‫جامعه فاروس‬
‫كلية الهندسة‬
‫قسم البتروكيماويات‬
Pharos University
Faculty of Engineering
Petrochemical Department
MASS TRANSFER
LECTURE (2)
1. FLUXES:
The mass (or molar) flux of a given species is a vector quantity denoting the amount of
the particular species, in either mass or molar units, that passes per given increment of
time through a unit area normal to the vector.
Fick's first law defines the diffusion of component A in an isothermal, isobaric
system, for diffusion in only the z direction, the Fick,s rate equation is:
Where JA,Z is the molar flux in the z direction relative to the molar-average velocity,
dcA/dz is the concentration gradient in the z direction, and DAB, is the mass diffusivity
or diffusion coefficient for component A diffusing through component B. A more
general flux relation that is not restricted to isothermal, isobaric systems:
flux = — (overall density) (diffusion coefficient) (concentration gradient)
An equivalent expression for jA,z, the mass flux in the z direction relative to the massaverage velocity, is:
Where dωA/dz is the concentration gradient in terms of the mass fraction. When the
density is constant, this relation simplifies to:
Initial experimental investigations of molecular diffusion were unable to verify Fick's
law of diffusion. This was apparently due to the fact that mass is often transferred
simultaneously by two possible means: (1) as a result of the concentration differences
as postulated by Fick and (2) by convection differences induced by the density
differences that resulted from the concentration variation.
For a binary system with a constant average velocity in the z direction, the molar flux
in the z direction relative to the molar-average velocity may also be expressed by:
By rearranging above equations:
As the component velocities, vA,Z and vB,Z are velocities relative to the fixed z axis, the
quantities cAvA,Z and cBvB,Z are fluxes of components A and B relative to a fixed z
coordinate; accordingly, we symbolize this new type of flux as:
Substituting these symbols into above equation, we obtain a relation for the flux of
component A relative to the z axis:
This relation may be generalized and written in vector form as:
It is important to note that the molar flux, NA is a resultant of the two vector quantities:
- c DAB∇yA the molar flux, JA, resulting from the concentration gradient. This term is
referred to as the concentration gradient contribution; and yA (NA + NB) = cAV the
molar flux resulting as component A is carried in the bulk flow of the fluid. This flux
term is designated the bulk motion contribution.
The mass flux, nA relative to a fixed spatial coordinate system, is defined for a binary
system in terms of mass density and mass fraction by:
nA = -ρ DAB
+ wA (nA+ nB)
This relation may be generalized and written in vector form as:
Where:
Under isothermal, isobaric conditions, this relation simplifies to:
As previously noted, the flux is a resultant of two vector quantities:
- DAB ∇ρA the mass flux, jA, resulting from a concentration gradient; the concentration
gradient contribution.
ωA ( nA + nB ) = ρA v, the mass flux resulting as component A is carried in the bulk flow
of the fluid; the bulk motion contribution.
2. DIFFUSION COEFFICIENT:
Fick's law of proportionality, DAB, is known as the diffusion coefficient, which may be
obtained from equation:
The mass diffusivity has been reported in cm2/s: the SI units are m2/s, which is a factor
10-4smaller.
The diffusion coefficient depends upon the pressure, temperature, and composition of the
system. Experimental values for the diffusivities of gases, liquids, and solids are
tabulated in Appendix Tables J. 1, J.2, and J.3, respectively. As one might expect from
the consideration of the mobility of the molecules, the diffusion coefficients are generally
higher for gases (in the range of 5 x 10-6 to 1 x 10-5 m2/s), than for liquids (in the range of
10-10 to 10-9 m2/s), which are higher than the values reported for solids (in the range of 1014
to 10-10 m2/s). In the absence of experimental data, semi theoretical expressions have
been developed which give approximations, sometimes as valid as experimental values
due to the difficulties encountered in their measurement.
2.1 GAS MASS DIFFUSIVITY:
Hirschfelder et al. presented an equation for the diffusion coefficient for gas pairs of
non-polar, non-reacting molecules:
Where DAB is the mass diffusivity of A through B, in cm2/s; T is the absolute temperature,
in K; MA, MB are molecular weights of A and B, respectively; P is the absolute pressure,
in atmospheres; ϬAB is the "collision diameter," in A; and ΩD is the "collision integral" for
molecular diffusion, a dimensionless function . Appendix Table K.1 lists ΩD as a function
of kT/εAB , k is the Boltzmann constant, which is 1.38 x 10-16 ergs/K, and εAB is the energy
of molecular interaction for the binary system A and B, in ergs. Unlike the other two
molecular transport coefficients, viscosity and thermal conductivity, the diffusion
coefficient is dependent on pressure as well as on a higher order of the absolute
temperature.
This information is available for only a very few pure gases. Appendix Table K.2
tabulates these values.
For a binary system composed of non-polar molecular pairs, the parameters of the pure
component may be combined empirically by the following relations:
The Hirschfelder equation is often used to extrapolate experimental data. For moderate
ranges of pressure, up to 25 atm, the diffusion coefficient varies inversely with the
pressure. Higher pressures apparently require dense gas corrections; unfortunately, no
satisfactory correlation is available for high pressures. Equation also states that the
diffusion coefficient varies with the temperature as T 3/2/ ΩD varies. Simplifying
Hirschfelder, we can predict the diffusion coefficient at any temperature and at any
pressure below 25 atm from a known experimental value by:
DAB,T2 = DAB,T1*
3/2
*
*
In Appendix Table J.l, experimental values of the product DABP are listed for several gas
pairs at a particular temperature. Using this equation, we may extend these values to other
temperatures.
EXAMPLE (1):
Evaluate the diffusion coefficient of carbon dioxide in air at 20 °C and atmospheric
pressure. Compare this value with the experimental value reported in appendix table J.I.
SOLUTION:
From Appendix Table K.2, the values of Ϭ and εA/k are obtained:
Substance
CO2
Air
Ϭ in A
3.996
3.617
εA/k
190
97
T = 20+273=293K, P= 1 atm
ΩD = 1.047
M CO2 = 44 g/gmol, Mair =29 g/gmol
Substituting these values in Hirschfelder equation:
From Appendix Table J.l for CO2 in air at 273 K, 1 atm, we have DAB = 0.136 cm2/s
To correct for the differences in temperature, the following equation is applied:
DAB,T2 = DAB,T1*
3/2
*
*
DAB,T2 = 0.136*(
3/2
At T= 273 K:
ΩD = 1.074
By substituting this value in the previous equation:
DAB,T2 = 0.136*(
3/2
= 0.155 cm2/s
We readily see that the temperature dependency of the "collision integral" is very small.
Accordingly, most scaling of diffusivities relative to temperature includes only the ratio
(T1/T2)3/2.