CHAPTER 4 RESULTS AND DISCUSSION
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.