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Aeration efficiency over stepped cascades: Better predictions from flow regimes
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Aeration efficiency over stepped cascades: Better predictions from flow regimes Hatem Khdhiri*, Olivier Potier, Jean-Pierre Leclerc Laboratoire Re´actions et Ge´nie des Proce´de´s, UMR 7274 CNRS, Universite´ de Lorraine, 1 rue Grandville, 54001 Nancy, France
article info
abstract
Article history:
Stepped cascades are recognized as high potential airewater gas exchangers. In natural
Received 19 September 2013
rivers, these structures enhance oxygen transfer to water by creating turbulence at inter-
Received in revised form
face with increasing air entrainment in water and airewater surface exchange. Stepped
3 February 2014
cascades could be really useful to improve the natural self-purification process by
Accepted 6 February 2014
providing oxygen to aerobic micro-organisms. The aeration performance of these struc-
Available online 17 February 2014
tures depends on several operating and geometrical parameters. In the literature, several empirical correlations for aeration efficiency prediction on stepped cascades exist. Most of
Keywords:
these correlations are only applicable for operating and geometrical parameters in the
Oxygen transfer
range of which they have been developed. In this paper, 398 experimental sets of data
Stepped spillways design
(from our experiments and collected from literature) were used to develop a correlation for
Modeling
aeration prediction over stepped cascades derived from dimensional analysis and
Self-purification
parameterized for each individual flow regime in order to consider change in flow regime
River
effect on oxygen transfer. This new correlation allowed calculating the whole set of data
Cascade
obtained for cascades with steps heights between 0.05 m and 0.254 m, cascade total height between 0.25 m and 2.5 m, for discharges per unit of width ranging from 0.28 103 m2/s to 600 103 m2/s and for cascade steps number between 3 and 25. In these ranges of parameters, standard deviation for aeration efficiency estimation was found to be less than 17%. Finally, advices were proposed to help and improve the structure design in order to improve aeration. ª 2014 Elsevier Ltd. All rights reserved.
1.
Introduction
Dissolved oxygen concentration is an important indicator of water quality in natural environment. Oxygen deficit in aquatic systems occurs when water is highly polluted or in calm or stagnant canal or river. Natural water systems could have a potential to eliminate the pollution and restore ecological properties by self-purification processes. The main
process requires dissolved oxygen for chemical and mainly biological mechanisms involving aerobic microorganisms. In case of O2 deficit, biological processes and chemical oxidations are limited. Then, natural system is not able to reduce pollution concentration with harmful consequences on water quality and aquatic species. Thus, enhancing oxygenation in watercourses represents a good solution to improve selfpurification process. If oxygen is spontaneously transferred by diffusion phenomenon, which depend essentially on
* Corresponding author. Tel.: þ33383175170. E-mail addresses: [email protected] (H. Khdhiri), [email protected] (O. Potier), [email protected] (J.-P. Leclerc). 0043-1354/$ e see front matter ª 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.watres.2014.02.022
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Nomenclature a a b C Cs CU CD Ci Do2 D, Di E20 Fr* F g h H: He hc KL Ka K
cascade slope airewater interfacial area (m2) probability of nullity dissolved oxygen concentration (mg/L). dissolved oxygen saturation concentration (mg/L) upstream dissolved oxygen concentration (mg/L) downstream dissolved oxygen concentration (mg/L) oxygen concentration at airewater interface (mg/L) diffusion constant of oxygen in the water (m2.s1) diffusivity of oxygen at T, Ti temperatures (m2.s1) aeration efficiency at 20 C roughness Froude number as defined by Baylar et al. (2006) Fr* ¼ qW/(g.sin a.h3)1/2 oxygen transfer rate (mg.L1.s1) gravity constant (m/s2) step height (m) total cascade height (m) Henry constant for O2 (Pa. L. mg1) critical water depth (m) water side global transfer coefficient of oxygen (m.s1) air side global transfer coefficient of oxygen (m.s1) global transfer coefficient of oxygen (m.s1)
concentration gradient between the two phases as well as exchange surface, this process is slow in calm water. However, oxygen transfer could be particularly accelerated by hydraulic structures such as stepped cascades. Generally, oxygen transferred on a stepped cascade is quantitatively equivalent to a transfer on several kilometers of linear streams (Baylar et al., 2006). Artificial stepped cascades have been used for more than 3500 years. These structures were used in aqueducts in some antic Roman cities (Chanson, 2001). In civil engineering, these structures are recognized as efficient energy dissipaters used to prevent erosion and damage of dikes and dams. Moreover, stepped cascades could be used to eliminate chlorine, tastes and odors of drinkable waters (Baylar et al., 2010) and for volatile organic compounds stripping (Toombes and Chanson, 2000). The aeration potential of these structures has been extensively studied by several authors. Gameson (1957) is one of the first authors to be interested in the aeration by stepped cascades. Since then, several authors studied oxygen transfer by these hydraulic structures. The aeration efficiency was measured by Tebbutt (1972) on a laboratory stepped cascade essentially for discharges below 4 L/s, cascade total height was of 1.8 m and a maximum steps number of 25 were tested. Essery et al. (1978) proposed a correlation to predict cascades aeration efficiency with steps height between 0.025 and 0.5 m and for discharges between 1.5 L/s and 22 L/s. Toombes and Chanson (2005) studied the oxygenation on a stepped waterway with low chute slope (about 3.4 , the steps length was about 17 time more important than steps heights) and for high discharges between 19 and 300 L/s. Baylar et al. 2006, 2007b, 2007c, 2010); Baylar and
l L La Li n Po2 q qw r, ri RH
r Re s, si T t m, mi V W x Xi, Yi
195
step length (m) cascade length (m) length of aerated flow on the stepped cascade (m) length of non aerated flow on the stepped cascade (m) number of steps partial pressure of O2 in the air (Pa) water flowrate (m3/s) water flowrate per unit of width (m2/s) density of water at T, Ti temperatures (Kg.m3) inflow hydraulic radius (m) ¼ the ratio of the cross sectional area to the wetted perimeter of inflow channel aeration deficit ratio Reynolds number airewater surface tension at T, Ti temperatures (N.m1) water temperature ( C) the Student test’s parameter dynamic viscosities of water at T, Ti temperatures (Kg. m1.s1) water volume on the cascade (m3) channel width (m) pseudo-roughness height (m) Baylar et al. (2007b) correlation’s coefficients
Bagatur (2007); Baylar and Emiroglu (2003) conducted a large number of water aeration experiments on laboratory stepped cascade. They characterized the oxygenation by several empirical correlations for discharges between 5 L/s and 50 L/s; cascade total height was between 1.2 and 2.5 m and maximum steps number of 50. Generally, previous correlations neglect the effects of some significant parameters and are only applicable on restricted conditions. So, Essery’s correlation ignores viscosity and steps length effects. Baylar et al. correlations are purely empirical and neglect the effects of steps number on aeration. The aim of this paper is to develop a more general correlation for aeration efficiency prediction on stepped cascade, useful for design in wide range of parameters. The experimental measures as well as the data collected in the literature are used. Dimensional analysis was used to determine the type and the number of parameters to be considered in aeration efficiency modeling.
2.
Bibliographic review
2.1.
Air water mass transfer
According to Whitman and Lewis theory of double film, the interface between two fluids can be considered as two stagnant thin films where solute transfer takes place; one on the liquid side described by the coefficient KL and the other on the gas side (Roustan, 2003). For slightly soluble gases in water, the resistance to the transfer liquid side is much more important than the one on the gas side.
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Fig. 1 e Flow regimes over stepped cascade: (a) nappe flow, (b) transition flow, (c) skimming flow. The transfer rate of oxygen from air to water can be expressed as:
E20 ¼ 1 ð1 EÞ1=f
(5)
a a Po2 C 4 ¼ kL $ $ðCi CÞ ¼ KL $ $ V V He
f ¼ 1 þ 2:1$102 ðT 20Þ þ 8:26$ 105 ðT 20Þ2
(6)
(1)
Po2/.He represents the oxygen concentration in water which is in equilibrium with oxygen concentration on gaseous phase. This quantity represents the saturation oxygen concentration in water. Then: a 4 ¼ KL $ $ðCS CÞ V
(2)
Transfer coefficient KL determination requires informations about the system hydrodynamics to reach the value of oxygen concentration within the studied system. However, aeration can be evaluated by the aeration efficiency E (Gameson, 1957) without knowledge of the system hydrodynamics by measuring three concentrations: CU, CD and CS (respectively upstream, downstream and saturation concentrations). For a given aeration system, with no oxygen consumption, aeration efficiency E could be calculated between upstream and downstream as: CD CU 1 ¼E¼1 r CS CU
(3)
The value of aeration efficiency E is between 0 and 1. When total transfer occurs E ¼ 1. E ¼ 0 in the case of no transfer. r is the oxygen deficit.
2.2.
Temperature impact on aeration efficiency
The diffusion coefficient of dissolved oxygen and oxygen saturation concentration in the water depend on temperature. Thus, aeration efficiency E is dependent on the temperature (Gameson et al., 1958; Demars and Manson, 2013). Gulliver et al. (1990) developed a relation involving the influence of viscosity, surface tension, water density and gas diffusivity in water effects to calculate the aeration efficiency for different temperature/compounds (indexed by i): Ln ð1 Ei Þ ¼ Ln ð1 EÞ$
1=2 3=4 3=5 17=20 Di m s r $ $ $ i mi si D r
(4)
Gulliver et al. (1990) showed a good agreement of this relation with the experimental data of Gameson (Gulliver et al., 1993). By using the relation (4), if E20 is the oxygenation efficiency in 20 C, the efficiency E at a temperature T can be calculated by the following relation:
2.3.
Flow hydrodynamic on stepped cascade
Hydrodynamic regime has an important impact on oxygen transfer. In a stepped cascade, there are three main flow regimes: nappe regime, transition regime and a skimming regime (Chanson, 1994): *Nappe regime: The flow is a series of small consecutive falls (Fig. 1a). Generally, this flow type occurs for low water flowrates and/or important steps lengths. *Skimming regime: In this flow regime, recirculation zones with horizontals axes take place between steps outer edges (Fig. 1c). These vortices are maintained between steps by shear stress. Generally, this flow type occurs for high flowrates and/or small steps lengths (Chanson, 1998). *Transition regime: It is the intermediate regime which marks the transition between nappe regime and skimming regime (see Fig. 1b). This regime was introduced for the first time by Ohtsu and Yasuda (1997). At the approach of the first step edge, the flow conditions change from sub-critical flow in a critical flow. The water depth passes by a critical value hc, which corresponds to a minimum of specific energy and to a Froude number equal to 1 (Chanson, 1994). hc ¼
2.4.
q2=3 W2=3 $g1=3
(7)
Aeration efficiency correlations
In the literature, there are several correlations of oxygenation prediction over stepped cascades. Several geometrical and hydrodynamic parameters were tested as the Table 1 shows. From equation (3) and equation (5) we have: 1=f CD CU E20 ¼ 1 1 CS CU
(8)
Essery et al. (1978) developed an empirical correlation from measures on a laboratory prototype with various heights of cascades/steps and different water flowrates. The relationship between these parameters and E20 is expressed as:
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Table 1 e Summary of aeration experiments on stepped cascades. Authors
Number of used data
Gameson (1957) Tebbutt (1972) Essery et al. (1978) Baylar and Emiroglu (2003); Baylar et al. (2006) Toombes and Chanson (2005) This paper
E20
Geometric parameters
Hydrodynamic parameters
h (m)
l (m)
W (m)
H (m)
q (10-3 m3/s)
hc (m)
e 55 e 126
e 0.05e0.254 0.025e0.5 0.05e0.15
e 0.07e0.254 e 0.07e0.6
e 0.3e0.15 0.15 0.3
0.9-2.2 1.8 2 1.2e2.5
e 0.084e3.9 1.5e21.75 5e50
e 0.002e0.025 0.0047e0.465 0.03e0.15
12 205
0.1433 0.05e0.10
2.4 0.10e0.14
0.5 0.15e0.3
1.72 0.25e0.5
19e300 0.3e2.5
0.086e0.132 0.007e0.03
! H hc ¼ 1 exp pffiffiffiffiffiffi 0; 427 þ 0; 31 h gh
(9)
hc is the critical depth: the flow depth at a Froude number equal to 1. Calculated E20 increases with cascade total height H and water discharge. However, Baylar et al. (2006) experiments in skimming flow showed an aeration efficiency decrease with discharge increase. For a given cascade total height H, increasing steps height h implies a decreasing in calculated E20. Besides, this model does not consider the steps length as a parameter influencing oxygen transfer. Baylar et al. (2006, 2007b, 2010) have developed several aeration efficiency correlations. Baylar et al. (2006) have calculated the aeration as a function of chute slope, step height h and of a Froude number defined in terms of macroroughness (the cascade steps are considered as roughness). Froude number Fr* is defined as: Fr* ¼ qW/(q$ina$ h3)1/2 Then, aeration efficiency at 20 C is:
Correlations (12) and (13) do not consider number of cascade’s steps effect on oxygen uptake. Yi and Xi are functions of the flow regimes as Table 2 shows. The experiments of Tebbutt (1972) (Table 1) were realized at 9 C and 14 C. These data were corrected using the relation of Gulliver et al. (1990) (equations (5) and (6)) in order to estimate the aeration efficiency at 20 C.
3.
Experimental section
La represents the length of aerated flow which is the difference between cascade length L and the length of nonaerated flow Li (equation (11)).
The experimental set-up design used to collect experimental data is given Fig. 2. Compared to a real water stream, it offers two main advantages: firstly, the possibility to vary height, length, width and number of cascade steps in a given range of dimensions, and secondly to remove dissolved oxygen; it’s afterwards possible to observe the re-aeration process. The dissolved oxygen contained in tap water is first partially consumed by adding sodium sulfite and Cobalt dichloride as catalyst in a 1 m3 volume agitated tank. 50e55 g/ m3 of Na2SO3 were used to deoxygenate water in order to set oxygen concentration in the storage tank between 1 mg/L and 2.5 mg/L for temperatures between 9 and 21 C. Dissolved oxygen in water is consumed according to the following reaction (Bo et al., 2005):
Li 2:98 ¼ cos a 1:88Fr0:35 þ 0:17Fr h
SO2 3 þ
1;34 La E20 ¼ 1 8:24 104 Fr1:65 þ 0:50 F1:34 r h 0;50 0;28 L a þ2; 23 102 h
(10)
(11)
Later, Baylar et al. (2007c) developed an empirical correlation for E20 prediction on a single step in nappe flow depending on 3 parameters (qw, h, a): i h 1:594 E1ð20Þ ¼ 1 exp 5:730 q0:035 h0:998 ðcosðaÞÞ12:042 ðsinðaÞÞ w
(12)
with 14 < a < 30 . According to Baylar et al. (2007b), the sensibility of the aeration to the various parameters varies with flow regimes. They developed a correlation with 3 sets of coefficients considering the flow regime effect. The aeration efficiency can be given by the following relation: " X
E20 ¼ 1 exp 1 þ X1 $ðsin aÞ 2 $
X3 hc $ h !#
X4 X5
L Y1 $hY2 $qY3 $ðcos aÞY4
(13)
1 Co2þ O2 / SO2 4 2
(14)
The variation of oxygen Henry’s constant at this amount of sulfite remains negligible and then has not significant effect on oxygen solubility in water (Watson et al., 1998). The partially deoxygenated water flows above the cascade. Dissolved oxygen concentrations measured in upstream and downstream, as well as saturation concentration by METTLER TOLEDO SG9 oximeters provided with “Inlab Optiox” optical electrodes allowing aeration efficiency E calculation. In our experiments, several configurations have been tested: *cascade with 3, 4 and 5 steps h ¼ l ¼ 10 cm. *cascade with 3, 4 and 5 steps h ¼ 10 cm and l ¼ 14 cm *cascade with 5, 8 and 10 steps h ¼ 5 cm and l ¼ 10 cm Low cascades heights (between 0.25 and 0.5 m) were chosen to study oxygen transfer enhancement in small water streams. So, as mentioned above, number of steps is varying between 3 and 10. This choice was made to obtain maximum
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Table 2 e Xi, Yi variation with flow regimes (Baylar’s, 2007b) (equation (14)). Flow regime Skimming Transition Nappe
Y1
Y2
Y3
Y4
X1
X2
X3
X4
X5
2.643 3.256 6.834
0.508 0.043 0.749
0.896 0.504 0.205
1.708 0.991 0.915
1.704 0.250 0.265
0.448 3.292 2.661
0.078 0.063 0.007
0.419 2.705 2.057
0.992 1.387 1.575
experimental data to investigate number of steps effect with a minimum value of height of 5 cm. Channel was large enough to avoid edge effects.
4.
Results and discussion
4.1.
Aeration efficiency correlation determination model
Aeration in stepped cascades was modeled referring to dimensional analysis by Buckingham theorem application (Le Moullec et al., 2008). It stipulates that any physical quantity can be expressed with peq dimensionless numbers (p is the number of parameters and q is the number of their physical dimensions). In the case of cascade, oxygen transfer depends on several parameters. After studying different experiments, we propose to use 8 significant parameters in the model. They can be classified as proposed in Table 3. (L, M and T symbolize respectively the fundamental physical dimensions of length, mass and time). Temperature effect on the aeration is included on the equation (8). Gravity effect is implicitly included in hc expression. These 8 parameters are expressed by 3 dimensions. Therefore, 5 dimensionless numbers are necessary and sufficient to model aeration efficiency over stepped cascade:
RH ¼
The resulting equation of oxygen uptake on cascade is a function of the chosen dimensionless numbers and 5 coefficients as: a2 a3 a4 H hc h $ $ E20 ¼ A$Rae 1 $ h l H
(15)
W$hc 2$hc þ W
This number represents a comparison between inertia forces and viscosity forces. It characterizes the turbulence level.
(16)
A, a1, a2, a3 and a4 were determined by fitting available data (from literature and our experimental data) with the equation (16). From a design point of view, the aeration efficiency can be also expressed as: E20 ¼ A$Rae 1 $na2 $
a3 hc $ðtan aÞa4 H
(17)
n is the number of steps, a is the global cascade chute slope.
4.2.
The inflow Reynolds number defined as: q$RH $r Re ¼ hc $W$m
The ratio H/h represents the number of cascade steps effect, when steps have uniform height. The ratio hc/H related the water critical depth and the total cascade height. It expresses the gravity and inflow conditions effects on aeration. The ratio h/l expresses the effect of steps geometry on aeration. The efficiency E20 is the ratio between the transferred quantity of oxygen and the maximal quantity susceptible to be transferred at 20 C (equation (3)).
Flow regime prediction
As described below, the flow regimes are identified by simple visual observation according to water behavior on the cascade. It is a function of inflow conditions (critical water depth), and steps dimensions (h, l). Several authors determined the limits between the flow regimes by empirical equations. Increasing steps height and decreasing the ratio h/l and flow rate q further nappe flow. Inversely, decreasing steps height and increasing flow rate q and (h/l) ratio furthers skimming regime.
Fig. 2 e Scheme of experimental apparatus.
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2.5
Table 3 e Parameters influencing stepped cascade aeration. Properties Physical properties
Flow properties Geometric properties
Frx
Parameters
Fundamental units
r water density m water dynamic viscosity Q water flowrate h step height hc flow critical height l steps length W cascade width H stepped cascade total height
M.L3 M.L1.T1
1.5
1
L3.T1 L L L L L
0.5 this paper Baylar et al (2003, 2006) 0 0
For their 126 experiments, Baylar and Emiroglu (2003); Baylar et al. (2006) mentioned the flow regimes. Under their experimental conditions, the 3 regimes were observed. In our case, skimming flow was not reached under our experimental conditions because it is not efficient as nappe/transition for aeration. Only nappe and transition flows were observed. The twelve measurements of efficiency from Toombes and Chanson (2005) were realized in a nappe flow. On the other hand, there was no indication about flow regimes for Tebbutt’s (1972) data. According to Baylar and Emiroglu (2003); Baylar et al. (2006) observations and our study, flow regimes can be distinguished by a dimensionless number Frx calculation. It compares inflow kinetic energy to gravity forces at the scale of “macro-roughness” represented by the steps. qw =hc Frx ¼ pffiffiffiffiffiffiffiffi ¼ g$x
h c h$cos a tan hl
!1=2 (18)
x represents the height of the pseudo-roughness, which is the distance between the internal edge and plan passing by the steps outer edges For Baylar and Emiroglu (2003); Baylar et al. (2006) observations, nappe flow occurred for Frx lower than 0.8 and skimming flow took place when Frx was over 1. For our experiments, the limit between nappe and transition flows was observed for Frx about 0.5. For both experiments, nappe flow took place for Frx under 0.5e0.8 and skimming flow when Frx is over 1.1. Thus, Tebbutt (1972) data were classified on the basis of the number Frx (nappe flow if Frx 0.5; transition flow if 0.5 Frx 1.1 and skimming flow when Frx 1.1) as showed in the Fig. 3. These results are valid for stepped channel slope between 14 and 50 and not applicable to Toombes and Chanson (2005) experiments with too low channel slope about 3.5 .
4.3.
2
Nappe
Transition
Skimming
Fig. 3 e Variation of flow regimes over stepped cascade with Frx.
flow regime. In another terms, the effects of operational and geometric parameters on the aeration varies from one regime to another. Preliminary simulations have shown that it is possible to have a good representation of all the data with only one correlation as illustrated Fig. 4. However, it clearly appears that the data issue from each regime is not statistically distributed around the correlation straight line. Therefore, taking into account the process, we considered it would be better to determine three different sets of parameters based on the same correlation to characterize the aeration of each regime. Table 4 represents the coefficients A, a1, a2, a3 and a4 for each flow regime. Statistics based on the calculation of student parameter t allowed testing the sensibility of oxygenation efficiency to the used dimensionless number. In this context, the calculated coefficients were compared to their standard deviations to test whether they are significantly different from 0 by estimating the probability of nullity b. In transition flow, the coefficient a1 shows a probability of 0.02. In this case, the Reynolds number effect cannot be neglected. In skimming regime, the coefficient a4 and effect of
Aeration efficiency correlation
The coefficients A, a1, a2, a3 and a4 were determined by a multi-linear regression using least square method. 193 collected data and 205 experiments realized on our pilot were used. The interfaces between three phases (solid represented by the cascade, liquid and gaseous) changes considerably with
Fig. 4 e Comparison between experimental and calculated E20 with the global correlation developed for all flow regimes.E20 [0:211$RL0:033 $n0:445 $ðtan aÞ0:083 e
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Table 4 e Correlation coefficients, student test. Global model
Model with 3 flow regimes Nappe
A a1 a2 a3 a4
Transition
Skimming
Value
t
b
Value
t
b
Value
t
b
Value
t
b
0.211 0.033 0.445 3.091E-05 0.083
16.7751107 5.02750138 24.3891692 0.00185722 3.46394616
0.000 0.000 0.000 0.998 0.000
0.331 0.048 0.687 0.169 0.234
14.394007 9.46332575 29.695529 9.29440977 9.92339927
0.000 0.000 0.000 0.000 0.000
0.403 0.042 0.670 0.297 0.211
3.007 2.420 11.308 3.861 2.623
0.004 0.020 0.000 0.000 0.012
0.14 103 0.536 0.348 1.186 0.031
5.247 4.390 5.764 9.042 0.397
0.000 0.000 0.000 0.000 0.692
cascade slope a can be neglected since the nullity probability is b ¼ 0.692. This coefficient characterizes the weight of the steps dimensions on the aeration efficiency. This ratio (h/l) has not much impact on the aeration since step’s corners are fulfilled with water in skimming flow. Otherwise, turbulence has more impact on aeration in skimming regime: a1 reach maximum value in this regime. Between nappe and transition regimes, there are a few changes of turbulence effects, the number of steps and the cascade slope a. In skimming flow, it was observed the higher hc the higher water volume over the cascade steps and the less interfacial area. Thus, aeration decreases when hc increases. For a given total cascade height H, in skimming flow, aeration is less efficient with increasing number of steps unlike the cases of nappe and transition flows. Figs. 5e7 illustrate the agreement between the calculated values and the experimental efficiencies E20 for nappe, transition and skimming flows. These correlations could be applied for wide range of geometrical and hydrodynamic parameters. In order to analyze the effect of each dimensionless number, aeration efficiency response to the unique variation of each number is represented in supplementary informations (Figures S1 to S5). Therefore, it will be possible to see different flow regimes in the same figure. In skimming flow, aeration efficiency decreases when Reynolds number, hc/H and steps number increase. In these flow conditions, E20
is insensible to chute slope variation but remains at the higher level. In nappe and transition flow, aeration efficiency increases with increasing all dimensionless numbers. The chute slope a effect on aeration efficiency illustration was separated (Figures S2 and S3 in supplementary data) because it was impossible to fix all the parameters (except a) simultaneously for the three flow regimes. It is difficult to separate Re and (hc/H) effects since they both depends on water flowrate. In nappe flow, increasing Re and (hc/H) induces an improvement in aeration efficiency unlike skimming flow. Despite the Reynolds number exponent reaches maximum value for skimming flow, aeration efficiency decreases when this number increases. This can be explained by the preponderant effect of (hc/H) on Reynolds number. Slightly over estimation of aeration efficiency can be noticed on Fig. 7 for E20 over 0.7 and under estimation for E20 under 0.3. Nevertheless, aeration efficiency estimation was found accurate enough.
Fig. 5 e Comparison between experimental and calculated E20 with correlation developed in this paper for the nappe flow.E20 [0:331$RL0:048 $n0:687 $ðhc =HÞ0:169 $ðtan aÞ0:234 e
Fig. 6 e Comparison between experimental and calculated E20 with correlation developed in this paper for the transition $n0:670 $ðhc =HÞ0:297 $ðtan aÞ0:211 flow.E20 [0:403$RL0:042 e
4.4.
Correlations comparison
Fig. 8 shows a good agreement between aeration efficiency values calculated by the relation proposed in this study (with the 3 different regimes) and the experimental values with a relative error about 16.5%. The correlation of Baylar et al. (2006) is not applicable to the data from Toombes and Chanson (2005), the values of Li
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201
parameters and overcome the scale effects since local aeration prediction is not calculated for each step.
4.5.
Aeration on non uniform steps cascade
In order to restore ecological properties of some watercourses, stepped cascade could be constructed for oxygenation improvement. Cascade can be composed of steps at different height/length. In this case, aeration efficiency can be estimated by the following relation (Gameson, 1957): Ecascade ¼ 1 ð1 E1 Þ$ð1 E2 Þ.ð1 EN Þ
Fig. 7 e Comparison between experimental and calculated E20 with correlation developed in this paper for skimming $nL0:348 $ðhc =HÞL1:186 flow.E20 [1:4 $10L4 $R0:536 e
where Ei (i Є {1, n}) are the aeration efficiency of blocks with uniform h and l. By applying the correlation obtained on these blocks, the total aeration obtained on the stepped cascade can be estimated as: Ecascade ¼ 1
5. calculated for the stepped cascade of Toombes and Chanson leads to negative values. Baylar et al. (2010) correlation is only applicable with one step. Highly turbulent flows on stepped cascade are difficult to model at the local scale because the behavior varied from the upper step to the lower one. Moreover, the relative invariance of bubble size, and scale effects are difficult to predict due to the great number of coupling parameters. One of the solutions would be to simulate in detail the interface using Volume Of Fluid simulation. This is time consuming and the solution for one configuration does not allow an easy scale-up or scaledown and it requires simulating each new configuration. This is why, as explained in paragraph 4.1, we developed a global correlation based on aeration efficiency prediction regardless to multiphase flow details. It provides a reasonable accuracy of the global aeration prediction for a large range of
(19)
a2 a3 a4 N Y hi hci hi $ $ 1 A$Raei1 $ H H li i i i¼1
(20)
Conclusion
A model of aeration efficiency in stepped cascades with horizontal steps was elaborated by using dimensional analysis and 398 experimental data. These data have been taken from the literature but also measured on an experimental set-up design. The obtained correlation is composed of five dimensionless numbers that characterized the aeration behavior in stepped cascades. It allows predicting the experimental data issued from a wide range of operating conditions with a satisfactory average relative error of 16.5%. Three hydrodynamic regimes were found on stepped cascades: nappe, transition and Skimming regimes. The effect of the studied parameters on the aeration strongly differs according to the flow regime. Also, three different sets of coefficients were optimized for each flow regime in order to better consider the physic of the aeration prediction. This correlation gives a satisfactory estimation of oxygen transfer in existing cascades and could be used to design new cascades to improve oxygenation and therefore self-purification processes. Since cascades provide aeration without operating cost, the proposed correlation could be also useful to design cascades downstream WWTPs to eliminate residual pollution or even in wetlands.
Acknowledgments This work was funded by the French National Research Agency (EPEC ANR-10-ECOT-007-01). Authors would like to thank the mechanical workshop of the LRGP for construction and installation of the laboratory pilot.
Fig. 8 e Comparison between experimental and calculated E20 with the 3 correlations taking into account the 3 different flow regimes developed in this paper.
Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.watres.2014.02.022.
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references
Baylar, A., Bagatur, T., 2007. Aeration efficiency with nappe flow over stepped cascade. Proc. Inst. Civ. Eng. ewater Manag. 160 (1), 43e50. Baylar, A., Bagatur, T., Emiroglu, E., 2007b. Prediction of oxygen content of nappe, transition and skimming flow regimes in stepped-channel chutes. J. Environ. Eng. Sci. 6, 201e208. Baylar, A., Emiroglu, E., Bagatur, T., 2006. An experimental investigation of aeration performance in stepped spillways. Water Environ. J. 20, 35e42. Baylar, A., Emiroglu, M.E., 2003. Study of aeration efficiency at stepped channels. Proc. Inst. Civ. Eng. Water Marit. Eng. 156 (WM3), 257e263. Baylar, A., Unsal, M., Ozkan, F., 2007c. GEP modeling of oxygen transfer efficiency prediction in aeration cascades. KSCE J. Civ. Eng. 15 (5), 799e804. Baylar, A., Unsal, M., Ozkan, F., 2010. Hydraulic structures in water aeration process. Water Air Soil. Pollut. 210, 87e100. Bo, Z., Yan, L., Huiling, T., Yuqun, Z., Long, Z., Jian, S., Changhe, C., 2005. Study on the reaction rate of sulfite oxidation with cobalt ion catalyst. Chem. Eng. Sci. 60, 863e868. Chanson, H., 1994. Hydraulics of nappe flow regime above stepped chutes and spillways. Aust. Civ. Eng. Trans. IE Aust. CE36 (1), 69e76. Chanson, H., 1998. Review of studies on stepped channel flows. In: Workshop on Flow Characteristics Around Hydraulic Structures and River Environment. Nihon University, Tokyo, Japan (November). Chanson, H., 2001. Historical development of stepped cascades for dissipation of hydraulic energy. Trans. Newcom. Soc. 72, 295e318.
Demars, B.O.L., Manson, J.R., 2013. Temperature dependence of stream aeration coefficients and the effect of water turbulence: a critical review. Water Res. 47, 1e15. Essery, I.T.S., Tebbutt, T.H.Y., Rajaratnam, S.K., 1978. Design of Spillways for Reaeration of Polluted Waters. CIRIA Report No. 72, Jan. London, UK, p. 36. Gameson, A.L.H., 1957. Weirs and aeration of rivers. J. Inst. Water Eng. 11, 477e490. Gameson, T.K., VanDyke, K., Ogden, C., 1958. The effect of temperature on aeration at weirs. Water Water Eng. 62 (5), 489e492. Gulliver, J.S., Thene, J.R., Rindels, A.J., 1990. Indexing gas transfer in self aerated flow. J. Environ. Eng. ASCE 116, 503e523. Gulliver, J., Rindels, A., 1993. Measurement of air-water oxygen transfer at hydraulic structures. J. of Hydraulic Structures 119, 327e349. Le Moullec, Y., Potier, O., Gentric, C., Leclerc, J.-P., 2008. A general correlation to predict axial dispersion coefficients in aerated channel reactors. Water Res. 42, 1767e1777. Ohtsu, I., Yasuda, Y., 1997. Characteristics of flow conditions of stepped channels. In: Proc. 27th IAHR Congress, San Francisco, USA, pp. 583e588. Roustan, M., 2003. Transfert gaz-liquide dans les proce´de´s de traitement des eaux et des effluents gazeux. Tech&Doc, Lavoisier: Paris. Tebbutt, T.H.Y., 1972. Some studies on reaeration in cascades. Water Res. 6, 297e304. Toombes, L., Chanson, H., 2000. Air water flow and gas transfer at aeration cascades a comparative study of smooth and stepped chutes. Hydraul. Stepped Spillw. 1, 77e84. Toombes, L., Chanson, H., 2005. Air water mass transfer on a stepped waterway. J. Environ. Eng. ASCE 131, 1377e1386. Watson, C.C., Walters, R.W., Hogan, S.A., 1998. Aeration performance of low drop weirs. J. Hydraul. Eng. 124, 65e71.
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Aeration efficiency over stepped cascades: Better predictions from flow regimes Hatem Khdhiri*, Olivier Potier, Jean-Pierre Leclerc Laboratoire Re´actions et Ge´nie des Proce´de´s, UMR 7274 CNRS, Universite´ de Lorraine, 1 rue Grandville, 54001 Nancy, France
article info
abstract
Article history:
Stepped cascades are recognized as high potential airewater gas exchangers. In natural
Received 19 September 2013
rivers, these structures enhance oxygen transfer to water by creating turbulence at inter-
Received in revised form
face with increasing air entrainment in water and airewater surface exchange. Stepped
3 February 2014
cascades could be really useful to improve the natural self-purification process by
Accepted 6 February 2014
providing oxygen to aerobic micro-organisms. The aeration performance of these struc-
Available online 17 February 2014
tures depends on several operating and geometrical parameters. In the literature, several empirical correlations for aeration efficiency prediction on stepped cascades exist. Most of
Keywords:
these correlations are only applicable for operating and geometrical parameters in the
Oxygen transfer
range of which they have been developed. In this paper, 398 experimental sets of data
Stepped spillways design
(from our experiments and collected from literature) were used to develop a correlation for
Modeling
aeration prediction over stepped cascades derived from dimensional analysis and
Self-purification
parameterized for each individual flow regime in order to consider change in flow regime
River
effect on oxygen transfer. This new correlation allowed calculating the whole set of data
Cascade
obtained for cascades with steps heights between 0.05 m and 0.254 m, cascade total height between 0.25 m and 2.5 m, for discharges per unit of width ranging from 0.28 103 m2/s to 600 103 m2/s and for cascade steps number between 3 and 25. In these ranges of parameters, standard deviation for aeration efficiency estimation was found to be less than 17%. Finally, advices were proposed to help and improve the structure design in order to improve aeration. ª 2014 Elsevier Ltd. All rights reserved.
1.
Introduction
Dissolved oxygen concentration is an important indicator of water quality in natural environment. Oxygen deficit in aquatic systems occurs when water is highly polluted or in calm or stagnant canal or river. Natural water systems could have a potential to eliminate the pollution and restore ecological properties by self-purification processes. The main
process requires dissolved oxygen for chemical and mainly biological mechanisms involving aerobic microorganisms. In case of O2 deficit, biological processes and chemical oxidations are limited. Then, natural system is not able to reduce pollution concentration with harmful consequences on water quality and aquatic species. Thus, enhancing oxygenation in watercourses represents a good solution to improve selfpurification process. If oxygen is spontaneously transferred by diffusion phenomenon, which depend essentially on
* Corresponding author. Tel.: þ33383175170. E-mail addresses: [email protected] (H. Khdhiri), [email protected] (O. Potier), [email protected] (J.-P. Leclerc). 0043-1354/$ e see front matter ª 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.watres.2014.02.022
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Nomenclature a a b C Cs CU CD Ci Do2 D, Di E20 Fr* F g h H: He hc KL Ka K
cascade slope airewater interfacial area (m2) probability of nullity dissolved oxygen concentration (mg/L). dissolved oxygen saturation concentration (mg/L) upstream dissolved oxygen concentration (mg/L) downstream dissolved oxygen concentration (mg/L) oxygen concentration at airewater interface (mg/L) diffusion constant of oxygen in the water (m2.s1) diffusivity of oxygen at T, Ti temperatures (m2.s1) aeration efficiency at 20 C roughness Froude number as defined by Baylar et al. (2006) Fr* ¼ qW/(g.sin a.h3)1/2 oxygen transfer rate (mg.L1.s1) gravity constant (m/s2) step height (m) total cascade height (m) Henry constant for O2 (Pa. L. mg1) critical water depth (m) water side global transfer coefficient of oxygen (m.s1) air side global transfer coefficient of oxygen (m.s1) global transfer coefficient of oxygen (m.s1)
concentration gradient between the two phases as well as exchange surface, this process is slow in calm water. However, oxygen transfer could be particularly accelerated by hydraulic structures such as stepped cascades. Generally, oxygen transferred on a stepped cascade is quantitatively equivalent to a transfer on several kilometers of linear streams (Baylar et al., 2006). Artificial stepped cascades have been used for more than 3500 years. These structures were used in aqueducts in some antic Roman cities (Chanson, 2001). In civil engineering, these structures are recognized as efficient energy dissipaters used to prevent erosion and damage of dikes and dams. Moreover, stepped cascades could be used to eliminate chlorine, tastes and odors of drinkable waters (Baylar et al., 2010) and for volatile organic compounds stripping (Toombes and Chanson, 2000). The aeration potential of these structures has been extensively studied by several authors. Gameson (1957) is one of the first authors to be interested in the aeration by stepped cascades. Since then, several authors studied oxygen transfer by these hydraulic structures. The aeration efficiency was measured by Tebbutt (1972) on a laboratory stepped cascade essentially for discharges below 4 L/s, cascade total height was of 1.8 m and a maximum steps number of 25 were tested. Essery et al. (1978) proposed a correlation to predict cascades aeration efficiency with steps height between 0.025 and 0.5 m and for discharges between 1.5 L/s and 22 L/s. Toombes and Chanson (2005) studied the oxygenation on a stepped waterway with low chute slope (about 3.4 , the steps length was about 17 time more important than steps heights) and for high discharges between 19 and 300 L/s. Baylar et al. 2006, 2007b, 2007c, 2010); Baylar and
l L La Li n Po2 q qw r, ri RH
r Re s, si T t m, mi V W x Xi, Yi
195
step length (m) cascade length (m) length of aerated flow on the stepped cascade (m) length of non aerated flow on the stepped cascade (m) number of steps partial pressure of O2 in the air (Pa) water flowrate (m3/s) water flowrate per unit of width (m2/s) density of water at T, Ti temperatures (Kg.m3) inflow hydraulic radius (m) ¼ the ratio of the cross sectional area to the wetted perimeter of inflow channel aeration deficit ratio Reynolds number airewater surface tension at T, Ti temperatures (N.m1) water temperature ( C) the Student test’s parameter dynamic viscosities of water at T, Ti temperatures (Kg. m1.s1) water volume on the cascade (m3) channel width (m) pseudo-roughness height (m) Baylar et al. (2007b) correlation’s coefficients
Bagatur (2007); Baylar and Emiroglu (2003) conducted a large number of water aeration experiments on laboratory stepped cascade. They characterized the oxygenation by several empirical correlations for discharges between 5 L/s and 50 L/s; cascade total height was between 1.2 and 2.5 m and maximum steps number of 50. Generally, previous correlations neglect the effects of some significant parameters and are only applicable on restricted conditions. So, Essery’s correlation ignores viscosity and steps length effects. Baylar et al. correlations are purely empirical and neglect the effects of steps number on aeration. The aim of this paper is to develop a more general correlation for aeration efficiency prediction on stepped cascade, useful for design in wide range of parameters. The experimental measures as well as the data collected in the literature are used. Dimensional analysis was used to determine the type and the number of parameters to be considered in aeration efficiency modeling.
2.
Bibliographic review
2.1.
Air water mass transfer
According to Whitman and Lewis theory of double film, the interface between two fluids can be considered as two stagnant thin films where solute transfer takes place; one on the liquid side described by the coefficient KL and the other on the gas side (Roustan, 2003). For slightly soluble gases in water, the resistance to the transfer liquid side is much more important than the one on the gas side.
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Fig. 1 e Flow regimes over stepped cascade: (a) nappe flow, (b) transition flow, (c) skimming flow. The transfer rate of oxygen from air to water can be expressed as:
E20 ¼ 1 ð1 EÞ1=f
(5)
a a Po2 C 4 ¼ kL $ $ðCi CÞ ¼ KL $ $ V V He
f ¼ 1 þ 2:1$102 ðT 20Þ þ 8:26$ 105 ðT 20Þ2
(6)
(1)
Po2/.He represents the oxygen concentration in water which is in equilibrium with oxygen concentration on gaseous phase. This quantity represents the saturation oxygen concentration in water. Then: a 4 ¼ KL $ $ðCS CÞ V
(2)
Transfer coefficient KL determination requires informations about the system hydrodynamics to reach the value of oxygen concentration within the studied system. However, aeration can be evaluated by the aeration efficiency E (Gameson, 1957) without knowledge of the system hydrodynamics by measuring three concentrations: CU, CD and CS (respectively upstream, downstream and saturation concentrations). For a given aeration system, with no oxygen consumption, aeration efficiency E could be calculated between upstream and downstream as: CD CU 1 ¼E¼1 r CS CU
(3)
The value of aeration efficiency E is between 0 and 1. When total transfer occurs E ¼ 1. E ¼ 0 in the case of no transfer. r is the oxygen deficit.
2.2.
Temperature impact on aeration efficiency
The diffusion coefficient of dissolved oxygen and oxygen saturation concentration in the water depend on temperature. Thus, aeration efficiency E is dependent on the temperature (Gameson et al., 1958; Demars and Manson, 2013). Gulliver et al. (1990) developed a relation involving the influence of viscosity, surface tension, water density and gas diffusivity in water effects to calculate the aeration efficiency for different temperature/compounds (indexed by i): Ln ð1 Ei Þ ¼ Ln ð1 EÞ$
1=2 3=4 3=5 17=20 Di m s r $ $ $ i mi si D r
(4)
Gulliver et al. (1990) showed a good agreement of this relation with the experimental data of Gameson (Gulliver et al., 1993). By using the relation (4), if E20 is the oxygenation efficiency in 20 C, the efficiency E at a temperature T can be calculated by the following relation:
2.3.
Flow hydrodynamic on stepped cascade
Hydrodynamic regime has an important impact on oxygen transfer. In a stepped cascade, there are three main flow regimes: nappe regime, transition regime and a skimming regime (Chanson, 1994): *Nappe regime: The flow is a series of small consecutive falls (Fig. 1a). Generally, this flow type occurs for low water flowrates and/or important steps lengths. *Skimming regime: In this flow regime, recirculation zones with horizontals axes take place between steps outer edges (Fig. 1c). These vortices are maintained between steps by shear stress. Generally, this flow type occurs for high flowrates and/or small steps lengths (Chanson, 1998). *Transition regime: It is the intermediate regime which marks the transition between nappe regime and skimming regime (see Fig. 1b). This regime was introduced for the first time by Ohtsu and Yasuda (1997). At the approach of the first step edge, the flow conditions change from sub-critical flow in a critical flow. The water depth passes by a critical value hc, which corresponds to a minimum of specific energy and to a Froude number equal to 1 (Chanson, 1994). hc ¼
2.4.
q2=3 W2=3 $g1=3
(7)
Aeration efficiency correlations
In the literature, there are several correlations of oxygenation prediction over stepped cascades. Several geometrical and hydrodynamic parameters were tested as the Table 1 shows. From equation (3) and equation (5) we have: 1=f CD CU E20 ¼ 1 1 CS CU
(8)
Essery et al. (1978) developed an empirical correlation from measures on a laboratory prototype with various heights of cascades/steps and different water flowrates. The relationship between these parameters and E20 is expressed as:
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Table 1 e Summary of aeration experiments on stepped cascades. Authors
Number of used data
Gameson (1957) Tebbutt (1972) Essery et al. (1978) Baylar and Emiroglu (2003); Baylar et al. (2006) Toombes and Chanson (2005) This paper
E20
Geometric parameters
Hydrodynamic parameters
h (m)
l (m)
W (m)
H (m)
q (10-3 m3/s)
hc (m)
e 55 e 126
e 0.05e0.254 0.025e0.5 0.05e0.15
e 0.07e0.254 e 0.07e0.6
e 0.3e0.15 0.15 0.3
0.9-2.2 1.8 2 1.2e2.5
e 0.084e3.9 1.5e21.75 5e50
e 0.002e0.025 0.0047e0.465 0.03e0.15
12 205
0.1433 0.05e0.10
2.4 0.10e0.14
0.5 0.15e0.3
1.72 0.25e0.5
19e300 0.3e2.5
0.086e0.132 0.007e0.03
! H hc ¼ 1 exp pffiffiffiffiffiffi 0; 427 þ 0; 31 h gh
(9)
hc is the critical depth: the flow depth at a Froude number equal to 1. Calculated E20 increases with cascade total height H and water discharge. However, Baylar et al. (2006) experiments in skimming flow showed an aeration efficiency decrease with discharge increase. For a given cascade total height H, increasing steps height h implies a decreasing in calculated E20. Besides, this model does not consider the steps length as a parameter influencing oxygen transfer. Baylar et al. (2006, 2007b, 2010) have developed several aeration efficiency correlations. Baylar et al. (2006) have calculated the aeration as a function of chute slope, step height h and of a Froude number defined in terms of macroroughness (the cascade steps are considered as roughness). Froude number Fr* is defined as: Fr* ¼ qW/(q$ina$ h3)1/2 Then, aeration efficiency at 20 C is:
Correlations (12) and (13) do not consider number of cascade’s steps effect on oxygen uptake. Yi and Xi are functions of the flow regimes as Table 2 shows. The experiments of Tebbutt (1972) (Table 1) were realized at 9 C and 14 C. These data were corrected using the relation of Gulliver et al. (1990) (equations (5) and (6)) in order to estimate the aeration efficiency at 20 C.
3.
Experimental section
La represents the length of aerated flow which is the difference between cascade length L and the length of nonaerated flow Li (equation (11)).
The experimental set-up design used to collect experimental data is given Fig. 2. Compared to a real water stream, it offers two main advantages: firstly, the possibility to vary height, length, width and number of cascade steps in a given range of dimensions, and secondly to remove dissolved oxygen; it’s afterwards possible to observe the re-aeration process. The dissolved oxygen contained in tap water is first partially consumed by adding sodium sulfite and Cobalt dichloride as catalyst in a 1 m3 volume agitated tank. 50e55 g/ m3 of Na2SO3 were used to deoxygenate water in order to set oxygen concentration in the storage tank between 1 mg/L and 2.5 mg/L for temperatures between 9 and 21 C. Dissolved oxygen in water is consumed according to the following reaction (Bo et al., 2005):
Li 2:98 ¼ cos a 1:88Fr0:35 þ 0:17Fr h
SO2 3 þ
1;34 La E20 ¼ 1 8:24 104 Fr1:65 þ 0:50 F1:34 r h 0;50 0;28 L a þ2; 23 102 h
(10)
(11)
Later, Baylar et al. (2007c) developed an empirical correlation for E20 prediction on a single step in nappe flow depending on 3 parameters (qw, h, a): i h 1:594 E1ð20Þ ¼ 1 exp 5:730 q0:035 h0:998 ðcosðaÞÞ12:042 ðsinðaÞÞ w
(12)
with 14 < a < 30 . According to Baylar et al. (2007b), the sensibility of the aeration to the various parameters varies with flow regimes. They developed a correlation with 3 sets of coefficients considering the flow regime effect. The aeration efficiency can be given by the following relation: " X
E20 ¼ 1 exp 1 þ X1 $ðsin aÞ 2 $
X3 hc $ h !#
X4 X5
L Y1 $hY2 $qY3 $ðcos aÞY4
(13)
1 Co2þ O2 / SO2 4 2
(14)
The variation of oxygen Henry’s constant at this amount of sulfite remains negligible and then has not significant effect on oxygen solubility in water (Watson et al., 1998). The partially deoxygenated water flows above the cascade. Dissolved oxygen concentrations measured in upstream and downstream, as well as saturation concentration by METTLER TOLEDO SG9 oximeters provided with “Inlab Optiox” optical electrodes allowing aeration efficiency E calculation. In our experiments, several configurations have been tested: *cascade with 3, 4 and 5 steps h ¼ l ¼ 10 cm. *cascade with 3, 4 and 5 steps h ¼ 10 cm and l ¼ 14 cm *cascade with 5, 8 and 10 steps h ¼ 5 cm and l ¼ 10 cm Low cascades heights (between 0.25 and 0.5 m) were chosen to study oxygen transfer enhancement in small water streams. So, as mentioned above, number of steps is varying between 3 and 10. This choice was made to obtain maximum
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Table 2 e Xi, Yi variation with flow regimes (Baylar’s, 2007b) (equation (14)). Flow regime Skimming Transition Nappe
Y1
Y2
Y3
Y4
X1
X2
X3
X4
X5
2.643 3.256 6.834
0.508 0.043 0.749
0.896 0.504 0.205
1.708 0.991 0.915
1.704 0.250 0.265
0.448 3.292 2.661
0.078 0.063 0.007
0.419 2.705 2.057
0.992 1.387 1.575
experimental data to investigate number of steps effect with a minimum value of height of 5 cm. Channel was large enough to avoid edge effects.
4.
Results and discussion
4.1.
Aeration efficiency correlation determination model
Aeration in stepped cascades was modeled referring to dimensional analysis by Buckingham theorem application (Le Moullec et al., 2008). It stipulates that any physical quantity can be expressed with peq dimensionless numbers (p is the number of parameters and q is the number of their physical dimensions). In the case of cascade, oxygen transfer depends on several parameters. After studying different experiments, we propose to use 8 significant parameters in the model. They can be classified as proposed in Table 3. (L, M and T symbolize respectively the fundamental physical dimensions of length, mass and time). Temperature effect on the aeration is included on the equation (8). Gravity effect is implicitly included in hc expression. These 8 parameters are expressed by 3 dimensions. Therefore, 5 dimensionless numbers are necessary and sufficient to model aeration efficiency over stepped cascade:
RH ¼
The resulting equation of oxygen uptake on cascade is a function of the chosen dimensionless numbers and 5 coefficients as: a2 a3 a4 H hc h $ $ E20 ¼ A$Rae 1 $ h l H
(15)
W$hc 2$hc þ W
This number represents a comparison between inertia forces and viscosity forces. It characterizes the turbulence level.
(16)
A, a1, a2, a3 and a4 were determined by fitting available data (from literature and our experimental data) with the equation (16). From a design point of view, the aeration efficiency can be also expressed as: E20 ¼ A$Rae 1 $na2 $
a3 hc $ðtan aÞa4 H
(17)
n is the number of steps, a is the global cascade chute slope.
4.2.
The inflow Reynolds number defined as: q$RH $r Re ¼ hc $W$m
The ratio H/h represents the number of cascade steps effect, when steps have uniform height. The ratio hc/H related the water critical depth and the total cascade height. It expresses the gravity and inflow conditions effects on aeration. The ratio h/l expresses the effect of steps geometry on aeration. The efficiency E20 is the ratio between the transferred quantity of oxygen and the maximal quantity susceptible to be transferred at 20 C (equation (3)).
Flow regime prediction
As described below, the flow regimes are identified by simple visual observation according to water behavior on the cascade. It is a function of inflow conditions (critical water depth), and steps dimensions (h, l). Several authors determined the limits between the flow regimes by empirical equations. Increasing steps height and decreasing the ratio h/l and flow rate q further nappe flow. Inversely, decreasing steps height and increasing flow rate q and (h/l) ratio furthers skimming regime.
Fig. 2 e Scheme of experimental apparatus.
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2.5
Table 3 e Parameters influencing stepped cascade aeration. Properties Physical properties
Flow properties Geometric properties
Frx
Parameters
Fundamental units
r water density m water dynamic viscosity Q water flowrate h step height hc flow critical height l steps length W cascade width H stepped cascade total height
M.L3 M.L1.T1
1.5
1
L3.T1 L L L L L
0.5 this paper Baylar et al (2003, 2006) 0 0
For their 126 experiments, Baylar and Emiroglu (2003); Baylar et al. (2006) mentioned the flow regimes. Under their experimental conditions, the 3 regimes were observed. In our case, skimming flow was not reached under our experimental conditions because it is not efficient as nappe/transition for aeration. Only nappe and transition flows were observed. The twelve measurements of efficiency from Toombes and Chanson (2005) were realized in a nappe flow. On the other hand, there was no indication about flow regimes for Tebbutt’s (1972) data. According to Baylar and Emiroglu (2003); Baylar et al. (2006) observations and our study, flow regimes can be distinguished by a dimensionless number Frx calculation. It compares inflow kinetic energy to gravity forces at the scale of “macro-roughness” represented by the steps. qw =hc Frx ¼ pffiffiffiffiffiffiffiffi ¼ g$x
h c h$cos a tan hl
!1=2 (18)
x represents the height of the pseudo-roughness, which is the distance between the internal edge and plan passing by the steps outer edges For Baylar and Emiroglu (2003); Baylar et al. (2006) observations, nappe flow occurred for Frx lower than 0.8 and skimming flow took place when Frx was over 1. For our experiments, the limit between nappe and transition flows was observed for Frx about 0.5. For both experiments, nappe flow took place for Frx under 0.5e0.8 and skimming flow when Frx is over 1.1. Thus, Tebbutt (1972) data were classified on the basis of the number Frx (nappe flow if Frx 0.5; transition flow if 0.5 Frx 1.1 and skimming flow when Frx 1.1) as showed in the Fig. 3. These results are valid for stepped channel slope between 14 and 50 and not applicable to Toombes and Chanson (2005) experiments with too low channel slope about 3.5 .
4.3.
2
Nappe
Transition
Skimming
Fig. 3 e Variation of flow regimes over stepped cascade with Frx.
flow regime. In another terms, the effects of operational and geometric parameters on the aeration varies from one regime to another. Preliminary simulations have shown that it is possible to have a good representation of all the data with only one correlation as illustrated Fig. 4. However, it clearly appears that the data issue from each regime is not statistically distributed around the correlation straight line. Therefore, taking into account the process, we considered it would be better to determine three different sets of parameters based on the same correlation to characterize the aeration of each regime. Table 4 represents the coefficients A, a1, a2, a3 and a4 for each flow regime. Statistics based on the calculation of student parameter t allowed testing the sensibility of oxygenation efficiency to the used dimensionless number. In this context, the calculated coefficients were compared to their standard deviations to test whether they are significantly different from 0 by estimating the probability of nullity b. In transition flow, the coefficient a1 shows a probability of 0.02. In this case, the Reynolds number effect cannot be neglected. In skimming regime, the coefficient a4 and effect of
Aeration efficiency correlation
The coefficients A, a1, a2, a3 and a4 were determined by a multi-linear regression using least square method. 193 collected data and 205 experiments realized on our pilot were used. The interfaces between three phases (solid represented by the cascade, liquid and gaseous) changes considerably with
Fig. 4 e Comparison between experimental and calculated E20 with the global correlation developed for all flow regimes.E20 [0:211$RL0:033 $n0:445 $ðtan aÞ0:083 e
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Table 4 e Correlation coefficients, student test. Global model
Model with 3 flow regimes Nappe
A a1 a2 a3 a4
Transition
Skimming
Value
t
b
Value
t
b
Value
t
b
Value
t
b
0.211 0.033 0.445 3.091E-05 0.083
16.7751107 5.02750138 24.3891692 0.00185722 3.46394616
0.000 0.000 0.000 0.998 0.000
0.331 0.048 0.687 0.169 0.234
14.394007 9.46332575 29.695529 9.29440977 9.92339927
0.000 0.000 0.000 0.000 0.000
0.403 0.042 0.670 0.297 0.211
3.007 2.420 11.308 3.861 2.623
0.004 0.020 0.000 0.000 0.012
0.14 103 0.536 0.348 1.186 0.031
5.247 4.390 5.764 9.042 0.397
0.000 0.000 0.000 0.000 0.692
cascade slope a can be neglected since the nullity probability is b ¼ 0.692. This coefficient characterizes the weight of the steps dimensions on the aeration efficiency. This ratio (h/l) has not much impact on the aeration since step’s corners are fulfilled with water in skimming flow. Otherwise, turbulence has more impact on aeration in skimming regime: a1 reach maximum value in this regime. Between nappe and transition regimes, there are a few changes of turbulence effects, the number of steps and the cascade slope a. In skimming flow, it was observed the higher hc the higher water volume over the cascade steps and the less interfacial area. Thus, aeration decreases when hc increases. For a given total cascade height H, in skimming flow, aeration is less efficient with increasing number of steps unlike the cases of nappe and transition flows. Figs. 5e7 illustrate the agreement between the calculated values and the experimental efficiencies E20 for nappe, transition and skimming flows. These correlations could be applied for wide range of geometrical and hydrodynamic parameters. In order to analyze the effect of each dimensionless number, aeration efficiency response to the unique variation of each number is represented in supplementary informations (Figures S1 to S5). Therefore, it will be possible to see different flow regimes in the same figure. In skimming flow, aeration efficiency decreases when Reynolds number, hc/H and steps number increase. In these flow conditions, E20
is insensible to chute slope variation but remains at the higher level. In nappe and transition flow, aeration efficiency increases with increasing all dimensionless numbers. The chute slope a effect on aeration efficiency illustration was separated (Figures S2 and S3 in supplementary data) because it was impossible to fix all the parameters (except a) simultaneously for the three flow regimes. It is difficult to separate Re and (hc/H) effects since they both depends on water flowrate. In nappe flow, increasing Re and (hc/H) induces an improvement in aeration efficiency unlike skimming flow. Despite the Reynolds number exponent reaches maximum value for skimming flow, aeration efficiency decreases when this number increases. This can be explained by the preponderant effect of (hc/H) on Reynolds number. Slightly over estimation of aeration efficiency can be noticed on Fig. 7 for E20 over 0.7 and under estimation for E20 under 0.3. Nevertheless, aeration efficiency estimation was found accurate enough.
Fig. 5 e Comparison between experimental and calculated E20 with correlation developed in this paper for the nappe flow.E20 [0:331$RL0:048 $n0:687 $ðhc =HÞ0:169 $ðtan aÞ0:234 e
Fig. 6 e Comparison between experimental and calculated E20 with correlation developed in this paper for the transition $n0:670 $ðhc =HÞ0:297 $ðtan aÞ0:211 flow.E20 [0:403$RL0:042 e
4.4.
Correlations comparison
Fig. 8 shows a good agreement between aeration efficiency values calculated by the relation proposed in this study (with the 3 different regimes) and the experimental values with a relative error about 16.5%. The correlation of Baylar et al. (2006) is not applicable to the data from Toombes and Chanson (2005), the values of Li
w a t e r r e s e a r c h 5 5 ( 2 0 1 4 ) 1 9 4 e2 0 2
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parameters and overcome the scale effects since local aeration prediction is not calculated for each step.
4.5.
Aeration on non uniform steps cascade
In order to restore ecological properties of some watercourses, stepped cascade could be constructed for oxygenation improvement. Cascade can be composed of steps at different height/length. In this case, aeration efficiency can be estimated by the following relation (Gameson, 1957): Ecascade ¼ 1 ð1 E1 Þ$ð1 E2 Þ.ð1 EN Þ
Fig. 7 e Comparison between experimental and calculated E20 with correlation developed in this paper for skimming $nL0:348 $ðhc =HÞL1:186 flow.E20 [1:4 $10L4 $R0:536 e
where Ei (i Є {1, n}) are the aeration efficiency of blocks with uniform h and l. By applying the correlation obtained on these blocks, the total aeration obtained on the stepped cascade can be estimated as: Ecascade ¼ 1
5. calculated for the stepped cascade of Toombes and Chanson leads to negative values. Baylar et al. (2010) correlation is only applicable with one step. Highly turbulent flows on stepped cascade are difficult to model at the local scale because the behavior varied from the upper step to the lower one. Moreover, the relative invariance of bubble size, and scale effects are difficult to predict due to the great number of coupling parameters. One of the solutions would be to simulate in detail the interface using Volume Of Fluid simulation. This is time consuming and the solution for one configuration does not allow an easy scale-up or scaledown and it requires simulating each new configuration. This is why, as explained in paragraph 4.1, we developed a global correlation based on aeration efficiency prediction regardless to multiphase flow details. It provides a reasonable accuracy of the global aeration prediction for a large range of
(19)
a2 a3 a4 N Y hi hci hi $ $ 1 A$Raei1 $ H H li i i i¼1
(20)
Conclusion
A model of aeration efficiency in stepped cascades with horizontal steps was elaborated by using dimensional analysis and 398 experimental data. These data have been taken from the literature but also measured on an experimental set-up design. The obtained correlation is composed of five dimensionless numbers that characterized the aeration behavior in stepped cascades. It allows predicting the experimental data issued from a wide range of operating conditions with a satisfactory average relative error of 16.5%. Three hydrodynamic regimes were found on stepped cascades: nappe, transition and Skimming regimes. The effect of the studied parameters on the aeration strongly differs according to the flow regime. Also, three different sets of coefficients were optimized for each flow regime in order to better consider the physic of the aeration prediction. This correlation gives a satisfactory estimation of oxygen transfer in existing cascades and could be used to design new cascades to improve oxygenation and therefore self-purification processes. Since cascades provide aeration without operating cost, the proposed correlation could be also useful to design cascades downstream WWTPs to eliminate residual pollution or even in wetlands.
Acknowledgments This work was funded by the French National Research Agency (EPEC ANR-10-ECOT-007-01). Authors would like to thank the mechanical workshop of the LRGP for construction and installation of the laboratory pilot.
Fig. 8 e Comparison between experimental and calculated E20 with the 3 correlations taking into account the 3 different flow regimes developed in this paper.
Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.watres.2014.02.022.
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