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MASS DIFFUSION
Mass Transfer ................................................................................................................................................ 1
What it is and what it isn't ......................................................................................................................... 1
What it is for. Applications ..................................................................................................

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Mass diffusion page 1
M
ASS DIFFUSION
Mass Transfer ................................................................................................................................................ 1
What it is and what it isn't ......................................................................................................................... 1
What it is for. Applications ....................................................................................................................... 2
How to study it. Similarities and differences between Mass Transfer and Heat Transfer ........................ 2
Forces and fluxes .................................................................................................................................. 4
Specifying composition. Nomenclature ................................................................................................ 4
Specifying boundary conditions for composition ................................................................................. 6
Species balance equation .......................................................................................................................... 9
Diffusion rate: Fick's law .................................................................................................................... 10
The diffusion equation for mass transfer ................................................................................................ 14
Some analytical solutions to mass diffusion ........................................................................................... 15
Instantaneous point-source .................................................................................................................. 15
Semi-infinite planar diffusion ............................................................................................................. 16
Diffusion through a wall ..................................................................................................................... 17
Summary table of analytical solutions to diffusion problems............................................................. 19
Evaporation rate ...................................................................................................................................... 22
References ............................................................................................................................................... 23
M
ASS
T
RANSFER
W
HAT IT IS AND WHAT IT ISN
'
T
The subject of Mass Transfer studies the relative motion of some chemical species with respect to others (i.e. separation and mixing processes), driven by concentration gradients (really, an imbalance in chemical potential, as explained in Entropy). Fluid flow without mass transfer is not part of the Mass
Transfer field but of Fluid Mechanics. Heat transfer and mass transfer are kinetic processes that may occur and be studied separately or jointly. Studying them apart is simpler, but it is most convenient (to optimise the effort) to realise that both processes are modelled by similar mathematical equations in the case of diffusion and convection (there is no mass-transfer similarity to heat radiation), and it is thus more efficient to consider them jointly. On the other hand, the subject of Mass Transfer is directly linked to Fluid Mechanics, where the single-component fluid-flow is studied, but the approach usually followed is more similar to that used in Heat Transfer, where fluid flow is mainly a boundary condition empirically modelled; thus, the teaching of Mass Transfer traditionally follows and builds upon that of Heat Transfer (and not upon Fluid
Mechanics). In fact, development in mass-transfer theory closely follows that in heat transfer, with the pioneering works of Lewis and Whitman in 1924 (already proposing a mass-transfer coefficient
h
m
similar to the thermal convection coefficient
h
), and Sherwood's book of 1937 on Absorption and extraction . Even more, since the milestone book on Transport phenomena by Bird et al. (1960), heat transfer, mass transfer, and momentum transfer, are often jointly considered as a new discipline.
Mass diffusion page 2 As usual, the basic study first focuses on homogeneous non-reacting systems with well-defined boundaries (not only in Mass Transfer, but in Heat Transfer and in Fluid Mechanics), touching upon moving-boundary problems and reacting processes only afterwards. As for the other subjects, it is based on the continuum media theory, i.e. without accounting for the microscopic motion of the molecules (so that field theory and the fluid-particle concept are applied here too). Diffusion theory only applies to molecular mixtures (
d
<10
-8
m); for colloids and suspensions (10
-8
..10
-5
m), Brownian theory must be applied, and for larger particles (>10
-5
m) Newtonian mechanics applies. Notice that we only consider here mass diffusion due to a concentration gradient, what might be called concentration-phoresis in analogy to other mechanisms of mass diffusion like thermo-phoresis (Soret effect), piezo-phoresis (diffusion due to a pressure gradient), or electrophoresis (diffusion due to a gradient of electrical potential applied to ionic media). Traditionally, the field of Mass Transfer has been studied only within the Chemical Engineering curriculum, except for humid-air applications (evaporation) and thermal desalination processes, which has been always studied in Mechanical Engineering. But mass transfer problems are proliferating in so many circumstances, especially at high temperatures (drying, combustion, materials treatment, pyrolysis, ablation...), that the subject should be covered on different grounds to encourage effective interdisciplinary team-work
W
HAT IT IS FOR
.
A
PPLICATIONS
Applications of Mass Transfer include the dispersion of contaminants, drying and humidifying, segregation and doping in materials, vaporisation and condensation in a mixture, evaporation (boiling of a pure substance is not mass transfer), combustion and most other chemical processes, cooling towers, sorption at an interface (adsorption) or in a bulk (absorption), and most living-matter processes as respiration (in the lungs and at cell level), nutrition, secretion, sweating, etc. A common process to separate a gas from a gaseous mixture is to selectively dissolve it in an appropriate liquid (this way, carbon dioxide from exhaust gases can be trapped in aqueous lime solutions, and hydrogen sulfide is absorbed from natural-gas sources; when water vapour is removed, the absorption process is called drying. Stripping is the reverse of absorption, i.e. the removal of dissolved components in a liquid mixture. Distillation is the most important separation technique.
H
OW TO STUDY IT
.
S
IMILARITIES AND DIFFERENCES BETWEEN
M
ASS
T
RANSFER AND
H
EAT
T
RANSFER
Mass Transfer education traditionally follows and builds upon that of Heat Transfer because, on the one hand, mass diffusion due to a concentration gradient is analogous to thermal-energy diffusion due to a temperature gradient, and thus the mathematical modelling practically coincides, and there are many cases where mass diffusion is coupled to heat transfer (as in evaporative cooling and fractional distillation); on another hand, Heat Transfer is mathematically simpler and of wider engineering interest than Mass Transfer, what dictates the precedence. But there are important differences between both subjects.
Mass diffusion page 3
ã
Radiation. First of all, from the three heat transfer modes (conduction, convection, and radiation), only the two first are considered in mass transfer (diffusion and convection), radiation of material particles (as neutrons and electrons) being studied apart (in Nuclear Physics). Notice, by the way, that the word diffusion can be applied to the spreading of energy (heat diffusion), or species (mass diffusion), or even momentum in a fluid or electric charges in conductors, but the word conduction is more commonly used than heat diffusion (whereas mass conduction is rarely used).
ã
Solids versus fluids. Heat Transfer starts with, and focuses on, heat diffusion in solids, which have higher thermal conductivities than fluids, the latter being considered globally through empirical convective coefficients, whereas Mass Transfer focuses on gases and liquids, which have higher mass diffusivities than solids. The explanation for such a difference is that heat conduction propagates by particle contact (for the same type of particles, the shortest separation the better), whereas mass diffusion propagates by particles moving through the material medium (for the same type of particles, the largest voids the better). Moreover, Heat Transfer problems in solids are simple and relevant to many applications, whereas Mass Transfer problems in solids are of much lesser relevance, and Mass Transfer problems in fluids are much more complicated because the simplest mass-diffusion problems are of little practical interest, convection within fluids being the rule (fluids tend to flow). When diffusion in solids is wanted, as in doping silicon substrates in microelectronics, or in surface diffusion of carbon or nitrogen in steel hardening, high temperature operation is the rule (diffusion coefficients show an Arrhenius' type dependence with temperature).
ã
Slowness. Thermal diffusivities decrease from solids to fluids, with typical values of
a
≈
10
-4
m
2
/s for metals and
a
≈
10
-5
m
2
/s for non-metals, down to
a
≈
10
-5
m
2
/s for gases and
a
≈
10
-7
m
2
/s for liquids. On the contrary, mass diffusivities decrease from fluids to solids, with typical values of
D
i
≈
10
-5
m
2
/s for gases and
D
i
≈
10
-9
m
2
/s for liquids, down to
D
i
≈
10
-12
m
2
/s for solids.
ã
Bulk flow. There is no bulk flow in heat diffusion (either within solids or fluids), whereas there is always some bulk flow associated to diffusion of species (except in the rare event of counter-diffusion of similar species); i.e. mass diffusion generates mass convection, in general.
ã
Number of field variables. One may consider just one heat-transfer function, the temperature field
T
(the heat flux is basically the gradient field), but several mass-transfer functions must be considered, one mass fraction,
y
i
, for each species
i
=1..
C
(
C
being the number of distinct chemical species), although most problems are modelled as a binary system of just one species of interest diffusing in a background mixture of averaged properties.
ã
Continuity at interfaces. Mass-transfer boundary conditions at interfaces are more complex than thermal boundary conditions, because there are always concentration discontinuities, contrary to the continuous temperature dictated by local equilibrium (chemical potentials are continuous at an interface, not concentrations).
ã
Diffusion 'uphill'. Besides the effect of coupled fluxes, it is important to realise that mass diffusion can be from a low concentration within a condensed medium towards a high concentration within a more disperse medium, because, as said, it is not concentration-gradient but chemical-potential-gradient, what drives mass diffusion (e.g. see Diffusion through a wall, below).
Mass diffusion page 4
Forces and fluxes
Mixing, i.e. decreasing differences in composition (really, in chemical potential) or temperature, is a natural process (i.e. it does not require an energy expenditure), driven by the gradients of temperature, relative speed and chemical composition (with the natural stratification in the presence of gravity or another force field). It is interesting to realise that the thermal and mechanical forces towards equilibrium have been harnessed to yield useful power (heat engines, wind and water turbines), but the chemical forces that drive mass transfer have not yet been rendered useful as energy source, no doubt because of its low specific energy (there has been proposals to built power plants driven by the difference in salt concentration at river mouths). The gradient of temperature, momentum and concentrations, give rise to corresponding fluxes in thermal energy, momentum and amount of species. The relation between forces and fluxes are the transport constitutive equations: Fourier law for Heat Transfer, Newton (or Stokes) law for Fluid Mechanics, and Fick law for Mass Transfer (to be presented below), and the purpose of the subject is to solve generic field balance equations (energy balance, momentum balance, and species balance), with the help of constitutive equations, and the particular boundary conditions and initial conditions. But before developing the theory, it must be understood that mixing is a slow physical process, if not forced by convection and turbulence, and even so. Many practical processes are limited by the difficulty to increase the mass transfer rate. An order of magnitude analysis shows that the relaxation time for diffusion-controlled phenomena (thermal, momentum, species) across a distance
L
is
t
relax
=
L
2
/
a
, where
a
is the diffusivity that, as explained below, is of order 10
-5
m
2
/s in gases, what teaches that diffusion across a 1 m distance takes some 10
5
s, i.e. one whole day. Of course, everybody knows that heating one metre of air doesn't take one day, neither it takes so long for odours to travel one metre, or for putting in motion or arresting a gas; the explanation is that fluids are very difficult to keep at rest when perturbed, and the convection that develops greatly increases the mixing rate and lowers the required time. Thermodynamics teaches that, within an isolated system in absence of external forces, temperature, relative motion and chemical potential tend to get uniform, by establishing a thermal-energy flux, a momentum flux and a mass-diffusion flux, proportional (to a first approximation) to the gradients of temperature, velocity and concentration, that tend to equilibrate the system. Notice however that, besides those direct fluxes, other smaller cross-coupling fluxes may appear, as mass-diffusion due to a temperature gradient in a uniform concentration, or heat transfer due to a concentration gradient in an isothermal field, which, in the linear approximation, are related among them by Onsager's reciprocal relations.
Specifying composition. Nomenclature
Mass transfer may take place within gases, liquids, solids or through their interfaces, always involving a mixture, but mass diffusion in a gas is of main interest for two reasons: first, it is the best understood, and

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