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Mass transfer in hollow fiber vacuum membrane distillation process based on membrane structure
Journal of Membrane Science 532 (2017) 115–123
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Mass transfer in hollow fiber vacuum membrane distillation process based on membrane structure
MARK
Jun Liua,b,c,d, Hong Guoa,b,c,d, Meiling Liua,b,c,d, Wei Zhanga,b,c,d, ⁎ Kai Xua,b,c,d, Baoan Lia,b,c,d, a
School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), Tianjin 300072, China c State Key Laboratory of Chemical Engineering (Tianjin University), Tianjin 300072, China d Tianjin Key Laboratory of Membrane Science and Desalination Technology, Tianjin 300072, China b
A R T I C L E I N F O
A BS T RAC T
Keywords: Vacuum membrane distillation Mass transfer resistance Knudsen-viscous flow Membrane character
The mass transfer process within the homogenous and the heterogeneous membranes with experiments and simulation works were investigated and compared in this paper. The combined Knudsen-viscous regime of flow was adopted to describe the mass transfer process in the membrane pores. The mass transfer resistance distribution in the heterogeneous membrane were considered and the sensitivity of membrane characters including the pore size, the porosity, the membrane thickness and the pore tortuosity on permeate flux were investigated. The results show that the mass transfer regime of the vacuum membrane distillation (VMD) process is dominated by the combined Knudsen-viscous flow within large temperature and vacuum pressure ranges. It is found that the pore size is the most sensitive parameter affecting the permeate flux. The simulation work indicates that the mass transfer resistance model (heterogeneous model or multi-layer model) is more precise than the original combined Knudsen-viscous model (homogeneous model or single-layer model) within the heterogeneous membrane, and the mass transfer resistance mainly exists on the layers with small pore size.
1. Introduction The thermal driven separation process, membrane distillation (MD), has drawn much attention in water treatment and desalination industries since better performed membranes and modules become available in 1980s [1]. Four major MD patterns have been well developed, the DCMD (direct contact MD), the VMD (vacuum MD), the AGMD (air gap MD) and the SGMD (sweeping gas MD). Although more than 60% studies in MD are focus on DCMD due to the possibility of internal heat recovery and the simplicity of configuration design, the VMD process with higher permeate flux and less conductive heat loss is more attractive in desalination and some coupled processes [2–4]. Further, the VMD configuration has no possibility of pore wetting from permeate side and light temperature polarization phenomenon [5]. It should be pointed out that there is an increasing risk that the membrane pores may easily be wetted at the feed side because of the vacuum applied on the permeate side. However this risk is worthy when the high permeate flux with 100% salt rejection can be obtained by VMD process. Furthermore, the pore wetting problem can be controlled by limiting the operation vacuum pressure be lower than
⁎
the least entrance pressure (LEP) of the membrane pores. In the MD literatures, the membrane permeate flux has been investigated by both experimental methods and theoretical derivations. The dusty gas model (DGM) was thought to be the most general model for permeate flux calculation in MD process. Some review papers have reported the detailed discusses about membrane properties, heat and mass transport theories in VMD process and the module design [2,6]. The mass transfer regime of Knudsen-limited diffusion is accepted in VMD process because the collisions between gas molecules and pore walls are dominant in porous membranes. A different theoretical calculation method of mass and heat transfer in VMD process was presented by M. Khayet et al. [7]. They developed the heat transfer equations in terms of Nusselt, Reynolds and Prandtl numbers when the coupling transmembrane mass transfer and heat transfer were considered in the VMD process. Some other modelling approaches developed for VMD process were analyzed and discussed by L. Hitsov et al. [8], such as Schofield's model [9] and Structural network model (Monte Carlo model) [10]. Based on the dusty gas model, the mass transfer regime can be divided into Knudsen diffusion, viscous flow and/or their combination
Corresponding author at: School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China. E-mail address: [email protected] (B. Li).
http://dx.doi.org/10.1016/j.memsci.2017.03.018 Received 30 December 2016; Received in revised form 14 March 2017; Accepted 15 March 2017 Available online 16 March 2017 0376-7388/ © 2017 Elsevier B.V. All rights reserved.
Journal of Membrane Science 532 (2017) 115–123
J. Liu et al.
Nomenclature
B Bk Bt Bv b d f GK GV ∆Hv J kb km kg kp
M P ∇P Qc
permeability of the membrane (kg/(m2 h Pa)) membrane permeability of Knudsen diffusion (kg/ (m2 h Pa)) membrane permeability of transition flow (kg/(m2 h Pa)) membrane permeability of viscous flow (kg/(m2 h Pa)) mass transfer resistance, which is the reciprocal of the membrane permeability (m2 h Pa/kg) the diameter of the membrane pore (m) the fraction of pores size the effective Knudsen diffusion coefficient (m2/s) the effective viscous diffusion coefficient (m2/s) latent heat of water vaporization (kJ/kg) mass flow rate of permeate flux (kg/(m2 h)) Boltzmann constant thermal conductivity of the membrane (w/(m K)) thermal conductivity of the vapor (w/(m K)) thermal conductivity of the membrane material (w/(m K))
Qm Qv
r T Vs
relative molecular mass of vapor vapor pressure (Pa) pressure difference (Pa) heat transfer through conduction across the membrane matrix (J/(m2 s)) total heat transfer (J/(m2 s)) the latent heat associated with the evaporation process (J/ (m2 s)) membrane pore radius (m) temperature (K) dimensionless sensitivity of the membrane character
Greek letters
τ ε δ σ λ
tortuosity of the membrane pores porosity of the fiber membrane the thickness of membrane layer (μm) collision diameter, for water vapor is 2.641 Å mean free path (m)
MD membrane is defined as the ‘homogeneous membrane’. If the pore size on the membrane cross-section varies greatly and its influence on the mass transfer process cannot be neglected, then this kind of membrane is defined as the ‘heterogeneous membrane’. For example, the cross-section of the polyvinylidene fluoride (PVDF) hollow fiber membrane through TIPS method prepared by Lin et al. [19] has the bicontinuous structure and the pore size changes little along the pore axis direction on the cross-section. This kind of membranes can be considered as the homogeneous membrane as we defined. The PVDF composite hydrophobic hollow fiber membranes fabricated through non-solvent induced phase inversion method as reported by Hou et al. [16] have the heterogeneous structure on the cross-section, the pore size on the cross-section has a large variation range as they reported. That is to say, the membrane pores have a size distribution not only on the membrane surface but also on the membrane cross-section along the pore axis. In order to distinguish the pore size distribution on the membrane surface (surface pore size distribution), the pore size variation on the cross-section along the pore axis direction can be defined as interior pore size distribution. Many investigators considered the pore size distribution in calculating the permeate flux just on the membrane surface with different mass transfer regimes. However, the mass transfer mechanisms may also change when the vapor molecules transport from the feed side to the permeate side across the membrane matrix. Therefore in the theoretical calculation of mass transfer process, the different membrane layers should be considered based on the pore structure characteristics especially when dealing with the heterogeneous membrane. Further, the mass transfer resistance should be analyzed to guide for preparing high performance membrane. However, literatures concerning on this specific discussions of mass transfer process are hard to search out. With this in mind, the mechanisms of mass transfer process associating with operation conditions during the hollow fiber VMD process were discussed in this work. The simulation work of mass transfer process in both homogenous and heterogeneous membranes were investigated. The combination of Knudsen diffusion and viscous flow were engaged to calculate the permeate fluxes. The mass transfer resistances in different pore structure layers as an analogy to the typical heat transfer process were analyzed in the multi-layer model. The influences on the permeate flux of membrane characters including pore size, porosity, tortuosity and the membrane thickness were considered and their sensitivities to the permeate flux were discussed as well.
according to the Knudsen Number (Kn ), which is defined as the ratio of the vapor mean free path (λ ) and the pore diameter (dp ). If Kn < 0.01 (dp > 100 λ ), the molecule-molecule collisions will dominate the vapor transport process and the viscous flow occurs [11,12]. If Kn is greater than 10 (dp < 0.1 λ ), the molecule-pore wall collisions dominate the mass transfer process and the Knudsen diffusion happens in the membrane pores. While Kn is between 0.01 and 10 (0.1 λ < dp < 100 λ ), both molecule-molecule collisions and molecule-wall interactions have to be considered [13]. In this case, the viscous flow and the Knudsen diffusion mechanism are combined to describe the permeability of vapor transport through the porous membranes. This division method was mostly accepted by researchers as summarized by E. Drioli et al. [5] and Mostafa Abd El-Rady Abu-Zeid et al. [2]. In fact, when the Knudsen diffusion dominates, the vicious flow regime also occurs and vice versa. That is to say, the Knudsen diffusion and the viscous flow may exist simultaneously no matter how Kn varies in VMD process. In the literatures the different mass transfer mechanisms of VMD process based on the dusty gas model are stated, but few of them clearly point out the ranges of mass transfer regime associating with the operation conditions. The studies on the membrane pore size characteristics are important since the pore size is such a significant parameter in the mass transfer process as discussed above. The pore characters on the membrane surface such as the pore size distribution has been well studied in many literatures. However, the pore size variation on the cross-section of the MD membrane receives little discussions even though many researchers have reported in their work [14–18] – the SEM figures of the membrane cross-section. The single hydrophobic layer membranes and multi-layered membranes were both reported as Khayet et al. summarized [1]. The pore size along the pore axis on the cross-section may change greatly and different pore-character layers should be discussed separately. For example, both figure-like and sponge-like voids exist on the cross-section of the hollow fibers which was prepared from suspensions with different ceramic powders/PESF ratios as Zhang et al. reported [15]. The mass transfer mechanism in the figure-like and the sponge-like voids layers may be different according to their distinguish pore sizes. According to pore size distribution of the membrane layer structure, the homogeneous membrane and the heterogeneous membrane were defined here. If the membranes with narrow pore size variation on the membrane cross-section, or the pores on the cross-section can be considered as uniform pores, then this kind of
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2. Theory
Bv =
2.1. Mass transfer model
kb T 2 πP σ 2
Bt =
where kb is the Boltzmann constant, P is the mean pressure within the membrane pores and T is the absolute temperature. The value of collision diameter σ for water vapor is 2.641 Å. In VMD, the application of vacuum dramatically decreases the transmembrane hydrostatic pressure, therefore a high value of λ may be achieved. In other words, in the VMD process the mean free path of the gas molecule is largely dependent on the vacuum pressure and may be much larger than the mean membrane pore size. The membrane permeate flux J can be expressed as the combined influence of membrane permeability and the process driven force [21,22]:
When the homogenous membrane is utilized, the pores across the membrane section are within a small size range. The mass transfer process in this condition can be calculated by the theoretical model as stated in Eq. (8) without considering the interior pore size distribution. However when the heterogeneous structured membrane is utilized in MD process, the interior pore size of the MD membrane has a large distribution range. In this case, the mass transfer rate may be controlled by a specific pore size range or a specific membrane layer. So it is necessary to separate the membrane matrix into several transfer zones according to the variation of the interior pore size. The total permeate flux is equal in every zone based on the continuous flow process:
(2)
⎛ m (d =0.1λ) N ⎜ ∑ G kf j dj3 + 8τδ ⎜⎝ j =1 n (d = dmax )
+
∑ j = p (d =100λ )
J = B1 * Δp1 = B2 * Δp2 =…= Bi * Δpi
(G kf j dj3+G vf j d j4 P )
J=
j = m (d =0.1λ )
⎞ G vf j d j4 P ⎟⎟ ⎠
Gk =
Gv =
(3)
B= (4)
π 8ηRT
Δp Δp1 Δp2 Δp = =…= i = i = n b1 b2 bi ∑1=1 bi
(10)
where n is the number of membrane layers. Compare Eq. (2) and Eq. (10), it is obvious that the membrane permeability B can be expressed as:
where
32π 9MRT
(9)
where i is the sequence of different pore range layers. If we define the mass transfer resistance b as the reciprocal of B , the equation above can be rewritten as the driven force divided by the process resistance formula:
p (d =100λ )
∑
(8)
2.2. Mass transfer process
where B is the permeability of the membrane, which is seriously affected by the membrane characters such as the porosity and the pore size. The driven force is usually expressed as the water vapor difference (Δp ) between the feed and permeate side. The local temperature and concentration are considered to be the main factors that can influence the vapor pressure difference greatly. The permeability of the membrane (B) is calculated by the full dusty gas model as can be found in Refs [1,12,13]:
B=
1/2 εdˆ ⎛ 1 ⎛ 8RT ⎞ dˆ ⎞ ⎜⎜ ⎜ P ⎟⎟ ⎟ + ⎠ ⎝ τδRT ⎝ 3 πM 32η ⎠
As stated before, the Knudsen diffusion and the viscous flow may happen simultaneously no matter what the range of Kn is. Therefore the transition flow mechanism shown in Eq. (8) is accepted and used in the following study. It is worth noticing that the permeate flux calculation is based on the columnar pore hypothesis, the membrane pores are defined as the combination of the straight columnar pores and the socalled tortuosity. However, the tortuosity is hard to measure and the experience value is applied.
(1)
J = B*Δp
(7)
when 0.01 < Kn < 10, Knudsen diffusion and viscous flow decide the membrane permeability together, and also this mechanism is defined as the transition flow,
The heat and mass transfer in MD have been extensively studied and several theoretical models have been developed for vapor transporting through porous media to predict the performance of MD configurations. The dusty gas model is widely accepted by many investigators in MD field as stated before. The Knudsen number (Kn ), which is defined as the ratio of mean free path (λ) to the membrane pore size (dp), is accepted to divide the vapor transport patterns into Knudsen diffusion, viscous flow and their combination. For a specific membrane, the Knudsen number depends on the mean free path λ , which is calculated by the following expression [20]:
λ=
2 εdˆ P 32RTδτη
⎛ i = n 1 ⎞−1 1 = ⎜⎜ ∑ ⎟⎟ = b1 + b 2 +…+ bn ⎝ i =1 Bi ⎠
i=n
∑ Bi i =1
(11)
Base on the equations above, the total permeate flux of the membrane can be calculated by the following equation:
(5)
i=n
J = B * Δp = Δp * ∑ Bi
and N is the total number of pores per unit area, m is the last class of pores in Knudsen region, p is the last class of pores in transition region, n is the last class of pores in viscous flow region, f j is the fraction of pores with pore diameter dj , M is the relative molecule mass of water and η is the viscosity of vapor, τ is the pore tortuosity and δ is the porosity of the membrane. When the uniform pore diameter dˆ is assumed, the equations above can be simplified and can be rewritten as the specific formulas with different conditions. When Kn > 10, the Knudsen diffusion dominates the mass transfer process within the membrane pores,
εdˆ ⎛ 8RT ⎞ ⎟ ⎜ 3τδRT ⎝ πM ⎠
i =1
This equation comprehensively considers the mass transfer process in the heterogeneous membrane, and the mass transfer resistance in different membrane layers can be analyzed by this model. 2.3. Heat balance The MD is a combined process of mass transfer and heat transfer. Two mechanisms of heat transfer in MD have been developed, heat transfer through conduction across the membrane matrix (Qc ) and the latent heat associated with the evaporation process (Qv ).
1/2
Bk =
(12)
(6)
Qm = Qc + Qv
and when Kn < 0.01 the viscous flow will dominant, 117
(13)
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found elsewhere [24–27]. The turbulent flow with standard k − ε model was selected to solve the flow equations and SIMPLE algorithm was selected for the velocity and pressure coupled calculation. Second Order Upwind algorithm for all other terms were settled. In the simulation models, the membrane material zone was set to be porous zone, and the parameters were set to be the same as the experimental tests. The phase change of liquid water turning into vapor and the vapor transporting in the membrane layer(s) were realized by user defined functions (UDFs) which were based on the mass transfer equations as stated before in VMD process. The energy for water evaporation on the membrane surface of the feed side was also included in the UDFs because in the software Fluent@ only the heat conduction can be computed automatically under the selected calculation model. During the simulation process, the water flux of VMD process was realized by solving the mass transfer equations together with the heat transfer process. The calculation procedure is shown in Fig. 3. The flow equations were solved first until the calculation become convergent. The UDFs for water evaporation were engaged in solving the energy equations to realize the energy transfer process by evaporation. And then the mass transfer UDFs were loaded to calculate the water flux by solving both energy equations and the mass transfer equations. This is different from the reality that the mass transfer and the energy transfer process happening simultaneously during the VMD process. One should note that the influence of temperature and concentration polarization on the model flux can be calculated by the software automatically because the evaporation rate is calculated based on the membrane surface temperature and concentration of the feed side and the local pressure of the membrane pores. The temperature and concentration values and the pressure values in simulating the flux can be read by the macros within the software. Therefore the model flux can fit the experimental data without considering the influence of the temperature and concentration polarization even though the temperature and concentration polarization phenomenon exist both in experiments and the simulations.
It must be pointed out that the conducted heat by the membrane (Qc ) is considered to be the heat loss during the MD process. The useful heat in MD is the latent heat (Qv ) for the permeate flux. Therefore, a low thermal conductivity material of the MD membrane is helpful in minimizing the heat loss during the MD process. According to the research of Fine et al. [23], only 50–80% of the total heat is consumed as the useful heat in MD and the remainder is lost by thermal conduction of the membrane. However, in VMD process the heat conduction through the membrane matrix (Qc ) can be neglected due to the large thermal resistance in the boundary layer of the permeate side [2], and the vacuum applied on the permeate side also hinders the heat conduction process across the membrane matrix [6]. So the heat balance in the VMD process from the hot feed side to the permeate side at an equilibrium state can be expressed as
Qm = Qv = BΔHv
(14)
where ΔHv is the evaporation enthalpy of water at the absolute temperature T with the transmembrane flux B . 2.4. Experiments The cross-flow hollow fiber VMD experiments were carried out with self-made homogenous and heterogeneous membranes. The schematic of VMD experimental apparatus is presented in Fig. 1. The right side is the feed circulation process, the hot feed is bumped into the membrane configuration M4 and the concentrated feed flows back to the feed tank V1. The feed velocity can read from the rotameter R3. The produced vapor in M4 is extracted out by the vacuum pump P8 and condensed through the heat exchanger H5. The liquid pure water is collected by tanks V6 and V7. The feed flows in the shell side of the VMD configuration and the vapor passes through the membrane layer into the tube side, then the vapor is extracted out by the applied vacuum. The operation temperature in the experiments varied from 323 K to 363 K and the feed inlet velocity was kept at 0.05 m/s. The 1% (mass fraction) sodium chloride solution was supplied as the inlet feed in order to weaken the influence of concentration polarization on the mass transfer process. The cross-flow VMD configurations were well checked before the experiments and no leakage problems were found. The parameters of the experimental fibers are listed in Table 1. The overall parameters of fiber C and D were measured by the aperture analyzer. The detail parameters of the different membrane layers in fiber D were determined by the SEM pictures and the analyzer testing results. Cross-flow VMD configurations with homogenous and heterogeneous fibers were built separately. The effective length of the fibers are both 150 mm and the total number of fibers in each configuration is 50. The fibers use regular triangle arrangement and the spaces are about 4 mm from the fiber center to the adjacent fiber center.
3. Results and discussion 3.1. Mass transport model of VMD process As shown in Eq. (1), the mean free path λ is related to the local temperature and the mean pressure of the membrane pores. When a vacuum is applied on the permeate side, the mean pressure in the membrane pores can be very low and the value of λ may be very high. Fig. 4 shows the distribution of mean free path when the temperature range is 273–373 K and the mean pore pressure (absolute pressure) range is 0–100 kPa within a closed system. The saturation vapor
2.5. Simulation work The simulation work based on the experimental material data was carried out. Two different calculation models were built up as shown in Fig. 2. The simulation model A is based on the real hollow fiber C with homogenous membrane, and the simulation model B is based on the real hollow fiber D which has three different porous layers. The layers in the model B are different in porosity ε and thickness δ , and the pores in these layers are also different in radius r and tortuosity τ . Model A can be defined as the homogenous membrane model (single-layer model) and model B can be called the heterogeneous membrane model (multi-layer model) according to the characters of the membrane matrix. The Fluent (ANSYS) (version 15.0) program package was engaged to simulate the hollow fiber VMD progress in this study. 3D simulation models of both homogenous membrane and heterogeneous membrane were built to investigate the influence of different pore layers on membrane performance. The governing equations used to compute the fluid flow and the heat conduction in the calculation domain can be
Fig. 1. Schematic of experimental apparatus. V1—feed tank with heating apparatus, P2—pump, R3—rotameter, M4—cross-flow hollow fiber configuration, H5—heat exchanger, V6 and V7—pure water collection tanks, P8—vacuum pump.
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control the permeate flux together. According to the results in Fig. 4, the transition flow occurs when the membrane pore size is 0.2–0.5 µm with the operation condition of temperature range from 328 K to 373 K and the mean pore pressure range from 15 kPa to 100 kPa. These are the most common operation conditions within the VMD process. It can be drawn out that the mass transfer of VMD process is dominated by the Knudsen diffusion regime according to the Dusty-gas model only when the vacuum degree applied on the permeate side is very high as shown in Fig. 4 (absolute pressure is lower than 3 kPa), and the transition regime occurs when the feed temperature is high and the vacuum on the permeate side is low (mean pore pressure is high). The mass transfer mechanism in the remaining range is the mixing of Knudsen diffusion and viscous flow. When using the transition flow regime to calculate the permeate flux of VMD process, the Knudsen flow and the viscous flow are included simultaneously. This is also suitable when the mass transfer process is strictly dominated by the Knudsen flow or the viscous flow according to the dusty gas model. Therefore, in the following simulation work, the transition flow regime (the combination of Knudsen flow and viscous flow) was applied to simulate the mass transfer process both in the homogenous and heterogeneous membrane models.
Table 1 Parameters of hollow fibers and the experimental configurations. Parameters
Porosity ε Pore radius r Thickness δ Tortuosity τ Fiber Inner diameter Fiber outer diameter
Fiber C.
Fiber D. (Heterogeneous model)
(Homogeneous model)
Layer 1
Layer 2
Layer 3
Overall
55% 0.13 µm 0.27 mm 2.1 1.31 mm
60% 0.15 µm 0.12 mm 2.0 1.24 mm
80% 15 µm 0.14 mm 1.2
75% 0.20 µm 0.05 mm 1.8
60% 0.18 0.31 mm 1.5
1.85 mm
1.86 mm
pressure line (Psv line) is also included in this figure. The value of λ decreases sharply when the mean pore pressure increases from 0 kPa to 10 kPa. When the mean pore pressure is low enough (which means the vacuum degree applied on the permeate side is very high), the value of λ can be greater than 50 µm. In this case, the Knudsen number could be higher than 100 (The range of membrane pore is 0.1–0.5 µm) and the Knudsen diffusion will dominate the mass transfer process in the membrane pores as defined by the dusty gas model. However, the case of viscous flow dominated process will never occur in the studied range according to the definition (Kn < 0.01). As is known to all that when the vapor pressure is higher than the saturation vapor pressure at the certain temperature, the water will not evaporate into vapor. Therefore when the operation condition point is above the Psv line as shown in Fig. 4, no permeate flux could be measured theoretically. And when the operation condition point is under the Psv line (the shaded area in Fig. 4), the MD process could be carried out normally. If the operation temperature decreases with an isothermal process or the mean pore pressure increases with an isobaric process, the operation point will meet the Psv line and further the VMD process will unable to continue. According to the definition in Eq. (8), the transition flow (0.01 < Kn < 10) occurs when the temperature and the mean pore pressure both rise up, the Knudsen diffusion regime and viscous flow mechanism will
3.2. Homogenous membrane 3.2.1. Validation of simulation model The simulation model A was validated with the experiment fiber C (homogeneous membrane) and the mass transfer regime was set to be the combination of Knudsen flow and the viscous flow as stated before. The feed inlet velocity was kept at 0.05 m/s both in the experiments and the simulation work. The comparison of the experimental results and the simulation data was shown in Fig. 5. Good accuracy was achieved with the temperature range from 323 K to 363 K as can be observed. It means that the single-layer model of calculation model A works well with the real experimental fiber C under the same operation conditions. When the operation vacuum on the permeate side rises up from 0.09MPa to 0.096 MPa (S-0.09 MPa, S-0.092 MPa, S-0.094 MPa and S-0.096 MPa), the simulation results show that the variation trends of the permeate fluxes are the same with
Fig. 2. Computing models and experimental materials (A. simulation model of hollow fiber with uniform membrane layer, the homogenous membrane model or the single-layer model; B. simulation model of hollow fiber with three different membrane layers, the heterogeneous membrane model or the multi-layer model; C. cross-section of PVDF hollow fiber with uniform pore structure; D. cross-section of PVDF hollow fiber with non-uniform pore structure).
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Fig. 3. The procedure of simulation work.
Fig. 4. The mean free path distribution of water vapor. Temperature range is from 273K to 373 K, mean pore pressure range is from 0 kPa to 100 kPa, the unite of λ is micron (μm).
Fig. 5. Flux comparison of simulation model A and experiment fiber C (single-layer model).
the experimental values (Exp-0.09 MPa), and the permeate flux increases when the vacuum degree rises up.
permeate flux of the pore parameters including the pore size, the porosity, the tortuosity and the thickness of the membrane were studied here. In order to simplify the calculation process, the simulated membrane characters were considered in the single layer membrane which is stated as the calculation model A. The single factor analysis
3.2.2. Influences of membrane characters Based on the simulation model above, the influence on the 120
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parameter was changed each time and the other parameters remained unchanged. All of these calculations were carried out under the same operation conditions with feed inlet velocity at 0.05 m/s and the vacuum degree applied on the permeate side at 0.09 MPa (absolute pressure at 10 kPa). Within the research scope, the variation range of the membrane character was divided into 10 even scales to save calculation costs. The permeate flux increment in each scale was calculated as the absolute value Ji−Ji+1 and the sensitivity of the character was defined as the ratio of the absolute incremental flux and the former permeate flux
method was applied here and the coupled influences of these membrane parameters were not considered in this work. In all of these simulation process, the feed inlet velocity was set to be 0.05 m/s and the absolute pressure on the permeate side was 10 kPa (vacuum degree of 0.09 MPa). As shown in Fig. 6A and B, the permeate fluxes increase with the enlargement of the pore size and the porosity. The flux growth rate enlarges when the feed inlet temperature rises up. It is not difficult to understand this flux variation according to Eq. (8) that the membrane permeate coefficient is positively related to the pore size and the porosity, and when the temperature rises up, the saturation vapor pressure in the membrane pores lifts remarkably especially at a high temperature range as shown in Fig. 4, the Psv line. However the flux decreases sharply when the membrane layer becomes thicker until 0.25 mm, and then decreases slowly when the thickness of the membrane layer continues to increase as shown in Fig. 6C. The influence of the tortuosity on the permeate flux has a similar tendency with the thickness of the membrane as indicated in Fig. 6D. The decreases of flux caused by the membrane thickness and the tortuosity of the membrane pores can make sense with the permeate coefficient calculation model in Eq. (8), the membrane flux varies inversely with the membrane thickness and the pore tortuosity. Fig. 6 also shows that high feed temperature contributes to high membrane flux because the saturation vapor pressure increases with the rise of temperature and the vapor pressure difference.
Vs =
Ji − Ji +1 Ji
(15)
where Vs is the dimensionless sensitivity of the membrane characters, i stands for the number of the scales and the range is from 0 to 10, J is the permeate flux. For example, the research scope of the tortuosity is from 1.0 to 3.0, then J0 means the permeate flux with the tortuosity at 1.0 and J1 is the permeate flux with tortuosity at 1.3 (the increment of the tortuosity in the research scope is 0.3 because it is divided into 10 even scales). Then the sensitivity value of tortuosity on scale 1 can be calculated as J0 − J1 and the other scales could be calculated out by the J0
same method, as well as the sensitivities of other membrane parameters. The average sensitivity of temperature range from 323 K to 363 K in each scale was used as the final value. With this definition the influences of different membrane parameters on the permeate flux could be compared and their comparisons are shown in Fig. 7. The permeate flux is largely dependent on the membrane characters as shown in Fig. 6. In Fig. 7, the relative influences of these membrane parameters are well demonstrated. Among all of these four membrane characters, the pore size affects the membrane flux most and the
3.2.3. Sensitivity of membrane characters The sensitivities of the membrane characters were also investigated here to find out the major factor affecting the membrane performance. The single factor analysis method was applied that only one membrane
Fig. 6. The influence of pore parameters on permeate flux (A: pore radius, B: porosity, C: membrane thickness, D: tortuosity).
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multi-layer model of simulation model B considered the pore size variation across the membrane matrix and the changes of mass transfer process in different membrane layers were calculated individually, therefore the more accuracy results could be obtained. One should note that when the operation condition point is under the Psv line as shown in Fig. 4 the VMD process could be carried out normally and the reasons were explained before. The least operation temperature is 323 K here and only when the mean pore pressure is below the saturate vapor pressure at 323 K (12.31 kPa) can the VMD process carried out with measurable permeate flux. We choose 10 kPa (absolute pressure) as the vacuum pressure both in the experiments and the simulation work here. The influence of permeate pressure on the fluxes of model A and B were not discussed because the flux variation trend of the influence on model B is the same as on model A which was shown in Fig. 5. One more conclusion can be drawn from Fig. 5 and Fig. 8 that the mass transfer process of homogenous membrane can be simulated by the Dusty-gas model directly and good accuracy may be achieved. But the mass transfer process in the heterogeneous membrane should be considered individually according to the membrane interior pore size distribution. The multi-layer model is more appropriate than the single-layer model in simulating the VMD process with heterogeneous membrane as the results shown in our work, and a more precise result can be obtained based on the experimental data.
Fig. 7. The sensitivities of the membrane characters.
defined sensitivity factor is 0.37 at the first scale. The influence of the membrane thickness is also very striking and the sensitivity factor is 0.24, ranking in the second place. The pore tortuosity and the membrane porosity are not as important as the pore size and the membrane thickness as depicted in Fig. 7 and the influence of the porosity on permeate flux is the least among them. It is also found that the sensitivities of the membrane characters are constant with the max permeate fluxes in the research domains as shown in Fig. 6, 128 kg/ (m2 h) for the pore size, 91.5 kg/(m2 h) for the membrane thickness, 71.6 kg/(m2 h) for the pore tortuosity and 62 kg/(m2 h) for the membrane porosity, respectively. These results show that the membrane performance can change greatly when the membrane characters varied, and the membrane performance could be optimized by designing a membrane with large pore size, high porosity, thin membrane thickness and straight pores. However, it must be pointed out that membrane with large pore size is more likely to be contaminated than the small pore size membranes and pore wetting problems may happen easily.
3.3.2. Mass transfer resistance distribution According to the interior pore size distribution across the multiple membrane layers, the mass transfer resistances in these layers were studied to analyse the mass transfer resistance distributions of the VMD process in the heterogeneous membrane. In Eqs. (9)–(12), the mass transfer resistances were discussed individually according to the interior pore size distribution, and the total mass transfer resistance can be influenced by different membrane layers as described in Eq. (12). The mean pressure and the local temperature in different layers can be obtained from the simulation model. The mass transfer resistances in layer 1 (R-L1) and layer 3 (R-L3) in model B were studied and the results were shown in Fig. 9. As can be observed in Fig. 9, the mass transfer resistance in layer 1 (RL1) decreases when the temperature rises up, but in layer 3 (R-L3) the mass transfer resistance almost unchanged. The reasons may be that the temperature is more sensitive to the mass transfer process in membrane layer 1 with smaller mean pore size and thicker layer matrix than in membrane layer 3 as shown in Table 1. The low porosity of layer 1 also contributes to the high mass transfer resistance in the membrane pores.
3.3. Heterogeneous membrane The heterogeneous membrane of simulation model B (multi-layer model) was validated with the experiment fiber D and compared to the homogenous simulation model A. For the homogeneous model A (single-layer model) the membrane parameters were the same as the overall parameters of the real fiber D. And the parameters of the three layers of fiber D were used for the heterogeneous model B (multi-layer model). The mass transfer regime was set to be the combination of Knudsen flow and the viscous flow as stated before. The mass transfer resistance model as deduced in Eqs. (9)–(12) was applied here to calculate the permeate flux and analysis the resistance distribution in different pore size layers. 3.3.1. Comparison of homogenous model and heterogeneous model The permeate fluxes of homogenous model and heterogeneous model were compared in Fig. 8. The homogenous model A was found to be less accuracy when compared to simulation model B although these two simulation models can both fit well with the experimental results. The relative error of simulation model A can reach higher than 7% especially when the temperature is high, while this value of simulation model B is below 4%. This may be that high temperature changes the mass transfer regime in the membrane pores according to Eq. (1). The high temperature increases the mean free path and the mass transfer regime in the small pore size area may be controlled by Knudsen diffusion. However in the simulation model A the uniform pore size across the membrane matrix was considered and the interior pore size distribution was neglected. The
Fig. 8. Flux comparison of simulation model A and B to the experimental fiber D (heterogeneous membrane layer), with permeate pressure at 10 kPa and feed inlet velocity at 0.05 m/s.
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tortuosity on the permeate flux. The multi-layer model constructed in our work is more accurate than the typical dusty gas model (singlelayer model with transition flow regime) in calculating the permeate flux with heterogeneous membrane. The mass transfer resistance mainly exists in the membrane layers with small pore size. These results could be used to design MD membrane with high performance. Acknowledgement The authors gratefully acknowledge the Projects of Science & Technology Plan Program of Tianjin (Grant no.: 12ZCZDSF02200). References [1] M. Khayet, Membranes and theoretical modeling of membrane distillation: a review, Adv. Colloid Interface Sci. 164 (2011) 56–88. [2] M.A.E.-R. Abu-Zeid, Y. Zhang, H. Dong, L. Zhang, H.-L. Chen, L. Hou, A comprehensive review of vacuum membrane distillation technique, Desalination 356 (2015) 1–14. [3] E. El-Zanati, K.M. El-Khatib, Integrated membrane –based desalination system, Desalination 205 (2007) 15–25. [4] M. Khayet, T. Matsuura, Membrane Distillation: Principles and Applications, Elsevier, Great Britain, 2011. [5] E. Drioli, A. Ali, F. Macedonio, Membrane distillation: recent developments and perspectives, Desalination 356 (2015) 56–84. [6] A. Hassan, M. Darwish, H. Fath, H. Abdulrahim, Vacuum membrane distillation: state of the art, in: Euromed 2015 Desalination for Clean Water and Energy, 2015. [7] J.I. Mengual, M. Khayet, M.P. Godino, Heat and mass transfer in vacuum membrane distillation, Int. J. Heat. Mass Transf. 47 (2004) 865–875. [8] I. Hitsov, T. Maere, K. De Sitter, C. Dotremont, I. Nopens, Modelling approaches in membrane distillation: a critical review, Sep. Purif. Technol. 142 (2015) 48–64. [9] R.W. Schofield, A.G. Fane, C.J.D. Fell, Gas and vapour transport through microporous membranes. II. Membrane distillation, J. Membr. Sci. 53 (1990) 173–185. [10] A.O. Imdakm, M. Khayet, T. Matsuura, A Monte Carlo simulation model for vacuum membrane distillation process, J. Membr. Sci. 306 (2007) 341–348. [11] A. Alkhudhiri, N. Darwish, N. Hilal, Membrane distillation: a comprehensive review, Desalination 287 (2012) 2–18. [12] M. Khayet, K. Khulbe, T. Matsuura, Characterization of membranes for membrane distillation by atomic force microscopy and estimation of their water vapor transfer coefficients in vacuum membrane distillation process, J. Membr. Sci. 238 (2004) 199–211. [13] M. Khayet, T. Matsuura, Pervaporation and vacuum membrane distillation processes: modeling and experiments, AlChE J. 50 (2004) 1697–1712. [14] C. Feng, B. Shi, G. Li, Y. Wu, Preparation and properties of micro-porous membrane from poly(vinylidene fluoride-co-tetrafluoroethylene) (F2.4) for membrane distillation, J. Membr. Sci. 237 (1–2) (2004) 15–24. [15] J.-W. Zhang, H. Fang, J.-W. Wang, L.-Y. Hao, X. Xu, C.-S. Chen, Preparation and characterization of silicon nitride hollow fiber membranes for seawater desalination, J. Membr. Sci. 450 (2014) 197–206. [16] D. Hou, J. Wang, X. Sun, Z. Ji, Z. Luan, Preparation and properties of PVDF composite hollow fiber membranes for desalination through direct contact membrane distillation, J. Membr. Sci. 405–406 (2012) 185–200. [17] M.C. García-Payo, M. Essalhi, M. Khayet, Preparation and characterization of PVDF–HFP copolymer hollow fiber membranes for membrane distillation, Desalination 245 (2009) 469–473. [18] B. Wu, K. Li, W.K. Teo, Preparation and characterization of poly(vinylidene fluoride) hollow fiber membranes for vacuum membrane distillation, J. Appl. Polym. Sci. 106 (2007) 1482–1495. [19] L. Lin, H. Geng, Y. An, P. Li, H. Chang, Preparation and properties of PVDF hollow fiber membrane for desalination using air gap membrane distillation, Desalination 367 (2015) 145–153. [20] N.N. Li, A.G. Fane, W.W. Ho, T. Matsuura, Advanced Membrane Technology and Applications, John Wiley & Sons, New Jerzey, 2008. [21] K.W. Lawson, D.R. Lloyd, Membrane distillation, J. Membr. Sci. 124 (1997) 1–25. [22] M.S. El-Bourawi, Z. Ding, R. Ma, M. Khayet, A framework for better understanding membrane distillation separation process, J. Membr. Sci. 285 (2006) 4–29. [23] A.G. Fane, R.W. Schofield, C.J.D. Fell, The efficient use of energy in membrane distillation, Desalination 64 (1987) 231–243. [24] A.L. Ahmad, K.K. Lau, M.Z.A. Bakar, Impact of different spacer filament geometries on concentration polarization control in narrow membrane channel, J. Membr. Sci. 262 (2005) 138–152. [25] H. Yu, X. Yang, R. Wang, A.G. Fane, Analysis of heat and mass transfer by CFD for performance enhancement in direct contact membrane distillation, J. Membr. Sci. 405–406 (2012) 38–47. [26] M.M.A. Shirazi, A. Kargari, A.F. Ismail, T. Matsuura, Computational fluid dynamic (CFD) opportunities applied to the membrane distillation process: state-of-the-art and perspectives, Desalination 377 (2016) 73–90. [27] B. Lian, Y. Wang, P. Le-Clech, V. Chen, G. Leslie, A numerical approach to module design for crossflow vacuum membrane distillation systems, J. Membr. Sci. 510 (2016) 489–496.
Fig. 9. The mass transfer resistances through the membrane layers based on the heterogeneous membrane D, resistances (R) and their percentages (P) of membrane layers (layer 1, L1 and layer 3, L3, respectively) with different pressures at permeate side (10 kPa and 5 kPa).
The influence of the permeate side pressure on the mass transfer resistance of the membrane layer is not remarkable as shown in Fig. 9. When the absolute pressure changes from 10 kPa to 5 kPa, the resistances in layer 1 and layer 3 both indicate a little increase. The pressure at the permeate side influences the pressure drop across the membrane pores, but only the viscous flow regime is affected by this pressure change as shown in Eq. (8). Furthermore, the mass transfer process is dominated by the Knudsen diffusion with the operation conditions of pressure at 5–10 kPa and temperature range 323–368 K as seen in Fig. 4. A low absolute pressure on the permeate side indicates that the mean pressure across the membrane pores is small, and the mass transfer resistance increases as the definition in Eq. (11). Therefore the vacuum pressure in this variation range just affects the mass transfer resistance slightly. The comparison of the mass transfer resistances with different permeate pressure also reveals that the resistance in the membrane layers is mainly affected by the membrane characteristic parameters. The influences of the operating conditions such as the vacuum pressure and the feed temperature are not as remarkable as the membrane characters. The percentage of the mass transfer resistance in different membrane layers shown in Fig. 9 reveals that the main mass transfer resistance across the membrane matrix is focus on layer 1 and layer 3. Approximately two thirds of the total resistance is distributed in membrane layer 1, and the percentage of mass transfer resistance of layer 1 reduces from 68% to 63% when the temperature changes from 323 K to 363 K. When the pressure on the permeate side changes, the percentage of mass transfer resistance almost keeps the same. This value of layer 3 rises up from 31% to 36% as the temperature changes from 323 K to 363 K. This also indicates that layer 2 with large pore size almost has no mass transfer resistance when compared to layer 1 and layer 3. It may be helpful in improving the permeate performance by decreasing the mass transfer resistance in layer 1 and layer 3. 4. Conclusion The mass transfer of hollow fiber VMD process with homogenous membrane model and the heterogeneous membrane model were both investigated in this study. The combination of Knudsen diffusion and viscous flow regimes without considering the Knudsen number in calculating the permeate flux can fit well with the experiments. In large ranges of vacuum pressure and temperature the mass transfer mechanism can be described by the transition flow, which is the combination of Knudsen diffusion and viscous flow. The membrane characters of pore size and the thickness are more sensitive than the porosity and the
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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Mass transfer in hollow fiber vacuum membrane distillation process based on membrane structure
MARK
Jun Liua,b,c,d, Hong Guoa,b,c,d, Meiling Liua,b,c,d, Wei Zhanga,b,c,d, ⁎ Kai Xua,b,c,d, Baoan Lia,b,c,d, a
School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), Tianjin 300072, China c State Key Laboratory of Chemical Engineering (Tianjin University), Tianjin 300072, China d Tianjin Key Laboratory of Membrane Science and Desalination Technology, Tianjin 300072, China b
A R T I C L E I N F O
A BS T RAC T
Keywords: Vacuum membrane distillation Mass transfer resistance Knudsen-viscous flow Membrane character
The mass transfer process within the homogenous and the heterogeneous membranes with experiments and simulation works were investigated and compared in this paper. The combined Knudsen-viscous regime of flow was adopted to describe the mass transfer process in the membrane pores. The mass transfer resistance distribution in the heterogeneous membrane were considered and the sensitivity of membrane characters including the pore size, the porosity, the membrane thickness and the pore tortuosity on permeate flux were investigated. The results show that the mass transfer regime of the vacuum membrane distillation (VMD) process is dominated by the combined Knudsen-viscous flow within large temperature and vacuum pressure ranges. It is found that the pore size is the most sensitive parameter affecting the permeate flux. The simulation work indicates that the mass transfer resistance model (heterogeneous model or multi-layer model) is more precise than the original combined Knudsen-viscous model (homogeneous model or single-layer model) within the heterogeneous membrane, and the mass transfer resistance mainly exists on the layers with small pore size.
1. Introduction The thermal driven separation process, membrane distillation (MD), has drawn much attention in water treatment and desalination industries since better performed membranes and modules become available in 1980s [1]. Four major MD patterns have been well developed, the DCMD (direct contact MD), the VMD (vacuum MD), the AGMD (air gap MD) and the SGMD (sweeping gas MD). Although more than 60% studies in MD are focus on DCMD due to the possibility of internal heat recovery and the simplicity of configuration design, the VMD process with higher permeate flux and less conductive heat loss is more attractive in desalination and some coupled processes [2–4]. Further, the VMD configuration has no possibility of pore wetting from permeate side and light temperature polarization phenomenon [5]. It should be pointed out that there is an increasing risk that the membrane pores may easily be wetted at the feed side because of the vacuum applied on the permeate side. However this risk is worthy when the high permeate flux with 100% salt rejection can be obtained by VMD process. Furthermore, the pore wetting problem can be controlled by limiting the operation vacuum pressure be lower than
⁎
the least entrance pressure (LEP) of the membrane pores. In the MD literatures, the membrane permeate flux has been investigated by both experimental methods and theoretical derivations. The dusty gas model (DGM) was thought to be the most general model for permeate flux calculation in MD process. Some review papers have reported the detailed discusses about membrane properties, heat and mass transport theories in VMD process and the module design [2,6]. The mass transfer regime of Knudsen-limited diffusion is accepted in VMD process because the collisions between gas molecules and pore walls are dominant in porous membranes. A different theoretical calculation method of mass and heat transfer in VMD process was presented by M. Khayet et al. [7]. They developed the heat transfer equations in terms of Nusselt, Reynolds and Prandtl numbers when the coupling transmembrane mass transfer and heat transfer were considered in the VMD process. Some other modelling approaches developed for VMD process were analyzed and discussed by L. Hitsov et al. [8], such as Schofield's model [9] and Structural network model (Monte Carlo model) [10]. Based on the dusty gas model, the mass transfer regime can be divided into Knudsen diffusion, viscous flow and/or their combination
Corresponding author at: School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China. E-mail address: [email protected] (B. Li).
http://dx.doi.org/10.1016/j.memsci.2017.03.018 Received 30 December 2016; Received in revised form 14 March 2017; Accepted 15 March 2017 Available online 16 March 2017 0376-7388/ © 2017 Elsevier B.V. All rights reserved.
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Nomenclature
B Bk Bt Bv b d f GK GV ∆Hv J kb km kg kp
M P ∇P Qc
permeability of the membrane (kg/(m2 h Pa)) membrane permeability of Knudsen diffusion (kg/ (m2 h Pa)) membrane permeability of transition flow (kg/(m2 h Pa)) membrane permeability of viscous flow (kg/(m2 h Pa)) mass transfer resistance, which is the reciprocal of the membrane permeability (m2 h Pa/kg) the diameter of the membrane pore (m) the fraction of pores size the effective Knudsen diffusion coefficient (m2/s) the effective viscous diffusion coefficient (m2/s) latent heat of water vaporization (kJ/kg) mass flow rate of permeate flux (kg/(m2 h)) Boltzmann constant thermal conductivity of the membrane (w/(m K)) thermal conductivity of the vapor (w/(m K)) thermal conductivity of the membrane material (w/(m K))
Qm Qv
r T Vs
relative molecular mass of vapor vapor pressure (Pa) pressure difference (Pa) heat transfer through conduction across the membrane matrix (J/(m2 s)) total heat transfer (J/(m2 s)) the latent heat associated with the evaporation process (J/ (m2 s)) membrane pore radius (m) temperature (K) dimensionless sensitivity of the membrane character
Greek letters
τ ε δ σ λ
tortuosity of the membrane pores porosity of the fiber membrane the thickness of membrane layer (μm) collision diameter, for water vapor is 2.641 Å mean free path (m)
MD membrane is defined as the ‘homogeneous membrane’. If the pore size on the membrane cross-section varies greatly and its influence on the mass transfer process cannot be neglected, then this kind of membrane is defined as the ‘heterogeneous membrane’. For example, the cross-section of the polyvinylidene fluoride (PVDF) hollow fiber membrane through TIPS method prepared by Lin et al. [19] has the bicontinuous structure and the pore size changes little along the pore axis direction on the cross-section. This kind of membranes can be considered as the homogeneous membrane as we defined. The PVDF composite hydrophobic hollow fiber membranes fabricated through non-solvent induced phase inversion method as reported by Hou et al. [16] have the heterogeneous structure on the cross-section, the pore size on the cross-section has a large variation range as they reported. That is to say, the membrane pores have a size distribution not only on the membrane surface but also on the membrane cross-section along the pore axis. In order to distinguish the pore size distribution on the membrane surface (surface pore size distribution), the pore size variation on the cross-section along the pore axis direction can be defined as interior pore size distribution. Many investigators considered the pore size distribution in calculating the permeate flux just on the membrane surface with different mass transfer regimes. However, the mass transfer mechanisms may also change when the vapor molecules transport from the feed side to the permeate side across the membrane matrix. Therefore in the theoretical calculation of mass transfer process, the different membrane layers should be considered based on the pore structure characteristics especially when dealing with the heterogeneous membrane. Further, the mass transfer resistance should be analyzed to guide for preparing high performance membrane. However, literatures concerning on this specific discussions of mass transfer process are hard to search out. With this in mind, the mechanisms of mass transfer process associating with operation conditions during the hollow fiber VMD process were discussed in this work. The simulation work of mass transfer process in both homogenous and heterogeneous membranes were investigated. The combination of Knudsen diffusion and viscous flow were engaged to calculate the permeate fluxes. The mass transfer resistances in different pore structure layers as an analogy to the typical heat transfer process were analyzed in the multi-layer model. The influences on the permeate flux of membrane characters including pore size, porosity, tortuosity and the membrane thickness were considered and their sensitivities to the permeate flux were discussed as well.
according to the Knudsen Number (Kn ), which is defined as the ratio of the vapor mean free path (λ ) and the pore diameter (dp ). If Kn < 0.01 (dp > 100 λ ), the molecule-molecule collisions will dominate the vapor transport process and the viscous flow occurs [11,12]. If Kn is greater than 10 (dp < 0.1 λ ), the molecule-pore wall collisions dominate the mass transfer process and the Knudsen diffusion happens in the membrane pores. While Kn is between 0.01 and 10 (0.1 λ < dp < 100 λ ), both molecule-molecule collisions and molecule-wall interactions have to be considered [13]. In this case, the viscous flow and the Knudsen diffusion mechanism are combined to describe the permeability of vapor transport through the porous membranes. This division method was mostly accepted by researchers as summarized by E. Drioli et al. [5] and Mostafa Abd El-Rady Abu-Zeid et al. [2]. In fact, when the Knudsen diffusion dominates, the vicious flow regime also occurs and vice versa. That is to say, the Knudsen diffusion and the viscous flow may exist simultaneously no matter how Kn varies in VMD process. In the literatures the different mass transfer mechanisms of VMD process based on the dusty gas model are stated, but few of them clearly point out the ranges of mass transfer regime associating with the operation conditions. The studies on the membrane pore size characteristics are important since the pore size is such a significant parameter in the mass transfer process as discussed above. The pore characters on the membrane surface such as the pore size distribution has been well studied in many literatures. However, the pore size variation on the cross-section of the MD membrane receives little discussions even though many researchers have reported in their work [14–18] – the SEM figures of the membrane cross-section. The single hydrophobic layer membranes and multi-layered membranes were both reported as Khayet et al. summarized [1]. The pore size along the pore axis on the cross-section may change greatly and different pore-character layers should be discussed separately. For example, both figure-like and sponge-like voids exist on the cross-section of the hollow fibers which was prepared from suspensions with different ceramic powders/PESF ratios as Zhang et al. reported [15]. The mass transfer mechanism in the figure-like and the sponge-like voids layers may be different according to their distinguish pore sizes. According to pore size distribution of the membrane layer structure, the homogeneous membrane and the heterogeneous membrane were defined here. If the membranes with narrow pore size variation on the membrane cross-section, or the pores on the cross-section can be considered as uniform pores, then this kind of
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2. Theory
Bv =
2.1. Mass transfer model
kb T 2 πP σ 2
Bt =
where kb is the Boltzmann constant, P is the mean pressure within the membrane pores and T is the absolute temperature. The value of collision diameter σ for water vapor is 2.641 Å. In VMD, the application of vacuum dramatically decreases the transmembrane hydrostatic pressure, therefore a high value of λ may be achieved. In other words, in the VMD process the mean free path of the gas molecule is largely dependent on the vacuum pressure and may be much larger than the mean membrane pore size. The membrane permeate flux J can be expressed as the combined influence of membrane permeability and the process driven force [21,22]:
When the homogenous membrane is utilized, the pores across the membrane section are within a small size range. The mass transfer process in this condition can be calculated by the theoretical model as stated in Eq. (8) without considering the interior pore size distribution. However when the heterogeneous structured membrane is utilized in MD process, the interior pore size of the MD membrane has a large distribution range. In this case, the mass transfer rate may be controlled by a specific pore size range or a specific membrane layer. So it is necessary to separate the membrane matrix into several transfer zones according to the variation of the interior pore size. The total permeate flux is equal in every zone based on the continuous flow process:
(2)
⎛ m (d =0.1λ) N ⎜ ∑ G kf j dj3 + 8τδ ⎜⎝ j =1 n (d = dmax )
+
∑ j = p (d =100λ )
J = B1 * Δp1 = B2 * Δp2 =…= Bi * Δpi
(G kf j dj3+G vf j d j4 P )
J=
j = m (d =0.1λ )
⎞ G vf j d j4 P ⎟⎟ ⎠
Gk =
Gv =
(3)
B= (4)
π 8ηRT
Δp Δp1 Δp2 Δp = =…= i = i = n b1 b2 bi ∑1=1 bi
(10)
where n is the number of membrane layers. Compare Eq. (2) and Eq. (10), it is obvious that the membrane permeability B can be expressed as:
where
32π 9MRT
(9)
where i is the sequence of different pore range layers. If we define the mass transfer resistance b as the reciprocal of B , the equation above can be rewritten as the driven force divided by the process resistance formula:
p (d =100λ )
∑
(8)
2.2. Mass transfer process
where B is the permeability of the membrane, which is seriously affected by the membrane characters such as the porosity and the pore size. The driven force is usually expressed as the water vapor difference (Δp ) between the feed and permeate side. The local temperature and concentration are considered to be the main factors that can influence the vapor pressure difference greatly. The permeability of the membrane (B) is calculated by the full dusty gas model as can be found in Refs [1,12,13]:
B=
1/2 εdˆ ⎛ 1 ⎛ 8RT ⎞ dˆ ⎞ ⎜⎜ ⎜ P ⎟⎟ ⎟ + ⎠ ⎝ τδRT ⎝ 3 πM 32η ⎠
As stated before, the Knudsen diffusion and the viscous flow may happen simultaneously no matter what the range of Kn is. Therefore the transition flow mechanism shown in Eq. (8) is accepted and used in the following study. It is worth noticing that the permeate flux calculation is based on the columnar pore hypothesis, the membrane pores are defined as the combination of the straight columnar pores and the socalled tortuosity. However, the tortuosity is hard to measure and the experience value is applied.
(1)
J = B*Δp
(7)
when 0.01 < Kn < 10, Knudsen diffusion and viscous flow decide the membrane permeability together, and also this mechanism is defined as the transition flow,
The heat and mass transfer in MD have been extensively studied and several theoretical models have been developed for vapor transporting through porous media to predict the performance of MD configurations. The dusty gas model is widely accepted by many investigators in MD field as stated before. The Knudsen number (Kn ), which is defined as the ratio of mean free path (λ) to the membrane pore size (dp), is accepted to divide the vapor transport patterns into Knudsen diffusion, viscous flow and their combination. For a specific membrane, the Knudsen number depends on the mean free path λ , which is calculated by the following expression [20]:
λ=
2 εdˆ P 32RTδτη
⎛ i = n 1 ⎞−1 1 = ⎜⎜ ∑ ⎟⎟ = b1 + b 2 +…+ bn ⎝ i =1 Bi ⎠
i=n
∑ Bi i =1
(11)
Base on the equations above, the total permeate flux of the membrane can be calculated by the following equation:
(5)
i=n
J = B * Δp = Δp * ∑ Bi
and N is the total number of pores per unit area, m is the last class of pores in Knudsen region, p is the last class of pores in transition region, n is the last class of pores in viscous flow region, f j is the fraction of pores with pore diameter dj , M is the relative molecule mass of water and η is the viscosity of vapor, τ is the pore tortuosity and δ is the porosity of the membrane. When the uniform pore diameter dˆ is assumed, the equations above can be simplified and can be rewritten as the specific formulas with different conditions. When Kn > 10, the Knudsen diffusion dominates the mass transfer process within the membrane pores,
εdˆ ⎛ 8RT ⎞ ⎟ ⎜ 3τδRT ⎝ πM ⎠
i =1
This equation comprehensively considers the mass transfer process in the heterogeneous membrane, and the mass transfer resistance in different membrane layers can be analyzed by this model. 2.3. Heat balance The MD is a combined process of mass transfer and heat transfer. Two mechanisms of heat transfer in MD have been developed, heat transfer through conduction across the membrane matrix (Qc ) and the latent heat associated with the evaporation process (Qv ).
1/2
Bk =
(12)
(6)
Qm = Qc + Qv
and when Kn < 0.01 the viscous flow will dominant, 117
(13)
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found elsewhere [24–27]. The turbulent flow with standard k − ε model was selected to solve the flow equations and SIMPLE algorithm was selected for the velocity and pressure coupled calculation. Second Order Upwind algorithm for all other terms were settled. In the simulation models, the membrane material zone was set to be porous zone, and the parameters were set to be the same as the experimental tests. The phase change of liquid water turning into vapor and the vapor transporting in the membrane layer(s) were realized by user defined functions (UDFs) which were based on the mass transfer equations as stated before in VMD process. The energy for water evaporation on the membrane surface of the feed side was also included in the UDFs because in the software Fluent@ only the heat conduction can be computed automatically under the selected calculation model. During the simulation process, the water flux of VMD process was realized by solving the mass transfer equations together with the heat transfer process. The calculation procedure is shown in Fig. 3. The flow equations were solved first until the calculation become convergent. The UDFs for water evaporation were engaged in solving the energy equations to realize the energy transfer process by evaporation. And then the mass transfer UDFs were loaded to calculate the water flux by solving both energy equations and the mass transfer equations. This is different from the reality that the mass transfer and the energy transfer process happening simultaneously during the VMD process. One should note that the influence of temperature and concentration polarization on the model flux can be calculated by the software automatically because the evaporation rate is calculated based on the membrane surface temperature and concentration of the feed side and the local pressure of the membrane pores. The temperature and concentration values and the pressure values in simulating the flux can be read by the macros within the software. Therefore the model flux can fit the experimental data without considering the influence of the temperature and concentration polarization even though the temperature and concentration polarization phenomenon exist both in experiments and the simulations.
It must be pointed out that the conducted heat by the membrane (Qc ) is considered to be the heat loss during the MD process. The useful heat in MD is the latent heat (Qv ) for the permeate flux. Therefore, a low thermal conductivity material of the MD membrane is helpful in minimizing the heat loss during the MD process. According to the research of Fine et al. [23], only 50–80% of the total heat is consumed as the useful heat in MD and the remainder is lost by thermal conduction of the membrane. However, in VMD process the heat conduction through the membrane matrix (Qc ) can be neglected due to the large thermal resistance in the boundary layer of the permeate side [2], and the vacuum applied on the permeate side also hinders the heat conduction process across the membrane matrix [6]. So the heat balance in the VMD process from the hot feed side to the permeate side at an equilibrium state can be expressed as
Qm = Qv = BΔHv
(14)
where ΔHv is the evaporation enthalpy of water at the absolute temperature T with the transmembrane flux B . 2.4. Experiments The cross-flow hollow fiber VMD experiments were carried out with self-made homogenous and heterogeneous membranes. The schematic of VMD experimental apparatus is presented in Fig. 1. The right side is the feed circulation process, the hot feed is bumped into the membrane configuration M4 and the concentrated feed flows back to the feed tank V1. The feed velocity can read from the rotameter R3. The produced vapor in M4 is extracted out by the vacuum pump P8 and condensed through the heat exchanger H5. The liquid pure water is collected by tanks V6 and V7. The feed flows in the shell side of the VMD configuration and the vapor passes through the membrane layer into the tube side, then the vapor is extracted out by the applied vacuum. The operation temperature in the experiments varied from 323 K to 363 K and the feed inlet velocity was kept at 0.05 m/s. The 1% (mass fraction) sodium chloride solution was supplied as the inlet feed in order to weaken the influence of concentration polarization on the mass transfer process. The cross-flow VMD configurations were well checked before the experiments and no leakage problems were found. The parameters of the experimental fibers are listed in Table 1. The overall parameters of fiber C and D were measured by the aperture analyzer. The detail parameters of the different membrane layers in fiber D were determined by the SEM pictures and the analyzer testing results. Cross-flow VMD configurations with homogenous and heterogeneous fibers were built separately. The effective length of the fibers are both 150 mm and the total number of fibers in each configuration is 50. The fibers use regular triangle arrangement and the spaces are about 4 mm from the fiber center to the adjacent fiber center.
3. Results and discussion 3.1. Mass transport model of VMD process As shown in Eq. (1), the mean free path λ is related to the local temperature and the mean pressure of the membrane pores. When a vacuum is applied on the permeate side, the mean pressure in the membrane pores can be very low and the value of λ may be very high. Fig. 4 shows the distribution of mean free path when the temperature range is 273–373 K and the mean pore pressure (absolute pressure) range is 0–100 kPa within a closed system. The saturation vapor
2.5. Simulation work The simulation work based on the experimental material data was carried out. Two different calculation models were built up as shown in Fig. 2. The simulation model A is based on the real hollow fiber C with homogenous membrane, and the simulation model B is based on the real hollow fiber D which has three different porous layers. The layers in the model B are different in porosity ε and thickness δ , and the pores in these layers are also different in radius r and tortuosity τ . Model A can be defined as the homogenous membrane model (single-layer model) and model B can be called the heterogeneous membrane model (multi-layer model) according to the characters of the membrane matrix. The Fluent (ANSYS) (version 15.0) program package was engaged to simulate the hollow fiber VMD progress in this study. 3D simulation models of both homogenous membrane and heterogeneous membrane were built to investigate the influence of different pore layers on membrane performance. The governing equations used to compute the fluid flow and the heat conduction in the calculation domain can be
Fig. 1. Schematic of experimental apparatus. V1—feed tank with heating apparatus, P2—pump, R3—rotameter, M4—cross-flow hollow fiber configuration, H5—heat exchanger, V6 and V7—pure water collection tanks, P8—vacuum pump.
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control the permeate flux together. According to the results in Fig. 4, the transition flow occurs when the membrane pore size is 0.2–0.5 µm with the operation condition of temperature range from 328 K to 373 K and the mean pore pressure range from 15 kPa to 100 kPa. These are the most common operation conditions within the VMD process. It can be drawn out that the mass transfer of VMD process is dominated by the Knudsen diffusion regime according to the Dusty-gas model only when the vacuum degree applied on the permeate side is very high as shown in Fig. 4 (absolute pressure is lower than 3 kPa), and the transition regime occurs when the feed temperature is high and the vacuum on the permeate side is low (mean pore pressure is high). The mass transfer mechanism in the remaining range is the mixing of Knudsen diffusion and viscous flow. When using the transition flow regime to calculate the permeate flux of VMD process, the Knudsen flow and the viscous flow are included simultaneously. This is also suitable when the mass transfer process is strictly dominated by the Knudsen flow or the viscous flow according to the dusty gas model. Therefore, in the following simulation work, the transition flow regime (the combination of Knudsen flow and viscous flow) was applied to simulate the mass transfer process both in the homogenous and heterogeneous membrane models.
Table 1 Parameters of hollow fibers and the experimental configurations. Parameters
Porosity ε Pore radius r Thickness δ Tortuosity τ Fiber Inner diameter Fiber outer diameter
Fiber C.
Fiber D. (Heterogeneous model)
(Homogeneous model)
Layer 1
Layer 2
Layer 3
Overall
55% 0.13 µm 0.27 mm 2.1 1.31 mm
60% 0.15 µm 0.12 mm 2.0 1.24 mm
80% 15 µm 0.14 mm 1.2
75% 0.20 µm 0.05 mm 1.8
60% 0.18 0.31 mm 1.5
1.85 mm
1.86 mm
pressure line (Psv line) is also included in this figure. The value of λ decreases sharply when the mean pore pressure increases from 0 kPa to 10 kPa. When the mean pore pressure is low enough (which means the vacuum degree applied on the permeate side is very high), the value of λ can be greater than 50 µm. In this case, the Knudsen number could be higher than 100 (The range of membrane pore is 0.1–0.5 µm) and the Knudsen diffusion will dominate the mass transfer process in the membrane pores as defined by the dusty gas model. However, the case of viscous flow dominated process will never occur in the studied range according to the definition (Kn < 0.01). As is known to all that when the vapor pressure is higher than the saturation vapor pressure at the certain temperature, the water will not evaporate into vapor. Therefore when the operation condition point is above the Psv line as shown in Fig. 4, no permeate flux could be measured theoretically. And when the operation condition point is under the Psv line (the shaded area in Fig. 4), the MD process could be carried out normally. If the operation temperature decreases with an isothermal process or the mean pore pressure increases with an isobaric process, the operation point will meet the Psv line and further the VMD process will unable to continue. According to the definition in Eq. (8), the transition flow (0.01 < Kn < 10) occurs when the temperature and the mean pore pressure both rise up, the Knudsen diffusion regime and viscous flow mechanism will
3.2. Homogenous membrane 3.2.1. Validation of simulation model The simulation model A was validated with the experiment fiber C (homogeneous membrane) and the mass transfer regime was set to be the combination of Knudsen flow and the viscous flow as stated before. The feed inlet velocity was kept at 0.05 m/s both in the experiments and the simulation work. The comparison of the experimental results and the simulation data was shown in Fig. 5. Good accuracy was achieved with the temperature range from 323 K to 363 K as can be observed. It means that the single-layer model of calculation model A works well with the real experimental fiber C under the same operation conditions. When the operation vacuum on the permeate side rises up from 0.09MPa to 0.096 MPa (S-0.09 MPa, S-0.092 MPa, S-0.094 MPa and S-0.096 MPa), the simulation results show that the variation trends of the permeate fluxes are the same with
Fig. 2. Computing models and experimental materials (A. simulation model of hollow fiber with uniform membrane layer, the homogenous membrane model or the single-layer model; B. simulation model of hollow fiber with three different membrane layers, the heterogeneous membrane model or the multi-layer model; C. cross-section of PVDF hollow fiber with uniform pore structure; D. cross-section of PVDF hollow fiber with non-uniform pore structure).
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Fig. 3. The procedure of simulation work.
Fig. 4. The mean free path distribution of water vapor. Temperature range is from 273K to 373 K, mean pore pressure range is from 0 kPa to 100 kPa, the unite of λ is micron (μm).
Fig. 5. Flux comparison of simulation model A and experiment fiber C (single-layer model).
the experimental values (Exp-0.09 MPa), and the permeate flux increases when the vacuum degree rises up.
permeate flux of the pore parameters including the pore size, the porosity, the tortuosity and the thickness of the membrane were studied here. In order to simplify the calculation process, the simulated membrane characters were considered in the single layer membrane which is stated as the calculation model A. The single factor analysis
3.2.2. Influences of membrane characters Based on the simulation model above, the influence on the 120
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parameter was changed each time and the other parameters remained unchanged. All of these calculations were carried out under the same operation conditions with feed inlet velocity at 0.05 m/s and the vacuum degree applied on the permeate side at 0.09 MPa (absolute pressure at 10 kPa). Within the research scope, the variation range of the membrane character was divided into 10 even scales to save calculation costs. The permeate flux increment in each scale was calculated as the absolute value Ji−Ji+1 and the sensitivity of the character was defined as the ratio of the absolute incremental flux and the former permeate flux
method was applied here and the coupled influences of these membrane parameters were not considered in this work. In all of these simulation process, the feed inlet velocity was set to be 0.05 m/s and the absolute pressure on the permeate side was 10 kPa (vacuum degree of 0.09 MPa). As shown in Fig. 6A and B, the permeate fluxes increase with the enlargement of the pore size and the porosity. The flux growth rate enlarges when the feed inlet temperature rises up. It is not difficult to understand this flux variation according to Eq. (8) that the membrane permeate coefficient is positively related to the pore size and the porosity, and when the temperature rises up, the saturation vapor pressure in the membrane pores lifts remarkably especially at a high temperature range as shown in Fig. 4, the Psv line. However the flux decreases sharply when the membrane layer becomes thicker until 0.25 mm, and then decreases slowly when the thickness of the membrane layer continues to increase as shown in Fig. 6C. The influence of the tortuosity on the permeate flux has a similar tendency with the thickness of the membrane as indicated in Fig. 6D. The decreases of flux caused by the membrane thickness and the tortuosity of the membrane pores can make sense with the permeate coefficient calculation model in Eq. (8), the membrane flux varies inversely with the membrane thickness and the pore tortuosity. Fig. 6 also shows that high feed temperature contributes to high membrane flux because the saturation vapor pressure increases with the rise of temperature and the vapor pressure difference.
Vs =
Ji − Ji +1 Ji
(15)
where Vs is the dimensionless sensitivity of the membrane characters, i stands for the number of the scales and the range is from 0 to 10, J is the permeate flux. For example, the research scope of the tortuosity is from 1.0 to 3.0, then J0 means the permeate flux with the tortuosity at 1.0 and J1 is the permeate flux with tortuosity at 1.3 (the increment of the tortuosity in the research scope is 0.3 because it is divided into 10 even scales). Then the sensitivity value of tortuosity on scale 1 can be calculated as J0 − J1 and the other scales could be calculated out by the J0
same method, as well as the sensitivities of other membrane parameters. The average sensitivity of temperature range from 323 K to 363 K in each scale was used as the final value. With this definition the influences of different membrane parameters on the permeate flux could be compared and their comparisons are shown in Fig. 7. The permeate flux is largely dependent on the membrane characters as shown in Fig. 6. In Fig. 7, the relative influences of these membrane parameters are well demonstrated. Among all of these four membrane characters, the pore size affects the membrane flux most and the
3.2.3. Sensitivity of membrane characters The sensitivities of the membrane characters were also investigated here to find out the major factor affecting the membrane performance. The single factor analysis method was applied that only one membrane
Fig. 6. The influence of pore parameters on permeate flux (A: pore radius, B: porosity, C: membrane thickness, D: tortuosity).
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multi-layer model of simulation model B considered the pore size variation across the membrane matrix and the changes of mass transfer process in different membrane layers were calculated individually, therefore the more accuracy results could be obtained. One should note that when the operation condition point is under the Psv line as shown in Fig. 4 the VMD process could be carried out normally and the reasons were explained before. The least operation temperature is 323 K here and only when the mean pore pressure is below the saturate vapor pressure at 323 K (12.31 kPa) can the VMD process carried out with measurable permeate flux. We choose 10 kPa (absolute pressure) as the vacuum pressure both in the experiments and the simulation work here. The influence of permeate pressure on the fluxes of model A and B were not discussed because the flux variation trend of the influence on model B is the same as on model A which was shown in Fig. 5. One more conclusion can be drawn from Fig. 5 and Fig. 8 that the mass transfer process of homogenous membrane can be simulated by the Dusty-gas model directly and good accuracy may be achieved. But the mass transfer process in the heterogeneous membrane should be considered individually according to the membrane interior pore size distribution. The multi-layer model is more appropriate than the single-layer model in simulating the VMD process with heterogeneous membrane as the results shown in our work, and a more precise result can be obtained based on the experimental data.
Fig. 7. The sensitivities of the membrane characters.
defined sensitivity factor is 0.37 at the first scale. The influence of the membrane thickness is also very striking and the sensitivity factor is 0.24, ranking in the second place. The pore tortuosity and the membrane porosity are not as important as the pore size and the membrane thickness as depicted in Fig. 7 and the influence of the porosity on permeate flux is the least among them. It is also found that the sensitivities of the membrane characters are constant with the max permeate fluxes in the research domains as shown in Fig. 6, 128 kg/ (m2 h) for the pore size, 91.5 kg/(m2 h) for the membrane thickness, 71.6 kg/(m2 h) for the pore tortuosity and 62 kg/(m2 h) for the membrane porosity, respectively. These results show that the membrane performance can change greatly when the membrane characters varied, and the membrane performance could be optimized by designing a membrane with large pore size, high porosity, thin membrane thickness and straight pores. However, it must be pointed out that membrane with large pore size is more likely to be contaminated than the small pore size membranes and pore wetting problems may happen easily.
3.3.2. Mass transfer resistance distribution According to the interior pore size distribution across the multiple membrane layers, the mass transfer resistances in these layers were studied to analyse the mass transfer resistance distributions of the VMD process in the heterogeneous membrane. In Eqs. (9)–(12), the mass transfer resistances were discussed individually according to the interior pore size distribution, and the total mass transfer resistance can be influenced by different membrane layers as described in Eq. (12). The mean pressure and the local temperature in different layers can be obtained from the simulation model. The mass transfer resistances in layer 1 (R-L1) and layer 3 (R-L3) in model B were studied and the results were shown in Fig. 9. As can be observed in Fig. 9, the mass transfer resistance in layer 1 (RL1) decreases when the temperature rises up, but in layer 3 (R-L3) the mass transfer resistance almost unchanged. The reasons may be that the temperature is more sensitive to the mass transfer process in membrane layer 1 with smaller mean pore size and thicker layer matrix than in membrane layer 3 as shown in Table 1. The low porosity of layer 1 also contributes to the high mass transfer resistance in the membrane pores.
3.3. Heterogeneous membrane The heterogeneous membrane of simulation model B (multi-layer model) was validated with the experiment fiber D and compared to the homogenous simulation model A. For the homogeneous model A (single-layer model) the membrane parameters were the same as the overall parameters of the real fiber D. And the parameters of the three layers of fiber D were used for the heterogeneous model B (multi-layer model). The mass transfer regime was set to be the combination of Knudsen flow and the viscous flow as stated before. The mass transfer resistance model as deduced in Eqs. (9)–(12) was applied here to calculate the permeate flux and analysis the resistance distribution in different pore size layers. 3.3.1. Comparison of homogenous model and heterogeneous model The permeate fluxes of homogenous model and heterogeneous model were compared in Fig. 8. The homogenous model A was found to be less accuracy when compared to simulation model B although these two simulation models can both fit well with the experimental results. The relative error of simulation model A can reach higher than 7% especially when the temperature is high, while this value of simulation model B is below 4%. This may be that high temperature changes the mass transfer regime in the membrane pores according to Eq. (1). The high temperature increases the mean free path and the mass transfer regime in the small pore size area may be controlled by Knudsen diffusion. However in the simulation model A the uniform pore size across the membrane matrix was considered and the interior pore size distribution was neglected. The
Fig. 8. Flux comparison of simulation model A and B to the experimental fiber D (heterogeneous membrane layer), with permeate pressure at 10 kPa and feed inlet velocity at 0.05 m/s.
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tortuosity on the permeate flux. The multi-layer model constructed in our work is more accurate than the typical dusty gas model (singlelayer model with transition flow regime) in calculating the permeate flux with heterogeneous membrane. The mass transfer resistance mainly exists in the membrane layers with small pore size. These results could be used to design MD membrane with high performance. Acknowledgement The authors gratefully acknowledge the Projects of Science & Technology Plan Program of Tianjin (Grant no.: 12ZCZDSF02200). References [1] M. Khayet, Membranes and theoretical modeling of membrane distillation: a review, Adv. Colloid Interface Sci. 164 (2011) 56–88. [2] M.A.E.-R. Abu-Zeid, Y. Zhang, H. Dong, L. Zhang, H.-L. Chen, L. Hou, A comprehensive review of vacuum membrane distillation technique, Desalination 356 (2015) 1–14. [3] E. El-Zanati, K.M. El-Khatib, Integrated membrane –based desalination system, Desalination 205 (2007) 15–25. [4] M. Khayet, T. Matsuura, Membrane Distillation: Principles and Applications, Elsevier, Great Britain, 2011. [5] E. Drioli, A. Ali, F. Macedonio, Membrane distillation: recent developments and perspectives, Desalination 356 (2015) 56–84. [6] A. Hassan, M. Darwish, H. Fath, H. Abdulrahim, Vacuum membrane distillation: state of the art, in: Euromed 2015 Desalination for Clean Water and Energy, 2015. [7] J.I. Mengual, M. Khayet, M.P. Godino, Heat and mass transfer in vacuum membrane distillation, Int. J. Heat. Mass Transf. 47 (2004) 865–875. [8] I. Hitsov, T. Maere, K. De Sitter, C. Dotremont, I. Nopens, Modelling approaches in membrane distillation: a critical review, Sep. Purif. Technol. 142 (2015) 48–64. [9] R.W. Schofield, A.G. Fane, C.J.D. Fell, Gas and vapour transport through microporous membranes. II. Membrane distillation, J. Membr. Sci. 53 (1990) 173–185. [10] A.O. Imdakm, M. Khayet, T. Matsuura, A Monte Carlo simulation model for vacuum membrane distillation process, J. Membr. Sci. 306 (2007) 341–348. [11] A. Alkhudhiri, N. Darwish, N. Hilal, Membrane distillation: a comprehensive review, Desalination 287 (2012) 2–18. [12] M. Khayet, K. Khulbe, T. Matsuura, Characterization of membranes for membrane distillation by atomic force microscopy and estimation of their water vapor transfer coefficients in vacuum membrane distillation process, J. Membr. Sci. 238 (2004) 199–211. [13] M. Khayet, T. Matsuura, Pervaporation and vacuum membrane distillation processes: modeling and experiments, AlChE J. 50 (2004) 1697–1712. [14] C. Feng, B. Shi, G. Li, Y. Wu, Preparation and properties of micro-porous membrane from poly(vinylidene fluoride-co-tetrafluoroethylene) (F2.4) for membrane distillation, J. Membr. Sci. 237 (1–2) (2004) 15–24. [15] J.-W. Zhang, H. Fang, J.-W. Wang, L.-Y. Hao, X. Xu, C.-S. Chen, Preparation and characterization of silicon nitride hollow fiber membranes for seawater desalination, J. Membr. Sci. 450 (2014) 197–206. [16] D. Hou, J. Wang, X. Sun, Z. Ji, Z. Luan, Preparation and properties of PVDF composite hollow fiber membranes for desalination through direct contact membrane distillation, J. Membr. Sci. 405–406 (2012) 185–200. [17] M.C. García-Payo, M. Essalhi, M. Khayet, Preparation and characterization of PVDF–HFP copolymer hollow fiber membranes for membrane distillation, Desalination 245 (2009) 469–473. [18] B. Wu, K. Li, W.K. Teo, Preparation and characterization of poly(vinylidene fluoride) hollow fiber membranes for vacuum membrane distillation, J. Appl. Polym. Sci. 106 (2007) 1482–1495. [19] L. Lin, H. Geng, Y. An, P. Li, H. Chang, Preparation and properties of PVDF hollow fiber membrane for desalination using air gap membrane distillation, Desalination 367 (2015) 145–153. [20] N.N. Li, A.G. Fane, W.W. Ho, T. Matsuura, Advanced Membrane Technology and Applications, John Wiley & Sons, New Jerzey, 2008. [21] K.W. Lawson, D.R. Lloyd, Membrane distillation, J. Membr. Sci. 124 (1997) 1–25. [22] M.S. El-Bourawi, Z. Ding, R. Ma, M. Khayet, A framework for better understanding membrane distillation separation process, J. Membr. Sci. 285 (2006) 4–29. [23] A.G. Fane, R.W. Schofield, C.J.D. Fell, The efficient use of energy in membrane distillation, Desalination 64 (1987) 231–243. [24] A.L. Ahmad, K.K. Lau, M.Z.A. Bakar, Impact of different spacer filament geometries on concentration polarization control in narrow membrane channel, J. Membr. Sci. 262 (2005) 138–152. [25] H. Yu, X. Yang, R. Wang, A.G. Fane, Analysis of heat and mass transfer by CFD for performance enhancement in direct contact membrane distillation, J. Membr. Sci. 405–406 (2012) 38–47. [26] M.M.A. Shirazi, A. Kargari, A.F. Ismail, T. Matsuura, Computational fluid dynamic (CFD) opportunities applied to the membrane distillation process: state-of-the-art and perspectives, Desalination 377 (2016) 73–90. [27] B. Lian, Y. Wang, P. Le-Clech, V. Chen, G. Leslie, A numerical approach to module design for crossflow vacuum membrane distillation systems, J. Membr. Sci. 510 (2016) 489–496.
Fig. 9. The mass transfer resistances through the membrane layers based on the heterogeneous membrane D, resistances (R) and their percentages (P) of membrane layers (layer 1, L1 and layer 3, L3, respectively) with different pressures at permeate side (10 kPa and 5 kPa).
The influence of the permeate side pressure on the mass transfer resistance of the membrane layer is not remarkable as shown in Fig. 9. When the absolute pressure changes from 10 kPa to 5 kPa, the resistances in layer 1 and layer 3 both indicate a little increase. The pressure at the permeate side influences the pressure drop across the membrane pores, but only the viscous flow regime is affected by this pressure change as shown in Eq. (8). Furthermore, the mass transfer process is dominated by the Knudsen diffusion with the operation conditions of pressure at 5–10 kPa and temperature range 323–368 K as seen in Fig. 4. A low absolute pressure on the permeate side indicates that the mean pressure across the membrane pores is small, and the mass transfer resistance increases as the definition in Eq. (11). Therefore the vacuum pressure in this variation range just affects the mass transfer resistance slightly. The comparison of the mass transfer resistances with different permeate pressure also reveals that the resistance in the membrane layers is mainly affected by the membrane characteristic parameters. The influences of the operating conditions such as the vacuum pressure and the feed temperature are not as remarkable as the membrane characters. The percentage of the mass transfer resistance in different membrane layers shown in Fig. 9 reveals that the main mass transfer resistance across the membrane matrix is focus on layer 1 and layer 3. Approximately two thirds of the total resistance is distributed in membrane layer 1, and the percentage of mass transfer resistance of layer 1 reduces from 68% to 63% when the temperature changes from 323 K to 363 K. When the pressure on the permeate side changes, the percentage of mass transfer resistance almost keeps the same. This value of layer 3 rises up from 31% to 36% as the temperature changes from 323 K to 363 K. This also indicates that layer 2 with large pore size almost has no mass transfer resistance when compared to layer 1 and layer 3. It may be helpful in improving the permeate performance by decreasing the mass transfer resistance in layer 1 and layer 3. 4. Conclusion The mass transfer of hollow fiber VMD process with homogenous membrane model and the heterogeneous membrane model were both investigated in this study. The combination of Knudsen diffusion and viscous flow regimes without considering the Knudsen number in calculating the permeate flux can fit well with the experiments. In large ranges of vacuum pressure and temperature the mass transfer mechanism can be described by the transition flow, which is the combination of Knudsen diffusion and viscous flow. The membrane characters of pore size and the thickness are more sensitive than the porosity and the
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