Environmental Fluid Mechanics - Chapter 12 ppt

12 Exchange Processes at the Air/Water Interface 12.1 INTRODUCTION One of the major boundaries in most natural surface water bodies is the inter- face with the atmosphere. In order to describe the distribution of various properties of the water body it is necessary to specify boundary conditions for the transport equations for those properties at this (as well as other) boundaries. The usual kinds of boundary conditions are applicable here, i.e., specifica- tion of either the values for the property of interest or its gradients. For the air/water interface it is more common to prescribe fluxes or transport rates for the property of interest. A significant feature of this transport is the wind, which strongly affects property fluxes. Because of limited fetch, wind effects in open channel flows are less significant than in lakes and reservoirs, which have much larger surface areas. In this chapter we consider transport of momentum, heat, gases, and volatile organic chemicals across the air/water interface. 12.2 MOMENTUM TRANSPORT The main driving force for momentum is shear stress exerted as a result of velocity gradients across the air/water interface. The main effects are genera- tion of surface drift currents, waves, setup and seiche motions, as illustrated in Fig. 12.1. In lakes and reservoirs wind is a primary driving force for general circulation. Shear stress at the air/water interface is exerted as a result of differences between the wind speed and direction, and the water surface velocity. Part of this stress works to develop the wave field and part is used for generation of surface drift currents. In the present section the focus is on generation of drift currents and circulation — see Chap. 8 for a discussion of surface water Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 12.1 Illustration of setup and circulation (side view, top) and horizontal circu- lation (plan view, bottom) generated by wind over a lake. waves. For the present discussion, it is assumed that a steady wind is blowing over a water surface and that a fully developed wave field is present, i.e., wind/wave interactions are in equilibrium and there is no further partitioning of the surface stress into wave development. Figure 12.2 illustrates a possible velocity profile for wind and water, where W z is the wind speed measured at position z above the water surface and u d is the surface drift velocity in the water. The mean surface level is at z D 0, and z increases upwards (it is convenient to work with z increasing upwards when describing the wind velocity profile; however, when the main concern is with properties of the water body, it may be more convenient to work with z increasing downwards from the surface — see the following section, for example). The shear stress at the surface is given by  s D c z  a W z  u d  2 ¾ D c z  a W 2 z 12.2.1 where c z is a drag coefficient and  a is air density. The approximation of the second part of this equation results from the assumption that u d /W z − 1. The Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Figure 12.2 Velocity profiles for wind over a water surface. shear stress also is defined in terms of a friction velocity,  s D  a u 2 Ła 12.2.2 where u Ła is the friction velocity for the wind profile. It is commonly assumed that the wind velocity distribution follows a boundary layer logarithmic profile, W z D u Ła Ä ln  z z 0  12.2.3 where Ä is the von Karman constant, equal to about 0.4, and z 0 is the virtual origin of the profile (note that W z ! 0forz ! z 0 ). In order for this assump- tion to be valid, the measurement height (z) must be chosen so that the logarithmic profile is valid. According to Wu (1971), this is satisfied when z D    10 cm, Re < 5 ð10 7 7.35 Re 2/3 ð 10 5 cm, 5 ð 10 7 < Re < 5 ð10 10 10 m, Re > 5 ð 10 10 12.24 where Re D W z L/ is a fetch Reynolds number, with L D fetch D distance over which wind has blown over the surface of the water and  D kinematic viscosity. When Eq. (12.1.4) is satisfied, z is greater than the significant wave amplitude and less than about four-tenths of the total boundary Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. layer thickness, so it is well within the range where the logarithmic profile should be valid. The magnitude of z 0 depends on the roughness of the surface and is assumed to be a linear function of roughness, similar to a turbulent boundary layer in pipe or open channel flow. In a fully developed wave field, the dynamic roughness length scale is estimated by (u 2 Ła /g), so z 0 D ˛ c u 2 Ła g 12.2.5 where ˛ c is known as the Charnock coefficient and has a value between 0.011 and 0.035, with most reported values falling in the range 0.011 to 0.016. Upon substituting Eqs. (12.2.1), (12.2.2), and (12.2.5) into (12.2.3), an expression for c z is obtained, 1 p c z D 1 Ä ln  gz ˛ c c z W 2 z  12.2.6 which must be solved iteratively since c z is found in different terms on both sides of the equation. Normally, c z has a value on the order of 10 3 .Oncec z is found, the surface shear stress is calculated from Eq. (12.2.1). For drift current, again assuming a fully developed wave field, it is assumed that the surface shear stress in the water is the same as that in the air,  0 ¾ D  s ,where  0 D  w u 2 Ł 12.2.7 and  w is water density at the surface. (Note that  0 < s when the wave field is not fully developed, since part of the surface shear is used in developing the waves.) By combining Eqs. (12.2.2) and (12.2.7), we find u Ł D u Ła   a  w 12.2.8 Then, assuming that the general shapes of the velocity profiles in air and water are similar (see Fig. 12.2, where the profile in water is inverted and reversed but has the same general shape), it follows that the drag coefficient should be the same on both sides of the air/water interface. This gives  0 D c z  w u 2 d D c z  a W 2 z 12.2.9 from which it is found that u d D W z   a  w 12.2.10 The ratio of air to water density is approximately 10 3 ,sothislast result implies that surface drift velocity is about 3% of wind speed, which Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. may be used as a rough rule of thumb. Also, if c z is estimated as being about 10 3 , then from Eqs. (12.2.1) and (12.2.2), uŁ a ¾ D 0.03W z , and, combined with Eqs. (12.2.8) and (12.2.10), we find uŁ ¾ D 0.03u d . This gives an alternative relation that may be used to estimate the velocities, at least for the conditions of steady wind, long fetch, and fully developed wave field. A more realistic relationship for lakes and reservoirs, with limited fetch, is u Ł /u d  ¾ D 0.1–1. Finally, the kinetic energy flux across the surface is KE F D u d  s D C w u 3 Ł 12.2.11 where the coefficient C incorporates various assumptions noted above, including the relationship between u d and uŁ. This last expression is needed to calculate possible mixing induced by wind (see Sec. 13.5). 12.2.1 Seiches When fetch is limited, the possibility of wind-generated seiche motions must be considered. These are in essence wavelike motions with a half-wave length given by the fetch L. Seiche motions are primarily of interest when winds are relatively constant in speed and direction over a long period of time, as sketched in Fig. 12.3. The surface shear stress exerted by the wind causes the water surface to tilt, establishing a pressure gradient to balance this stress. The tilted water surface position is given by Áx, and the difference between the tilted surface and the horizontal equilibrium position is referred to as the wind setup. If the lake is large enough that the dynamics are affected by the Earth’s rotation, then the position of maximum setup moves in the counterclockwise direction (northern hemisphere). For the present discussion we neglect this effect. If the wind suddenly stops, the tilted water surface moves back towards a level condition. As the water reaches this condition, however, it still has momentum and overshoots, resulting in a setup on the Figure 12.3 Wind-generated seiche motion. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. opposite side of the lake. This motion is eventually arrested as the kinetic energy of the moving water is transferred to the potential energy of the setup. The flow then reverses and the surface “rocks” back towards the original setup condition. This oscillatory motion continues until eventually viscous effects cause it to die out. Depending on the lake geometry and relative wind direction, seiche motions can be quite substantial. The frequency of the oscillations is known as the natural or inertial frequency of the lake. The tilted water surface profile is found by considering a force balance on a small control volume, as shown in Fig. 12.4. Due to circulation of water resulting from the surface shear, there is some motion along the bottom, which is estimated to generate an additional shear stress approximately equal to 10% of  s . Under steady-state conditions, the momentum fluxes into and out of the control volume are equal, and a force balance in the horizontal direction (per unit width) gives 1.1 s dx D 1 2    h C ∂h ∂x dx  2  h 2  12.2.12 where hydrostatic pressure is assumed for the two sides, h is depth, and  D g is specific gravity. In general, the wind shear acts at an angle to the horizontal, once the water surface tilts. However, this angle is very small, and its cosine is assumed to be approximately 1. Also neglecting the second-order Figure 12.4 Control volume for analysis of water surface profile. Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. term in dx, the term in brackets in Eq. (12.2.12) is simplified as ∂h ∂x D ∂Á ∂x D 1.1 s h 12.2.13 If  s is assumed constant, or at least is known as a function of x, this equation can be integrated, along with a boundary condition such as h0[orÁ0], to calculate the water surface profile Áx. A simple approximation for the water surface is a linear profile, using mean water depth H: Á ¾ D 1.1 s H  x  L 2  12.2.14 12.3 SOLAR RADIATION AND SURFACE HEAT TRANSFER 12.3.1 Temperature Equation Before discussing surface heat transfer, it is helpful to review formally the derivation of the temperature equation. This is directly related to the thermal energy equation, which was introduced in Sec. 2.9.3. The temperature equation is derived from the general conservation of energy statement, Eq. (2.8.7), which is repeated here for convenience: dQ dt  dW s dt D ∂ ∂t  U e dU C  S  V 2 2 C gz C u C p     V Ð  ndS 12.3.1 where Q D heat added, W D work done by the fluid on its surroundings, e D energy per unit mass, the first integral on the right-hand side is the rate of change of energy in the control volume, and the second integral is the net rate at which energy is transported across the control surface. Note that thermo- dynamic convention is followed here in writing the work term as a positive quantity when the fluid does work on its surroundings. The heat added can be represented as a surface integral of heat flux,  ϕ (energy transport per unit time and per unit area), which in turn is the sum of radiation   ϕ r  and conduction   ϕ c  terms, dQ dt D  S  ϕ Ð  ndSD  S   ϕ r C  ϕ c  Ð  ndS D  8 rÐ  ϕ r C  ϕ c d8 12.3.2 where S indicates the control surface, dS is an elemental area of the control surface,  n is a unit normal vector pointing out of the control volume (refer to Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. Chap. 2 for further discussion), and the divergence theorem has been used to write the surface integral as a volume integral in the last part of Eq. (12.3.2). The negative sign is added since the flux, by convention, is defined as being positive when directed outwards from the control volume. The work rate term is considered to consist of gravity work and surface work. The gravity work rate is given by a volume integral,  8   g Ð  Vd8 and the surface work rate, from Eq. (2.8.5), is  S p  V Ð  ndS  S Q Ð  n Ð  VdSD  S Q S Ð  n Ð  VdS where Q is the deviatoric stress tensor and Q S is the full stress tensor, as intro- duced in Chap. 2. Rewriting the surface integrals as a volume integral, again using the divergence theorem, the total work rate is  dW dt D  8    g Ð  V CrÐ Q S Ð  V  d8 D  8    g CrÐ Q S Ð  V C  Q S Ðr Ð  V  d8 12.3.3 The first term in the last expression in Eq. (12.3.3) may be rewritten using the momentum equation (2.7.1),   g CrÐ Q S D  D  V Dt 12.3.4 Then, noting also that D  V Dt Ð  V D D Dt  V 2 2  12.3.5 we substitute Eqs. (12.3.2) and (12.3.3) into Eq. (12.3.1), rewriting the surface integral in Eq. (12.3.1) as a volume integral, and combine all volume integrals to obtain ∂ ∂t    V 2 2 C u  DrÐ  ϕ C  D Dt  V 2 2  C  Q S Ðr Ð  V rÐ    V 2 2 C u   V  12.3.6 where u is internal energy, from Eq. (2.8.4). Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. This last result is further simplified using the continuity equation (2.5.6) and noting that the terms in (V 2 /2) cancel, so that  Du Dt DrÐ  ϕ C  Q S Ðr Ð  V12.3.7 The left-hand side of Eq. (12.3.7) is the time rate of change in internal energy of a fluid element, and the right-hand side expresses the heat flux and work done by stresses. Usually the contribution of stresses is negligible in affecting temperature, as shown by the scaling arguments for viscous heating discussed in Sec. 2.9.4. Internal energy in liquids is assumed to be a function of temper- ature only, i.e., u D uT,whereT D temperature, and from the definition of specific heat (Eq. 2.8.9), a change in energy is related to a change in temperature by du D cdT 12.3.8 where c D c p D c v since in liquids the specific heats for constant pressure and constant volume are nearly equal. Again considering the heat flux term to consist of conduction and radiation terms, the conduction term is expressed using Fourier’s law of heat conduction, which states that the conductive heat flux is proportional to the temperature gradient,  ϕ c DÄ T rT12.3.9 where Ä T is thermal conductivity. Making the foregoing simplifications and substitutions, Eq. (12.3.7) becomes c DT Dt D c  ∂T ∂t C  V ÐrT  DrÐÄ T rT rÐ  ϕ r 12.3.10 where constant c has been assumed. This is the temperature equation,which has the general form of an advection–diffusion equation expressing conserva- tion of thermal energy, with a main source term due to radiative heat flux. If Ä T is constant, this equation becomes ∂T ∂t C  V Ð  rT D k T r 2 T  1 c rÐ  ϕ r 12.3.11 where k T D Ä T /c is thermal diffusivity. This result is similar to Eq. (2.9.28), but without the viscous or compression heating terms, which have been neglected here. In order to solve Eq. (12.3.11), initial and boundary conditions are needed, and possible source terms for radiation must be specified. For natural water bodies, the primary considerations are the heat transfer rate at the Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. air/water interface (which is normally specified as a boundary condition) and the solar heating rate due to absorption of solar radiation (an internal source term). We examine this latter term first. 12.3.2 Solar Radiation Absorption Solar radiation consists of a large range of wavelengths, that have different absorption properties in water. For light of a given frequency, Beer’s law states that the radiation intensity decreases exponentially with depth (z is assumed positive downwards in the following):  0 s ω, z D  sn ωe Á s ω,zz 12.3.12 where  0 s ω, z is the solar radiation intensity at depth z for frequency ω,  sn ω is the net (after reflection) radiation intensity at the water surface for frequency ω,andÁ s ω, z is the extinction, or absorption coefficient, as a function of ω and z. The total radiation intensity at any depth z is then the sum of  0 s ω, z over all ω. However, in practice insufficient data are available to evaluate such an integral. A simpler approach has been found to give adequate results, in which the range of light frequencies is divided into one or more subranges, with surface radiation intensity and extinction coefficient values defined for each subrange, rather than for each individual frequency. The total is then the sum over all subranges,  0 s z D n  iD1  i  sn e Á si z 12.3.13 where n is the number of subranges. Longer wave radiation (infrared range) tends to be absorbed strongly in water, relative to shorter wave radiation, and has correspondingly higher values for Á s . Thus when a small number of terms is used in the summation of Eq. (12.3.13), an adjustment must be made to account for this stronger absorption near the surface and to provide a better fit for the exponential model. This is usually done by defining a fraction, ˇ s , of the surface radiation as that portion of the radiation intensity with relatively high extinction coef- ficients, absorbing nearly completely within a shallow depth near the surface. In general, different values of ˇ s should be defined for each of the subranges used in Eq. (12.3.13), but a simple common approach is to use a single term in the summation. In that case, a single value is needed, as well as a single value for the extinction coefficient (Á s ), to calculate the exponential decay of the remaining fraction of radiation that is not absorbed near the surface. The resulting equation for a one-term model (n D 1) is  0 s z  so D 1 ˇ s e Á s z 12.3.14 Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved. [...]... Bowen ratio, Rb D c b Ts es Ta ea 12. 3.37 so that, substituting into Eq (12. 3.35), c D Rb e D cb f W Ts Ta 12. 3.38 Total Heat Flux, Linearized Approach The total net heat flux is the sum of each of the above processes (as illustrated in Fig 12. 6), n D sn C an b e c 12. 3.39 where n is positive when the water body is being heated The first two terms on the right-hand side represent sources of heat... DO is used up, any time-dependent model for DO generally requires solution of a coupled set of equations, one for BOD and one for DO The various source and sink terms in Eq (12. 4.19) are usually evaluated using first-order kinetics For example, if M represents reaeration, we may write (see Eq 12. 4.4) M D K2 Cs 12. 4.20 C where Cs is the saturated concentration, based on the cross-sectional average temperature... Also, the decay of BOD due to oxygen uptake is ∂CBOD D ∂t 12. 4.21 K1 CBOD where CBOD is the cross-sectional average BOD concentration and K1 is the first-order rate constant, which also applies to the rate of depletion of DO The solution to Eq (12. 4.21) is CBOD D CBOD0 e K1 t 12. 4.22 where CBOD 0 is the initial value This is the same as the steady-state solution for CBOD , if dispersion is neglected and... solution for BOD is Eq (12. 4.24) For DO, the governing equation for these conditions is U ∂C D ∂x K1 CBOD C K2 D 12. 4.25 Now, assuming Cs is constant in x, D is substituted for C to obtain U ∂D D ∂x Figure 12. 12 K1 CBOD C K2 D 12. 4.26 Source flow into a river and initial conditions for DO calculation Copyright 2001 by Marcel Dekker, Inc All Rights Reserved Upon substituting Eq (12. 4.24) for CBOD and... Marcel Dekker, Inc All Rights Reserved Figure 12. 16 Mass transfer coefficients (kSF6 ), adjusted to 20° C value, as a function of wind speed at 10 m, from measurements at two well-mixed lakes reported by Upstill-Goddard et al (1990) to derive Eq (12. 4.16), for example Following Upstill-Goddard et al (1990), the appropriate function is kSF6 D kg ScSF6 Scg 1/2 12. 5.5 where the subscript g indicates any other... ac D 1.2 ð 10 13 Ta C 460 6 Btu/ft2 -day 12. 3.18a where ac is the clear-sky value for atmospheric radiation and Ta is air temperature in ° F, measured 2 m above ground level In SI units, ac D 5.35 ð 10 13 Ta C 273 6 W/m2 12. 3.18b where Ta here is in ° C Similar to the formulation for solar radiation, a cloud cover correction is usually added, a D ac 1 C KC2 12. 3.19 where K has a value between 0.04... assumed to be turbulent (Fig 12. 11) Figure 12. 11 Definition sketch for two-film model Copyright 2001 by Marcel Dekker, Inc All Rights Reserved Since the films are laminar, transport is governed by Fickean diffusion acting on the gradients in each film layer (see Sec 10.2) Across the liquid layer, the flux (Eq 12. 4.1) must be equal to the diffusive flux, J D Dm dC ¾ D Dm dz Cs C 12. 4.6 υl where the gradient... layer, J D Kg Cg Csg ¾ Dmg D Cg Csg υg 12. 4.7 where Kg is a bulk mass transfer coefficient for the gas layer, Dmg is the molecular diffusivity for the gas phase, Cg is the gas phase concentration, and υg is the gas film thickness (Fig 12. 11) Using Eqs (12. 4.6) and (12. 4.7), and the definition of Henry’s constant [here it is convenient to use the dimensionless form, Eq (12. 4.3)], Cs and Csg may be eliminated... the water (see also Sec 12. 5) 12. 4.1 Dissolved Oxygen in Open Channel Flow Open channel flow water quality problems are often solved using a oneor two-dimensional framework In the case of DO modeling in rivers, the classic analysis involves variations in the longitudinal direction only, assuming well-mixed conditions at any cross section The general transport equation is the one-dimensional advection–dispersion... 3% of incident, so the net atmospheric radiation is an D 0.97 1.2 ð 10 13 Ta C 460 6 1 C KC2 Btu/ft2 -day Copyright 2001 by Marcel Dekker, Inc All Rights Reserved 12. 3.20a or an D 0.97 5.35 ð 10 13 Ta C 273 6 1 C KC2 W/m2 12. 3.20b where, as before, Ta is in ° F in Eq (12. 3.20a) and in ° C in Eq (12. 3.20b) Back Radiation Back radiation is longwave radiation from the water surface to the atmosphere . also that D  V Dt Ð  V D D Dt  V 2 2  12. 3.5 we substitute Eqs. (12. 3.2) and (12. 3.3) into Eq. (12. 3.1), rewriting the surface integral in Eq. (12. 3.1) as a volume integral, and combine. written as (incor- porating the value for )  ac D 1.2 ð 10 13 T a C 460 6 Btu/ft 2 -day 12. 3.18a where  ac is the clear-sky value for atmospheric radiation and T a is air temper- ature in ° F,. c z is estimated as being about 10 3 , then from Eqs. (12. 2.1) and (12. 2.2), uŁ a ¾ D 0.03W z , and, combined with Eqs. (12. 2.8) and (12. 2.10), we find uŁ ¾ D 0.03u d . This gives an alternative relation