water Article
Numerical Simulation of the Interaction between Phosphorus and Sediment Based on the Modified Langmuir Equation Pengjie Hu 1,2 , Lingling Wang 1,2, *, Zhiwei Li 1,2 , Hai Zhu 1,2 and Hongwu Tang 1,2 1
2
*
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China;
[email protected] (P.H.);
[email protected] (Z.L.);
[email protected] (H.Z.);
[email protected] (H.T.) College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China Correspondence:
[email protected]; Tel.: +86-153-8078-9343
Received: 23 May 2018; Accepted: 22 June 2018; Published: 25 June 2018
Abstract: Phosphorus is the primary factor that limits eutrophication of surface waters in aquatic environments. Sediment particles have a strong affinity to phosphorus due to the high specific surface areas and surface active sites. In this paper, a numerical model containing hydrodynamics, sediment, and phosphorus module based on improved Langmuir equation is established, where the processes of adsorption and desorption are considered. Through the statistical analysis of the physical experiment data, the fitting formulas of two important parameters in the Langmuir equation are obtained, which are the adsorption coefficient, k a , and the ratio k between the adsorption coefficient and the desorption coefficient. In order to simulate the experimental flume and get a constant and uniform water flow, a periodical numerical flume is built by adding a streamwise body force, Fx . The adsorbed phosphorus by sediment and the dissolved phosphorus in the water are separately added into the Advection Diffusion equation as a source term to simulate the interaction between them. The result of the numerical model turns out to be well matched with that of the physical experiment and can thus provide the basis for further analysis. With the application of the numerical model to some new and relative cases, the conclusion will be drawn through an afterwards analysis. The concentration of dissolved phosphorus proves to be unevenly distributed along the depth and the maximum value approximately appears in the 3/4 water depth because both the high velocity in the top layer and the high turbulence intensity in the bottom layer can promote sediment adsorption on phosphorus. Keywords: sediment; phosphorus; numerical simulation; Langmuir equation; Huaihe River
1. Introduction Phosphorus (P) is one of the main limiting factors for eutrophication in most lakes and rivers [1]. Therefore, the migration and transformation processes of phosphorus play an important role in aquatic environments [2]. Sediment particles have a strong affinity to phosphorus due to the high specific surface areas and surface active sites [3]. Phosphorus absorbed by the sediment accumulates in the riverbed as sediment settles [4] and may be released by resuspension [5]. The main interactions between sediment and phosphorus are adsorption and desorption [6]. There are many factors affecting these interactions such as the physical [7] and chemical [8] properties of sediment, the water environment chemical properties [9], the adsorbate and sorbent concentration, and the hydrodynamic characteristics. Usually, the amount of adsorption per unit mass of sediment in the quasi-equilibrium state increases as the initial phosphorus concentration in water increases or as
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the sediment concentration decreases [10]. The increase of velocity can greatly affect the adsorption and desorption due to flow turbulence [11]. Besides the physical experiments, a great number of water quality models have been developed over recent decades [12]. Early models often ignored the effect of sediment on phosphorus transport [13]. Subsequently, lots of models considering the influence of sediment were proposed [14]. Most of these models considered that with empirical parameters such as a linear distribution coefficient k d of adsorption thermodynamics [15], or adsorption rate k1 and desorption rate k2 of adsorption kinetics [16]. Other simplifications included a sedimentation coefficient and a suspension coefficient, or a constant phosphorus release rate at the sediment-water surface [14]. It is very difficult to determine these parameters reasonably for lacking fundamental mechanistic analysis, and they are site-specific and not easily extended. This paper develops a model regarding hydrodynamics and the interactions of suspended sediment and phosphorus based on the Langmuir equation. Sediment samples are collected from the Wujiadu gauging station of the main stream in Huaihe River. After finishing the dynamic water physical experiments, the adsorption coefficient in the Langmuir adsorption kinetic model, k a , and the ratio k between k a and the desorption coefficient k b are obtained. By separately fitting these values, two formulas are deduced. One is the relationship between k and suspended sediment S, cross-sectional average velocity v, initial phosphorus concentration C0 . The other is the relationship between k a and v, C0 . Then k and k a are applied to the Langmuir equation in the numerical model for verification and application. The purpose of this study is to get the distribution between adsorbed phosphorus on the suspended sediment and dissolved phosphorus in the water, and find out how hydrodynamic condition affects the phosphorus transport. 2. Materials and Methods 2.1. Sediment Collection and Dynamic Water Experiments Sediment samples were collected from a depth of >5 cm from the bank of Huaihe River at the Wujiadu gauging station in October 2013. In order to get uniformly distributed particle sized and clean sediment, all points were far away from scoured riverbeds and pollutant discharge ports. Sediments were immersed in deionized water for a month and then were air-dried in a ventilated environment. After being screened through 200 mesh sieves, particles finer than 90 µm were analyzed in the physical experiment. The median diameter of the samples was 22.7 µm, and the other two typical particle size in the sediment distribution curve D10 and D90 were 3.3 µm and 60 µm, respectively. The dynamic water experiment was carried out in an elongated annular flume, which did not break up the sediment flocs when the flow is propelled by a Plexiglas gear instead of a water pump. A defined amount of sediment and KH2 PO4 solution of known concentration was added to the water at the beginning of the experiment, as shown in Table 1. The rotational speed was set at a certain value and was kept running for 24 h. Samples were filtered through a 0.45 µm filter membrane and non-absorbed phosphorus was determined using a molybdenum blue method on ultraviolet-visible spectrophotometer (TU-1810PC, Persee Co., Beijing, China). The amount of adsorbed phosphorus per unit mass sediment, Ne, at an equilibrium state of different cases were also shown in Table 1. During the experiments, the indoor temperature and the water pH were 21 ± 2 ◦ C and 6.5–7.0, respectively. The detailed description of the physical experiment can be found in the literature [10].
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Table 1. C0 , S, v and the balanced adsorbed phosphorus amount per unit mass sediment Ne of various cases in the physical experiments. Case
C0 (mg/L)
S (g/L)
v (m/s)
Ne (mg/g)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0.5 0.5 0.5 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5
1 1 1 1 1 1 0.5 0.5 0.5 0.5 1 1 1 1 1.5 1.5 1.5 1.5 2 2 2 2 1 1 1
0.515 0.48 0.44 0.515 0.48 0.44 0.515 0.48 0.44 0.39 0.515 0.48 0.44 0.39 0.515 0.48 0.44 0.39 0.515 0.48 0.44 0.39 0.515 0.48 0.44
0.0614 0.062 0.0662 0.072 0.0787 0.0817 0.1431 0.1289 0.1084 0.1017 0.112 0.102 0.0885 0.0587 0.0907 0.0783 0.065 0.041 0.0702 0.0629 0.054 0.0332 0.1272 0.1102 0.1052
2.2. Adsorption Parameters There are two kinds of classical adsorption theories. One is adsorption isotherm model for studying the phosphorus distribution during the equilibrium, such as Langmuir model and Freundlich model [17]. Another is adsorption kinetic model for studying the dynamic phosphorus distribution with time increasing, such as Elovich equation [18], parabolic equation [6] and Langmuir model. According to the theory proposed by Huang S.L. in 1997 [19], the adsorbed phosphorus amount per unit mass by sediment at an equilibrium state relates to C0 , S, maximum amount of adsorption Bm and k. Among them, Bm is only related to the characteristic of specific sediment. In this paper, Bm is given 0.15 mg/g according to the batch reactor experiments [10]. Thus we can predicate the adsorbed phosphorus amount per unit mass by sediment if k is given. And another parameter k a represents the speed of the reaction. 2.2.1. The Ratio k between k a and k b In this paper, the formula of k is given by the fitting method according to the results of the physical experiments of our research group. Firstly, in order to study the relationships between k and S, v, C0 , the single factor analysis method is used. Samples of Figure 1a–c are obtained from the results of different physical experiments. All the blue lines are fitting values. The author analyzes the effect of every single factor by keeping the others same, such as the different S and equal v, C0 . Figure 1a shows that the logarithms of k decrease monotonically as logarithms of S increase. Figure 1b shows that most of the logarithms of k increase linearly with the increase of logarithms of v apart from these red samples when C0 is smaller than 3 mg/L. In Figure 1c, it should be said that the fit is not very good. The R2 based on all scattered points is 0.7. However, the R2 based on cases of “v = 0.44 mg/L, S = 1 g/L” and “v = 0.48 mg/L, S = 1 g/L” are 0.96 and 0.99, respectively. While the R2 based on the rest case of “v = 0.515 mg/L,
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0.515 mg/L, S = 1 g/L” is small. The possible reason is that the corresponding physical experiment may have some errors. On the whole, we still think that the logarithms of k have a linear correlation S = 1the g/L” is small. of The reason is that the corresponding physical experiment may with logarithms . Secondly, the multivariate nonlinear regression analysis method is have used C0possible some errors. On the whole, we still think that the logarithms of k have a linear correlation with the to obtain the comprehensive impact of all variables and get the Equation (1). Obviously, v and S are logarithms of C0 . affecting Secondly,kthe multivariate regression analysis is used the major factors as their absolutenonlinear exponential values are largermethod than 1.0, and to is the C0 obtain the comprehensive impact of all variables and get the Equation (1). Obviously, v and S are the major secondary factor. Figure 1d lists the original values of Ne from the physical experiments and the factorsvalues affecting k ason their values cases are larger than and C0 1.0 is the secondary fitting based theabsolute Equationexponential (1) with different except = 0.5 and mg/L. This is C01.0, factor. Figure 1d lists the original values of Ne from the physical experiments and the fitting values consistent with the literature [20]. The result shows that the fitting equation can be used in the next based on the Equation (1) with different cases except C = 0.5 and 1.0 mg/L. This is consistent with the 0 numerical simulation or even for further analysis. literature [20]. The result shows that the fitting equation can be used in the next numerical simulation 27v 4.6 or even for further analysis. − 0.006 k = 0.3 4.6 (1) C027vS1.343 − 0.006 (1) k= C0 0.3 S1.343 1.2
0.4
0.4
0.0
0.0
-0.4
-0.4
-0.8 -1.2 -1.6
-0.8 -1.2 -1.6
-2.0
-2.0
-2.4 -0.8
C0=3mg/l, S=0.5g/l C0=3mg/l, S=0.5g/l C0=3mg/l, S=0.5g/l C0=3mg/l, S=0.5g/l C0=0.5mg/l,S=1g/l C0=1mg/l, S=1g/l C0=5mg/l, S=1g/l
0.8
lnk (l/mg)
lnk (l/mg)
1.2
v=0.515m/s,C0=3mg/l v=0.48m/s, C0=3mg/l v=0.44m/s, C0=3mg/l v=0.39m/s, C0=3mg/l
0.8
-2.4 -0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
-0.95
(a) lnS (g/l) 0.6
-0.85
-0.80
-0.75
-0.70
-0.65
(b) lnv (m/s)
v=0.515mg/l,S=1g/l v=0.48mg/l, S=1g/l v=0.44mg/l, S=1g/l
0.4
-0.90
Experimental Ne Fitting Ne
0.14
0.12
0.0
Ne (mg/g)
lnk (l/mg)
0.2
-0.2
0.10
0.08
-0.4 0.06
-0.6 0.04
-0.8 -0.5
0.0
0.5
1.0
(c) lnC0 (mg/l)
1.5
6
8
10
12
14
16
18
20
22
24
26
(d) Case
Figure (a–c) and and the the comparison comparison between between the the fitting fitting and C0 (a–c) Figure 1. 1. Relationships Relationships between betweenkkand andS,S,v,v, C and 0 experimental with cases cases 7–25 7–25 (d). (d). experimental ratio ratio Ne Ne with
2.2.2. 2.2.2. The The Adsorption Adsorption Coefficient Coefficient kka a Another which represents represents the Another important important parameter parameterin inthe theadsorption-desorption adsorption-desorptionprocess processisis kkaa which the speed of ofthe thereaction. reaction. adopting method as above, the author theaffecting factors ByBy adopting the the samesame method as above, the author analyzesanalyzes the factors ka formula affecting and gives of ka . k a and gives of formula ka. Firstly, in order to study the relationships between k a and C0 , v, theCsingle factor analysis method ka v,and Firstly, in order to study the relationships between 0 , the single factor analysis is used. Samples Figure 2a,b obtained the resultsfrom of different physical when method is used.ofSamples of are Figure 2a,b from are obtained the results of experiments different physical suspended sediment S equals to 1.0 g/L. All the blue lines are fitting values. The author uses the experiments when suspended sediment S equals to 1.0 g/L. All the blue lines are fitting values. The variable control approach to analyze the effect of every single factor, such as v and C . In Figure 2a, author uses the variable control approach to analyze the effect of every single factor,0such as v and there a logarithmically trend ofdecreasing k a as the initial increases. Figure 2b ka as the initial 2a, there is decreasing a logarithmically trend Pofconcentration P concentration C0 . InisFigure shows that k a decreases linearly with the increase of velocity when the initial P concentrations were increases. Figure 2b shows that ka decreases linearly with the increase of velocity when the initial low, while the values of k a are similar at higher P concentrations. Secondly, the multivariate nonlinear
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P concentrations were low, while the values of ka are similar at higher P concentrations. Secondly, the multivariate nonlinear regression analysis method is used to obtain the comprehensive impact regression analysis method is used to obtain the comprehensive impact of all variables and get the of all variables and get the Equation (2). Obviously, both of v and S greatly affect ka . Figure 2c lists Equation (2). Obviously, both of v and S greatly affect k a . Figure 2c lists the original values of k a ka fromcases. the values of ka fromand thethe physical experiments andthe theEquation fitting values the fromoriginal the physical experiments fitting values of k a from (1) withofdifferent 2 2 based2con scattered in Figure 2c literature is 0.88.This is Equation (1) with different cases. TheinRFigure The R based on all scattered points is all 0.88. This is points consistent with the [10]. consistent the literature [10].equation The result thatinthe equation can be used or in even the next The result with shows that the fitting canshows be used thefitting next numerical simulation for numerical simulation or even for further analysis. further analysis. −1.089C0 kka == 3.713e 11.584v ++5.909 5.909 3.713e −1.089C0 −−11.584v a
(2) (2)
3.5
v=0.45m/s v=0.48m/s v=0.51m/s
3
Co=0.5mg/l Co=1mg/l Co=3mg/l Co=5mg/l
3.0
2
ka (l/mg/h)
ka (l/mg/h)
2.5
1
2.0 1.5 1.0 0.5
0 0
1
2
3
4
0.44
5
0.46
0.48
0.52
experimental(v=0.45) fitting (v=0.45) experimental(v=0.48) fitting (v=0.48) experimental(v=0.51) fitting (v=0.51)
3.0 2.5
Ka (l/mg·h)
0.50
(b) v (m/s)
(a) C0 (mg/l) 3.5
2.0 1.5 1.0 0.5 0.0 0
2
4
(c) C0 (mg/l)
ka v,and Figure 2.2.Relationships Relationshipsof of v, Cand and the comparison theexperimental fitting and 0 (a,b) k a and C0 (a,b) the comparison between thebetween fitting and ratio k a with different experimental ratio ka cases with(c). different cases (c). 2.3. Numerical Model In the physical experiments, experiments, an elongated elongated annular annular flume flume propelled propelled by a Plexiglas gear is used to get a circulatory flow. While in the numerical model, a periodical numeric flume with the stream to get a constant and uniform flow. Based Equations (1) and (2), interaction body force forceisisbuilt built to get a constant and uniform flow. on Based on Equations (1)the and (2), the between sediment and phosphorus is also considered. Then, the numerical model is built which interaction between sediment and phosphorus is also considered. Then, the numerical model is consists of hydrodynamic module, sediment and phosphorus transport module. built which consists of hydrodynamic module, sediment and phosphorus transport module. 2.3.1. Hydrodynamic Hydrodynamic Module Module 2.3.1. The three-dimensional module is based on continuity Equation (3) and momentum The three-dimensionalhydrodynamic hydrodynamic module is based on continuity Equation (3) and Equation (4). momentum Equation (4). ∇·u = 0 (3) (3) ∇⋅u = 0
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∂u 1 + (u · ∇)u = − ∇ p + ν · ∇2 u + f ∂t ρ
(4)
where u is the velocity vector, ∇ is the Laplace operator, t is the time, p is the pressure, ν is the kinematic viscosity, which is equal to 10−6 m2 /s at the normal temperature of 20 ◦ C, and f is the body force, which is equal to the resultant force of the gravitational acceleration in the vertical direction and the stream body force, Fx . In order to get a constant and uniform water flow in the numerical study, a streamwise body force, Fx , is added to counteract the kinetic energy loss caused by resistances on solid boundaries and water viscosity. 2.3.2. Sediment Transport Module The sediment transport is described using an equilibrium approach, assuming that all sediment are suspended in the given hydrodynamic condition. The governing equation can be written in Equation (5). ∂S ∂(uS) ∂(vS) ∂(wS) ∂2 S ∂2 S ∂(ωS) ∂2 S + + + = εx 2 + εy 2 + εz 2 + ∂t ∂x ∂y ∂z ∂y ∂x ∂y ∂z
(5)
where S is the concentration of suspended sediment; t is time; x, y, z are coordinate directions, respectively; u, v, w are velocities of direction x, y, and z; εx , εy , εz are diffusion coefficient of direction x, y and z; ω is the settling velocity. 2.3.3. Phosphorus Transport Module Figure 3 shows a conceptual model of the phosphorus transport. The red circles represent dissolved or adsorbed phosphorus, the irregular khaki circles represent suspended sediment particles. Usually, the advection-diffusion equation is used for phosphorus transport to simulate the evolution processes of spatiotemporal concentration. By using the classical Langmuir equation (Equation (6)), the interaction of adsorption and desorption between dissolved phosphorus and adsorbed phosphorus is considered. Then the Equation (6) is added into the Advection-Diffusion equation (Equation (7)) as a source term which is shown in Equation (8).
∂N ∂t ∂C ∂t
dN = k a · C · (Bm − N) − k d · N dt
(6)
∂N ∂N ∂N ∂N ∂2 N ∂2 N ∂2 N +u +v +w = Dx 2 + Dy 2 + Dz 2 + S0 ∂t ∂x ∂y ∂z ∂x ∂y ∂z
(7)
2
2
2
∂N ∂N ∂ N ∂ N ∂ N + u ∂N ∂x + v ∂y + w ∂z = Dx ∂x2 + Dy ∂y2 + Dz ∂z2 + k a · C · (Bm − N) − k d · N 2
2
2
∂C ∂C ∂ C ∂ C ∂ C + u ∂C ∂x + v ∂y + w ∂z = Dx ∂x2 + Dy ∂y2 + Dz ∂z2 − ( k a · C · (Bm − N) − kd · N) · S
(8)
where N is the adsorbed phosphorus amount per unit mass by sediment (mg/g); C is the concentration of dissolved P in the water (mg/L); t is time; S0 is the source term; S is sediment concentration (g/L); k a , k d (= ka /k) are separately the coefficients of the adsorption and the desorption (l/mg/h, 1/h); Bm is the maximum adsorption of adsorption (mg/g).
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Figure 3. 3. Conceptual Conceptual model model of the phosphorus phosphorus transport transport (the (the red red circles circles represent represent dissolved dissolved or or Figure 3. Figure Conceptual model of of the the phosphorus transport (the red circles represent dissolved or adsorbedphosphorus, phosphorus,the theirregular irregularkhaki khakicircles circlesrepresent representsuspended suspendedsediment sedimentparticles). particles). adsorbed the irregular khaki circles represent suspended sediment particles). adsorbed phosphorus,
3. Model Model Verification Verification 3. 3. Model Verification 3.1. Hydrodynamic Hydrodynamic 3.1. 3.1. Hydrodynamic As is isdemonstrated demonstratedin inFigure Figure4,4, 4,a aperiodical a periodical periodical numerical flume is built built to simulate simulate the flume of As is demonstrated in Figure numerical flume is to flume of As numerical flume is built to simulate thethe flume of the the previous previous physical experiment where the recirculating recirculating water is propelled propelled by gear Plexiglas gear the physical experiment the is by Plexiglas gear previous physical experiment wherewhere the recirculating water iswater propelled by Plexiglas driven by driven by an AC frequency conversion electric machine. In order to get a constant and uniform driven by an AC frequency conversion electric machine. In order to get a constant and uniform an AC frequency conversion electric machine. In order to get a constant and uniform water flow in the water flow flowstudy, in the theanumerical numerical study, streamwise bodytoforce, force, added to counteract counteract the kinetic FFxx,, isisthe water in study, streamwise body added to kinetic numerical streamwise bodyaaforce, Fx , is added counteract kinetic energy lossthe caused by energy loss losson caused by resistances onwater solid viscosity. boundaries and and water water viscosity. viscosity. energy caused resistances on solid boundaries resistances solidby boundaries and
Figure4. 4.Schematic Schematicdiagram diagramof ofthe theperiodical periodicalnumerical numericalflume. flume. Figure Figure 4. Schematic diagram of the periodical numerical flume.
The numerical numerical simulation simulation is is designed designed in in aa 22 m m ×× 0.3 0.3 m m ×× 0.2 0.2 m m (L (L ×× W W ×× H) H) periodical periodical numerical numerical The The numerical simulation is designed in a 2 m × 0.3 m × 0.2 m (L × W × H) periodical numerical flume, and and the the resolution resolution of of computational computational grids grids is is 200 200 ×× 30 30 ×× 20, 20, which which is is sufficiently sufficiently fine fine to to obtain obtain flume, flume, and the resolution of computational grids is 200 × 30 × 20, which is sufficiently fine to obtain a stable numerical solution. The velocities after the flow of laboratory experiment reaching a steady a stable numerical solution. The velocities after the flow of laboratory experiment reaching a steady a stable numerical solution. The velocities after the flow of laboratory experiment reaching a steady state are are used used as as the the initial initial condition condition of of the the numerical numerical simulation. simulation. The The initial initial fluid fluid level level of of all all state state are used as the initial condition of the numerical simulation. The initial fluid level of all different different cases cases is is 0.2 0.2 m. m. All All the the solid solid walls walls including including the the sidewalls sidewalls and and the the flume flume bed bed are are considered considered different cases is 0.2 m. All the solid walls including the sidewalls and the flume bed are considered as the as the the no-slip no-slip wall wall boundary boundary conditions. conditions. In In this this study, study, the the turbulence turbulence is is predicted predicted by by the the as no-slip wall boundary conditions. In this study, the turbulence is predicted by the renormalization renormalization group group (RNG) (RNG) kk −− εε turbulent turbulent model model (turbulent (turbulent kinetic kinetic energy energy kk and and its its dissipation dissipation renormalization group (RNG) k − ε turbulent model (turbulent kinetic energy k and its dissipation ratio ε). ratio εε ).). ratio To compare the numerical and experimental hydrodynamic conditions, two cases are set with To compare compare the the numerical numerical and and experimental experimental hydrodynamic hydrodynamic conditions, conditions, two two cases cases are are set set with with To the mean cross-sectional velocities of 0.515 m/s and 0.48 m/s, respectively. Shown in Figure 5, the mean mean cross-sectional cross-sectional velocities velocities of of 0.515 0.515 m/s m/s and and 0.48 0.48 m/s, m/s, respectively. respectively. Shown Shown in in Figure Figure 5, 5, the the the the streamwise velocities along the water depth at the bold black line AB (Figure 4) of these two cases streamwise velocities velocities along along the the water water depth depth at at the the bold bold black black line line AB AB (Figure (Figure 4) 4) of of these these two two cases cases streamwise are in good agreement with experimental velocities. are in in good good agreement agreement with with experimental experimental velocities. velocities. are
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experimental numerical experimental numerical
0.15
0.20
experimental numerical experimental numerical
0.10
Distance upon bed(m)
Distance upon bed(m)
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velocity(v=0.48m/s) velocity(v=0.48m/s) velocity(v=0.515m/s) velocity(v=0.515m/s)
0.05 0.10
0.00 0.05 0.40
0.45
0.50
0.55
velocity(m/s)
Figure 5. Comparison0.00 between the numerical and experimental streamwise velocity at the line AB Figure 5. Comparison between the velocity at the line AB 0.40numerical and 0.45 experimental 0.50 streamwise 0.55 with two cases of cross-sectional average velocity 0.48 and 0.515 m/s. with two cases of cross-sectional average velocityvelocity(m/s) 0.48 and 0.515 m/s. Figure and 5. Comparison the numerical and experimental streamwise velocity at the line AB 3.2. Sediment Phosphorusbetween Transport 3.2. Sediment Transportaverage velocity 0.48 and 0.515 m/s. withand two Phosphorus cases of cross-sectional According to the result of the batch reactor experiments [10], the maximum adsorption of AccordingB toisthe the initial batch phosphorus reactor experiments [10], and the maximum adsorption of adsorption 0.15result mg/g.of The concentration sediment concentration 3.2. Sedimentmand Phosphorus Transport adsorption 0.15 The initialphysical phosphorus concentration andin sediment m isof equal to B that themg/g. corresponding experiments are given Table 2.concentration The variationequal of to the result of the experiments batch reactor are experiments [10], the2. maximum adsorption of to adsorbed that of According the corresponding physical given in Table The variation of adsorbed phosphorus amount per unit mass sediment with time is shown in Figure 6. Although the adsorption Bm is 0.15 mg/g. The initial phosphorus concentration and sediment concentration phosphorus amount per unit sediment with timeresults, is shown Figureof6.them Although the experimental experimental N changes a mass little around numerical thein values are almost the same equal to that of the corresponding physical experiments are given in Table 2. The variation of N changes a little an around numerical when reaching equilibrium state.results, the values of them are almost the same when reaching an adsorbed phosphorus amount per unit mass sediment with time is shown in Figure 6. Although the equilibrium state.
experimental N changes a little around numerical results, the values of them are almost the same Table 2. The initial phosphorus concentration C0, initial sediment concentration S, and the when reaching an equilibrium state. phosphorus C0the , initial sediment concentration S, and the Table 2. The initial cross-sectional average velocity v ofconcentration different cases in numerical simulations. cross-sectional average velocity v of different cases in the numerical simulations. Table 2. Case The initial phosphorus concentration C0, initial sediment concentration C0 (mg/L) S (g/L) v (m/s)S, and the cross-sectional average velocity v of different cases in the numerical simulations. A1 0.5 1 0.515
Case
A2 Case
A1 A2
A1 A20.10
C0 (mg/L)
1 C0 (mg/L) 0.5 0.5 1 1
S (g/L) 1 1
v (m/s)
1 S (g/L) 0.515 1 0.515 1
0.515 v (m/s) 0.515 0.515
0.10
0.08 0.08
0.06
N (mg/g)
N (mg/g)
0.06
0.04
experimental(case A1) numerical (case A1) experimental(case A1) experimental(case A2) numerical (case A1) numerical (case A2)
0.04
0.02 0.02
experimental(case A2) numerical (case A2)
0.00 0
4
0.00
0
4
8
12
8
12 (h) time
16 16
20 20
24 24
time (h) Figure 6. Variation of the adsorbed phosphorus amount per unit mass sediment with time and the comparison between numerical and experimental value ofper case A1mass and sediment A2. Figure 6. Variation of the adsorbed phosphorus amount unit with time and the Figure 6. Variation of the adsorbed phosphorus amount per unit mass sediment with time and the comparison between numerical and experimental value of case A1 and A2. comparison between numerical and experimental value of case A1 and A2.
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4. 4. Model Application Model Application In In order to to further study thethe interaction processes between phosphorus and sediment, another order further study interaction processes between phosphorus and sediment, another four cases are considered (listed in Table 3). Derived k and k based on Equation (1) and (2)(2) areare four cases are considered (listed in Table 3). Derived k and a ka based on Equation (1) and applied to new cases (B1–B4). Figure 7 shows the variation of the amount of adsorbed phosphorus applied to new cases (B1–B4). Figure 7 shows the variation of the amount of adsorbed phosphorus amount per unit mass sediment with time for a sediment concentration of 1 g/L with different initial amount per unit mass sediment with time for a sediment concentration of 1 g/L with different P concentrations. Obviously, N reaches to a stable state different from dynamic equilibrium of the initial P concentrations. Obviously, N reaches to a stable state different from dynamic equilibrium physical experiment after several hours because Equation (5) is used to consider the interaction in the of the physical experiment after several hours because Equation (5) is used to consider the numerical simulation. N increases with the initial phosphorus concentration by comparing of A1, A2, interaction in the numerical simulation. N increases with the initial phosphorus concentration by B1 and B3 from Figures 6 and 7. And N also increases with the velocity by comparing B1 and B2 or B3 comparing of A1, A2, B1 and B3 from Figures 6 and 7. And N also increases with the velocity by and B4. comparing B1 and B2 or B3 and B4. Table 3. The initial phosphorus concentration C0 , initial sediment concentration S, and the Table 3. The initial phosphorus concentration C0, initial sediment concentration S, and the cross-sectional average velocity v of different cases in the numerical simulations. cross-sectional average velocity v of different cases in the numerical simulations. Case C0C(mg/L) 0 (mg/L) 2 2 B1 B2 2 2 B3 4 4 B4 4 4
Case B1 B2 B3 B4
S (g/L) 1 1 1 1
v (m/s) S (g/L) 1 0.515 1 0.48 1 0.515 1 0.48
v (m/s) 0.515 0.48 0.515 0.48
0.12
0.10
N (mg/g)
0.08
0.06
case B1 case B2 case B3 case B4
0.04
0.02
0.00 0
2
4
6
8
10
12
time (h)
Figure 7. Variation ofadsorbed the adsorbed phosphorus amount per unitsediment mass sediment time of case Figure 7. Variation of the phosphorus amount per unit mass with timewith of case B1–B4. B1–B4.
Figure 8 shows that the concentration of dissolved phosphorus in the water changes little within Figure 8 shows that the concentration of dissolved phosphorus in the water changes little the whole water depth and its maximum approximately appears in the 3/4 water depth. One reason within the whole water depth and its maximum approximately appears in the 3/4 water depth. One is that the strong turbulence intensity of the water flow and the high sediment concentration at the reason is that the strong turbulence intensity of the water flow and the high sediment concentration bottom of the tank can result in the large amount of phosphorus absorbed by the sediment (shown at the bottom of the tank can result in the large amount of phosphorus absorbed by the sediment in Figure 9). Another is that the large flow rate of the upper water body leads to the increase of the (shown in Figure 9). Another is that the large flow rate of the upper water body leads to the adsorption of phosphorus by sediment (Figure 5). So in the final upper water body and the lower increase of the adsorption of phosphorus by sediment (Figure 5). So in the final upper water body water body, the amount of dissolved phosphorus in the water is relatively low, and the concentration and the lower water body, the amount of dissolved phosphorus in the water is relatively low, and of phosphorus in the 3/4 water depth shows a relatively large value. In stable natural rivers or the concentration of phosphorus in the 3/4 water depth shows a relatively large value. In stable reservoirs, dissolved phosphorus shows an increasing trend from the bottom to the surface of the water. natural rivers or reservoirs, dissolved phosphorus shows an increasing trend from the bottom to the However, this is not always the case. Both sediment and hydrodynamic forces can cause changes in surface of the water. However, this is not always the case. Both sediment and hydrodynamic forces the distribution of dissolved phosphorus along the depth. can cause changes in the distribution of dissolved phosphorus along the depth.
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case B1 caseB2 B1 case case B2
0.15 0.15
water depth h (m) water depth h (m)
water depth h (m) water depth h (m)
0.20 0.20
0.10 0.10
0.05 0.05
0.00 0.00
1.88 1.88
1.89 1.89
1.90 1.90
(a) C (mg/l) (a) C (mg/l)
1.91 1.91
1.92 1.92
case B3 caseB4 B3 case case B4
0.15 0.15
0.10 0.10
0.05 0.05
0.00 0.00
3.86 3.86
3.87 3.87
3.88 3.88
(b) C (mg/l) (b) C (mg/l)
3.89 3.89
3.90 3.90
Figure 8. Variation of the concentration of dissolved P in the water of case B1~B4 at an Figure 8. Variation of the concentration of dissolved P in the water of case B1~B4 at an Figure 8. Variation equilibrium state.of the concentration of dissolved P in the water of case B1–B4 at an equilibrium state. equilibrium state.
Distance upon bed(m) Distance upon bed(m)
0.20 0.20
v=0.515m/s v=0.515m/s v=0.48m/s v=0.48m/s
0.15 0.15
0.10 0.10
0.05 0.05
0.00 0.00
0.0005 0.0005
0.0010 0.0010
Turbulent intensity (m/s) Turbulent intensity (m/s)
0.0015 0.0015
Figure Comparison the turbulent intensity distribution along water depth with two cases Figure 9. 9. Comparison of of the turbulent intensity distribution along thethe water depth with two cases of of Figure 9. Comparison of the turbulent intensity distribution along the water depth with two cases of cross-sectional average velocity 0.48 and 0.515m/s. cross-sectional average velocity 0.48 and 0.515m/s. cross-sectional average velocity 0.48 and 0.515m/s.
5. Conclusions 5. Conclusions 5. Conclusions In this paper, a numerical model of hydrodynamics, sediment, and phosphorus based on In this paper, a numerical model model of hydrodynamics, sediment,sediment, and phosphorus based on improved In this paper, a numerical of hydrodynamics, and phosphorus based on improved Langmuir equation was established where the processes of adsorption and desorption Langmuir equation was established where the processes of adsorption and of desorption were considered. improved Langmuir equation was established where the processes adsorption and desorption were considered. The main conclusions can be summarized as follows: Thewere main conclusions canmain be summarized follows: considered. The conclusions as can be summarized as follows: (1) The influence of both hydraulic and environmental factors on phosphorus sorption to The influence of hydraulic both hydraulic and environmental on phosphorus sorption to (1) (1)The influence of both and environmental factors onfactors phosphorus sorption to suspended suspended sediments was quantitatively investigated by fitting analysis of ka , and the ratio k k suspended sediments was quantitatively investigated by fitting analysis of , and the ratio sediments was quantitatively investigated by fitting analysis of k a , and the ratioak between the k between the adsorption coefficient and the desorption coefficient in flume experiments. adsorption coefficient and coefficient the desorption coefficient in flume experiments. between the adsorption and the desorption coefficient in flume experiments. (2) The concentration of dissolved phosphorus was unevenly distributed along the depth, and the (2) (2)The concentration of of dissolved phosphorus was unevenly distributed along thethe depth, and thethe The concentration dissolved phosphorus was unevenly distributed along depth, and maximum value approximately appeared in the 3/4 water depth because both the high velocity maximum value approximately appeared in the water depth because both high velocity maximum value approximately appeared in the 3/43/4 water depth because both thethe high velocity in the top layer and the high turbulence intensity in the bottom layer can promote sediment layer and high turbulence intensity bottom layer promote sediment in in thethe toptop layer and thethe high turbulence intensity in in thethe bottom layer cancan promote sediment adsorption on phosphorus. adsorption phosphorus. adsorption onon phosphorus. (3) Derived k and ka based on equation can well be applied to new cases. However, it would be ka based Derived k and on equation canbewell be applied newHowever, cases. However, (3) (3)Derived k and k a based on equation can well applied to new to cases. it woulditbewould much be much more meaningful to establish a general formula for k based on a large quantity of more meaningful to establish general formula for formula k based on quantity experiments much more meaningful to aestablish a general fora klarge based on a of large quantity of experiments with sediment of different origins. with sediment of different origins. experiments with sediment of different origins. (4) This paper hasn’t taken bed sediment into consideration. However, bed sediment widely exists This paper hasn’t taken bed sediment into consideration. However, bed sediment widely exists (4) (4)This paper hasn’t taken bed sediment into consideration. However, bed sediment widely exists in natural rivers and has great impaction on the adsorption and desorption of phosphorus. So natural rivers great impaction onadsorption the adsorption and desorption of phosphorus. in in natural rivers andand hashas great impaction on the and desorption of phosphorus. So the So the next step is to further consider the sedimentation and suspension between suspended thestep next step is to consider further consider the sedimentation and suspension between sediment suspended next is to further the sedimentation and suspension between suspended sediment and bed sediment, and the adsorption and phosphorus processes in the bed sediment sediment and bedand sediment, and theand adsorption and phosphorus in the bed sediment and bed sediment, the adsorption phosphorus processes in processes the bed sediment layer. layer. layer. (5) Natural rivers are very different from experimental flumes because of complex terrain and (5) Natural rivers are very different from experimental flumes because of complex terrain and hydrodynamic conditions. So, it is of great importance to build a model based on typical hydrodynamic conditions. So, it is of great importance to build a model based on typical
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Natural rivers are very different from experimental flumes because of complex terrain and hydrodynamic conditions. So, it is of great importance to build a model based on typical riverbed and real dynamic conditions with the data of on-site water samples and sand samples, especially in the river seriously affected by eutrophication.
Author Contributions: Z.L. carried out the physical experiment and gave some basic data. P.H. analyzed the correlation of various parameters, carried out the numerical simulations and the data treatment, and participated in the writing. L.W. and H.Z. carried out the analysis of the methodology and participated in the writing. All five authors reviewed and contributed to the final manuscript. Funding: This work was funded by the National Key Research and Development Program of China (2017YFC0405605, 2016YFC0401503), the State Key Program of National Natural Science of China (Grant No. 51239003), the 111 Project (Grant No. B17015), the Fundamental Research Funds for the Central Universities (2016B40614). Conflicts of Interest: The authors declare no conflicts of interest.
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