Multimedia Environmental Models - Chapter 7 ppsx

If only 40 mol/h could evaporate,the evaporated benzene may react in the air phase at 40 mol/h, but it will tend tobuild up in the water phase to a higher concentration and fugacity unti

McKay, Donald "Intermedia Transport"

Multimedia Environmental Models

Edited by Donald McKay

Boca Raton: CRC Press LLC,2001

Suppose we have air and water media as illustrated in Figure 7.1, with emissions

of 100 mol/h of benzene into the water There is only slow reaction in the water(say, 20 mol/h), but there is rapid reaction (say, 80 mol/h) in the air This impliesthat benzene is evaporating from water to air at a rate of 80 mol/h The questionarises: is benzene capable of evaporating at 80 mol/h, or will there be a resistance

to transfer that prevents evaporation at this rate? If only 40 mol/h could evaporate,the evaporated benzene may react in the air phase at 40 mol/h, but it will tend tobuild up in the water phase to a higher concentration and fugacity until the rate ofreaction in the water increases to 60 mol/h The benzene fugacity in the air will thus

be lower than the fugacity in water, and a nonequilibrium situation will have oped The ability to calculate how fast chemicals can migrate from one phase toanother is the challenging task of this chapter The topic is one in which there stillremain considerable uncertainty and scope for scientific investigation and innovation

devel-We begin it by listing and categorizing all the transport processes that are likely tooccur

7.2.1 Nondiffusive Processes

The first group of processes consists ofnondiffusive, or piggyback, or advective

processes A chemical may move from one phase to another by piggybacking on

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material that has decided, for reasons unrelated to the presence of the chemical, tomake this journey Examples include advective flows in air, water, or particulatephases, as discussed in Chapter 6; deposition of chemical in rainfall or sorbed toaerosols from the atmosphere to soil or water; and sedimentation of chemical inassociation with particles that fall from the water column to the bottom sediments.These are usually one-way processes The rate of chemical transfer is simplythe product of the concentration C mol/m3 of chemical in the moving medium, andthe flowrate of that medium, G, m3/h We can thus treat all these processes asadvection and calculate the D value and rate as follows:

N = GC = GZf = Df mol/h

The usual problem is to measure or estimate G and the corresponding Z value

or partition coefficient We examine these rates in more detail later, when we focus

on individual intermedia transfer processes

7.2.2 Diffusive Processes

The second group of processes are diffusive in nature If we have water containing

1 mol/m3 of benzene and add some octanol to it as a second phase, the benzene will

Figure 7.1 Illustration of nonequilibrium behavior in an air-water system In the lower diagram,

the rate of reaction in air is constrained by the rate of evaporation.

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diffuse from the water to the octanol until it reaches a concentration in octanol that

is KOW, or 135, times that in the water We could rephrase this by stating that, initially,the fugacity of benzene in the water was (say) 500 Pa, and the fugacity in the octanolwas zero The benzene then migrates from water to octanol until both fugacitiesreach a common value of (say) 200 Pa At this common fugacity, the ratio CO/CW

is, of course, ZO/ZW or KOW We argue that diffusion will always occur from highfugacity (for example, fW in water) to low fugacity (fO in octanol) Therefore, it istempting to write the transfer rate equation from water to octanol as

a departure from equilibrium group, just as a temperature difference represents adeparture from thermal equilibrium It quantifies the diffusive driving force.

Other areas of science provide good precedents for using this approach Ohm’slaw states that current flows at a rate proportional to voltage difference timeselectrical conductivity Electricians prefer to use resistance, which is simply thereciprocal of conductivity The rate of heat transfer is expressed by Fourier’s law as

a thermal conductivity times a difference in temperature Again, it is occasionallyconvenient to think in terms of a thermal resistance (the reciprocal of thermalconductivity), especially when buying insulation These equations have the generalform

rate = (conductivity) ¥ (departure from equilibrium)

or

rate = (departure from equilibrium)/(resistance)

Our task is to devise recipes for calculating D as an expression of conductivity orreciprocal resistance for a number of processes involving diffusive interphase trans-fer These include the following:

1 Evaporation of chemical from water to air and the reverse process of absorption Note that we consider the chemical to be in solution in water and not present as

a film or oil slick, or in sorbed form.

2 Sorption from water to suspended matter in the water column, and the reverse desorption

3 Sorption from the atmosphere to aerosol particles, and the reverse desorption.

4 Sorption of chemical from water to bottom sediment, and the reverse desorption.

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5 Diffusion within soils, and from soil to air.

6 Absorption of chemical by fish and other organisms by diffusion through the gills, following the same route traveled by oxygen.

7 Transfer of chemical across other membranes in organisms, for example, from air through lung surfaces to blood, or from gut contents to blood through the walls of the gastrointestinal tract, or from blood to organs in the body.

Armed with these D values, we can set up mass balance equations that are similar

to the Level II calculations but allow for unequal fugacities between media

To address these tasks, we return to first principles, quantify diffusion processes

in a single phase, then extend this capability to more complex situations involvingtwo phases Chemical engineers have discovered that it is possible to make a greatdeal of money by inducing chemicals to diffuse from one phase to another Examplesare the separation of alcohol from fermented liquors to make spirits, the separation

of gasoline from crude oil, the removal of salt from sea water, and the removal ofmetals from solutions of dissolved ores They have thus devoted considerable effort

to quantifying diffusion rates, and especially to accomplishing diffusion processesinexpensively in chemical plants We therefore exploit this body of profit-orientedinformation for the nobler purpose of environmental betterment

7.3 MOLECULAR DIFFUSION WITHIN A PHASE

7.3.1 Diffusion As a Mixing Process

In liquids and gases, molecules are in a continuous state of relative motion If

a group of molecules in a particular location is labeled at a point in time, as shown

in the upper part of Figure 7.2, then at some time later it will be observed that theyhave distributed themselves randomly throughout the available volume of fluid.Mixing has occurred

Since the number of molecules is large, it is exceedingly unlikely that they willever return to their initial condition This process is merely a manifestation of mixing

in which one specific distribution of molecules gives way to one of many otherstatistically more likely mixed distributions This phenomenon is easily demonstrated

by combining salt and pepper in a jar, then shaking it to obtain a homogeneousmixture It is the rate of this mixing process that is at issue

We approach this issue from two points of view First is a purely mathematicalapproach in which we postulate an equation that describes this mixing, or diffusion,process Second is a more fundamental approach in which we seek to understandthe basic determinants of diffusion in terms of molecular velocities

Most texts follow the mathematical approach and introduce a quantity termed

diffusivity or diffusion coefficient, which has dimensions of m2/h, to characterize thisprocess It appears as the proportionality constant, B, in the equation expressingFick’s first law of diffusion, namely

N = –B A dC/dy

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Here, N is the flux of chemical (mol/h), B is the diffusivity (m2/h), A is area (m2),

C is concentration of the diffusing chemical (for example, benzene in water)(mol/m3), and y is distance (m) in the direction of diffusion The group dC/dy isthus the concentration gradient and is characteristic of the degree to which thesolution is unmixed or heterogeneous The negative sign arises because the direction

of diffusion is from high to low concentration, i.e., it is positive when dC/dy isnegative Here, we use the symbol B for diffusivity to avoid confusion with D values.Most texts sensibly use the symbol D The equation is really a statement that therate of diffusion is proportional to the concentration gradient and the proportionalityconstant is diffusivity When the equation is apparently not obeyed, we attribute thismisbehavior to deviations or changes in the diffusivity, not to failure of the equation

As was discussed earlier, there are differences of opinion about the word flux

We use it here to denote a transfer rate in units such as mol/h Others insist that itshould be area specific and have units of mol/m2h We ignore their advice Occa-

Figure 7.2 The fundamental nature of molecular diffusion.

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sionally, the term flux rate is used in the literature This is definitely wrong, becauseflux contains the concept of rate just as does speed Flux rate is as sensible as speed rate.

It is worthwhile digressing to examine how the mixing process leads to diffusionand eventually to Fick’s first law This elucidates the fundamental nature of diffu-sivity and the reason for its rather strange units of m2/h Much of the pioneeringwork in this area was done by Einstein in the early part of this century and arosefrom an interest in Brownian movement—the erratic, slow, but observable motion

of microscopic solid particles in liquids, which is believed to be due to multiplecollisions with liquid molecules

7.3.2 Fick’s Law and Diffusion at Steady State

We consider a square tunnel of cross-sectional area A m2 containing a nonuniformsolution, as shown in the middle of Figure 7.2, having volumes V1, V2, etc., separated

by planes 1–2, 2–3, 3–4, etc., each y metres apart

We assume that the solution consists of identical dissolved particles that moveerratically, but on the average travel a horizontal distance of y metres in t hours Intime t, half the particles in volume V3 will cross the plane 2–3, and half the plane3–4 They will be replaced by (different) particles that enter volume V3 by crossingthese planes in the opposite direction from volumes V2 and V4 Let the concentration

of particles in V3 and V4 be C3 and C4 mol/m3 such that C3 exceeds C4 The nettransfer across plane 3–4 will be the sum of the two processes: C3 yA/2 moles fromleft to right, and C4 yA/2 moles from right to left The net amount transferred intime t is then

C3yA/2 – C4 yA/2 = (C3 – C4) yA/2 mol

Note that CyA is the product of concentration and volume and is thus an amount(moles)

The concentration gradient that is causing this net diffusion from left to right is(C3 – C4)/y or, in differential form, dC/dy The negative sign below is necessary,because C decreases in the direction in which y increases It follows that

(C3 – C4) = –ydC/dy

The flux or diffusion rate is then N or

N = (C3 – C4) yA/2t = –(y2A/2t) dC/dy = –BAdC/dy mol/h

which is referred to as Fick’s first law The diffusivity B is thus (y2/2t), where y isthe molecular displacement that occurs in time t

In a typical gas at atmospheric pressure, the molecules are moving at a velocity

of some 500 m/s, but they collide after traveling only some 10–7 m, i.e., after 10–7/500

or 2 ¥ 10–10 s It can be argued that y is 10–7 m, and t is 2 ¥ 10–10; therefore, we

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expect a diffusivity of approximately 0.25 ¥ 10–4 m2/s or 0.25 cm2/s or 0.1 m2/h,

which is borne out experimentally The kinetic theory of gases can be used to

calculate B theoretically but, more usefully, the theory gives a suggested structure

for equations that can be used to correlate diffusivity as a function of molecular

properties, temperature, and pressure

In liquids, molecular motion is more restricted, collisions occur almost every

molecular diameter, and the friction experienced by a molecule as it attempts to

“slide” between adjacent molecules becomes important This frictional resistance is

related to the liquid viscosity m (Pa s) It can be shown that, for a liquid, the group

(Bm/T) should be relatively constant and (by the Stokes-Einstein equation)

approx-imately equal to R/(6pNr), where N is Avogadro’s number, R is the gas constant,

and r is the molecular radius (typically 10–10 m) B is therefore T R/(m6pNr), where

the viscosity of water m is typically 10–3 Pa s Substituting values of R, T, µ, and r

suggests that B will be approximately 2 ¥ 10–9 m2/s or 2 ¥ 10–5 cm2/s or 7 ¥ 10–6

m2/h, which is also borne out experimentally Again, this equation forms the

foun-dation of correlation equations

The important conclusion is that, during its diffusion journey, a molecule does

not move with a constant velocity related to the molecular velocity On average, it

spends as much time moving backward as forward, thus its net progress in one

direction in a given time interval is not simply velocity/time In t seconds, the distance

traveled (y) will be m Taking typical gas and liquid diffusivities of 0.25 ¥

10–4 m2/s and 2 ¥ 10–9 m2/s respectively, a molecule will travel distances of 7 mm

in a gas and 0.06 mm in a liquid in one second To double these distances will

require four seconds, not two seconds It thus may take a considerable time for a

molecule to diffuse a “long” distance, since the time taken is proportional to the

square of the distance The most significant environmental implication is that, for a

molecule to diffuse through, for example, a 1 m depth of still water requires (in

principle) a time on the order of 3000 days A layer of still water 1 m deep can thus

effectively act as an impermeable barrier to chemical movement In practice, of

course, it is unlikely that the water would remain still for such a period of time

The reader who is interested in a fuller account of molecular diffusion is referred

to the texts by Reid et al (1987), Sherwood et al (1975), Thibodeaux (1996), and

Bird et al (1960) Diffusion processes occur in a large number of geometric

con-figurations from CO2 diffusion through the stomata of leaves to large-scale diffusion

in ocean currents There is thus a considerable literature on the mathematics of

diffusion in these situations The classic text on the subject is by Crank (1975), and

Choy and Reible (2000) have summarized some of the more environmentally useful

equations

7.3.3 Mass Transfer Coefficients

Diffusivity is a quantity with some characteristics of a velocity but,

dimension-ally, it is the product of velocity and the distance to which that velocity applies In

many environmental situations, B is not known accurately, nor is y or Dy; therefore,

the flux equation in finite difference form contains two unknowns, B and Dy Ignoring

the negative sign,

2tB

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N = ABDC/Dy mol/hCombining B and Dy in one term kM, equal to B/Dy, with dimensions of velocity

thus appears to decrease our ignorance, since we now do not know one quantity

instead of two Hence we write

N = AkMDC mol/hTerm kM is termed a mass transfer coefficient, has units of velocity (m/h), and is

widely used in environmental transport equations It can be viewed as the net

diffusion velocity The flux N in one direction is then the product of the velocity,

area, and concentration

For example, if, as in the lower section of Figure 7.2, diffusion is occurring in

an area of 1 m2 from point 1 to 2, C1 is 10 mol/m3, C2 is 8 mol/m3, and kM is 2.0

m/h, we may have diffusion from 1 to 2 at a velocity of 2.0 m/h, giving a flux of

kMAC1 of 20 mol/h There is an opposing flux from 2 to 1 of kMAC2 or 16 mol/h

The net flux is thus the difference or 4 mol/h from 1 to 2, which of course equals

kMA(C1 – C2) The group kMA is an effective volumetric flowrate and is equivalent

to the term G m3/h, introduced for advective flow in Chapter 6

7.3.4 Fugacity Format, D Values for Diffusion

The concentration approach is to calculate diffusion fluxes N as ABdC/dy or

ABDC/Dy or kMADC In fugacity format, we substitute Zf for C and define D values

as BAZ/Dy or kMAZ, and the flux is then DDf, since DC is ZDf Note that the units

of D are mol/Pa h, identical to those used for advection and reaction D values

D = BAZ/Dy or D = kMAZ

N = Df1 – Df2 = D(f1 – f2)

Worked Example 7.1

A chemical is diffusing through a layer of still water 1 mm thick, with an area

of 200 m2 and with concentrations on either side of 15 and 5 mol/m3 If the diffusivity

is 10–5 cm2/s, what is the flux and the mass transfer coefficient?

y = 10–3 m, B = 10–5 cm2/s ¥ 10–4 m2/cm2 = 10–9 m2/s

Thus,

kM is B/Dy = 10–6 m/sThe flux N is thus

kMA(C1–C2) = 10–6 (200(15 – 5)) = 0.002 mol/s

cm2/s, how long will the water take to evaporate completely?

B is 0.25 cm 2 /s or 0.09 m 2 /h

Dy is 0.002 m

DC is 15 g/m 3

N = ABDC/Dy = 675 g/h

To evaporate 10000 g will take 14.8 hours

Note that the “amount” unit in N and C need not be moles It can be another quantitysuch as grams, but it must be consistent in both In this example, the 2 mm thickfilm is controlled by the air speed over the pan Increasing the air speed could reducethis to 1 mm, thus doubling the evaporation rate This Dy is rather suspect, so it ismore honest to use a mass transfer coefficient, which, in the example above is0.09/0.002 or 45 m/h This is the actual net velocity with which water moleculesmigrate from the water surface into the air phase

7.3.5 Sources of Molecular Diffusivities

Many handbooks contain compilations of molecular diffusivities The text byReid et al (1987) contains data and correlations, as does the text on mass transfer

by Sherwood, Pigford, and Wilke (1975) The handbook by Lyman et al (1982) andthe text by Schwarzenbach et al (1994) give correlations from an environmentalperspective The correlations for gas diffusivity are based on kinetic theory, whilethose for liquids are based on the Stokes–Einstein equation In most cases, onlyapproximate values are needed In some equations, the diffusivity is expressed indimensionless form as the Schmidt number (Sc) where

Sc = m/rBwhere m is viscosity and r is density

So far, we have assumed that diffusion is entirely due to random molecularmotion and that the medium in which diffusion occurs is immobile or stagnant, with

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no currents or eddies In practice, of course, the environment is rarely stagnant, therebeing currents and eddies induced by the motion of wind, water, and biota such asfish and worms This turbulent motion, illustrated in Figure 7.3, also promotes mixing

by conveying an element or eddy of fluid from one region to another The eddiesmay vary in size from millimetres to kilometres, and a large eddy may contain afine structure of small eddies Intuitively, it is unreasonable for an eddy to penetrate

an interface, thus in regions close to interfaces, eddies tend to be damped, and onlyslippage parallel to the interface is possible There may, therefore, be a thin layer

of relatively quiescent fluid close to the interface that can be referred to as a laminar

sublayer In this layer, movement of solute to and from the interface may occur only

by molecular diffusion

Under certain conditions, eddies in fluids may be severely damped, or theirgeneration may be prevented This occurs in a layer of air or water when the fluiddensity decreases with increasing height This may be due to the upper layers beingwarmer or, in the case of sea water, less saline An eddy that is attempting to moveupward immediately finds itself entering a less dense fluid and experiences a hydro-static “sinking” force Conversely, a companion eddy moving downward experiences

a “floating” force, which also tends to restore it to its original position This inherentresistance to eddy movement damps out most fluid movement, and stable, stagnantconditions prevail Thermoclines in water and inversions in the atmosphere areexamples of this phenomenon These stagnant or near-stagnant layers may act asdiffusion barriers in which only molecular diffusion or slight eddy diffusion canoccur Conversely, situations in which density increases with height tend to beunstable, and eddy movement is enhanced and accelerated by the density field

An attractive approach is to postulate the existence of an eddy diffusivity, or a

turbulent diffusivity, BT, which is defined identically to the molecular diffusivity,

BM The flux equation within a phase then becomes

N = –A(BM + BT)dC/dy

The task is then to devise methods of estimating BT for various environmentalconditions We expect that, in many situations, such as in winds or fast rivers, BT

Figure 7.3 The nature of turbulent or eddy diffusion in which chemical is conveyed in eddies

within a fluid to a surface.

A complicating factor is that we have no guarantee that BT is isotropic, i.e., thatthe same value applies vertically and horizontally In Figure 7.3, we postulate thatsome eddies may be constrained to form elongated “roll cells.” The horizontal BTwill therefore exceed the vertical value In practice, this nonisotropic situation iscommon and even leads to conditions in rivers where three BT values must beconsidered: vertical, upstream-downstream, and cross-stream.

To give an order of magnitude appreciation of turbulent diffusivities, it isobserved that a vertical eddy diffusivity in air is typically 3600 m2/h plus or minus

a factor of 3, thus the time for moving a distance of 1 m is of the order of 1 s.Molecular diffusion is clearly negligible in comparison In lakes, a vertical eddydiffusivity may be 36 m2/h near the surface, corresponding to a velocity over adistance of 1 m of 1 cm/s At greater depths, diffusion is much slower, possibly by

a factor of 100 To estimate eddy diffusivities, one can watch a buoyant particle andtime its transport over a given distance The diffusivity is then that distance squared,divided by the time

Turbulent processes in the environment are thus quite complex and difficult todescribe mathematically The interested reader can consult Thibodeaux (1996) orCsanady (1973) for a review of the mathematical approaches adopted We sidestepthis complex issue here, but certain generalizations that emerge from the study ofturbulent diffusion are worth noting

In the bulk of most fluid masses (air and water) that are in motion, turbulentdiffusion dominates We can measure and correlate these diffusivities Generally,vertical diffusion is slower than horizontal diffusion Often, diffusion is so fast thatnear-homogeneous conditions exist, which is fortunate, because it eliminates theneed to calculate diffusion rates

In the atmosphere and oceans, there is a spectrum of eddies of varying size andvelocity The larger eddies move faster Consequently, when a plume in the atmo-sphere or a dye patch in an ocean expands in size, it becomes subject to dispersion

by larger, faster eddies, and the diffusivity increases If the velocity of expansion ofthe plume or patch is constant, this implies that diffusivity increases as the square

of distance

At phase interfaces (e.g., air-water, water-bottom sediment), turbulent diffusion

is severely damped or is eliminated, thus only molecular diffusion remains One caneven postulate the presence of a “stagnant layer” in which only molecular diffusionoccurs and calculate its diffusion resistance This model is usually inherently wrong

in that no such layer exists It is more honest (and less trouble) to avoid the use ofdiffusivities and stagnant layer thicknesses close to the phase interfaces and invokemass transfer coefficients that combine the varying eddy diffusivities, the molecular

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diffusivity, and some unknown layer thickness, into one parameter, kM We thenmeasure and correlate kM as a function of fluid conditions (e.g., wind speed) andseek advice from the turbulent transport theorists as to the best form of the correlationequations

In some diffusion situations, such as bottom sediments, the eddy diffusion may

be induced by burrowing worms or creatures that “pump” water This is termed

bioturbation and is difficult to quantify Its high variability and unpredictability is

a source of delight to biologists and irritation to physical scientists

The study of turbulent diffusion in the atmosphere includes aspects such as themicrometeorology of diffusion near the ground as it influences evaporation of pes-ticides, the uptake of contaminants by foliage, and the dispersion of plumes fromstacks, in which case the plume is treated by the Gaussian dispersion equations Inlakes, rivers, and oceans it is important to calculate concentrations near sewage andindustrial outfalls and in intensively used regions such as harbors In each case, abody of specialized knowledge and calculation methods has evolved

There are now three variables: concentration (C), position (y), and time (t) If

we consider a volume of ADy, as shown in Figure 7.4, then the flux in is –BA dC/dy,and the flux out is –BA(dC/dy + Dyd2C/dy2), while the accumulation is ADyDC inthe time increment Dt It follows that

–BA dC/dy + BA(dC/dy + Dyd2C/dy2) = ADyDC/Dt

or as Dy and Dt tend to zero,

Bd2C/dy2 = dC/dt

This is Fick’s second law Solution of this partial differential equation requires twoboundary conditions, usually initial concentrations at specified positions A partic-ularly useful solution is the “penetration” equation, which describes diffusion into

a slab of fluid that is brought into contact with another slab of constant concentration

CS The boundary conditions are

It can be shown that

C = CS (1 – (2/ ) 0ÚX exp(–X2) dX) = Cs [1–erf(X)]

Figure 7.4 Unsteady-state or penetration diffusion.

4Bt

p

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where

X = y/

Unfortunately, this integral, which is known as the Gauss Error integral or

probability function or error function, cannot be solved analytically, thus tabulated

values must be used The error function has the property that it is zero when X iszero, and it approaches unity when X is 3 or larger Its value can be found in tables

of mathematical functions, or it can be evaluated using built-in approximations inspreadsheet software A convenient approximation is

erf(X) = 1 – exp(–0.746X – 1.101 X2)

which is quite accurate when X exceeds 0.75 When X is less than 0.5, erf(X) isapproximately 1.1X The penetration solution shown in Figure 7.4 illustrates thevery rapid initial transfer close to the interface, followed by slower penetration thatoccurs later as the concentration gradient becomes smaller Now the transfer rate atthe boundary (y = 0) can be shown to be

The mass transfer coefficient, kM, under these transient conditions, thus depends

on the time of exposure (short exposures giving a large kM) and on the square root

of diffusivity This contrasts with the steady-state solution, in which kM is dent of time and proportional to diffusivity The reason for this behavior is that kM

indepen-is apparently very large initially, because the concentration gradient indepen-is large It falls

in inverse proportion to , thus the average also falls in this proportion The lowerdependence on diffusivity (to the power of 0.5 instead of 1.0) arises, because notall the transferring mass has to diffuse the total distance; much of it goes into

“storage” during the transient concentration buildup

A problem now arises in environmental calculations: which definition of kMapplies, B/Dy or ? Contact time is the key determinant If the contact timebetween phases is long, and the amount transferred exceeds the capacity of thephases, it is likely that a steady-state condition applies, and we should use B/Dy

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Conversely, if the contact time is short, we can expect to use If we measurethe transfer rates at several temperatures, and thus different diffusivities, or measurethe transfer rate of different chemicals of different B, then plot kM versus B on log-log paper, the slope of the line will be 1.0 if steady-state applies, and 0.5 if unsteady-state applies In practice, an intermediate power of about 2/3 often applies, suggestingthat we have mostly penetration diffusion followed by a period of near-steady-statediffusion

7.6 DIFFUSION IN POROUS MEDIA

When a solute is diffusing in air or water, its movement is restricted only bycollisions with other molecules If solid particles or phases are also present, the solidsurfaces will also block diffusion and slow the net velocity Environmentally, this

is important in sediments in which a solute may have been deposited at some time

in the past, and from which it is now diffusing back to the overlying water It is alsoimportant in soils from which pesticides may be volatilizing It is therefore essential

to address the question, “By how much does the presence of the solid phase retarddiffusion?” We assume that the solid particles are in contact, but there remains atortuous path for diffusion (otherwise, there is no access route, and the diffusivitywould be zero)

The process of diffusion is shown schematically in Figure 7.5, in which it isapparent that the solute experiences two difficulties First, it must take a moretortuous path, which can be defined by a tortuosity factor, FY, the ratio of tortuousdistance to direct distance Second, it does not have available the full area fordiffusion, i.e., it is forced to move through a smaller area, which can be definedusing an area factor, FA This area factor FA, is equal to the void fraction, i.e., the

4B/p t

Figure 7.5 Diffusion in a porous medium in which only part of the area is accessible, and the

diffusing molecule must take a longer, tortuous path.

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fraction of the total volume that is fluid, and thus is accessible to diffusion It can

be argued that the tortuousity factor, FY, is related to void fraction, v, raised possibly

to the power –0.5; therefore, in total, we can postulate that the effective diffusivity

in the porous medium, BE, is related to the molecular diffusivity, B, by

BE = BFA/FY = Bv1.5

Such a relationship is found for packings of various types of solids, as discussed

by Satterfield (1970) This equation may be seriously in error since (1) the effectivediffusivity is sensitive to the shape and size distribution of the particles, (2) theremay also be “surface diffusion” along the solid surfaces, and (3) the solute maybecome trapped in “cul de sacs” or become sorbed on active sites At least theequation has the correct property that it reduces to intuitively correct limits that BEequals B when v is unity, and BE is zero when v is zero There is no substitute foractual experimental measurements using the soil or sediment and solute in question.For soils, it is usual to employ the Millington–Quirk (MQ) expression for dif-fusivity as a function of air and water contents An example is in the soil diffusionmodel of Jury et al (1983)

The MQ expression uses air and water volume fractions vA and vW and calculateseffective air and water diffusivities as follows:

1000 The problem is that diffusivity is then apparently controlled by the extent ofsorption

In sediments, it is suspected that much of the chemical present in the pore orinterstitial water, and therefore available for diffusion, is associated with colloidalorganic material These colloids can also diffuse; consequently, the diffusing chem-ical has the option of diffusing in solution or piggy backing on the colloid Fromthe Stokes–Einstein equation, the diffusivity B is approximately inversely propor-tional to the molecular radius A typical chemical may have a molecular mass of

200 and a colloid an equivalent molar mass of 6000 g/mol, i.e., it is a factor of 30

7.7 DIFFUSION BETWEEN PHASES: THE TWO-RESISTANCE CONCEPT 7.7.1 Derivation Using Concentrations

So far in this discussion, we have treated diffusion in only one phase, but inreality, we are most interested in situations where the chemical is migrating fromone phase to another It thus encounters two diffusion regimes, one on each side ofthe interface Environmentally, this is discussed most frequently for air-waterexchange, but the same principles apply to diffusion from sediment to water, soil towater and to air, and even to biota-water exchange

An immediate problem arises at the interface, where the chemical must undergo

a concentration “jump” from one equilibrium value to another The chemical mayeven migrate across the interface from low to high concentration Clearly, whereas

concentration difference was a satisfactory “driving force” for diffusion within one phase, it is not satisfactory for describing diffusion between two phases When

diffusion is complete, the chemical’s fugacities on both sides of the interface will

be equal Thus, we can use fugacity as a “driving force” or as a measure of “departurefrom equilibrium.” Indeed, fugacity is the fundamental driving force in both cases,but it was not necessary to introduce it for one-phase systems, because only one Zapplies, and the fugacity difference is proportional to the concentration difference.Traditionally, interphase transfer processes have been characterized using theWhitman Two-Resistance mass transfer coefficient (MTC) approach (Whitman,1923), in which departure from equilibrium is characterized using a partition coef-ficient, or in the case of air-water exchange, a Henry’s law constant We derive theflux equations for air-water exchange using the Whitman approach and followingLiss and Slater (1974), who first applied it to transfer of gases between the atmo-sphere and the ocean, and Mackay and Leinonen (1975), who applied the sameprinciples to other organic solutes We will later derive the same equations in fugacityformat Unfortunately, the algebra is lengthy, but the conclusions are very important,

so the pain is justified

Figure 7.6 illustrates an air-water system in which a solute (chemical) is diffusing

at steady-state from solution in water at concentration CW (mol/m3) to the air atconcentration CA mol/m3, or at a partial pressure P (Pa), equivalent to CART Weassume that the solute is transferred relatively rapidly in the bulk of the water byeddies, thus the concentration gradient is slight As it approaches the interface,however, the eddies are damped, diffusion slows, and a larger concentration gradient

is required to sustain a steady diffusive flux A mass transfer coefficient, kW, appliesover this region The solute reaches the interface at a concentration CWI, then abruptlychanges to CAI, the air phase value The question arises as to whether there is a

Figure 7.6 Mass transfer at the interface between two phases as described by the two-resistance concept Note the concentration

discontinuity on the right, whereas, in the equivalent fugacity profile on the left, there is no discontinuity.

©2001 CRC Press LLC

©2001 CRC Press LLC

significant resistance to transfer at the interface It appears that if it does exist, it issmall and unmeasurable In any event, we do not know how to estimate it, so it isconvenient to ignore it and assume that equilibrium applies We thus argue that there

is no interfacial resistance, and CWI and CAI are in equilibrium

N = kWA(CW – CWI) = kAA(CAI – CA) mol/h

The term kOW is an “overall” mass transfer coefficient that contains the individual

kW and kA terms and KAW It should not be confused with KOW, the octanol-water