Environmental Fluid Mechanics - Chapter 10 doc

10 Environmental Transport Processes 10.1 INTRODUCTION The field of environmental fluid mechanics spans a broad range of topics. Though based primarily on fluid mechanics and hydraulics concepts, as descri- bed in Part 1 of this text, the area has grown in the past few decades to encompass many applications in water quality modeling. A major connection between this latter field and pure fluid mechanics lies in the determination of terms needed to specify the transport and mixing rates for a given parameter of interest. This will be seen in the present chapter, in which the classic advec- tion–diffusion equation is derived to express a mass balance statement for a dissolved chemical species distributed in a fluid flow field. In later chapters this idea is expanded to include transport of suspended sediment particles, and several important classes of environmental flows are discussed. Typical parameters of interest might include concentrations of dissolved gases (partic- ularly oxygen), nutrients such as phosphorus or nitrogen, various chemical contaminants, both organic and inorganic, salinity, suspended solids, temper- ature, biological species, and others. In order to fully describe the fate and transport of a particular species, a knowledge of specific source and sink terms, including interactions with other species, must be incorporated in the general conservation equation. The present text, with several exceptions, generally does not deal directly with these terms, but rather concentrates on the physical transport mechanisms. 10.1.1 Water, Heat, and Solute Transport Earlier we saw that certain quantities control the rate at which different prop- erties of a flow are transported, either by mean motions or by diffusion. For example, kinematic viscosity may be thought of as a molecular diffusivity for momentum. Thermal diffusivity represents a similar transport term for heat energy, and solute diffusivity represents a corresponding transport mechanism Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. for a dissolved species. Mean motions generally carry all properties of a flow at the same rate, but molecular diffusivities can vary widely. For instance, kine- matic viscosity of water is about 10 2 cm 2 /s, thermal diffusivity is of order 10 3 cm 2 /s, and salt diffusivity is of order 10 5 cm 2 /s. These differences are associated with molecular activity, as shown below, and can be related by values of the Prandtl and Schmidt numbers, Pr D /k  Sc D /k 10.1.1 where  is kinematic viscosity, k  is thermal diffusivity, and k is molec- ular diffusivity of a dissolved material. Given the above values, Pr for water is around 10 and Sc is around 10 3 . However, when turbulent diffusion is considered, it is normally assumed that the eddies responsible for transporting the properties of the flow are effective in transporting all properties at about the same rate (this is generally referred to as the Reynolds analogy —see Chap. 5). Of course, the net transport depends on mean gradients, as described previously, but the diffusion coefficients or diffusivities are the same. This implies that the turbulent Prandtl and Schmidt numbers (defined similar to Eqn. 10.1.1, but using turbulent or eddy diffusivities) are both around 1, which is a basic result of the Reynolds analogy. Another transport term of interest is that of dispersion. Many authors have used the terms diffusion and dispersion interchangeably, since the net results of these processes are similar in causing spreading or mixing of material fluid properties. In fact, both terms are often represented mathematically in the same way in the conservation equations. However, dispersion arises from a completely different process than diffusion, as described hereinafter. The effect of dispersion on transport of different properties of a flow is similar to that of turbulent diffusion, in the sense that dispersion causes mixing of a fluid property about a mean position, and dispersion coefficients for momentum, heat and mass all tend to be similar. Various transport processes of interest may be summarized as follows: Advection. These motions are associated with mean flow or currents, such as rivers, streams, or tidal motions. They are normally driven by gravity or pressure forces and are usually thought of as primarily horizontal motions. Convection. This term usually refers to vertical motions induced by hydrostatic instability, i.e., they are buoyancy driven. Examples of this type of motion include heating a pot of water on a stove, or fall and spring lake overturns occurring when the surface temperature on a lake passes through 4 ° C (temperature of maximum density). Molecular diffusion. Molecules of a fluid are naturally in random motion, relative to other molecules (Brownian motion), and this leads to a mixing or spreading of fluid properties, consistent with the second law of thermody- namics. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Turbulent diffusion. This is a type of mixing similar to molecular diffu- sion but with a much stronger effect. Mixing in this case derives from the larger scale movement of packets of fluid (rather than individual molecules) by turbulent eddies. Shear. Shear exists when there is a variation of advection (mean flow velocity) at different locations in a flow field, so that a gradient exists for flow velocity. This produces a variation in the rate of advective transport of a fluid property, with associated spreading of the average concentration of that property, as illustrated in Fig. 10.1. Dispersion. This is spreading of a fluid property by the combined effects of shear and transverse diffusion. The main distinction between advection (or convection) and diffusion or dispersion is that advection represents a net movement of the center of mass of a packet of fluid or fluid property, while diffusion and dispersion represent a spreading about the center of mass. This is illustrated in Fig. 10.2. As will be seen later, dispersion is normally included in the conservation equations in a similar manner as diffusion, but the effects of dispersion are normally much greater than those of diffusion. In this text the term mixing will refer to either diffusion or dispersion. Figure 10.1 Effect of shear on spreading of mean (here, depth-averaged) concentration. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 10.2 Illustration of advective and diffusive transport; advection moves the center of mass while diffusion spreads mass relative to the center, whether it is moving or not. 10.2 BASIC DEFINITIONS, ADVECTIVE TRANSPORT The basic balance equation for dissolved mass may be derived following procedures similar to those used in the derivations for water mass, momentum, and energy balances from Chap. 2. However, it will first be important to define the various transport properties involved, especially diffusion, and this will lead to a derivation of the classic advection–diffusion equation. It will be important to keep in mind the definition of a flux,whichis the transport of a given property across a surface, per unit time and per unit area of that surface, and the definition of mass concentration, or density, C or  1 (see below), which is the amount of mass of dissolved solids per unit volume of fluid. In the following discussion, it is also helpful to define a pure concentration, C Ł , which is a dimensionless ratio of dissolved solid mass in a given volume of fluid to total mass in that volume. C is related to C Ł by C D C Ł 10.2.1 where  is the total density of the solution. C Ł also is referred to as mass fraction or relative concentration. Concentrations are often listed in units of ppm (parts per million), referring to C Ł , or as mg/L (milligrams per liter), referring to C or  1 . In aqueous systems, the numerical values for both C Ł 1 and  1 turn out to be the same, since the density of water is typically  2 ¾ D 1gm/cm 3 D 10 6 mg/L. Thus for chemical concentrations on the order of 1 mg/L, both C Ł 1 and  1 would have values of about 10 6 (1 mg/L or 1 ppm). In addition, we will be concerned primarily with binary systems, i.e., water plus one other component (the extension to conditions with more than one additional component is straightforward). Thus let  1 D mass of species Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. 1 per unit volume of solution and  2 D mass of species 2 per unit volume of solution. The total density of the solution is  D  1 C  2 . Normally there is a relatively small amount of one species compared with the other, such as with a pollutant dissolved in water. If species 2 is water, then  2 ×  1 and  ¾ D  2 . In other words, the addition of the pollutant, or tracer, does not significantly affect the water (solution) density. The (nondimensional) concentration of species 1 is C Ł 1 D  1  10.2.2 and in binary systems, C Ł 1 C C Ł 2 D 1. The advective flux of species 1, for a velocity field  V D u, v,w is defined as  A D  1  V10.2.3 Thus the advective fluxes in each Cartesian coordinate direction are A x D u 1 , A y D  1 ,andA z D w 1 . 10.3 DIFFUSION 10.3.1 Molecular Diffusion, Fick’s Law The basic form of diffusion is molecular diffusion, which is due to the random motions all molecules undergo (Brownian motion). From considerations of nonequilibrium thermodynamics, the simplest assumption about diffusion of mass that is consistent with the second law of thermodynamics (increasing entropy), is that this diffusion is proportional to the gradient of the chemical potential of the system ( c ). This statement is analogous to Fourier’s law for heat conduction and can be written as  F 1 Dk  r c 10.3.1 where  F 1 is the diffusive flux of species 1, k is a constant, and the negative sign indicates the flux is in the opposite direction to the gradient of  c .Now,  c is generally a function of system properties, mostly temperature (Â), mass density ( 1 ), and pressure (p). Then (in one dimension for simplicity), F 1 x Dk  ∂ c ∂ ∂ ∂x  C  ∂ c ∂ 1 ∂ 1 ∂x  C  ∂ c ∂p ∂p ∂x  10.3.2 The main contribution to the flux is the middle term on the right-hand side. The first term is called the Soret effect and indicates the possible diffusion of mass due to a temperature gradient. This term may become important Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. under some circumstances, usually involving high salinity concentrations, but is ignored in most applications. Not much is known about the third term, but it is assumed to be negligible. Considering only the density gradient term, then, F 1 x D  k ∂ c ∂ 1  ∂ 1 ∂x Dk 1 ∂ 1 ∂x 10.3.3 where k 1 is defined as the molecular diffusivity for species 1. This result, extended to all three directions, becomes  F 1 Dk 1  r 1 10.3.4 which is known as Fick’s law or Fickean diffusion. The concept of diffusivity can also be derived using mixing length theory, illustrated in Fig. 10.3 for a simple one-dimensional stratification (z direction) of species 1. On the left-hand plot is shown a general density profile for species 1, while the right-hand plot is a magnified view of a small part of the profile. The average velocity of molecules in the z-direction, due to inherent Brownian motion, is w 1 ,andl m is the mixing length, which is assumed to be related to the average distance the molecules travel before colliding with other molecules (molecular free path). Considering a small “window” in the fluid, perpendicular to the z axis, the flux of species 1 through the window in the positive z-direction is  1 w 1  1 , and in the negative z-direction it is  1 w 1  2 , where the subscripts outside the parentheses indicate the levels at which the flux terms are evaluated. If w 1 is assumed constant, the net flux across this window is F z 1 D [ 1  1   1  2 ]w 1 10.3.5 For small l m , the density gradient is approximately constant, so  1  1   1  2 ¾ D l m ∂ 1 ∂z 10.3.6 Figure 10.3 Mixing length concept. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. where the minus sign is used since (∂ 1 /∂z) is negative. Substituting Eq. (10.3.6) into Eq. (10.3.5), F z 1 Dw 1 l m ∂ 1 ∂z 10.3.7 The product of mean molecular velocity and mean free path (w 1 l m ) is a property of the system and is given the symbol k 1 and defined as the diffusivity for species 1, similar to the term in Eq. (10.3.3). Similar analyses may be applied for the x and y directions, to arrive at the same result as before (Eq. 10.3.4). In general, k 1 is a function of temperature and density ( 1 ), and possibly pressure. It is interesting to note that k 1 is defined here as the product of a characteristic length and a characteristic velocity scale — this idea also is applied when discussing turbulent diffusion, though of course the velocity and length scales are different. 10.3.2 Turbulent Diffusion The concept of turbulent diffusive transport is analogous to the turbulent trans- port of momentum, discussed in Chap. 5. In particular, the Reynolds transport terms are derived in exactly the same way as the Reynolds transport terms for momentum, (i.e., Eqs. 5.4.7 and 5.4.8). Here, however, fluctuations of concentration are used instead of a second velocity component. In other words, the turbulent or Reynolds transport of dissolved mass in the i direction is  u 0 i c 0 D E i ∂C Ł ∂x i 10.3.8 where, as before, the overbar indicates a time-averaged quantity and the primes denote fluctuating terms. This equation serves to define the turbulent diffu- sivity, E i , which in general is anisotropic and inhomogeneous. That is, in a general sense, turbulent diffusivity may depend on orientation and on location (as will be seen later, it is a function of stratification, for instance), so that E x , E y ,andE z could all have different values (anisotropic) and they might all be functions of (x, y, z) (inhomogeneous). Often turbulent diffusivities are not very well known and must be estimated, unless direct measurements of the terms in Eq. (10.3.8) are available. Instrumentation must be capable of measuring the fluctuating quantities making up the Reynolds transport term (left-hand side), and mean concentration gradient must be measured. In many cases, the turbulent diffusivities are treated as fitting parameters, chosen to optimize the results of a particular model, compared to observations. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. 10.3.3 Statistical Theory of Diffusion One example of the use of observations to estimate diffusive-type spreading characteristics is illustrated by the use of dye release experiments in a natural system. This approach is based on the observation that average dye concen- tration distributions are often closely approximated by Gaussian profiles (i.e., they are normally distributed). Figure 10.4 shows a conceptual sketch of the outer extent of spreading of dye in the surface layer of a lake, following a concentrated “instantaneous” release over a small area. The dye patch is shown at three different times after this release. For simplicity, mean velocity isassumedtobeinthex-direction. Figure 10.5 shows corresponding concen- tration profiles measured across the patch at one of these times. The ensemble average profile is approximately normally distributed. At earlier times it would be more peaked, and at later times it would be more flattened. In other words, the variance of the distribution increases with time. Assuming the Gaussian distribution is appropriate, the concentration can be described in terms of the variances in each of the three coordinate Figure 10.4 Spreading of a dye patch at three times: t 3 >t 2 >t 1 . Figure 10.5 Measurements of concentration across the patch at one time, with en- semble average. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. directions as C Ł D Me Kt 2 3/2  x  y  z exp   x  Ut 2 2 2 x  y 2 2 2 y  z 2 2 2 z  10.3.9 where M is the mass of dye released,  x ,  y ,and z are the standard deviations (square roots of variances) in each of the coordinate directions,  is density, K is a first-order decay rate (see Sec. 10.4), and t is time following the dye release. The exponential term with K incorporates any loss of the original mass M due to physical, biological, or chemical processes. In general, the mass remaining in the patch at any time t is given by Me Kt  D   C Ł dx dy dz 10.3.10 The variances are measures of the degree of spreading along each of the coordinate directions. For example, variance in the x-direction is calculated from  2 x D  Me Kt  x Ut 2 cdxdydz 10.3.11 and similar expressions may be defined for the y and z directions. It will be shown later how the variances are related to the turbulent diffusivities. 10.4 THE ADVECTION–DIFFUSION EQUATION Having developed the basic transport terms, we are now in position to write a conservation equation for dissolved mass of a tracer in an aqueous system. First, a total flux of dissolved mass of species 1 is defined as the sum of the advective (Eq. 10.2.2) and diffusive terms (Eq. 10.3.4),  N 1 D  1  V C  F 1 D  1  V  k 1  r 1 10.4.1 where  N 1 is the total flux of mass of species 1. For now, a molecular diffu- sion term is used (recall that k 1 is molecular diffusivity), though the following development holds equally well for turbulent diffusion, by substituting a turbu- lent diffusivity for k 1 . Sometimes it is convenient to define a representative velocity for species 1,  q 1 , such that the total flux may be written as  N 1 D  1  q 1 10.4.2 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Although this velocity cannot be measured directly, it is defined by equating Eqs. (10.4.1) and (10.4.2). If  q 2 is a representative velocity for species 2, then the total momentum per unit volume is   q D  1  q 1 C  2  q 2 10.4.3 where  q is the bulk velocity, or total momentum per unit mass,  q D  1  q 1 C  2  q 2  D  N 1 C  N 2  D  V10.4.4 Note that this result indicates that diffusion does not contribute to the total momentum (both Eqs. (10.4.4) and (10.4.1) can be satisfied together only when  F 1 D  F 2 D  0 ), which is consistent with the idea that diffusion causes spreading about the center of mass and not net transport of the center of mass. The mass balance for species 1 may be formulated in several ways, but the most direct way is to consider a differential fluid element and incorporate the transport of mass of species 1 across the boundaries of the element. The approach is similar to the development leading to the continuity equation (conservation of fluid mass — see Chap. 2), except diffusive flux must also be included here. A general statement of dissolved mass conservation for the fluid element (volume 8) sketched in Fig. 10.6 is [rate of change of mass in 8 per unit time] D [rate at which mass moves across the boundaries by net flux] š [rate at which mass is produced C or consumed  by chemical and biological reactions] For simplicity, only the fluxes in the x-direction are shown in Fig. 10.6, but similar fluxes may be defined for the y-andz-directions. Using a truncated Figure 10.6 Fluid element, showing fluxes of species 1 in the x-direction. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. [...]... t1 ) CD q0 exp xU/2Ex 4 Ex Ey 1/2 U2 t/4Ex CKt 1 exp 1 0 1 2 ˇ2 d 1 10. 7.21 1 where U2 C K t t1 4Ex [ Ey x 2 C Ex y 2 U2 Ey C 4Ex Ey K ]1/2 ˇ2 D 4Ex Ey 1 D 10. 7.22 10. 7.23 and q0 is the injection rate (per unit length) The steady-state solution is CD q0 exp xU/2Ex Äo 2ˇ2 2 Ex Ey 1/2 10. 7.24 where Ä0 Á ¾ D 10. 7.3 1/2 2Á e Á for Á > 1 10. 7.25 Plane Source Instantaneous Plane Source Next we consider an... kt 10. 5.6 where M is the original mass of dye injected Taylor showed that this solution was a good approximation to the exact solution, using EL D 10. 1RuŁ 10. 5.7 This value may be compared with the radial or longitudinal diffusivities, estimated from 10. 5.8 Er ¾ Ex ¾ 0.07RuŁ D D which are several orders of magnitude smaller The different effects of diffusion and dispersion are illustrated in Fig 10. 10,... ∂t ∂x ∂ 00 00 u c ∂x 10. 5.16 This last expression may be written using a dispersion term, by defining the dispersion coefficient as EL D u00 c00 ∂C/∂x 10. 5.17 so that ∂C ∂2 C ∂C CU D EL 2 ∂t ∂x ∂x 10. 5.18 This result is the (one-dimensional) advection–dispersion equation for this flow The equation is mathematically much simpler to solve than the original two-dimensional equation (10. 5.12) and provides... 1 CŁ dx 10. 4.16 Substituting for CŁ from Eq (10. 4.15), MD A D 2 AB C1 1 p x2 4kt B p exp t p B dx D A p 2 kt t 10. 4.17 k Note that M is independent of time, as it should be for a conservative dye Rearranging this last result to solve for B, and substituting into Eq (10. 4.15), the final solution is CD M p exp 2 A kt x2 4kt Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 10. 4.18 Figure 10. 8 Concentration... concentration of the chemical species of interest The most common reaction terms are either zero order or first order A zero-order source/sink term does not depend on concentration at all but would be a constant (possibly time-varying) term added to the right-hand side of Eq (10. 4 .10) An example of this type of reaction is a municipal waste stream discharging into a river The loading of a contaminant... Rights Reserved Figure 10. 10 Effect of velocity shear on the spreading of dye, with a longitudinal dispersion approach: (a) spreading due to diffusion only; (b) spreading due to diffusion and velocity shear; and (c) resulting cross-sectional average concentration profiles, according to the dispersion model For purposes of demonstration, the dye is assumed to be conservative In Fig 10. 10a a uniform velocity... velocity U Figure 10. 10b shows a more realistic situation that accounts for the no-slip Copyright 2001 by Marcel Dekker, Inc All Rights Reserved conditions at the tube wall, although the mean velocity is unchanged The variations in velocity cause a stretching of the dye distribution, effectively spreading the concentration more quickly than by diffusion alone, as shown in Fig 10. 10c In this case,... in Fig 10. 11, where arbitrary distributions are assumed The important point to note is the decomposition of both velocity u and concentration c into mean and fluctuating Figure 10. 11 Two-dimensional velocity and concentration profiles, with depth-averaged values and spatial deviations shown Copyright 2001 by Marcel Dekker, Inc All Rights Reserved components, i.e., u D U C u00 z c D C C c00 z 10. 5.9... ∂t ∂x ∂x Ex ∂c ∂x C ∂ ∂z Ez ∂c ∂z 10. 5.12 where Ex and Ez are diffusivities in the x and z directions, respectively Following the same approach as in Taylor’s analysis of dispersion in a tube, the diffusive transport in the x-direction is assumed to be negligible Substituting Eq (10. 5.9) into Eq (10. 5.12), ∂ ∂ ∂ ∂ C C c00 C U C u00 C C c00 ¾ C C c00 Ez D ∂t ∂x ∂z ∂z 10. 5.13 Again following a procedure... similar to Reynolds averaging, these terms are depth-averaged to obtain ∂C ∂c00 ∂C 1 CU C u00 D ∂t ∂x ∂x h Ez ∂C ∂z h 10. 5.14 0 where the double overbar indicates a spatial average and Eq (10. 5.11) has been used to set the average of any single fluctuating term to zero The Copyright 2001 by Marcel Dekker, Inc All Rights Reserved right-hand side (RHS) of Eq (10. 5.14) is the difference between the vertical . in the x-direction are shown in Fig. 10. 6, but similar fluxes may be defined for the y-andz-directions. Using a truncated Figure 10. 6 Fluid element, showing fluxes of species 1 in the x-direction. Copyright. 10 Environmental Transport Processes 10. 1 INTRODUCTION The field of environmental fluid mechanics spans a broad range of topics. Though based primarily on fluid mechanics and hydraulics. first order. A zero-order source/sink term does not depend on concentration at all but would be a constant (possibly time-varying) term added to the right-hand side of Eq. (10. 4 .10) . An example