Environ. Sci. Technol. 1993, 27, 2086-2097
Gas Exchange at River Cascades: Field Experiments and Model Calculations Olaf Clrpka,t Peter Relchert,’ Oskar Wanner, Stephan R. Muller, and Ren6 P. Schwarzenbach
Swiss Federal Institute for Water Resources and Water Pollution Control (EAWAG), Dubendorf, Switzerland, and Swiss Federal Institute of Technology (ETH), Zurich, Switzerland
A model describing gas exchange, at river cascades, which considers direct gas exchange over the free surface as well as gas exchange via air bubbles entrained into the water is presented. The superiority of this model over simpler models that consider only one of the two gas-exchange processes is demonstrated by an application to experimental data for five compounds at four different cascades. Using the calibrated model, it is shown that the relative importance of the two gas-exchange processes depends strongly on the physicochemical properties (especially on the Henry coefficient) of the compound exchanged. A special consequence of this finding is that, in rivers in which the water-level difference over hydraulic controls contributes significantlyto the total water-leveldifference, gas transfer of volatile organic compounds cannot be estimated from oxygen exchange and evaporation data alone using the two film models of gas exchange. Introduction
Quantification of gas exchange between rivers and the atmosphere is of great interest for describing the loss of volatile organic pollutants from rivers and for assessing rates of oxygen transfer (especiallyfor reaeration). Various models using the penetration theory (I) or the surface renewal theory ( 2 , 3 )have been applied with reasonable success to describe gas exchange between rivers and the atmosphere, at least for river reaches without important hydraulic controls. In such simple cases, prediction of volatilization rates of organic pollutants with large Henry coefficients based on reaeration rates and vice versa is possible using the ratio of the diffusion coefficients of the pollutant and of oxygen, respectively (4). At hydraulic controls such as weirs, falls, or even cascades, however, gas exchange may be strongly influenced or even dominated by the entrainment of air bubbles into the water jet penetrating into the receiving basin. Due to the limiting volume of the rising air bubbles, this latter process depends strongly on the Henry coefficient of the compound undergoing exchange. Thus, for such situations, the approach involving the conversion of volatilization rates using only diffusion coefficients, as recommended by Gulliver et al. (5), is only applicable for substances with very large Henry coefficients. For substances with smaller Henry coefficients, the dependence on Henry coefficient has to be considered. Measured data on volatilization rates of organic pollutants at hydraulic controls, which demonstrate this dependence, are very scarce. In this paper, we present the results of a study in which the volatilization of five model compounds (SFe, R113, CzHC13, CHC13, CHBr3) with very different Henry coefficients was investigated at four river cascades as shown
in Figures 1and 2. After a short introduction on measures for gas exchange at hydraulic controls, in the theoretical part of the paper, a model for gas exchange at river cascades is developed that combines both direct gas exchange to the atmosphere over the water surface and exchange via air bubbles. This model describes the exchange efficiency of a cascade as a function of properties of the compound (Le.,Henry coefficient, molecular diffusion coefficients in air and water) and of cascade-specific empirical parameters. In the experimental section, a very sensitive analytical method is described that allows us to obtain accurate mesurements of even small concentration differences of the model compounds over a river cascade. Finally, the model is calibrated with these data for the cascades, and the gas exchange for the model compounds is discussed. Theory
Measures for Gas Exchange at Hydraulic Controls. Gas exchange at hydraulic controls can be quantified by the total gas transport between water and air, i.e., as the mass of the compound exchanged per unit of time. Usually, the retention time of water within a hydraulic control is short, and therefore, transformation processes of compounds of moderate to low reactivity may be neglected. In this case, the total gas transport J is given by = Q(Cl,up - Cl,dn)
(1) where Q is the river discharge (volume of water per unit of time), qUp and Cl,dn are the dissolved concentrations of the compound upstream and downstream of the control, and positive flux is directed out of the water. This measure of gas exchange depends on the compound concentrations in water and air. The driving force of gas exchange is the difference c1- cg/Hbetween actual compound concentration in water, c1, and the equilibrium concentration cg/H,where cg is the actual concentration in air and H is the nondimensional Henry coefficient. Gas exchange always leads to a reduction of the magnitude of the difference c1- cg/H,i.e., equilibrium is approached. The factor, r, by which this difference is reduced at the control J
r=
c1,up ‘1,dn
(cg/H)
- (‘g/W
(2)
+ Present address: Institut fur Wasserbau, Universitat Stuttgart, Stuttgart, Germany.
is an alternative measure for quantifying gas exchange at a hydraulic control (6). This measure is nondimensional, always larger than or equal to 1(1implies no gas exchange) and does not usually depend on the compound concentration. This makes it a more universal measure of gas exchange than the total transport J . For the organic pollutants treated in this paper, usually C I > cg/H, an appropriate name for r would be “excess ratio”. Since eq 2 was originally introduced for oxygen, where more often c1 < cg/H,the name “deficit ratio” is more usual (6) and is also used in this paper.
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0013-936X/93/0927-2086$04.00/0
Envlron. Sci. Technol., Voi. 27, No. 10, 1993
0 1993 American Chemical Society
Flgure 1. Vlew of a river cascade in the River Glan (cascade 1) consisting of a ramp wlth randomized blocks foilowed by a reCBiVlng basin.
As can easily he verified using eqs 1 and 2, the two measures J and r are related hy J=~ [ c ~, ,( c d m
+
= Q[C,
- ( q m i (r - 1) (3)
This equation leads to the introduction of another nondmensional measure of gas exchange,the gas-exchange efficiency (4)
[originally introduced by Gameson (6)with an additional factor of 100,hut now more often used in the form of eq 4 (31. The gas-exchange efficiency E is the fraction by which the saturation deficit or excess is reduced a t the hydraulic control. The values of E are between 0 and 1, Orepresentsnogasexchangeand1impliestotalelimiition of the saturation deficit or excess. The deficit ratio r has the simpler mathematical form, and it is easier to use for calculating the combined effect of several hydraulic controls in series as compared to the gas-exchange efficiency E. Therefore, r is used for the derivation of the gas-exchange models. Because the interpretation of the gas-exchange efficiency is simpler, this quantity is used for the discussion of model results and field data. Gas Exchangeover the Free Surface. In this section we derive a formula for the deficit ratio at a hydraulic control for which direct gas exchange at the water surface is the dominant process. To distinguish this exchange process from indirect exchange via entrained air bubbles, it is called "free surface" exchange. Note that this is also
the usual model approach for river reaches without hydraulic controls. With the assumptions that (1)transformation processes are negligible and (2) longitudinal dispersion is negligible, the temporal change of the concentration in a parcel of water moving with the mean flow velocity is given by dc,/dt = -(weg/A)j
(5)
where w.g is the effective surface area per unit of river length, A is the croes-sectionalarea, and j is the gas flux per unit of surface area. For a smooth surface,w a is equal to the surface width, whereas rough hydraulic structures can cause w.g to become many times larger. Several theories have been proposed to calculate the gas fluxj . All of these assume gas exchange to he limited hy two thin boundary layers in air and water, where molecular diffusion becomes important, and all lead to the expression
where and rEare the resistances (time per unit of length) ofthe water and air boundary layers, respectively (8). The first approach hy Whitman (9) and Lewis and Whitman (10)assumes sharp transitions between the well-mixed water and air bodies and the corresponding boundary layers and considers only molecular diffusion within the layers. Later theories revise this concept by allowing turbulence events to occasionally destroy the molecular layers (I) or by assuming a continuous transition from turbulent to molecular diffusion near the interface (11, Envtm. Sd.Techrol.. Vd. 27. NO. 10, 1883 1081
Flpure 2. View of a river cascade in the River Glan (cascade 3) consisting of a Succession of steps followed by a receiving basin
12). The formal differences among the outcome of these models are the exponents with which the resistances n and r8 depend on the molecular diffusion coefficients DI and De In this paper, we use the expressions
rl = rl*/D:I2
(7) where
and
rg= r;lDg
Application of eq 3 and use of the resistances given by eqs 7 and 8 lead to the corresponding deficit ratio:
(the corresponding gas-exchange efficiency E s d is given Envton. Scl. Tllchnol.. Vol. 27.
Qrl*
213
(8) supported by Hunt (10,Ledwell (12),and Mackay and Yeun (13). In eqs 7 and 8 n* and rg* are coefficients depending on the turbulent structure of the boundary layers which usually have to be determined empirically. With the additional assumptions that (3) water flow is uniform and (4) the concentration c8 of gas in the air above the control is constant, integration of eq 5 with the gas flux given by eq 6 for a given length Ax of the hydraulic control leads to the following expression for the total transport over the surface:
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by the insertion of eq 10 into eq 4). Equation 10 can be rewritten as
No. 10, 1993
P l = q
and
are two model parameters that depend on the type of the hydraulic control and on the hydraulic conditions. Although these parameters are given as functions of more fundamental parameters. because of the empirical nature of the parameters wee, n*, and rg*, they have to be determined experimentally for practical applications. In principle, this is possible using measured deficit ratios for two compoundswith different physicochemicalproperties. The use of more than two compounds is, however, advisable since this allows a statistical analysis and a test of the reliability of the model to be conducted. Figure 3 shows the dependence of the gas-exchange efficiencyE 8 d according to eqs 4 and 11on the molecular diffusioncoefficientin water and on the Henry coefficient as a contour plot for typical values of the parameters p1 and p2. Since the diffusion coefficient of a compound in a fluid depends on the size and shape of its molecules, the diffusion coefficients in water and air are not mutually
coefficient in water:
Note however,that eq 13used for plotting Figure 3 destroys this independence of EsWffrom D1.
Figure 3. Contour plot of the gas-exchange efflclency of the free surface gas-exchange model according to eqs 4, 11, and 13 as a function of Henry coefflclent Hand molecular diffusion coefficient LJ in water for p1 = 2.2 X ms-112and p2 = 1.1 X lo-' m4/3s 2 I 3 . The diagram Is dlvlded into three parts: zones I and 111 are the zones of approxlmate valldlty of eqs 15 and 14, respectively, whereas zone I1 is the transitional zone where the full model (eq 11) must be used. The white square marks the positionof oxygen, and the other symbols represent the posltions of the compounds listed In Table I (triangle = CHBr3, diamond = CHCI3, cross = C2HCi3,black square = R113, clrcle = SF8).
Table I. Henry Coefficients (If)and Diffusion Coefficients in Air (Dg) and Water (9) and Octanol-Water Partition at 4 OC for All Model Compounds Used Coefficients (KOw) compound sulfur hexafluoride (SFd 1,1,2-trichlorotrifluoroethane (R113) trichloroethene (CzHCls) trichloromethane (CHCld tribromomethane (CHBrd a
H 82.4
DI(m2/s) D, (m2/s) KO, 6.23 X 10-l0 1.87 X 106 > @ J & ~ / ~ ) / @ J ~becomes Henry coefficient and of the diffusion coefficient in air:
-
rsWf exp(D11/2/pl) (14) For very small Henry coefficients, i.e., for H > gas exchange becomes independent of the Henry coefficient: ‘bubble
4- p3p4D,1‘2
(22)
This approximation corresponds to the limiting case where the change in concentration within a bubble has no significant effect on the gas flux through its surface. For the deficit small Henry coefficients, i.e., for H
