Predicting the performance of reverse osmosis membranes using optimization, mathematical modeling and solving the model with the help of numerical methods

Academic Thesis for receiving a master's degree in environmental chemical engineering

Abstract

Among the models that are able to make successful predictions for the performance of reverse osmosis membranes, that is, the separation rate and flow flux in a wide range of physical conditions (membrane and solution) and laboratory conditions (pressure and concentration), the modification model surface force-cavity flow model and its more complete state i.e. modified generalized model of surface force-cavity flow.  In this research, efforts have been made to find the potential function and the friction function, which are two very important and effective functions in the mentioned models as well as other models of porous membranes, and valuable and new results have been obtained in this field. Examining the results showed that the best model for predicting the laboratory data is the proposed model with potential and friction functions, i.e. the Ex-P3-F3 model with the value of the target function equal to 0.0335. On the other hand, the investigation of the effect of the potential function on the results of the modified generalized model of surface force - pore flow, showed that in these conditions, replacing the proposed potential function in the mentioned model, the Ex-P4-F1 model is obtained, which, having the lowest value of the target function equal to 0.0337, predicts the laboratory data well.  On the other hand, today, due to the progress of mathematics and the invention of new numerical solution methods such as the finite difference method and the finite volume method, it has become possible to solve complex and non-linear equations. In this research, using these methods, the non-linear equations of speed and concentration governing the aforementioned models have been solved and finally, the performance of reverse osmosis membranes has been predicted based on these models.

Key words: reverse osmosis membranes, modified generalized model of surface force - pore flow, potential function, friction function, numerical method.

-1- Introduction

In the late 1950s, Reed and Britton [1] at the University of Florida discovered that the membrane made of Cellulose acetate has the ability to remove salt. The flow of water passing through this dense membrane [2] was so low that it was impossible to use it in practice [1 and 2].

The turning point of membrane phenomena can be considered to be around 1958, when the first asymmetric cellulose acetate membrane was made at the University of Los Angeles California (UCLA).

Leob and Surirajan[3] was able to produce an asymmetric membrane of cellulose acetate by making a thin layer (0.1-1.0 µm thick) on a porous structure behind it]1 [.

Cellulose acetate membranes have two major limitations. First, they are prone to attack by biological substances, in which case the incoming water must be chlorinated. Secondly, under different acidic and basic conditions, Cellulose acetate is hydrolyzed to acetate, so the pH of the system must be controlled between 5.4 and 5.7.

One ??of the ways to develop asymmetric membranes was to produce the shell layer and the porous protective layer separately and then somehow put them on top of each other. In a new type of membrane, called thin film composite [4] (TFC), this process is performed in two steps. First, the thick layer is covered with a layer consisting of very small holes[5] and then a thin layer is covered over it. One of the merits of this method is that it is possible to optimize the performance of each layer separately. Nowadays, these types of membranes (TFC) are economically very affordable and have high performance [4].

1-2- Introduction of the topic

There are different points of view in membrane research. The science of modeling tries to describe and predict the performance of membranes at a higher level, and in this way, both to expand the knowledge related to the transfer mechanism and to use this knowledge in the design of membranes and reverse osmosis devices. Materials science, using physical and chemical criteria, strives to accelerate membrane process technology by preparing and synthesizing newer membranes. Process design researches in the field of general design and optimization of reverse osmosis systems on an industrial scale [6, 7].

Among this, modeling and predicting the performance of reverse osmosis membranes [6] is particularly important. In order to adequately describe the performance of reverse osmosis membranes, very accurate and suitable mathematical models are needed. This need has led to the development of several transport models.

The general purpose of transport models is to relate membrane performance, which is usually expressed in terms of separation (the fraction of solute that is separated from the feed) and flux, to operating conditions (such as pressure and feed concentration) or driving force (which are usually concentration and pressure gradients), by coefficients (known as transport coefficients and including model parameters). to give Coefficients or parameters should be determined from laboratory data. The complete success of a model means the power of that model to mathematically describe data with coefficients (or parameters) that remain constant within the range of practical conditions (parameters independent of driving forces). Finally, the model with specific transfer coefficients (or parameters) can predict the performance of a membrane in a wide range of operating conditions. This performance prediction power is the real power of the transmission model. The transfer model can sometimes eliminate high experimental costs, or it can be coupled with a research program in membrane manufacturing to lead to better design criteria, or it can be used together with a process design program and lead to a logical industrial scale for the reverse osmosis system [1]. So far, many models have been presented to express the performance and also predict the performance of reverse osmosis membranes, and some of them have had relatively good success. Over time, these models have become more complete and provided better results by clarifying the new facts about the separation mechanism and the relationships governing the model [8]. Among the models that have provided good results for describing reverse osmosis processes with very high accuracy, there is a model called the modified surface force-pore flow model [7] or the MD-SF-PF model and its more complete model, i.e. Ex-MD-SF-PF model [8], both of which were presented by Mahdizadeh and his colleagues] 10-9[.

The beauty of these two models is that they have been able to predict the performance of membranes in different operating conditions with high accuracy based on a physical mechanism, in addition to determining the relationships governing reverse osmosis processes.

The main problem in using these models Solving their nonlinear differential equations. To solve the equations of these two models, the numerical method of orthogonal superposition [9] has been used [11].

In this research, first, several friction and potential functions were evaluated and selected from among the various functions, and after replacing them in the Ex-MD-SF-PF model with the help of programming in Matlab software and optimization, the ability and power of the new models in predicting the performance of reverse osmosis membranes has been examined, which can be seen later in the results section. It was found that not only the new models are able to predict the performance of reverse osmosis membranes, but sometimes some of the presented models have better results than the MD-SF-PF and Ex-MD-SF-PF models.

On the other hand, the nonlinear equations governing the model were solved using finite difference [10] and finite volume [11] numerical methods, and then the results obtained for the membrane performance were compared with the laboratory results.

Abstract:

In order to explain and predict the behavior of reverse osmosis membranes and also design of reverse osmosis operation units, a good understanding of principals of transport in membranes is inevitable.

This means that a powerful and efficient transport model has to be prepared.