Mass transfer

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When there is a species concentration difference in a multicomponent mixture, mass transfer occurs. There are two modes of mass transfer: diffusion and convection. Diffusion results from random molecular motion at the microscopic level, and it can occur in a solid, liquid or gas. Similar to convective heat transfer, convective mass transfer is due to a combination of random molecular motion at the microscopic level and bulk motion at the macroscopic level. It can occur only in a liquid or gas.


Species Concentration

The species concentration in a mixture ρi, is defined as the mass of species i per unit volume of the mixture (kg/m3). The density of the mixture equals the sum of the concentrations of all N species, i.e.,

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

The concentration of the ith species can also be represented by the mass fraction of the ith species, defined as

{\omega _i} = \frac{{{\rho _i}}}{\rho } \qquad \qquad(2)

It follows from eq. (1) that

\sum\limits_{i = 1}^N {{\omega _i}}  = 1 \qquad \qquad(3)

The concentration of the ith species can also be represented as molar concentration, defined as the number of moles of the ith species per unit volume, ci (kmol/m3), which is related to the mass concentration by

 {c_i} = \frac{{{\rho _i}}}{{{M_i}}} \qquad \qquad(4)

The molar concentration of the mixture equals the sum of the molar concentrations of all N species, i.e.,

c = \sum\limits_{i = 1}^N {{c_i}} \qquad \qquad(5)

The molar fraction of the ith species is defined as

{x_i} = \frac{{{c_i}}}{c} \qquad \qquad(6)

which is identical to the molecular number fraction – the fraction of the number of molecules of the ith species to the number of molecules of all species in a given volume. This concept is essential when kinetic theory is used to describe the mass transfer process. Equation (6) leads to

\sum\limits_{i = 1}^N {{x_i}}  = 1 \qquad \qquad(7)

The mean molecular mass of the mixture can be expressed as

\bar M = \frac{\rho }{c} = \sum\limits_{i = 1}^N {{x_i}{M_i}} \qquad \qquad(8)

The mass fraction is related to the molar fraction by

{\omega _i} = \frac{{{x_i}{M_i}}}{{\sum\limits_{j = 1}^N {{x_j}{M_j}} }} = \frac{{{x_i}{M_i}}}{{\bar M}} \qquad \qquad(9)

The molar fraction is related to the mass fraction by

{x_i} = \frac{{{\omega _i}/{M_i}}}{{\sum\limits_{j = 1}^N {{\omega _j}/{M_j}} }} \qquad \qquad(10)

Mass diffusion of the ith component in the mixture will result in a velocity, Vi, of the ith component relative to the stationary coordinate axes. The local mass-averaged velocity of all species, {\mathbf{\tilde V}}, is defined as

 {\mathbf{\tilde V}} = \frac{{\sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{V}}_i}} }}{{\sum\limits_{i = 1}^N {{\rho _i}} }} = \frac{{\sum\limits_{i = 1}^N {{\rho _i}{V_i}} }}{\rho } = \sum\limits_{i = 1}^N {{\omega _i}{{\mathbf{V}}_i}} \qquad \qquad(11)

which demonstrates that the local mass flux due to diffusion, \rho {\mathbf{\tilde V}} , is equal to the summation of mass flux for each species,  \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{V}}_i}} .

The molar-averaged velocity can be defined in a similar manner:

 {{\mathbf{\tilde V}}^*} = \frac{{\sum\limits_{i = 1}^N {{c_i}{{\mathbf{V}}_i}} }}{c} = \sum\limits_{i = 1}^N {{x_i}{{\mathbf{V}}_i}} \qquad \qquad(12)

The velocity of the Vi species relative to the mass or molar-averaged velocity, {{\mathbf{V}}_i} - {\mathbf{\tilde V}} or {{\mathbf{V}}_i} - {{\mathbf{\tilde V}}^*} is defined as diffusion velocity. The mass flux and molar flux relative to stationary coordinate axes are defined as

 {{\mathbf{\dot m''}}_i} = {\rho _i}{{\mathbf{V}}_i} \qquad \qquad(13)
 {{\mathbf{\dot n''}}_i} = {c_i}{{\mathbf{V}}_i} \qquad \qquad(14)

The fluxes defined in eqs. (13) and (14) are related by

	{{\mathbf{\dot n''}}_i} = \frac{{{{{\mathbf{\dot m''}}}_i}}}{{{M_i}}} \qquad \qquad(15)

Applying eqs. (13) and (14) into eqs. (11) and (12), the total mass flux and molar flux are obtained.

 {\mathbf{\dot m''}} = \sum\limits_{i = 1}^N {{{{\mathbf{\dot m''}}}_i}}  = \rho {\mathbf{\tilde V}} \qquad \qquad(16)
 {\mathbf{\dot n''}} = \sum\limits_{i = 1}^N {{{{\mathbf{\dot n''}}}_i}}  = c{{\mathbf{\tilde V}}^*} \qquad \qquad(17)

The mass flux relative to the mass-averaged velocity is

	{{\mathbf{J}}_i} = {\rho _i}({{\mathbf{V}}_i} - {\mathbf{\tilde V}}) \qquad \qquad(18)

and the molar flux relative to the molar averaged velocity is

 {\mathbf{J}}_i^* = {c_i}({{\mathbf{V}}_i} - {{\mathbf{\tilde V}}^*}) \qquad \qquad(19)

According to eqs. (11) and (12), we have

\sum\limits_{i = 1}^N {{{\mathbf{J}}_i}}  = \sum\limits_{i = 1}^N {{\mathbf{J}}_i^*}  = 0 \qquad \qquad(20)

Although any one of the four fluxes defined in eqs. (13) – (14) and (18) – (19) are adequate to describe mass diffusion under all circumstances, there is usually a preferred definition of flux that can lead to less algebraic complexity. When mass diffusion is coupled with advection, eq. (18) is preferred because the mass-averaged velocity, {\mathbf{\tilde V}}, is the velocity used in the momentum and energy equations. On the other hand, eq. (19) is preferred for the multicomponent system with constant molar density c, resulting from constant pressure and temperature. According to eqs. (13)–(19), the following relationships between different fluxes are valid:

{{\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} \qquad \qquad(21)
{{\mathbf{\dot n''}}_i} = {c_i}{{\mathbf{\tilde V}}^*} + {\mathbf{J}}_i^* = {x_i}\sum\limits_{i = 1}^N {{{{\mathbf{\dot n''}}}_i}}  + {\mathbf{J}}_i^* \qquad \qquad(22)

Binary Systems

For a binary system that is uniform in all aspects except concentration, i.e., no temperature or pressure gradient, the diffusive mass flux can be obtained by Fick’s law:

One-dimensional mass diffusion
One-dimensional mass diffusion.
 {{\mathbf{J}}_1} =  - \rho {D_{12}}\nabla {\omega _1} \qquad \qquad(23)

or alternatively

{\mathbf{J}}_1^* =  - c{D_{12}}\nabla {x_1} \qquad \qquad(24)

where D12 is binary diffusivity (m2/s).

The mass and molar-flux relative to a stationary coordinate axes are

 {{\mathbf{\dot m''}}_1} = {\omega _1}({{\mathbf{\dot m''}}_1} + {{\mathbf{\dot m''}}_2}) - \rho {D_{12}}\nabla {\omega _1} \qquad \qquad(25)
{{\mathbf{\dot n''}}_1} = {x_1}({{\mathbf{\dot n''}}_1} + {{\mathbf{\dot n''}}_2}) - c{D_{12}}\nabla {x_1} \qquad \qquad(26)

Equation (52) is widely applied in binary systems with constant density, while eq. (26) is more appropriate for systems with constant molar concentration. It should be noted that from eqs. (25) and (26) the absolute fluxes of species ({{\mathbf{\dot m''}}_1} or  {{\mathbf{\dot n''}}_1} for a binary system can always be presented as a summation of two parts: one part due to convection [the first term in eqs. (25) and (26)], and another part due to diffusion [the second term in eqs. (25) and (26)]. For an isothermal and isobaric steady-state one-dimensional binary system shown in the figure on the right, in which surface (y =0) is impermeable to species 2, the mass flux of species 1 at the surface (y = 0) is

{\dot m''_{1y}} =  - \frac{{\rho {D_{12}}}}{{1 - {\omega _1}}}\frac{{\partial {\omega _1}}}{{\partial y}} \qquad \qquad(27)

Multicomponent System and Maxwell-Stefan relation

For a system with more than two components, Fick’s law is no longer appropriate and one must find other approaches to relate mass flux and concentration gradient. For a multicomponent low-density gaseous mixture, the following Maxwell-Stefan relation can be used to relate the molar fraction gradient of the ith component and the molar flux:

 \nabla {x_i} =  - \sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{{D_{ij}}}}({{\mathbf{V}}_i} - {{\mathbf{V}}_j})}  \\ 
  =  - \sum\limits_{j = 1(j \ne i)}^N {\frac{1}{{c{D_{ij}}}}} \left( {{x_j}{{{\mathbf{\dot n''}}}_i} - {x_i}{{{\mathbf{\dot n''}}}_j}} \right)\begin{array}{*{20}{c}}
   , & {i = 1,2, \cdots ,N}  \\
\end{array} - 1 \\
\qquad \qquad(28)

where Dij is the binary diffusivity from species i to species j. Equation (28) was originally suggested by Maxwell for a binary mixture based on kinetic theory and was extended to diffusion of gaseous mixtures of N species by Stefan. For an N-component system, N(N–1)/2 diffusivities are required. The diffusion in a multicomponent system is different from diffusion in a binary system, because the movement of the ith species is no longer proportional to the negative concentration gradient of the ith species. It is possible that (1) a species moves against its own concentration gradient, referred to as reverse diffusion; (2) a species can diffuse even when its concentration gradient is zero, referred to as osmotic diffusion; or (3) a species does not diffuse although its concentration gradient is favorable to such diffusion, referred to as diffusion barrier (Bird et al., 2002).

The Maxwell-Stefan relation can also be rewritten in terms of mass fraction and mass flux

 \nabla {\omega _i} + {\omega _i}\nabla (\ln \bar M) = \frac{{\bar M}}{\rho }\sum\limits_{j = 1(j \ne i)}^N {\frac{{{\omega _i}{{\mathbf{J}}_j} - {\omega _j}{{\mathbf{J}}_i}}}{{{D_{ij}}{M_j}}}} \begin{array}{*{20}{c}}
   , & {i = 1,2, \cdots ,N}  \\
\end{array} - 1
\qquad \qquad(29)

where \bar M is the molar-averaged molecular mass of the mixture. Although eqs. (28) and (29) were originally developed for low-density gaseous mixtures, it has been shown that they are also valid for dense gases, liquids, and polymers, except that Dij should be replaced by the multicomponent Maxwell-Stefan diffusivity \mathfrak{D_{ij}}.

The total diffusive mass flux vector of species i can be expressed as a linear form (Curtiss and Bird, 1999)

{{\mathbf{J}}_i} =  - D_i^T\nabla \ln T + {\rho _i}\sum\limits_{j = 1}^N {{\mathbb{D}_{ij}}{{\mathbf{d}}_j}} \begin{array}{*{20}{c}}
   , & {i = 1,2, \cdots ,N}  \\

\qquad \qquad(30)

which is referred to as the generalized Fick equation, which is applicable to a system with concentration, temperature, and pressure gradients. The first term on the right-hand side represents thermal diffusion, where D_i^T is thermal diffusion coefficient. The second term is diffusion caused by all other driving forces, including concentration gradient, pressure, and body force, where {\mathbb{D}_{ij}} is the multicomponent Fick diffusivity. {\mathbb{D}_{ij}} is obtained from

{\omega _i}{\mathbb{D}_{ij}} = {\hat D_{ij}} - \sum\limits_{n = 1(n \ne i)}^N {{\omega _n}{{\hat D}_{in}}}\qquad \qquad(31)


{\hat D_{ij}} = \frac{{{\omega _i}{\omega _j}}}{{{x_i}{x_j}}}{D_{ij}}\qquad \qquad(32)

The multicomponent Fick diffusivities {\mathbb{D}_{ij}} are symmetric ({\mathbb{D}_{ij}} = {\mathbb{D}_{ji}}) and satisfy \sum\limits_{i = 1}^N {{\omega _i}{\mathbb{D}_{ij}}}  = 1. Equation (30) can be rearranged to express the driving force, di, in terms of mass flux, i.e.,

 {{\mathbf{d}}_i} =  - \sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{\mathfrak{D_{ij}}}}} \left( {\frac{{D_i^T}}{{{\rho _i}}} - \frac{{D_j^T}}{{{\rho _j}}}} \right)\nabla (\ln T) \\ 
 {\rm{      }} - \sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{\mathfrak{D_{ij}}}}} \left( {\frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}} - \frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}}} \right)\begin{array}{*{20}{c}}
   , & {i = 1,2, \cdots ,N}  \\
\end{array} \\ 
 \end{array}\qquad \qquad(33)

which are referred to as generalized Maxwell-Stefan equations. The diffusional driving force di is obtained by

\frac{{c{R_u}T}}{{{\rho _i}}}{{\mathbf{d}}_i} = T\nabla \left( {\frac{1}{T}\frac{{{{\bar g}_i}}}{{{M_i}}}} \right) + \frac{{{{\bar h}_i}}}{{{M_i}}}\nabla \ln T - \frac{1}{\rho }\nabla p - {{\mathbf{X}}_i} + \frac{1}{\rho }\sum\limits_{j = 1}^N {{\rho _j}{{\mathbf{X}}_j}}\qquad \qquad(34)

where {\bar g_i} is partial molar Gibbs free energy (J/kmol), {\bar h_i} is partial molar enthalpy (J/kmol), and Xj is the body force per unit mass (m/s2) for the ith component. For an ideal gas mixture, the generalized driving force becomes

c{R_u}T{{\mathbf{d}}_i} = \nabla {p_i} - {\omega _i}\nabla p - {\rho _i}{{\mathbf{X}}_i} + {\omega _i}\sum\limits_{j = 1}^N {{\rho _j}{{\mathbf{X}}_j}} \qquad \qquad(35)

where the first two terms on the right-hand side are driving forces for ordinary diffusion and pressure diffusion. The last two terms are driving force for body force diffusion. If gravity is the only body force, the body force diffusion is zero because {{\mathbf{X}}_j} = {\mathbf{g}} for any component. The thermal (Soret) diffusion was included in the first term in eq. (30).

Substituting eq. (35) into eq. (30) and after some manipulations, the final form of mass flux in the Fick form is obtained as (Curtiss and Bird, 1999)

{{\mathbf{J}}_i} =  - D_i^T\nabla \ln T + \frac{{{\rho _i}}}
{{c{R_u}T}}\sum\limits_{j = 1}^N {{\mathbb{D}_{ij}}\left( {\nabla {p_i} - {\omega _j}\nabla p - {\rho _j}{{\mathbf{X}}_j} + {\omega _j}\sum\limits_{n = 1}^N {{\rho _n}{{\mathbf{X}}_n}} } \right)}
\qquad \qquad(36)

Similarly, substituting eq. (35) into eq. (32), the final form of the Maxwell-Stefan equations becomes

\begin{array}{l} c{R_u}T\sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{\mathfrak{D_{ij}}}}\left( {\frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}} - \frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}}} \right)}  = \nabla {p_i} - {\omega _i}\nabla p - {\rho _i}{{\mathbf{X}}_i} \\ + {\omega _i}\sum\limits_{j = 1}^N {{\rho _j}{{\mathbf{X}}_j}}  - c{R_u}T\sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{\mathfrak{D_{ij}}}}} \left( {\frac{{D_j^T}}{{{\rho _j}}} - \frac{{D_i^T}}{{{\rho _i}}}} \right)\nabla (\ln T)\begin{array}{*{20}{c}}, & {i = 1,2, \cdots ,N}  \\
\end{array} \\ 
\end{array}\qquad \qquad(37)

For dilute monatomic gas mixtures, the Maxwell-Stefan diffusivities can be approximated by binary diffusivity, i.e., \mathfrak{D_{ij}} \approx {D_{ij}} and eq. (37) is preferred over eq. (36) because {\mathbb{D}_{ij}} strongly depends on the mass concentrations. Equation (37) reduces to eq. (28) for the gaseous mixture with uniform temperatures and pressures. The multicomponent thermodiffusivity for species i is expressed as

D_i^T = \sum\limits_{j = 1(j \ne i)}^N {\frac{{\rho {M_i}{M_j}}}{{{{\bar M}^2}}}} {D_{ij}}k_{ij}^T \qquad \qquad(38)

where k_{ij}^T is the thermal diffusion ratio and it is related to the thermal diffusion factor, \alpha _{ij}^T by k_{ij}^T = \alpha _{ij}^T{x_i}{x_j} (Bird et al., 2002) where xi and xj are molar fractions of component i and j, respectively. For applications that involve mass diffusion in a binary mixture containing species 1 and 2, the multicomponent thermodiffusivity becomes

D_i^T = \frac{{\rho {M_1}{M_2}}}{{{{\bar M}^2}}}{D_{12}}k_{12}^T \qquad \qquad(39)

The thermal diffusion ratio in a binary system depends on both temperature and concentration, and some selected values for liquids and gases can be found in Bird et al.(2002). For transport phenomena in applications such as biotechnology, fuel cells and many others, it is usually assumed that the gas is ideal, the only body force is gravity, and the mixture pressure gradient is negligible. Although eq. (36) can be simplified to get the diffusive mass flux in this case, the multicomponent Fick diffusivities {\mathbb{D}_{ij}} are strongly dependent on the concentration as evidenced by eqs. (31) and (32). An alternative and simpler approach that utilizes all effects was developed in Faghri and Zhang (2006).

\left[ {\mathbf{J}} \right] = \frac{\rho }{{\bar M}}{{\mathbf{A}}^{ - 1}}{{\mathbf{B}}^{ - 1}}\left[ {\nabla \omega } \right] - \left[ {{D^T}} \right]\nabla \ln T \qquad \qquad(40)

The components of matrices A and B can be found below.

{A_{ii}} =  - \left[ {\frac{{{\omega _i}}}{{{M_i}{M_N}{D_{iN}}}} + \sum\limits_{j = 1(j \ne i)}^N {\frac{{{\omega _j}}}{{{M_i}{M_j}{D_{ij}}}}} } \right] \qquad \qquad(41)
{A_{ij}} = {\omega _i}\left( {\frac{1}{{{M_i}{M_j}{D_{ij}}}} - \frac{1}{{{M_i}{M_N}{D_{iN}}}}} \right)  \qquad \qquad(42)
{B_{ii}} = \left( {1 - {\omega _i}} \right){M_i} + {\omega _i}{M_N}  \qquad \qquad(43)
{B_{ij}} = {\omega _i}\left( {{M_N} - {M_j}} \right)  \qquad \qquad(44)

Equation (40) is in a form that can be very easily programmed and it can be simplified for a variety of cases. For a binary mixture – a simplest mixture, A and B are both a single value.

A =  - \frac{1}{{{M_1}{M_2}{D_{12}}}} \qquad \qquad(45)
B = {\omega _2}{M_1} + {\omega _1}{M_2} = \frac{{{x_2}{M_2}}}{M}{M_1} + \frac{{{x_1}{M_1}}}{M}{M_2} = \frac{{{M_1}{M_2}}}{M}  \qquad \qquad(46)

Therefore, the diffusion mass flux of a binary mixture is:

{J_i} =  - \rho {D_{12}}\nabla {\omega _i} - D_i^T\nabla \left( {\ln T} \right)  \qquad \qquad(47)

For a mixture of several species that are all very dilute in species N ({\omega _i} \ll 1,{\rm{ for }}i = 1,2,...,N - 1{\rm{ and }}{\omega _N} \approx 1), the B matrix is approximately

{\mathbf{B}} \approx \left[ {\begin{array}{*{20}{c}}
   {{M_1}} & 0 &  \ldots  & 0  \\
   0 & {{M_2}} &  \ddots  &  \vdots   \\
    \vdots  &  \ddots  &  \ddots  & 0  \\
   0 &  \ldots  & 0 & {{M_{N - 1}}}  \\
\end{array}} \right]
  \qquad \qquad(48)

and the A matrix is approximately

A \approx \left[ {\begin{array}{*{20}{c}}
   {{{\left( {{M_1}{M_N}{D_{1N}}} \right)}^{ - 1}}} & 0 &  \ldots  & 0  \\
   0 & {{{\left( {{M_2}{M_N}{D_{2N}}} \right)}^{ - 1}}} &  \ddots  &  \vdots   \\
    \vdots  &  \ddots  &  \ddots  & 0  \\
   0 &  \ldots  & 0 & {{{\left( {{M_{N - 1}}{M_N}{D_{N - 1,N}}} \right)}^{ - 1}}}  \\
\end{array}} \right]
  \qquad \qquad(49)

The mixture molecular mass is also approximately equal to the molecular mass of the Nth component, M \approx {M_N}. Therefore the diffusion flux for each component is:

{J_i} =  - \rho {D_{1N}}\nabla {\omega _i} - D_i^T\nabla \left( {\ln T} \right)  \qquad \qquad(50)

While the above discussions provided the generalized descriptions for mass transfer in a multicomponent system, the mass flux in a nonisothermal and nonisobaric binary system has a much simpler expression. The mass flux of species 1 due to ordinary pressure, body force, and thermal diffusion for a binary system becomes

 {{\mathbf{J}}_1} =  - \rho {D_{12}}\nabla {\omega _1} + \frac{{{M_1}{M_2}{D_{12}}{\omega _1}}}{p}\nabla p \\ 
 {\rm{       }} + \frac{{{M_1}{M_2}{D_{12}}{\omega _1}{\omega _2}}}{{{R_u}T}}({{\mathbf{X}}_1} - {{\mathbf{X}}_2}) - D_1^T\nabla (\ln T) \\ 
  \qquad \qquad(51)

where D12 is the binary diffusion coefficient (mass diffusivity, m2/s) of species 1 in a mixture of species 1 and 2. The unit for the mass diffusivity, D12, is the same as the kinematic viscosity, ν = μ / ρ, and the thermal diffusivity, α= k/(ρcp). The binary diffusivity at 1 atm for selected gases is listed in Tables E.1-E.3, Appendix E. It should be pointed out that the pressure diffusion, body force diffusion, and Soret diffusion represented by the second to fourth terms in eq. (51) is negligible for most applications.

Similarly one can reduce the generalized Maxwell-Stefan equation (37) to the following form by neglecting pressure, body force, and thermal diffusion effects and assuming the mean molecular mass is constant [see. eq. (29)]:

\nabla {\omega _i} = \frac{{\bar M}}{\rho }\sum\limits_{j = 1(j \ne i)}^N {\frac{{{\omega _i}{{\mathbf{J}}_j} - {\omega _j}{{\mathbf{J}}_i}}}{{{D_{ij}}{M_j}}}} \begin{array}{*{20}{c}}
   , & {i = 1,2, \cdots ,N}  \\
\end{array} - 1
\qquad \qquad(52)

In order to solve for the mass flux J, eq. (52) can be rearranged to get

{{\mathbf{J}}_i} =  - \sum\limits_{j = 1}^{N - 1} {\rho {D_{eff,ij}}\nabla {\omega _j}} \begin{array}{*{20}{c}}
   , & {i = 1,2, \cdots ,N}  \\
\end{array} - 1
\qquad \qquad(53)


[{D_{eff,ij}}] = {{\mathbf{F}}^{ - 1}} \qquad \qquad(54)
{F_{ii}} = \frac{{{\omega _i}\bar M}}{{{D_{iN}}{M_N}}} + \sum\limits_{k = 1(k \ne i)}^N {\frac{{{\omega _k}\bar M}}{{{D_{ik}}{M_k}}}} \qquad \qquad(55)
{F_{ij}} = {\omega _i}\bar M\left( {\frac{1}{{{D_{ij}}{M_j}}} - \frac{1}{{{D_{iN}}{M_N}}}} \right),{\rm{   }}i \ne j  \qquad \qquad(56)

The second mode of mass transfer, convective mass transfer, may be expressed in a manner analogous to eq. (19):

 {\dot m''_1} = {h_m}({\rho _{1,w}} - {\rho _{1,\infty }}) = \rho {h_m}({\omega _{1,w}} - {\omega _{1,\infty }}) \qquad \qquad(57)

where the species mass flux, {\dot m''_1} is again exemplified by transport from a flat surface to a vapor stream flowing over that surface. The term hm (m/s) in eq. (57) is the convection mass transfer coefficient, ρ1,w is the species mass concentration at the surface, {\rho _{1,\infty }} is the species mass concentration in the free stream, and ω1,w and {\omega _{1,\infty }} are the species mass fractions at the surface and in the free stream, respectively.

As is the case for the convective heat transfer coefficient, the convective mass transfer coefficient is a function of fluid properties, the flow field characteristics, and the geometric configuration. Results are frequently expressed in a dimensionless form that also reflects the analogy between heat and mass transfer, the Sherwood number:

Sh = \frac{{{h_m}L}}{{{D_{12}}}} \qquad \qquad(58)

For convenience and physical analogy between various diffusion transport processes, the three rates of diffusion transport equations for mass, momentum, and heat for one-dimensional, constant properties are summarized below:

Fick's law (binary diffusion):
{\dot m''_{1,y}} =  - \rho {D_{12}}\frac{{d{\omega _1}}}{{dy}}  \qquad \qquad(59)

Newton's law viscosity (viscous fluid shear):

{\tau _{yx}} =  - \nu \frac{{d(\rho u)}}{{dy}}  \qquad \qquad(60)

Fourier's law of heat conduction:

{q''_y} =  - \alpha \frac{{d(\rho {c_p}T)}}{{dy}}  \qquad \qquad(61)

It is apparent that all the rate equations are of the same form, where flux equals a constant times potential gradient. The proportionality constant is a function of materials involved in the transport process. This analogous relationship can be used to predict a transport phenomenon on the basis of knowledge of another transport phenomenon. For example, the empirical correlation of turbulent heat transfer can be obtained by applying correlation of friction or similarly mass transfer coefficients. In general, when an analogy exists and it does not always, information obtained from a simple experimental setup can be applied to more complex physical experimental setups. In subsequent chapters, we see circumstances in which two or more processes are governed by the same dimensionless governing equations and boundary conditions. In these circumstances, one can develop these analogous relations from a more accurate basis. As a first approximation, the convective heat and mass transfer coefficients for laminar and turbulent heat transfer can be related by the following equation:

\frac{h}{{{h_m}}} = \frac{{\bar h}}{{{{\bar h}_m}}} = \rho {c_p}{\rm{L}}{{\rm{e}}^{2/3}} = \frac{k}{{{D_{12}}{\rm{L}}{{\rm{e}}^{1/3}}}} \qquad \qquad(62)

Various fluxes in terms of the transport properties in multicomponent systems are summarized in the following table.

Summary of fundamental laws in momentum, heat and mass transfer

Flux Equations Requirements Comments/Assumptions
MomentumNewton’s law of viscosity (Introduction to Momentum Transfer)
{\mathbf{\tau '}} = - p{\mathbf{I}} + \mu \left[ {\nabla |{\mathbf{V}} + {{\left( {\nabla {\mathbf{V}}} \right)}^T}} \right] \frac{2}{3}\mu (\nabla  \cdot {\mathbf{V}}){\mathbf{I}}
Newtonian fluid LaminarSingle or multicomponent
Simplified Newton’s law of viscosity
{\mathbf{\tau '}} =  - p{\mathbf{I}} + \mu \left[ {\nabla {\mathbf{V}} + {{\left( {\nabla {\mathbf{V}}} \right)}^T}} \right]
Newtonian fluid LaminarSingle or multicomponent Incompressible
Energy Fourier’s law (Introduction to Heat Transfer)
{\mathbf{q''}} =  - k\nabla T
Single componentIsotropic
Fourier’s law for anisotropic material
{\mathbf{q''}} =  - {\mathbf{k}} \cdot \nabla T
Single componentAnisotropic
Fourier’s law for anisotropic material with interdiffusion convection and thermodiffusion effects
\begin{array}{l} {\mathbf{q''}} =  - {\mathbf{k}} \cdot \nabla T + \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}}  \\  + c{R_u}T\sum\limits_{i = 1}^N {\sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{{\rho _i}}}\frac{{D_i^T}}{{\mathfrak{D_{ij}}}}} \left( {\frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}} - \frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}}} \right)}  \\ \end{array}
Multi-component Anisotropic The second and third terms account for, respectively, the interdiffusion convection and thermodiffusion effects
MassFick’s law, eqs. (23) and (24)
{{\mathbf{J}}_1} =  - \rho {D_{12}}\nabla {\omega _1}
{\mathbf{J}}_1^* =  - c{D_{12}}\nabla {x_1}
Binary onlyNo temperature and pressure gradients. Same body force for both components
Maxwell-Stefan equation (28)
\begin{array}{l}\nabla {x_i} =  - \sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{{D_{ij}}}}({{\mathbf{V}}_i} - {{\mathbf{V}}_j})}  \\ =  - \sum\limits_{j = 1(j \ne i)}^N {\frac{1}{{c{D_{ij}}}}} \left( {{x_j}{{{\mathbf{\dot n''}}}_i} - {x_i}{{{\mathbf{\dot |n''}}}_j}} \right) \\ \end{array}
Multi-component gaseous mixtureNo temperature and pressure gradients.
Same body forces for all N components
Maxwell-Stefan equation (56)
\begin{array}{l} \nabla {\omega _i} + {\omega _i}\nabla (\ln \bar M) \\ = \frac{{\bar M}}{\rho }\sum\limits_{j = 1(j \ne i)}^N {\frac{{{\omega _i}{{\mathbf{J}}_j} - {\omega _j}{{\mathbf{J}}_i}}}{{{D_{ij}}{M_j}}}}  \\ \end{array}
Multi-component gaseous mixtureNo temperature and pressure gradients.
Same body forces for all N components
Maxwell-Stefan equation (37) \begin{array}{l} c{R_u}T\sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}}{{\mathfrak{D_{ij}}}}\left( {\frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}} - \frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}}} \right)}  = \nabla {p_i} - {\omega _i}\nabla p - {\rho _i}{{\mathbf{X}}_i} \\ + {\omega _i}\sum\limits_{j = 1}^N {{\rho _j}{{\mathbf{X}}_j}}  - c{R_u}T\sum\limits_{j = 1(j \ne i)}^N {\frac{{{x_i}{x_j}}} {{\mathfrak{D_{ij}}}}} \left( {\frac{{D_j^T}}{{{\rho _j}}} - \frac{{D_i^T}}{{{\rho _i}}}} \right)\nabla (\ln T) \\ \end{array} Multi-component systemIncluding ordinary, pressure,
body force, and thermal diffusion
Mass flux in binary system, eq. (51) \begin{array}{l}{{\mathbf{J}}_1} =  - \rho {D_{12}}\nabla {\omega _1} + \frac{{{M_1}{M_2}{D_{12}}{\omega _1}}}{p}\nabla p \\ {\rm{       }} + \frac{{{M_1}{M_2}{D_{12}}{\omega _1}{\omega _2}}}{{RT}}({{\mathbf{X}}_1} - {{\mathbf{X}}_2}) - D_1^T\nabla (\ln |T) \\ \end{array} Binary onlyIncluding ordinary, pressure,
body force, and thermal diffusion

The transport properties are presented in Thermophysical properties: mass transfer properties. Various empirical equations for transport properties of gases and liquids can also be found in Poling et al. (2000).


Bird, R.B., Stewart, W.E., and Lightfoot, E.N., 2002, Transport Phenomena, 2nd edition, John Wiley & Sons, New York.

Curtiss, C. F., and Bird, R. B., 1999, “Multicomponent Diffusion,” Industrial and Engineering Chemistry Research, Vol. 38, pp. 2115-2522.

Curtiss, C. F., and Bird, R. B., 2001, “Errata,” Industrial and Engineering Chemistry Research, Vol. 40, p. 1791.

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.

Kays, W.M., Crawford, M.E., and Weigand, B., 2004, Convective Heat Transfer, 4th ed., McGraw-Hill, New York, NY.

Poling, B.E., Prausnitz, J. M., and O’Connell, J.P., 2000, The Properties of Gases and Liquids, 5th edition, McGraw-Hill, New York, NY.

Further Reading

External Links