Environmental Fluid Mechanics - Chapter 12 ppt

Shear stress at the air/water interface is exerted as a result of differencesbetween the wind speed and direction, and the water surface velocity.. Figure 12.2 illustrates a possible vel

inter-which strongly affects property fluxes Because of limited fetch, wind effects in

open channel flows are less significant than in lakes and reservoirs, which havemuch larger surface areas In this chapter we consider transport of momentum,heat, gases, and volatile organic chemicals across the air/water interface

The main driving force for momentum is shear stress exerted as a result ofvelocity 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 generalcirculation

Shear stress at the air/water interface is exerted as a result of differencesbetween 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 ofdrift currents and circulation — seeChap 8for a discussion of surface water

Figure 12.1 Illustration of setup and circulation (side view, top) and horizontal lation (plan view, bottom) generated by wind over a lake.

circu-waves For the present discussion, it is assumed that a steady wind is blowingover 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 Wz is the wind speed measured at position z above the water surfaceand ud 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 increasingupwards when describing the wind velocity profile; however, when the mainconcern is with properties of the water body, it may be more convenient

to work with z increasing downwards from the surface — see the followingsection, for example) The shear stress at the surface is given by

sD cz
aWz
ud2 ¾D c

where cz is a drag coefficient and
ais air density The approximation of thesecond part of this equation results from the assumption that ud/Wz − 1 The

Figure 12.2 Velocity profiles for wind over a water surface.

shear stress also is defined in terms of a friction velocity,

where uŁa is the friction velocity for the wind profile

It is commonly assumed that the wind velocity distribution follows aboundary layer logarithmic profile,

where Re D WzL/ is a fetch Reynolds number, with L D fetch D distance

over which wind has blown over the surface of the water and Dkinematic 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

layer thickness, so it is well within the range where the logarithmic profileshould be valid.

The magnitude of z0 depends on the roughness of the surface and isassumed to be a linear function of roughness, similar to a turbulent boundarylayer in pipe or open channel flow In a fully developed wave field, the dynamicroughness length scale is estimated by (u2Ła/g), so

z0D ˛c

u2 Ła

where ˛cis 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 Uponsubstituting Eqs (12.2.1), (12.2.2), and (12.2.5) into (12.2.3), an expressionfor cz is obtained,

1p



12.2.6

which must be solved iteratively since cz is found in different terms on bothsides of the equation Normally, cz has a value on the order of 10
3 Once cz

is found, the surface shear stress is calculated from Eq (12.2.1)

For drift current, again assuming a fully developed wave field, it isassumed that the surface shear stress in the water is the same as that in theair, 0 ¾D 

s, where

and
wis 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 developingthe waves.) By combining Eqs (12.2.2) and (12.2.7), we find

may be used as a rough rule of thumb Also, if cz is estimated as being about

10
3, then from Eqs (12.2.1) and (12.2.2), uŁa¾D 0.03W

z, and, combined withEqs (12.2.8) and (12.2.10), we find uŁ ¾D 0.03ud This gives an alternativerelation 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 realisticrelationship for lakes and reservoirs, with limited fetch, is u Ł /ud ¾D 0.1–1.Finally, the kinetic energy flux across the surface is

When fetch is limited, the possibility of wind-generated seiche motions must

be considered These are in essence wavelike motions with a half-wave lengthgiven by the fetch L Seiche motions are primarily of interest when windsare relatively constant in speed and direction over a long period of time, assketched inFig 12.3.The surface shear stress exerted by the wind causes thewater surface to tilt, establishing a pressure gradient to balance this stress.The tilted water surface position is given by x, and the difference betweenthe 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 thecounterclockwise direction (northern hemisphere) For the present discussion

we neglect this effect If the wind suddenly stops, the tilted water surfacemoves 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.

opposite side of the lake This motion is eventually arrested as the kineticenergy 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 setupcondition This oscillatory motion continues until eventually viscous effectscause 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 inFig 12.4.Due to circulation of waterresulting 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 thecontrol volume are equal, and a force balance in the horizontal direction (perunit width) gives

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 thehorizontal, once the water surface tilts However, this angle is very small, andits cosine is assumed to be approximately 1 Also neglecting the second-order

Figure 12.4 Control volume for analysis of water surface profile.

term in dx, the term in brackets in Eq (12.2.12) is simplified as

 ¾D 1.1s

H



x
L2



12.2.14

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:

e Denergy per unit mass, the first integral on the right-hand side is the rate ofchange 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 dynamic convention is followed here in writing the work term as a positivequantity when the fluid does work on its surroundings

thermo-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 ofradiation ϕrand conduction ϕcterms,

Chap 2for 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 beingpositive when directed outwards from the control volume

The work rate term is considered to consist of gravity work and surfacework The gravity work rate is given by a volume integral,

8
g ÐVd8and the surface work rate, from Eq (2.8.5), is

This last result is further simplified using the continuity equation (2.5.6)and noting that the terms in (V2/2) cancel, so that

Du

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 workdone by stresses Usually the contribution of stresses is negligible in affectingtemperature, 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 ature only, i.e., u D uT, where T D temperature, and from the definition

temper-of specific heat (Eq 2.8.9), a change in energy is related to a change intemperature by

where c D cpD cv since in liquids the specific heats for constant pressure andconstant volume are nearly equal Again considering the heat flux term toconsist 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,



where Tis thermal conductivity.

Making the foregoing simplifications and substitutions, Eq (12.3.7)becomes

where constant c has been assumed This is the temperature equation, which

has the general form of an advection–diffusion equation expressing tion of thermal energy, with a main source term due to radiative heat flux If

conserva-T is constant, this equation becomes

∂T

∂t C

V ÐrT D kTr2T
1

where kTD T/
cis thermal diffusivity This result is similar to Eq (2.9.28),

but without the viscous or compression heating terms, which have beenneglected here

In order to solve Eq (12.3.11), initial and boundary conditions areneeded, and possible source terms for radiation must be specified For naturalwater bodies, the primary considerations are the heat transfer rate at the

air/water interface (which is normally specified as a boundary condition) andthe solar heating rate due to absorption of solar radiation (an internal sourceterm) We examine this latter term first.

12.3.2 Solar Radiation Absorption

Solar radiation consists of a large range of wavelengths, that have differentabsorption properties in water For light of a given frequency, Beer’s law statesthat the radiation intensity decreases exponentially with depth (z is assumedpositive downwards in the following):

where 0sω, z is the solar radiation intensity at depth z for frequency ω,

snω is the net (after reflection) radiation intensity at the water surfacefor frequency ω, and sω, zis the extinction, or absorption coefficient, as afunction of ω and z The total radiation intensity at any depth z is then the sum

of 0sω, zover all ω However, in practice insufficient data are available toevaluate such an integral A simpler approach has been found to give adequateresults, in which the range of light frequencies is divided into one or moresubranges, with surface radiation intensity and extinction coefficient valuesdefined for each subrange, rather than for each individual frequency The total

is then the sum over all subranges,

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 highervalues for 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 strongerabsorption near the surface and to provide a better fit for the exponentialmodel This is usually done by defining a fraction, ˇs, of the surface radiation

as that portion of the radiation intensity with relatively high extinction ficients, absorbing nearly completely within a shallow depth near the surface

coef-In general, different values of ˇs should be defined for each of the subrangesused 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 singlevalue for the extinction coefficient (s), to calculate the exponential decay ofthe remaining fraction of radiation that is not absorbed near the surface Theresulting equation for a one-term model (n D 1) is

0sz

so

Figure 12.5 Variation of solar radiation intensity with depth, compared with model approximation (Eq 12.3.14).

where so is now the total radiation intensity at the surface, including allfrequencies This model is sketched in Fig 12.5, where it can be seen thatpredictions very close to the surface are not very good This region is typicallyonly 5 or 10 cm deep at most, and the problem is usually not significant whenmodeling large water bodies, where variations within this region close to thesurface are not of major concern Also note that the horizontal axis in thisfigure is a log scale, so the magnitude of the slope of the model curve is given

by s The offset at the top (at z D 0) is the fraction assumed to be absorbednear the surface (ˇs)

In general, sis a function of z in Eq (12.3.14) and may be affected byturbidity gradients in the water, though it is often considered a constant for

a particular water body It is related to the secchi depth, which is a simple measurement used to describe water clarity The secchi disk is circular, with

alternating black and white sections painted on top The disk is loweredinto the water until it is no longer visible, and the depth at which that

occurs is the secchi depth Larger secchi depths correspond with greater waterclarity Typical values for the parameters in Eq (12.3.4) are ˇs¾D 0.5 and

s¾D 0.5 m
1(higher values correspond with higher turbidity)

12.3.3 Surface Heat Exchange

In addition to solar radiation absorption, surface heat transfer represents amajor source or sink of heat in determining the thermal structure of a waterbody It is incorporated as a boundary condition in models formulated to solvefor vertical temperature distribution, or as an internal source for verticallyaveraged models The basic elements of surface heat transfer include solar radi-ation (s), atmospheric radiation (a), back radiation from the water surface(b), evaporation (e), and conduction (c) These elements are sketched in

Fig 12.6and discussed in each of the subsections below

Solar Radiation

Ideally, solar radiation intensity is measured directly at a particularsite of interest It consists of direct and diffuse radiation components,

usually measured with a pyrheliometer or pyrenometer Unfortunately, direct

measurements are not very common At many weather stations only the

“percent possible sunshine” or some other measure of sunlight is reported.Solar radiation at the edge of the earth’s atmosphere is, however, well known,and values of solar radiation intensity have been tabulated as a function oflocation and time of year (see, for example, Kreith and Kreider, 1979) Thesevalues must be modified according to pollution or water vapor content (clouds,fog, smog, etc.) of the air Another measure of possible sunshine is the percent

of cloud cover Cc, and if the clear-sky radiation intensity is known, the net

Figure 12.6 Components of surface radiative heat transfer.

solar radiation can be estimated from

where sc is the clear-sky value and Cc is expressed in decimal units.The reflected solar radiation, sr, depends on angle of incidence and onwater surface roughness (a smooth surface will reflect a higher proportion).The reflected radiation ranges between about 3 and 10% of the incident radia-tion, and an average of 6% is reasonable Thus the total net radiation passinginto the water surface is estimated as

where ε is emissivity (with values ranging between about 0.7 for clear sky and

Ta Ł is air temperature, on an absolute scale The Swinbank formula suggests

that emissivity for a clear sky is proportional to T2aŁ and is written as

(incor-acD 1.2 ð 10
13TaC 4606 Btu/ft2-day 12.3.18awhere acis the clear-sky value for atmospheric radiation and Tais air temper-ature in°F, measured 2 m above ground level In SI units,

acD 5.35 ð 10
13TaC 2736 W/m2 12.3.18bwhere Ta here is in°C

Similar to the formulation for solar radiation, a cloud cover correction

is usually added,

where K has a value between 0.04 and 0.25, with an average around 0.17.Reflected atmospheric radiation is approximately 3% of incident, so the netatmospheric radiation is

anD 0.971.2 ð 10
13TaC 46061 C KC2

anD 0.975.35 ð 10
13TaC 2736 1 C KC2 W/m2 12.3.20bwhere, 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

It represents a loss of heat from the water body The intensity is calculated in amanner similar to atmospheric radiation, with an emissivity of about 0.97, so

water molecules from the liquid to the gaseous phase, so mass transfer is the

primary consideration Heat transfer occurs due to the latent heat of ization associated with this phase change Molecular diffusive-type transport

vapor-represents a limiting rate, while free and forced convection usually play a

much more significant role in determining evaporative fluxes Free convection

is due to buoyancy effects related to a heated water surface (relative to the airtemperature), and forced convection is related to wind Both of these transportmechanisms are turbulent in nature

Evaporative mass transfer is proportional to the difference between rated vapor pressure (as a function of temperature) and the actual atmosphericvapor pressure and may be expressed as

where E is mass flux,
v is vapor density, esis saturated vapor pressure, ea isatmospheric vapor pressure, and f0Wis a function of wind speed that musttake into account the effects of both free and forced convection The effect

of wind or buoyant (free) convection is to introduce turbulence, which movesvapor away from the interface, thus maintaining a higher difference between

Figure 12.7 Illustration of forced and free convection effects on evaporation.

vapor pressures, with a corresponding increase in evaporation, as illustrated

in Fig 12.7 The evaporation rate driven by wind is highest near the shore,where the vapor boundary layer thickness is smallest, with a correspondinglyhigher vapor pressure gradient In this case, the evaporation rate is a function

of fetch, L Free convection, on the other hand, does not depend on L, since it

is a function of heating of the water surface, relative to the air The resultingbuoyant convective motions serve to transport vapor away from the interfacialregion and increase the gradient driving E

The heat flux associated with E is

where Lv is the latent heat of vaporization and fW is a wind speed function

that incorporates
vand Lvinto f’(W) Many forms of the wind speed functionhave been proposed, usually as a simple constant or first- or second-order

polynomial in W An example of a first-order function of W is the Lake

Hefner formula,

where W2is the wind speed in mph, measured at a height 2 m above the watersurface This formula was developed to predict evaporation over a medium-size lake and was found to give reasonable estimates for wind speeds betweenabout 3 and 20 mph This formula, however, does not account for free convec-tion effects

To develop an expression for free convection, consider first a heatedflat plate, held at temperature Ts, which is assumed to be greater than the

Figure 12.8 Convective heat transfer above a heated flat plate.

air temperature Ta (Fig 12.8).Due to heating from the plate, the air near theplate is warmed, becomes unstable, and begins to rise (seeChap 13for furtherdiscussion of convective instability) As the heated air rises, it is replaced bycooler air, thus maintaining a relatively high temperature gradient near theplate surface The convective motions are turbulent, and the heat transfer rate

is assumed to be proportional to the temperature difference (Ts
Ta),

1/3

12.3.27

where ˇ is the thermal expansion coefficient and is kinematic viscosity, asbefore

These results may be used to estimate water vapor mass transfer,

assuming that the vapor is transported at the same rate as heat (i.e., Reynolds’

analogy applies, so that the heat transfer coefficient is the same as a mass

transfer coefficient) Similar to the previous development for wind-inducedevaporation (Eq 12.3.13), the mass flux due to convection is written as afunction of the difference between saturated vapor density (
vs) and the actualvapor density in the air (
v),

where Km is a mass transfer coefficient equal to Kh This equation can bewritten in terms of vapor pressure by introducing the perfect gas law,

eaD
vRvTŁD
v

R

Mv

where Rv is the specific gas constant, R is the universal gas constant, and Mv

is the molecular weight of the vapor Substituting Eqs (12.3.27) and (12.3.29)into Eq (12.3.28) and multiplying by latent heat of vaporization to convertmass transport to heat transport, we have

and Tsv
Tav1/3takes the place of fW (compare with Eq 12.3.14)

As noted previously, there are many forms for f(W) suggested in theliterature Other than modeling natural water bodies, a specific engineeringapplication for these calculations is in the design of cooling ponds for disposal

of waste heat from power plants There the effects of free convection aremuch more important than for a natural water body, and research has focusedmore on developing expressions for buoyancy effects For any application, thechoice of fW depends on the relative importance of forced and free convec-tion effects Ryan and Harleman (1973) suggested a formula that incorporatesboth effects,

Figure 12.9 Qualitative comparison of Lake Hefner formula and Eq (12.3.33) (for different temperature differences); the vertical dashed line distinguishes approximately between regions where free convection (low W) and forced convection (high W) dominate the evaporation flux.

where W2 has the same meaning as in the Lake Hefner formula and atures are in°F A qualitative comparison of formulas for fW is shown in

temper-Fig 12.9.For small W it is expected that free convection effects are relativelymore important, while for large W, forced convection dominates A formulasuch as the Lake Hefner formula performs better for higher W but may under-predict evaporation at low W Formulas such as Eq (12.3.33), which depend

on temperature, are better able to simulate fluxes for low to medium W butmay underpredict evaporation for large W

Conduction

Conduction is driven by temperature differences between the water and theatmosphere It can be either a source or a sink of heat for the water body,depending on the relative magnitudes of Ta and Ts Conduction also is rela-tively small, compared with the turbulent convection effects associated with

evaporation, or the radiative transfer rates from solar, atmospheric, and backradiation The approach for estimating convective heat flux is similar to thatfor evaporation, except here the driving force is temperature difference ratherthan vapor pressure difference,

cD fŁ

where fŁWis a wind function for conduction Conduction is usually related

to evaporation By taking the ratio of conductive to evaporative heat flux,

Total Heat Flux, Linearized Approach

The total net heat flux is the sum of each of the above processes (as illustrated

Eq (12.3.39) represents the boundary condition for a thermal energy tion for a water body

calcula-In some studies it may be preferable to use a linearized approach, inwhich n is expressed as a linear function of temperature,

Figure 12.10 Linear approach for calculating  n

where Knis a net heat exchange coefficient (in general, a function of Ts
Te)and Te is the equilibrium temperature, i.e., the water surface temperature at

which nD 0 (Fig 12.10).In other words, for a given Te, Kn is the rate ofchange of heat flux rate with temperature,

KnD
dn

dTs

12.3.41This is a useful concept for some analytical solutions but is not of greatinterest when using numerical modeling approaches in which the various heatflux terms can be calculated directly (otherwise, Kn must be calculated from

Eq (12.3.41), which requires evaluation of the various terms in n, anyway)

On the other hand, Eq (12.3.40) may be used for quick estimates even innumerical studies, as long as a reasonable value for Kn is known

Historically, most of the work done on surface gas transfer has involvedoxygen, and the present text also will focus on this parameter Other gasescould be treated similarly as in the following development In fact, asdiscussed in this and the following sections, bulk mass transfer coefficientsare usually assumed to be independent of properties of the specific gas under

consideration — they are more a function of physical driving mechanismssuch as wind speed The emphasis on oxygen transport is primarily due to the

role dissolved oxygen (DO) plays as a water quality parameter DO levels are

critical in determining the general health of a system, the diversity of species,and the number of organisms that can be supported For example, most fishrequire a concentration of at least 4 to 5 ppm (or mg/L) DO for survival Amajor factor in modeling DO in surface waters is in defining the reaerationrate, or the rate at which oxygen is transported across the air/water interface,since the atmosphere is the primary source of oxygen A large number of DOmodels have been developed, with perhaps the best known due to Streeter andPhelps (1925), as described briefly later in this chapter

In this section it will be convenient to define gas concentrations in terms

of mass densities (i.e., mass per unit volume) The oxygen mass flux across the

air/water interface is assumed to be proportional to the DO deficit in the water

body, which is defined as the difference between saturated DO concentration,

as a function of temperature, and the actual concentration,

where J is oxygen mass flux, C is DO concentration, Cs is saturated DOconcentration, and KL is a bulk mass transfer coefficient The value for Cs

is temperature dependent, decreasing from 16.4 ppm at T D 10°C to 7.8 ppm

at T D 30°C It represents the concentration in water that is in equilibriumwith the partial pressure of the gas in the atmosphere This equilibrium rela-

tionship may be expressed by the Henry’s law constant, defined either in

For a well-mixed water body of average depth H, the rate of change of

DO due to atmospheric flux is

dC

dt D JA

8 D KLA