CFD-based genetic programming model for liquid entry pressure estimation of hydrophobic membranes
Graphical abstract
Introduction
More than two-billion individuals all over the world do not have access to safe drinking water and this number is prone to increase as a result of population growth, increased living standards, and development of industrial and agricultural activities [1,2]. One of the methods which can mitigate water shortage is to increase freshwater production via desalination of saline waters [3]. Seawater and saline aquifer sources represent 97.5% of all water on Earth. Hence, treating even a small portion of saline water could significantly reduce water shortage [4]. Although reverse osmosis (RO) is one of the state-of-art pressure driven membrane desalination technologies, it is incapable of desalinating high-salinity streams (70–300 g salt/kg solution) due to the very high osmotic pressure to overcome and the only means of treating hypersaline waters is desalination by thermal evaporation [5,6]. Membrane distillation (MD) is one of the emerging methods, which has attracted much attention for desalinating highly saline brines [7].
MD is a thermally driven process in which only vapor molecules pass through the pores of a microporous hydrophobic membrane [8]. This process, however, has not been fully commercialized due to a number of challenges, including “pore wetting” [[9], [10], [11], [12], [13]]. Pore wetting refers to the penetration of liquid feed, instead of just water vapor, into the membrane pores, causing a significant decrease in MD flux [[14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]] and/or deterioration of distillate quality [10,11,28,29]. Liquid entry pressure (LEP), as a parameter showing wetting characteristic of a membrane, can be defined as the minimum transmembrane pressure which causes liquid feed to overcome the hydrophobic forces and enter the pores [14]. It is noteworthy that the Smolder's procedure is the experimental procedure applied in most cases for the determination of LEP [30]. Fig. 1 illustrates a schematic of a setup used for LEP measurement in which the membrane sample is mounted in the cell filled with water and the pressure is increased gradually using nitrogen gas. The pressure at which the first droplet of water appears on the permeate side of the membrane is recorded as LEP [31,32].
Since exceedance of LEP is one of the principal reasons of wetting phenomena [28], precise estimation of LEP is essential in order to assess the membrane wetting potential. Hitherto, several models have been presented to estimate LEP. The Young-Laplace model is the classic model for estimation of LEP for cylindrical pores, given as:where σ (N/m), θ (o) and rmax (m) are surface tension, contact angle between wetting liquid and membrane, and maximum pore radius, respectively. Franken's model [33], which considers an additional parameter, B (−), as a pore geometry coefficient, is shown in Eq. (2).where the value of B depends on the shape of the pore cross-section and for a circular cross-section pore it is equal to unity.
Kim et al. [34] developed a model for LEP, in which membranes were considered as arrays of uniform fibers which intersect at constant angles. In fact, the pores are described as tori instead of cylinders that are assumed in the Young-Laplace model (see Fig. 2) [35]. They presented Eq. 3 as the difference of pressure across the liquid–gas interface at the membrane surface [36]:where σ (N/m) is surface tension, θ (o) is contact angle and r (m) is pore radius. Furthermore, R (m) and θeff (o) are radius of fiber and effective contact angle, respectively. α (o) is the structural angle which accounts for the axial deviation of the pores and can be obtained using the following equation [36]:
As a drawback for this model, the effective contact angle (θeff) cannot be lower than 90o even for materials possessing low contact angle near zero and based on this model, spontaneous wetting occurs when contact angle is below 90o and r/R is very small, which contradicts the experimental results that showed the occurrence of spontaneous wetting at a contact angle of greater than zero, regardless of r/R [36].
Servi et al. modified Kim et al.'s model to improve its ability for prediction of the experimental data by introducing a new parameter, h, the distance between the floor and the bottom of the fibers (see Fig. 3) [35]. Their model accounts for interactions between the pores below the initially wetted surface and the liquid [35]. However, in this model estimation of r/R is difficult. Moreover, this model was tested only for fibrous nylon membrane and the effect of membrane thickness on LEP was neglected in all previous equations.
In order to predict LEP precisely, recently a model was developed using computational fluid dynamics (CFD) tools and validation was carried out using experimental LEP data [37]. The unique feature of the model is that the effect of membrane thickness was considered. Although this model allows very accurate prediction of LEP, it is impossible to put the model into few equations since it is based on CFD. If a mathematical formula can be built allowing the same prediction as that of the CFD, it would be beneficial for the user of the model.
Genetic programming (GP), as a genetic algorithm branch, has been proved powerful to capture the nonlinear relationship existing among the variables in different fields [38]. The principal idea behind the GP is evolution theory according to which the population is improved progressively by disregarding the not-so-fit population and generating new children from better populations [39]. GP has been used for modeling of membrane processes in a few studies. Okhovat et al. [40] employed GP to deliver a mathematical function for the membrane rejection of nanofiltration process, considering the influence of feed concentration as well as transmembrane pressure. The proposed model exhibited a satisfactory accuracy and proved the ability of GP as a powerful prediction tool for the nanofiltration process. Fouladitajar et al. [41] compared the GP model to the individual and combined blocking laws for assessing membrane fouling of the microfiltration process with oil-in-water emulsion. As a result, the process was predicted more accurately by the GP model. Suh et al. [42] developed a model in order to estimate the membrane damage degree by applying GP. The model predicted satisfactorily the area of membrane breach as a function of mass of the permeated particle, particle concentration, transmembrane pressure, and permeate water flux. Shokrkar et al. [43] used GP for estimating the membrane flux in the separation process of oily wastewaters by ceramic microfiltration membranes. Moreover, GP has been applied for estimation of pervaporation separation index with high accuracy (R2 = 0.9722) [44].
The objective of this work is to present a single equation by which LEP can be predicted under various conditions. It should be noted that the effect of membrane thickness on LEP is predicted for the first time by a single equation in this work. To this end, the CFD tool is combined with the GP model for the evaluation of LEP. In this attempt, LEP modeling is carried out by the CFD model for different sets of variables including contact angle, pore radius and membrane thickness. Then, an explicit formula is generated by applying GP to reproduce the results of CFD model as accurately as possible. Moreover, after the validation of the newly developed formula, the effect of different parameters on LEP is assessed.
Section snippets
CFD model
As mentioned, the LEP values were calculated for the sets of variables (contact angle, pore radius and membrane thickness) by the CFD model, the details of which are given in the literature [37]. For this purpose, a two dimensional CFD model was developed by applying the volume of fluid (VOF) approach with the following assumptions:
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The pore structure is straight and cylindrical.
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The pore tortuosity is negligible.
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Heat does not transfer through the pore wall.
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Water and air are considered
Model development
A two dimensional solver mesh was generated through an automated mesh operation in Star-CCM+ software and optimization of the mesh size was performed by changing the mesh size gradually until LEP no longer was changed. Fig. 8 illustrates the mesh structure of a pore as well as the schematic of the CFD model geometry for a single pore. The feed pressure was increased linearly with a constant rate of 10−8 bar s−1 and as soon as the liquid entered the pore, the way of pressure increase was changed
Conclusion
In this study, a new equation was presented for estimation of LEP using a CFD-based GP model. After validation of the model, the influence of effective parameters including contact angle, pore radius and membrane thickness on LEP was discussed. Moreover, the difference between the estimated LEP using the newly proposed model and Young-Laplace model at different conditions was studied. Considerable differences in the LEP values were noticed between these two models. In particular, even though
Abbreviations
- CFD
Computational fluid dynamics
- GP
Genetic programming
- LEP
Liquid entry pressure
- MD
Membrane distillation
- I
Internal nodels
- RO
Reverse osmosis
- T
Terminal nodes
- VFW
Volume fraction of water
- VOF
Volume of fluid
CRediT authorship contribution statement
Hooman Chamani: Conceptualization, Methodology, Software, Validation, Investigation, Writing - original draft, Writing - review & editing, Visualization. Pelin Yazgan-Birgi: Methodology, Software, Investigation, Writing - original draft. Takeshi Matsuura: Supervision, Writing - review & editing. Dipak Rana: Supervision. Mohamed I. Hassan Ali: Supervision. Hassan A. Arafat: Supervision, Writing - review & editing. Christopher Q. Lan: Supervision.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors greatly acknowledge the support of Natural Sciences and Engineering Research Council of Canada (NSERC) individual discovery grant RGPIN-2014-03753. The work at Khalifa University was supported through the Center for Membrane and Advanced Water Technology (CMAT), under Award No. RC2-2018-009.
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