following relation

{{x}_{A,g}}=\frac{{{\left. {{p}_{A}} \right|}_{y={{0}^{-}}}}}{p}


{{\omega }_{A,g}}=\frac{{{x}_{A,g}}{{M}_{A}}}{{{x}_{A,g}}{{M}_{A}}+{{x}_{B,g}}{{M}_{B}}}


where MA and MB are molecular weights of species A and B. A conventional example for the case in Fig. 1(a) is the evaporation of water to a water-air mixture; obviously, one can imagine that the water absorbed a small amount of air. Sublimation of iodine in air is an application for the case of Fig. 1(b).

Mass transfer from a solid to a gaseous state sometimes requires the specification of diffusion molar flux, rather than concentration, at the solid surface.

J_{A}^{*}=-c{{D}_{AB}}\frac{\partial {{x}_{A}}}{\partial y}


where J_{A}^{*} can be a function of concentration and not necessarily constant. One application is related to catalytic surface reaction. For this case, {\dot{m}}''=0 because the mass flux of reactants equals the mass flux of products. Therefore, from eq. (34), {{{\dot{m}}''}_{i}}={{J}_{k,i}}\cdot n. Catalytic surfaces are used to promote heterogeneous reactions (see Chapter 3), which occur at the surface; the appropriate boundary condition is

Figure 2 Species concentration and mass transfer from gas mixture to liquid or solid
Figure 2 Species concentration and mass transfer from gas mixture to liquid or solid

J_{A}^{*}=-{{{k}'}_{1}}{{\left( {{\left. {{c}_{A}} \right|}_{y=0}} \right)}^{n}}


where k'1 is the reaction rate constant and n is the order of reaction.

Figure 3 Species concentration and mass transfer from solid to liquid and liquid to solid
Figure 3 Species concentration and mass transfer from solid to liquid and liquid to solid

b. Mass transfer from gases to liquids or solids

There are two major forms of mass transfer from gas to liquid, shown in Fig. 2. Case I is for condensable species in the liquid phase, or for species that can deposit vapor into the solid phase. In this case, a species of low concentration in the gas phase can condense at a higher concentration in the liquid phase. This phenomenon is important in distillation processes. Case II is for species that are weakly soluble in a liquid or a solid. In the liquid ({{x}_{A,\ell }} is small), Henry’s law will relate the mole fraction of species A in the liquid to the partial vapor pressure of A in the gas mixture at the interface by following relation [Fig. 2(a)]

{{\left. {{x}_{A,\ell }} \right|}_{y={{0}^{+}}}}=\frac{{{\left. {{p}_{A}} \right|}_{y={{0}^{-}}}}}{H}


Henry’s constant, H, is dependent primarily on the temperature of the aqueous solution and the absorption of two species. The pressure dependence is usually small and in fact, negligible up to pressure equal to +5 bar. Table B.77 provides H for selected aqueous solutions. For binary soluble gases, Henry’s law is not appropriate, and solubility data are usually presented in term of gas-phase partial pressure to liquid-phase mole fraction. Such data are given in Appendix B for NH3-water and SO2-water systems.

In chemical engineering applications, there are many cases in which gas is absorbed into a liquid; two examples are the absorption of hydrogen sulfide from an H2S-air mixture into liquid water and the absorption of O2 or chlorine into liquid water.

The concentration of gas in a solid at the interface is usually obtained by the use of a property known as the solubility, S, defined below

{{\left. {{c}_{A,s}} \right|}_{y={{0}^{+}}}}=S{{\left. {{p}_{A,g}} \right|}_{y={{0}^{-}}}}


where cA,s is the mole concentration of species A in the solid at the interface and {{\left. {{p}_{A,g}} \right|}_{y={{0}^{-}}}} is the vapor partial pressure of species A in the gas at the interface. The values of S (kmol/Pa-m3) for vapor gas-solid combinations are given in chemistry handbooks; some are in Appendix B (Tables B.76, B.78 and B.79). Special care should be exercised when using eq. (49) for the variation in form and units – two different versions of S are presented in Appendix B. An example of this case is the diffusion of helium in a glass. The dissolution of gas in metals is much more complex and depends on the type of metal used. The dissolution of gas in some metals can be reversed, such as the case of hydrogen into titanium. In contrast to hydrogen, oxygen dissolution in titanium is irreversible but is complicated because it forms a layer of scale TiO2 on the surface. This topic is beyond the scope of this book, and interested readers should refer to a metallurgy or materials handbook to consult appropriate phase diagrams.

c. Mass transit from solid to liquid or liquid to solid Two cases are also presented for the case of mass transfer between a solid and a liquid (Fig. 3). Case I represents a pure solid dissolving into a liquid, or a pure liquid diffusing into a solid. Case II represents a solid mixture melting into a liquid mixture, or a liquid mixture solidifying into a solid mixture. For the first case, the mass transfer is related to the solubility, which can be found in chemistry handbooks (Lide, 2004) and some are reproduced in Appendix B. A conventional example for this case is the dissolution of salt into water.

For a liquid/solid interface during melting and solidification, the ratio of the species concentration of the solid in liquid phases is called the partition ratio, Kp.

{{K}_{p}}=\frac{{{c}_{A,s}}}{{{c}_{A,\ell }}}


The partition coefficient is a function of temperature. Furthermore, when the liquid and solid lines are nearly straight, it is a constant. This information can be obtained from phase diagrams for different solid/liquid mixtures.

Additional Equations

{{p}_{cap}}=\Delta p=\sigma \left( \frac{1}{{{R}_{I}}}+\frac{1}{{{R}_{II}}} \right)=\sigma ({{K}_{1}}+{{K}_{2}})



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

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