Rate-Limited Mass Transfer
ESM 222
Equilibrium conditions are valid when the volumes of the various phases are relatively small
Rate-Limited Mass Transfer Between Phases
in experimental conditions
surface area of the phases in contact is large relative to the volume of one of the phases raindrops falling through the atmosphere
1
© Arturo A. Keller
2
Rate-Limited Mass Transfer
Rate-Limited Mass Transfer
In many cases, equilibrium conditions do not occur, because a certain amount of time is needed to transfer mass between the phases. In some cases (e.g. sorption of pollutants to the soil), we assume that the groundwater flow is so slow that equilibrium conditions are nearly achieved.
Mass-transfer limitations are also important when considering the transfer of gases (e.g. O2 , CO2) from the atmosphere to a water body.
3
© Arturo A. Keller
Examples: slow transport of CO2 from the atmosphere to the deep oceans (centuries) O2 to a river or lake which has become depleted due to excessive biological activity 4
Air-Water Exchange
© Arturo A. Keller
Air-Water Exchange Mixing of a pollutant in the bulk of the phase (e.g. in atmosphere, river or shallow lake) may be fast if there is significant flow and turbulence Mixing in the bulk phase is slow when flow is slow and without turbulence (e.g. groundwater), or the bulk phase is very large (e.g. deep lakes or oceans)
Source: Schwarzenbach et al., 1993
~1 mm
~ 0.1 mm
5
© Arturo A. Keller
© Arturo A. Keller
6
© Arturo A. Keller
Air-Water Exchange
Air-Water Exchange
Reduced mixing in the boundary layers due to: too much friction at the boundary, difficult to form currents in either the air or the water surface tension reduces movement of water molecules in boundary
What drives the molecules from one phase to the other? Thermodynamic equilibrium Molecules of pollutant go from high concentration (polluted phase) to low concentration (“clean” phase)
7
© Arturo A. Keller
8
Air-Water Exchange
Air-Water Exchange Stagnant Two-Film Model Molecules move through each boundary layer mostly by diffusion Diffusion is the random movement of molecules (Brownian motion) due to their interactions At the interface, we have equilibrium conditions
The limiting step for mass transfer from one bulk phase to the other bulk phase is the movement through the two layers Several models exist to describe this “rate-limiting” step. We will discuss the stagnant two-film model.
9
© Arturo A. Keller
© Arturo A. Keller
10
Stagnant Two-Film Model
© Arturo A. Keller
Stagnant Two-Film Model
Source: Schwarzenbach et al., 1993
Movement of mass from one phase to another is measured as a flux mass moving per unit area per unit time
Flux through the water phase is:
Fw = -Dw
11
© Arturo A. Keller
12
(Cw/a - Cw) zw
Dw = diffusivity in water, zw = thickness of film © Arturo A. Keller
Stagnant Two-Film Model
Stagnant Two-Film Model We can use Henry’s law to relate the concentrations at the interface Ca/w KH = Cw/a
Similarly, the flux of pollutant through the air phase is:
Fa = -Da
(Ca- Ca/w )
The flux through the two layers is then
za
(after some algebra and using Henry’s law)
Flux through the water layer must be equal to the flux through the air layer 13
F = vtot (Cw © Arturo A. Keller
va = (Da / za)
The total velocity through the two layers is: 1 vtot =
( (z /D ) + (z /(D K )) )
=
w
( (v ) w
a
a H
1
-1
+ (va KH)-1
)
15
) © Arturo A. Keller
Stagnant Two-Film Model
The “velocity” through each layers is:
w
KH
14
Stagnant Two-Film Model
vw = (Dw / zw)
Ca
© Arturo A. Keller
Which layer determines the rate of mass transfer? if vw >> vaKH, then vtot ~ vaKH and transfer is determined by the flux through the air phase if vw supersaturation
pH concentration of other ions
Excess ions form solid compounds (salts) which “precipitate” out of solution 57
© Arturo A. Keller
58
Precipitation
© Arturo A. Keller
Energy of Activation
There is an energy barrier to overcome before the first solid crystals form Microscopic organisms and suspended solids provide a site (nucleus) for the formation of crystals, acting like a catalyst and reducing the energy barrier
59
© Arturo A. Keller
60
From Langmuir (1997) © Arturo A. Keller
Precipitation
Precipitation Inorganic crystals, skeletal particles, clays, sand and biocolloids serve as seeds for the formation of precipitates The more affinity between the two solids the more likely nucleation will occur Once nucleation begins, it may proceed rapidly, removing large amounts of ions from the aqueous body
61
From Langmuir (1997) © Arturo A. Keller
62
Precipitation
Precipitation
Growth of crystals proceeds in several steps:
Some other environmental constituents may inhibit or retard the formation of crystals (crystal “poisons”), such as orthophosphates and dissolved organic matter, which themselves may sorb on crystallization sites There may also be competition between ions for available sites (e.g. Pb2+, Mg+2,Ca+2, …)
transport of dissolved pollutant (ion) to stagnant layer near crystal surface diffusion to crystal surface sorption of the ion to surface incorporation into solid matrix
63
© Arturo A. Keller
© Arturo A. Keller
Dissolution
64
© Arturo A. Keller
Dissolution
From Stumm & Morgan (1990)
From Langmuir (1997)
Sorption kinetics Sorption is usually modeled as an equilibrium process, but there are some cases where this is not valid:
Rate-limited Sorption
67
storm-related resuspension of sediments organic particles falling through the water column too fast to allow equilibrium air or water flow is faster than the time required to establish equilibrium
© Arturo A. Keller
68
Sorption kinetics
© Arturo A. Keller
Sorption kinetics
Two types of limitations may occur: the pollutant attaches chemically to the solid phase, and a reaction is needed to sorb or desorb the pollutant there is physically not enough time for the pollutant to travel to the sorption site
69
71
© Arturo A. Keller
70
Source: Schwarzenbach et al., 1993 © Arturo A. Keller
Sorption kinetics
Sorption kinetics
In many cases, most of the sites available for sorption are in the internal pores of a grain, which present a large surface area compared to the outside of the grain (e.g. GAC) In organic matter, the large “macromolecules” may present a very viscous path for the sorbing pollutant, which is then slowed during desorption
Sometimes a diffusing pollutant finds a “deadend” pore, where the water is essentially immobile: it has to diffuse back out to find a sorption site In other cases, the inner pores are almost the same size as the pollutant, in which case it has to find the right orientation to move in and sorb (or desorb)
© Arturo A. Keller
72
© Arturo A. Keller
Sorption kinetics
Sorption kinetics Effect of rate-limited sorption is most important during remediation of a site: as we pump water out for treatment, the sorbed pollutants begin to slowly desorb, contaminating fresh water in the aquifer in other cases, we determine a site has been cleaned up, only to find later that the pollutant is desorbing again
73
© Arturo A. Keller
74
Source: Schwarzenbach et al., 1993 © Arturo A. Keller