Journal of Process Control 22 (2012) 218–227
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Periodic control of a reverse osmosis desalination process Ali Emad ∗ , A. Ajbar, I. Almutaz Department of Chemical Engineering, King Saud University, P.O. Box 800, Riyadh 11421 Saudi Arabia. Tel.: +96 614678650; fax: +96 614678770
a r t i c l e
i n f o
Article history: Received 2 April 2011 Received in revised form 2 August 2011 Accepted 7 September 2011 Available online 1 October 2011 Keywords: Reverse osmosis Cyclic operation Nonlinear model predictive control Water desalination
a b s t r a c t This paper addresses the forced periodic operation of a tubular reverse osmosis process for improved performance. The investigation is carried out through simulation of a previously validated model for the RO process. The feed pressure and feed flow rate are transformed into periodic behavior in the form of sinusoidal functions. A nonlinear model predictive control algorithm is utilized to regulate the amplitude and period of the sinusoidal functions that formulate the input signal. The control system managed to generate the cyclic inputs necessary to enhance the closed-loop performance in terms of higher permeate production and lower salt concentration. The proposed control algorithm can attain its objective with and without defining a set point for the controlled outputs. In the latter case, the process is driven to the best achievable performance. Similar successful results in terms of improved production and quality of water were also obtained in the presence of modeling errors and external disturbances. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Reverse osmosis (RO) process is an important filtration process that is used extensively for the desalination of sea and brackish water all over the world. Concentration polarization [1] and membrane fouling are the major problems faced in the any membrane-based separation operation. The ultimate effect of these factors is to reduce the permeate flux and consequently loss of productivity. Fouling effects are characterized by an irreversible decrease in flux. In real practice, fouling is handled by shutting the process down and cleaning the membrane by chemical or physical methods. Concentration polarization on the other hand results in reversible decline in water flux through the membrane. Usually, concentration polarization can be controlled via two main methods [2]: (i) changes in the characteristics of the membrane [2], (ii) modification of flow rates and flow regime. The latter requires introducing instabilities in the flow of the RO process [3,4]. Examples of such approach includes backwashing and periodic operation of the module, through forcing some of the process variables [5]. Periodic forcing improves the mixing of the solution on the feed side and, hence, reduces the build-up of solute ions near the membrane wall. The analysis of periodically forced reverse osmosis has received considerable attention in the literature [2,5–10]. One of the first works in this area was carried out by Kennedy et al. [11] who varied harmonically the flow rate of sucrose solution in an RO
∗ Corresponding author. E-mail address:
[email protected] (A. Emad). 0959-1524/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2011.09.001
unit. For a frequency of 1 Hz, the authors reported an enhancement in the permeation flow rate by 70% over the corresponding steady state operation. Ilias and Govind [12] followed the experimental work in [10] and developed a mathematical model to evaluate the unsteady state operation of tubular membranes through changing the inlet feed flow harmonically. They concluded that flow pulsation offered an alternative, cost-effective method to improve the trans-membrane flux. The enhancement effect of pulsed flow was also reported by Finnigan and Howell [2]. In addition, these authors found that such improvement is increasing with decreasing the flow rate. Al-Bastaki and Abbas [5] modified the model that was proposed by Kennedy et al. [11] so that it can be used for asymmetric square waves. The authors, further, studied the effect of using an asymmetric pressure square wave on the performance of an RO process. They found that cyclic operation lead to an increase in the permeate flux of about 6.5% over that obtained from steady-state operation. The increase in performance was attributed to the reduction in concentration polarization in the process. The same authors [6] used a simple model to predict the performance of periodically forced RO system. Abbas and Al-Bastaki [9] also, studied the periodic performance of a seawater desalination unit based on a small-scale commercial spiral-wound membrane. For the case of unsteady-state operation, the operating pressure was varied according to a symmetric square wave function around an average pressure of 50 bars. The production rate increased as the period of the wave decreased. Such an improvement was obtained at the expense of a marginal increase in the total energy consumed. Abufayad [8], on the other hand, reported the experience gained from the periodic operation of the Tajoura sea-water reverse
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Nomenclature A Am b Bj cb,i cF cp,i cpss cp cw,i Ds d F fF Jv,i ji Lp M P pb,i pF pp,i q qss q r R R j Re Rr Sc tk T Ts ui u x x y Greek k b p u U
Matrix of linear constraints in NLMPC Pulse amplitude Vector of constraints values for NLMPC Salt permeability [m/s] Salt concentration in brine in i-th increment [mol/m3 ] Salt concentration in the feed [mol/m3 ] Salt concentration in permeate in i-th increment [mol/m3 ] Steady state value of cp [mol/m3 ] Time averaged value for cp as ratio to its steady state value Salt concentration at the wall in i-th increment [mol/m3 ] Diffusion coefficient of salt in solution [m2 /s] Internal diameter of tube, also disturbance estimate [m] Feed flow rate [m3 /s] Fanning factor Volume flux density through membrane in i-th increment [m/s] Chilton–Colburn factor Hydraulic permeability [m/s atm] Control horizon Prediction horizon Brine pressure in i-th increment [N/m2 ] Feed pressure [N/m2 ] Permeate pressure in increment i [N/m2 ] Permeate production rate [m3 /s] Steady state value of q [m3 /s] Time averaged value of q as ratio to steady state value Set point Universal gas constant, also vector of set point [m3 atm/K mol] Intrinsic salt rejection coefficient Reynolds number Recovery ratio Schmidt number Sampling instant Operating temperature [K] Sampling time [S] Brine average velocity in i-th increment [m/s] Input vector State vector Differential tube length [M] Vector of outputs
Number of ions produced on complete dissociation of one molecule of electrolyte ( = 2 in this work) Kinematic viscosity [m2 /s] Brine density [kg/m3 ] Permeate density [kg/m3 ] Change in manipulated variable Vector of change in manipulated variable Matrix of weights on manipulated variables Matrix of weights on controlled variables
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osmosis plant in Libya. Different methods used to enhance membrane performance were also reviewed by Al-Bastaki and Abbas [13]. Ali et al. [14,15] studied the performance of an RO tubular membrane module under oscillatory feed conditions using a validated rigorous dynamic model. It is found that the performance of the RO operation in terms of higher permeate production and less salt concentration can be obtained by periodic forcing of the feed pressure and feed velocity. The previous work was based on a validated mathematical model for a tubular membrane unit [16]. Operating the RO process in a cyclic manner to maximize the performance requires applying a suitable control system. One way is to use feed forward control structure to force the process input to follow predefined trajectory in the form of a periodic function. However, feed forward control algorithms do not respond successfully to unmeasured disturbances that affect the output directly. Another approach is to implement regular feedback control systems that incorporate a sinusoidal reference signals for the controlled output. However, this is a complicated task because it requires pre-designing the output reference profile including its mean value, period of oscillation and amplitude. The design procedure becomes even intractable when the process operating condition or parameters vary intentionally for unforeseen reasons. The nonlinear model predictive control belongs to the family of model predictive controllers (MPC). The MPC algorithms differ from the other advanced controllers in that a dynamic optimization problem is solved on-line each control execution [17]. Due to the MPC appealing features such as constraints handling and superiority for processes with a large number of manipulated and controlled variables, it became the most widely used control system in the chemical industries [18,19]. Review of the nonlinear MPC theory and its industrial applications has been reported in the literature [20,21]. The appealing capabilities of NLMPC make it suitable for the present control objective. Its optimality features allow for maximizing the RO process performance. Its flexible structure permits modification to incorporate periodic control behavior. Ali and Ali [22] utilized NLMPC to control the molecular weight distribution of polyethylene product. Due to the predictive nature and dynamic optimization of NLMPC, the controller was able to recognize the need to operate the process in a periodic fashion in order to achieve the desired objectives. Lee et al. [23] have integrated the repetitive control concept with internal model control to handle control problems involving periodic reference signals. This type of approach is suitable for tracking periodic reference trajectories or operation requiring repeated runs which is also known as run-to-run control. They have suggested further testing of potential applications in chemical processes. In the previous work [14,15] the periodic forcing was carried out in open-loop mode. In real practice, a feedback control system is necessary to ensure the periodic operation. In this paper we compliment the work of Ali et al. [15] to incorporate a robust control algorithm such as NLMPC to achieve the desired periodic input necessary to optimize the process performance. Furthermore, the standard NLMPC is modified to allow for incorporating periodic input functions. The objective is not to make the output follow predefined periodic trajectories but to force the process to operate in a periodic fashion through input cycling to improve the overall performance. The contribution of this paper falls into enhancing the water production and quality of RO processes through periodic operation. Moreover, a modified NLMPC algorithm that generates the necessary periodic input signals is presented. The paper is organized as follows. Section 2 describes the process model used to test the proposed periodic control algorithm. Section 3 describes the proposed control algorithm and the process control objective. Section 4 presents the main results of the simulation and provides detailed discussion of the outlet. Finally Section 5 outlines the conclusions of the work.
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2. Process model The cycling study is based on a dynamic model of tubular membranes that was developed and validated in an earlier study [16]. Both steady state and dynamic behavior were validated against a lab-scale experimental unit. The model considers the unit as a series of single tubes with appropriate minor pressure losses introduced between them. Each tube is described by coupled algebraic-ordinary differential equations. The tubes are modeled and solved sequentially where the output of any tube becomes the input for next tube. Model equations are summarized in the Appendix A. 3. The on-line NLMPC algorithm In this work, the structure of the MPC version developed by Ali and Zafiriou [17] that utilizes directly the nonlinear model for output prediction is used. A usual MPC formulation solves the following on-line optimization problem: min u(tk ),....,u(tk+M−1 )
P M (y(tk+i ) − R(tk+i ))2 + u(tk+i−1 )2 i=1
A U(tk ) ≤ b
3.1. Control objective and implementation The control objective here is to maximize the RO performance by operating in cyclic mode via forcing the input to be in the form of periodic function. Specifically, the objective is to maximize the permeate production, q and minimize the salt concentration, Cp . In this case, the controlled outputs are the permeate (q) and the salt concentration (Cp ) while the manipulated variables (MV) are the feed pressure (Pf ) and the feed flow rate in terms of its velocity (uf ). Standard NLMPC regulates the manipulated variables in optimal fashion according to the control law given in Eq. (1). The generated control signals may not be necessarily periodic. Implementing the NLMPC algorithm to generate periodic behavior for the MV, the process inputs, i.e. the MVs are transformed into periodic function in discrete time fashion as follows: Pf (tk ) = Pfss + Am1 sin(ˇ)
(6)
(1)
uf (tk ) = ufss + Am2 sin(ˇ)
(7)
(2)
where Am is the period amplitude, tk is the sampling instant and ˇ is the argument of the sin function that includes the cycle period p as follows:
i=1
subject to T
a more stable feedback response. A small value for M is preferable to reduce the on-line computational requirements. More details on the best practices in tuning such controller are given elsewhere [24].
For nonlinear MPC, the predicted output, y over the prediction horizon P is obtained by the numerical integration of the state space equation:
ˇ=
dx = f (x, u, t) dt
(3)
The cycle period is defined as function of the sampling instant (Ts ) as follows:
y = g(x)
(4)
p = ˛Ts
in discrete time fashion from tk up to tk+P where x and y represent the states and the output of the model, respectively. f(·) is a general nonlinear function of the process states, inputs and time while g(·) is either a linear or nonlinear function that maps the states into the process measured outputs. The symbols ||·|| denotes the Euclidean norm, k is the sampling instant, and are diagonal weight matrices and R = [r(tk+1 ). . .r(tk+P )]T is a vector of the desired output trajectory. U (tk ) = [u (tk ). . .u (tk+M−1 )]T is a vector of M future changes of the manipulated variable vector u that are to be determined by the on-line optimization. The control horizon (M) and the prediction horizon (P) are used to adjust the speed of the response and hence to stabilize the feedback behavior. The parameter is usually used for trade-offs between different controlled outputs. The input move suppression parameter, , on the other hand, is used to penalize different inputs and thus to stabilize the feedback response. The objective function (Eq. (1)) is solved online to determine the optimum value of U (tk ). Only the current value of u, which is the first element of U (tk ), is implemented on the plant. At the next sampling instant, the whole procedure is repeated. In order to compensate for modeling errors and eliminate steady state offset, a regular feedback is incorporated on the output predictions, y(tk+1 ) through an additive disturbance term. Therefore, the output prediction is corrected by adding to it the disturbance estimates. The latter is set equal to the difference between plant and model outputs at present time k as follows: d(tk) = yp(tk) − y(tk)
(5)
The disturbance estimate, d is assumed constant over the prediction horizon due to the lack of explicit means for predicting the disturbance. M and P are fixed according to general tuning guidelines for MPC [24]. For example, a large value of P tends to provide
2 tk p
(9)
The periodic functions (Eqs. (6) and (7)) transform the primary MV, i.e. the feed pressure and the feed flow rate into sinusoidal functions. The period of the cyclic function is considered identical for both variables. In this case, the primary MVs are defined by surrogate three manipulated variables, which comprise the amplitude for each input and the unified cycle period. Therefore, NLMPC will manipulate the feed pressure and flow rate indirectly through regulating their input characteristics, i.e. the amplitude and period of oscillation. Note that the new formulation can be easily reset to the standard formulation by setting ˇ to a constant value of /2. The controlled outputs embedded in Eq. (1) include the permeate production and salt concentration as a point values at specific sampling instants. However, because the operation will be periodic, the point value will be replaced by the time–average value of the permeate flow rate and the salt concentration in permeate. These time-averaged outputs are defined as ratio to their corresponding steady state values as follows: q(tk ) =
tk q(t)dt 0 tk 0
qss dt
tk Cp (t)dt C p (tk ) = 0tk 0
Cpss dt
(10)
(11)
In discrete time formulation, the numerical integration can be approximated by summation over the predefined simulation time. For future prediction, the model equations can be numerically integrated over the future p horizon from t = 0 to t = tk+P to estimate the average value for the controlled outputs at tk+P. The model prediction can still be improved by adding the disturbance estimates in Eq. (5) to the calculated outputs. Because the control objective in this paper is to maximize the permeate production and reduce the
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salt content to an arbitrary values, the objective function in Eq. (1) can be written in a specific form as follows:
P 1 2 2 min q(tk+i ) + 2 C p (tk+i ) u(tk ),....,u(tk+M−1 ) i=1
M u(tk+i−1 )2
+
(12)
i=1
Note that the set point, r in Eq. (1) is set to zero which means the controlled output should follow an arbitrary value. Usually, the arbitrary value is the maximum achievable magnitude for the permeate production and the minimum achievable magnitude for the salt concentration. In fact, setting the set point to zero does not make the controller set point-free. However, it will make NLMPC, according to Eq. (12), to drive q towards infinity and cp to zero in order to satisfy the zero set point. However, this is impossible because of the constraints imposed by the mass and energy balances. Therefore, the controller will try to attain the possible maximum/minimum achievable values for q and cp . These best achievable values are defined as “arbitrary values” in this work. It should be noted the entire simulation including the numerical integration of the model and the optimization of the NLMPC objective function is carried out using MATLAB software. Specifically the built-in “ode23” Matlab routine is used to integrate the model ODE and the “fmincon” routine associated with the Matlab-optimization toolbox is used to solve the multivariable constrained optimization problem. A selfdeveloped MPC algorithm that utilizes both ODE23 and FMINCON routines is employed in this paper. 4. Results and discussion Implementation of the proposed NLMPC algorithm for servo problem is shown in Fig. 1. The objective here is to maximize the ratio of the average permeate production to its steady state value
and to minimize the ratio of the salt concentration to its steady state value. No set point is defined here, thus the process variables will be maximized/minimized to an arbitrary value, i.e. the best achievable. The signal amplitude for feed pressure is bounded by ±30 bar, and that for feed velocity by ±30 cm/s. The period per sampling time (˛) is constrained between 3 and 10. Note that 3 is the minimum value that allows for complete periodic behavior within the given sampling time and simulation interval. A sampling time of 1 s is used in the simulation. The MLPC parameter values are M = 1, P = 20, = [0 0 0] and = [11]. In the entire simulations, both controlled outputs are given the same weight by setting to one (or any other equal values). Giving different values for the elements of will affect the control performance because NLMPC will give more weight to the output with larger value. This was avoided in the simulation because the set point is arbitrary. If significant set point is defined for the outputs then it might be necessary to adjust to achieve offset-free response. The simulation outcome illustrated the ability of MPC to generate oscillatory response which resulted in a reasonable improvement of the process operation. For example 22% increase in the permeate production over the steady state is observed. Similarly 5% decrease in the salt mass concentration is reported. The interesting part is that the enhancement was achieved without additional increment in the feed conditions. In fact, the ratio of the time-average value of the feed pressure and the feed velocity to their corresponding steady state value is 1. This is the main goal after periodic operation. The sampling time affects the cycle period and consequently the feedback performance. To demonstrate the influence of the sampling time, the servo problem is repeated for different values for Ts . The response of the average scaled-value of the permeate flow and salt concentration is shown in Figs. 2 and 3. The reaction of the raw value for q and Cp and that for the manipulated variables are omitted for clearer plots. Based on the steady state part of the plots, it is indicated that Ts = 0.1 s and Ts = 0.25 s provide the best performance for both cases. The effect is more remarkable on the salt concentration where enhancement as much as 20% can be obtained at high 150
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Fig. 1. Closed-loop simulation using NLMPC with Ts = 1 s and arbitrary set point.
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Fig. 3. Response of average permeate flow using NLMPC at different values for the sampling time.
frequency sampling. This is also reported in [15] and is attributed to the fast dynamic of the process. The ambiguous response for the case Ts = 0.5 can be attributed to nonlinearity as the pulse period has a nonlinear effect especially when it approaches the settling time for one of the outputs. It should be noted though that small sampling instants may not be practical in real practice. The previous simulation illustrated successful results for improving the process performance when perfect process model is used to represent the plant dynamics. The control objective is repeated in the presence of parametric errors in the model. Specifically, −20% errors are introduced in the value of the salt permeability and hydraulic permeability and +20% errors are introduced in the water diffusivity. The closed-loop response under these circumstances is depicted in Fig. 4. The simulation and NLMPC parameters values are the same as before. The first row of Fig. 4
shows the model response in addition to the plant response for the raw value of the process variable to demonstrate the effect of uncertainty. The second row of Fig. 4 shows the average value of the same plant outputs for the case when perfect and imperfect models are used in the control algorithm. The corresponding response of the manipulated variables for the case of perfect and imperfect models is shown in Fig. 5. Despite the effect of the model-plant mismatch, NLMPC was able to move the process operation to an arbitrary condition that is much improved than the initial condition. However, the obtained process response does not match that resulted from utilizing perfect model. The feedback capability and the disturbance estimates algorithm of the NLMPC could not overcome this situation because no specific set point is employed. In fact, the NLMPC is implemented with arbitrary reference for the controlled outputs. Interestingly, the modeling error resulted in improved permeate
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Fig. 4. Process closed-loop response in the existence of modeling error of Bj = –20%, Lp = −20%, and Ds = +20% using NLMPC without set point.
A. Emad et al. / Journal of Process Control 22 (2012) 218–227
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row value of the process outputs for both the plant and the model to highlight the mismatch due to uncertainty. The controlled outputs, which are the average value of the process outputs, are also shown in the figure when imperfect and perfect model are used. The latter is included for comparison purposes. In this figure and similarly Fig. 7, the response of the manipulated variables in the case of perfect model is not shown for clarity and brevity reasons. The simulation results indicate that the modified NLMPC can still
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flow but degraded salt concentration. Generally, the permeate production is more important as long as the salt concentration is within acceptable range. To illustrate the robustness of NLMPC in the presence of modelplant mismatch, the previous test is repeated but with specific set points for the controlled outputs. The outcome is shown in Fig. 6. In this case, the set point for the ratio of average q to initial steady state value is 1.25 and that for Cp is 0.95. The figure shows the
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Fig. 6. Process closed-loop response in the existence of modeling errors of Bj = −20%, Lp = −20%, Ds = +20% using NLMPC with specific set point.
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Fig. 7. Process closed-loop response in the existence of modeling errors of Bj = +20%, Lp = −20%, Ds = −20% using NLMPC with specific set point.
track fixed set point for the controlled variables. Moreover, the consequence proves the effectiveness of the feedback features of NLMPC to reject the influence of the model uncertainty. To further confirm the previous finding, the control problem is repeated using different parametric errors. Indeed, the salt permeability is set 20% higher in the model, while the hydraulic permeability and water diffusivity are set 20% less in the model. Applying the same
NLMPC and simulation parameters, the feedback transient reaction is depicted in Fig. 7. Once again, successful tracking of the reference value and overcoming the adverse effect of model-plant mismatch are obtained. Nevertheless, influence of the model inaccuracy on the transient performance is obvious. It is equally important to examine the effectiveness of the regulatory performance of the NLMPC for the RO process. Fig. 8
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demonstrates the process closed-loop operation when the feed flow is suddenly decreased by 10%. Specifically, the row value of the process outputs and manipulated variables when the process is under disturbances is depicted in the figure. The average value of the process outputs is included in the figure and compared to that when the process is not influenced by disturbances. In this figure and subsequent ones, the MV response when the process is not under disturbance is omitted for clarity and briefness purposes.
Enhance process performance over the steady state operation in the sense of higher permeate production and lower salt concentration is still observed. It should be noted that exact match of the process response when no disturbance is in service is not necessarily because strict set point is not specified. Moreover, the impact of this type of disturbance on the process variables is marginal compared to that shown in Fig. 9. In the latter, a disturbance of −10% is injected in the feed pressure. Note that both q and Cp are
1
250
Cp (g/l)
with disturbance
3
q (cm /s)
with disturbance 0.6
0.2
0
20
40
60
80
150
50
100
0
20
Time (sec) 100
uf (cm/s)
P f (bar)
100
0
20
40
60
80
80
100
80
100
50
0
100
0
20
Time (sec)
40
60
Time (sec)
1.5
1.2
with disturbance
1.1
no disturbance
ss
1
Cp/Cp
q/qss
80
with disturbance
50
with disturbance no disturbance
0.5
60
100
with disturbance
0
40
Time (sec)
0
20
40
60
Time (sec)
1 0.9
80
100
0.8
0
20
40
60
Time (sec)
Fig. 10. Process closed-loop response for the rejection of disturbance of +20% in the feed salt concentration.
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A. Emad et al. / Journal of Process Control 22 (2012) 218–227
adversely influenced by the reduction of the feed pressure which is an expected phenomenon. Positive disturbance in the feed pressure and flow produces favorable process operation hence, it is not discussed here. Furthermore, disturbances in the feed parameters can be easily handled through feed forward control systems. However, these disturbances were examined here for demonstration purposes. Finally, the incorporation of 20% increase in the feed salt concentration is investigated. The simulation result is shown in Fig. 10. It can be seen that the response of the permeate flow is not affected while the salt concentration deteriorated. This situation is predictable because further minimization of the salt concentration is limited by its value at the RO entrance. Nevertheless, in all the regulatory control problems discussed earlier, reasonable propagation of the permeate production and lessening of salt concentration were observed. Moreover, the regulatory performance can be further improved by incorporating definite set point for the controlled outputs as illustrated earlier. However, this was not sought because it is meant to implement the control system with arbitrary reference value. In this case, the controller will always drive the process operation towards the best achievable situation regardless of any parameter/condition changes that may occur.
constant. cwi is the wall brine concentration at the i-th increment, it is calculated using the flowing equation: exp(Jv,i · Sc 2/3 /ji · ui )
cw,i = cb,i ·
with Sc is Schmidt number and ji is Chilton–Colburn factor. For turbulent flow in smooth circular tubes it is given by [25]: −1/4 ji ∼ = 0.0395 Rei
(A.3)
with Rei being the local Reynolds number in i-th increment. The dynamics of salt concentration in the brine leaving increment i, are given by the following equation dcb,i+1 = dt
u · c u · c i b,i i+1 b,i+1 −
x
−
x
(1 − Rj ) ·
4Jv,i d
· cw,i (A.4)
Here ui+1 is the velocity of brine leaving the i-th increment. It can be obtained from a volumetric balance about the i-th increment and is given by ui+1 = ui − 4Jv,i
5. Conclusions The operation of a tubular RO process under forced periodic inputs in order to improve its performance is investigated. The periodic forcing is imposed via feedback control. Specifically, nonlinear model predictive control (NLMPC) is implemented for this purpose. NLMPC regulates the feed pressure and flow indirectly through manipulating their transformation parameters. The feedback simulation indicated the effectiveness of NLMPC to generate periodic input functions that helped in enhancing the time-averaged value of the permeate production and salt concentration. Improvement of up to 40% increase in desalted water and up to 20% reduction in the salt concentration is observed. The promising outcome is maintained even in the presence of model uncertainty and in the sudden injection of input disturbances. The control system is implemented with arbitrary set point to make the regulator searches for the best achievable performance regardless of the process conditions.
(A.2)
Rj + (1 − Rj ) · exp(Jv,i · Sc 1/3 /ji · ui )
x d
(A.5)
with x and d being the increment length and tube diameter, respectively. The energy balance, for the i-th increment is used to calculate the pressure in the brine-side for different i-th increments. It can be proved that the pressure leaving the t-th increment can be calculated as follows: pb,i+1 =
u i ui+1
· pb,i + 0.5 · 10−7 · b,i · u2i − 2 · 10−7 · fF · x · u2i ·
−
4x · Jv,i d · ui+1
·b,i · u2i+1
b,i d
· (pb,i + 0.5 · 10−7 · p,i · Jv2,i ) − 0.5 · 10−7 pb,i = pb,i+1 − pb,i
(A.6)
with bi and pi being the brine and permeate density respectively and fF is Fanning friction factor, given by
Acknowledgments The Authors are thankful to the Deanship of Scientific Research at King Saud University (Project # RGP-VPP-118) for the financial support of this research.
fF ∼ =2·j
(A.7)
The permeate production rate for the i-th increment is given by qi = Jv,i · · d · x
(A.8)
The cumulative production rate from all increments is Appendix A. Dynamic model of the RO unit The model assumptions and the detailed derivation of model equations can be found in [16], in this Appendix A, the model is briefly presented. Because the process variables vary with the length of the RO tube, the membrane module is divided into n hypothetical increments. Therefore, mass and energy balances are made for each segment. The permeate flux (Jv,i ) through the membrane at, increment i, is described by the three parameter nonlinear Spiegler–Kedem (SK) model: Jv,i = Lp [(pb,i − pp,i ) − (Rj ) RTcw,i ] 2
(A.1)
where pb,i and pp,i are brine-side and permeate-side pressures respectively, Rj is the intrinsic salt rejection, and R is the ideal gas
q=
n
Jvi · · d · x
(A.9)
i=1
The cumulative average salt concentration of the product water can be obtained by multiplying the quantity of water produced in certain increment by the salt concentration in that increment. Rearranging the resulted equation gives
n−1
cp,i = cp,i−1 ·
i=1
q
qi
+
(1 − Rj ) · cw,i · Jv,i · · d · x q
(A.10)
When cyclic mode of operation is used, the ratio of the unsteady state to steady flow mass transfer coefficients was approximated by [12]:
kp up n up n = 0.5 · 1 + + 0.5 · 1 − us us ks
(A.11)
A. Emad et al. / Journal of Process Control 22 (2012) 218–227
with subscripts p and s referring to periodic and steady state operations, respectively. up is the periodic velocity. n = 1/3 for laminar flow and n = 0.8 for fully developed turbulent flow. The recovery of the permeate mass salt concentration is related to the mass flux of permeate as follows: cwi − cpi cpi
J = vi Bj
(A.12)
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