CHAPTER 3 METHODOLOGY OF PRESENT MODEL
3.6 Segment division in a pipe line at each hydraulic time step
3.6 Segment division in a pipe line at each hydraulic time step
At the first HTS, the length of segment in each pipe is equal to the pipe length; thus, the retention time of each segment equals the pipe length divided by the average flow velocity.
The dimensionless time at the end of the mth HTS in pipe j, T ′jm, is equal to th /PTjm. Consider that the water passing through the outlet node of pipe j at the end of mth HTS comes from v′th segment of pipe k′. The variables k′ and v′ can be determined by Eq. (36) when Xj
= 1 and τ = th. However, the water in v′th segment of pipe k′ may not reside completely in pipe j at the end of mth HTS because a part of the water may travel through the outlet node of pipe j. The flow time for the water in v′th segment of pipe k′ passing through the outlet node
of pipe j is equal to . The remaining part of the water staying in
pipe j becomes the first segment of pipe j at the (m+1)th HTS and its flow time is
. Therefore, the retention time for the first segment of
pipe j at the (m+1)th HTS, is where
m
uj and are the average flow velocities in pipe j at the mth and (m+1)th HTS, respectively.
m+1
uj
The water in the second segment of pipe j at the (m+1)th HTS may come from a whole segment situated at the upstream of the v′th segment in pipe k′. Therefore, the retention time for water in the 2nd segment is equal to if the v′th segment is the
last segment in pipe k′ or if the v′th segment is not the last one. Similarly, the retention times for succeeding segments in pipe j at the (m+1)th HTS except for the last one can then be determined.
m+1
The sum of the retention time for water in each segment within a pipe must equal the hydraulic retention time for water flowing through the pipe j. If the sum of the retention time for water from the first segment to the Nth segment is greater than or equal to the hydraulic retention time of pipe j, the Nth segment is indeed the last segment of pipe j. Thus, the retention time for water in the Nth segment STN, jm+1 is equal to jm+1 1 i m+1.
The initial concentration distribution in each segment at the (m+1)th HTS can be determined based on Eq. (33) or (35), once the origin of the water in each segment of pipe j at the (m+1)th HTS is known. For example, the concentration distribution in the first segment at the (m+1)th HTS can be determined with known T ′jm, v′, and k′ according to Eq. (33) if k′
= j or Eq. (35) if k′ ≠ j . 3.7 Network
In a water network, there may be more than one pipe connected to a node. Assume that the concentration at the confluent node is well mixed and its concentration, Cm, can be calculated as (Rossman et al., 1994)
i the total flow rate at confluent node i, and qi is the flow rate at confluent node i in a single flow path. Figure 3 shows an example network which comprises nine pipes, seven nodes, and two confluent nodes denoted as black circles. For the outlet node in pipe 9, there are three flow paths. Path 1 includes pipes 1, 2, 3, 5, 6 and 9; path 2 has pipes 1, 2, 4, 6, and 9;
path 3 contains pipes 1 and 7 to 9. Based on Eq. (37), the mixing concentration at the outlet node of pipe 9 is
3.8 Flowchart of water quality simulation
The flowchart for the simulation of chlorine concentration in a water network using the present model is illustrated in Figure 4. The flowchart includes three parts. The first part is to set the initial condition of the retention time and the initial concentration distribution in each segment for each flow path at the fist HTS. Then, the present model is proceeding to the second part for the estimation of the chlorine concentration at each WQTS. The second part, which includes the main solution algorithm of the present model, includes three steps.
The first step is to determine the vth segment of pipe k for each flow path in which the water passes through the location Xj at the current WQTS using Eq. (36). The second step is to calculate the chlorine concentration in each flow path using Eq. (33) or (35) and the last step is to calculate the mixed concentration for the current WQTS by Eq. (37). The last part is to determine the retention times and concentration distribution in each segment for all flow paths for the next HTS. Then, the algorithm of the present model will go back to the second part to estimate the chlorine concentration for the next HTS. When accomplishing all the HTSs, the analytical model will complete the network concentration simulations.
CHAPTER 4 RESULTS AND DISCUSSION
4.1 Accuracy comparisons of the approximate solutions
The present approximate solution is compared with both the analytical and approximate solutions given in Biswas et al. (1993). Three figures are plotted to investigate the effect of the parameters D, K, and W on the corresponding predicted chlorine concentration. Figures 5(a), 5(b) and 5(c) show the curves for the chlorine concentration distribution of, Cav with initial concentration equal to one, at the outlet (X = 1) versus the dimensionless wall decay rate (W) with different values of dimensionless radial diffusivity (D) for the dimensionless water decay rate (K) equal to 0.001, 0.1 and 1. The solid line, dotted line, and dashed line represent the analytical solution, present approximate solution, and Biswas et al.’s approximate solution (1993), respectively.
As indicated in Figures 5(a) - (c) for different K values, the value of Cav based on Biswas et al.'s approximate solution (1993), starts to be different from the analytical solution for W at 0.003, 0.03 and 0.3 when D = 100, 10, and 1, respectively. The present approximate solution is in good agreement with the analytical solution for K ranging from 0.001 to 1 with D equals 100, 10, and 1 except in the region where W > 0.5 and K = 1. Those results indicate that the parameters of D and W have an apparent influence on the accuracy of those two approximate solutions. In addition, the present approximate solution generally gives better prediction for the chlorine concentration than that of Biswas et al.’s approximate solution (1993).
The poor accuracy of the Biswas et al.’s approximate solution (1993) stems from the fact that the expressions of fractional factor, ε, in terms of D and Wwere developed using the regression techniques as shown in Eq. (25). On the other hand, the error of the present approximate solution is made mainly by neglecting the higher order terms of the Bessel
functions in Eqs. (27) and (28). If the first eigenvalue λ1 is small, the errors of neglecting the high order terms in the Bessel functions of Eqs. (27) and (28) will be very small.
Figure 6 shows the plots of the true and approximate values of λ1 against W. The solid line represents the true λ1 obtained from Eq. (16) by Newton’s method and the dashed line denotes the approximate λ1 calculated from Eq. (29). This figure indicates that both the value of λ1 and the difference in λ1 increase with W. In addition, Table 1 shows the relative errors of the approximate λ1 to the true λ1 for W ranging from 0.001 to 0.5 and the relative error is about 1.2 % at W = 0.1. Accordingly, the present approximate solution gives accurate results when W< 0.1 and is thus appropriate for most of field cases.
4.2 Two case studies for the approximate solution
The first-order reaction kinetics is usually used to represent the chlorine decay in the bulk liquid of the pipe and at the pipe wall. The decay parameters can be determined based on an appropriate mathematical model and measured chlorine concentration data. Two cases are chosen to demonstrate the application of the present approximate solution. The wall surface reaction constant estimated based on the approximate solution is compared with those obtained from Biswas et al. (1993) and Rossman (2006) in Case 1 and from the water network of SCCRWA (Biswas et al., 1993) in Case 2.
4.2.1 Case 1
Rossman (2006) used a distribution system simulator which consisted of a 27 m long loop with 0.15 m diameter unlined ductile iron pipe, a recirculation pump and a heat exchanger cooling system. An experiment was made to measure the reaction rate of chlorine in a simulated pipe for water treated by different forms of advanced treatment at US EPA’s Test and Evaluation Facility in Cincinnati, Ohio. In Case 1, the present approximate solution is used to determine the pipe wall surface reaction constants wd for water applied by three different treatments. The initial chlorine was about 6 mg/L and the values of kd shown in the second column of Table 2 (Rossman, 2006) for lab-tested water under different
treatments were determined based on the analysis of the kinetic test data. The wall surface reaction constants, wd, for three sorts of lab-tested water can then be determined from the experiment-observed data based on the present approximate solution. In the experiment, the flow velocity was maintained constant, so steady-state flow condition was considered. Note that the axial distance from the inlet along the pipe, x, is equal to the flow velocity multiplied by the flow time in pipe. Under the turbulent condition, the eddy diffusion is greater than the molecular diffusion. Thus, the effective diffusivity in the radial direction, Dr, is only considered the eddy diffusivity which can be obtained from Edwards et al. (1979) because of turbulent flow. With the known values of pipe radius, pipe length, flow velocity, and chlorine bulk decay constant, wd can be determined based on the present approximate solution, Eq. (31), when minimizing the objective function defined as the sum of square errors between the observed and predicted chlorine concentrations. In order to determine the optimal value of wd for three sorts of lab-tested water, simulated annealing (SA) is applied. The SA is a generic probabilistic meta-algorithm for the global optimization problem based on the annealing concept, namely locating a good approximation to the global optimum of an objective function in a large search space. The initial temperature of the SA is chosen as 100 and the temperature is decreased by the temperature reduction factor (0.85) after 8100 calculations. The annealing process will be terminated if the absolute differences between two successive objective function values are all less than 10-10 within 20 iterations or the number of evaluations is greater than 107. The SA has been successfully applied in forecasting THM Species (Lin and Yeh, 2005), parameter estimations (e.g., Yeh and Chen, 2007; Yeh et al., 2007a), and source identifications (e.g., Linand Yeh, 2007; Yeh et al., 2007b).
As suggested by Rossman (2006), a first-order reaction model for describing the first-order decay of chlorine in bulk flow and at the pipe wall was expressed as
d approximate solution, the first-order reaction model (Rossman, 2006), and Biswas et al.’s approximate solution (1993) for three sorts of lab-tested water. The table shows that the values of wd estimated by these three models are close for the same lab-tested water. Note that the first-order reaction model neglects the radial diffusion and thus the estimated wd
slightly differs from those given by the other two solutions. Figure 7 shows the experiment-observed data (Rossman, 2006) and the simulated results by the present approximate solution and Biswas et al.’s approximate solution (1993). The solid line represents the result of the present approximate solution and the dashed line represents the result of Biswas et al.’s approximate solution (1993). The symbols of circle, rhombus, and triangle displayed in Figure 7 denote experiment-observed values for lab-tested water treated by reverse osmosis, conventional treatment, and ozonation, respectively. This figure indicates the simulated results of the present approximate solution are in good agreement with experiment-observed values from Rossman (2006). In contrast, the simulated results of Biswas et al.’s approximate solution (1993) are discordant in the case of the lab-tested water treated by ozonation with greater wd value. This problem may be attributed to the fact that the chlorine concentration is inversely proportional to the quadratic of wall decay constant in Biswas et al.’s approximate solution (1993) as expressed in Eqs. (24) and (25). Chlorine concentration is considered to decay exponentially with increasing retention time as indicted in Eq. (39) and thus the chlorine concentration is inversely proportional to the exponent of kd
and wd. The present approximate solution expressed as Eq. (31) conforms to the form of the first-order decay reaction. This may be the reason why the simulated results of the present approximate solution are better than those of Biswas et al.’s approximate solution (1993) when wd is large.
4.2.2 Case 2
In Case 2, the present approximate solution is employed to determine the wd in a field test conducted by the SCCRWA in New Haven, Connecticut. The service area of this network covers 5.2 km2 and the network is composed of a storage tank, a pump station, 40 pipes ranging from 76 to 731 m with constant diameters of 20.3 and 30.5 cm, and totally 36 nodes. The schematic of this network is shown in Figure 8 which includes eight sample points denoted by the words “SP”. The sampling results of chlorine concentration at the inlet and outlet points of these pipe segments are presented in the second and third columns of Table 3 (Biswas et al., 1993). The geometrical and flow parameter including pipe length, pipe radius, flow velocity (Biswas et al., 1993) and diffusion coefficients for all the pipes are listed in Table 4. The diffusion coefficients were determined by the eddy diffusivity, as obtained from Edwards et al. (1979). In addition, the chlorine bulk decay constant, kd, was 6.4×10-6 (1/s) (Biswas et al., 1993) obtained by bench kinetic tests performed with the water sample taken at the inlet to the network.
In pipes 1 to 3, 6 to 16, 21, and 26 to 28 of this network (Figure 8), those pipes numbered 3, 10 and 21 are dead end pipes while the other pipes are main branch. The present approximate solution expressed in Eq. (31) is used to determine the wall surface reaction constant, wd, in this network. The average value of wd for the main pipes such as 7, 9, 11 to 15, and 26 to 28 in this network can be determined first. Assume that the wall surface reaction constant for the main pipes are all the same because those pipes were made by the same material. Based on Eq. (31), the average concentrations at X = 1, the outlet of pipe 7, can be expressed as
The subscript in each variable represents the pipe number and Cin 7 is the inlet concentration of pipe 7. The average concentrations at the outlet of other main pipes can be expressed in a
similar manner. The chlorine concentrations are only measured at the inlet and outlet nodes of the segment and a segment usually contains several pipes. The outlet concentration of a pipe is in fact the inlet concentration of the next pipe. Thus, the concentration Cav at the outlet of a segment is equal to the product of Cav of each pipe within the segment. The concentration at the outlet of pipe 28, Cout 28 in the segment containing main pipes 7, 9, 11 to 15 and 26 to 28 can be written as
Furthermore, Eq. (41) can be simplified and expressed in term of W as
i i
where D and K are known dimensionless parameters and W is a function of wd. Note that W equals wd Dr/r0 and is the only unknown in Eq. (42). Solving Eq. (42) by Newton’s method, the average value of wd for the main branches including pipes 7, 9, 11 to 15 and 26 to 28 is obtained as 3.47×10-7 (m/s). This value presents the average wall surface reaction constant in the main branch. The same approach and the main branch wd are then employed to further evaluate the wd for the dead ends pipes 3, 10, and 21 as shown in Figure 8. The estimated wd
including those three pipes are in the range from 3.47×10-7 to 1.01×10-5 (m/s) and listed in the last column of Table 4. Note that the value of wd ranges from 0 to 7.06×10-5 (m/s) reported from field or experimental studies (Biswas et al., 1993; Rossman et al., 1994; Vasconcelos et al., 1997; Ozdemir and Ger, 1998; Munavalli and Kumar, 2006; Rossman, 2006). With the known geometrical and flow parameters and the estimated values of wd, the concentration Cav
for all the segments in this network predicted by Biswas et al.’s analytical solution (1993), Eq.
(21) are listed in the fifth column of Table 3.
The sampled concentrations in the inlet and outlet of segments in the network are listed in the second and third columns of Table 3, respectively. In addition, the dimensionless
sampled concentration at each segment is listed in the fourth column. The segments of pipes 7, 9, 11 and 12 to 15, 26 to 28 consist of the main branch. Table 3 lists the Cav/Cin predicted based on the analytical solution and the sampling concentration given in Biswas et al. (1993) at some inlets and outlets of the pipe segments. Table 3 indicates that the predicted Cav/Cin
for the segments containing dead-end pipe agrees with the dimensionless sampling concentration (Cout/Cin) for the same segment. This demonstrates that the present approximate solution can be applied to determine wd for the field application problem. Table 4 indicates that the wd’s in the dead-end pipes 3, 10, and 21 are much greater than those in the main branch pipes. High value of wddenotes that wall decay is significant. Biswas et al.
(1993) also mentioned that significant biofilm growth occurs in the dead-end pipe where the flow velocity is relatively low if compared with that in main branch pipe. Low water flow velocity causes more retention time in the pipe, and consequently, yields lower chlorine concentration due to the decay reaction. Once the chlorine concentration is lowered, the microorganisms formed as biofilm on the pipe wall is then increased. Consequently, the high values of pipe wall surface reaction may result in large values of wd for the dead-end pipes 3, 10, and 21 as demonstrated in the last column of Table 4.
4.3 Network simulation
The present model is used to simulate the chlorine distribution in the water distribution system of the SCCRWA as indicated in Figure 8. The water supply network of the SCCRWA was employed as a study site many times in the past to test various water quality models (e.g., Clark et al., 1993, 1994; Rossman et al., 1994). The water flows into the network from the pump station (denoted as node 1) after the water level in the storage tank is lower than the lowest standard of water level. When the storage tank is filled, the water supply from pump station is stopped and then the water is supplied from the storage tank (denoted as node 26). The hydraulic simulation is performed using EAPNET program to determine the flow rate and velocity for all pipes in this network within 1 hr time interval over
53 sampling period with the pipe and nodal demand data given by Rossman (personal communication). For water quality simulation, each nodal data for the sampling chlorine concentration provided by Rossman (personal communication) is used as the initial concentration of the present model for its upstream pipe.
A value of 6.4 × 10-6 from the bottle test (Rossman et al., 1994) is assigned as the first-order reaction constant kd for chlorine in the network. The flow velocities are high in pipes 1 to 7, 9 to 15 and 26 to 28 and their variations are very small within the first three HTSs. Therefore, the flow is considered steady in those pipes within the first three HTSs.
There are two flows into node 25 at time equal to 3 hr. One is the treated water from pump station, which has flowed through the node 25; the other is from node 4 to node 25 at the end of third HTS, as shown in Figure 8. Note that the flow rate and chorine concentration in pipe 17, i.e., from node 17 to node 15, are very small and negligible. Initial observed concentrations at node 4 is 1.15 mg/L; the chorine concentration at the pump station is kept as
There are two flows into node 25 at time equal to 3 hr. One is the treated water from pump station, which has flowed through the node 25; the other is from node 4 to node 25 at the end of third HTS, as shown in Figure 8. Note that the flow rate and chorine concentration in pipe 17, i.e., from node 17 to node 15, are very small and negligible. Initial observed concentrations at node 4 is 1.15 mg/L; the chorine concentration at the pump station is kept as