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Analysis of water quality in shallow lakes with a two-dimensional flow-sediment model
501
Ser.B, 2007,19(4):501-508
ANALYSIS OF WATER QUALITY IN SHALLOW LAKES WITH A TWO-DIMENSIONAL FLOW-SEDIMENT MODEL* JI Yong, ZhANG Jie Nanchang Institute of Technology, Nanchang 330099, China, E-mail:[email protected] YAO Qi College of Environmental Science and Engineering, Hohai University, Nanjing 210098, China ZHAO Di-hua Nanjing Hydraulic Research Institute, Nanjing 210098, China
(Received December 2, 2005; Revised March 24, 2006)
ABSTRACT: The governing equation for sediment pollutions was derived based on the turbulent diffusion of pollutants in shallow lakes. Coupled with shallow water equations, a depth-averaged 2-D flow and water quality model was developed. By means of the conservation law, a proposed differential equation for the change of sediment pollutants was linked to the 2-D equations. Under the framework of the finite volume method, the Osher approximate Riemann solver was employed to solve the equations. An analytical resolution was used to examine the model capabilities. Simulated results matched the exact solutions especially well. As an example, the simulation of CODMn in the Wuli Lake, a part of the Taihu lake, was conducted, which led to reasonable results. This study provides a new approach and a practical tool for the simulation of flow and water quality in shallow lakes. KEY WORDS：sediment, falling velocity, turbulence, flow and water quality model, finite volume method，approximate Riemann solver, shallow water equation，advection-dispersion equation
1. INTRODUCTION A lot of research on the flows in deep lakes has been conducted by a lot of scholars, but the simulation study on shallow lakes is still not mature[1]. As far as the Taihu lake is concerned, the water current and the mechanism of pollutant
transfer are different from those in deep lakes. For shallow water, any outside functions can disturb the body of water, causing the pollution to come in from the sludge. Since the 1940s, there have been a great many researches on the oxygen consumption in the sludge, but few using the equations of flow and water quality, and sometimes the impact of sludge is simplified as a constant. In this article, based on turbulence diffusion theory, a water quality equation including the sediment pollutions will be derived, which is coupled with the shallow water equations and an equation describing the sediment pollutions, resulting in a complete model of two-dimensional flow-water quality with semimetal pollution. Under the framework of the finite volume method[2-4], the Osher approximate Riemann solver is employed to solve the system of equations. The Osher approximate Riemann solver is a two-dimensional model first used in aerodynamics. Now it has been applied in the simulation of water quality[5-9]. The model is employed to simulate a complex water body, including steady and unsteady flows, jet and torrent flow, and a water body with a complex surrounding terrain. And the result is rather perfect. Finally, the accuracy of the model is examined, and then by
* Project supported by the National Natural Science Foundation of China (Grant No. 50239093) and Nanjing Construction Commission (Grant No. 20050176). Biography: JI Yong (1979- ), Male, Master, Lecturer
502
taking the Xuanwu Lake in Nanjing as an example, the effectiveness of the model is verified.
2.THE EQUATION OF WATER QUALITY WITH SEDIMENT POLLUTION 2.1 The basic equations The sediment pollution is related to the movement of contamination in a vertical direction. It is a three-dimensional problem. According to the turbulent diffusion theory[10-12], the 3-D convection-diffusion equation for the water quality including sediment pollution is expressed as
unchanged in the z direction and the velocity component in the z direction is zero. The pollution of sediment comes from the turbulence diffusion and contamination settlement happening at the bottom. Therefore, the 2-D horizontal equation of water quality with sediment pollution is
∂ (hC ) ∂ (huC ) ∂ (hvC ) ∂ ∂ (hc) ]+ + + = [ Dx ∂t ∂x ∂y ∂x ∂x ∂ ∂ (hc) ∂C [ Dy ] + ( Dz |+ − ∂y ∂y ∂z 0
∂C ∂ ∂ ∂ + (Cu ) + (Cv) + (Cw) = ∂t ∂x ∂y ∂z
ω c C0 ) − K c hC
In the equation, u and v are respectively the vertical average current velocity in the x and y direction, C is the vertical average
∂ ∂C ∂ ∂C ( Dx ) + ( Dy )+ ∂x ∂x ∂y ∂y ∂ ∂C ∂ ( Dz ) − (Cω c ) − K c C ∂z ∂z ∂z
concentration (1)
where u , v , w are the velocity components of flow in the x , y , and z directions respectively,
C is the concentration of the contamination, ωc is the sedimentation rate of contamination that stems from the integrated effect of its weight and the diffusion of water, Dx ， Dy ， and Dz are the diffusion coefficients of the contamination in x , y , and z directions respectively, Kc is integrated sedimentation coefficient of contamination. The following equation is integral of the above equation in the z direction:
∫
h
0
the the the an
h ∂ h ∂ ∂C dz + ∫ (Cu )dz + ∫ (Cv)dz + 0 ∂x 0 ∂y ∂t
∫
h
0
∫
h
0
(2)
+
h ∂ ∂ ∂C (Cw)dz = ∫ ( Dx )dz + 0 ∂x ∂z ∂x
h ∂ ∂ ∂C ∂C ( Dy )dz + ∫ ( Dz − 0 ∂z ∂y ∂y ∂z
h
Cωc )dz − ∫ Κ cCdz 0
For the adequate mixture in the shallow lake, it is considered that the dependent variables remain
of
the
pollutant,
C0+ is
the
concentration of pollutants in the near-bottom water, according to horizontal 2-D assumption C0+ = C ,
ωc , and K c are the same as mentioned earlier, Dz
∂C | + is the upward mobility of pollutants in ∂z 0
the sediment of bottom interface through turbulent motion. On the basis of the turbulent diffusion theory, if the concentrations of pollutants at different sections are different, the pollutant diffuses from the section with higher concentration to the section with lower concentration, and its diffusion intensity is related to the concentration gradient and diffusion coefficient, and the diffusion coefficient differs from the turbulent intensity. If the concentration of pollutants in the sediment is higher than that of water or there is outside force disturbing, the pollutants of sediment will diffuse into water. Meanwhile, under the action of gravity, pollutants drop to the bottom and increase the concentration of pollutants in the sediment. The concentration gradient of the bottom interface is
C −C ∂C , in which Cb and C are the |0+ = b ∂z Δz
concentration of pollutants in sediment and water respectively, and vary with space-time. Therefore, in Dz
∂C D |0+ = z (Cb − C ) = wc (Cb − C ) , wc is ∂z Δz
the diffusion coefficient of unit depth, and can be considered as the diffusion upward velocity of pollutants in the sediment and will be rated by field
503
data. The water quality control equation including the pollutants in sediment is
∂ (hC ) ∂ (huC ) ∂ (hvC ) + + = ∂t ∂x ∂y
(3)
It can be seen from Eq.(3) that the pollution to the water from sediments, depends on the release rate, wc (Cb − C ) , of the concentration of pollutants in the sediment, and the comprehensive descending rate, ω c C , of the concentration of pollutants in the water. When the release of pollutants in the sediment is larger than the drop of pollutants in the water, the concentration of pollutants in the water will keep increasing till it reaches a dynamic balance. 2.2 The determination of the pollution parameter in the sediment (1) Comprehensive sinking velocity ωc The determination of the comprehensive sinking velocity, ωc , can adopt the following two methods: (a) Calculating method of the suspension Ks of pollutants: Ks is coefficient back-calculated from the experimental data for release of the sediment. According to its physical meaning, and through dimensional analysis, gain
Ks =
ωc h
calculate ωc :
ωc = β
∂ ∂ (hC ) ∂ ∂ (hC ) [ Dx ]+ [ Dy ]+ ∂x ∂x ∂y ∂y
[ wc (Cb − C ) − ω c C ] − K c hC
action of gravity. This article adopts the Stokes falling velocity formula to be multiplied by β , to
, in which h is the depth of water.
Therefore, the comprehensive sink and float velocity is ω c = K s h . (b) Calculating method of the suspension formula for suspended matter: as the pollutants sink and suspend under the action of gravitation and outside forces, similar to the colloidal particles of mud, the existing sink and float formula for suspended matter can be adopted, which just multiplies the reduction coefficient. The reduction coefficient, β , depends on the factors that cause the suspension of pollutants, such as, the wind velocity, wind direction, water velocity, and turbulent motion, which reflects the contrasting relationship between turbulent motion and the
1 γs −γ d2 g ν 18 γ
where β is the reduction coefficient, representing the contrasting relations between the suspension effect of turbulent motion and the sink effect of the gravity, β ≤ 1 , γ s , and γ are respectively the colloidal particles and volume weight of water, as the pollutants can sink in single particles as well as in flock, d is the equivalent particle diameter, ν is the kinematic viscosity coefficient. The diameter d and reduction coefficient β will be rated according to the tested data. (2) The concentration of pollutants in sediment Cb The variation in the concentration of pollutants in the sediment can be affected by many factors, such as, absorption, desorption, and mineralization, and it is very complicated. It is assumed that the variation is in accord with the first-order reaction, considering the conservation of mass of exchanged pollutants at the lake bottom, the equation for the variation can be written as
dCb w w = − kbCb − c (Cb − C ) + c C dt h h
(4)
where h is the depth of water, Kb is the attenuation coefficient of pollutants in the sediment, which is related to the ambient conditions, such as, temperature, current, and sediment, and will be calculated and rated by the release coefficient of the sediment, other notations are the same as those in Eq.(3). The physical meaning of the equation is that the variation in the concentration Cb of pollutants in the sediment is composed of three parts: one is the attenuation of itself, one is the diffusion towards the water, and the other is the sedimentation of pollutants in water.
3. 2-D MODEL FOR CURRENT, WATER QUALITY, AND SEDIMENT POLLUTION The conservation form of the coupled shallow-water equation and water-quality equation with sediment pollution can be expressed as
504
∂h ∂(hu) ∂(hv) + + =0 ∂t ∂x ∂y
(5a)
1 τ wx ρ
(5b)
∂ (hv) ∂ (huv) ∂ (hv 2 + gh 2 / 2) + + = ∂t ∂x ∂y gh( s0 y − s fy ) +
1 τ wy ρ
(5c)
and ωci are the diffusion velocities of the sediment of the six pollutant compositions and the comprehensive sinking velocity, and Cbi and K bi are the concentrations of the six pollutant compositions in the sediment and attenuation coefficient. Now the problem is to solve the coupled equations. Equation (5) can be expressed as
∂q ∂f (q) ∂g (q ) + + = b(q ) ∂t ∂x ∂y
(7)
The expression of the limited volume after dispersement of the integral in Eq.(7) is, (for details, refer to Ref.[5])
∂ (hCi ) ∂ (huCi ) ∂ (hvCi ) + + = ∂t ∂x ∂y ∂ (hCi ) ∂ (hCi ) ∂ ∂ [ Dix ] + [ Diy ]+ ∂x ∂x ∂y ∂y
A
[Wci (Cbi − Ci ) − ω ci Ci ] − K ci hCi + Si (5d) The equation for the variation of the pollutants in sediment is
dCbi W W = − kbi Cbi − ci (Cbi − Ci ) + ci Ci dt h h
and Diy are the diffusion coefficients of each pollutant in the x and y directions respectively, Wci
∂ (hu ) ∂ (hu 2 + gh 2 / 2) ∂ (huv) + + = ∂t ∂x ∂y gh( s0 x − s fx ) +
collection of resources of each pollutant, and its contents will change with the composition, Dix
(6)
m Δq = −∑ T (Φ ) −1 f (q ) Lj + b* (q) Δt j =1
(8)
In the equation, A is the area of entity Ω , m is the total number of entity edges, L j is the length of j entity edge,
b* (q ) = [0, Agh( s0 x − s fx ), Agh( s0 y − s fy ), ΣDi (∇hCi ) n L j + Aωc (Cbi − Ci ) − ( AK ci hCi + Si )]T
where h is the depth of water, u and v are the components of the vertical average horizontal velocity in the x and y directions respectively, g
∇ is the gradient operator, q = T (Φ )q , and
is the gravity acceleration, s0 x and s fx are the
T (Φ)
bottom slope of ground and the friction gradient, respectively, in the x direction, s0 y and s fy are the bottom slope of ground and friction gradient in the y direction respectively, ρ is the density of
−1
and T (Φ ) are the rotation transformation and inverse transformation matrix of the coordinates. The Osher approximate Rieman solver refer to Ref.[5].
water, τ wx and τ wy are the wind stresses in the x and y directions respectively, Ci is the vertical average concentration of pollutants, including six compositions (i.e., COD， NBOD， CBOD， DO， TP and TN), K ci is the comprehensive degradation coefficient of each pollutant, Si is the
4. EXAMPLE 4.1 Transport of concentration with gaussian distribution in a uniform flow field The computational comain for the transport of concentration with Gaussian distribution in a uniform flow field is 5000 m ×5000 m, and the size
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Table 1 Computed results of concentration with Gaussian distribution Time
0
0.5
1.0
1.5
2.0
Exact solution
1.0
0.85
0.74
0.68
0.63
Numerical solution
1.0
0.84
0.73
0.66
0.61
Error (%)
0
1.18
2.70
2.94
3.17
of the grid is Δx = Δy = 100 m. It is assumed that the bottom of the field has no friction, the gradient is zero, the current is invariable and average, the current velocity is 0.5 m/s and the depth of water is 0.5 m. The borderline is determined as CR = CL , its peak concentration of initial distribution is 1.0 mg/L, and the diffusion coefficient in each direction is Dx = D y = 5 m2/s. Compared with the exact solution which is given in Table 1, the computation error is within 3.5%, and no negative value and shock occur during the computation (for details refer to Ref.[5]). 4.2 The calculation of COD in the Xuanwu Lake with the pollution of sediment (1) General description: The Xuanwu Lake is located in the northeast of the old city zone of Nanjing City, and is a small-sized natural shallow lake, which totally covers an area of 472 ha, in which the area of water is 368 ha, and the area of land is 104 ha. The shape of the lakeshore of the Xuanwu Lake is similar to a diamond, the perimeter of the lake is about 10 km, the average water level is lower than 10 m, the average depth of water is 1.43 m, the largest depth of the lake is 2.8 m, and the perennial water level is between 9.8 m and 10.2 m. The flux entering the lake is controlled manually to a small extent. In the Xuanwu Lake there are five oases, Huanzhou, Yingzhou, Lingzhou, Liangzhou, and Cuizhou, which are connected and unified by bridges and banks. With the development of economy and the growth of population, the sanitary sewage from the city and surface runoff has polluted the Xuanwu Lake to a serious extent, and the water quality has been in a state of eutrophication. The main pollutants of the lake are TP, TN, NH3-N, and COD. Therefore, it is planned to introduce the water from the Yangtze River to dilute the concentration of pollutants and improve the water quality of the lake.
(2) Calculation parameters: In consideration of the landform, the location of the sewage drain outlet, Peclet value, Courant value, grid rate, and so on, the computational domain for the Xuanwu Lake is composed of a nonstructural grid with 3032 units and 3301 nodal points (Fig.1), and the side length is 10 m - 40 m. The Manning coefficient n is taken as 0.022. The diffusion coefficients of CODMn in both longitudinal and horizontal directions are 1.0 m2/s. As the pollution is serious, the degradation coefficient is taken as 0.0 d, according to the demands of the computational stability and accuracy, respectively. Time steps of 1 s and 2 s are selected for computing current and water quality.
Fig.1 Grid in Xuanwu Lake
(3) Parameters of the sediment: According to the experimental data, the COD releasing coefficient of the sediment in the Xuanwu Lake is 0.032mg/L.d. On the basis of this, the comprehensive CODMn coefficient Kb is rated as 0.0075 d, and the comprehensive suspension coefficient is 0.10 d. By referring to the
506
Table 2 Computed results of the CODMn in selected cells Unit No.
365
372
379
382
386
393
400
407
No sediment pollution
9.81
9.23
6.37
6.02
6.00
7.05
9.40
10.11
Method of suspension coefficient of sediment pollution
10.18
9.60
6.95
6.21
6.04
7.92
10.01
10.53
Calculating method of the velocity of sediment pollution
10.18
9.61
6.95
6.20
6.04
7.91
9.96
10.47
classification table of mud and sand, the particle diameter of the colloidal particle of pollutants is rated as 0.0022 mm. (4) Initial and boundary conditions: The four entering boundaries for the current flux in the computational domain in turn, from left to right are, 0.9 m3/s, 0.1 m3/s, 0.1 m3/s and 0.9 m3/s, and the two outflow boundary fluxes are: for the Dashugen Gate 0.5 m3/s and for the Wu Temple Gate 1.5 m3/s. The initial water level of each unit in the computational domain takes the relative level of the lower boundary, the primary velocity of current is 0 m/s, and the introduced water quality CODMn takes the type-Ⅱ water standard 4 mg/L, and the primary concentration of CODMn in water and sediment both take the value of 8 mg/L.
current flows from the entering boundaries to both the upper and lower boundaries respectively, the velocity distribution is reasonable, and the relative CODMn values gradually increase outwards from 4 mg/L of the upper boundary. As the entering current flux is not large, the velocity in the lake area is also low (less than 0.03 m/s), the convection diffusion of pollutants is relatively slow. After introducing water for 9 d, the concentration at the outlet of the Dashugen Gate reaches the standard of water type-Ⅲ (6 mg/L), and after introducing water for 29 d, the concentration at the outlet of the Wu Temple Gate reaches the standard of water type-Ⅲ (6 mg/L). Its contour distribution is reasonable and consistent with the flow field and underwater topography. The computed results of the CODMn with or without pollution from the sediment are given in Table 2. The data show that the pollution from the sediment enhances the CODMn concentration of water, the variation is related to the suspension velocity and the CODMn concentration of sediment and water, and is reasonable and trustable.
Fig.2 Velocity field in Xuanwu Lake at 48 h
(5) Results of calculation and analysis: The stable velocity field is shown in Fig.2, and the CODMn contours at 9 d and 29 d of introducing water to clean the pollution in Figs. 3 and 4. It can be seen from the figures that the
Fig.3 Computed CODMn contours at 9 d
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Table 3 Sensitivity analysis for sediment pollution parameters of CODMn Sink and float coefficient
Particle diameter of mud and sand
Sediment CODMn concentration
(1/ d)
(mm)
(mg/L)
Name of parameters
Parameter value
0.18
0.144
0.216
0.0032
0.0026
0.0038
12.0
10.0
14.0
CODMn
10.27
10.22
10.32
10.24
10.16
10.32
10.27
10.02
10.53
Variation (mg/L)
0
-0.05
0.05
0
-0.08
0.08
0
-0.25
0.26
Variation (%)
0
-0.49
0.49
0
-0.78
0.78
0
-2.43
2.53
second order, the absolute value of the 20% changed initial sediment CODMn concentration is relatively big, and causes a relatively large variation in the water CODMn.
Fig.4 Computed CODMn contours at 29 d
(6) Sensitivity analysis for parameters: Table 3 shows different parameter values of the sediment, including the particle diameter of mud and sand, suspension coefficient, and the sensitivity of initial CODMn concentration of the sediment on the variation of the concentration of water pollution. The Table shows that when the suspension coefficient changes at 20%, the largest variation in CODMn concentration of water is 0.5%, when the particle diameter of mud and sand changes at 20%, the largest variation in CODMn concentration of water is 0.8%, and when the sediment CODMn concentration changes for 2.0 mg/L the largest variation in water pollution is 0.26 mg/L, with a sensitivity of about 2.5% of the suspension coefficient and the particle diameter of mud and sand being not so high, and the variation of water CODMn concentration caused by the variation of the former one being much smaller. Because the relation between the comprehensive sink velocity ω c and the suspension coefficient in the equation is of the first order, and that between the velocity and the particle diameter is of the
5. CONCLUSIONS On the basis of the theory of turbulent diffusion, the water quality equation was derived for the sediment pollution, which was then coupled with the shallow water equation and formed a 2-D current-water quality model. Under the framework of the finite volume method, with the Osher approximate Rieman solver, the coupled equations had been solved. The model can be used to simulate the water quality status of steady and unsteady flows and to catch it’s fast and slow variation, and can well fit the water body with complicated ambient conditions, and the computation accuracy is rather high. But the procedure is constrained by the Courant condition, and the step length is relatively small. The model fits well with the computed results and the exact solution of concentration with Gaussian distribution, and the water-quality simulation for the water of the Xuanwu Lake with sediment pollution is reasonable and trustable. The presented results provide the water-quality simulation of the shallow lake with different approaches and a practical numerical model, although, for the parameters of sediment pollution, more research experiments must be conducted.
REFERENCES [1]
WANG Zhi-li, GENG Yan-fen, JIN Sheng. An unstructured finite volume algorithm for nonlinear two-dimensional shallow-water equation [J]. Journal of
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[2]
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Hydrodynamics, Ser.B, 2005, 17(3):306-312. FU Guo-wei. The numerical models and application of water quality in rivers [M]. Beijing: Environmental Press, 1987, 25-33(in Chinese). LI Da-ming, ZHANG Hong-ping, LI Bing-fei. Basic theory and mathematical modeling of urban rainstorm water logging [J]. Journal of Hydrodynamics, Ser. B, 2004, 16(1): 17-27. GENG Yan-fen. FVS Scheme for severe transient flow in pipe networks [J]. Journal of Hydrodynamics, Ser. B, 2005, 17(5):621-628. ZHAO D. H., SHEN H. W., TABIOS III G. Q. et al. Finite-volume two-dimensional unsteady-flow model for river basins[J]. Journal of Hydraulic Engineering, ASCE, 1994, 120 (7): 863-883. ZHAO D. H., SHEN H. W., LAI J. S. et al. Approximate Riemann Solvers in FVM for 2D hydraulic shock waves modeling [J]. Journal of Hydraulic Engineering, ASCE, 1996, 122 (12): 693-702. ZHAO Di-hua. The finite volume method and the approximate Riemann solver of 2D flow-water quality model [J]. Progress in Water Science, 2000, 11(4):
368-373(in Chinese). ZHAO Di-hua. The simulation of 2D flow-water quality with the cleavage format of the flux and vector [J]. Progress in Water Science, 2002, 13(6): 701-706(in Chinese). [9] ZHAO Di-hua. The 2D flow-water quality simulation of Jiangsu segment in the Yangtze River [J]. Journal of Hydraulic Engineering, 2003, (6): 72-77(in Chinese). [10] DOU Guo-ren. Mechanics of turbulence flows [M]. Beijing: Higher Education Press, 1985, 102-119(in Chinese). [11] LIN P. N. and SHEN H. W. Two-dimensional flow with sediment by characteristics method [J]. Journal of Hydraulic Engineering, ASCE, 1984, 110 (5): 615-626. [12] SWEBY P. K. High resolution schemes using flux limiters for hyperbolic conservation laws [J]. SIAM J. Numer. Anal., 1988, 21 (5): 995-1011. [8]
Ser.B, 2007,19(4):501-508
ANALYSIS OF WATER QUALITY IN SHALLOW LAKES WITH A TWO-DIMENSIONAL FLOW-SEDIMENT MODEL* JI Yong, ZhANG Jie Nanchang Institute of Technology, Nanchang 330099, China, E-mail:[email protected] YAO Qi College of Environmental Science and Engineering, Hohai University, Nanjing 210098, China ZHAO Di-hua Nanjing Hydraulic Research Institute, Nanjing 210098, China
(Received December 2, 2005; Revised March 24, 2006)
ABSTRACT: The governing equation for sediment pollutions was derived based on the turbulent diffusion of pollutants in shallow lakes. Coupled with shallow water equations, a depth-averaged 2-D flow and water quality model was developed. By means of the conservation law, a proposed differential equation for the change of sediment pollutants was linked to the 2-D equations. Under the framework of the finite volume method, the Osher approximate Riemann solver was employed to solve the equations. An analytical resolution was used to examine the model capabilities. Simulated results matched the exact solutions especially well. As an example, the simulation of CODMn in the Wuli Lake, a part of the Taihu lake, was conducted, which led to reasonable results. This study provides a new approach and a practical tool for the simulation of flow and water quality in shallow lakes. KEY WORDS：sediment, falling velocity, turbulence, flow and water quality model, finite volume method，approximate Riemann solver, shallow water equation，advection-dispersion equation
1. INTRODUCTION A lot of research on the flows in deep lakes has been conducted by a lot of scholars, but the simulation study on shallow lakes is still not mature[1]. As far as the Taihu lake is concerned, the water current and the mechanism of pollutant
transfer are different from those in deep lakes. For shallow water, any outside functions can disturb the body of water, causing the pollution to come in from the sludge. Since the 1940s, there have been a great many researches on the oxygen consumption in the sludge, but few using the equations of flow and water quality, and sometimes the impact of sludge is simplified as a constant. In this article, based on turbulence diffusion theory, a water quality equation including the sediment pollutions will be derived, which is coupled with the shallow water equations and an equation describing the sediment pollutions, resulting in a complete model of two-dimensional flow-water quality with semimetal pollution. Under the framework of the finite volume method[2-4], the Osher approximate Riemann solver is employed to solve the system of equations. The Osher approximate Riemann solver is a two-dimensional model first used in aerodynamics. Now it has been applied in the simulation of water quality[5-9]. The model is employed to simulate a complex water body, including steady and unsteady flows, jet and torrent flow, and a water body with a complex surrounding terrain. And the result is rather perfect. Finally, the accuracy of the model is examined, and then by
* Project supported by the National Natural Science Foundation of China (Grant No. 50239093) and Nanjing Construction Commission (Grant No. 20050176). Biography: JI Yong (1979- ), Male, Master, Lecturer
502
taking the Xuanwu Lake in Nanjing as an example, the effectiveness of the model is verified.
2.THE EQUATION OF WATER QUALITY WITH SEDIMENT POLLUTION 2.1 The basic equations The sediment pollution is related to the movement of contamination in a vertical direction. It is a three-dimensional problem. According to the turbulent diffusion theory[10-12], the 3-D convection-diffusion equation for the water quality including sediment pollution is expressed as
unchanged in the z direction and the velocity component in the z direction is zero. The pollution of sediment comes from the turbulence diffusion and contamination settlement happening at the bottom. Therefore, the 2-D horizontal equation of water quality with sediment pollution is
∂ (hC ) ∂ (huC ) ∂ (hvC ) ∂ ∂ (hc) ]+ + + = [ Dx ∂t ∂x ∂y ∂x ∂x ∂ ∂ (hc) ∂C [ Dy ] + ( Dz |+ − ∂y ∂y ∂z 0
∂C ∂ ∂ ∂ + (Cu ) + (Cv) + (Cw) = ∂t ∂x ∂y ∂z
ω c C0 ) − K c hC
In the equation, u and v are respectively the vertical average current velocity in the x and y direction, C is the vertical average
∂ ∂C ∂ ∂C ( Dx ) + ( Dy )+ ∂x ∂x ∂y ∂y ∂ ∂C ∂ ( Dz ) − (Cω c ) − K c C ∂z ∂z ∂z
concentration (1)
where u , v , w are the velocity components of flow in the x , y , and z directions respectively,
C is the concentration of the contamination, ωc is the sedimentation rate of contamination that stems from the integrated effect of its weight and the diffusion of water, Dx ， Dy ， and Dz are the diffusion coefficients of the contamination in x , y , and z directions respectively, Kc is integrated sedimentation coefficient of contamination. The following equation is integral of the above equation in the z direction:
∫
h
0
the the the an
h ∂ h ∂ ∂C dz + ∫ (Cu )dz + ∫ (Cv)dz + 0 ∂x 0 ∂y ∂t
∫
h
0
∫
h
0
(2)
+
h ∂ ∂ ∂C (Cw)dz = ∫ ( Dx )dz + 0 ∂x ∂z ∂x
h ∂ ∂ ∂C ∂C ( Dy )dz + ∫ ( Dz − 0 ∂z ∂y ∂y ∂z
h
Cωc )dz − ∫ Κ cCdz 0
For the adequate mixture in the shallow lake, it is considered that the dependent variables remain
of
the
pollutant,
C0+ is
the
concentration of pollutants in the near-bottom water, according to horizontal 2-D assumption C0+ = C ,
ωc , and K c are the same as mentioned earlier, Dz
∂C | + is the upward mobility of pollutants in ∂z 0
the sediment of bottom interface through turbulent motion. On the basis of the turbulent diffusion theory, if the concentrations of pollutants at different sections are different, the pollutant diffuses from the section with higher concentration to the section with lower concentration, and its diffusion intensity is related to the concentration gradient and diffusion coefficient, and the diffusion coefficient differs from the turbulent intensity. If the concentration of pollutants in the sediment is higher than that of water or there is outside force disturbing, the pollutants of sediment will diffuse into water. Meanwhile, under the action of gravity, pollutants drop to the bottom and increase the concentration of pollutants in the sediment. The concentration gradient of the bottom interface is
C −C ∂C , in which Cb and C are the |0+ = b ∂z Δz
concentration of pollutants in sediment and water respectively, and vary with space-time. Therefore, in Dz
∂C D |0+ = z (Cb − C ) = wc (Cb − C ) , wc is ∂z Δz
the diffusion coefficient of unit depth, and can be considered as the diffusion upward velocity of pollutants in the sediment and will be rated by field
503
data. The water quality control equation including the pollutants in sediment is
∂ (hC ) ∂ (huC ) ∂ (hvC ) + + = ∂t ∂x ∂y
(3)
It can be seen from Eq.(3) that the pollution to the water from sediments, depends on the release rate, wc (Cb − C ) , of the concentration of pollutants in the sediment, and the comprehensive descending rate, ω c C , of the concentration of pollutants in the water. When the release of pollutants in the sediment is larger than the drop of pollutants in the water, the concentration of pollutants in the water will keep increasing till it reaches a dynamic balance. 2.2 The determination of the pollution parameter in the sediment (1) Comprehensive sinking velocity ωc The determination of the comprehensive sinking velocity, ωc , can adopt the following two methods: (a) Calculating method of the suspension Ks of pollutants: Ks is coefficient back-calculated from the experimental data for release of the sediment. According to its physical meaning, and through dimensional analysis, gain
Ks =
ωc h
calculate ωc :
ωc = β
∂ ∂ (hC ) ∂ ∂ (hC ) [ Dx ]+ [ Dy ]+ ∂x ∂x ∂y ∂y
[ wc (Cb − C ) − ω c C ] − K c hC
action of gravity. This article adopts the Stokes falling velocity formula to be multiplied by β , to
, in which h is the depth of water.
Therefore, the comprehensive sink and float velocity is ω c = K s h . (b) Calculating method of the suspension formula for suspended matter: as the pollutants sink and suspend under the action of gravitation and outside forces, similar to the colloidal particles of mud, the existing sink and float formula for suspended matter can be adopted, which just multiplies the reduction coefficient. The reduction coefficient, β , depends on the factors that cause the suspension of pollutants, such as, the wind velocity, wind direction, water velocity, and turbulent motion, which reflects the contrasting relationship between turbulent motion and the
1 γs −γ d2 g ν 18 γ
where β is the reduction coefficient, representing the contrasting relations between the suspension effect of turbulent motion and the sink effect of the gravity, β ≤ 1 , γ s , and γ are respectively the colloidal particles and volume weight of water, as the pollutants can sink in single particles as well as in flock, d is the equivalent particle diameter, ν is the kinematic viscosity coefficient. The diameter d and reduction coefficient β will be rated according to the tested data. (2) The concentration of pollutants in sediment Cb The variation in the concentration of pollutants in the sediment can be affected by many factors, such as, absorption, desorption, and mineralization, and it is very complicated. It is assumed that the variation is in accord with the first-order reaction, considering the conservation of mass of exchanged pollutants at the lake bottom, the equation for the variation can be written as
dCb w w = − kbCb − c (Cb − C ) + c C dt h h
(4)
where h is the depth of water, Kb is the attenuation coefficient of pollutants in the sediment, which is related to the ambient conditions, such as, temperature, current, and sediment, and will be calculated and rated by the release coefficient of the sediment, other notations are the same as those in Eq.(3). The physical meaning of the equation is that the variation in the concentration Cb of pollutants in the sediment is composed of three parts: one is the attenuation of itself, one is the diffusion towards the water, and the other is the sedimentation of pollutants in water.
3. 2-D MODEL FOR CURRENT, WATER QUALITY, AND SEDIMENT POLLUTION The conservation form of the coupled shallow-water equation and water-quality equation with sediment pollution can be expressed as
504
∂h ∂(hu) ∂(hv) + + =0 ∂t ∂x ∂y
(5a)
1 τ wx ρ
(5b)
∂ (hv) ∂ (huv) ∂ (hv 2 + gh 2 / 2) + + = ∂t ∂x ∂y gh( s0 y − s fy ) +
1 τ wy ρ
(5c)
and ωci are the diffusion velocities of the sediment of the six pollutant compositions and the comprehensive sinking velocity, and Cbi and K bi are the concentrations of the six pollutant compositions in the sediment and attenuation coefficient. Now the problem is to solve the coupled equations. Equation (5) can be expressed as
∂q ∂f (q) ∂g (q ) + + = b(q ) ∂t ∂x ∂y
(7)
The expression of the limited volume after dispersement of the integral in Eq.(7) is, (for details, refer to Ref.[5])
∂ (hCi ) ∂ (huCi ) ∂ (hvCi ) + + = ∂t ∂x ∂y ∂ (hCi ) ∂ (hCi ) ∂ ∂ [ Dix ] + [ Diy ]+ ∂x ∂x ∂y ∂y
A
[Wci (Cbi − Ci ) − ω ci Ci ] − K ci hCi + Si (5d) The equation for the variation of the pollutants in sediment is
dCbi W W = − kbi Cbi − ci (Cbi − Ci ) + ci Ci dt h h
and Diy are the diffusion coefficients of each pollutant in the x and y directions respectively, Wci
∂ (hu ) ∂ (hu 2 + gh 2 / 2) ∂ (huv) + + = ∂t ∂x ∂y gh( s0 x − s fx ) +
collection of resources of each pollutant, and its contents will change with the composition, Dix
(6)
m Δq = −∑ T (Φ ) −1 f (q ) Lj + b* (q) Δt j =1
(8)
In the equation, A is the area of entity Ω , m is the total number of entity edges, L j is the length of j entity edge,
b* (q ) = [0, Agh( s0 x − s fx ), Agh( s0 y − s fy ), ΣDi (∇hCi ) n L j + Aωc (Cbi − Ci ) − ( AK ci hCi + Si )]T
where h is the depth of water, u and v are the components of the vertical average horizontal velocity in the x and y directions respectively, g
∇ is the gradient operator, q = T (Φ )q , and
is the gravity acceleration, s0 x and s fx are the
T (Φ)
bottom slope of ground and the friction gradient, respectively, in the x direction, s0 y and s fy are the bottom slope of ground and friction gradient in the y direction respectively, ρ is the density of
−1
and T (Φ ) are the rotation transformation and inverse transformation matrix of the coordinates. The Osher approximate Rieman solver refer to Ref.[5].
water, τ wx and τ wy are the wind stresses in the x and y directions respectively, Ci is the vertical average concentration of pollutants, including six compositions (i.e., COD， NBOD， CBOD， DO， TP and TN), K ci is the comprehensive degradation coefficient of each pollutant, Si is the
4. EXAMPLE 4.1 Transport of concentration with gaussian distribution in a uniform flow field The computational comain for the transport of concentration with Gaussian distribution in a uniform flow field is 5000 m ×5000 m, and the size
505
Table 1 Computed results of concentration with Gaussian distribution Time
0
0.5
1.0
1.5
2.0
Exact solution
1.0
0.85
0.74
0.68
0.63
Numerical solution
1.0
0.84
0.73
0.66
0.61
Error (%)
0
1.18
2.70
2.94
3.17
of the grid is Δx = Δy = 100 m. It is assumed that the bottom of the field has no friction, the gradient is zero, the current is invariable and average, the current velocity is 0.5 m/s and the depth of water is 0.5 m. The borderline is determined as CR = CL , its peak concentration of initial distribution is 1.0 mg/L, and the diffusion coefficient in each direction is Dx = D y = 5 m2/s. Compared with the exact solution which is given in Table 1, the computation error is within 3.5%, and no negative value and shock occur during the computation (for details refer to Ref.[5]). 4.2 The calculation of COD in the Xuanwu Lake with the pollution of sediment (1) General description: The Xuanwu Lake is located in the northeast of the old city zone of Nanjing City, and is a small-sized natural shallow lake, which totally covers an area of 472 ha, in which the area of water is 368 ha, and the area of land is 104 ha. The shape of the lakeshore of the Xuanwu Lake is similar to a diamond, the perimeter of the lake is about 10 km, the average water level is lower than 10 m, the average depth of water is 1.43 m, the largest depth of the lake is 2.8 m, and the perennial water level is between 9.8 m and 10.2 m. The flux entering the lake is controlled manually to a small extent. In the Xuanwu Lake there are five oases, Huanzhou, Yingzhou, Lingzhou, Liangzhou, and Cuizhou, which are connected and unified by bridges and banks. With the development of economy and the growth of population, the sanitary sewage from the city and surface runoff has polluted the Xuanwu Lake to a serious extent, and the water quality has been in a state of eutrophication. The main pollutants of the lake are TP, TN, NH3-N, and COD. Therefore, it is planned to introduce the water from the Yangtze River to dilute the concentration of pollutants and improve the water quality of the lake.
(2) Calculation parameters: In consideration of the landform, the location of the sewage drain outlet, Peclet value, Courant value, grid rate, and so on, the computational domain for the Xuanwu Lake is composed of a nonstructural grid with 3032 units and 3301 nodal points (Fig.1), and the side length is 10 m - 40 m. The Manning coefficient n is taken as 0.022. The diffusion coefficients of CODMn in both longitudinal and horizontal directions are 1.0 m2/s. As the pollution is serious, the degradation coefficient is taken as 0.0 d, according to the demands of the computational stability and accuracy, respectively. Time steps of 1 s and 2 s are selected for computing current and water quality.
Fig.1 Grid in Xuanwu Lake
(3) Parameters of the sediment: According to the experimental data, the COD releasing coefficient of the sediment in the Xuanwu Lake is 0.032mg/L.d. On the basis of this, the comprehensive CODMn coefficient Kb is rated as 0.0075 d, and the comprehensive suspension coefficient is 0.10 d. By referring to the
506
Table 2 Computed results of the CODMn in selected cells Unit No.
365
372
379
382
386
393
400
407
No sediment pollution
9.81
9.23
6.37
6.02
6.00
7.05
9.40
10.11
Method of suspension coefficient of sediment pollution
10.18
9.60
6.95
6.21
6.04
7.92
10.01
10.53
Calculating method of the velocity of sediment pollution
10.18
9.61
6.95
6.20
6.04
7.91
9.96
10.47
classification table of mud and sand, the particle diameter of the colloidal particle of pollutants is rated as 0.0022 mm. (4) Initial and boundary conditions: The four entering boundaries for the current flux in the computational domain in turn, from left to right are, 0.9 m3/s, 0.1 m3/s, 0.1 m3/s and 0.9 m3/s, and the two outflow boundary fluxes are: for the Dashugen Gate 0.5 m3/s and for the Wu Temple Gate 1.5 m3/s. The initial water level of each unit in the computational domain takes the relative level of the lower boundary, the primary velocity of current is 0 m/s, and the introduced water quality CODMn takes the type-Ⅱ water standard 4 mg/L, and the primary concentration of CODMn in water and sediment both take the value of 8 mg/L.
current flows from the entering boundaries to both the upper and lower boundaries respectively, the velocity distribution is reasonable, and the relative CODMn values gradually increase outwards from 4 mg/L of the upper boundary. As the entering current flux is not large, the velocity in the lake area is also low (less than 0.03 m/s), the convection diffusion of pollutants is relatively slow. After introducing water for 9 d, the concentration at the outlet of the Dashugen Gate reaches the standard of water type-Ⅲ (6 mg/L), and after introducing water for 29 d, the concentration at the outlet of the Wu Temple Gate reaches the standard of water type-Ⅲ (6 mg/L). Its contour distribution is reasonable and consistent with the flow field and underwater topography. The computed results of the CODMn with or without pollution from the sediment are given in Table 2. The data show that the pollution from the sediment enhances the CODMn concentration of water, the variation is related to the suspension velocity and the CODMn concentration of sediment and water, and is reasonable and trustable.
Fig.2 Velocity field in Xuanwu Lake at 48 h
(5) Results of calculation and analysis: The stable velocity field is shown in Fig.2, and the CODMn contours at 9 d and 29 d of introducing water to clean the pollution in Figs. 3 and 4. It can be seen from the figures that the
Fig.3 Computed CODMn contours at 9 d
507
Table 3 Sensitivity analysis for sediment pollution parameters of CODMn Sink and float coefficient
Particle diameter of mud and sand
Sediment CODMn concentration
(1/ d)
(mm)
(mg/L)
Name of parameters
Parameter value
0.18
0.144
0.216
0.0032
0.0026
0.0038
12.0
10.0
14.0
CODMn
10.27
10.22
10.32
10.24
10.16
10.32
10.27
10.02
10.53
Variation (mg/L)
0
-0.05
0.05
0
-0.08
0.08
0
-0.25
0.26
Variation (%)
0
-0.49
0.49
0
-0.78
0.78
0
-2.43
2.53
second order, the absolute value of the 20% changed initial sediment CODMn concentration is relatively big, and causes a relatively large variation in the water CODMn.
Fig.4 Computed CODMn contours at 29 d
(6) Sensitivity analysis for parameters: Table 3 shows different parameter values of the sediment, including the particle diameter of mud and sand, suspension coefficient, and the sensitivity of initial CODMn concentration of the sediment on the variation of the concentration of water pollution. The Table shows that when the suspension coefficient changes at 20%, the largest variation in CODMn concentration of water is 0.5%, when the particle diameter of mud and sand changes at 20%, the largest variation in CODMn concentration of water is 0.8%, and when the sediment CODMn concentration changes for 2.0 mg/L the largest variation in water pollution is 0.26 mg/L, with a sensitivity of about 2.5% of the suspension coefficient and the particle diameter of mud and sand being not so high, and the variation of water CODMn concentration caused by the variation of the former one being much smaller. Because the relation between the comprehensive sink velocity ω c and the suspension coefficient in the equation is of the first order, and that between the velocity and the particle diameter is of the
5. CONCLUSIONS On the basis of the theory of turbulent diffusion, the water quality equation was derived for the sediment pollution, which was then coupled with the shallow water equation and formed a 2-D current-water quality model. Under the framework of the finite volume method, with the Osher approximate Rieman solver, the coupled equations had been solved. The model can be used to simulate the water quality status of steady and unsteady flows and to catch it’s fast and slow variation, and can well fit the water body with complicated ambient conditions, and the computation accuracy is rather high. But the procedure is constrained by the Courant condition, and the step length is relatively small. The model fits well with the computed results and the exact solution of concentration with Gaussian distribution, and the water-quality simulation for the water of the Xuanwu Lake with sediment pollution is reasonable and trustable. The presented results provide the water-quality simulation of the shallow lake with different approaches and a practical numerical model, although, for the parameters of sediment pollution, more research experiments must be conducted.
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