反渗透膜法海水淡化过程最优化设计的研究
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摘要
水是生命的源泉,是人类宝贵的、不可替代的自然资源。水资源危机已经成为继石油危机、电力危机之外又一制约中国经济发展的瓶颈。长远来看,发展淡化技术,从海洋获取淡水是解决水资源短缺的重要途径。反渗透膜法海水淡化技术由于其能耗低,系统安装维护相对简单,目前已得到了广泛的应用。但其技术经济指标与社会的期盼值之间,还存在一定的距离,淡化水的成本偏高。这说明当前的反渗透工艺、工程设计以及系统操作等方面仍有很多值得改进的地方。根据反渗透技术的特点,以及它在水资源开发和保护中的作用,本文对反渗透淡化过程进行了分析,采用最优化方法对反渗透淡化系统的设计进行了研究。在综合考虑了多种水源合理配置和有效利用的基础上,提出了同时考虑多水质输入、多水质输出的反渗透淡化系统的设计问题。该系统不仅可以将海水转变成淡水,而且也可以同时用来处理苦咸水和工业废水,提高了水的利用率。同时根据不同的用途,输出多种水质,实现了一水多用,重复利用,充分发挥了水资源的多种功能。本文主要内容有以下几点:
1.对反渗透海水淡化过程进行了模拟和设计研究,首先对过程的每个单元给出了单元操作模型和相关的经济模型。通过一定的变量将这些模型相互关联,组成系统模型。系统模型主要考虑了一级流程,一级两段流程和二级流程。以年费用最小为目标,对反渗透海水淡化系统进行设计,将设计问题表达为一个混合整数非线性规划。将膜组件分为高脱盐膜,高通量膜,普通膜,低脱盐膜四种类型。当产品水的设计要求不同,进料水浓度不同时,采用本文的设计方法分别得出了不同的最优设计方案。当进料水浓度逐渐降低时,所用膜的类型逐渐由高脱盐膜转变为高通量膜,低脱盐膜,这些膜的脱盐率降低,水通量增大,这说明了进料水浓度高时,脱盐是主要矛盾,进料水浓度低时,水通量是主要矛盾。
当进料水浓度较高,反渗透系统需要较高的操作压力,能量消耗大,因此换能器效率对年费用影响较大。如能量回收效率较高时,采用结构简单的一级流程;进料水浓度低时,能量费用在年费用中占的比例小,投资费用对年费用的影响较大,采用水回收率高,膜组件的选用合理的二级流程。这个方法可用于海水、苦咸水淡化的工程设计,也可用于高浓度含盐废水的脱盐工艺设计。本文对一个实例进行了设计,得到的吨水费用,能耗低于文献报道值,结果合理,表明这种方法是有效的。
2.本文对在不同进料水水质条件下,不同产品水需求时,反渗透系统的优化设计进行了研究。提出了一种系统化的、集成化的方法对反渗透系统进行设计。首先,采用状态空间法给出一个网状的超结构描述,这个超结构包括了所有可行的设计方案。它和其它结构相比,不同之处在于:1).假定反渗透网络只有一个进料水入口,进料水从第一个反渗透级进入网络。2).将物流分布集合简化为物流混合节点。淡水、盐水物流在增压级进行混合,在反渗透级进行分离。3).超结构数学模型中采用了物流分率,等压混合约束等技术手段,简化了模型中物流混合、分离的数学描述。4).反渗透网状超结构的描述中以功压交换器作为能量回收装置。
这个超结构数学模型中采用了物流分率,等压混合约束等技术手段,减少了整数变量的数目和算法的搜索空间,使得模型的求解更容易。当进料水水质发生变化时,可通过改变多级反渗透系统的结构,选择恰当的操作条件和膜组件的型号,获得具有最大经济效益的产品水。
本文同时给出改进的卷式膜组件模型。该模型考虑了膜元件中的压降和浓差极化,可以近似地描述膜分离单元中压力容器的行为。由此就可以将系统中膜组件的设计和整个反渗透系统的设计结合起来,同时进行优化。设计的结果可以确定最优的压力容器排布方式,每个压力容器中膜元件的个数以及元件的选型。反渗透网络的优化设计问题最终可表达为一个混合整数非线性规划(MINLP),它以反渗透系统年费用最小为目标,满足过程热力学、单元操作、设计要求的约束。提出的系统化的反渗透网络优化设计方法被应用到海水、苦咸水淡化实例中。结果表明,采用最优化设计方案可以明显地降低单位产品水的费用,同时这种设计方法可以加深对整个反渗透系统的了解,适合应用于实际的反渗透淡化工程设计中。
3.在综合考虑了多种水源合理配置和有效利用的基础上,根据反渗透淡化技术的特点,提出了同时考虑多水质输入、多水质输出的反渗透淡化系统的设计问题。该系统不仅可以将海水转变成淡水,而且也可以同时用来处理苦咸水和工业废水,提高了水的利用率。同时根据不同的用途,输出多种水质,充分发挥了水资源的多种功能。本文采用过程综合优化的方法,对多水质输入、多水质输出的反渗透系统的优化设计进行了研究。首先,在前面提出的网状超结构的基础上,给出这个问题的超结构描述。改进之处在于:该反渗透网络由Nps个增压级和NRO个反渗透级构成。网络中有四套物流节点,分别是Nw个进料物流节点;Nps个物流混合节点;NRO个反渗透单元节点;Np个产品水节点。这个结构描述简单,易于表达。在它的数学模型表达中采用了物流分率,物流混合逻辑表达等技术手段,减少了整数变量的数目和算法的搜索空间,使得模型的求解更容易。
这个系统结构中的膜分离单元也采用了上一章给出的卷式膜组件模型,在系统优化的同时也对膜组件的设计进行优化,设计的结果可以确定最优的压力容器排布方式,每个压力容器中膜元件的个数以及元件的选型。
反渗透系统的优化设计问题最终表达为一个混合整数非线性规划(MINLP)。当输入的水源发生变化或输出的水质需求不同时,可通过改变多级反渗透系统的结构,选择恰当的操作条件和膜组件的型号,恰当的进料水源搭配,进料的位置,获得具有最大经济效益的产品水。这种过程综合优化的方法可以用较低的成本,实现水资源的合理配置和有效利用,充分发挥了水资源的多种功能。
4.膜组件的定期清洗是防治膜污染的主要措施之一。本文在反渗透海水淡化过程的模拟和工艺设计基础上,进一步对一级反渗透系统的清洗时机问题进行了研究。本文以规划期内总操作费用最小为目标,提出了新的污染模型,给出了清洗的判据,建立了清洗规划模型,将清洗策略问题表达为一个混合整数非线性规划(MINLP)。用这种方法对一个一级反渗透系统的操作、清洗维护进行优化,计算结果给出了该系统在五年规划期内的最优操作和清洗策略,最优单元数为两个单元。在不同的反渗透系统中,最优单元数依赖于清洗的固定费用,每个膜组件的清洗费用,以及整个反渗透系统膜组件的数量。实例计算结果表明,这种策略对于优化寿命为3-5年的膜组件的清洗维护是有效的。
5.在工艺优化和操作优化的基础上,对考虑膜组件清洗及更换的反渗透系统的优化设计进行了研究。由于影响一个反渗透系统正常运行的主要因素是膜污染,它引起的操作压力上升、水质恶化,从而使操作费用大增。本文在设计阶段考虑了膜组件的清洗和更换对整个反渗透系统的工艺设计和运行操作的影响,采取了以下步骤对反渗透过程进行设计:1)给出过程单元数学模型,用这些模型模拟反渗透过程,预测不同条件下操作参数对过程的影响;2)建立系统模型对反渗透结构进行描述;3)建立描述膜组件污染的数学模型,提出膜组件清洗和更换的判据,给出规划期内膜组件的最优清洗和更换策略。4)给出相关的经济模型,将投资费用、操作费用和设计变量相关联。5)在前四步的基础上以年费用最小为目标,将设计问题表达为一个混合整数非线性规划(MINLP)。在GAMS集成开发环境中求解以上混合整数非线性规划,可得出最优的设计工艺及操作策略。
按照本文提出的优化设计模型,它假设系统的单元数是个变量,每个单元所包含的膜组件数目也是个变量,反渗透系统中不同单元可根据自身受污染的情况,操作上可进行相互补偿,以满足最终产水量,产水水质的要求。这种设计使得反渗透系统的操作更灵活,消耗的费用更低。对三个实例进行了设计,分别为在设计阶段只考虑膜清洗情况的例子,膜组件使用五年后统一更换。及在设计阶段同时考虑膜清洗和更换的例子。从设计结果来看,得到的吨水费用,能耗低于文献报道值,表明这种优化设计方法是有效的。
The fresh water is an important resource to human. It is becoming increasingly scarcity. Therefore, the problem of fresh water resource should be solved for the sustainable develop in China. One of the more promising alternative is desalination of seawater and brackish water. In the last few year, the reverse osmosis (RO) desalination process have found widespread application in the field of water resource protect and develop. The rapid growth of RO process is because it is able to produce fresh water with lower cost . The other attractive feature of RO process is that the plant design and operation is simplicity and modularity. At present, the technical economical indices of RO don’t reach the public desired value, and the cost of desalination is on the high side. It shows that the current RO system can be improved in some aspects-the RO technology, the engineering design, and the operation etc.
In this paper, the optimization method based on process synthesis has been applied to design the RO system. The reverse osmosis desalination process with multiple-feed and multiple-product required has been studied. The adoption of this approach can provide an economically attractive desalination scheme. It can lead to significant energy and raw-materials saving and provide income from the multiple product water sales. The main contents are extracted and listed below:
1. The simulation and design of RO desalination process has been investigated. From the beginning, the detailed mathematical models of each process units are given, and the relevant economic models to the RO desalination process are introduced. Then the interdependence of the above mathematical models via some corresponding variables constitutes a system model. Single-stage and two-stage processed are considered. The membrane module can be classed four type, i.e. higher salt-rejection membrane, high-flux membrane, normal membrane, lower salt-rejection membrane. Finally, the optimum design problem can be solved as a mixed-integer non-linear programming (MINLP), which incorporates thermodynamic, technical and flexibility constraints and minimizes the total annualized cost. The different optimal design options are obtained for different design specification and feed water quality and the corresponding design rule are derived. Two examples in different cases are solved to illustrate the advantage and effectiveness of the new method.
2. The design of various multistage RO systems under different feed concentration and product specification is presented in this work. An optimization method using the process synthesis approach to design an RO system has been developed. First, a simplified superstructure that contains all the feasible design in present desalination process has been presented. It offers extensive flexibility towards optimizing various types of RO system and thus may be used for the selection of the optimal structural and operating schemes. A pressure vessel model that takes into account the pressure drop and concentration changes in the membrane channel has also been given to simulate multi-element performance in the pressure vessel. The solution to the problem includes optimal arrangement of the RO modules, pumps, energy recovery devices, the optimal operating conditions, and the optimal selection of types and number of membrane elements.
3. The reverse osmosis (RO) desalination process with multiple-feed and multiple-product required has been studied in this work. The optimization method based on process synthesis has been applied to design the RO system. The adoption of this approach can provide an economically attractive desalination scheme. Membrane separation units employing the spiral wound reverse osmosis elements were described by using the pressure vessel model presented in this paper, which takes into account the pressure drop and concentration changes in the membrane channel. A simplified superstructure that contains all the feasible design for this desalination problem has also been presented. In this structural representation, the stream split ratios and logical expressions of stream mixing were employed which can make the mathematical model to be easily handled. The solution of the problem includes the optimal system structure and operating conditions, and the optimal streams distribution. The design method could also be used for the optimal selection of the types of membrane elements in each stages and the optimal number of membrane elements in each pressure vessel.
4. In the reverse osmosis desalination process, the membrane fouling and scaling is a big problem that must be solved. The regular cleaning is one of main measures that prevent and cure the membrane fouling and scaling. Cleaning can eliminate the surfacial contamination of membranes and recover the water flux. On the condition that the water production is ensured, the optimal cleaning frequency and the minimum cleaning cost can be obtained by choosing the appropriate cleaning strategy and time. This paper makes a more deep research on the problem of membrane module cleaning and replacing base on the previous study, and addresses the new fouling model and the criterion of cleaning. The mathematical formulation of the optimum design problem is a mixed-integer non-linear programming (MINLP), which targets the minimum total operation cost during the whole project period. One example was solved and the corresponding optimal time and cleaning strategy was acquired during the five-year project period, which illustrates the effectiveness of the new method for the optimal maintenance cost of the 3-5 year life-span membrane modules.
5. Based on the studies of technology and operation optimization, this paper made a more deep research into optimization design of reverse osmosis seawater desalination system. In the design process, the optimal cleaning and replacing schedule of membrane module during the five-year project period is consided simultaneously. RO system process design is carried out in the following steps: 1) The mathematical model of each operation unit is given and used to simulate and predict the effects in different conditions of operation; 2) RO system model is built up to describe the overall RO process; 3) RO system model describing the pollution of membrane module is built up and the criterion of membrane washing and repairing are presented; 4) The economic model of RO system is given, which correlates the investment and operation cost via some design variables; 5) Based on the above 4 steps, the RO process is designed using the mathematical programming method. Two examples in different cases are solved to illustrate the advantage and effectiveness of the new method.
1.对反渗透海水淡化过程进行了模拟和设计研究,首先对过程的每个单元给出了单元操作模型和相关的经济模型。通过一定的变量将这些模型相互关联,组成系统模型。系统模型主要考虑了一级流程,一级两段流程和二级流程。以年费用最小为目标,对反渗透海水淡化系统进行设计,将设计问题表达为一个混合整数非线性规划。将膜组件分为高脱盐膜,高通量膜,普通膜,低脱盐膜四种类型。当产品水的设计要求不同,进料水浓度不同时,采用本文的设计方法分别得出了不同的最优设计方案。当进料水浓度逐渐降低时,所用膜的类型逐渐由高脱盐膜转变为高通量膜,低脱盐膜,这些膜的脱盐率降低,水通量增大,这说明了进料水浓度高时,脱盐是主要矛盾,进料水浓度低时,水通量是主要矛盾。
当进料水浓度较高,反渗透系统需要较高的操作压力,能量消耗大,因此换能器效率对年费用影响较大。如能量回收效率较高时,采用结构简单的一级流程;进料水浓度低时,能量费用在年费用中占的比例小,投资费用对年费用的影响较大,采用水回收率高,膜组件的选用合理的二级流程。这个方法可用于海水、苦咸水淡化的工程设计,也可用于高浓度含盐废水的脱盐工艺设计。本文对一个实例进行了设计,得到的吨水费用,能耗低于文献报道值,结果合理,表明这种方法是有效的。
2.本文对在不同进料水水质条件下,不同产品水需求时,反渗透系统的优化设计进行了研究。提出了一种系统化的、集成化的方法对反渗透系统进行设计。首先,采用状态空间法给出一个网状的超结构描述,这个超结构包括了所有可行的设计方案。它和其它结构相比,不同之处在于:1).假定反渗透网络只有一个进料水入口,进料水从第一个反渗透级进入网络。2).将物流分布集合简化为物流混合节点。淡水、盐水物流在增压级进行混合,在反渗透级进行分离。3).超结构数学模型中采用了物流分率,等压混合约束等技术手段,简化了模型中物流混合、分离的数学描述。4).反渗透网状超结构的描述中以功压交换器作为能量回收装置。
这个超结构数学模型中采用了物流分率,等压混合约束等技术手段,减少了整数变量的数目和算法的搜索空间,使得模型的求解更容易。当进料水水质发生变化时,可通过改变多级反渗透系统的结构,选择恰当的操作条件和膜组件的型号,获得具有最大经济效益的产品水。
本文同时给出改进的卷式膜组件模型。该模型考虑了膜元件中的压降和浓差极化,可以近似地描述膜分离单元中压力容器的行为。由此就可以将系统中膜组件的设计和整个反渗透系统的设计结合起来,同时进行优化。设计的结果可以确定最优的压力容器排布方式,每个压力容器中膜元件的个数以及元件的选型。反渗透网络的优化设计问题最终可表达为一个混合整数非线性规划(MINLP),它以反渗透系统年费用最小为目标,满足过程热力学、单元操作、设计要求的约束。提出的系统化的反渗透网络优化设计方法被应用到海水、苦咸水淡化实例中。结果表明,采用最优化设计方案可以明显地降低单位产品水的费用,同时这种设计方法可以加深对整个反渗透系统的了解,适合应用于实际的反渗透淡化工程设计中。
3.在综合考虑了多种水源合理配置和有效利用的基础上,根据反渗透淡化技术的特点,提出了同时考虑多水质输入、多水质输出的反渗透淡化系统的设计问题。该系统不仅可以将海水转变成淡水,而且也可以同时用来处理苦咸水和工业废水,提高了水的利用率。同时根据不同的用途,输出多种水质,充分发挥了水资源的多种功能。本文采用过程综合优化的方法,对多水质输入、多水质输出的反渗透系统的优化设计进行了研究。首先,在前面提出的网状超结构的基础上,给出这个问题的超结构描述。改进之处在于:该反渗透网络由Nps个增压级和NRO个反渗透级构成。网络中有四套物流节点,分别是Nw个进料物流节点;Nps个物流混合节点;NRO个反渗透单元节点;Np个产品水节点。这个结构描述简单,易于表达。在它的数学模型表达中采用了物流分率,物流混合逻辑表达等技术手段,减少了整数变量的数目和算法的搜索空间,使得模型的求解更容易。
这个系统结构中的膜分离单元也采用了上一章给出的卷式膜组件模型,在系统优化的同时也对膜组件的设计进行优化,设计的结果可以确定最优的压力容器排布方式,每个压力容器中膜元件的个数以及元件的选型。
反渗透系统的优化设计问题最终表达为一个混合整数非线性规划(MINLP)。当输入的水源发生变化或输出的水质需求不同时,可通过改变多级反渗透系统的结构,选择恰当的操作条件和膜组件的型号,恰当的进料水源搭配,进料的位置,获得具有最大经济效益的产品水。这种过程综合优化的方法可以用较低的成本,实现水资源的合理配置和有效利用,充分发挥了水资源的多种功能。
4.膜组件的定期清洗是防治膜污染的主要措施之一。本文在反渗透海水淡化过程的模拟和工艺设计基础上,进一步对一级反渗透系统的清洗时机问题进行了研究。本文以规划期内总操作费用最小为目标,提出了新的污染模型,给出了清洗的判据,建立了清洗规划模型,将清洗策略问题表达为一个混合整数非线性规划(MINLP)。用这种方法对一个一级反渗透系统的操作、清洗维护进行优化,计算结果给出了该系统在五年规划期内的最优操作和清洗策略,最优单元数为两个单元。在不同的反渗透系统中,最优单元数依赖于清洗的固定费用,每个膜组件的清洗费用,以及整个反渗透系统膜组件的数量。实例计算结果表明,这种策略对于优化寿命为3-5年的膜组件的清洗维护是有效的。
5.在工艺优化和操作优化的基础上,对考虑膜组件清洗及更换的反渗透系统的优化设计进行了研究。由于影响一个反渗透系统正常运行的主要因素是膜污染,它引起的操作压力上升、水质恶化,从而使操作费用大增。本文在设计阶段考虑了膜组件的清洗和更换对整个反渗透系统的工艺设计和运行操作的影响,采取了以下步骤对反渗透过程进行设计:1)给出过程单元数学模型,用这些模型模拟反渗透过程,预测不同条件下操作参数对过程的影响;2)建立系统模型对反渗透结构进行描述;3)建立描述膜组件污染的数学模型,提出膜组件清洗和更换的判据,给出规划期内膜组件的最优清洗和更换策略。4)给出相关的经济模型,将投资费用、操作费用和设计变量相关联。5)在前四步的基础上以年费用最小为目标,将设计问题表达为一个混合整数非线性规划(MINLP)。在GAMS集成开发环境中求解以上混合整数非线性规划,可得出最优的设计工艺及操作策略。
按照本文提出的优化设计模型,它假设系统的单元数是个变量,每个单元所包含的膜组件数目也是个变量,反渗透系统中不同单元可根据自身受污染的情况,操作上可进行相互补偿,以满足最终产水量,产水水质的要求。这种设计使得反渗透系统的操作更灵活,消耗的费用更低。对三个实例进行了设计,分别为在设计阶段只考虑膜清洗情况的例子,膜组件使用五年后统一更换。及在设计阶段同时考虑膜清洗和更换的例子。从设计结果来看,得到的吨水费用,能耗低于文献报道值,表明这种优化设计方法是有效的。
The fresh water is an important resource to human. It is becoming increasingly scarcity. Therefore, the problem of fresh water resource should be solved for the sustainable develop in China. One of the more promising alternative is desalination of seawater and brackish water. In the last few year, the reverse osmosis (RO) desalination process have found widespread application in the field of water resource protect and develop. The rapid growth of RO process is because it is able to produce fresh water with lower cost . The other attractive feature of RO process is that the plant design and operation is simplicity and modularity. At present, the technical economical indices of RO don’t reach the public desired value, and the cost of desalination is on the high side. It shows that the current RO system can be improved in some aspects-the RO technology, the engineering design, and the operation etc.
In this paper, the optimization method based on process synthesis has been applied to design the RO system. The reverse osmosis desalination process with multiple-feed and multiple-product required has been studied. The adoption of this approach can provide an economically attractive desalination scheme. It can lead to significant energy and raw-materials saving and provide income from the multiple product water sales. The main contents are extracted and listed below:
1. The simulation and design of RO desalination process has been investigated. From the beginning, the detailed mathematical models of each process units are given, and the relevant economic models to the RO desalination process are introduced. Then the interdependence of the above mathematical models via some corresponding variables constitutes a system model. Single-stage and two-stage processed are considered. The membrane module can be classed four type, i.e. higher salt-rejection membrane, high-flux membrane, normal membrane, lower salt-rejection membrane. Finally, the optimum design problem can be solved as a mixed-integer non-linear programming (MINLP), which incorporates thermodynamic, technical and flexibility constraints and minimizes the total annualized cost. The different optimal design options are obtained for different design specification and feed water quality and the corresponding design rule are derived. Two examples in different cases are solved to illustrate the advantage and effectiveness of the new method.
2. The design of various multistage RO systems under different feed concentration and product specification is presented in this work. An optimization method using the process synthesis approach to design an RO system has been developed. First, a simplified superstructure that contains all the feasible design in present desalination process has been presented. It offers extensive flexibility towards optimizing various types of RO system and thus may be used for the selection of the optimal structural and operating schemes. A pressure vessel model that takes into account the pressure drop and concentration changes in the membrane channel has also been given to simulate multi-element performance in the pressure vessel. The solution to the problem includes optimal arrangement of the RO modules, pumps, energy recovery devices, the optimal operating conditions, and the optimal selection of types and number of membrane elements.
3. The reverse osmosis (RO) desalination process with multiple-feed and multiple-product required has been studied in this work. The optimization method based on process synthesis has been applied to design the RO system. The adoption of this approach can provide an economically attractive desalination scheme. Membrane separation units employing the spiral wound reverse osmosis elements were described by using the pressure vessel model presented in this paper, which takes into account the pressure drop and concentration changes in the membrane channel. A simplified superstructure that contains all the feasible design for this desalination problem has also been presented. In this structural representation, the stream split ratios and logical expressions of stream mixing were employed which can make the mathematical model to be easily handled. The solution of the problem includes the optimal system structure and operating conditions, and the optimal streams distribution. The design method could also be used for the optimal selection of the types of membrane elements in each stages and the optimal number of membrane elements in each pressure vessel.
4. In the reverse osmosis desalination process, the membrane fouling and scaling is a big problem that must be solved. The regular cleaning is one of main measures that prevent and cure the membrane fouling and scaling. Cleaning can eliminate the surfacial contamination of membranes and recover the water flux. On the condition that the water production is ensured, the optimal cleaning frequency and the minimum cleaning cost can be obtained by choosing the appropriate cleaning strategy and time. This paper makes a more deep research on the problem of membrane module cleaning and replacing base on the previous study, and addresses the new fouling model and the criterion of cleaning. The mathematical formulation of the optimum design problem is a mixed-integer non-linear programming (MINLP), which targets the minimum total operation cost during the whole project period. One example was solved and the corresponding optimal time and cleaning strategy was acquired during the five-year project period, which illustrates the effectiveness of the new method for the optimal maintenance cost of the 3-5 year life-span membrane modules.
5. Based on the studies of technology and operation optimization, this paper made a more deep research into optimization design of reverse osmosis seawater desalination system. In the design process, the optimal cleaning and replacing schedule of membrane module during the five-year project period is consided simultaneously. RO system process design is carried out in the following steps: 1) The mathematical model of each operation unit is given and used to simulate and predict the effects in different conditions of operation; 2) RO system model is built up to describe the overall RO process; 3) RO system model describing the pollution of membrane module is built up and the criterion of membrane washing and repairing are presented; 4) The economic model of RO system is given, which correlates the investment and operation cost via some design variables; 5) Based on the above 4 steps, the RO process is designed using the mathematical programming method. Two examples in different cases are solved to illustrate the advantage and effectiveness of the new method.
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