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Statistical regression and modeliing analysis for reverse osmosis desalination process
Desalination 351 (2014) 120–127
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Desalination journal homepage: www.elsevier.com/locate/desal
Statistical regression and modeliing analysis for reverse osmosis desalination process Sobana Subramani, Rames C. Panda ⁎ Department of Chemical Engineering, CSIR-CLRI, Adyar, Chennai 600 020, India
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Correlations indicate that the permeate characteristics depend on feed flowrate. • The model is also validated using plant data. • The model for permeate flow-rate supports the use as a 2nd degree prediction model. • Modeling 2-input, 3-output will help in developing control strategies.
a r t i c l e
i n f o
Article history: Received 22 May 2014 Received in revised form 23 July 2014 Accepted 26 July 2014 Available online xxxx Keywords: Desalination Reverse osmosis Statistical model Regression Multivariate ANOVA
a b s t r a c t Experiment is conducted, across different units of a typical desalination plant, to build correlations between inputs/ outputs. Steady-state statistical models of reverse osmosis (RO) are developed using stream characteristic data (flow rate, concentration and pH) over a period of time. Data-driven models are useful for deciding real time operational control strategies of desalination plant. The statistical analysis of correlations obtained indicates that permeate characteristics depend on feed stream flow rate by a second degree polynomial. Significance of regression was evaluated based on multivariate ANOVA analysis, on visual standardized residuals distribution and their means for confidence levels of 95% and 99%, clearly validating these models. Sensitivity of parameters is found from interaction and co-relation studies. These models will help in safe operation and control of RO-desalination plant. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Desalination is used to separate salts from raw water for use in boiler feed, thermal power generation, electronic industries, chemical industries, textile industries, and leather industry and also for production of portable water. The process is carried out through distillation, multiple ⁎ Corresponding author. Fax: +91 44 24911589. E-mail address: [email protected] (R.C. Panda).
http://dx.doi.org/10.1016/j.desal.2014.07.038 0011-9164/© 2014 Elsevier B.V. All rights reserved.
effect vapor compression, evaporation, or membrane processes such as electro-dialysis reversal, nano-filtration, and reverse osmosis (RO). Being advantageous over other separation techniques, reverse osmosis is basically a pressure driven process where no energy phase change (or) potentially expensive solvents (or) adsorbents are needed. It is simple to design and operate compared to other traditional separation processes. Separation of inorganic, organic and microbiological impurities is simultaneously done by RO. A schematic of the process is shown in Fig. 1. There are mainly two inputs, namely, pump pressure (ΔP) and
S. Subramani, R.C. Panda / Desalination 351 (2014) 120–127
121
ΔP Pump
Sea water
RO Module-
Fp Cp
RO Module-
r
pH
RO ModuleProduct
Equalisation Tank
RO Module-
Portable water Conc Brine Fig. 1. Schematic of a reverse osmosis desalination process.
recycle ratio (ratio of flow rates of recycled brine to that of raw sea water) entering to mixing/equalization tank. Total dissolved solid (TDS) concentration and flow rate of feed (sea water) act as a disturbance or load variable. The exit stream from equalization tank enters the RO module through a high pressure pump. There are two exit streams (permeate and brine) from RO. The characteristic variables of permeate stream are: concentration (Cp), flow rate (Fp) and pH. Part of the brine is recycled during the process while the rest may be treated to recover salts (value added product). The portable water comes out from radial directions of cylindrical membranes while brine comes out from horizontal direction of the membranes. The process is still under research as the transport and separation of salts through membranes are not well established and characterization of membranes (development of concentration polarization) is difficult. It is indeed necessary to know if there exist interactions between measured output and input variables or not which can be found out by statistical correlation analysis. In case of interactions, decoupler has to be designed to make the decoupled-process free from interactions for designing the controllers easily. Centralized and decentralized configuration using MPC had been designed and their performances were compared by Sobana and Panda [10]. Experiments have been carried out to understand the mechanism of separation at laboratory scale. As it is difficult to conduct experiments at commercial scales, mathematical models of the integrated process may be helpful in design and operation of the plant at various levels. In the literature [2], many models derived from first principle have been reported which are later validated. Many of these models are found to deviate from real time operations. Hence, phenomenological models derived from real time plant data will be helpful in calculating the characteristics of permeate and brine. Sobana and Panda [10] formulated mathematical models for desalination using (spiral) RO systems. They linearized the material balance equations for each section/unit and presented in input–output form for facilitating controller design. A review [1] citing several works related to phenomenological, steady-state and transient mass balances of solute incorporating concentration polarization in RO has been presented. But the behavior of a real plant needs to be known by analyzing its input–output data through statistical tools. Models relating COD to phenolic compounds in an olive mill waste water plant were reported by Luisa et al. [3]. They analyzed pH, COD, BOD etc. of waste water from different olive plants/mills through statistical tools and presented a model for the prediction of unknown waste water characteristics and for future planning and operation of those olive mills. Miyamoto et al. [4] found new fouling index β as the regression coefficient of linear regression model. They claimed this to be more reliable and feasible than the silt density index (SDI). They analyzed statistically the relationships between the amount of filtered water, elapsed time and environmental factors and to gain new insight into the performance and deficiencies of using SDI from a statistical point of view. Khajet et al. [5] reported on the optimization of RO desalination plant (driven be solar power) by response
surface methodology (central composite experimental design of orthogonal type) and obtained salt rejection coefficient and specific permeate flux. They used ANOVA to test the significance of the RSM polynomial model. Performances of RO desalination process in terms of recovery, permeate flow/RO skid, normalized permeate flow during the erratic period, normalized differential pressure across RO system and salt rejection have been evaluated by Mohsen and Salem [6] for 150 days to bring out the state of its operation and maintenance. Modeling and optimization of air gap membrane distillation process using response surface methodology was reported by Khajet and Cojocaru [7]. They used regression models to predict the performance index that takes into account energy consumption as the function of different variables. Dewei et al. [9] presented modeliing and control techniques for desalination and implementation of FPGA systems. Modeling and optimization of air gap membrane distillation process using response surface methodology was reported by Khajet and Cojocaru [7]. They used regression models to predict the performance index that takes into account energy consumption as the function of different variables. Jose et al. [8] represented all the cost data of RO desalination plant in bar diagram and box and whisker diagram. The outlier's values of RO desalination study were determined, and the Kolmogorov–Smirov & Shalor– Wilk tests were carried out based on the Hubera's M Tukey's biweight. Hampelson and Andrews values were estimated. All the above models are specific to solve a certain purpose. Desalination processes work under (experiences) a rugged condition and it can handle a wide range of feed condition. Enormous data are available (in the industry) which can be used to understand the behavior of the process. There is a lack of data-driven models that use regression and other statistical methods on RO desalination plant data. In the present study, ANOVA model is used to understand the effect of the interaction of the parameters on the output of the process. Thus data characterizing outputs from different streams (entering and exiting mixing tank, brine tank and RO module) will be helpful in analyzing and modeling the behavior of desalination plant. These models will also help in planning, operation and monitoring of water treatment plant. The objective of the present study is to establish statistical correlational models between input (feed to equalization tank) and output (permeate stream) variables. The rest of the paper is organized as follows: Section 2 discusses experimental set-up and procedures. Data are summarized and correlations between the most commonly measured parameters are sought and are presented in Section 3. Results of regressions and different tests (significance, residuals etc.) are provided in this section. Regressional analysis using multivariate ANOVA is presented in Section 4. Conclusion is drawn at the end. 2. Experimental set-up A sea water based desalination plant has been installed in Narippaiyur village of Ramanathapuram district, Tamil Nadu, India, with a capacity
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of 3.80 MLD of drinking water. This is the first desalination plant for the production of potable water in South Asia. The arrangement of the experimental set up is shown in Fig. 2. 2.1. Filters and pumps A well with an inner diameter of 1000 mm is dug in the sea and is covered with a basket with small holes and mesh (100 × 15 mm GRP material) so that fish and algae are not allowed to enter inside the well which is connected to an equalization tank of 70 m2 × 7 m by two pipes (GRP material) with an inner diameter of 450 mm and a length of 450 meters. The pipe is connected to the equalization tank with a certain slope so that the height of water in the equalization tank is equal to the height of sea. The tank is equipped with three vertical turbines and two horizontal centrifugal pumps (capacity 450–500 m3/hr) that deliver water to filter section with a pressure of 3–3.5 kg/cm2. Three dual media filters of 3 m diam and 7 m height, with a capacity of 120–170 m3/hr filled with sand and gravels, are employed in a series as primary filters to remove coarse particles and suspended solids from water. The dirt and turbidity are also reduced to some extent at first and 2nd stages of filters. Water is then allowed to enter in the 3rd stage of filter; in polished dual media filters (PDMF), were suspended solids, dirt and turbidity are reduced to yield colorless and odorless water. After PDMF, water enters to cartridge filter to eliminate particles more than 5 μm after which it is pumped (discharge pressure N 50 kg/cm2) to RO sections.
the membrane. The feed water is allowed to enter in the innermost radius of the RO. The permeate comes out through the outermost layer of the RO. The ions are attracted by the polyamide material of the membrane. The TDS reduces from 40,000 ppm to 500 ppm for a running (operation) time of 12 hours through 168 ROs (spiral bound) of 1 m length each. An RO consists of 30 membrane leafs. Each leaf is made up of two membrane sheets glued together back to back with a permeate spacer in between them. The consistent glue line of about 1.5 inches wide seals the inner (permeate) side of the leaf against the outer (feed/concentrated) side. The leaves are rolled up with a sheet of feed spacer between each of them, which provide the channel for feed and concentration flow. The permeate (p) and brine (b) from all ROs are collected and passed to the opposite direction of the feed water entering the section of RO. The brine from the RO section is collected that amounts to approx 50% of feed. A schematic process flow sheet (Fig. 3) describes flow rates of different streams in the entire plant. About 70% of brine is recirculated to feed-mixing tank. The rest of the brine can be used for recovery of salts. Ten percent of feed goes to precipitate or forms scale that gets adhered to the membrane. An amount of about 35%–40% of feed goes to permeate tank and can be used for potable purpose. To boost the feed water pressure from 51 kg/cm2 to 60 kg/cm2, a centrifugal type of energy recovery devices (electromechanically operated butterfly dump valve and a flow control valve) are used by taking (energy conservation/recovered) the energy from the brine stream. The percentage of salt rejection is found to be 99.5%. Designed recovery is calculated as 50%.
2.2. RO section 3. Results and discussion ROs operate over a pH range of 2 to11 with excellent performance in terms of flux, salt, organic rejection, and microbiological resistance and with free chlorine tolerance of less than 0.1 ppm. Water permeability and solute permeability characteristics depend on the performance of
Salt concentration and flow rate are important parameters to characterize the streams. The present modeliing work has been done in two stages; first, collecting data through experiments and development
Fig. 2. Process flow scheme of desalination plant at TWAD, Ramanathapuram, Tamil Nadu.
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Fig. 3. Schematic liquid flow and pressure flow schemes in desalination process.
of model using part of the collected data, and finally, validating the model with the rest of the data. It is interesting to note that TDS, ΔP and flow-rate of streams at the entrance and exit of each unit have been collected and are shown in Table 1. The first step is to compile all the data after which some useful analytical relations are determined based on average values of TDS, ΔP and flow-rates of each streams as shown in Table 1. It can be observed that the average TDS across the entrance of the mixing tank is 45,287 while that across the exit from this unit is 45,283 ppm. The average TDS in the feed stream (CRO) of RO is 45,283 ppm. ΔP average is 60.2 kg/cm2 while that of flow rate is 188.9 m3/hr. Average TDS of permeate (Cp) is 845.76 ppm and flow-rate (Fp) is 83.1 m3/hr. The mean TDS at the exit of brine stream is 66149.7 ppm and its average flow rate becomes 95.52 m3/hr. As brine is not being recycled in this study, recycle ratio (r) at the inlet of mixing tank is = flow rate of feed from sea water. Standardized values of RO feed stream variables with correlation coefficients and effect terms are given in Table 2a. Mean, standard deviation, coefficient of variance and R-squared values of RO exit stream after standardization for Cp, Fp and pH with test terms as ΔP and r are provided in Tables 3a, 3b and 3c respectively. 3.1. Model development The experimental values across RO unit are collected and are considered for model development. They are fitted with a second-degree polynomial function. The correlation parameter (R2) is 0.0551 for Cp.
In order to define the boundaries of the model, tables of absolute frequencies of CRO values are constructed using a class interval of 499.1035 kg m−3. These intervals are shown in Fig. 4 as bar charts. The distribution in Fig. 4 is of Gaussian type with CRO varying from 44,919.34 to 47,215.19 kg m−3 occurring most frequently. This information is crucial for the conception, design, scale up and optimization of the RO unit. We find from this chart that values are concentrated between 44,500 and 47,500 kg m−3. The distribution between 0 and 800 kg m−3 appears to be approximately uniform, considering expected statistical fluctuation of 1/(N)0.5, where N is the number of values in the interval. This would define the normal working limits of RO units. There is a clear, but small, decrease in the number of values in the range 800 to 1600 kg m−3 followed by a very sharp fall: of the total N values only 12 stand in the range 1600 to 35,000 kg m− 3. To explain these higher values, correlations between input and output are sought, and it may be attributed to a statistical fluctuation. There are noticeable peaks in the range of 600–700 kg m−3 however only the lower peak may be considered statistically significant. There are two exits from the RO unit, one delivers permeate and the other gives brine. A matrix is constructed by combining the TDS at entrance and exit absolute frequencies. This clearly illustrates that there is a correlation between ΔP and r inlet and outlet values (Cp, Fp, pH). For the lower values the correlation is clearly linear, whereas for the higher inlet r values, exit TDS increases very slowly. Low frequency of inlet r values together with the absence of values in some intervals for high inlet ΔP led us to base the model only on Cp concentration up to 800 kg m− 3 as presented in Fig. 5. Main feature in these
Table 1 Variable data collected from inlet and exit streams of RO. Date
01.05.2009 02.05.2009 03.05.2009 04.05.2009 05.05.2009 06.05.2009 07.05.2009 08.05.2009 09.05.2009 10.05.2009 11.05.2009 12.05.2009 13.05.2009 14.05.2009 15.05.2009 16.05.2009 17.05.2009 Mean value
RO inlet parameters
RO outlet parameters (permeate)
Feed pressure (kg/cm2)
Feed flowrate (m3/hr)
Feed TDS (ppm)
Feed pH
Permeate pressure (kg/cm2)
Permeate flowrate (m3/hr)
Permeate TDS (ppm)
Permeate pH
59.00 59.50 60.00 59.00 61.00 60.00 61.00 59.00 60.00 60.50 60.00 60.00 60.00 60.50 61.50 61.00 61.50 60.2059
186.00 190.00 193.00 187.00 185.00 190.00 186.00 192.00 190.00 186.00 187.00 195.00 186.00 192.00 198.00 186.00 187.50 186.912
45,000.00 46,000.00 46,100.00 46,600.00 46,400.00 46,700.00 46,300.00 46,500.00 46,100.00 46,400.00 46,700.00 46,500.00 46,500.00 46,300.00 46,600.00 47,300.00 47,100.00 46,417.65
7.15 7.13 7.25 6.93 7.32 7.28 7.20 7.15 6.83 7.17 7.18 7.15 7.28 7.25 6.98 7.10 7.19 7.15
.51 .47 .49 .53 .48 .50 .52 .49 .50 .51 .54 .48 .49 .51 .50 .50 .48 0.5
88.00 80.00 81.00 80.00 78.00 81.00 80.00 78.00 79.00 78.00 79.00 80.00 82.00 78.00 77v00 80.00 83.00 80.11
842.00 840.00 839.00 857.00 868.00 870.00 842.00 835.00 846.00 845.00 842.00 841.00 842.00 842.00 837.00 842.00 848.00 845.76
6.13 6.09 6.12 6.10 6.18 6.15 6.10 6.11 6.14 6.12 6.11 6.13 6.20 6.24 6.02 6.08 6.22 6.16
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Table 2a Standardized values of RO feed stream variables with correlation coefficients and effects. Serial no.
a1 (ΔP)
a2 (Feed flowrate)
a1 · a1 (Feed pressure)2
a1 · a2 (Feed flow * Feed pres)
a2 · a2 (Feedflowrate)2
1 2 3 4 5 6 7 8 9 10 11 12
−0.95 0.35 1.31 −0.62 −1.28 0.35 −0.95 0.99 0.35 −0.95 −0.62 1.95
−1.31 1.31 −0.85 −0.38 −0.15 −0.61 −0.38 0.07 0.53 1.00 1.46 1.91
1.25 −0.46 −1.11 0.24 0.19 −0.21 0.36 0.07 0.18 −0.95 −0.91 3.72
0.90 0.12 1.71 0.39 1.64 0.12 0.90 0.98 0.12 0.90 0.39 3.77
1.73 1.73 0.72 0.14 0.02 0.38 0.14 0.00 0.29 1.00 2.13 3.68
Fp (m3/hr)
1 2 3 4 5 6 7 8 9 10 11 12
−1.54 −0.55 0.13 −0.13 0.41 1.52 0.97 1.24 0.97 −0.69 −1.12 −1.26
−0.34 −0.34 1.03 −1.74 −1.74 −0.34 −0.34 1.03 1.03 1.03 −0.34 1.03
Mean
Std deviation
CV, %
R-squared
1 = a1 2 = a2 1.2 = a1 * a2 1.1 = a21 2.2 = a22 Y = Fp
−0.01 −0.01 −0.2800 1.0 1.0 4.4218
1 1 0.98 1.43 1.4 0.0001
−∞ −∞ −350 143 140 0
0.0025 0.3844 0.0061 0.1764 0.0196 0.0064
C p=exit; ppm ¼ 6:7488 þ 0:0018 logΔP þ 0:000776 logr−0:0044 2
logΔPr þ 0:0014 logΔP −0:0029 logr
Table 2b Standardized values of RO exit stream variables. Cp (ppm)
Test term
the model. Let r be recycle-ratio at the entrance of RO unit. The regressions obtained in this study are
Standardized values of RO exit streams are provided in Table 2b.
Serial no.
Table 3b Mean, Std deviation, coefficient of variance and R-squared values of RO exit stream after standardization for Fp with test terms as ΔP (=1 or a1) and r (=2 or a2).
pH 0.27 −1.39 −0.13 −0.97 2.36 1.11 −0.97 −0.55 0.69 −0.13 −0.55 0.27
representations is that for this range both second-degree polynomial and linear functions fit well, with R2 being very close to 1. However when inlet r increases above the range considered here there is no corresponding increase in exit Cp and a saturation point is reached. Although linearity is observed for low inlet r or ΔP, the second degree polynomial model is preferable because it fits well for high and low r or ΔP.
F p=exit; m3 =hr ¼ 4:4223−3:5632 10 −4
þ 3:4526 10
þ 0:0025 logr
−4
10
logr
ð1Þ
logΔP þ 0:0056 logr
logΔPr−0:0046 logΔP
2
2
ð2Þ
pHExit ¼ 1:8118 þ 0:0016 logΔP−4:3220 10 −4
2
−4
logΔPr þ 1:7565 10
−4
logr−9:1125 2
logΔP þ 9:5627
2
ð3Þ
The right hand side of the above three equations reveal the significance or importance of particular output variable based on the magnitude of coefficients of respective terms. A significance test was performed for this experiment. For a regression of 2nd degree polynomial type Y = β2X2 + β1X + β0 this test tries to determine whether a linear relationship exists between the response variable Y(Cp)exit and a subset of the regressor variable X [(ΔP)inlet and (r)2inlet] as described in Eq. (7). The appropriate hypotheses are verified For H0 : β1 ¼ β2 ¼ 0
ð4Þ
For H1: β j ≠0 for at least one j
ð5Þ
The value of R2 gives the amount of variability in the data explained or accounted by the model regressions. The regressions presented in Fig. 5 account for a variability of 0.0236 for Fp, 0.0207 for Cp and 0.0114 for pH in the data. However a large value of R2 does not necessarily imply that the model is a good one, because this parameter does not measure the statistical significance of a regression. In order to asses these models adequately an ANOVA table for each type of data and a residual analysis are prepared and studied. The results on comparison of models are summarized in Fig. 5. There is good agreement between the regression for the literature values and for values computed from
In the present case, we reject H0: as the results from table reveal that β1 or β2 is non-zero. There is a possibility of at least one of the regressor variables to contribute significantly to the model giving rise to interactions. The parameters obtained from an ANOVA analysis are given in Tables 4a, 4b and 4c. It can be found (from Table 4a) from the numerical values of autocorrelation term (a1 * a2) that the Cp is affected by ΔP and r mostly as the interactive term (SS column, a1 * a2) is of least value. Similarly, Table 4b reveals (in SS column, least value is in row a2) that Fp will be mostly affected by changes in recycle ratio, r. Table 4c shows that pH may be affected mostly by change in ΔP.
Table 3a Mean, Std deviation, coefficient of variance and R-squared values of RO exit stream after standardization for Cp with test terms as ΔP (=1 or a1) and r (=2 or a2).
Table 3c Mean, Std deviation, coefficient of variance and R-squared values of RO exit stream after standardization for pH with test terms as ΔP (=1 or a1) and r (=2 or a2).
3.2. Test for significance of the regression
Test term
Mean
Std deviation
CV, %
R-squared
Test term
Mean
Std deviation
CV, %
R-squared
1 = a1 2 = a2 1.2 = a1 * a2 1.1 = a21 2.2 = a22 Y = Cp
−0.01 −0.01 −0.2834 1 1 6.7488
1.0 1.0 0.98 1.43 1.40 0.000066
−∞ −∞ −350 143 140 0
0.0036 0.0121 0.0861 0.0121 0.0529 0
1 = a1 2 = a2 1.2 = a1 * a2 1.1 = a21 2.2 = a22 Y = pH
−0.01 −0.01 −0.28 1 1 1.8121
1.0 1.0 0.98 1.43 1.40 0.0
−∞ −∞ −350 143 140 0
0.2025 0.1089 0.1453 0.0961 0.0483 0
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125
Fig. 4. Frequency distribution data of input and output variables: where inputs are: feed pressure and feedflow rate to RO system while outputs are: permeate flow rate, TDS and pH.
Since the P values for Cp and Fp are considerably smaller than 0.05, we reject the null hypothesis and conclude that Cp exit is linearly related to either ΔPinlet or (r)inlet or both. Further tests (residual tests) of model adequacy are required before we can use this model in real time. 3.2.1. Residual analysis Standardized residuals from the multiple regression model are di ¼
ei ðMSE Þ0:5
ð6Þ
3.3. Calculation of correlation coefficient Correlation coefficient is calculated by establishing a relationship between the two variables (input and output) in such a way that with an increase in the value of one variable (X) the value of the other variable (Y) increases or decreases in a fixed proportion. The R2 value can be calculated by n X ðX i −XÞðY i −YÞ i¼1
nσ x σ y
2 2 Y^ ¼ β1 x1 þ β2 x2 þ β1 β2 x1 x2 þ β3 x1 þ β4 x2 þ C
ð7Þ
where C = c1 + c2 + c1c2
where ei is (Cp)exit, measured − (Cp)exit, calculated and MS is mean square error. Standardized residuals are calculated and plotted against (Cp)exit, calculated as shown in Fig. 5. It can be seen that residuals are independently distributed. The mean residual value is 9.73 for a confidence level of 95%. In all the cases, the mean residual values are almost zero, validating the use of a second order polynomial. Both residual distribution analysis and the mean residual values with errors calculated for confidence level of 95%, indicate that the model is more accurate for all values thus equation for model (3) is to be considered.
2 R ¼
and output vector (Y1, Y2 and Y3) by fitting curve through polynomial regression. We fit the data statistically using equations given as:
ð6Þ
Polynomial regression coefficients for multi input output systems can be calculated to establish relation between input vector (X1, X2)
βi ¼
n
X X xi y− xi y X X 2 2 xi n xi −
X
X and
ci ¼
y−β i
X
n
xi
ð8Þ
Using these fittings Eqs. (1)–(3) have been obtained. 4. Conclusion Application of statistical techniques to real/physical problems has been carried out in this article. A reverse osmosis system has been characterized by its input streams as pump pressure and recycle feed ratio, defined as recycle ratio (r) of flow rates between raw water (Fs) to that of raw-water and recycled brine (Fs + b · Fb), and output streams as permeate flow rate, concentration and pH. A good correlation between inputs and output of RO section in a desalination plant is formulated using 2nd degree polynomial. The novelty in the present analysis is to understand the mechanism/interaction behavior between input and output variables of the desalination plant by formulating regression models. The ANOVA analysis provides information that permeate TDS is mostly affected by change in feed recycle ratio, r. The model developed here is useful for planning, monitoring and analysis of the present separation system. The model is obtained after multivariable analysis resulting to P values being smaller than α b 0.052 indicating independently distributed residuals with mean residual values for confidence level of 95% and 99% being negligible. This validation supports the present model for analyzing the sensitivity of parameters towards output.
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Fig. 5. Comparison of experimental data with standardized calculation models: (a) permeate flowrate (Fp) (m3/hr), (b) permeate concentration (ppm), (Cp) and (c) permeate pH.
Table 4a ANOVA results for concentration of permeate, Cp. Serial no.
Source
DF
SS
1 2 3 4 5
a1 a2 a1 * a2 a21 a22 Model Residual Total
1 1 1 1 1 5 6 11
8.0972 8.4221 6.4291 8.2564 7.5427 0.0039 7.3758 0.0039
Table 4b ANOVA results for flow-rate of permeate, Fp.
MSS = SS/DF * * * * *
10−4 10−4 10−4 10−4 10−4
* 10−6
8.0972 8.4221 6.4291 8.2564 7.5427 0.0033 1.2293 3.5292
* * * * *
10−4 10−4 10−4 10−4 10−4
* 10−6 * 10−4
F-ratio = MSS model/ MSS residual
Serial no. Source
DF
SS
MSS = SS/DF
658.68 685.11 522.98 671.63 613.57 2.6611 * 103
1 2 3 4 5
1 1 1 1 1 5 6 11
9.0363 * 10−4 5.3080 * 10−4 9.0389 * 10−4 6.4179 * 10−4 8.3089 * 10−4 0.0038 4.74 * 10−4 0.0043
9.0363 5.3080 9.0389 6.4179 8.3089 0.0031 7.9001 3.8955
a1 a2 a1 * a2 a21 a22 Model Residual Total
* * * * *
10−4 10−4 10−4 10−4 10−4
* 10−5 * 10−4
F-ratio = MSS model/ MSS residual 11.43 6.71 11.44 8.12 10.51 39.82
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Tamil-Nadu for providing experimental data for carrying-out this study.
Table 4c ANOVA results for pH of permeate. Serial no. Source 1 2 3 4 5
DF SS
1 a1 1 a2 1 a1 * a2 2 a1 1 2 1 a2 Model 5 Residual 6 Total 11
1.5261 * 10−4 1.8031 * 10−4 1.73688 * 10−4 1.8215 * 10−4 1.7183 * 10−4 8.6058 * 10−4 1.7297 * 10−4 0.010
127
MSS = SS/DF
F-ratio = MSS model/ MSS residual
1.5261 * 10−4 5.29 1.8031 * 10−4 6.25 1.73688 * 10−4 6.02 −4 1.8215 * 10 6.31 −4 1.7183 * 10 5.96 −4 7.2312 * 10 25.08 2.8828 * 10−5 9.3959 * 10−5
Nomenclature b constant, 0.6 concentration of permeate stream, ppm (mg/lit) Cp TDS in feed stream of RO, ppm CRO brine flow-rate, m3/hr Fb permeate flow-rate, m3/hr Fp sea water flow-rate, m3/hr Fs ΔP pressure of pump, bar pH pH of permeate stream r recycle ratio defined as Fs/(Fs + b · Fb) TDS total dissolved solids regression model coefficients βi
Acknowledgment The authors are willing to express their sincere thanks to Desalination plant, TWAD board, Ramanathapuram district,
References [1] S. Sobana, Rames C. Panda, Identification, modelling and control of continuous reverse osmosis desalination system—a review, Sep. Sci. Technol. 46 (2011) 551–560. [2] S. Sobana, Rames C. Panda, Development of a transient model for the desalination of sea/brackish water through reverse osmosis, Desalination Water Treat. 51 (13–15) (2013) 2755–2767. [3] C.D. Luisa, M.V. Andre, M.N. Jullo, M.D. Susete, Olive mill wastewater characteristics: modelling and statistical analysis, Grasas Aceites 55 (3) (2004) 233–241. [4] Junya Miyamoto, Kozo Nakamura, Tsuyoshi Nakamura, Searching for an efficient fouling index for reverse osmosis desalination, A Statistical Approach, Proceeding of the 2010 International Conference on Industrial Engineering and Operators Management, IEOM Society, Dhaka,Bangladesh, January 9–10, 2010. [5] M. Khajet, M. Essalh, A. Tmenta-Deu, Optimization of solar powered reverse osmosis desalination pilot plant using response surface methodology, Desalination 261 (3) (Oct 31 2010) 284–292. [6] Mousa S. Mohsen, Gammoh Salem, Performance evaluation of reverse osmosis desalination plant. A case study of Wadi Ma'in, Zara and Mujib plant, Desalin. Water Treat. 14 (2010) 265–272. [7] M. Khajet, C. Cojocaru, Air gap membrane distillation desalination modelling and optimization, Desalination 287 (2012) 138–145. [8] Jose Feo, J. Jaime Sadhwani, Luis Alvarez, Cost analysis in RO desalination plants production lines. Mathematical model and simulation, Desalin. Water Treat. 51 (2013) 4800–4805. [9] Li De-wei, Nan Yang, Hai Qui, Yugeng Xi, FPGA based QDMC control for reverseosmosis water desalination system, Desalination 285 (1) (2012) 83–90. [10] S. Sobana, Rames C. Panda, Modeling and control of reverse osmosis desalination process using centralized and decentralized techniques, Desalination 344 (2014) 243–251.
Contents lists available at ScienceDirect
Desalination journal homepage: www.elsevier.com/locate/desal
Statistical regression and modeliing analysis for reverse osmosis desalination process Sobana Subramani, Rames C. Panda ⁎ Department of Chemical Engineering, CSIR-CLRI, Adyar, Chennai 600 020, India
H I G H L I G H T S
G R A P H I C A L
A B S T R A C T
• Correlations indicate that the permeate characteristics depend on feed flowrate. • The model is also validated using plant data. • The model for permeate flow-rate supports the use as a 2nd degree prediction model. • Modeling 2-input, 3-output will help in developing control strategies.
a r t i c l e
i n f o
Article history: Received 22 May 2014 Received in revised form 23 July 2014 Accepted 26 July 2014 Available online xxxx Keywords: Desalination Reverse osmosis Statistical model Regression Multivariate ANOVA
a b s t r a c t Experiment is conducted, across different units of a typical desalination plant, to build correlations between inputs/ outputs. Steady-state statistical models of reverse osmosis (RO) are developed using stream characteristic data (flow rate, concentration and pH) over a period of time. Data-driven models are useful for deciding real time operational control strategies of desalination plant. The statistical analysis of correlations obtained indicates that permeate characteristics depend on feed stream flow rate by a second degree polynomial. Significance of regression was evaluated based on multivariate ANOVA analysis, on visual standardized residuals distribution and their means for confidence levels of 95% and 99%, clearly validating these models. Sensitivity of parameters is found from interaction and co-relation studies. These models will help in safe operation and control of RO-desalination plant. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Desalination is used to separate salts from raw water for use in boiler feed, thermal power generation, electronic industries, chemical industries, textile industries, and leather industry and also for production of portable water. The process is carried out through distillation, multiple ⁎ Corresponding author. Fax: +91 44 24911589. E-mail address: [email protected] (R.C. Panda).
http://dx.doi.org/10.1016/j.desal.2014.07.038 0011-9164/© 2014 Elsevier B.V. All rights reserved.
effect vapor compression, evaporation, or membrane processes such as electro-dialysis reversal, nano-filtration, and reverse osmosis (RO). Being advantageous over other separation techniques, reverse osmosis is basically a pressure driven process where no energy phase change (or) potentially expensive solvents (or) adsorbents are needed. It is simple to design and operate compared to other traditional separation processes. Separation of inorganic, organic and microbiological impurities is simultaneously done by RO. A schematic of the process is shown in Fig. 1. There are mainly two inputs, namely, pump pressure (ΔP) and
S. Subramani, R.C. Panda / Desalination 351 (2014) 120–127
121
ΔP Pump
Sea water
RO Module-
Fp Cp
RO Module-
r
pH
RO ModuleProduct
Equalisation Tank
RO Module-
Portable water Conc Brine Fig. 1. Schematic of a reverse osmosis desalination process.
recycle ratio (ratio of flow rates of recycled brine to that of raw sea water) entering to mixing/equalization tank. Total dissolved solid (TDS) concentration and flow rate of feed (sea water) act as a disturbance or load variable. The exit stream from equalization tank enters the RO module through a high pressure pump. There are two exit streams (permeate and brine) from RO. The characteristic variables of permeate stream are: concentration (Cp), flow rate (Fp) and pH. Part of the brine is recycled during the process while the rest may be treated to recover salts (value added product). The portable water comes out from radial directions of cylindrical membranes while brine comes out from horizontal direction of the membranes. The process is still under research as the transport and separation of salts through membranes are not well established and characterization of membranes (development of concentration polarization) is difficult. It is indeed necessary to know if there exist interactions between measured output and input variables or not which can be found out by statistical correlation analysis. In case of interactions, decoupler has to be designed to make the decoupled-process free from interactions for designing the controllers easily. Centralized and decentralized configuration using MPC had been designed and their performances were compared by Sobana and Panda [10]. Experiments have been carried out to understand the mechanism of separation at laboratory scale. As it is difficult to conduct experiments at commercial scales, mathematical models of the integrated process may be helpful in design and operation of the plant at various levels. In the literature [2], many models derived from first principle have been reported which are later validated. Many of these models are found to deviate from real time operations. Hence, phenomenological models derived from real time plant data will be helpful in calculating the characteristics of permeate and brine. Sobana and Panda [10] formulated mathematical models for desalination using (spiral) RO systems. They linearized the material balance equations for each section/unit and presented in input–output form for facilitating controller design. A review [1] citing several works related to phenomenological, steady-state and transient mass balances of solute incorporating concentration polarization in RO has been presented. But the behavior of a real plant needs to be known by analyzing its input–output data through statistical tools. Models relating COD to phenolic compounds in an olive mill waste water plant were reported by Luisa et al. [3]. They analyzed pH, COD, BOD etc. of waste water from different olive plants/mills through statistical tools and presented a model for the prediction of unknown waste water characteristics and for future planning and operation of those olive mills. Miyamoto et al. [4] found new fouling index β as the regression coefficient of linear regression model. They claimed this to be more reliable and feasible than the silt density index (SDI). They analyzed statistically the relationships between the amount of filtered water, elapsed time and environmental factors and to gain new insight into the performance and deficiencies of using SDI from a statistical point of view. Khajet et al. [5] reported on the optimization of RO desalination plant (driven be solar power) by response
surface methodology (central composite experimental design of orthogonal type) and obtained salt rejection coefficient and specific permeate flux. They used ANOVA to test the significance of the RSM polynomial model. Performances of RO desalination process in terms of recovery, permeate flow/RO skid, normalized permeate flow during the erratic period, normalized differential pressure across RO system and salt rejection have been evaluated by Mohsen and Salem [6] for 150 days to bring out the state of its operation and maintenance. Modeling and optimization of air gap membrane distillation process using response surface methodology was reported by Khajet and Cojocaru [7]. They used regression models to predict the performance index that takes into account energy consumption as the function of different variables. Dewei et al. [9] presented modeliing and control techniques for desalination and implementation of FPGA systems. Modeling and optimization of air gap membrane distillation process using response surface methodology was reported by Khajet and Cojocaru [7]. They used regression models to predict the performance index that takes into account energy consumption as the function of different variables. Jose et al. [8] represented all the cost data of RO desalination plant in bar diagram and box and whisker diagram. The outlier's values of RO desalination study were determined, and the Kolmogorov–Smirov & Shalor– Wilk tests were carried out based on the Hubera's M Tukey's biweight. Hampelson and Andrews values were estimated. All the above models are specific to solve a certain purpose. Desalination processes work under (experiences) a rugged condition and it can handle a wide range of feed condition. Enormous data are available (in the industry) which can be used to understand the behavior of the process. There is a lack of data-driven models that use regression and other statistical methods on RO desalination plant data. In the present study, ANOVA model is used to understand the effect of the interaction of the parameters on the output of the process. Thus data characterizing outputs from different streams (entering and exiting mixing tank, brine tank and RO module) will be helpful in analyzing and modeling the behavior of desalination plant. These models will also help in planning, operation and monitoring of water treatment plant. The objective of the present study is to establish statistical correlational models between input (feed to equalization tank) and output (permeate stream) variables. The rest of the paper is organized as follows: Section 2 discusses experimental set-up and procedures. Data are summarized and correlations between the most commonly measured parameters are sought and are presented in Section 3. Results of regressions and different tests (significance, residuals etc.) are provided in this section. Regressional analysis using multivariate ANOVA is presented in Section 4. Conclusion is drawn at the end. 2. Experimental set-up A sea water based desalination plant has been installed in Narippaiyur village of Ramanathapuram district, Tamil Nadu, India, with a capacity
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of 3.80 MLD of drinking water. This is the first desalination plant for the production of potable water in South Asia. The arrangement of the experimental set up is shown in Fig. 2. 2.1. Filters and pumps A well with an inner diameter of 1000 mm is dug in the sea and is covered with a basket with small holes and mesh (100 × 15 mm GRP material) so that fish and algae are not allowed to enter inside the well which is connected to an equalization tank of 70 m2 × 7 m by two pipes (GRP material) with an inner diameter of 450 mm and a length of 450 meters. The pipe is connected to the equalization tank with a certain slope so that the height of water in the equalization tank is equal to the height of sea. The tank is equipped with three vertical turbines and two horizontal centrifugal pumps (capacity 450–500 m3/hr) that deliver water to filter section with a pressure of 3–3.5 kg/cm2. Three dual media filters of 3 m diam and 7 m height, with a capacity of 120–170 m3/hr filled with sand and gravels, are employed in a series as primary filters to remove coarse particles and suspended solids from water. The dirt and turbidity are also reduced to some extent at first and 2nd stages of filters. Water is then allowed to enter in the 3rd stage of filter; in polished dual media filters (PDMF), were suspended solids, dirt and turbidity are reduced to yield colorless and odorless water. After PDMF, water enters to cartridge filter to eliminate particles more than 5 μm after which it is pumped (discharge pressure N 50 kg/cm2) to RO sections.
the membrane. The feed water is allowed to enter in the innermost radius of the RO. The permeate comes out through the outermost layer of the RO. The ions are attracted by the polyamide material of the membrane. The TDS reduces from 40,000 ppm to 500 ppm for a running (operation) time of 12 hours through 168 ROs (spiral bound) of 1 m length each. An RO consists of 30 membrane leafs. Each leaf is made up of two membrane sheets glued together back to back with a permeate spacer in between them. The consistent glue line of about 1.5 inches wide seals the inner (permeate) side of the leaf against the outer (feed/concentrated) side. The leaves are rolled up with a sheet of feed spacer between each of them, which provide the channel for feed and concentration flow. The permeate (p) and brine (b) from all ROs are collected and passed to the opposite direction of the feed water entering the section of RO. The brine from the RO section is collected that amounts to approx 50% of feed. A schematic process flow sheet (Fig. 3) describes flow rates of different streams in the entire plant. About 70% of brine is recirculated to feed-mixing tank. The rest of the brine can be used for recovery of salts. Ten percent of feed goes to precipitate or forms scale that gets adhered to the membrane. An amount of about 35%–40% of feed goes to permeate tank and can be used for potable purpose. To boost the feed water pressure from 51 kg/cm2 to 60 kg/cm2, a centrifugal type of energy recovery devices (electromechanically operated butterfly dump valve and a flow control valve) are used by taking (energy conservation/recovered) the energy from the brine stream. The percentage of salt rejection is found to be 99.5%. Designed recovery is calculated as 50%.
2.2. RO section 3. Results and discussion ROs operate over a pH range of 2 to11 with excellent performance in terms of flux, salt, organic rejection, and microbiological resistance and with free chlorine tolerance of less than 0.1 ppm. Water permeability and solute permeability characteristics depend on the performance of
Salt concentration and flow rate are important parameters to characterize the streams. The present modeliing work has been done in two stages; first, collecting data through experiments and development
Fig. 2. Process flow scheme of desalination plant at TWAD, Ramanathapuram, Tamil Nadu.
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123
Fig. 3. Schematic liquid flow and pressure flow schemes in desalination process.
of model using part of the collected data, and finally, validating the model with the rest of the data. It is interesting to note that TDS, ΔP and flow-rate of streams at the entrance and exit of each unit have been collected and are shown in Table 1. The first step is to compile all the data after which some useful analytical relations are determined based on average values of TDS, ΔP and flow-rates of each streams as shown in Table 1. It can be observed that the average TDS across the entrance of the mixing tank is 45,287 while that across the exit from this unit is 45,283 ppm. The average TDS in the feed stream (CRO) of RO is 45,283 ppm. ΔP average is 60.2 kg/cm2 while that of flow rate is 188.9 m3/hr. Average TDS of permeate (Cp) is 845.76 ppm and flow-rate (Fp) is 83.1 m3/hr. The mean TDS at the exit of brine stream is 66149.7 ppm and its average flow rate becomes 95.52 m3/hr. As brine is not being recycled in this study, recycle ratio (r) at the inlet of mixing tank is = flow rate of feed from sea water. Standardized values of RO feed stream variables with correlation coefficients and effect terms are given in Table 2a. Mean, standard deviation, coefficient of variance and R-squared values of RO exit stream after standardization for Cp, Fp and pH with test terms as ΔP and r are provided in Tables 3a, 3b and 3c respectively. 3.1. Model development The experimental values across RO unit are collected and are considered for model development. They are fitted with a second-degree polynomial function. The correlation parameter (R2) is 0.0551 for Cp.
In order to define the boundaries of the model, tables of absolute frequencies of CRO values are constructed using a class interval of 499.1035 kg m−3. These intervals are shown in Fig. 4 as bar charts. The distribution in Fig. 4 is of Gaussian type with CRO varying from 44,919.34 to 47,215.19 kg m−3 occurring most frequently. This information is crucial for the conception, design, scale up and optimization of the RO unit. We find from this chart that values are concentrated between 44,500 and 47,500 kg m−3. The distribution between 0 and 800 kg m−3 appears to be approximately uniform, considering expected statistical fluctuation of 1/(N)0.5, where N is the number of values in the interval. This would define the normal working limits of RO units. There is a clear, but small, decrease in the number of values in the range 800 to 1600 kg m−3 followed by a very sharp fall: of the total N values only 12 stand in the range 1600 to 35,000 kg m− 3. To explain these higher values, correlations between input and output are sought, and it may be attributed to a statistical fluctuation. There are noticeable peaks in the range of 600–700 kg m−3 however only the lower peak may be considered statistically significant. There are two exits from the RO unit, one delivers permeate and the other gives brine. A matrix is constructed by combining the TDS at entrance and exit absolute frequencies. This clearly illustrates that there is a correlation between ΔP and r inlet and outlet values (Cp, Fp, pH). For the lower values the correlation is clearly linear, whereas for the higher inlet r values, exit TDS increases very slowly. Low frequency of inlet r values together with the absence of values in some intervals for high inlet ΔP led us to base the model only on Cp concentration up to 800 kg m− 3 as presented in Fig. 5. Main feature in these
Table 1 Variable data collected from inlet and exit streams of RO. Date
01.05.2009 02.05.2009 03.05.2009 04.05.2009 05.05.2009 06.05.2009 07.05.2009 08.05.2009 09.05.2009 10.05.2009 11.05.2009 12.05.2009 13.05.2009 14.05.2009 15.05.2009 16.05.2009 17.05.2009 Mean value
RO inlet parameters
RO outlet parameters (permeate)
Feed pressure (kg/cm2)
Feed flowrate (m3/hr)
Feed TDS (ppm)
Feed pH
Permeate pressure (kg/cm2)
Permeate flowrate (m3/hr)
Permeate TDS (ppm)
Permeate pH
59.00 59.50 60.00 59.00 61.00 60.00 61.00 59.00 60.00 60.50 60.00 60.00 60.00 60.50 61.50 61.00 61.50 60.2059
186.00 190.00 193.00 187.00 185.00 190.00 186.00 192.00 190.00 186.00 187.00 195.00 186.00 192.00 198.00 186.00 187.50 186.912
45,000.00 46,000.00 46,100.00 46,600.00 46,400.00 46,700.00 46,300.00 46,500.00 46,100.00 46,400.00 46,700.00 46,500.00 46,500.00 46,300.00 46,600.00 47,300.00 47,100.00 46,417.65
7.15 7.13 7.25 6.93 7.32 7.28 7.20 7.15 6.83 7.17 7.18 7.15 7.28 7.25 6.98 7.10 7.19 7.15
.51 .47 .49 .53 .48 .50 .52 .49 .50 .51 .54 .48 .49 .51 .50 .50 .48 0.5
88.00 80.00 81.00 80.00 78.00 81.00 80.00 78.00 79.00 78.00 79.00 80.00 82.00 78.00 77v00 80.00 83.00 80.11
842.00 840.00 839.00 857.00 868.00 870.00 842.00 835.00 846.00 845.00 842.00 841.00 842.00 842.00 837.00 842.00 848.00 845.76
6.13 6.09 6.12 6.10 6.18 6.15 6.10 6.11 6.14 6.12 6.11 6.13 6.20 6.24 6.02 6.08 6.22 6.16
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Table 2a Standardized values of RO feed stream variables with correlation coefficients and effects. Serial no.
a1 (ΔP)
a2 (Feed flowrate)
a1 · a1 (Feed pressure)2
a1 · a2 (Feed flow * Feed pres)
a2 · a2 (Feedflowrate)2
1 2 3 4 5 6 7 8 9 10 11 12
−0.95 0.35 1.31 −0.62 −1.28 0.35 −0.95 0.99 0.35 −0.95 −0.62 1.95
−1.31 1.31 −0.85 −0.38 −0.15 −0.61 −0.38 0.07 0.53 1.00 1.46 1.91
1.25 −0.46 −1.11 0.24 0.19 −0.21 0.36 0.07 0.18 −0.95 −0.91 3.72
0.90 0.12 1.71 0.39 1.64 0.12 0.90 0.98 0.12 0.90 0.39 3.77
1.73 1.73 0.72 0.14 0.02 0.38 0.14 0.00 0.29 1.00 2.13 3.68
Fp (m3/hr)
1 2 3 4 5 6 7 8 9 10 11 12
−1.54 −0.55 0.13 −0.13 0.41 1.52 0.97 1.24 0.97 −0.69 −1.12 −1.26
−0.34 −0.34 1.03 −1.74 −1.74 −0.34 −0.34 1.03 1.03 1.03 −0.34 1.03
Mean
Std deviation
CV, %
R-squared
1 = a1 2 = a2 1.2 = a1 * a2 1.1 = a21 2.2 = a22 Y = Fp
−0.01 −0.01 −0.2800 1.0 1.0 4.4218
1 1 0.98 1.43 1.4 0.0001
−∞ −∞ −350 143 140 0
0.0025 0.3844 0.0061 0.1764 0.0196 0.0064
C p=exit; ppm ¼ 6:7488 þ 0:0018 logΔP þ 0:000776 logr−0:0044 2
logΔPr þ 0:0014 logΔP −0:0029 logr
Table 2b Standardized values of RO exit stream variables. Cp (ppm)
Test term
the model. Let r be recycle-ratio at the entrance of RO unit. The regressions obtained in this study are
Standardized values of RO exit streams are provided in Table 2b.
Serial no.
Table 3b Mean, Std deviation, coefficient of variance and R-squared values of RO exit stream after standardization for Fp with test terms as ΔP (=1 or a1) and r (=2 or a2).
pH 0.27 −1.39 −0.13 −0.97 2.36 1.11 −0.97 −0.55 0.69 −0.13 −0.55 0.27
representations is that for this range both second-degree polynomial and linear functions fit well, with R2 being very close to 1. However when inlet r increases above the range considered here there is no corresponding increase in exit Cp and a saturation point is reached. Although linearity is observed for low inlet r or ΔP, the second degree polynomial model is preferable because it fits well for high and low r or ΔP.
F p=exit; m3 =hr ¼ 4:4223−3:5632 10 −4
þ 3:4526 10
þ 0:0025 logr
−4
10
logr
ð1Þ
logΔP þ 0:0056 logr
logΔPr−0:0046 logΔP
2
2
ð2Þ
pHExit ¼ 1:8118 þ 0:0016 logΔP−4:3220 10 −4
2
−4
logΔPr þ 1:7565 10
−4
logr−9:1125 2
logΔP þ 9:5627
2
ð3Þ
The right hand side of the above three equations reveal the significance or importance of particular output variable based on the magnitude of coefficients of respective terms. A significance test was performed for this experiment. For a regression of 2nd degree polynomial type Y = β2X2 + β1X + β0 this test tries to determine whether a linear relationship exists between the response variable Y(Cp)exit and a subset of the regressor variable X [(ΔP)inlet and (r)2inlet] as described in Eq. (7). The appropriate hypotheses are verified For H0 : β1 ¼ β2 ¼ 0
ð4Þ
For H1: β j ≠0 for at least one j
ð5Þ
The value of R2 gives the amount of variability in the data explained or accounted by the model regressions. The regressions presented in Fig. 5 account for a variability of 0.0236 for Fp, 0.0207 for Cp and 0.0114 for pH in the data. However a large value of R2 does not necessarily imply that the model is a good one, because this parameter does not measure the statistical significance of a regression. In order to asses these models adequately an ANOVA table for each type of data and a residual analysis are prepared and studied. The results on comparison of models are summarized in Fig. 5. There is good agreement between the regression for the literature values and for values computed from
In the present case, we reject H0: as the results from table reveal that β1 or β2 is non-zero. There is a possibility of at least one of the regressor variables to contribute significantly to the model giving rise to interactions. The parameters obtained from an ANOVA analysis are given in Tables 4a, 4b and 4c. It can be found (from Table 4a) from the numerical values of autocorrelation term (a1 * a2) that the Cp is affected by ΔP and r mostly as the interactive term (SS column, a1 * a2) is of least value. Similarly, Table 4b reveals (in SS column, least value is in row a2) that Fp will be mostly affected by changes in recycle ratio, r. Table 4c shows that pH may be affected mostly by change in ΔP.
Table 3a Mean, Std deviation, coefficient of variance and R-squared values of RO exit stream after standardization for Cp with test terms as ΔP (=1 or a1) and r (=2 or a2).
Table 3c Mean, Std deviation, coefficient of variance and R-squared values of RO exit stream after standardization for pH with test terms as ΔP (=1 or a1) and r (=2 or a2).
3.2. Test for significance of the regression
Test term
Mean
Std deviation
CV, %
R-squared
Test term
Mean
Std deviation
CV, %
R-squared
1 = a1 2 = a2 1.2 = a1 * a2 1.1 = a21 2.2 = a22 Y = Cp
−0.01 −0.01 −0.2834 1 1 6.7488
1.0 1.0 0.98 1.43 1.40 0.000066
−∞ −∞ −350 143 140 0
0.0036 0.0121 0.0861 0.0121 0.0529 0
1 = a1 2 = a2 1.2 = a1 * a2 1.1 = a21 2.2 = a22 Y = pH
−0.01 −0.01 −0.28 1 1 1.8121
1.0 1.0 0.98 1.43 1.40 0.0
−∞ −∞ −350 143 140 0
0.2025 0.1089 0.1453 0.0961 0.0483 0
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125
Fig. 4. Frequency distribution data of input and output variables: where inputs are: feed pressure and feedflow rate to RO system while outputs are: permeate flow rate, TDS and pH.
Since the P values for Cp and Fp are considerably smaller than 0.05, we reject the null hypothesis and conclude that Cp exit is linearly related to either ΔPinlet or (r)inlet or both. Further tests (residual tests) of model adequacy are required before we can use this model in real time. 3.2.1. Residual analysis Standardized residuals from the multiple regression model are di ¼
ei ðMSE Þ0:5
ð6Þ
3.3. Calculation of correlation coefficient Correlation coefficient is calculated by establishing a relationship between the two variables (input and output) in such a way that with an increase in the value of one variable (X) the value of the other variable (Y) increases or decreases in a fixed proportion. The R2 value can be calculated by n X ðX i −XÞðY i −YÞ i¼1
nσ x σ y
2 2 Y^ ¼ β1 x1 þ β2 x2 þ β1 β2 x1 x2 þ β3 x1 þ β4 x2 þ C
ð7Þ
where C = c1 + c2 + c1c2
where ei is (Cp)exit, measured − (Cp)exit, calculated and MS is mean square error. Standardized residuals are calculated and plotted against (Cp)exit, calculated as shown in Fig. 5. It can be seen that residuals are independently distributed. The mean residual value is 9.73 for a confidence level of 95%. In all the cases, the mean residual values are almost zero, validating the use of a second order polynomial. Both residual distribution analysis and the mean residual values with errors calculated for confidence level of 95%, indicate that the model is more accurate for all values thus equation for model (3) is to be considered.
2 R ¼
and output vector (Y1, Y2 and Y3) by fitting curve through polynomial regression. We fit the data statistically using equations given as:
ð6Þ
Polynomial regression coefficients for multi input output systems can be calculated to establish relation between input vector (X1, X2)
βi ¼
n
X X xi y− xi y X X 2 2 xi n xi −
X
X and
ci ¼
y−β i
X
n
xi
ð8Þ
Using these fittings Eqs. (1)–(3) have been obtained. 4. Conclusion Application of statistical techniques to real/physical problems has been carried out in this article. A reverse osmosis system has been characterized by its input streams as pump pressure and recycle feed ratio, defined as recycle ratio (r) of flow rates between raw water (Fs) to that of raw-water and recycled brine (Fs + b · Fb), and output streams as permeate flow rate, concentration and pH. A good correlation between inputs and output of RO section in a desalination plant is formulated using 2nd degree polynomial. The novelty in the present analysis is to understand the mechanism/interaction behavior between input and output variables of the desalination plant by formulating regression models. The ANOVA analysis provides information that permeate TDS is mostly affected by change in feed recycle ratio, r. The model developed here is useful for planning, monitoring and analysis of the present separation system. The model is obtained after multivariable analysis resulting to P values being smaller than α b 0.052 indicating independently distributed residuals with mean residual values for confidence level of 95% and 99% being negligible. This validation supports the present model for analyzing the sensitivity of parameters towards output.
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Fig. 5. Comparison of experimental data with standardized calculation models: (a) permeate flowrate (Fp) (m3/hr), (b) permeate concentration (ppm), (Cp) and (c) permeate pH.
Table 4a ANOVA results for concentration of permeate, Cp. Serial no.
Source
DF
SS
1 2 3 4 5
a1 a2 a1 * a2 a21 a22 Model Residual Total
1 1 1 1 1 5 6 11
8.0972 8.4221 6.4291 8.2564 7.5427 0.0039 7.3758 0.0039
Table 4b ANOVA results for flow-rate of permeate, Fp.
MSS = SS/DF * * * * *
10−4 10−4 10−4 10−4 10−4
* 10−6
8.0972 8.4221 6.4291 8.2564 7.5427 0.0033 1.2293 3.5292
* * * * *
10−4 10−4 10−4 10−4 10−4
* 10−6 * 10−4
F-ratio = MSS model/ MSS residual
Serial no. Source
DF
SS
MSS = SS/DF
658.68 685.11 522.98 671.63 613.57 2.6611 * 103
1 2 3 4 5
1 1 1 1 1 5 6 11
9.0363 * 10−4 5.3080 * 10−4 9.0389 * 10−4 6.4179 * 10−4 8.3089 * 10−4 0.0038 4.74 * 10−4 0.0043
9.0363 5.3080 9.0389 6.4179 8.3089 0.0031 7.9001 3.8955
a1 a2 a1 * a2 a21 a22 Model Residual Total
* * * * *
10−4 10−4 10−4 10−4 10−4
* 10−5 * 10−4
F-ratio = MSS model/ MSS residual 11.43 6.71 11.44 8.12 10.51 39.82
S. Subramani, R.C. Panda / Desalination 351 (2014) 120–127
Tamil-Nadu for providing experimental data for carrying-out this study.
Table 4c ANOVA results for pH of permeate. Serial no. Source 1 2 3 4 5
DF SS
1 a1 1 a2 1 a1 * a2 2 a1 1 2 1 a2 Model 5 Residual 6 Total 11
1.5261 * 10−4 1.8031 * 10−4 1.73688 * 10−4 1.8215 * 10−4 1.7183 * 10−4 8.6058 * 10−4 1.7297 * 10−4 0.010
127
MSS = SS/DF
F-ratio = MSS model/ MSS residual
1.5261 * 10−4 5.29 1.8031 * 10−4 6.25 1.73688 * 10−4 6.02 −4 1.8215 * 10 6.31 −4 1.7183 * 10 5.96 −4 7.2312 * 10 25.08 2.8828 * 10−5 9.3959 * 10−5
Nomenclature b constant, 0.6 concentration of permeate stream, ppm (mg/lit) Cp TDS in feed stream of RO, ppm CRO brine flow-rate, m3/hr Fb permeate flow-rate, m3/hr Fp sea water flow-rate, m3/hr Fs ΔP pressure of pump, bar pH pH of permeate stream r recycle ratio defined as Fs/(Fs + b · Fb) TDS total dissolved solids regression model coefficients βi
Acknowledgment The authors are willing to express their sincere thanks to Desalination plant, TWAD board, Ramanathapuram district,
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