Phone:
0086 28 87573647
+86 19150187139
[Abstract] This paper proposes the original mathematical model and the classical mathematical model for the optimization design of small reverse osmosis membrane systems; discusses the indirect solution algorithm of the classical model using existing design software; derives the mapping relationship between the three-dimensional design index of the system and the three-dimensional optimization solution; and proposes the "table lookup-correction" optimization design mode. This mode is a fundamental correction to the existing "trial calculation-optimization" design mode.
[Keywords] Reverse osmosis system, optimization design, optimization mathematical model
In the past decade, reverse osmosis technology has developed rapidly in China [1], and the design and operation technologies of reverse osmosis pretreatment and membrane systems have been rapidly improved and widely popularized. However, as a rapidly developing new technology, there are still many major problems that need to be studied. The design mode, design tools and their usage methods of membrane systems are one of the typical problems.
In the reverse osmosis process, the membrane, shell, pump and auxiliary equipment are collectively referred to as the membrane system. For many years, the tools for membrane system design have been dedicated design software provided by foreign membrane manufacturers. These design software have provided a design platform for domestic reverse osmosis technology and have powerfully promoted the development of this technology. However, these design software and their corresponding design modes have obvious limitations. They all employ the same design pattern: given feedwater quality, permeate flow rate, recovery rate, number of membranes, and their arrangement, they solve for the system's operating pressure and permeate quality, which can be called the "trial and error optimization" design optimization pattern. The shortcomings of this design pattern are:
(1) It cannot guarantee that every design scheme is a feasible solution, let alone the optimal solution. When designing membrane systems, designers need to rely on their experience, conducting numerous trial calculations and comparing results to arrive at a satisfactory design scheme. Inexperienced designers often struggle with this requirement and may even feel overwhelmed.
(2) The original engineering design problem faced by designers is an optimization design problem, namely, solving for the optimal number and arrangement of membranes, the maximum recovery rate, and the corresponding operating pressure based on specific feedwater quality conditions, permeate quality, and permeate flow requirements. This differs significantly from the current "trial and error optimization" design pattern.
This paper attempts to demonstrate a theoretically rigorous and practically simple engineering design pattern to improve the design level and calculation speed of current engineering companies. This paper focuses on the optimization design of small-scale reverse osmosis membrane systems consisting of 18 or fewer 4040 membrane elements.
Based on systems engineering and operations research, the optimization design problem of reverse osmosis membrane systems has the following characteristics:
The optimization problem is an economic and technical comparison problem. If technical indicators are treated as limiting factors, the goal of optimization design is to minimize the total cost of the system, which includes three aspects: minimum investment cost, minimum operating cost, and minimum water consumption cost.
As a complete chemical and physical system, the reverse osmosis system has its own inherent laws, mainly manifested in the direct proportionality between the system's permeate flow rate and the pure driving pressure, and the direct proportionality between the system's salt permeability and the salt concentration difference across the membrane. The balance of these inherent relationships among system parameters is expressed as certain mathematical equations, constituting the system constraints in the optimization model.
The system design involves numerous parameter constraints, such as the maximum limit for the system concentration polarization index; the maximum limit for the sparingly soluble salt saturation; the maximum limit for the feed water flow rate of a single membrane to prevent excessive flow resistance; and the minimum limit for the concentrate flow rate of a single membrane to prevent insufficient membrane surface scouring. These constraints constitute the limit constraints of the optimization model.
Concentration polarization is symbiotic with system operation. A higher upper limit for its index can improve the system recovery rate; however, it will exacerbate system performance degradation, accelerate scaling, increase cleaning costs, require more frequent membrane replacements, and increase system operating costs. The upper limit for sparingly soluble salt saturation is another sensitive issue in system design, directly affecting the system recovery rate and membrane arrangement, and deserves careful discussion. Currently, the following approach is commonly used in engineering design: the upper limit for the concentration polarization index is set to 1.2, and the upper limit for the sparingly soluble salt saturation is set to 100%.
The problems to be solved by the optimization design are represented as optimization variables in the model. These include the type, quantity, and arrangement of membrane elements; the specifications and quantity of membrane housings; the tonnage and head of the pump; and the system recovery rate, a crucial system design indicator.
In the original engineering design problem, the requirements for feed water quality, product water quality, and product water quantity are collectively referred to as design constraints.
(6) Setting the average product water quantity per membrane element is a sensitive issue in system design. A higher value for this parameter reduces the number of membranes used in the system and lowers the system investment cost; however, it increases cleaning costs, requires more frequent membrane replacements, and increases system operating costs. The setting of this value can even constitute another specific optimization problem. Once this parameter is set, the number of membranes in the system is solely determined by the permeate production requirement. In this discussion, the permeate production of a single 4040 membrane element is assumed to be 0.2 tons per hour.
The above analysis of the membrane system optimization design can be summarized as the following original mathematical model for optimization design:
Optimization Objectives:
Minimum investment cost
Minimum operating cost
Minimum water consumption
System Constraints:
Permeate production is directly proportional to the pure driving pressure across the membrane
Salt permeability is directly proportional to the salt concentration difference across the membrane
Limit Constraints:
Upper limit of feedwater flow rate
Upper limit of concentration polarization index
Lower limit of concentrate flow rate
Upper limit of sparingly soluble salt saturation
Design Constraints:
Feedwater quality
Permeate quality
Permeate production
Optimization Variables:
Number of elements
Membrane housing specifications
Pump specifications
Element type
Membrane arrangement
System recovery rate
In the optimization variables, the membrane arrangement can be considered an integer variable, while the recovery rate is a continuous variable. Therefore, this problem is a typical multivariate, multi-objective, nonlinear, mixed-integer programming problem. Due to the complexity of this model, it is difficult to solve directly, but it defines the basic problem of system optimization design and provides a basis for model simplification.
Many factors in the original model can be simplified to make optimization calculations feasible.
Handling Optimization Objectives
Handling Water Costs
Water cost divided by water price equals the water supply. Therefore, the objective of minimizing water costs implies maximizing the system recovery rate.
Handling Investment Costs
Investment in reverse osmosis membrane systems can be divided into fixed investment and variable investment. Fixed investment refers to the costs of structure, piping, valves, instrumentation, and control, which are largely irrelevant to the optimization design. Variable investment mainly includes the investment in membranes, membrane housings, and pumps. Since the permeate output of a single membrane is given, the number of membranes and their cost depend on the system's designed permeate output and are still irrelevant to the optimization design. Optimization calculations targeting the highest recovery rate will minimize the system's water supply and pump tonnage requirements, while increasing the system's operating pressure and pump head requirements. However, the increase in pump price due to increased head is generally less than the decrease in pump price due to decreased tonnage. Therefore, it can be assumed that as the recovery rate increases, the overall price of pump specifications decreases, meaning that pump investment cost optimization can be naturally achieved in system optimization targeting the highest recovery rate. Regarding membrane housing costs, the more membranes installed per housing, the lower the price per unit length of the membrane housing, and the lower the total cost of the membrane housing. Under the objective of maximizing system recovery rate, the optimization solution naturally maximizes the system flow and the number of membranes per housing; therefore, membrane housing cost optimization is also naturally achieved in system optimization targeting the highest recovery rate.
Handling operating costs.
Membrane system operating costs can also be divided into fixed costs for normal maintenance, semi-fixed costs such as membrane cleaning and replacement, and variable costs such as pump power consumption. Fixed costs are irrelevant to system optimization. The system optimization process naturally pushes parameters such as concentration polarization and sparingly soluble salt saturation to their upper limits, at which point semi-fixed costs naturally become fixed values. As the system recovery rate increases, system operating pressure and permeate power consumption increase, but the increase in electricity costs is generally less than the decrease in water costs; that is, the impact of variable costs is overwhelmed by the impact of the recovery rate target.
In short, the goal of minimizing all investment and operating costs—whether irrelevant to system optimization, naturally satisfied under the goal of maximizing recovery rate, or overwhelmed by the impact of maximizing recovery rate—can be neglected from the objective function of the mathematical model. The system optimization design goal can ultimately be simplified to the single objective of maximizing system recovery rate.
Handling Limit Constraints
Concentration polarization and sparingly soluble salt saturation are fundamental limit constraint terms. It is worth noting that the most severe concentration polarization occurs at the end of each segment, and the highest sparingly soluble salt saturation occurs at the membrane surface at the system's end. Currently, the design software that only provides the sparingly soluble salt saturation in the system concentrate is a serious deficiency.
System design schemes generally do not involve upper limits for feedwater flow rate or lower limits for concentrate flow rate of membrane elements. Special cases can be handled specially, and to ensure clarity of the mathematical model, this constraint may be omitted from the model.
Elimination of Independent Variables
In the original model's optimization variables, the number of membranes is determined solely by the system's permeate flow rate. The specifications and number of membrane housings are determined by the membrane arrangement. Once the system recovery rate, membrane type, and arrangement are determined, the feed water flow rate and pressure can be determined, and consequently, the pump specifications can be determined. Therefore, the first three types of variables in the model can be determined by other variables or factors and do not need to exist as independent variables.
Decomposition of Independent Variables
Given known feed water quality, the membrane type, i.e., the membrane desalination rate, is the determining factor for permeate water quality and system operating pressure. However, the effect of the membrane type on the system is independent of the influence of the membrane arrangement and recovery rate. Therefore, the optimization design problem can be decomposed into two problems: system recovery rate optimization and membrane arrangement optimization, and membrane type optimization. First, optimize the recovery rate and arrangement of various membrane elements separately. Then, compare the optimization results for each membrane element to find the result that meets the permeate quality requirements and has the lowest operating pressure. This leads to the optimal membrane type, the corresponding optimal arrangement, and the system recovery rate.
Since the latter problem is easily solved, the model discussion can be limited to the optimization design of membrane systems with specific membrane types.
Classical Model for Design Optimization
Based on the above analysis, the original model can be simplified to a classic optimization design model under given membrane type and quantity conditions:
Optimization Objective: Maximize system recovery rate
System Constraints: Permeate flow rate is proportional to the pure driving pressure across the membrane
Salt permeability is proportional to the salt concentration difference across the membrane
Limit Constraints: Upper limit of concentration polarization index
Upper limit of sparingly soluble salt saturation
Design Constraints: Feed water quality conditions
Permeate water quality requirements
Permeate flow rate requirements
Optimization Variables: Membrane arrangement
System recovery rate
The system and limit constraints in the model are general terms, common to all design problems. Design constraints and optimization variables are specific terms, representing computational conditions and results unique to a particular design problem.
The classic mathematical model is a bivariate, single-objective, nonlinear, mixed-integer programming problem. There are two solution methods for this problem. One is to directly calculate and solve using traditional mathematical programming methods, achieving system optimization design in the original operations research sense; the difficulty lies mainly in establishing analytical expressions for system constraints. The other method is to indirectly solve this programming problem using existing design software; the engineering concept of this solution process is clearer, facilitating flexible adjustments based on actual engineering conditions. This paper focuses solely on the discussion of the indirect solution method.
4. Indirect Solution Method for Classical Models
Dynamic Algorithm for System Design
In the classic model, the membrane arrangement can be considered as integer variables. For each integer value, there exists a maximum system recovery rate that satisfies all constraints, leading to a local optimum for the optimization objective. The process of finding the local optimum recovery rate is called the integerization process. If the integer variable values and the corresponding maximum values of continuous variables are considered as a whole, the above mixed-integer programming can be viewed as an integer programming problem. The integer programming calculation here only seeks a specific solution that is optimal overall for the planning objective value among a finite number of feasible solutions. Therefore, the indirect solution process of mixed integer programming can be divided into two steps: integerization and integer solution comparison.
When the membrane arrangement takes a specific form, the traditional design calculation mode using design software is: given two known quantities, permeate flow rate and recovery rate, the values of two variables, permeate pressure and permeate quality, are statically calculated. The integerization calculation mode using design software is: gradually increasing the recovery rate from a low value, ultimately obtaining the maximum system recovery rate constrained by concentration polarization index or sparingly soluble salt saturation, as well as the corresponding permeate pressure and permeate quality. During the integerization process:
Permeate flow rate remains constant and is always a known quantity.
Operating pressure changes with the recovery rate and is always a variable.
As the recovery rate is optimized, it evolves from a known quantity in the original sense into a variable.
Permeate quality changes with the system recovery rate and should be classified as a variable. However, since the membrane desalination rate is given, the variation in permeate water quality is very limited, and therefore can be considered a known quantity.
It is precisely because of the gradual optimization process of recovery rate during integerization that the system design is transformed from static to dynamic, ensuring that the configuration of known quantities and variables to be solved in dynamic calculations meets the needs of the original engineering design problem.
Two Criteria for Optimization of Membrane Arrangement
In the indirect solution process of mixed integer programming, for each feasible membrane arrangement of a specific system, the maximum system recovery rate that satisfies the constraints is first calculated, and this is called the inherent recovery rate of that arrangement. The maximum value of each inherent recovery rate is the maximum recovery rate of the specific system.
If the maximum recovery rate is unique, the membrane arrangement corresponding to this recovery rate is the optimal membrane arrangement for the specific system. Therefore, the highest recovery rate can be called the first criterion for arrangement optimization. In actual calculations, multiple equivalent maximum recovery rates often occur. Among the multiple membrane arrangements corresponding to these rates, the membrane arrangement with the lowest operating pressure and the highest desalination rate is the optimal membrane arrangement for the specific system. Therefore, the lowest operating pressure or the highest system desalination rate can be called the second criterion for arrangement optimization.
Thus, based on the three design constraints of the original engineering design problem, and using classical models and indirect solution methods, the optimal design process for a specific membrane system can be completed.
During integer calculations, if the saturation of sparingly soluble salts on the membrane surface at the system's end remains consistently low, without affecting the system recovery rate, the limit constraint is actually only the concentration polarization index. The feedwater conditions under these conditions are called low-sprinkler-soluble salt feedwater conditions. Maintaining the definition of low-sprinkler-soluble salts, the highest saturation of each sparingly soluble salt in the feedwater is called the critical saturation of low-sprinkler-soluble salts. Since there are various types of sparingly soluble salts, the critical saturation of low-sprinkler-soluble salts is a set of values.
Corresponding to different system permeate flow requirements, the number of membranes, system flow length, system recovery rate, and the optimal arrangement of low-sprinkler-soluble salts in the system will vary.
Critical saturation also varies. Generally, the larger the system, the longer the flow path, the higher the maximum recovery rate, and the lower the critical saturation value for low sparingly soluble salts in the feed water.
Optimization calculations show that for systems with 18 4040 membranes or smaller, the optimal membrane arrangement under low sparingly soluble salt conditions is the so-called "six-segment saturated structure." The main characteristics of this arrangement are: when the number of membranes is 6 or less, the arrangement is one segment in series; when the number of membranes is 8 to 18, the second segment consists of 6 membranes in series, while the first segment consists of two parallel membranes of equal length.
When the sparingly soluble salt saturation in the feed water exceeds the low sparingly soluble salt critical saturation, the main factor limiting the system flow path length and determining the membrane arrangement becomes the sparingly soluble salt saturation in the feed water, and the system flow path will be shortened accordingly. The system optimization design problem, where the constraint is essentially transformed into sparingly soluble salt saturation, is a typical broad-spectrum optimization design problem.
Although classical models, indirect solution methods, and design software can be used to complete the optimization design of a specific membrane system,… However, to meet the demands of engineering companies' daily consulting, bidding, and design calculations, and especially to accommodate the limited technical capabilities of small and medium-sized enterprises, improvements should be made to create a completely new design model, making it a clear, easy-to-learn, fast, and accurate design tool.
Broad-spectrum Optimization Design
The basic idea of this new design model is to pre-establish a mapping table between the design constraint set and the optimal solution set through extensive calculations. Designers only need to start from the design constraints and directly look up the corresponding optimal design solution using the existing mapping table. Because this design model is applicable to almost all design conditions and requirements, it is called broad-spectrum optimization design, and the optimal solution set is called the broad-spectrum optimization structure.
In the optimization design, the real-valued three-dimensional design constraints of feedwater quality, product water quality, and product water quantity constitute a real-valued three-dimensional constraint set. Optimization calculations using each element in the set as constraint data yield rigid parameters such as the number, type, and arrangement of membranes (integer values), and flexible parameters such as recovery rate, operating pressure, and desalination rate (in real-valued values). The rigid parameters constitute the integer-valued three-dimensional solution set, while the flexible parameters are subordinate to the rigid parameters. When the design constraints and rigid parameter solutions are known, integer-based calculations can easily reproduce all system flexibility parameter values. System flexibility parameters are highly sensitive to many atypical feedwater quality and membrane performance parameters, such as feedwater TDS, ion composition, temperature, pH, and membrane performance degradation. The determination of flexibility parameters can or needs to be corrected based on actual data during integer-based calculations. It is precisely this correctability of flexibility parameters that embodies the flexibility of the indirect solution method.
Due to the mathematical characteristics of the real number constraint set and the integer solution set, the optimization solution mapping is a mapping from the real number set to the integer set, and the finiteness of the elements in the integer set determines the feasibility of the "table lookup" model.
Product Water Quality and Membrane Element Types
System product water quality is generally expressed by soluble solids content, i.e., TDS value (or conductivity). One of the feedwater quality indicators is also the TDS value. The ratio of the product water TDS value to the feedwater TDS value is the system salinity. The salt permeability of three ESPA series membrane elements from Hyster Corporation, USA, under test conditions was 0.5%, 1.0%, and 2.0%, respectively, representing a four-fold variation. Therefore, under specific permeate TDS requirements, selecting different membrane types will yield a four-fold range of feedwater TDS values. In other words, if the one-dimensional constraint on permeate quality is changed to the difference between feedwater and permeate TDS values, a finite number of continuous intervals in the real domain of this difference will map to a finite integer sequence of membrane element types.
Permeate Flow Rate and Number of Membrane Elements
Once the permeate flow rate of a single membrane is determined, the system permeate flow rate determines the number of membranes. For example, if the permeate flux of a single 4040 membrane is 0.2 tons per hour, to adjust the system permeate flow rate within the range of 0.2 to 3.6 tons per hour, this can be achieved by selecting 1 to 18 different numbers of membrane elements. Therefore, a finite number of continuous intervals in the real domain of permeate flow rate will map to a finite integer sequence of the number of membrane elements.
Saturation of Sparingly Soluble Salts and Membrane Alignment
When the saturation of sparingly soluble salts in the system feedwater exceeds the critical saturation of low-saturation sparingly soluble salts, among the various water quality indicators of the system feedwater, only the saturation concentration of sparingly soluble salts (calcium carbonate is represented by the Langri index LSI) is the main indicator limiting the system recovery rate and membrane alignment. The saturation concentration of a certain sparingly soluble salt on the membrane surface at the system's end is approximately the product of the saturation concentration of that sparingly soluble salt in the concentrate and the concentration polarization index. When this value reaches the limit of 100%, the sparingly soluble salt saturates and precipitates. The system recovery rate at the critical saturation precipitation of a certain sparingly soluble salt is the system's specific recovery rate for that sparingly soluble salt, and the minimum of the specific recovery rates for each sparingly soluble salt is the system's maximum recovery rate.
For specific feedwater conditions, if the system recovery rate is continuously increased, one type of sparingly soluble salt on the membrane surface at the system's end will always saturate and precipitate first, thus limiting the system recovery rate; while other types of sparingly soluble salts are not yet saturated and do not have a substantial impact on the recovery rate; therefore, a specific feedwater system can be defined as having a problem with this type of sparingly soluble salt. Because different sparingly soluble salts exhibit varying degrees of saturation precipitation, various feedwater conditions can be categorized as finite-class sparingly soluble salt problems. The membrane arrangement corresponding to the highest system recovery rate depends on the type of sparingly soluble salt in the feedwater and its saturation level.
Due to the discrete nature of membrane element arrangements, a finite number of continuous intervals in the real domain of the saturation levels of various sparingly soluble salts in the system feedwater will be mapped to a finite integer sequence of membrane arrangements.
In summary, in the broad-spectrum optimization design of membrane systems, if the one-dimensional constraint on product water quality is replaced with the difference between the feedwater and product water TDS values, the optimization solution mapping becomes a mapping between a finite number of uninterrupted, non-overlapping subsets of three-dimensional constraints and a finite number of integer solution subsets. It is a mapping of multiple three-dimensional cuboid subspaces in the continuous three-dimensional constraint space to points in the three-dimensional integer solution space, and a correspondence between each three-dimensional interval and the three-dimensional numerical solution in the continuous three-dimensional constraint value interval table. It is precisely because of the existence and finite number of such relationships that the "table lookup" design mode is valid.
"Look-up-and-Correction" Optimization Design Mode for Broad-Spectrum Optimization
Because there is a correspondence between the three-dimensional interval values and the three-dimensional solutions in the continuous value interval table of three-dimensional constraints, the three-dimensional design constraint values of any membrane system's original engineering design problem must fall into a certain three-dimensional constraint interval and must correspond to a certain three-dimensional rigid parameter solution in the broad-spectrum optimization structure. This is the optimal solution for the membrane system's optimization design problem. System designers only need to find the corresponding position in the table based on the three-dimensional design constraint data in the original engineering design problem to obtain the optimal three-dimensional rigid parameter solutions for the system, such as the number, type, and arrangement of membranes.
By performing an integer-based calculation using the three-dimensional design constraints, the three-dimensional rigid parameter solutions, and various atypical feedwater quality parameters and membrane parameters such as TDS, ion composition, temperature, pH, and membrane performance degradation, the system's flexible parameters can be reproduced or corrected. This completes the entire optimization design process. Due to the different solutions for rigid and flexible parameters, the entire optimization design mode is called the "look-up-and-correction" design optimization mode.
In the process of applying the "table lookup-correction" method for system design, the system's permeate water quality and carbonate scaling should be tested under summer (high temperature) conditions, and the system's permeate flow and non-carbonate scaling should be tested under winter (low temperature) conditions. Attention should also be paid to the changes in system flexibility parameters before and after membrane performance degradation.
This paper proposes a primitive model for the optimal design of a reverse osmosis membrane system from the perspective of mathematical programming. Based on engineering characteristics, it is processed into a classical model. After completing the verification of the indirect algorithm, a complete optimization theory and feasible calculation method for specific system design optimization problems are formed, thus solving the problem of what is optimal and how to find the optimal solution in the field of membrane system design. However, this method requires designers to perform a large number of simulation calculations.
To simplify the design tools, this paper proposes a "table lookup-correction" design optimization mode. Under the guidance of broad-spectrum optimization design theory, through extensive preliminary analysis and calculation, a three-dimensional data table can be obtained, with system feed water quality, permeate water quality, and permeate flow as continuous independent variables, and membrane quantity, type, and arrangement as discrete dependent variables. This allows the system optimization design to form a design mode that combines simple data table lookup with a small amount of correction calculation. This solves the problem of how to simplify the optimization process in the field of system design. The specific implementation of the optimization design mode discussed in this paper will greatly facilitate the design work of engineering enterprises. Since the broad spectrum optimization design problem of membrane system is difficult to be quantitatively described by mathematical expressions, this paper adopts a textual description method. Due to the limited space, the actual three-dimensional data table and optimization calculation examples are not given in the paper. For relevant content, please refer to the reference document [2]. This paper does not involve the concentrate reflux process and its related optimization design problems.
References:
(1) Jing Dawei, Song Jing. Current status, development and research of reverse osmosis pure water technology [J]. Industrial Water Treatment, 2002.22(10): P16-18
(2) Jing Dawei, Xu Lamei, Wang Ting. Optimized structural design of small reverse osmosis membrane system [J]. Industrial Water Treatment, 2003.23(03): P65-68
