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Direct contact membrane distillation: An experimental and analytical investigation of the effect of membrane thickness upon transmembrane flux
Journal of Membrane Science 470 (2014) 257–265
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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Direct contact membrane distillation: An experimental and analytical investigation of the effect of membrane thickness upon transmembrane flux Ho Yan Wu a, Rong Wang b, Robert W. Field a,n a b
University of Oxford, Department of Chemical Engineering, UK Singapore Membrane Technology Centre, Nanyang Technological University, Singapore
art ic l e i nf o
a b s t r a c t
Article history: Received 9 January 2014 Received in revised form 30 May 2014 Accepted 2 June 2014 Available online 9 July 2014
Polyvinylidene fluoride (PVDF) electrospun nanofibrous membranes (ENMs), consolidated with a heatpress process and of various thicknesses were fabricated and tested in a DCMD cell at five different operating conditions. Membranes as thin as 27 μm were sufficiently robust for evaluation and gave a transmembrane flux as high as 60 L/h m2. An analytical model was created to estimate the optimal membrane thickness for DCMD operations. It is found that the value of optimal thickness increases with reduced heat transfer coefficients; decreased feed inlet temperature; increased membrane permeability; and increased salinity. Even for 10% NaCl the predicted optimum was estimated to be 13 μm which was too thin for experimental confirmation. Based upon this analysis but with due allowance for the variation of heat transfer coefficients with temperature dependent physical properties a single Matlab model was created to fit the five sets of data. With the introduction of a structural derivation factor, which reflected the experimentally determined variation of porosity and pore size with thickness, the model was found to fit the data very well. & 2014 Elsevier B.V. All rights reserved.
Keywords: Membrane distillation Electrospun membranes Structural parameter Effect of thickness
1. Introduction Membrane distillation (MD) has become a popular research area since it has the potential to tackle the shortage of water while using a relatively small amounts of high grade energy. MD is a thermal driven process that can be coupled with solar thermal systems to produce fresh water from brackish or seawater. Direct contact membrane distillation (DCMD) is the most popularly explored configuration due to its simple design that makes it particularly attractive and suitable for application in rural and less developed areas. One of the major disadvantages of MD is its relatively low transmembrane flux. A number of membrane parameters, including membrane material, porosity, tortuosity, pore size and distribution, are factors that affect the production rate in DCMD operation [1]. Most commercial MD membranes available are made of polytetrafluoroethylene (PTFE) because of its very low surface energy [2]. However PTFE membranes require complicated extrusion, rolling and stretching or sintering procedures. Polypropylene (PP) membranes prepared by phase inversion has also been used for MD membranes [3]. While PTFE and PP membranes
n
Corresponding author. E-mail address: robert.fi[email protected] (R.W. Field).
http://dx.doi.org/10.1016/j.memsci.2014.06.002 0376-7388/& 2014 Elsevier B.V. All rights reserved.
exhibit good hydrophobicity and promising MD performance, the structural parameters of the membranes produced are difficult to control. In recent years, the electrospinning technique has been explored extensively to produce nano-fibrous membranes. Previous research studies have shown that electrospun hydrophobic membranes are suitable for DCMD application [4–7]. Being hydrophobic with a good thermal resistivity and readily dissolvable at room temperature in a variety of solvents, poly(vinylidene fluoride) (PVDF) in particular has been adopted for the production of electrospun nano-fibrous membranes for MD applications [13]. Although the production rate from a conventional single needle electrospinning system is very low (which makes it difficult to employ this process in industrial settings) techniques which promote multiple jetting have been developed for large production of electrospun mats from polymer solutions [8]. Membrane thickness plays a significant role in DCMD systems. With a thicker membrane, less heat is conducted away from the feed side to the permeate side, giving a higher difference in transmembrane temperature which results in a higher driving force. However the membrane structure reduces the ease of permeation thus impeding the transmembrane flux. Since the membrane thickness is inversely proportional to both the rate of mass and heat transfer across the membrane, provided other membrane parameters remain the same, many researchers suggested the thickness of the single layer membrane might be
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expected to have an optimal value [9,10]. Previously for DCMD configurations, experiments have been carried out using available commercial membranes with pure water as feed and for various membrane thicknesses. For similar pore size and porosity, thicker membranes resulted in flux reduction [11,12]. Gostoli et al. [13] reported that for thin membranes, the transmembrane flux depends on the salt concentration whilst for thicker membranes, salt concentration does not play an important role for the flux; Song et al. [14] found that salt concentrations up to 10% led to only small flux reduction. Schofield showed that with the membranes he used, the permeability of the membrane was virtually independent of membrane thickness [15]. Using a computer simulated counter current hollow fibre DCMD module, Laganà suggested that the optimal thickness of a single hydrophobic layer membrane lies between 30 and 60 μm [10]. Whilst Al-Obaidani et al. identified an optimal thickness value of 700 μm for his model when operating with a low temperature gradient ( o5 1C) [16]. In this paper, the emphasis is upon the effect of membrane thickness and assessing whether there is an optimal value both analytically and experimentally. The membranes were fabricated using electrospinning and the thinnest membrane used in the experiment is about 27 μm. All membranes were fragile and thinner membranes fractured inside the module before water production rates could be determined. The experimental data collected is compared with model simulations based upon an analytical model that indicated an optimal thickness much thinner than that which could be achieved in practice.
2. Theory and analytical approach To assess optimal thickness, firstly an expression of dN/dδ is obtained where N is the transmembrane flux (kg m 2 s 1) and δ is membrane thickness. For unit area, the heat and mass transfer process of DCMD are generally expressed as Q ¼ hf ðT f T f m Þ
ð1Þ
Q ¼ hp ðT pm T p Þ
ð2Þ
Q ¼ NH þ
km
δ
ðT f m T pm Þ
N ¼ CðP f m P pm Þ
ð3Þ ð4Þ
These are a standard set of equations except that herein the enthalpy change across the membrane, H, is not taken to be independent of temperature. Other terms have their usual meaning and are defined in Nomenclature. Now provided the porosity and pore size do not vary with thickness: C¼
K
δ
ð5Þ
whilst P f m ¼ αw e23:238 ð3841=ðT f m 45ÞÞ
ð6Þ
P pm ¼ e23:238 ð3841=ðT pm 45ÞÞ
ð7Þ
H ¼ mT f m þ g
ð8Þ
where m and g are constants and have the value of 2400 (J/kg K) and 3,200,000 (J/kg). These coefficients were obtained by fitting a line of best fit to the values of specific enthalpy of water phase change given in a standard Oxford Engineering Data book [17], corresponding to water temperature in the range of 293–383 K. The term αw in (6) is activity of water and is less than unity in the presence of salt.
2.1. Determination of theoretical optimal thickness We have referred to theoretical optimal thickness in the subheading as the determined thickness may not be practical in most engineering applications. In Appendix A an equation relating the optimal thickness (the thickness at dN/dδ) to two other unknowns Tfm and Tpm is obtained in terms of known parameters including αw, km and the heat transfer coefficients. A second equation relating Tfm and Tpm can be obtained straight from Eqs. (1) and (2) by eliminating Q. A third equation relating δ and Tfm is obtained from Eqs. (2)–(5). Thus one has three equations and three unknowns one of which is the optimal thickness. Further mathematical details are in Appendix A and results are given later.
3. Experiments 3.1. Fabrication of the electrospun nanofibre membranes (ENMs) 3.1.1. Materials and chemicals Commercial polymer PVDF Kynar HSV 900 was purchased from Arkeme Inc., Singapore. Prior to use, the polymer was dried at 323 K under vacuum condition for at least 24 h. Solvents, acetone and N,N-Dimethylformamide (DMF) were obtained from Fisher, Singapore. Lithium chloride (LiCl) was obtained from Merck, Singapore. 3.1.2. Electrospinning dope solution The PVDF polymer dope solution was prepared by dissolving 5 wt% PVDF into a mixture of DMF and acetone with a weight ratio of 6 to 4. LiCl (0.004 wt% of total spinning dope solution) was added to the solution to enhance the conductivity of the dope. The dope solution was mechanically stirred for 24 h at 333 K and cooled at room temperature for 24 h prior to electrospinning. 3.1.3. Electrospinning of PVDF membranes The electrospinning conditions were set at: 27 kV electric voltage, 1800 mm3/h polymer flowrate, 120 mm between the spinning tip and the top of the rotating drum collector, rotational speed of the drum, with a radius of 38 mm is 2 rpm/s and the horizontal movement of the spin tip is 0.1 mm/s across a length of 80 mm. After electrospinning, the membranes were then dried in an incubator, which was set at 323 K, for 24 h, to ensure that all the solvents were evaporated. For the present work, the recorded electrospun membranes were fabricated by spinning for 4–10 h (Fig. 1). 3.1.4. Heat press post-treatment The dried PVDF membranes were pressed between two flat glass panes, at 423 K, for 1 h. The heat press was operated at just above atmospheric pressure. The melting temperature of PVDF used is around 443 K under ambient pressure.
Fig. 1. Schematic of the electrospinning setup used for membrane fabrication.
H.Y. Wu et al. / Journal of Membrane Science 470 (2014) 257–265
3.2. Characterization of electrospun membrane A scanning electron microscope (SEM) (EVO, 50, Carl Zeiss AG, Singapore) was used to evaluate the fibre size of the membrane and observe the morphology of the membranes. Platinum dust was sprayed onto the membrane in order to charge the surface and cross-section for effective imaging in the SEM. The porosity of the membrane was obtained using a gravimetric method with the membranes being immersed into isopropyl alcohol (IPA) which penetrates into the pores of the membrane. The membrane porosity is calculated using equation
ε¼
ðwwet wdry Þ=ρIPA ðwwet wdry Þ=ρIPA þ wdry =ρPVDF
ð9Þ
The pore size of membranes was determined by a capillary flow porometer (model CFP 1500A, from Porous Material. Inc. (PMI) in Singapore). The contact angles of the membranes were measured using goniometer (Contact Angle System OCA, from Data Physics Instruments GmbH in Singapore). 3.3. DCMD tests The electrospun membranes were tested with a DCMD setup as shown below. The size of the membrane module has an effective membrane area of 3600 mm2. The dimensions of the feed and permeate channels are identical with a length of 90 mm, a width of 40 mm and a depth of 2 mm. While deionized water was used on the permeate side throughout all the tests, 10 wt% NaCl concentration was used as feed solution and the mass flowrates were maintained at 1.5 kg min 1. The membrane module was placed vertically with hot feed water and cold deionized water running counter-currently. Fig. 2 shows the schematic setup of the experiment. Fluxes were determined by collecting permeate from runs that lasted for between 3 and 5 h, while the permeate temperature were kept at 293 K, the feed temperature differ for each set of runs, ranging from 318 to 338 K. At each condition at least three evaluations were made, and as the membranes were fragile new membranes were used for each testing. Fluxes were
259
calculated to an accuracy of better than 75% and therefore the the variation in the fluxes at a given thickness reflects the variation in porosity and pore size that arise naturally from the electrospinning process.
3.4. Model for estimation of performance A counter current DCMD flatsheet membrane module model was constructed with Matlab. The modules were divided into many thin strips and elements. For each element, Eqs. (1)–(4) were solved using Matlab function, fsolve. Outputs from each element including outlet temperature and flowrate, were used as input for the next element. The overall fitting later only had two adjustable parameter for the whole of the data set: one was membrane permeability and the other was the value of the heat transfer at coefficient at 293 K. The flowrates were the same for each channel and invariant and so the heat transfer coefficients would be the same if the temperatures were the same. However temperature changed but from the use of standard correlations, assuming turbulent flow along the channel, one can relate the heat transfer coefficients as the correlations suggest that the ratio of hf and hp should be (μf/μp) 0.5. The values obtained were within the estimates obtained from the data itself using the method given in [18]. Due to the uncertainties of flow geometry, entrance effects and roughness of the membrane surfaces it was difficult to accurately estimate values of hf and hp. Now instead of using common standard correlations found in the literature [19], the heat transfer coefficients were estimated using Eq. (17) in [18]. This was done for each set of data with the same inlet temperatures. The overall MD mass-transfer coefficient (denoted by Keff in [18]) is found from our experimental data and the graph of 1/Keff vs δ plotted. The membrane permeability (denoted by C in [18]) can be evaluated from the gradient of the graph. Then given that the thermal conductivity of the membrane matrix and the latent heat of vaporization are known, the combined heat-transfer coefficient (denoted by UL in [18]) can be obtained from intercept and split into hf and hp. This confirmed that values of around
Fig. 2. Schematic of the DCMD test setup used for membrane evaluation.
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Fig. 3. Graphical determinations of optimal thickness for pure water (A, top); 3.5% NaCl solution (B, middle); and 10% NaCl soltion (C, bottom). Simulation conditions are given in Table 1.
4000 W m 2 K 1 were justifiable for the experimental conditions used.
Table 1 Operation and membrane structural parameters used for simulation. Stimulation conditions
4. Results and Discussions 4.1. Results from analytical analysis In Fig. 3A, the curves obtained from expressions [A3] and [A5] do not meet, which suggests an optimal thickness does not exist in the absence of salinity in the feed. On the other hand, when using aqueous NaCl as feed solution, Fig. 3B and C suggests that optimal thickness does exist, the value of which increases with increasing salinity. With an activity coefficient of 0.98, using seawater as feed, it is estimated that the optimal thickness is about 5 μm, for the chosen simulation conditions as detailed in Fig. 3. When using 10% NaCl as feed solution, which has an activity coefficient of 0.94, it is estimated that for the chosen simulation condition, the optimal thickness is about 13 μm. The parameters values that were chosen for the simulation in Fig. 3 are close to the actual experimental condition (Table 1). Salinity plays an important part in the determination of optimal membrane thickness. The presence of salt reduces the water vapour partial pressure in the feed through the term αw in Eq. (6). Below optimal thickness, as the membrane thickness increases the high transmembrane temperature difference results in a higher, driving force, Pfm Ppm. In this region this increases more rapidly than the membrane resistance to mass transfer which increases linearly with thickness. However, above optimal thickness, the driving force increases less rapidly than the membrane resistance. For any given conditions, the value of membrane thickness corresponding to zero or negative flux is always smaller than the corresponding value for maximum flux. These values are
b hf K km Tf Tp
1 2500 4.5 10 11 0.045 338 293
– W/m2 K kg/m s Pa W/m K K K
too small to be of any practical interest; the thicknesses of commercially available membranes are about 100 μm. Indeed such thicknesses are significantly larger than the optimal thicknesses determined below. Fig. 4(A) shows that the optimal membrane thickness increases by 30% for a reduction in the feed temperature from 338 K to 313 K. A lower Tf results in a lower Tfm. Due to the temperature sensitivity of water vapour partial pressure, in order to achieve an effective driving force across the membrane, when Tf is lowered; a thicker membrane is required to sustain an effective temperature difference; (and hence an effective partial pressure difference) across the membrane. From Fig. 4(B), it can be seen that as the permeability of the membrane increases, the optimal membrane thickness increases. Owing to a higher permeate flux resulting in more heat transfer via convection; a thicker membrane is required to provide the optimal transmembrane temperature difference. Fig. 4(C) and (D) shows that the value of optimal membrane thickness is very sensitive to the heat transfer coefficients. This can be explained by the temperature polarization effect. Temperature polarization refers to the system characteristic wherein the fluid temperatures at the membrane boundaries are different to their
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261
Fig. 4. Variation in optimal thickness with changes in: (A) feed temperature; (B) membrane permeability; (C) heat transfer coefficient; and (D) ratio of permeate heat transfer coefficient to feed heat transfer coefficient. Other simulation parameters remain identical as those demonstrated in Fig. 3 for 10% NaCl solution.
Table 2 Properties of the membranes under different electrospinning times. Electrospinning time (h) Without heat treatment Porosity (%) Ave. fibre diameter (μm)
Ave. pore diameters (μm)
4
6
8
10
92.3 7 0.4 0.147 0.06
90.6 7 0.7 0.147 0.05
90.5 7 0.8 0.157 0.07
89.2 7 1 0.147 0.03
0.337 0.17
0.317 0.17
0.29 7 0.15
0.25 7 0.11
36.4 7 2.5
46.4 7 2.5
64.37 2.5
77.0 7 2.5
139.4 7 1.2
141.0 7 0.5
140.8 71.8
139.0 7 0.3
Ave. fibre diameter (μm)
83.6 7 0.4 0.167 0.05
80.8 7 0.2 0.157 0.04
80.0 7 1 0.157 0.03
80.5 7 0.4 0.167 0.03
0.217 0.09
0.187 0.06
0.177 0.08
0.157 0.07
Structural deviation factor:Φ, given by: porosity pore size/(porosity pore size)@10 h
1.45
1.20
1.13
1
27.9 7 2.5
35.67 2.5
45.7 7 2.5
58.4 7 2.5
130.6 7 1.6
129.9 7 1.5
131.3 7 2.5
130.0 7 2
Thickness (μm) Contact angle (deg) With heat treatment Porosity (%)
Ave. pore diameters (μm)
Thickness (μm) Contact angle (deg)
corresponding bulk temperatures. Due to thermal boundary layer resistance and the relatively large amount of heat required for vaporization of the feed liquid, the heat transfer across the boundary layers is often the major rate determining step. With higher heat transfer coefficients on either or both sides of the membrane, the overall boundary layer resistance is reduced, giving a reduction in the temperature polarization effect. As a result, a thinner membrane would be able to provide enough relative heat transfer resistance for optimal balance between the transmembrane temperature difference (that determines the water vapour pressure difference) and the membrane resistance. Thus in Figure 4(C) as both hp and hf increase simultaneously, where hp ¼bhf (see Appendix) and b ¼1, the optical thickness decreases. Fig. 4(D) shows the change in optimal thickness when only one of the heat transfer coefficients is changed. When the ratio is low (hp is low and the thermal resistance at the permeate side is high), the permeate-side flow is ineffective in the transferring of heat
from the membrane interface to the bulk fluid and so the consequence is that thicker membranes are required to maintain a sufficient transmembrane temperature difference, Tfm Tpm. However the corresponding optimal thickness reduces rapidly as b increases and the steepness of the curve reduces rapidly as b approaches 1. Similar behaviour (dashed line) can be observed if hp is kept constant and only hf is changed. From the steepness of the curve in Fig. 4(D) it is evident that the optimal thickness is relatively more sensitive to the change in hf then changes in hp. 4.2. Characterization of the membranes fabricated and DCMD experiment It is shown in Table 2 that the PVDF membrane properties and characteristics change significantly after the heat-press process. Since the heat is applied at a temperature that is below PVDF's melting temperature and the pressure applied is relatively
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Fig. 7. Experimental data and the estimated production rate of our DCMD experiment for feed temperature of 338 K. Fig. 5. SEM image showing surface morphology of the post-heat pressed membranes; magnification of 10,000.
Fig. 8. Experimental data and the estimated production rate of our DCMD experiment for feed temperature of 333 K.
Fig. 6. SEM image showing cross-section of the post-heat pressed membranes; magnification of 5000.
moderate, the nano-fibrous structure of the electrospun membranes were preserved as shown in Fig. 5. Fig. 6 shows the cross section of the membrane. It can be seen that the layers are closely packed. The porosities and average pore diameters of the membranes were reduced while the change in surface morphology reduces the contact angle of the membranes. It is worth highlighting that after the heat-press process, the thickness of all the membranes were reduced roughly by a factor of 0.75. From Table 2, it is clear that as the electrospun time increases, the product of porosity and pore size decreases. This has to be considered when estimating the permeability of the electrospun membranes. As a result, a structural deviation factor, Φ, is included in our simulations. The function of Ф that was used in the simulations was computed by fitting a polynomial to the experimental values given in Table 2. The value of K was assumed to be Φ Ko, where Ko is the value for the membrane with a thickness of 58.4 μm. Using Matlab function, fmincon, the values
Ko that best fits all the data is 4.5 10 11 kg/m s Pa, and the respective values hf and hp are shown in Figs. 7–11. From Figs. 7–11, it is evidenced that with for the combination of membranes and process conditions evaluated there was a clear influence of membrane thickness: the thinner the membranes, the higher the transmembrane flux. Due to the nonlinear dependence of partial pressure on temperature, a higher feed temperature results in a much higher flux. Experimentally, with the thinnest membrane, a flux of 60 L/h m2 is obtained when the input temperature is 338 K. In contrast, when the feed temperature is 323 K, the obtained permeate flux is only about a third of this value. In our model, the possible variations in tortuosity of the ENMs were not included. A value of about 2 is often assumed for membrane tortuosity factor for MD studies while it is also used as an adjusting parameter in models [3] but a lower value possibly the inverse of porosity would be more appropriate. Some literature suggest a relationship between tortuosity factor and porosity [20–22], however these correlations are not widely adopted. However due to the difficulty to measure and model the actual tortuosity for nanoporous membranes it was decided that its inclusion as a separate factor was not justified.
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Fig. 9. Experimental data and the estimated production rate of our DCMD experiment for feed temperature of 328 K.
Fig. 10. Experimental data and the estimated production rate of our DCMD experiment for feed temperature of 323K.
It is apparent that the error bars in Figs. 7 and 8 are significantly large, compared to other figures. It is suspected that due to the very thin nature of the membranes, the uncertainty in tortuosity and the non-uniformity imperfections of individual membrane and variations among each the membrane samples became more evidential as the driving force, the transmembrane partial pressure difference, increases. No experimental results were recorded for thinner membranes (with electrospinning time of 2 and 3 h). Whilst they could be fabricated, handled and mounted onto the test module, they were not robust enough to withstand the vibrations caused by the fluid flow and were broken and wetted within an hour of testing. Thus no results were obtained for membranes with a thickness close to the predicted optimal thickness.
5. Conclusion Obtaining solutions from our analytical model, it was shown that the value of optimal membrane thickness varies with salinity. While it was suggested that there is no optimal membrane
263
Fig. 11. Experimental data and the estimated production rate of our DCMD experiment at feed temperature of 318K.
thickness for the DCMD system in the absence of salinity, when the salinity was relatively low (below 10%), the typical values of membrane thickness was found to be around 13 μm. The values of optimal thickness were found to depend on operational conditions and the membrane properties. The value increases with lower heat transfer coefficients (especially lower feed-side heat transfer coefficients); lower feed inlet temperature; and increased membrane permeability. Electrospun membranes were produced by spinning from a PVDF dope. For spinning times of 2 and 3 h, the membrane were sufficiently robust to be handled but insufficiently robust to withstand the first hour of testing. Membranes fabricated with spinning time of 4–10 h had thickness of 27–58 μm after being consolidated in a heatpress. Fluxes over 60 L/h m2 were obtained with the 27 μm when inlet feed temperature exceeded 338 K. The bulk of the data lies in the range of 10–30 L/h m2. With the introduction of structural derivation factor in our estimations, the model was found to fit the data very well including the thinner membranes operated in a system with high inlet feed temperature. It had been hoped to produce sufficiently thin membranes to demonstrate experimentally the effect of optimal thickness. However from the experimental work, it is found that such thin unsupported PVDF membranes were not robust enough. Nevertheless the excellent fit to the model suggests that the maxima shown in Figs. 7–11 would be the experimental maxima if only sufficiently thin membranes had been made.
Acknowledgement We acknowledge the financial support from the National Research Foundation and the Economic Development Board – Singapore (SPORE, COY-15-EWI-RCFSA/N197-1).
Appendix A. Determination of expression for optimal thickness To solve the problem it is useful to express the feed and permeate heat transfer coefficients relatively: let hp ¼bhf and substitute Eqs. (5)–(8) into (3) and (4). With the assumptions that C and αw are independent of temperature and the effect of concentration polarization is neglected, then substitutions and
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conditions. The variation of the predicted optimal thickness with salinity is shown in Fig. A1.
Nomenclature Symbol b
Fig. A1. Variation of theoretical optimal thickness with respect to changes in salt concentration.
differentiations will yield: dN K dP f m dT f m dP pm dT pm K ¼ 2 ðP f m P pm Þ dT pm dδ dδ δ dT f m dδ δ " ! # dT f m 1 K 3841 ¼ e23:238 ð3841=ðT f m 45ÞÞ αw δ δ ðT f m 45Þ2 dδ " ! # dT pm 1 K 3841 e23:238 ð3841=ðT pm 45ÞÞ δ δ ðT pm 45Þ2 dδ Now from Eqs. [A1–A4]: dQ dN dH v km dT f m dT pm km ¼ Hv þ Nþ 2 ðT f m T pm Þ dδ dδ dδ δ dδ dδ δ
C g k K H h M m N P Q T w
α ε μ ρ Φ δ
½A1
½A2
Combining Eqs. [A1] and [A2] and setting dN/dδ ¼0. An expression for the optimal thickness is obtained as a function of Tfm and Tpm:
ratio of permeate to feed heat transfer coefficient (dimensionless) membrane coefficient (kg/m2 s Pa) constant defined by Eq. (8) (dimensionless) thermal conductivity (W/m K) membrane permeability (kg/m s Pa) heat of vaporization (J/kg) heat transfer coefficient (W/m2 K) molar mass (kg/mol) constant defined by Eq. (8) (dimensionless) mass flux across membrane (kg/m2 s) saturated vapour pressure (Pa) heat flux (W/m2) temperature (K) weight (kg) activity coefficient (dimensionless) membrane porosity (dimensionless) viscosity (Pa s) density (kg/m3) structural derivation factor (dimensionless) membrane thickness (m)
Subscript f m p w
feed condition adjacent to membrane permeate condition water
( km ðT f m T pm Þ½αw e ð3841=ðT f m 45ÞÞ ð3841=ðT f m 45Þ2 Þ þ e ð3841=ðT pm 45ÞÞ ð3841=ðT pm 45Þ2 Þð1=bÞ : hf ðe ð3841=ðT f m 45ÞÞ e ð3841=ðT pm 45ÞÞ Þ ) km 1 mK 1þ ðαw e23:238 ð3841=ðT f m 45ÞÞ e23:238 ð3841=ðT pm 45ÞÞ Þ b hf hf
δ¼
The above expression for the optimal thickness includes the unknown interface temperatures Tfm and Tpm. Now one of these can be eliminated in terms of the other by combining Eqs. (2) and (3). hp ðT pm T p Þ ¼ hf ðT f T f m Þ Moreover, from Eqs. (2)–(5) a second equation relating Tfm is obtained.
δ¼
KðmT f m þ cÞðP f m P pm Þ þ km ðT f m T pm Þ hf ðT f T f m Þ
½A4
δ and ½A5
The ratio of permeate to feed heat transfer coefficient is defined as ‘b’. By applying suitable values for b, hf, K, km, which are all experimental parameters, and setting the input temperatures Tf and Tp, the two expressions given by [A3] and [A5] can be plotted for a suitable range of Tfm. A solution can be obtained graphically from an intersection of the plots. The intersection gives the optimal thickness of the membrane for the specified operating
½A3
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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Direct contact membrane distillation: An experimental and analytical investigation of the effect of membrane thickness upon transmembrane flux Ho Yan Wu a, Rong Wang b, Robert W. Field a,n a b
University of Oxford, Department of Chemical Engineering, UK Singapore Membrane Technology Centre, Nanyang Technological University, Singapore
art ic l e i nf o
a b s t r a c t
Article history: Received 9 January 2014 Received in revised form 30 May 2014 Accepted 2 June 2014 Available online 9 July 2014
Polyvinylidene fluoride (PVDF) electrospun nanofibrous membranes (ENMs), consolidated with a heatpress process and of various thicknesses were fabricated and tested in a DCMD cell at five different operating conditions. Membranes as thin as 27 μm were sufficiently robust for evaluation and gave a transmembrane flux as high as 60 L/h m2. An analytical model was created to estimate the optimal membrane thickness for DCMD operations. It is found that the value of optimal thickness increases with reduced heat transfer coefficients; decreased feed inlet temperature; increased membrane permeability; and increased salinity. Even for 10% NaCl the predicted optimum was estimated to be 13 μm which was too thin for experimental confirmation. Based upon this analysis but with due allowance for the variation of heat transfer coefficients with temperature dependent physical properties a single Matlab model was created to fit the five sets of data. With the introduction of a structural derivation factor, which reflected the experimentally determined variation of porosity and pore size with thickness, the model was found to fit the data very well. & 2014 Elsevier B.V. All rights reserved.
Keywords: Membrane distillation Electrospun membranes Structural parameter Effect of thickness
1. Introduction Membrane distillation (MD) has become a popular research area since it has the potential to tackle the shortage of water while using a relatively small amounts of high grade energy. MD is a thermal driven process that can be coupled with solar thermal systems to produce fresh water from brackish or seawater. Direct contact membrane distillation (DCMD) is the most popularly explored configuration due to its simple design that makes it particularly attractive and suitable for application in rural and less developed areas. One of the major disadvantages of MD is its relatively low transmembrane flux. A number of membrane parameters, including membrane material, porosity, tortuosity, pore size and distribution, are factors that affect the production rate in DCMD operation [1]. Most commercial MD membranes available are made of polytetrafluoroethylene (PTFE) because of its very low surface energy [2]. However PTFE membranes require complicated extrusion, rolling and stretching or sintering procedures. Polypropylene (PP) membranes prepared by phase inversion has also been used for MD membranes [3]. While PTFE and PP membranes
n
Corresponding author. E-mail address: robert.fi[email protected] (R.W. Field).
http://dx.doi.org/10.1016/j.memsci.2014.06.002 0376-7388/& 2014 Elsevier B.V. All rights reserved.
exhibit good hydrophobicity and promising MD performance, the structural parameters of the membranes produced are difficult to control. In recent years, the electrospinning technique has been explored extensively to produce nano-fibrous membranes. Previous research studies have shown that electrospun hydrophobic membranes are suitable for DCMD application [4–7]. Being hydrophobic with a good thermal resistivity and readily dissolvable at room temperature in a variety of solvents, poly(vinylidene fluoride) (PVDF) in particular has been adopted for the production of electrospun nano-fibrous membranes for MD applications [13]. Although the production rate from a conventional single needle electrospinning system is very low (which makes it difficult to employ this process in industrial settings) techniques which promote multiple jetting have been developed for large production of electrospun mats from polymer solutions [8]. Membrane thickness plays a significant role in DCMD systems. With a thicker membrane, less heat is conducted away from the feed side to the permeate side, giving a higher difference in transmembrane temperature which results in a higher driving force. However the membrane structure reduces the ease of permeation thus impeding the transmembrane flux. Since the membrane thickness is inversely proportional to both the rate of mass and heat transfer across the membrane, provided other membrane parameters remain the same, many researchers suggested the thickness of the single layer membrane might be
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expected to have an optimal value [9,10]. Previously for DCMD configurations, experiments have been carried out using available commercial membranes with pure water as feed and for various membrane thicknesses. For similar pore size and porosity, thicker membranes resulted in flux reduction [11,12]. Gostoli et al. [13] reported that for thin membranes, the transmembrane flux depends on the salt concentration whilst for thicker membranes, salt concentration does not play an important role for the flux; Song et al. [14] found that salt concentrations up to 10% led to only small flux reduction. Schofield showed that with the membranes he used, the permeability of the membrane was virtually independent of membrane thickness [15]. Using a computer simulated counter current hollow fibre DCMD module, Laganà suggested that the optimal thickness of a single hydrophobic layer membrane lies between 30 and 60 μm [10]. Whilst Al-Obaidani et al. identified an optimal thickness value of 700 μm for his model when operating with a low temperature gradient ( o5 1C) [16]. In this paper, the emphasis is upon the effect of membrane thickness and assessing whether there is an optimal value both analytically and experimentally. The membranes were fabricated using electrospinning and the thinnest membrane used in the experiment is about 27 μm. All membranes were fragile and thinner membranes fractured inside the module before water production rates could be determined. The experimental data collected is compared with model simulations based upon an analytical model that indicated an optimal thickness much thinner than that which could be achieved in practice.
2. Theory and analytical approach To assess optimal thickness, firstly an expression of dN/dδ is obtained where N is the transmembrane flux (kg m 2 s 1) and δ is membrane thickness. For unit area, the heat and mass transfer process of DCMD are generally expressed as Q ¼ hf ðT f T f m Þ
ð1Þ
Q ¼ hp ðT pm T p Þ
ð2Þ
Q ¼ NH þ
km
δ
ðT f m T pm Þ
N ¼ CðP f m P pm Þ
ð3Þ ð4Þ
These are a standard set of equations except that herein the enthalpy change across the membrane, H, is not taken to be independent of temperature. Other terms have their usual meaning and are defined in Nomenclature. Now provided the porosity and pore size do not vary with thickness: C¼
K
δ
ð5Þ
whilst P f m ¼ αw e23:238 ð3841=ðT f m 45ÞÞ
ð6Þ
P pm ¼ e23:238 ð3841=ðT pm 45ÞÞ
ð7Þ
H ¼ mT f m þ g
ð8Þ
where m and g are constants and have the value of 2400 (J/kg K) and 3,200,000 (J/kg). These coefficients were obtained by fitting a line of best fit to the values of specific enthalpy of water phase change given in a standard Oxford Engineering Data book [17], corresponding to water temperature in the range of 293–383 K. The term αw in (6) is activity of water and is less than unity in the presence of salt.
2.1. Determination of theoretical optimal thickness We have referred to theoretical optimal thickness in the subheading as the determined thickness may not be practical in most engineering applications. In Appendix A an equation relating the optimal thickness (the thickness at dN/dδ) to two other unknowns Tfm and Tpm is obtained in terms of known parameters including αw, km and the heat transfer coefficients. A second equation relating Tfm and Tpm can be obtained straight from Eqs. (1) and (2) by eliminating Q. A third equation relating δ and Tfm is obtained from Eqs. (2)–(5). Thus one has three equations and three unknowns one of which is the optimal thickness. Further mathematical details are in Appendix A and results are given later.
3. Experiments 3.1. Fabrication of the electrospun nanofibre membranes (ENMs) 3.1.1. Materials and chemicals Commercial polymer PVDF Kynar HSV 900 was purchased from Arkeme Inc., Singapore. Prior to use, the polymer was dried at 323 K under vacuum condition for at least 24 h. Solvents, acetone and N,N-Dimethylformamide (DMF) were obtained from Fisher, Singapore. Lithium chloride (LiCl) was obtained from Merck, Singapore. 3.1.2. Electrospinning dope solution The PVDF polymer dope solution was prepared by dissolving 5 wt% PVDF into a mixture of DMF and acetone with a weight ratio of 6 to 4. LiCl (0.004 wt% of total spinning dope solution) was added to the solution to enhance the conductivity of the dope. The dope solution was mechanically stirred for 24 h at 333 K and cooled at room temperature for 24 h prior to electrospinning. 3.1.3. Electrospinning of PVDF membranes The electrospinning conditions were set at: 27 kV electric voltage, 1800 mm3/h polymer flowrate, 120 mm between the spinning tip and the top of the rotating drum collector, rotational speed of the drum, with a radius of 38 mm is 2 rpm/s and the horizontal movement of the spin tip is 0.1 mm/s across a length of 80 mm. After electrospinning, the membranes were then dried in an incubator, which was set at 323 K, for 24 h, to ensure that all the solvents were evaporated. For the present work, the recorded electrospun membranes were fabricated by spinning for 4–10 h (Fig. 1). 3.1.4. Heat press post-treatment The dried PVDF membranes were pressed between two flat glass panes, at 423 K, for 1 h. The heat press was operated at just above atmospheric pressure. The melting temperature of PVDF used is around 443 K under ambient pressure.
Fig. 1. Schematic of the electrospinning setup used for membrane fabrication.
H.Y. Wu et al. / Journal of Membrane Science 470 (2014) 257–265
3.2. Characterization of electrospun membrane A scanning electron microscope (SEM) (EVO, 50, Carl Zeiss AG, Singapore) was used to evaluate the fibre size of the membrane and observe the morphology of the membranes. Platinum dust was sprayed onto the membrane in order to charge the surface and cross-section for effective imaging in the SEM. The porosity of the membrane was obtained using a gravimetric method with the membranes being immersed into isopropyl alcohol (IPA) which penetrates into the pores of the membrane. The membrane porosity is calculated using equation
ε¼
ðwwet wdry Þ=ρIPA ðwwet wdry Þ=ρIPA þ wdry =ρPVDF
ð9Þ
The pore size of membranes was determined by a capillary flow porometer (model CFP 1500A, from Porous Material. Inc. (PMI) in Singapore). The contact angles of the membranes were measured using goniometer (Contact Angle System OCA, from Data Physics Instruments GmbH in Singapore). 3.3. DCMD tests The electrospun membranes were tested with a DCMD setup as shown below. The size of the membrane module has an effective membrane area of 3600 mm2. The dimensions of the feed and permeate channels are identical with a length of 90 mm, a width of 40 mm and a depth of 2 mm. While deionized water was used on the permeate side throughout all the tests, 10 wt% NaCl concentration was used as feed solution and the mass flowrates were maintained at 1.5 kg min 1. The membrane module was placed vertically with hot feed water and cold deionized water running counter-currently. Fig. 2 shows the schematic setup of the experiment. Fluxes were determined by collecting permeate from runs that lasted for between 3 and 5 h, while the permeate temperature were kept at 293 K, the feed temperature differ for each set of runs, ranging from 318 to 338 K. At each condition at least three evaluations were made, and as the membranes were fragile new membranes were used for each testing. Fluxes were
259
calculated to an accuracy of better than 75% and therefore the the variation in the fluxes at a given thickness reflects the variation in porosity and pore size that arise naturally from the electrospinning process.
3.4. Model for estimation of performance A counter current DCMD flatsheet membrane module model was constructed with Matlab. The modules were divided into many thin strips and elements. For each element, Eqs. (1)–(4) were solved using Matlab function, fsolve. Outputs from each element including outlet temperature and flowrate, were used as input for the next element. The overall fitting later only had two adjustable parameter for the whole of the data set: one was membrane permeability and the other was the value of the heat transfer at coefficient at 293 K. The flowrates were the same for each channel and invariant and so the heat transfer coefficients would be the same if the temperatures were the same. However temperature changed but from the use of standard correlations, assuming turbulent flow along the channel, one can relate the heat transfer coefficients as the correlations suggest that the ratio of hf and hp should be (μf/μp) 0.5. The values obtained were within the estimates obtained from the data itself using the method given in [18]. Due to the uncertainties of flow geometry, entrance effects and roughness of the membrane surfaces it was difficult to accurately estimate values of hf and hp. Now instead of using common standard correlations found in the literature [19], the heat transfer coefficients were estimated using Eq. (17) in [18]. This was done for each set of data with the same inlet temperatures. The overall MD mass-transfer coefficient (denoted by Keff in [18]) is found from our experimental data and the graph of 1/Keff vs δ plotted. The membrane permeability (denoted by C in [18]) can be evaluated from the gradient of the graph. Then given that the thermal conductivity of the membrane matrix and the latent heat of vaporization are known, the combined heat-transfer coefficient (denoted by UL in [18]) can be obtained from intercept and split into hf and hp. This confirmed that values of around
Fig. 2. Schematic of the DCMD test setup used for membrane evaluation.
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Fig. 3. Graphical determinations of optimal thickness for pure water (A, top); 3.5% NaCl solution (B, middle); and 10% NaCl soltion (C, bottom). Simulation conditions are given in Table 1.
4000 W m 2 K 1 were justifiable for the experimental conditions used.
Table 1 Operation and membrane structural parameters used for simulation. Stimulation conditions
4. Results and Discussions 4.1. Results from analytical analysis In Fig. 3A, the curves obtained from expressions [A3] and [A5] do not meet, which suggests an optimal thickness does not exist in the absence of salinity in the feed. On the other hand, when using aqueous NaCl as feed solution, Fig. 3B and C suggests that optimal thickness does exist, the value of which increases with increasing salinity. With an activity coefficient of 0.98, using seawater as feed, it is estimated that the optimal thickness is about 5 μm, for the chosen simulation conditions as detailed in Fig. 3. When using 10% NaCl as feed solution, which has an activity coefficient of 0.94, it is estimated that for the chosen simulation condition, the optimal thickness is about 13 μm. The parameters values that were chosen for the simulation in Fig. 3 are close to the actual experimental condition (Table 1). Salinity plays an important part in the determination of optimal membrane thickness. The presence of salt reduces the water vapour partial pressure in the feed through the term αw in Eq. (6). Below optimal thickness, as the membrane thickness increases the high transmembrane temperature difference results in a higher, driving force, Pfm Ppm. In this region this increases more rapidly than the membrane resistance to mass transfer which increases linearly with thickness. However, above optimal thickness, the driving force increases less rapidly than the membrane resistance. For any given conditions, the value of membrane thickness corresponding to zero or negative flux is always smaller than the corresponding value for maximum flux. These values are
b hf K km Tf Tp
1 2500 4.5 10 11 0.045 338 293
– W/m2 K kg/m s Pa W/m K K K
too small to be of any practical interest; the thicknesses of commercially available membranes are about 100 μm. Indeed such thicknesses are significantly larger than the optimal thicknesses determined below. Fig. 4(A) shows that the optimal membrane thickness increases by 30% for a reduction in the feed temperature from 338 K to 313 K. A lower Tf results in a lower Tfm. Due to the temperature sensitivity of water vapour partial pressure, in order to achieve an effective driving force across the membrane, when Tf is lowered; a thicker membrane is required to sustain an effective temperature difference; (and hence an effective partial pressure difference) across the membrane. From Fig. 4(B), it can be seen that as the permeability of the membrane increases, the optimal membrane thickness increases. Owing to a higher permeate flux resulting in more heat transfer via convection; a thicker membrane is required to provide the optimal transmembrane temperature difference. Fig. 4(C) and (D) shows that the value of optimal membrane thickness is very sensitive to the heat transfer coefficients. This can be explained by the temperature polarization effect. Temperature polarization refers to the system characteristic wherein the fluid temperatures at the membrane boundaries are different to their
H.Y. Wu et al. / Journal of Membrane Science 470 (2014) 257–265
261
Fig. 4. Variation in optimal thickness with changes in: (A) feed temperature; (B) membrane permeability; (C) heat transfer coefficient; and (D) ratio of permeate heat transfer coefficient to feed heat transfer coefficient. Other simulation parameters remain identical as those demonstrated in Fig. 3 for 10% NaCl solution.
Table 2 Properties of the membranes under different electrospinning times. Electrospinning time (h) Without heat treatment Porosity (%) Ave. fibre diameter (μm)
Ave. pore diameters (μm)
4
6
8
10
92.3 7 0.4 0.147 0.06
90.6 7 0.7 0.147 0.05
90.5 7 0.8 0.157 0.07
89.2 7 1 0.147 0.03
0.337 0.17
0.317 0.17
0.29 7 0.15
0.25 7 0.11
36.4 7 2.5
46.4 7 2.5
64.37 2.5
77.0 7 2.5
139.4 7 1.2
141.0 7 0.5
140.8 71.8
139.0 7 0.3
Ave. fibre diameter (μm)
83.6 7 0.4 0.167 0.05
80.8 7 0.2 0.157 0.04
80.0 7 1 0.157 0.03
80.5 7 0.4 0.167 0.03
0.217 0.09
0.187 0.06
0.177 0.08
0.157 0.07
Structural deviation factor:Φ, given by: porosity pore size/(porosity pore size)@10 h
1.45
1.20
1.13
1
27.9 7 2.5
35.67 2.5
45.7 7 2.5
58.4 7 2.5
130.6 7 1.6
129.9 7 1.5
131.3 7 2.5
130.0 7 2
Thickness (μm) Contact angle (deg) With heat treatment Porosity (%)
Ave. pore diameters (μm)
Thickness (μm) Contact angle (deg)
corresponding bulk temperatures. Due to thermal boundary layer resistance and the relatively large amount of heat required for vaporization of the feed liquid, the heat transfer across the boundary layers is often the major rate determining step. With higher heat transfer coefficients on either or both sides of the membrane, the overall boundary layer resistance is reduced, giving a reduction in the temperature polarization effect. As a result, a thinner membrane would be able to provide enough relative heat transfer resistance for optimal balance between the transmembrane temperature difference (that determines the water vapour pressure difference) and the membrane resistance. Thus in Figure 4(C) as both hp and hf increase simultaneously, where hp ¼bhf (see Appendix) and b ¼1, the optical thickness decreases. Fig. 4(D) shows the change in optimal thickness when only one of the heat transfer coefficients is changed. When the ratio is low (hp is low and the thermal resistance at the permeate side is high), the permeate-side flow is ineffective in the transferring of heat
from the membrane interface to the bulk fluid and so the consequence is that thicker membranes are required to maintain a sufficient transmembrane temperature difference, Tfm Tpm. However the corresponding optimal thickness reduces rapidly as b increases and the steepness of the curve reduces rapidly as b approaches 1. Similar behaviour (dashed line) can be observed if hp is kept constant and only hf is changed. From the steepness of the curve in Fig. 4(D) it is evident that the optimal thickness is relatively more sensitive to the change in hf then changes in hp. 4.2. Characterization of the membranes fabricated and DCMD experiment It is shown in Table 2 that the PVDF membrane properties and characteristics change significantly after the heat-press process. Since the heat is applied at a temperature that is below PVDF's melting temperature and the pressure applied is relatively
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Fig. 7. Experimental data and the estimated production rate of our DCMD experiment for feed temperature of 338 K. Fig. 5. SEM image showing surface morphology of the post-heat pressed membranes; magnification of 10,000.
Fig. 8. Experimental data and the estimated production rate of our DCMD experiment for feed temperature of 333 K.
Fig. 6. SEM image showing cross-section of the post-heat pressed membranes; magnification of 5000.
moderate, the nano-fibrous structure of the electrospun membranes were preserved as shown in Fig. 5. Fig. 6 shows the cross section of the membrane. It can be seen that the layers are closely packed. The porosities and average pore diameters of the membranes were reduced while the change in surface morphology reduces the contact angle of the membranes. It is worth highlighting that after the heat-press process, the thickness of all the membranes were reduced roughly by a factor of 0.75. From Table 2, it is clear that as the electrospun time increases, the product of porosity and pore size decreases. This has to be considered when estimating the permeability of the electrospun membranes. As a result, a structural deviation factor, Φ, is included in our simulations. The function of Ф that was used in the simulations was computed by fitting a polynomial to the experimental values given in Table 2. The value of K was assumed to be Φ Ko, where Ko is the value for the membrane with a thickness of 58.4 μm. Using Matlab function, fmincon, the values
Ko that best fits all the data is 4.5 10 11 kg/m s Pa, and the respective values hf and hp are shown in Figs. 7–11. From Figs. 7–11, it is evidenced that with for the combination of membranes and process conditions evaluated there was a clear influence of membrane thickness: the thinner the membranes, the higher the transmembrane flux. Due to the nonlinear dependence of partial pressure on temperature, a higher feed temperature results in a much higher flux. Experimentally, with the thinnest membrane, a flux of 60 L/h m2 is obtained when the input temperature is 338 K. In contrast, when the feed temperature is 323 K, the obtained permeate flux is only about a third of this value. In our model, the possible variations in tortuosity of the ENMs were not included. A value of about 2 is often assumed for membrane tortuosity factor for MD studies while it is also used as an adjusting parameter in models [3] but a lower value possibly the inverse of porosity would be more appropriate. Some literature suggest a relationship between tortuosity factor and porosity [20–22], however these correlations are not widely adopted. However due to the difficulty to measure and model the actual tortuosity for nanoporous membranes it was decided that its inclusion as a separate factor was not justified.
H.Y. Wu et al. / Journal of Membrane Science 470 (2014) 257–265
Fig. 9. Experimental data and the estimated production rate of our DCMD experiment for feed temperature of 328 K.
Fig. 10. Experimental data and the estimated production rate of our DCMD experiment for feed temperature of 323K.
It is apparent that the error bars in Figs. 7 and 8 are significantly large, compared to other figures. It is suspected that due to the very thin nature of the membranes, the uncertainty in tortuosity and the non-uniformity imperfections of individual membrane and variations among each the membrane samples became more evidential as the driving force, the transmembrane partial pressure difference, increases. No experimental results were recorded for thinner membranes (with electrospinning time of 2 and 3 h). Whilst they could be fabricated, handled and mounted onto the test module, they were not robust enough to withstand the vibrations caused by the fluid flow and were broken and wetted within an hour of testing. Thus no results were obtained for membranes with a thickness close to the predicted optimal thickness.
5. Conclusion Obtaining solutions from our analytical model, it was shown that the value of optimal membrane thickness varies with salinity. While it was suggested that there is no optimal membrane
263
Fig. 11. Experimental data and the estimated production rate of our DCMD experiment at feed temperature of 318K.
thickness for the DCMD system in the absence of salinity, when the salinity was relatively low (below 10%), the typical values of membrane thickness was found to be around 13 μm. The values of optimal thickness were found to depend on operational conditions and the membrane properties. The value increases with lower heat transfer coefficients (especially lower feed-side heat transfer coefficients); lower feed inlet temperature; and increased membrane permeability. Electrospun membranes were produced by spinning from a PVDF dope. For spinning times of 2 and 3 h, the membrane were sufficiently robust to be handled but insufficiently robust to withstand the first hour of testing. Membranes fabricated with spinning time of 4–10 h had thickness of 27–58 μm after being consolidated in a heatpress. Fluxes over 60 L/h m2 were obtained with the 27 μm when inlet feed temperature exceeded 338 K. The bulk of the data lies in the range of 10–30 L/h m2. With the introduction of structural derivation factor in our estimations, the model was found to fit the data very well including the thinner membranes operated in a system with high inlet feed temperature. It had been hoped to produce sufficiently thin membranes to demonstrate experimentally the effect of optimal thickness. However from the experimental work, it is found that such thin unsupported PVDF membranes were not robust enough. Nevertheless the excellent fit to the model suggests that the maxima shown in Figs. 7–11 would be the experimental maxima if only sufficiently thin membranes had been made.
Acknowledgement We acknowledge the financial support from the National Research Foundation and the Economic Development Board – Singapore (SPORE, COY-15-EWI-RCFSA/N197-1).
Appendix A. Determination of expression for optimal thickness To solve the problem it is useful to express the feed and permeate heat transfer coefficients relatively: let hp ¼bhf and substitute Eqs. (5)–(8) into (3) and (4). With the assumptions that C and αw are independent of temperature and the effect of concentration polarization is neglected, then substitutions and
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conditions. The variation of the predicted optimal thickness with salinity is shown in Fig. A1.
Nomenclature Symbol b
Fig. A1. Variation of theoretical optimal thickness with respect to changes in salt concentration.
differentiations will yield: dN K dP f m dT f m dP pm dT pm K ¼ 2 ðP f m P pm Þ dT pm dδ dδ δ dT f m dδ δ " ! # dT f m 1 K 3841 ¼ e23:238 ð3841=ðT f m 45ÞÞ αw δ δ ðT f m 45Þ2 dδ " ! # dT pm 1 K 3841 e23:238 ð3841=ðT pm 45ÞÞ δ δ ðT pm 45Þ2 dδ Now from Eqs. [A1–A4]: dQ dN dH v km dT f m dT pm km ¼ Hv þ Nþ 2 ðT f m T pm Þ dδ dδ dδ δ dδ dδ δ
C g k K H h M m N P Q T w
α ε μ ρ Φ δ
½A1
½A2
Combining Eqs. [A1] and [A2] and setting dN/dδ ¼0. An expression for the optimal thickness is obtained as a function of Tfm and Tpm:
ratio of permeate to feed heat transfer coefficient (dimensionless) membrane coefficient (kg/m2 s Pa) constant defined by Eq. (8) (dimensionless) thermal conductivity (W/m K) membrane permeability (kg/m s Pa) heat of vaporization (J/kg) heat transfer coefficient (W/m2 K) molar mass (kg/mol) constant defined by Eq. (8) (dimensionless) mass flux across membrane (kg/m2 s) saturated vapour pressure (Pa) heat flux (W/m2) temperature (K) weight (kg) activity coefficient (dimensionless) membrane porosity (dimensionless) viscosity (Pa s) density (kg/m3) structural derivation factor (dimensionless) membrane thickness (m)
Subscript f m p w
feed condition adjacent to membrane permeate condition water
( km ðT f m T pm Þ½αw e ð3841=ðT f m 45ÞÞ ð3841=ðT f m 45Þ2 Þ þ e ð3841=ðT pm 45ÞÞ ð3841=ðT pm 45Þ2 Þð1=bÞ : hf ðe ð3841=ðT f m 45ÞÞ e ð3841=ðT pm 45ÞÞ Þ ) km 1 mK 1þ ðαw e23:238 ð3841=ðT f m 45ÞÞ e23:238 ð3841=ðT pm 45ÞÞ Þ b hf hf
δ¼
The above expression for the optimal thickness includes the unknown interface temperatures Tfm and Tpm. Now one of these can be eliminated in terms of the other by combining Eqs. (2) and (3). hp ðT pm T p Þ ¼ hf ðT f T f m Þ Moreover, from Eqs. (2)–(5) a second equation relating Tfm is obtained.
δ¼
KðmT f m þ cÞðP f m P pm Þ þ km ðT f m T pm Þ hf ðT f T f m Þ
½A4
δ and ½A5
The ratio of permeate to feed heat transfer coefficient is defined as ‘b’. By applying suitable values for b, hf, K, km, which are all experimental parameters, and setting the input temperatures Tf and Tp, the two expressions given by [A3] and [A5] can be plotted for a suitable range of Tfm. A solution can be obtained graphically from an intersection of the plots. The intersection gives the optimal thickness of the membrane for the specified operating
½A3
References
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