.. Tr ,. ., 0 b. , .. . . ansP,()ftnc_. eDOinena;

4.A Basic Concepts and Mechanisms

4.A.l Contaminant Flux 4.A;2 Advection

'4.A.3 Molecular DiffusiOn

4.A.4 Dispersion

4.B Particle Motion

4.B.l Drag on Particles

4.B.2 Gravitational Settling

4.B.3 Brownian Diffusion

4.C Mass Transfer at Fluid Boundari~s 4.C.l Mass-Transfer Coefficient

4.C.2 TransPort across the Air-Water Interface 4.D Transport in PorqusMedia.

4.D.l Fluid Flow through Porous Media

4.D.2 Contaminant TransPort in Porous Media

. References

Problems

Transport phenomena are encountered in almost every aspect of environmental engi-neering science. In assessing the environmental impacts of waste discharges, we seek to predict the impact of emissions on contaminant concentrations in nearby air and water. Contaminant transport must be understood to evaluate the effect of wastewater discharge to a river on downstream w,ater qualir-y or the cff""'-t cf an incinerator on downwfild air pollutant levels. In waste treatment technologies~ co11taminant transport within the control device influences the overall efficiency. In many. instruments used to measure environmental contamination, performance depends on the effective trans-port of contaniitiants from the sampling point to the detector.

The physical scales of concern for contaminant transpOrt spaii ari enormous range. At the low end, we are interested in phenomena that occur over molecular di-mensions, such as the transport of a contaminant into the pore of an adsorbent. At the upper extreme, we consider the transport of air and waterborne contaminants o~er

159

l

t :; I 1: t

I ! I ! I ! .

I !

l I ! l

i.

1~_. ... - __ .' ,,,(_ }

.; ~_t60~r,:~~:~ ; .Transport Phenomena ) ',. \ ''tiobal distances. This range of linear dimensions encompasses approximately 15 -;;;..~;.;,.ders:of magnitude. Not surprising~, some phenomena that~-are -importantat one of the spectrum are irrelevant at-the other. "

:: , ; Transport phenomena also are important in other fields of study; -notably chemical and mechanical engineering. In these fields, the subject is often subdivided ~to fluid mechanics, heat transfer, and mass transfer. In environmental.engineering, where the focus is on the movement of contaminants in air 'and water, elements ofall three of these branches are conSidered. The essential ingredients fot understanding contami-nant transport processes are knowledge of the basic physical phenomena coupled with the analytical tools of engineering mathematics.

Because of the importance of both molecular and particulate impurities in envi ronmental fluids, the dominant transport processes of both molecules and particles

will be discussed in this chapter. We consider both the movement of impurities with fluids and their movement relative to fluids. In Chapter 5, we will explore how the transformation mechanisms studied in Chapter 3 and the transport mechanisms ex plored in this chapter are combined to predict system behavior using mathematical models.

4.A BASIC CONCEPTS AND MECHANISMS 4.A.l Contaminant Flux

Transport of both molecules and particles is commonly quantified in terms of flu.x density, or simply flux. Flux is a vector quantity, comprising both a magnitude and a direction. The flux vector points in the direction of net contaminant motion, and the magnitude indicates the rate at which the contaminant is moving.

For example, think of an environmental fluid as depicted in Figure 4.A.l. Imagine a small square frame suspended in this fluid and centered at a point of interest. The frame is oriented so that the transport of contaminants through -it is maximized. Then the flu~ vector points in a direction normal to the frame, aligned with the direction of contaminant transport. The magnitude of the flux vector is the net rate of contaminant transport per unit area of the frame. Since the flux can vary from one position to an other, the flux at a point represents the net transport through the frame per area in the limit of the frame becoming infinitesimally small. The common units of flux are mass or moles per area per time. We will use the symbol J to represent flux. An arrow is written above J when we wish to emphasize its vector character. Contributions to flux

--=-a

- .. .

urities in envi-s and particles impurities with \plore how the techanisms ex-

~ mathematical

1 terms of flux agnitude and a notion~ and the

lXimized. Then the direction of of conttarnltnaJBt position to an'"' per area in t~

,f flux are mass tx. An arrow is ibutions to flux

;.,~.,:~~P : . ~'{\~~ve~~t~~!!~~;: .. TranSport induced ~)'-~ty _~;: t;5 Ig = Cv,

~vdlrocivnamic dispersion media)

Elec:trostatl'c driftd

m, p Transport ~used by random .thermatmotion. -- _

m,p ..

m,p

m,p

p

Transport caused by apparently random fluid velocity-fluctuations in turbulent flow Transport caused by nonuniform flq.id flow with position Transport caused by nonuniform flow through porous material Movement of charged species in an electric field Transport associated with acceleration of a fluid

J = _ 0 dC.

4- dx

dC J =-c-t r dx

1 =-cdC s sdx

dC Jh = .-ehdx

: Je = Cve

""' species concentration, U = fluid velocity, v, = settling velocity, D =molecular or Brownian diffusivity, e, = /

~ \ -

t+ IlL u

Figure 4.A.2 Advective flux of contaminant through a tube. The two pictures represent the same tube at two points in time~ t and t + At, where At = ALl U.

per time:

J =CXA.LXA=CU a A(A.LIU)

In general, the three-dimensional advective flux :vector is the product of the con: taminant concentration and the fluid velocity:

~ -+ la (x, y, z) = C{x, y, z) X U (x, y, z)

4.A~3 Molecular Diffusion The molecules in air and wate.r are constantly moving. If it could be viewed at a lecular sca1e, this movement would appear random and chaotic. Molecular movemef!! in gases is particularly frenzied. In air at ordinary environmental conditions, a typic~l 'molecule, moving at a speed of -400 m s-1, collides with other molecules on the order of 10 billion times per second. Each collision involves an exchange of momentum tween the participants, causing them to change direction and speed. Every cubic centl meter of air has a phenomenally large number of molecules ( -25 million trillion at T = 298 K and P = 1 atm) participating in this dance. In water, where the molecules are packed a thousand times more densely, molecular motion is less fi:enetic,b,ut still extremely energetic.

Although the molecular-scale motion seems hopelessly disordered, the macro~ scopic effects are well underst'Ood and predictable. Qumitativdy, the random motion of fluid molecules causes a net movement of species from regions of high concentra tio.n to regions of low concentration .. This phenomenon is known as molecular dijfu-

. sion:, The rate of movement depends' on the 'concentration difference, with differences leading to higher rates of transport .. The rate also varies according to

. far the species must travel from high to low concentration: The longer the distance, the lower the flux. The rate of transport varies according to the molecular properties the species, particularly size and mass, with larger size and larger mass resulting in.,

~lower transport. The properties of the fluid itself play a key role: Molecular diffusion

"';

lfis;much slo nidter.molecul

nOurdi~ catalowmc

... , .. ::_vifonnienta .:1~lrations~ dif

Cussler, 19}

Fick~s Law

To better UI such as the a moderate The bulb is cules evapo bulb equal t ylbenzene n After a sho through the liquid ethyll in inverse p tube, and in teristics can

where Jd is time), ll.C h and llx is tht the diffusive

By intn to an equati(

The constan diffusing sp

x=L----

T

r=O~l

:'is muchslOWei' ia watet than in air~because:ofthe~osef!pacldiigiQ~t;Y~'Oft\va~. termolecwes .. .~-I:,";;;l~r{: .:

,._OUr discUSsio& is restricted to-_conmtto~:taAVmentlllCmnusmgs~~~ts ~nt __ ,,_ at a low mole fraction (

,:".'"'

1'.'"'..

sucltf:.S?:temperature ~and pressure:?liile~dilfusi

""re:)The ;Wttp~ii .... ,:reported in.

-1 rl 1 ~. ' s t. t n wa"er, ~ ,~ j;

tor is proportional n, and the propor-n the direction op-to its magnitude. ,

1, or simply Fick's 1 Figure 4.A.3.

t the characteristic is given by

1 to move a charac- 1 hour, and 1m ina 4

, will travel only''r.

)f Pick's law (equ. .f.A.1). If a speci 1en diffusion l the small values ;ive velocity is amount over a

ry important role mt at interfaces, tid interfaces. The l the surface. So ;rface~ Instead, 1all distance, 1purities, di or ions and part'""'""!ii: as electrostatic e important because transformation

4.A &sicCoii:ePts.wMechamsJA.~l~ . . -.:.::i~ ~~:

ceSSes that OCCUf 00 sUtfaces~IOften;'fue;rateOf.trafisport?ti'the,Wrface sfoVehlStl -~-overall removal :::....: ... ..:

r )~ ti. ;I

~eoc~=s~~~fii~-t~,~5 NH

f':]' . , ' ',},i\!iili~~i.ii

Cussler, 1984 )SUre = 1 bar)

beet al., 1993

.beet al., 1993

n air or water do not very _long. ~Coold.n_g hout an indoor a fire can be seen inant spreading is fundamentally it ~is. much more

.. . . . 8SIC . -.~ an . ............. ':, . ~-{~'f~~~~ F . 4.A B . ~ d:MeclJ~~:,~:~r;jf;"- -~

Species . . . Fonnula. T (K)"

. :168 tilhaftta.O'A "* 1'r . na.;:;._..; ~.- ,t_ ;,_ _.-:: ~::~fta:>iu.~;:rw;uu~-r;4~

) \ , .~,.

Side view

Top view

\if!_ . .

;,.. (

.. L

Peak concentration high,

but impact area low Peak concentration low,

but impact area high

Figure 4.A.S Effect of dispersion on pollutant concentrations downwind of a localized source. If the relea$e rate of pollutants and mean wind speed are the same in the left- and right-hand frames, then the total flow of pollutants is the same in the two cases. Rapid dispersion leads to smaller peak concentrations but a larger impact area.

dispersion has in all environmental settings: High dispersion rates reduce average concentrations at the expense of increasing the area or duration of impact. A common challenge in environmental transport modeling is~ predict concentrations in the 11 . gion downwind or downstream of a pollution source. To make these predictions in face of dispersion phenomena, we must solve two problems. First, we must have model that describe's pollutant concentration fields in the presence of dispersion. ond, we must be able to determine how the rate of dispersion varies with environmen-tal conditions.

In complex flow fields, it is impossible (or at least impractical) to describe veiocity exactly as a function of space and time. Without this information, dispersio cannot be accurately described in terms of the fundamental mechanisms of advecti"

and molecular diffusion. As an alternative, dispersion can be treated as a random cess, analogous to molecular diffusion, by applying Pick's first law of diffusion the molecular diffusion coefficient replaced by a dispersion coefficient .. So in one , mens ion, we would write

dC J dispersion = - e dx

where e is a dispersion coefficient, obtained through a combination of empirical and theoretical equations.

Alt..h.ough equation 4.A.l0 has the same form as Fick's first law, it is important bear in mind the great distinction between diffusion and dispersion. Diffusivities properties of the contaminant and the fluid~ depending weakly on environmental

.:ditions (such as temperature) but not at all on fluid flows. Dispersion c~fficients, the other hand, are primarily a function of the fluid flow field. Environmental

-are_ highly variable and very complex. Dispersion in environmental flows is far well understood than molecular diffusion. One should not expect a high degree of curacy froin any models that must account for dispersion, especially when applied environmental transport.

waste man< Seven ~neering. In turbulent a

Shear-Flo

In a shear 1 to the fluic tion of flo' from regio1

Shear-, contaminar large, such Let's consi laminar flo fluid flow i taminant is so that its c as it is adv shows the L L/U0 At th L, but it hru centerline o same time, ~ear the cer. ward veloci pulse along velocity. Tb rate of dispt

Position: x = 0 I I

--E I

.I I

"'Ir ----r-----

I

Figure 4.A.6 effects of she'

tes reduce impact. A cotn.mcM ;ntrations in the ;e predictions in st, we must have of dispersion. s with environmen-'

rmation, dispers misms of adveciliii !d as a random w of diffusion

.w, it is important Jn. Diffusivities environmental )ion co~fficients;:o ':nvironmental ttal flows is far a high degree tlly when applied .

4.A ~Basic~ad1M~iBrii: 169 ~ : . ' ::~ .: ', ,:.. . ~"

'-t"'

:DiSpersiOn ,phenomena aiise in i all; .tm.ee i>ianches::of:enviionmentil engineering t~f.; diseussed in this ;text ~Figure :4~A.5.' illusfrates;:one :.of ,man.yinstances~that:aiise in air

qualicy: engineering.; In .water qitality, the impact:of WllsteWater 'discliargesoncontami-:,~ .'.nantcoilcentration8 in:rivers;flakes, .or'oceaJisdepetld-on dispersion. IIi--hazardous

waste management, dispersion controls the movement of contamiriants in groundwater. Several types of dispersion phenomena are encountered in environmental engi-

neering. In this section, two common.~ are introduced: shear-flow dispersion and turbulent diffusion.

Shear-Flow Dispersion In a shear flow~ the fluid speed varies with position in a dirCctjon that is perpendicular to the fluid velocity. If contaminant concentrations in a shear flow vary in the direc-tion of flow, then dispersion will occur, leading to a net transport of contaminants from regions of high concentration to areas of low concentration.

Shear-flow dispersion is generally important when there is a short-term release of contaminants, such as a spill, in environments in which the velocity gradients are large, such as a river, an estuary influenced by tides, or the near-ground atmosphere. Let's consider a relatively simple example: shear-flow dispersion in fully developed laminar flow through a circular tube or pipe (Figure 4.A.6). The velocity profile of the fluid flow is parabolic because of wall friction. At .timet= 0, some mass, M, of con-taminant is injected into the tube at position x = 0 and instantaneously mixed laterally so that its concentration is uniform across the tube. We then observe the contaminant as it is advected past some position, L, downstream. The upper frame in the figure sh9ws the initial condition. The lower frame shows the tube at a subsequent time, t = L/U0 At the later time, the contaminant pulse is centered on downstream position x = L, but it has been substantially stretched because those molecules that lie close ~o the centerline of the tube are advected at a higher velocity than those near the wall. At tlle same time, contaminant molecules positioned near the leading edge of the pulse and ,pear the center of the tube tend to diffuse toward the walls. reducing their average for-ward velocity. Conversely, contaminant molecules located near the trailing edge of the pulse along the tube walls tend to diffuse toward the center, increasing their forward velocity. Therefore, surprisingly, the net effect of molecular diffusion is to slow the rate of dispersion, in this case.

Position: x = 0

----

x=L

L___ ,- Time I -; I ~----~~) --------r--------~ I t=O I

L t= Uo

Figure 4.A.6 Schematic of a model problem illustrating the _effects of shear-flow dispersion.

'

' I .

!~ h it; ~t HI F: ~'~i ;., ~i'

_,A

" ;l t H i ~ Hi t. ; ;

l'l,O; ~~4;tll'ranspqrttPhCnomena

) . . . . For the sifuatiOirdepicted in FigUre 4.A.6; the:contaminaptflux~~tne direcfi91 ' :1!> :: :s;.flow.:can be described as the sumoftwoool.llponen~:I(l)advection!~wbi~~~

rriH:~~ ne.the overallipulsetthrougll the tube. and {2)t&qear-flowtiii~persioil;!wbjc:Jr;~us~:;2 2::. 1 '~'1.

iq tliedirectjo~ ~~~whicbitratisport

wbJciF~uses' lf flowrcaused

.... ?;'t' ".l.t;.

. ~.::~2

.! transport of con-pipe or tube flow, rs and in estuarine

J.ows are turbulent ! the fluid velocity such a manner that ws in terms of their f the fluctuations. is characteristic of

ow concentrations. k's law:

s the time-averaged I'

; expresston can

or eddy diffusiv ansport in the atmo hat x and y lie in the lly differs from hori~ ::ty . lersion, ignite some tte or some incense.iiih~~~~fy~. permits good

lamp on the opposite, directthe_light at iume controls its distance on the _ . .dies. The flow in the >ortion is turbulent. If J be determined on a

~~

region

.A .. :~~~and.iMecbanii~;:~:.Jjl

Figure 4.A.7 Schematic representation of time-averaged smoke plume rising from a smoldering cigarette in still air.

time-averaged basis, over a minute or so, you should arrive at a picture like the one shown in Figure 4.A.7. The broadening of the plume in the turbulent region is a direct consequence of turbulent air motion. The mean fluid flow is still upward, but there are fluctuating horizontal components of the flow that cause rapid pollutant dispersion in the h~rizontal direction. For environmental engineering applications, this is .a key property of turbulence: In unbounded flow, turbulent diffusion controls the transport of pollutants in directions nom1al to the mean flow.

Figure 4.A.8 depicts a plume spreading from a point source discharging either to water or to air. A horizontal slice through the center of the plume is shown from above, and the mean fluid velocity is oriented from left to right. The coordinates are oriented so that x is aligned with the m.ean velocity andy represents the horizontal dis-tance from the plume centerline. The origin is positioned at the source.

Assume that the rate of pollutant discharge is constant and the mean fluid velocity is independent of position and time. Then Figure 4A.8 represents the average conditions over some period, such as an hour. On this basis, the plume spreads fairly smoothly and symmetrically from the source. The coordinate axes at right show the time-averaged concentration profile plotted against y at some distance away from the source. The peak concentration occurs at y ~ 0 and gradually approaches zero. as the distance from the centerline increases. The spreading of the plume in the y-direction is controlled by tur-bulent diffusion.

A practical challenge in the use of equation 4.A.13 is to determine the turbulent diffusivities. These parameters vary not only with flow conditions, but also with po-sition and direction. For. example, turbulent diffusivities diminish to zero at rigid fluid boundaries, because the fluid velocity itself goes to zero. Correlations exist for

y

-u

c Figure 4.A.8 Plume spreading from a point source. The pollutant discharge rate is constant and the mean fluid velocity is steady and uniform, but the flow is turbulent.

J

!\ I! ;:

.~~~' ';01~ic:::==itieS. but lhese should be used cautiously, as the j;t }. iing experimental data are limited. i\1 Thrbulent diffusivities are higher in rivers thanin regular channels because. !

1 regularities in the river channel create additional irregul~ties in the flow, which

-..

'to enhance mixing. Thrbulent diffusivities in rivers are ari order of magnitude than longitudinal (shear-flow) dispersion coefficients. Consequently, studies of port and mixing in rivers typically account for trarisport across the flow by diffusion and along the flow by advection and shear-flow dispersion. Vertical transOOt is assumed to occur very rapidly, since most rivers are much wider than deep. , : In the atmosphere, turbulent diffusion is especially important in the vertical tion. Most pollutants are emitted near the ground, and the rate and extent of turbulent diffusion ::il"ongly influences ground-level concentrations. Atmospheric bility, as will be discussed in 7 .D, has a very strong influence on turbulent diffusi especially in the vertical direction. In unstable conditions, which are characterized strong heating of the ground by the sun, vertical velocity fluctuations are strong and bulent diffusivities are high. .

Typical turbulent diffusivity values for both rivers and the atmosphere exceed many orders of magnitude molecular diffusivities of contaminants. This observatiol reinforces the point that pollutant dispersion away from fluid boundaries is controlled by a combination of shear-flow dispersion and turbulent diffusion, and not by moleal .. lar or Brownian motion. ..

4.B PARTICLE MOTION We now tum our attention to the movement of particles within fluids by mechanisms other than advection. The transport of particles within fluids is very strongly influ~ enced by their size and mass. Very small particles behave much like molecules, under going a diffusive-type process known as Brownian motion. For larger particles;; , transport mechanisms that depend on mass, such as gravitational settling and drift, dominate over diffusive _motion.

4.B.l Drag on Particles Mechanisms that cause particle movement relative to a fluid are always opposed drag force. The magnitude of the drag force increases as the particle velocity to the fluid increases. The drag force on a spherical particle. is usually computed the expression

'(1T 2 )(1 2 ) Fd = 4dP 2pfVoo Cd where dP represents particle diameter, p1 is the density of the fluid, V oo is the speed ?f the particle relative to t?e fl~id, and f!d is. the drag coefficient. As shown in ~ig}_;l d~~~~ ure 4.B .1, the drag coeffictent 1s a funcuon of the Reynolds number of the oarttcle.'"~f' which is defined by

Re = de V oo = dp V oo Pt . P v IL

where v and IL are the kinematic and dynamic viscosities of the fluid, respectively.

Figure 4.B.l number for s fluid flow. TI (1940). The c

Given a data in the f

'24 Cd= Re

p

24 1 c =-( d ReP

cd = o.445 1\vo intt

the particle force depen' 4.B.2 into 4.:

As a particle with the parti bution domir

At high 1 depends on tl

Work must h through a flu molecules ant particle Reyn viscous contr:

' th' .ly, as e SUPDOr

els because. the'-! flow; ,which nagnitude smauer J, studies of flow by turuuu:nt

. Vertical transoort .han deep . . the vertical diree~, l extent of vertical . Atmospheric mlent diffusivitie$,. e characterized by are strong and tur~

~-

jds by mechanisms very strongly e molecules, or larger p settling and lUfO'l uw

always opposed tcle velocity lsually computed

uid, V oo is the 'tt. As shown in mber of the oarticJC

fluid, respectively.

Figure 4.B.l Drag coefficient as a function of particle Reynolds num~ for smooth, spherical, nonaccelerating particles in a unifo!Jll fluid flow. The experimental data are from Lapple and Shepherd (1940). The correlation equations are described in the text.

Given a particle Reynoldsnumber, we can determine the drag coefficient from the data in the figure, or from the following equations (Perry and Green, 1984):

. . I 24 Cd = ;;- ReP < 0.3 Stokes's law . (4.B.3)

1.'\.eP

Cd = 24 (l + 0.14Re~7 ) 0.3 < ReP< 1000 ReP ,

Intermediate regime ( 4.B .4)

cd = OA45 1000 < ReP < 350,000 Newton's law (4.B.5) 'IWo intrinsic properties of fluids contribute to drag: viscosity and density. When

the particle Reynolds number is low (laminar flow, Stokes's law regime), the drag force depends only on viscosity, as can be seen by substituting equations 4.B.3 and 4.B.2 into4.B.l:

F d = 37T J.Ldp V 00 ReP< 0.3 (4.B.6) As a particle moves through a stationary fluid~ some fluid~ molecules are drawn along with the particle because of viscous interactions, causing drag. At low ReP~ this contri-bution dominates. .

At high particle Reynolds numbers, the drag force, as described by Newton's law, depends on the density of the flui~ but not on its viscosity.

2 2 Fd =0.113pfdpVoo 1000 < ReP < 350,000 (4.B.7) "'

Work must be done to accelerate mo~ecules out of the way of a particle as it passes through a fluid. This work an4 the drag that results depend on the mass of the fluid moleculesand therefore on the fluid density, but not on the fluid's viscosity. When the particle Reynolds number is large, this contribution to drag is much greater than the viscous contribution.

I; ~~ lt !,i m

'i

'

'

il 1 ~

-~; _. ~:'~~- .~ . ~U&Uo)~~UU. ~U\'a!Uwena . . ...

, -,~~~f~~~~;-:;.fuatleinconsideiingf,~oo:~ ) \ . Cle~ _(dP < -:lJ.t.m)' iniair~: Stokes's:'law ovetpredicts drag for these p~cles ~ui

does not acco!Jntforthe fact~~ aU.ris made up of molecules,,rather th~ .. ~ntinuously:smooth;tiuid. As ~~:~~~;~~;tjre drag on particl~~atmall ~eP is trolled by fluid viscosity.However;.-forprogressively smaller pat:ti~les, the molecul nature of the fluid becomes i.mportmlt. Consider a particle suspended 1n air at orA;ft .. .;;

t

environmental temperature and pressure. The particle is surrounded; !llostly by space. Very small gas molecules (

, liig.orf --- particleS betause rather than being lt small Rep is cles, the molecular !d in air at ordinary d mostly by empty lncentration (2.5 tlide with the parti-second). When the Neen collisions (the.. along with it as it

collisions with the 5" '."'

lowing expression:

path and represents ~ with another mole~ temperature and de-

the mean free path m

,th particle size in ait .e correction factor is

appr6x~y 1 and.s0''~' .. ~.11~1;~.~~.f~~.}~~-~~::. ~rrection factor can be'Cbmetquititwge~an'd~usibe":c~id~~;~Jnake even rea-sonably accurate predictio~,~~ ~,p8rti~I~,be~avior .. :Bec~:~er mo~~ules are packed so much more closelytog~ than. ga8 molecules, the sUp: correction fac-tor does not arise in the assessinenfof particle behavior in water. ~ f

Sometimes, in the air pollution:fiterature'where.Stokes's Ia~1i0lds. more coften than not, the expression for dragforce'~1s written ina differen(ro1111~thailequation 4.B.l: "":~;':~, ~

Fd=fV- ReP< 0.3 (4.B.10) The parameter f, known as the friction coefficient, is equal to 3-rrp,/tPc:-;1 (compare with equation 4.B.8). .

Gravitational Settling Particles experience a gravitational force downward that is proportional to their mass. They also experience a buoyancy force upward that is proportional to the mass of fluid that they displace. A particle whose density is greater than that of the surrounding fluid will drift downward relative to the fluid, and a particle whose density is less than the fluid's will drift upward.

Consider a particle suspended in a stationary fluid that has a density greater than the fluid's. If the particle is initially stationary, .it will accelerate downward. The downward velocity will be opposed by fluid drag, which increases with increasing ve-locity. The particle will continue: to accelerate until the drag force plus the buoyancy force balances the gravitational force. The particle has then attained its terminal set-tling velocity. (If the particle were less dense than the fluid, the same argument would hold in reverse, with the particle accelerating upward until it reaches its terminal rise velocity.) Because the terminal settling velocity is achieved very rapidly for contami-nant-sized particles, the details of acceleration and deceleration are of little interest in environmental engineering applications. The effect of gravity on particle motion is usually described only in terms of the terminal velocity.

Gravitational settling is used to remove unwanted particles from water and, to a lesser extent, air. Settling also affects the fate of particles discharged into water or the atmosphere. The rise of bubbles and the fall of liquid water drops through air also are important in certain air and water treatment technologies. The same general approach is used to detennine rise and settling velocities for bubbles and water drops as for par-ticulate contaminants.

The force balance in the case of a particle whose density is greater than the fluid density is (see Figure 4.B.3)

F8 =_Fb + F4 The gravitational force acting on a spherical particle can be written as

'1T 3 .Fg = mg = p6dpg

(4.B.ll)

(4.B.12)

where g is the acceleration of gravity (980 em s~2) and p is the particle density. The buoyancy force is similar, with the fluid density, pi' replacing_the particle density:

11' 3 Fb = Pt6dPg (4.B.l3)

,, ,j

I 1-

Figure 4.8.3 Force balance on a particle to determine gravitational settling velocity.

Table 4.B.l Equations and Data for Computing the Terminal Settling Velocity When the Stoke's Law or Newton's Law Expressions for Drag Apply

General information applies in all cases g = 980 em s-2

For air (assuming P = 1 atm) Usually set PJ = 0 since PJ 20 mm (rare in environmental engineering)

For water Newton's law applies if dP > 2.8 mm (p = 2.5 g cm-3)

'''The' drr ranging to s

The ev drag coeffic in tum, deJ Stokes's Ia\' simple enot provides eq' two cases. f (seeAppen< proximatior;

The ten is suspende< ance becom'

Figure 4.B.4 ' (P = 1 atm, T diameter from scale. For the 1 scale and the s<

gravitational settling

elocity When the

s

Jllll(p = 2.5 g cm-3)

~ring)

'

The~"force'u.ven1:r~;umm11i::B':I::COiiitiiDin2lliese'ei~;s""Bii..o:..rear.'" f~ag coefficient on velocity is simple enough to pennit equation 4.B.l4 to be solved explicitly for v,. Table 4.B.l provides equations and other information useful for carrying out calculations for these two cases. For the intermediate regime, a nonlinear algebraic equation must be solved (see Appendix D). The data in Figures 4.B.4 and 4.B.5 can be used to obtain a first ap-proximation of v1, as demonstrated in Example 4.B.l.

The tenninal rise velocity of a particle that is less dense than the fluid in which it is suspended is calculated by the same approach. For a rising particle, -the force bal-ance becomes Fg + Fd = Fb, yielding

vt = [4gde(Pt- p)]l/2 p < PJ (4.B.15) 3Cd. PJ

Particle density

~f I I 111111~#- X I 1111111 := ~.;;::;-3

Figure 4.B.4 Terininal settling velocity for rigid spherical particles in air (P = 1 atm, T = 298 K). For the upper pair of curves, read the particle diameter from the upper scale and the settling~elocity from the left-hand scale. For the lower pair of curves, read the particle diame~er from the lower scale and the settling velocity from the right-hand scale.

, 178 ~4 .~;;~port Phenomena

)'~

:, 1 l 111', Ill II II I ( l . l l 1. I tjJ 10- 3 . Part1cle denstty

I zr/, I I I IIIII 11----- ii!~~J

Figure 4.B.5 Tenninal settling velocity of rigid spherical particles (p = 2.5 g cm-3) and rise veloci~y of rigid spherical bubbles (p = 1.2 mg cm-3) !~water at T = 20 C. For the upper pairof curves, read the particle diameter from the upper scale and the velocity from the left-hand scale. For the lower pair of curves, read the particle diameter from the lower scale and the velocity from the right-hand scale.

i~EXAMPLE 4.B:t.. Particle Settling Velocity for Intermediate .Drag Regime :.e:::,~f..i'.:!..;;:~"~~~&.. -

-~ ~.; ;!:.-

)-2

o-3

10-4

2.5 g water >m the ,r of 1from

Regime tameter of 1 mm and a

lies in the intermediate -1 -1 p 4 B' g em s . 1gure . . ..

m s-1 With these values, .5 em s-1 X 1 g em-~ -~-ediate regime. tVe approach. Equations ,own parameters for t~i ese expressions): ,

. 4.B/ r Particle qJion ~ 79.

Ptll1kle Settling Velocity for Intermediate Drag Regime (continued) Now, we evaluate these equations in series, initially assuming v, = 15 em s-1 and

calculating ReP' then using that result to calculate Ct~t and then using Cd to calculate v,. We repeat this process until the numbers converge to a stable solution that doesn't change from one cycle to the next. The intermediate results are v1 = 11.3, 10.6, 10.4, 10.4, 10.4 em s-1 With a reasonable initial guess, a stable answer to three significant figures is achieved after three iterations.

Figure 4.B.4 shows the terminal settling-velocity for spherical particles in air as a fur.:tion of size, according to equation 4.B.l4. Figure 4.B.5 show~ the settling ve-locity for particles in water and the rise velocity of bubbles in water as a function of diameter, according to equations 4.B.l4 and 4.B.l5. Note that for the particle sizes considered, the settling velocity spans an enormous range: seven orders of magnitude in air and six orders of magnitude in water. As a general rule, settling is unimportant for particles smaller than about l p,m in diameter.

The particle flux caused by gravitational settling is analogous to the contaminant flux ~aused by advection (equations 4.A.1-2) .. The magnitude is given by

Jg = Cv1 The vector form of this relationship is

~ ~ 18 (x, y, z) = C(x, y, z)v1

(4.B.l6)

(4.B.l7) where C(x, y, z) is the concentration of particles at the point (x, y, z). The vector points downward, parallel to the gravitational settling velocity, vr Example 4.B.2 presents an illustration of how the prediction of particle flux can be applied to a practical environ-mental question.

~~LE 4Jl.2 Soiling Due to Particle Deposition Studies have shown that a white surface becomes noticeably soiled when 0.2 percent of its area is covered by black particles (i.e.~ soot) (Hancock et al., 1976). Assume that the same relationship applies for soil dust particles. Estimate the time required for an initially clean, horizontal surface to appear soiled if it is exposed to an atmosphere containing 10 l-lg m-3 of soil-dust particles of diameter 5 JLm. Assume that the parti-cles are spherical and have a density of 2.5 g cm-3

SOLUTION The particle settling velocity is determined by the Stokes's law expression (Ta-ble 4.B.1):

v = [Ccgd:(p- Pt)] = [1.03 X 980 X (5 X 10-4)2( 2.5 )] = O.l9 ems -1 1 18 p., 18 1.8 X 10-4 The number concentration of particles is obtained by dividing the mass concentration by the mass per particle:

10 10-6 . -3 ( 10-2 -1)3 C X g m X m em 0 061 rt. 1 -3 = =. ~~~~ 2.5g cm-3 x ( 11'/6) X (5 X 10-4 cm)3

!I i !

180 Chapter 4 '"Transport Phenomena

Soiling -:e to PilrtiCie DepiJsiiion ( ~oniiti~ti) The fl~ accumulating-Ontothe~ace isgiven by eqUation4.B~16: ". .

. , , . :. , ; ,~ 1; , . . : ~ ~ :i ~ , :. . . ~:: r: . . . ~, J8 =: Cv, = 0.061 partic!e.;;cm-:-~ x 0.19 cms"':"~ =0~012 par:ticles cm-:2 s-1

Each particle h~ a p~rijected are~ xp, equal to that of a ~ircle of diameter 5 &J.ffi. The fractional rateR, at which the surface is covered-is the produ~ of AP and 18:

R = id;J8 = ~(5 x 10-4 cm)2 x o.oi2 cm-2 s-1 = 2!3 x I0-9s-1 The time required to cover 0.2 percent of the surface is obtained directly as

t = 0

002 = 9 X 105 s = 10 days

R

Note that this is similar to the time scale over which household surfaces become per-ceptibly dusty.

4.8.3 Brownian Diffusion In 4.A.3 we discussed molecular diffusion, a transport mechanism caused by the ran-dom motion of fluid molecules. An analogous phenomenon applies to particles sus-pended in a fluid. This transport mechanism is known as Brownian diffusion (or Brownian motion)~ in recognition of the botanist Robert Brown, who in 1827 reported observing, through a microscope, the wiggling motion of pollen grains suspended in water. Brownian diffusion results from the random collisions of particles suspended in_ a fluid with surrounding molecules. The overall effect in a fluid with a population of suspended particles is to cause net transport from high concentration to low concen-tration. The rate of Brownian diffusion is much slower than the rate of molecular dif-fusion because particles are much larger and much more massive than molecules.

Fick's law effectively describes the flux of particles caused by Brownian diffusion (see equations 4.A.5-6). A key issue in applying Fick's law for particles is the evalua-tion of the diffusion coefficient, D.

Diffusivity of Particles in Air and Water The Brownian diffusivity of particles is detemrined by an equation known as the Stokes-Einstein relation,

D=kT f

(4.B.l8)

where k is Boltzmann's constant {1.38 X 10-16 erg K-1, where the erg is an energy unit equivalent to 1 g cm2 s-2), T is temperature (K), and f is the friction coefficient com-puted according to Stokes's law (37r,WPC~1), including the Cunningham slip correc-tion factor if the particle is in air (equations 4.B.9-10). Example 4.B.3 illustrates the application of equation 4.B .18. Figure 4.B.6 shows the Brownian diffusivity of particles in air and water as a function of particle size. As the particles become very small, their diffusivities approach those of large molecules, -0.1 cm2 s-1 in air and -10-5 cm2 s-1 in water. For large particles, where the slip correction factor approaches 1, the ratio of

. .J

'~""!

Fi T

B1 til

~EX~ c T

SOLUTIO

s(

c

It B te

. I

-f -1 n s

neter 5 IJ.ffi. md J8:

s-1

IS

'Jecome per-

l by the ran-articles sus-liffusion (or i27 reported :uspended in mspended in. opulation of low concen-olecular dif-)lecules. ian diffusion s the evalua-

nown as the

(4.B.18)

n energy unit fficient com-n slip correc-

~llustrates the :y of particles y small, their o-s cm2 s-l in , the ratio of

~ . .. -l

'

f

i i I ! 1

I ,_.;,Jl*, .

. ~~----

Figure 4.B.6 Brownian diffusivity for particles. in air and water; T = 20 C, P = 1 atm.

.4.B J1rarticle Motion 181

Brownian diffusivity in air to that in water approaches the inverse ratio of the viscosi-ties, approximately 56 at 20 C.

~;:E:X~~~~~~ Diffusivity of Particles in Air and Water ~-- . __ .... __ .": . __ :>-~-: .. :~:,;'-r~->.:~~-:- :'."-~-.~

Compute the Brownian diffusivity of a 0.1 J.LID particle in air and 'in water. Assume T = 293 K and P = 1 atm.

SOLUTION The slip correction factor for a 0.1 f.LID particle in air is

C = 1 + 0066(2.51 + 0.80 (-O.SS X O.l)]= 2.89 c 0.1 exp 0.066

So the diffusivity of this particle in air is

Dair = 1.38 xlo-16 x 293 x 2.89 = 6_9 x 10~ cm2 s-1 31T(l.8 X 10-4 X 0.1 X 10~)

In water, the slip correction factor doesn't apply. The diffusivity is

D = 1.38 X 10-16 X 293 = 4 3 X 10-8 cm2 s-1 water 317'(0.01 X 0.1 X 10-4) .

Comparing Particle Transport by Settling and Diffusion It is interesting to compare travel distances for gravitational settling with those for Brownian diffusion. Figure 4.B.7 makes this comparison for particles in air and wa-ter. Typical settling and diffusion distances are comparable for particle diameters of

n 'l . L I

181 ChaPter 4 Transport Phenomena

...

(== ~~ter}

, ,

, ,

, ,

, ,

........ 0 .............. ,

.,.. ................... ...

, ----,

, , ,

, ,

, ,

, ,

,

Figure 4.B.7 Distance traveled in 10 s by particles in air or water because of gravitational settling and Brownian diffusion. Particle settling velocities were determined from Stokes's law_(Table 4.8.1) with T = 293 K and P = l atm. Diffusion distances were estimated from expression 4.A.7, u8ing the Stokes-Einstein relation to compute diffusivity (equation 4.B.l8). Particle density is 2.5 g cm-3

-0.2 tJ.ffi for air and -0.7 ILm for water. (The exact crossing point depends on the as-sumptions about particle density and travel time.) For particles with a diameter of -1 tJ.ffi or larger, diffusion is a relatively unimportant transport mechanism in com-parison with gravitational settling (and other processes that depend on particle mass or inertia). For particles with a diameter of -0.1 ILm or less, the role of gravitational settling as a transport mechanism is negligible compared with Brownian diffusion.

4.C MASS TRANSFER AT FLUID BOUNDARIES Several important phenomena occur at the boundaries of fluids. The rate of contami-nant transport to and from these boundaries may have a strong influence on contami-nant . concentration. This influence occurs in both natural and engineered systems. Contaminants can enter or leave a fluid at a boundary, and they may also undergo chemical transformations there. The boundaries of interest include air-water interfaces as well as boundaries between either air or :water and (a) soil, (b) other solid surfaces, or (c) nonaqueous liquids such as fuels and solvents.

There is a close relationship between the transport phenomena explored in this section and the transformation processes involving phase change discussed in 3.B. In fact, the distinction between transport and transformation is somewhat blurred here. The kinetics of phase change processes depend, in general, both on the rate of trans-port to or from the fluid boundary and the kinetics of the transformation processes. Transport and transformation processes occur serially, so the slower process governs the overall rate~

Transport across air-water interfaces is of particular interest in environmental engi-neering. Several treatment technologies involve transferring pollutants from one phase

,!(, (

. -- ;.: '(' ( c

c 1: 1 p

a y

0 e

b

$1

a: fl k K

ii tl

4.C.l M T tt

H is th c th be cil

dil

Vo

- ~"-

-- ---'i"'""

!

on the as-ameter of n in com-dele mass .vitational iffusion.

: contami-l contami-i systems. o undergo interfaces

1 surfaces,

red in this in 3.B. In ured here. e of trans-processes. ss governs

ental engi-one phase

'-', ,,._ .. 4.C Mass~~eratFluidBo~es' 183

to the other. For example, scrubbers remove acid gases such as sulfur dioxide and hy-drochloric acid from waste air..streams by transferring them to. water, where they can easily be neu~ed. Air strippers remove volatile organic compounds from water. Once the organic molecules. are transfened to the gas phase,::they can be more easily captured by sorption or destroyed by oxidation processes. In nature, air-water ex-changes are also important. The transfer of oxygen from air to water supports aquatic life. The ultimate fate of many air pollutants involves transfer to cloud or rainwater fol-lowed by deposition to the earth's surface. The uptake of carbon dioxide by the oceans plays a key role in moderating the impact of fossil fuel combustion on climate.

Transport to a fluid boundary may occur by a Variety of mechanisms. Diffusion is always present and often is the rate-limiting process because, at rigid boundaries, flow velocities diminish to zero. Advection usually plays a role in controllillg the thickness of the layer through which diffusion must occur. For particles, gravitational settling, electrostatic drift, inertial drift, and other mechanisms may dominate transport near boundaries.

For engineering analysis of transport processes at fluid boundaries, we seek a de-scription that captures the overall effects~ that does not violate central principles such as mass conservation, and that is practical to apply. We will employ models that link flux to concentrations .. and may include an empirical parameter. In the case of turbu-lent diffusion, we used an equation inspired by Fick' s law of diffusion, and the empir-ical parameter was turbulent diffusivity. For mas& transfer at boundaries, similarly inspired flux equations will be written that in,corporate a mass-transfer coefficient as the empirical parameter.

4.C.l Mass-Transf~r Coefficient The net rate of mass flux between a fluid and its boundary is commonly expressed in this form:

Jb = km(C- C;) (4.C.l) Here Jb is the net flux to the boundary (amount of species per area per time). When Jb is greater than zero, the flux direction is from the fluid to the boundary; when J b is less than zero, the flux is directed from the boundary to the fluid. The concentration terms C and C;, respectively, represent the species concentration in the bulk fluid far from the boundary and the species concentration in the fluid immediately adjacent to the boundary. The other par~eter in this equation, km, is known as a mass-transfer coeffi-cient. The mass-transfer coefficient commonly has units of velocity (length per time).

Figure 4.C.l shows a simple system for which a mass-transfer coefficient can be directly derived. A glass cylinder is partly filled with a pure volatile liquid (with a sat-

~-L

Stagnant air t t:: .,g ;; &. Flux Volatile liquid C C;

Concentration

Figure 4.C.l A simple system in which a mass-transfer coefficient can be directly determined.

t

I I I I I I i i !

I

;,, ~ l' 'I

184 Chapter 4 Transport Phenomena

uration vapor pressure 1 atnt). 'Ibe remainder of the cylinder'contains stagnant air. Molecules of the liquid evaporate, diffuse through the cylinder, and escape Jrom the open top into the free air. Provided that diffusion is not too rapickthe :gu:..phase con-centration of the diffusing species at the liquid-gas interface will be detenmned di-rectly asP sa/(RT), whereP sat is the equilibrium vapor pressure of the liquid. Assume that the concentration at the top of the cylinder is maintained at some value,c, deter-mined by processes occurring in the open air. After an initial transient period, during which the gas-phase concentration profiles may change, a steady-state linear profile will be established, as depicted in Figure 4.C.l. The diffusive flux through the tube, Jd, is given by Pick's law as

Jd =Dei- c L

(4.C.2)

The net transport rate from the air to the liquid is equal to the negative of the diffusive flux, Jd. Comparing equation 4.C.2 with equation 4.C.l, we see that the mass-transfer coefficient for this system is

k =D m -L

(4.C.3)

This is a general result. provided that the concentration profile is steady: For pure dif-fusion through a stagnant layer, the mass-transfer coefficient is given by the diffusivity divided by the thickness of the layer.

A second case is illustrated in Figure 4.C.2. Particles are suspended in a fluid above a horizontal boundary. The fluid is motionless, the particle concentration is uni-form, and the particles migrate only because of gravitational settling. All particles are assumed to have the same settling velocity, v,. The concentration of particles in the fluid at the interface is taken to be zero. This may seem peculiar, since the particles accumulate on the bottom boundary. H.owever, provided resuspension does not occur (as it will not in a stagnant fluid), once the particles strike the boundary, they are no longer suspended in the fluid, and so it is reasonable to assign Ci = 0. The gravita-tional flux to the surface, 18 , is then given by

0 0 0

0 0

0

0

0

goo

cg 0

0

~

Jg = v1C

0 0

0 0

0 0 0 0

0 t V1 0

0 0

o'b 0 0

Figure 4.C.2 In a uniform suspension of monodisperse particles settling through a stagnant fluid onto a horizontal surface, the mass-transfer coefficient is equal to the settling velocity.

(4.C.4)

. .. .. . \ ~ .: I

.... :i, ..

c c~

T S{ tt

e' S<

P

e; lr' Ct is V< dt ic in b(

\\ rn

at.

rn

th rn

pl he b3 te1 an (fl

m Fe us

lat sy lal

Fi. Th Ac

fu~ to co

'

'stagnant air. tpe from the .;-phase con-termined di-

:~id. Assume ue~ C, deter-~riod, during inear profile gh the tube,

(4.C.2)

the diffusive nass-transfer

(4.C.3)

For pure dif-te diffusivity

~d in a fluid ration is uni-particles are rticles in the the particles

>es not occur ',they are no The gravita-

(4.C.4)

,.-... ,,'.!: ..

iii ~~

I _c_;.,L_

4.C Mass;~fel>al\Fluid Boundaries 185

Comparing this result with equation 4.C.t, we see that for this.case the mass-transfer coefficient is equal to the particle settling velocity:

km = V1 (4.C.5) This too is a general result. Whenever contaminant. transport to a surlace occurs solely because of a net migration velocity, the mass-transfer coefficient is equal to that velocity.

These examples yield exact expressions for the mass-transfer coefficient. How-ever, in most applications, the use of a mass-transfer coefficient is a simplified de-scription of what is a very complicated set of processes. Therefore~ a high degree :of precision should not be expected in problems involving mass transfer at boundaries.

Applyitig equation 4.C.l wisely requires a sound understanding of the meaning of each of the four variables. First, let's consider the concentration at the interface, C;. In many cases, C; can be taken to be zero. This is appropriate when a transformation pro-cess that is fast and irreversible occurs at the boundary. Particle deposition by settling is one example where C; is zero: When a particle strikes a surface with a low incident velocity, it adheres essentially immediately. Chemical transfofmations can also pro-duce an interface concentration of zero if transport, rather than surface-reaction kinet-ics, is the rate-limiting step. In some cases, such as that shown in Figure 4.C.l, the interface concentration can be determined by assuming local equilibrium at the boundary ..

An important issue arises concerning th~ species concentration in bulk fluid, C: Where should it be determined? The issue is easily resolved when the fluid is well mixed outside of a thin boundary layer (see Figure 4.C.3). Then C is the concentration anywhere outside the boundary layer. Howeve:r~ some circumstances arise in environ-mental engineering in which species concentrations v.ary strongly with position throughout the fluid. In such cases, the use of a mass-transfer coefficient may yield no more than a rough approximation of the mass-transfer rate at surfaces.

A comp~ication arises when the fluid boundary occurs at a surface that has a com-plex texture. Iflb represents the quantity of contaminant transferred per area per time, how do we define the area? Typically, the flux to rough surfaces is determined on the basis of a superficial or apparent area. fu other cases~ rather than determining the in-terface area and the mass-transfer coefficient separately, their product is determined and the mass-transfer equation is rewritten in terms of total net transfer of species (flux times area) rather than in tenns of flux.

The final issue to address is how an appropriate mass-transfer coefficient is deter-mined for a given situation. There are two main approaches, and both are widely used. For systems with fairly. regular geometries, the mass-transfer coefficient is calculated using existing correlations based on theory or experimental data. Some of these corre-lations are presented below. A second approach, commonly used for more complex systems, is to calculate mass-transfer coefficients based on experiments conducted in laboratory or field settings.

Film Theory

The simplest model system that includes fluid flow divides the fluid into two layers. Adjacent to the surface is a stagnant layer, or film, through which species must dif-fuse. The concentration profile within the film is assumed to.be linear, corresponding to steady diffusive flux. In the second layer, the. fluid is well mixed and the species concentration is uniform everywhe~e.

' y;

186 Chapter 4 Transport PhenOmena

.~ ~

~

C; Concentratioa

Flux

c

Well-mixed fl~d

Stagnant fluid film

Figure 4.C3 Schematic of mass transfer to a surface according to film theory.

Mass transfer from the fluid to a surface, according to film theory, is depicted in Figure 4.C.3. This situation is almost identical to that depicted in Figure 4.C.l~ and so, in analogy with equation 4.C.3, it should be clear that the mass-transfer coefficient for this case is

D k =-m L

'f (4.C.6)

where L1 is the film thickness. Film theory suggests that for given flow conditions, the mass-transfer coefficient should scale in direct proportion to species diffusivity (km oc D). Experiments consistently demonstrate that, in systems with fluid flow, the mass-transfer coefficient flow increases in proportion to Da._, with a < 1. The key weakness of film theory is its dependence on a specific value of the film thickness, 4- In most circumstances, there is no practical way to determine L1 that is independent of a mea-surement of mass transfer (or heat transfer) to a surface. Furthennore, the effective thickness of the fiim through which species must diffuse varies with species diffusiv-ity: A higher diffusion coefficient results in a larger film thickness. Still', although not entirely accurate, film theory provides a helpful conceptual picture of interfacial mass transfer. ,

Penetration Theory

Recall (equation 4.A.8} that the characteristic time required for a steady-state concen-tration profile to be established by diffusion is given by L 2(2Dt1 where L is the dis-tance over which diffusion occurs. In film theory, -the boundary must be in contact with the fluid for a minimum time ofT- L}(2D)-1 for the concentration profile to ap-proach the constant-slope condition depicted in Figure 4.C.3. In some circumstances, the contact time between a boundary and a fluid is not long enough for film theory to apply. A second conceptual model, known as penetration theory, has been developed to address this case.

The model is based on transient diffusion in one dimension. At timet= 0, the concentration in the fluid is assumed to be uniform everywhere except at the position of ,the boundary, where it is C~. Then diffusion to the surface begins and a boundary layer begins to grow (Figure 4.C.4). Fluid motion affects the contact time between the fluid and the boundary.

Analysis of time""dependent diffusion to a flat surface yields the following expres-sion for the mass-transfer coefficient:

km(t} = [~J'2 instantaneous (4.C.7)

~

..... J

.g !

11 th th cc

Fr ef cc

is at ne

di fu

Bt

Tr dil in. de co

ef

fa

.~L

s transfer to a

s depicted in -.C.l~ and so, oefficient for

(4.C.6)

mditions, the . usivity (km ex

,w, the mass-;.ey weakness s, Lf In most ent of a mea-the effective

:cies diffusiv-although not ~erfacial mass

-state concen-! L is the dis-be in contact

1 profile to ap-ircumstances, film theory to .!en developed

ime t = 0, the at the position nd a boundary 1e between the

lowing expres-

(4.C.7)

\ t

d .g I ' < '2 < t3

~

4.c :Massn:anst at Fluid Bound&rii;s 187 - ---

.Fluid

!!M'erS're~ Boundary Figure 4.C.4 Schematic of mass transfer to a surface according to penetratio1.1 theory.

This is the instantaneous mass-transfer coefficient at time t. Because the distance throughwhich species must diffuse increases with time, k, decreases as t increases. If the contact period is maintained for some interval t*, the time-averaged mass-transfer coefficient is obtained by integration:

t

km = .~1 km(t) dt = 2[_Q_] 112 0 1rt*

average (4.C.8)

From equation 4.C.8, we see that penetration theory predicts that the mass-transfer co-efficient should increase in proportion to the square root of species diffusivity. By contrast, we have just seen that film theory predicts that the mass-transfer coefficient is proportional to diffusivity .. The reason for the difference lies in the assumptions about the distance through which species must diffuse. In film theory, the film thick-ness is assumed to be constant and so is independent of D. In penetration theory, the diffusion distance increases with time, and the rate of growth depends on species dif-fusivity.

Boundary-Layer Theory: Laminar Flow along a Flat Surface Transport from moving fluids to boundaries is caused by simultaneous advection and diffusion, with advection being stronger far from the boundary and diffusion dominat-ing adjacent to the boundary. Film and penetration theories simplify the analysis by decoupling advection from diffusion. Boundary-layer theory can predict mass-transfer coefficients for simple flows and geometries while simultaneously accounting for the effects of both transport mechanisms.

Let's consider the case of laminar fluid flow at velocity U, parallel to a flat sur-face (see Figure 4.C.5). The species concentration is Ci at the surface and is C far from the surface. Within the boundary layer there is a concentration gradient, which gives rise to diffusion toward the surface. There are also advective velocity components in

u ~ ~

~

l-x

ci

~ , I

_r 188 Chapter 4 Transport Phenomena

', ..the x-direction (relatively strong} and the z-direction (relatively weak} that affect the concentration profile. The characteristic thickness of the boundary layer for a contam-inant species is the distance from the surface 'over which the concentration increases to approximately C. As shown in Figure 4.C.5, the boundary-layer thickness grows with-increasing distance downstream of the leading edge. The thickness of this bound-ary layer at any x-position is a function of species diffusivity, with higher diffusivity causing a thicker boundaly layer. For any given species, a thicker boundary layer means slower mass transfer to the surface. Because of species loss at the surface, the boundary-layer thickness grows with downstream distance and the rate of mass trans-fer correspondingly decreases.

This problem is analyzed using equations that describe the conservation of fluid mass, momentum, and species in the boundary layer {Cussler; 1984; Bejan, 1984). The result is a mass-transfer coefficient calculated at a specific position x downstream of the leading edge of the surface:

(u)u2 km(x) = 0.323 -:; v-116D213 local (4.C.9) where v is the kinematic fluid viscosity. (Recall that v = p/p, where p.. is the dynamic fluid viscosity and pis the fluid density.) The overall mass-transfer coefficient to the surface is obtained by averaging over the length of the smface:

L

km = i: L km(x) dx = 0,646(f2 v-116D213 average (4.C.10) Equation 4.C.l 0 gives the average mass-transfer coefficient for this flow system for a surface from the leading edge to a distance L downstream. Note that the mass-transfer coefficient, km, varies with the diffusion coefficient raised to the 2/3 power. It is gener-ally true that when advection and diffusion are combined, the mass-transfer coeffi.:. cient increases with diffusivity raised to some power between 0.5 (penetration theory) and 1 (film theory).

4.C.2 Transport across the Air-Water Interface . In en:vironmental engineering applications~ the rate of transport of a molecular species

across an air-water Interface is described by an expression similar to equation 4.C.l:

Jgl = kgl( Cs- C) (4.C.ll) where Jg1 is the net flux of a species from the gas phase to the liquid phase (mass per interfacial area per time), kg1 is a mass-transfer coefficient (length per time Y~ C is the concentration of the species in the bulk liquid phase, and Cs is the saturation (or equi-librium) concentration of the species in the liquid phase that corresponds to t.he given partial pressure of the species in the gas phase. 1Jpically, Cs is obtained from Henry~s law (see 3.B.2). When Cs exceeds the current aqueous concentration, Jg1 > 0 and net transfer occurs from the gas to the liquid. Conversely~ when the water is supersatu-rated with respect to the gas phase, C > Cs, so Jgt < 0, and equation 4.C.ll predicts a net rate of volatilization.

Ill general, the mass-transfer coefficient kg1 depends on fluid flow near the inter-face and on species diffusivity in both air and water. The film model introduced in the previous section can be extended to. a two-film model, as depicted in Figure 4.C.6, with the air-water interface as the boundary. In this model, stagnant film layers exist

1

1"

'

'

0

u a( e

that affect the for a contam-tion increases ckness grows Jf this bound-er diffusivity >nndary layer e surface, the lf mass trans-

ation of fluid Bejan, 1984). r downstream

(4.C.9)

; the dynamic fficient to the

(4.C.l0)

1 system for a mass-transfer er. It is gener-ansfer coeffi.:. ration theory)

-!cular species ,ation 4.C.l:

(4.C.ll) .ase (mass per

ime), Cis the 1tion (or equi-ls to the given from Henry's

:I > 0 and net : is supersatu-.C.ll predicts

near the inter-roduced in the Figure 4.C.6, m layers exist

J

Air ~

~ L_ -~

4.C Mass TrariSferat:Fluid BOundaries 189

Pi P Partial pressure.

Turbulently mixed zone

S~antfilnllayer

-~ !.~ ..

E----- 8 s ' -- ~~---~]j ___ ~----------~~~~~~~r=~

c:: .9

Water ~ ~

c ci

Turbulently mixed zone

Concentration

Figure 4.C.6 Schematic of the two-film model for estimating mass transport across an air-water interface.

on either side of t..lte boundary through which species transport occurs only by molec-ular diffusion. Outside of its stagnant film layer, each fluid is well mixed. Immediately adjacent to the interface, the partial pressure in the gas phase, Pi .. is assumed to be in equilibrium with the aqueous concentration, C;, as described by Henry's law. As in the film model, the concentration profiles in the stagnant film layers are assumed to be linear. Also, because of mass conservation, the flux through the air layer must be equal to the flux through the water layer.

Applying Fick's law, we can write the gas-side flux (from air to the interface) as _ . (P- P;)IRT

Jgl- Da (4.C.12) T.

a

where D a is the species diffusivity through air and La is the thickness of the stagnant film layer in the air. Likewise, the liquid-side flux (from the interface into the water) is

c.-c J gl = D w_.;..1 --

Lw (4.C.l3)

where D w is the species diffusivity in water and Lw is the film-layer thickness in the . water. Since we don ~t know either P; or C;, we would like to eliminate these parame-ters from the expressions. So far, we have two equations but three unknowns (Jg1, C;, andPJ

The third equation comes from the equilibrium relationship at the interface:

ci = K8 P; (4.C.l4) where KH is the Henry's law constant for the species (see Table 3.B.2). Now, we can use algebra to derive an expression for flux that is in the form of equation 4.C.ll. Use equation 4.C.l4 to replace Pi with CJKHin equation 4.C.l2. Then equate the right-hand sides of equations 4.C.12 and 4.C.13 and solve for Ci to obtain

C.= aCs + C ' 1 +a

(4.C.15)

where DaLw

a = ---"'---"'--DwLaK8RT

(4.C.16)

and . cs = KHP (4.C.l7)

,,

~

!~ ~~4J lfransport Phenomena Next substitute for C1 from equation,4.C.l5 into equation 4.C.l3. After some alge-

,braic manipulation, one arrives at equation 4.C.ll, where . ' .. ,( . . . ~-- .. ,

... , , kg1 = Dw(__!C) ::; 1 (4.C.l8)

Lw 1 +a L.,., La -+-KHRT Dw Da

The film thicknesses, L_.. and La, cannot be measured. So for practical application, we rewrite equation 4.C.18 in this form:

l 1 KHRT -=-+--kg/ kl kg

(4.C.l9)

where k1 and k8 are the respective mass-transfer coefficients through the liquid and gas boundary layers near the interface, each corresponding to the diffusivity divided by the film thickness (Dw/ Lw and D 0 /L0 , respectively), as in equation 4.C.6. The factor KH appears with kg in this expression because kg1 is used with the aqueous concentra-tions to determine flux (equation 4.C.ll)- The factor RT is needed to convert partial pressure to molar concentration, since we write Henry's law in a way that relates aqueous molar concentration to gaseous partial pressure.

Equations 4.C.ll and 4.C.l9 are together sometimes called the two-resistance model for interfacial mass transfer. Since species must be transported through fluid on both sides of the interfaCe, the total resistance (k-J) is the sum of the resistance in the liquid side of the interface (k[1) plus the resista~ce on the gas side (KHRTk;1).

From equation 4.C.l9 we see that the relative sizes of the gas and liquid resis-tances vary according to the Henry's law constant, KH. The diffusion coefficients, D, of molecular species in a given fluid vary byabout one order of magnitude (4.A.3). The film resistance tenns k1 and kg are expected to vary by no more than D to the frrst power (4.C.l ). The Henry's law constant, on the other hand, varies by at least eight orders of magnitude among species of interest in environmental engineering (see Ta-ble 3.B.2). Therefore, even for fixed flow conditions, the interfacial mass-transfer co-efficient, kgb may vary greatly from one species to another according to the value of the Henry's law constant. This.point is illustrated in Figure 4.C.7.

Liss and Slater (1974) reviewed available information on the mass transfer of species between the oceans and the atmosphere and concluded that liquid-side and gas-side mass-transfer coefficients of k1 = 0.2 m h~ 1 and kg = 30 m h-1 apply for average meteorological and ocean current conditions. These coefficients are sug-gested to be approximately correct for species with molecular weights between 15 and 65 g/mol. Figure 4.C. 7 shows how the overall mass-transfer coefficient varies with the Henry's law constant for molecular species exchange between the atmosphere and the seas. For sparingly soluble gases such as oxygen (KH = 0.0014 M atm-1), the resistance lies entirely on the liquid side and an overall average mass-transfer coeffi-cient of approximately kg1 = 0.2 1nlh applies. On the other hand, the resistance for a highly soluble species such as formaldehyde (K8 = 6300 M atm-1) lies entirely on

th~ gas side. The liquid-side and gas-side mass-transfer coefficients can be combined with values

of species diffusivities to estimate effective film thicknesses: La - D ak;1 - 2 mm and Lw - D wk )/ - 20 J.Lm. We see from these estimates that diffusive transport over very small length scales controls the overall rate of interfacial mass transfer. The time scales required for steady-state concentration profiles to be achieved in gas and liquid films of this size are approximately Tg- L:(2Da)-1 - 0.1 s and T1- L!C2Dw)-1 - 0.2 s.

'===- -c..~-~~~-----------

,-. ~

'f:} ~ ~~ ' li~:'\..

f;-

t;:r ~.

t,c

! ......_

Fi th w;

kg

ga sp ef

Fe

Fo

In SUl

me

mJ

ten in~ COl

the tra1

some alge-

(4.C.l8)

tpplication,

(4.C.l9)

uid and gas divided by The factor concentra-

tVert partial that relates

~-resistance rough fluid -!sistance in HRTk;1). iquid resis-~licients, D, le (4.A.3). >to the first t least eight ing (see Ta~ transfer co-.he value of

transfer of id-side and 1 apply for tts are sug-veen 15 and varies with atmosphere { atm-1), the asfer coeffi-stance for a entirely on

l with values -2 mm and 'rt over very :! time scales 1uid films of 0.2 s.

4.C M~~T~~;;~~~4.,~~~es 191

Sparingly soluble: Liquid film resistance

dominates

Highly soluble: Gas film resistance

dominates

Figure 4.C.7 Dependence of the overall mass-transfer coefficient on the Henry's law constant for average conditions in large bodies of water. The curve traces equation 4.C.l9 with k1 = 0.2 m h-1, kg = 30m h-1, and T = 293 K (Liss and Slater, 1974). '

For natural bodies of water, the following expressions can be applied to estimate gas-side and liquid-side mass-transfer coefficients in relation to environmental and species conditions (Schwarzenbach et al., 1993). For the gas-phase mass-transfer co-efficient,

~ D ]213 kg = a 2 _ 1 (7 U 10 + 11) .26 ems (4.C.20) For oceans, lakes, and other slowly flowing waters,

kl = [ Dw ]0.57 2.6 X I(Y-5 cm2 s-1 (0.0014Uio + 0.014) (4.C.21)

For rivers,

[ D ]o.s1(U )112 kl = 0.18 w ~

2.6 X 10-5 cm2 s-1 dw (4.C.22)

In these expressions, U10 is the mean wind speed measured at 10m above the water surface (units are rn!s). The term Uw is the mean water velocity in the river (rnls). The mean stream depth is dw (m). The equations are written so that k8 and k1 have units of mlh. Example 4.C.l illustrates how these equations are used.

In general, for interfacial transfer of molecular species in any air-water flow sys-tem, the overall mass-transfer coefficient for any species can be determined by mak- ing measurements of k81 for a minimum of two species with known Henry's law constants, provided one is sparingly soluble and the other is highly soluble. From these measurements, one can determine the values of k1 and kg~ Then the overall mass-transfer coefficient for any other species can be estimated using equation 4.C.l9.

r :191 lChapter 4 H !fnirispott Phenomena !1

,Mass-Transfer Coef!ieieittJO,.iJi:Yfeli.in;tiRwer: ..... , . ' . l . '. _ .. '~ .. : . .. . . . ... ,:

In Nebraska, the Missouri River has ~-meatf~pthof2:7 'in andfiows at a ntean veloc-ity of 1. 7 5 mls (Fischer et al., 1979)~1\ssume that the wind speed at l 0 m iS 4 m/s. Es-timate the overall mass-transfer coefficient for .. oxyge~ (02) from the atmosphere to the river. - -

SOLUTION The (diffusion coefficient of 0 2 in water is Dw = 2.6 X 10-5 cm2 s:-1 (Table 4.A.3). From equation 4.C.22, the liquid-side mass-transfer coefficient is estimated to be k1 = 0.14 mlh. The diffusion coefficient for 0 2 in air is D a = 0.178 cm2 s-1 (Table 4.A.2). From equation 4.C.20, the gas-side ~ass-transfer coefficient is estimated to be kg= 30 mlh. The Henry's law constant for oxygen is KH = 0.0014 M atm-1 (Table 3.B.2). With R = 0.0821 atm K-1 M-1 and T = 293 K, we predict from equation 4.C.l9 that kg1 = 0.14 mlh. As expected for a sparin-gly soluble gas such as oxygen, mass transfer through water adjacent to the interfacial boundary controls the overall rate of the process.

4.D 1RANSPORT IN POROUS MEDIA Porous materials are solids that contain distributed void spaces. Permeable porous ma-terials contain an interconnected network of voids or pores that permit bulk flow of fluid through the material. Soil is a common example of a permeable porous material. Usually, the pores are highly variable in shape and size, resulting in complex flow channels. :

Environmental engineers study the movement of fluids and contaminants through porous media for several reasons. Many treatment technologies for removing pollut-ants from water and air entail passing the fluid through a porous materiaL In munici-pal treatment plants. for example~ drinking water is passed through sand filters to remove small, suspended particles. Drinking water is also sometimes treated by pass-ing it through a column of granular activated carbon to remove dissolved organic mol-ecules that are harmful or that may cause taste and odor problems. Air used in industrial processes is often passed through fabric filters to remove suspended parti-cles. Filters of granular activated carbon are also used to remove volatile organic com-pounds from gas streams. Many hazardous waste treatment technologies also involve passing a fluid through a porous material. Much attention in hazardoUE waste manage-ment focuses on characterizing contaminant migration in subsurface soils, either to predict the threat to water quality or to evaluate a treatment strategy.

The aim of this section is to provide an introduction to transport in porous media, emphasizing . the movement of water and air and the contaminants dissolved or sus-pended within these fluids. The behavior of nonaqueous-phase liquids in subsurface environments is also important in environmental engineering, but the complexities that must be addressed render it beyond the scope of this book. Application of the ideas introduced here f~r filtering contaminants from water are discussed in Chapter 6. 'around water contamination is addressed in Chapter 8.

Before proceeding, we must define some . basic terms and concepts that arise when dealing with transpo~ through porous materials. One quantitative descriptor of a porous material is its porosity, here given the symbol f> and defined as

mean veloc-is"4 rnls. Es-mosphere to

12 s-1 (Table estimated to n2 s-1 (Table jmated to be atm-1 (Table Jm equation :1 as oxygen, 's the overall

e porous rna-bulk flow of ous material. omplex flow

1ants through oving pollut-tl. In munici-and filters to ated by pass-organic mol-Air used in

pended parti-organic corn-~ also involve 'aste manage-oils, either to

~orous media, ;olved or sus-in subsurface complexities

ication of the ed in Chapter

!pts that arise :iescriptor of a

(4.D.l) I I L

,AD Transport in Porous Media 193

The pores may contain both air and water (and, in general, other fluids). We define the air-filled porosity, .tz' and the water-filled porosity, .p.,, in analogy with equation 4.D.l, but with the numerator replaced by the pore volume-filled with air or water, re-spectively. If the air and water are the only fluids- contained in the pores, the porosities must satisfy this relationship:

cf> = cf>w + cf>a (4.D.2) A porous material is saturated with a particular fluid if that fluid entirely fills the pores. So, for example, a medium is saturated with water if t:f>w = q,.

Most porous materials encountered in environmental engineering are granular or fibrous. Granular materials, such as soils, typically have porosities in the range 0.3-0.7. The porosities of fibrous materials are usually higher, sometimes as high as 0.99. These materials may have two distinct classes of pores, those that are external to the grains or fibers and those that are internal. Bulk fluid flow occurs only in the external pores, but contaminants can migrate by diffusion into the internal pores and interact with the solid surfaces there. This ch~acteristic is especially important for porous sorbents, such as activated carbon, which have large internal porosities and enormous internal surface areas. However, in this section, we will emphasize fluid and contami-nant behavior in the pores that are external to grains and fibers.

Two densities are commonly defined for a porous solid The solids density~ Ps' represents the mass of solid per volume of solid (often including internal pores). The bulk density, ph, represents the mass of solid per total volume. These measures are re-lated by the total porosity:

Pb = Ps( 1 - 4>) (4.D.3) The solids density of soil grains is fairly constant at -2.65 g cm-3 Given a range of porosities of 0.3--0.7, the bulk density would be in the range 0.8-1.9 g cm-3 Soil bulk density varies with grain size, but the dependence is not strong. In soil, as in any gran-ular material, porosity tends to be higher when grain sizes are distributed over a nar-row range. With a broad distribution of grain sizes, smaller grains can fill the pores created by larger grains~ reducing overall porosity.

Porosity is an area characteristic as well as a volume characteristic of granular ma-terials. Imagine a plane slicing through a porous material. The area porosity is the ratio of the area of the plane that intersects pores to the total area that intersects the porous material. Conveniently, if pores are randomly distributed and the number of pores in a plane is large, then the area porosity is equal to the porosity measured on a volume ba-sis. This conclusion is reached by considering a porous material as a collection of infin-itesimally thin slices. The volume porosity of the whole is the average of the porosities of the individual slices. If these slices include a random distribution of a large number of pores, then the porosity of each slice will be close to the mean for all slices.

4.D.l Fluid Flow through Porous Media During the middle of the nineteenth century, Henri Darcy, a French engineer who was interested in the development of groundwater resources, studied the hydraulics of wa-ter flow through a sand column using an apparatus like that shown in Figure 4.D.l. With experiments conducted under steady flow c'onditions, he found that the follow-ing relationship described his results:

Q = KA!:ih L

(4.D.4)

J , 194 Chapter 4 - Tr~sport Phenomena

'\, ~~

area=A

1 Q llh

l

~ Figure 4.D.l Apparatus similar to that used by Darcy to study the hydraulics of water flow through

sand~

where Q is the volumetric flow rate (m3 s-1), llh is the change in fluid head from the inlet to the outlet of the column (m), A is the cross-sectional area of the column (m2), and L is the length of the column (m). The parameter K, which has units of velocity, was constant for a given sand but varied from one sand sample to another, increasing with increasing grain size.

This relationship has been generalized and is oow commonly known as Darcy's law. For water flow in one dimension through a water-saturated material, Darcy's law can be written as l

U = -Kdh dl

(4.D.5)

where U is called the Darcy velocity, filtration velocity, or superficial flow velocity, K is called the hydraulic conductivity; and dhldl is the rate of change of pressure head with distance. In relation to Darcy's experiment, U represents QIA, and the hydraulic gradient, dhldl, replaces .1.h/L. The filtration velocity (U) should not be confused with the local velocity of water through the pores. Rather, it represents the average volu-metric flow of water per unit total cross-sectional area of the porous material. Since solids occupy some area, and since water can flow only through pores, the average lo-cal velocity of water in the pores must be higher than the filtration velocity. The minus sign in Darcy's law, like the minus sign in Pick's law, reminds us that water flows from high to low pressure head, just as molecules diffuse from high to low concentra-tion. Since head has units of length, dh/dl is dimensionless, and so the hydraulic con-ductivity must have the same units as U, velocity.

Another form of Darcy's law applies to porous media saturated with any fluid:

U= _!dP p. dl

(4.D.6)

where k is called the intrinsic permeability, or simply permeability, of the porous ma-terial; p., is the viscosity of the fluid; and dP/dl is the derivative of dynamic pressure with respect to distance. This fonn of Darcy's law assumes that the fluid is incom-pressible, which, for air, would mean that the pressure drop across the medium must be much. smaller than the inlet pressure.

The hydraulic conductivity in Darcy's law (equation 4.D.5) depends primarily on two properties of the system: the viscosity of the fluid and the size of the pores. Fluids

thr flo Th pel

ve' me

ffiL

pn lll<

ce~

fie

cie in! ac' pe:

wl

H< da 4.1 th<

-r to

-r

10

_L

Cl gr.

AnubhaHighlight

tus similar to study the >wthrough

head from the column (m2),

,ts of velocity, 1er, increasing

.vn as Darcy's J, Darcy's law

(4.0.5)

ow velocity, K pressure head l the hydraulic confused with average volu-

naterial. Since he average lo-ity. The minus at water flows ~ow concentra-hydraulic con-

h any fluid:

(4.D.6)

~he porous ma-1amic pressure luid is incom-; medium must

is primarily on te pores. Fluids

L._

, 4D Trimsport in Porous Media 195

that are more viscous flow more slowly, and materials with smaller pores pennit less flow. The second form of Darcy's law (equation 4.0.6) separates these two properties. The intrinsic permeability is a function only.of the porous- material and does not de-pend on the fluid properties.

Given the units of viscosity (e.g., kg m-1 s-1), pressure derivative (kg m-2 s-2), and velocity (m s-1 ), we see that permeability has units of length squared. For a porous medium made up of uniform spherical grains, the permeability increases approxi-mately in proportion to the square of the grain diameter.

The dynamic pressure (P) in equation 4.0.6 is the difference between the total pressure and the hydrostatic pressure. In the absence of motion, fluid pressure must increase with depth to support the mass of the fluid suspended above it against the ac-celeration of gravity. This hydrostatic change in pressure with height does not induce flow.

The difference between the two forms of Darcy's law lies primarily in the coeffi-cients K and k. By noting that head loss corresponds to dynamic pressure drop accord-ing to p~~h = ~P, where Pw is the density of water and g is gravitational acceleration, we can show that the hydraulic conductivity is related to the intrinsic permeability by

K= kpwg ILw

where ILw is the dynamic viscosity of water.

(4.D.7)

Intrinsic permeability, k, may be expressed in ordinary area units such as m2 However, the permeabilities of ordinary materials are so low that a special unit, the darcy, has been defined for permeability, with 1 darcy = 0.987 X 1 o-12 m2 Figure 4.D.2 shows representative permeabilities and hydraulic conductivities of soils. Note the enormous range of values between homogeneous clays and coarse sand or gravel.

-o s:: C!l 0> v ~= s::-o~= ... ~~tb~~tb ~ e ~-g E e-g ]e

~ ~ = ~ ~= ~~ 13 ~-g ~ ~-g a ;:J u ~ ~ ::::: ~ Ui::>

>.

-n ~ >. 0

"0 >. s:: 4J

196 Chapter 4 'Transport Phenomena

Area=A v

~ I~ R

..--

Darcy's law Ohm's law Q K 1 -;r=TAh I=RV

Flow per area (QIA) Current(/) Head loss (M) Potential (V)

Column length to Figure 4.D.3 The analogy between hydraulic conductivity (UK) Resistance (R) Darcy~s law and Ohm~s law.

Fluid flow through a sand column is analogous to electrical current flow through a resistor. Figure 4.0.3 makes the comparison~ showing that the ratio of the column length to hydraulic conductivity acts like a resistor'~ to fluid flow driven by a loss in pressure head.

A widely used expression that relates permeability to grain size is known as the Carman-Kozeny equation. It was derived by analyzing flow through porous media in analogy to flow through a bundle of small capillaries~ but with varying size and orien~ tation (Dullien, 1979), giving

, /-. ) ~ '

between

flow through f the column 1 by a loss in

~own as the ous media in ze and orien:.

(4.0.8)

aced by tjJ deq ~iameter. The e as the grain l in granular-lt diameter is ample 4.0.1

[t has the fol-~nt (Cleasby,

lility:

I I t_

,4D. : ,Trimspot;t in Porous Media 197

"'l!'~l',ft!cb>

'" l 198 Chapter 4 Tr~sport Phenomena

. ----"'For three~imensional transpOrt applications, Fick's law~ bewtitteiiitfveetorform:

1-t = -D (ac ac ac) d ~ ox' oy' dZ (4.0.10}

Here J d representing diffusive flux through the porous medium, gives the quantity of contaminant transported per total area (not just pore area) per time. The parameter De is an effective diffusion coefficient, and Cis the species concentr3:tion in the pores (per pore volume, not per total volume). These equations have the same form as Fick's law for bulk fluids (equations 4.A.5-6), but each of the parameters has an altered meaning.

The effective diffusion coefficient through a porous material is smaller than the corresP,onding ordinary diffusion coefficient for the given contaminant/fluid combina-tion. For gas molecules in a granular material, such as soil, ilie following expression is widely used to estimate the effective diffusivity (Millington, 1959):

a = 0.2 and = 0.4. According to equation 4.D.ll, the effective diffusivity would be reduced to 0.03D in this case. For air-saturated pores, equation 4.0.11 reduces to

D = D,~..4t3 e 't'a a = cp {4.0.12)

So with a typical value for dry soil of 4>a = 0.4, De= 0.3D. An alternative expression, applied in the case of water- or air-saturated pores, uses

an empirical correction factor, the tortuosity, T, which accounts for (a) the reduced area through which diffusion can occur and (b) the longer average path a molecule must travel to move a certain distance through the pores. The effective diffusivity is related to the tortuosity by

D =D e -T

(4.0.13)

Typical values of tortuosity are in the range 2 to 6 (Cussler~ 1984), yielding effec-tive diffusivities that are approximately consistent with predictions based on 4.0.12.

Hydrodynamic Dispersion

Hydrodynamic dispersion in porous media is similar to the shear-flow dispersion that occurs in pipes and rivers, as discussed in 4.A.4. The dispersion of a nonreactive groundwater contaminant, or conserved tracer, is depicted in Figure 4.D.4. At some time .t 1, contamination occurs in the groundwater, as shown in th; left-hand porilun of the figure. The groundwater flows with a uniform filtration velocity, U, in the +x-direction. At some later time, t2, the center of mass of contamination has been trans-ported downstream to position x2, given by

Xz = x 1 + (t2 - t1 )U (4.0.14) In words, the center of mass of contamination moves at the same speed as the fluid. However, the areal extent of contaminated groundwater increases, and typically this increase is much greater than can be explained by molecular diffusion alone. Because

J Fif no; Sht tin is; ve1

to tra

tht tra m( m< an

ch

nu

cu

fie

tht ch av

w tht tht

in sh

wl thL

Ht: lei Sc

di~ fir an

dh th<

'ectorioim:~ (4.0.10)

quantity of U"ameter De n the pores mas Pick's . an altered

ler than the d combina-{pression is

(4.D.ll)

r-filled and 1e effective s, equation

(4.D.l2)

pores, uses he reduced a molecule iffusivity is

(4.D.l3)

ding effec-n4.D.12.

Jersion that 1onreactive 4. At some md portion , in the +x-been trans-

(4.0.14) :s the fluid. pically this te. Because

4D. Tmnsport in Porous Media 199

y

Xt

u ----..

'~ x3

Figure 4.D.4 Transport and dispersion of a fixed quantity of a nonreactive groundwater contaminant. The figure schematically shows the areal extent of contamination at three instants in time, with t3 > t2 > t1 The center of mass of contamination is advected in the x-direction at a rate given by the mean velocity, U. Hydrodynamic dispersion causes the contamination to spread more rapidly in the direction of flow than in the transverse direction.

the local velocity through the soil pores is not unif~ some of the contamination travels at an average speed greater than if~ while other contaminant molecules travel more slowly than U. There is also some spreading in the y-direction, even though the mean velocity in that direction is zero. Since the contaminant is conserved, the greater areal extent of contamination is offset by a lower average concentration. The same characteri