Temperature dependence of stream aeration coefficients and the effect of water turbulence: A critical review

Abstract

The gas transfer velocity (K_L) and related gas transfer coefficient (k₂ = K_LA/V, with A, area and V, volume) at the air–water interface are critical parameters in all gas flux studies such as green house gas emission, whole stream metabolism or industrial processes. So far, there is no theoretical model able to provide accurate estimation of gas transfer in streams. Hence, reaeration is often estimated with empirical equations. The gas transfer velocity need then to be corrected with a temperature coefficient θ = 1.0241. Yet several studies have long reported variation in θ with temperature and ‘turbulence’ of water (i.e. θ is not a constant). Here we re-investigate thoroughly a key theoretical model (Dobbins model) in detail after discovering important discrepancies. We then compare it with other theoretical models derived from a wide range of hydraulic behaviours (rigid to free continuous surface water, wave and waterfalls with bubbles). The results of the Dobbins model were found to hold, at least theoretically in the light of recent advances in hydraulics, although the more comprehensive results in this study highlighted a higher degree of complexity in θ’s behaviour. According to the Dobbins model, the temperature coefficient θ, could vary from 1.005 to 1.042 within a temperature range of 0–35 °C and wide range of gas transfer velocities, i.e. ‘turbulence’ condition (0.005 < K_L < 1.28 cm min⁻¹). No other theoretical models showed any significant variability in θ with change in ‘turbulence’, and only modest variability in θ with change in temperature. However, the other theoretical models did not have the same temperature coefficient θ (with 1.000 < θ < 1.056 within 0–35 °C). A model integrating turbulence and bubble mediated gas transfer velocities suggested a lower temperature dependence for bubble (1.013<θ < 1.017) than turbulence (1.023<θ < 1.031) mediated processes. As it stands, the effect of turbulence on the temperature dependence of gas transfer at the air–water interface has still to be clarified, although many models simulate different flow conditions which may explain some of the observed discrepancies. We suggest that the temperature dependence curves produced by the Dobbins model may be used tentatively as a simple theoretical guide for streams with free surface water but not self-aerated flows encountered in whitewater rapids, cascades or weirs. Greater awareness of the different models and conditions of applications should help choosing an appropriate correction. Three case studies investigated the effect of the temperature coefficient on reaeration and stream metabolism (photosynthesis and respiration). In practice, the temperature correction may be an important parameter under constant turbulence conditions, but as the range in turbulence increases, the role of temperature may become negligible in determining K_L, whatever the temperature correction. The theoretical models reviewed here are also useful references to correct K_L values determined using a reference tracer gas to a second species of interest.

Introduction

Reaeration is a critical parameter in gas flux studies/modelling at the air–liquid interface both in civil engineering and industrial processes (Gulliver et al., 1990; Vogelaar et al., 2000; Huisman et al., 2004), as well as environmental studies, such as dissolved gas concentration (e.g. oxygen partial pressure/concentration stress on animal), green house gas (GHG) emission, denitrification (open channel method), and stream metabolism (GPP, ER) – see e.g. Yongsiri et al. (2004); Schierholz et al. (2006); Battin et al. (2008); Baulch et al. (2010); Demars et al. (2011); Wallin et al. (2011). While the present study will focus on oxygen in stream, the same principles will hold for GHG (CO₂, CH₄, N₂O), N₂, and tracer gas (SF6, propane) in aquatic ecosystems (e.g. Jones and Mulholland, 1998; Huisman et al., 2004). Reaeration is also generally one of the least constrained parameter in models (e.g. Cox, 2003; Izagirre et al., 2007).

There is a plethora of theoretical models considering an interesting range of possible mechanisms but agreements among them are generally too broad (∼an order of magnitude) when applied to real systems (Owens et al., 1964; Wilson and Macleod, 1974; Genereux and Hemond, 1992; Aristegi et al., 2009). The theoretical models also have at least one constant that need to be fitted with empirical data (either from laboratory or stream data), except perhaps Lamont and Scott (1970). Lamont and Scott (1970) model, while performing broadly well across aquatic habitats (Zappa et al., 2007), cannot generally represent satisfactorily the many mechanisms at play in natural systems (see Jirka et al., 2010; MacIntyre et al., 2010; Vachon et al., 2010).

So, in practice, very accurate results (10% standard error) are obtained with direct measurements (e.g. Thyssen et al., 1987; Thene and Gulliver, 1990; Genereux and Hemond, 1992; Melching, 1999). Such measurements are fairly easy and inexpensive to carry out in small streams, but it becomes more challenging in large rivers (e.g. Richey et al., 2002). In lowland rivers, the reaeration coefficient of oxygen may also be derived indirectly from the recorded dissolved oxygen curves (Odum, 1956; Hornberger and Kelly, 1975; Chapra and Di Toro, 1991). Such indirect methods are not suitable for streams with high reaeration coefficient or/and very low biological activity (e.g. Hornberger and Kelly, 1975; Thyssen et al., 1987).

Direct measurements are also only possible for a discrete moment in time and space. The diel variability in reaeration coefficients has rarely been attempted (e.g. Tobias et al., 2009), and long term continuous estimation of reaeration coefficient need to be based on rating curves with discharge (e.g. Jones and Mulholland, 1998; Acuña et al., 2004; Roberts et al., 2007) but this is site specific and may not always work due to stream geomorphology (e.g. Genereux and Hemond, 1992; Wallin et al., 2011). The alternative use of sound in streams with standing broken waves (Morse et al., 2007) is ingenious as it relates perhaps better to stream turbulence at the air–water interface. Direct continuous measurements of turbulence at the water–air interface for gas transfer studies are at the core of several recent studies (e.g. Janzen et al., 2010; MacIntyre et al., 2010; Vachon et al., 2010).

With the need to scale up both in space, from river reach to river basin, and time, from days to years, the use of empirical relationships involving predictors such as depth, slope, and velocity (Owens et al., 1964; Moog and Jirka, 1995; Melching and Flores, 1999; Butman and Raymond, 2011; Raymond et al., 2012) have been popular. Those relationships, however, should not be used outside the predictors range and type of streams for which they were calibrated.

While the temperature dependence of the reaeration coefficients should reveal itself through the temperature variation of the physical properties involved in the theoretical models (e.g. Daniil and Gulliver, 1988; Urban and Gulliver, 2000; Gualtieri et al., 2002), empirical models of the reaeration coefficients need corrections for temperature.

The mass transfer of a slightly soluble non-reactive gas (e.g. oxygen) across an air–water interface is controlled by the liquid phase where gas molecular diffusion is about 10,000 times slower than in air and is generally written as follows:

$$\frac{\partial C_{\text{W}}}{\partial t}=k_2\left(C_{\text{S}}-C_{\text{W}}\right)$$

with C gas concentration (mg L⁻¹); C_s, saturated dissolved gas concentration (mg L⁻¹); C_W, dissolved gas concentration in the volume of water (mg L⁻¹); t, time (min) and k₂ gas exchange coefficient (min⁻¹). k₂ is determined by the product of gas transfer velocity K_L (cm min⁻¹) and specific surface area a (cm⁻¹), the latter expressed as air–water interface area (A, cm²) per unit volume of air and water (V, cm³). Hence, we have:

$$k_2=\frac{K_{L}A}{V}$$

In stream with smooth surface water, we have K_L = k₂h, with h average stream depth (cm). In self-aerated flows, measurements of specific surface area (A/V) become non-trivial due to bubbles and spray (see e.g. Toombes and Chanson, 2005; Wilhelms and Gulliver, 2005).

K_L and k₂ coefficients increase with increasing temperature, itself decreasing water viscosity and therefore facilitating the molecular diffusivity of dissolved gases (e.g. as formulated in the Stokes–Einstein equation). This rate of increase of K_L and k₂ with temperature (temperature coefficient) is the focus of the present study.

The coefficients (k₂ and K_L) are commonly temperature corrected relative to 20 °C with the simplified Arrhenius equation (see Appendix S1 in supplementary information for the full Arrhenius equation) as follows:

$$k_T=k_{20}{\theta }^{\left(T-20\right)}$$

with T, observed stream temperature; and θ, temperature coefficient, generally reported to be θ = 1.0241, i.e. an increase at the geometric rate of 2.41% per °C (Kilpatrick et al., 1989; Bott, 2007, p. 671; Stenstrom, 2007), no doubt based on the meticulous bottle experiments of Elmore and West (1961) but also most likely because some prominent theoretical models for free water surface confirmed those experimental results (e.g. King, 1966; Lamont and Scott, 1970; Wilson and Macleod, 1974; Gualtieri and Gualtieri, 2004). Other constant values for θ have been used, generally based on older experiments, probably unaware of the sharp criticisms by e.g. Elmore and West (1961); cf Kothandaraman and Evans (1969).

Although Elmore and West (1961) did not find significant differences in θ with change in turbulence, both earlier studies (Kishinevsky, 1954; Kishinevsky and Serebryansky, 1956), and later studies (Dobbins, 1964; Metzger and Dobbins, 1967; Metzger, 1968) contradicted this finding both theoretically and experimentally. It is worth noting that Elmore and West (1961) two experiments were run within a narrow range of turbulence: K_L = 0.02 cm min⁻¹ to 0.05 < K_L < 0.12 cm min⁻¹ (the latter depending on the size of the vortex that may have increased the water surface up to a very unlikely maximum of 2.4 times).

Limits in Dobbins (1964) new theoretical model were discussed (notably Thackston and Krenkel, 1965; Dobbins, 1965). In relation to the work of Kishinevsky, it was pointed out that Dobbins model applied for a wide range of turbulence with continuous water surface (free water surface), hence may be unable to representing gas transfer at extremely high mixing rates, as in Kishinevsky's model (Holley et al., 1970), river rapids (Hall et al., 2012), river cascades (Cirpka et al., 1993) or hydraulic structures (see below).

The work by Metzger (1968) was favourably accepted in the subsequent discussion by Rathbun and Bennett (1969) studying the limits of the model (see also Bennett and Rathbun, 1972; cf Daniil and Gulliver, 1988), although Kothandaraman and Evans (1969) warned about potential pitfalls in the comparison of experimental data and experimental design (see also Boyle, 1974). To our knowledge, no response from Metzger followed. Only Daniil and Gulliver (1988) reported some qualitative experimental discrepancies with the Dobbins–Metzger model and concluded that the effect of turbulence intensity on the temperature correction coefficient θ had still to be clarified. Generally, the Dobbins-Metzger results have been widely accepted contrary to other heavily criticised results such as Howe (1977; cf Brown and Stenstrom, 1980; Rathbun, 1981) and Chao et al., 1987a, Chao et al., 1987b; cf Daniil and Gulliver, 1989a,b; Rathbun, 1989a,b; Wilcock and McBride, 1989). Indeed the decreasing K_L with increasing temperature and θ < 1 in Howe and Chao et al. studies do not seem tenable.

Early experimental work at hydraulic structures (weirs) by Gameson et al. (1958) reported a temperature dependence of 1.4–2% per °C in the range 0–35 °C, independently of fall height and pollution effects. The independence of turbulence and mixing intensity on the temperature dependence was confirmed theoretically by Gulliver et al. (1990), which also reported that Eq. (3) above (with θ = 1.0241) did not describe as well the temperature dependence, likely because θ was determined in a stirred container experiment without bubbles. The theoretical model used by Gulliver et al. (1990) did not include explicitly however the role of gas solubility, which has been shown to be important (Asher et al., 1997) and known to change with temperature (Battino et al., 1983).

In their textbook, Thomann and Mueller (1987) recognised that θ may generally vary from 1.005 to 1.03 based on previous specialised synthesis reports ultimately referring to the work of Dobbins and Metzger. Of course, when θ→1, there is no need for a temperature correction (e.g. Genereux and Hemond, 1992; Demars et al., 2011). Hence, particularly for very turbulent systems θ = 1.0241 has not always been readily accepted (e.g. Eheart and Park, 1989; Baulch et al., 2010; Demars et al., 2011; Wallin et al., 2011). But the logic behind this choice may be questioned since Dobbins and Metzger model was limited to free surface water and contradicting results have been presented for highly turbulent systems (e.g. Kishinevsky, 1954 versus Gulliver et al., 1990).

Rather than using θ = 1.0241 indiscriminately for some work on whole stream metabolism, Demars et al. (2011) decided to calculate θ from the original formulae of Dobbins and Metzger (op. cit.). It soon appeared that while the original papers presented succinctly the theory, they did not provide all the necessary details to repeat the calculations and did not compute the temperature correction coefficient θ for a wide range of temperature and stream turbulence. Moreover, on closer inspection, while rL³ term (see below) should be constant under a given temperature (Dobbins, 1964; Metzger and Dobbins, 1967), this was not true from the values reported in the tables of the Metzger papers, hence the θ curves provided in Fig. 2 of Metzger (1968) might have been incorrect. Further, comparison between theory and empirical evidence was also biased by the selective use of data (cf Elmore and West, 1961; Downing and Truesdale, 1955; Truesdale and Van Dyke, 1958) and dubious comparability of old data (e.g. still cited or used θ = 1.016 from Streeter, 1926 and θ = 1.047 from Streeter et al., 1936; cf Elmore and West, 1961; Kothandaraman and Evans, 1969).

Hence the aim of the present study was to investigate whether the temperature coefficient θ is independent of temperature and turbulence through theoretical models. More specifically, the aims are (i) to provide the necessary details and repeat the calculations of the theory presented by Dobbins for a wider range of temperature; (ii) compare the results with other representative types of theoretical models; (iii) discuss the validity of the results for gas exchange studies in the light of recent advances in hydraulics; (iv) deduce the implications for the importance of θ temperature corrections for gas transfer at the air–water interface and whole stream metabolism estimates.