Review of Pervaporation and Vapor Permeation Process Factors Affecting the Removal of Water from Industrial Solvents

Abstract

A recent review article (J Chem Technol Biotechnol 94: 343–365 (2019)) identified several commercially-available permselective materials for drying organic solvents with pervaporation (PV) and vapor permeation (V·P) separation processes. The membrane materials included polymeric and inorganic substances exhibiting a range in the performance characteristics: water permeance, water/solvent selectivity, and maximum use temperature. This paper provides an overview of the factors affecting the design of PV/V·P processes utilizing these membranes to remove water from common organic solvents. Properties of the specific membrane and of the solvent substantially affect the PV/V·P separation. Equally important is the impact of operating parameters on the overall separation. To study these impacts, simplified process performance equations and detailed spreadsheet calculations were developed for single-pass and recirculating batch PV systems and for single-pass V·P systems. Estimates of membrane area, permeate concentration, solvent recovery, permeate condenser temperatures, and heating requirements were calculated. Process variables included: solvent type, water permeance, water/solvent selectivity, initial and final water concentrations, operating temperature (PV) or feed pressure (V·P), temperature drop due to evaporation (PV) or feed-side pressure drop (V·P), and permeate pressure. The target solvents considered were: acetonitrile, 1-butanol, N,N-dimethyl formamide, ethanol, methanol, methyl isobutyl ketone, methyl tert-butyl ether, tetrahydrofuran, acetone, and 2-propanol.

Introduction

The life cycle environmental impact of using organic solvents can be reduced by recovering and reusing the solvents, avoiding the impacts associated with producing the virgin solvents and disposing of the spent solvents^1–7. Solvent reuse/reprocessing requires separation technologies to recover and purify the solvents from mixtures with other materials they encounter in industrial applications. For hydrophilic solvents, and even some solvents not typically considered “water loving”, water is often a contaminant that must be removed prior to reuse. Unfortunately, many solvents form azeotropic mixtures with water, a property that makes separation by evaporation or simple distillation complicated. Alternative separation technologies, including the membrane processes of pervaporation (PV) and vapor permeation (V·P) have the potential to remove water from solvents, even under azeotropic conditions⁸. A recent article by this author provides a state-of-the-science review of materials available as the selective layer(s) of PV and V·P membranes for the removal of water from ten commonly used organic solvents that are promising targets for recovery and reuse⁹. Those solvents included acetonitrile, 1-butanol, N,N-dimethyl formamide (DMF), ethanol, methanol, methyl isobutyl ketone (MIBK), methyl tertiary butyl ether (MtBE), tetrahydrofuran (THF), acetone, and 2-propanol (isopropanol). The analysis concluded that a robust and varied PV/V·P membrane industry has developed, with membranes containing water-selective layers of poly(vinyl alcohol), polyimides, amorphous perfluoro polymers, sodium Linde Type A (NaA) zeolites, chabazite (CHA) zeolites, T-type zeolites, or hybrid silicas being commercially available. The objective of this work is to provide an overview of the factors that affect the design of PV/V·P processes that utilize these membranes to remove water from the target solvents.

Briefly, PV and V·P, depicted in Figure 1, involve the sorption and diffusion of molecules in and through a thin, dense, permselective membrane from a feed fluid into a permeate vapor. The differences in sorption and diffusion of the molecules present in the mixture result in a separation: the composition of the stream that is transported through the membrane (the “permeate”) is different than the composition of the fluid on the feed side of the membrane. The composition of the stream rejected by the membrane (the “retentate”) is also altered. The driving force for transport is the chemical potential difference between the feed-side mixture and the permeate, represented by the fugacity or, more conveniently, the partial pressure of each compound. In the context of this work on solvent dehydration, water preferentially permeates the membranes relative to the organic solvents, leaving a retentate depleted in water and a permeate enriched in water relative to the feed mixture. Thus, the retentate is the solvent to be reused while the permeate is either disposed of or processed for additional resource recovery.

Figure 1.

Illustrations of (a) vapor permeation (V·P) and (b) pervaporation (PV) processes with molar feed flow rate ($\dot{N}$), species i feed fluid mole fraction (x_i), feed temperature (T^F), feed pressure (p^F), permeate pressure (p^P), permeate mole fraction (y_i), and membrane permeance ($P_i/\mathcal{l}$). The mass transport model for pervaporation assumes a theoretical vapor phase is present in equilibrium with the feed-side liquid yielding hypothetical feed vapor mole fraction ($x_i^{*}$) and total feed vapor pressure (p^F*).

Simplified example flow diagrams for PV and V·P are illustrated in Figure 2 and Figure 3, respectively. The distinction between PV and V·P is the phase of the feed-side fluid: liquid for PV and vapor for V·P. In PV, the water/solvent liquid mixture is heated to a target feed temperature, processed through a membrane module or set of modules as a liquid, and then the reduced-water solvent retentate liquid is either returned to the feed source for a “batch” operation (Option 1 in Figure 2) or considered the solvent product for a “single-pass” operation (Option 2 in Figure 2). The permeate is a vapor, usually involving vacuum conditions where the vacuum is generated through the combination of a vapor condenser and a small vacuum pump to remove non-condensing gases, the latter originating from vacuum leaks or dissolved gases. The flow diagram for V·P looks very similar to that of PV. However, since most solvent-use applications require the solvent in liquid form, employing V·P for solvent drying would necessitate an evaporation step to create the vapor phase feed to the V·P system and then a condenser to recover the retentate as a liquid. The vapor generation unit could be a simple complete evaporator (as shown in Figure 3) or a distillation unit - taking advantage of the stripping and enrichment capabilities of distillation units as well as the ability of distillation units to handle dissolved and suspended solids¹⁰. Example flow diagrams of PV and V·P systems combined with a batch evaporator, evaporator with a rectification column, and a full distillation column are illustrated in Figures S-2 through S-7 of the Supporting Information.

Figure 2.

Simplified schematic diagram of a pervaporation (PV) process with the option of returning the water-depleted retentate liquid to the feed source tank (recirculating batch mode) or managing the retentate as the solvent product (single-pass mode). Common ancillary components, such as a feed-retentate recuperative heat exchanger, are not shown.

Figure 3.

Simplified schematic diagram of a vapor permeation (V·P) process with the option of returning the water-depleted retentate condensate to the feed source tank (recirculating batch mode) or managing the retentate as the solvent product (single-pass mode).

Evaporation-based separation technologies, especially distillation, predominate in the arena of solvent recovery due to the usually advantageous vapor-liquid equilibrium (VLE) behavior, relative ease of design, and potential for reduced incremental cost at larger scale. The ratio of vapor phase concentrations divided by the ratio of liquid phase concentrations when the two phases are in equilibrium is referred to as “relative volatility”. Relative volatility of species 1 compared to species 2 in a mixture, ${\alpha }_{12}^{\mathit{\text{VLE}}}$, is a measure of the separation produced by a single VLE stage:

$${\alpha }_{12}^{\mathit{\text{VLE}}}=\frac{\left({\omega }_1^V/{\omega }_2^V\right)}{\left({\omega }_1^L/{\omega }_2^L\right)}=\frac{\left(Y_1/Y_2\right)}{\left(X_1/X_2\right)}=\frac{{\gamma }_1{p}_1^{\mathit{\text{sat}}}}{{\gamma }_2{p}_2^{\mathit{\text{sat}}}}$$ Equation 1

Where ${\omega }_i^k$ is the mass fraction in phase k (either the liquid (L) or the vapor (V)), X_i and Y_i are the mole fractions in the liquid and vapor, respectively, γ_i is the activity coefficient, and $p_i^{\mathit{\text{sat}}}$ is the saturated vapor pressure, all of species i. The right side of the equation results by inserting the VLE equilibrium relationship: $Y_i=X_i{\gamma }_i{p}_i^{\mathit{\text{sat}}}$.

When species 1 is enriched in the vapor relative to species 2, ${\alpha }_{12}^{\text{VLE}}$ is greater than 1. Values of ${\alpha }_{12}^{\text{VLE}}$ less than 1 indicate the opposite. When multiple ideal VLE stages are strung together, as in a distillation column, the vapor composition ratio at the N^th stage is calculated from the liquid composition ratio at the 1^st stage and the product of the relative volatility values for the N stages as:

$${\left(\frac{Y_1}{Y_2}\right)}_N={\left(\frac{X_1}{X_2}\right)}_1\left[\prod _{n=1}^N{\left({\alpha }_{12}^{\mathit{\text{VLE}}}\right)}_n\right]$$ Equation 2

While the minimum reflux ratio required for a given separation can be calculated based on the Underwood method¹¹, the minimum number of stages required to yield a specific top vapor product and bottom liquid product from a distillation column can be estimated with the Fenske equation¹²:

$$N_{\mathit{\text{min}}}=\frac{\text{ln\hspace{0.17em}}\left[{\left(\raisebox{1ex}{$Y_1$}\!\left/ \!\raisebox{-1ex}{$Y_2$}\right.\right)}_N{\left(\raisebox{1ex}{$X_2$}\!\left/ \!\raisebox{-1ex}{$X_1$}\right.\right)}_1\right]}{\text{ln\hspace{0.17em}}{\overline{\alpha }}_{12}^{\mathit{\text{VLE}}}}$$ Equation 3

Where ${\overline{\alpha }}_{12}^{\mathit{\text{VLE}}}$ is the geometric mean of ${\alpha }_{12}^{\mathit{\text{VLE}}}$ over all stages, sometimes approximated as the geometric mean of ${\alpha }_{12}^{\text{VLE}}$ at the top and bottom of the column. Thus, larger relative volatility values result in fewer stages of VLE required to accomplish a desired separation. Conversely, as ${\alpha }_{12}^{\text{VLE}}$ approaches 1, the number of stages becomes infinite. If ${\alpha }_{12}^{\text{VLE}}$ drops below 1, the separation reverses with species 2 enriched in the vapor instead of species 1. When ${\alpha }_{12}^{\text{VLE}}$ is equal to 1, there is no difference between the vapor phase and the liquid phase concentrations, thus evaporation offers no separation of the two species. This condition is referred to as an azeotrope. To separate azeotropic mixtures, the VLE behavior must be altered by changing conditions, such as adding a third compound, or an alternative technology must be introduced. While the minimum number of stages can be estimated by determining the course of a distillation line¹³, the minimum energy demand can be calculated with the aid of pinch theory^{14, 15}.

The calculated volatility of water relative to that of a solvent in a binary mixture with each of the ten organic solvents highlighted in this work is shown in Figure 4 as a function of the weight fraction of water in the liquid phase over the full concentration range. MIBK, MtBE, and 1-butanol are only partially miscible with water and the curves in the figure are only shown in the regions where the liquid is a single phase. The point at which a relative volatility curve crosses the “no separation” line (${\alpha }_{\text{ws}}^{\text{VLE}}=1$) is the azeotrope of that mixture.

Figure 4.

Relative volatility of water/solvent mixtures (${\alpha }_{\mathit{\text{ws}}}^{\mathit{\text{VLE}}}$) as function of water weight fraction in the liquid phase ($w_w^L$). Values calculated with NRTL model at the normal boiling point temperature of each solvent. Solid gray “no separation” line (${\alpha }_{\mathit{\text{ws}}}^{\mathit{\text{VLE}}}$=1) added as a reference. Values for 1-Butanol, MIBK, and MtBE shown in the range of solubility of each with water^53–55.

Several observations can be made from Figure 4. First, the relative volatility curves for water in acetone, DMF, and methanol do not cross the no separation line, consistent with the fact that these solvents do not form an azeotrope with water. In particular, the water/acetone and water/methanol systems exhibit solvent enrichment in the vapor phase (${\alpha }_{\text{ws}}^{\text{VLE}}<1$) over the entire concentration range. Although, at low water concentrations, ${\alpha }_{\text{ws}}^{\text{VLE}}$ for water/acetone is practically equal to 1 and ${\alpha }_{\text{ws}}^{\text{VLE}}$ for water/methanol is only 0.45. Further, only the water/DMF system exhibits water enrichment (${\alpha }_{\text{ws}}^{\text{VLE}}>1$) in the vapor over the entire concentration range. Second, except for DMF, the solvents are strongly enriched in the vapor phase at low solvent concentrations with solvent/water relative volatility (i.e. $1/{\alpha }_{\text{ws}}^{\text{VLE}}$) ranging from 9.5 to 570 at infinite solvent dilution (i.e. the right edge of Figure 4). Thus, evaporation processes effectively strip solvents from binary aqueous mixtures into the vapor phase at low solvent concentrations. Third, for most of the solvents, the evaporative separation of mixtures containing less than 20 wt% water is challenging due to the presence of an azeotrope or low relative volatility – or both. The relative volatility behavior in this water concentration range is highlighted in Figure S-1 in the Supporting Information. The water/ethanol system presents particularly significant distillation challenges in this range because the relative volatility is close to 1 and exhibits an azeotrope at 4 wt% water.

Due to these observations, this paper will focus on the removal of water from mixtures containing less than 20 wt% water, using the water/ethanol system as the principal example. Although binary systems with a low relative volatility or an azeotrope cannot easily be separated by simple distillation, the same is not true for separation processes that operate on principles other than VLE. For example, PV and V·P leverage the sorption and diffusion differences of the mixture components in a dense membrane material to separate the mixture. Because of this, even azeotropic mixtures can be separated with PV or V·P with an appropriate membrane material and operating conditions.

PV and V·P membrane performance descriptors

In order to probe the variables that affect PV or V·P process performance, the relationships and parameters that define membrane performance should be established. The performance of a PV/V·P membrane for a given feed composition and temperature is most commonly reported in terms of total mass flux, or mass flux of water, and either separation factor of species 1 relative to species 2 (β₁₂) or permeate purity in terms of water weight fraction ($w_w^P$). The mass flux of a species (J_i, kg m⁻² s⁻¹) is defined as the mass of a species (m_i, kg) permeating the membrane per unit membrane area (A, m²) and unit time (t, s) as:

$$J_i=\frac{m_i}{At}$$ Equation 4

The molar flux, j_i (kmol m⁻² s⁻¹), is simply the mass flux divided by the molecular weight, M_i (kg kmol⁻¹).

The molar flux of a compound through a permselective membrane is a function of membrane transport properties and the partial vapor pressure driving force for transport, represented as¹⁶:

$$j_i=\frac{P_i}{\mathcal{l}}\left(p_i^F-p_i^P\right)$$ Equation 5

Where P_i is the molar permeability of species i in the permselective material (kmol m m⁻² s⁻¹ kPa⁻¹), ℓ is the thickness of the permselective layer (m), and $p_i^F$ and $p_i^P$ are the partial pressures (kPa) of species i in the feed-side fluid and permeate vapor, respectively. Permeability is defined as the product of the solubility and diffusivity of a mobile species in the dense permselective membrane. For a prepared membrane with fixed or unknown selective layer thickness, the ratio of $P_i/\mathcal{l}$, termed “permeance” (Π_i) (kmol m⁻² s⁻¹ kPa⁻¹), is often the most appropriate performance parameter. Common units for permeance are gas permeation units (GPU), defined as 10⁻⁶ cm³(STP) cm⁻² s⁻¹ cmHg⁻¹ with 1 kmol m⁻² s⁻¹ kPa⁻¹ = 2.99×10⁹ GPU. Common units for permeability are Barrer, defined as 10⁻¹⁰ cm³(STP)·cm cm⁻² s⁻¹ cmHg⁻¹ with 1 kmol·m m⁻² s⁻¹ kPa⁻¹ = 2.99×10¹⁵ Barrer.

The separation factor for PV or V·P is defined similarly to relative volatility, as a ratio of the permeate vapor compositions to a ratio of the feed-side compositions for two species:

$${\beta }_{12}=\frac{\left(w_1^P/w_2^P\right)}{\left(w_1^F/w_2^F\right)}=\frac{\left(y_1/y_2\right)}{\left(x_1/x_2\right)}=\frac{\left(j_1/j_2\right)}{\left(x_1/x_2\right)}$$ Equation 6

Where $w_i^k$ is the mass fraction of species i in stream k (feed (F), retentate (R), or permeate (P)), respectively, and x_i and y_i are the mole fractions in the feed fluid and permeate vapor, respectively. Lowercase x and y are used for PV or V·P to distinguish them from the VLE uppercase variables X and Y. The ratio of permeate mass fractions is equivalent to the ratio of mass fluxes and, as shown in Equation 6, the ratio of permeate mole fractions is equal to the ratio of molar fluxes. Thus, given the feed composition, the separation factor can be calculated from the permeate water purity and vice versa. Similarly, given the permeate composition, one can calculate total flux from water flux and vice versa.

For both PV and V·P, the permeate is a reduced pressure vapor where the partial pressure of a compound in the permeate, $p_i^P$, is simply calculated as the permeate mole fraction (y_i) times the total permeate pressure (i.e. $p_i^P=y_i{p}^P$). This is also the case for V·P feed-side streams with partial pressures calculated from the mole fraction x_i and total feed-side pressure p^F (i.e. $p_i^F=x_i{p}^F$). However, for PV, the feed-side partial pressure of a compound experienced by the membrane is calculated as if there was a hypothetical vapor phase (designated with an asterisk*) in equilibrium with both the feed-side liquid and the membrane¹⁶. The hypothetical feed-side partial vapor pressure ($p_i^{F*}$) is calculated from the actual feed-side liquid composition (x_i) using VLE behavior as:

$$p_i^{F*}=x_i{\gamma }_i{p}_i^{\mathit{\text{sat}}}$$ Equation 7

Such a calculation requires either experimental VLE data or thermodynamic models to calculate the activity coefficient (γ_i) for each compound in a solvent/water mixture and the saturated vapor pressure ($p_i^{\mathit{\text{sat}}}$) of each compound. The activity coefficients calculated with the NRTL model for water and solvents in binary mixtures at the normal boiling point temperature as a function of water in the liquid (0–20 wt%) are shown in Figure 5. The Antoine relationship was used to calculate $p_i^{\mathit{\text{sat}}}$. The NRTL and Antoine parameters used herein are given in Tables S-1 and S-2, respectively, in the Supporting Information. The activity coefficients for the solvents are in the narrow range of 1 to 1.6 while those of water range more widely from 1 to almost 28.

Figure 5.

Activity coefficients of (a) water and (b) solvent for binary water -solvent mixtures as a function of water weight fraction in the liquid phase ($w_w^L$) over the 0–20 wt% range. Values calculated with NRTL model at the normal boiling point temperature of each solvent. Values for MIBK and MtBE shown in the range of solubility of each with water^{53, 54}.

The total hypothetical feed-side vapor pressure for a PV liquid feed is then the sum of the hypothetical partial pressures:

$$p^{F*}={\sum }_i{x}_i{\gamma }_i{p}_i^{\mathit{\text{sat}}}$$ Equation 8

The hypothetical feed-side vapor mole fraction ($x_i^{*}$) is defined as the ratio of the hypothetical partial pressure to the hypothetical total pressure ($x_i^{*}=p_i^{F*}/p^{F*}$). Even though VLE behavior is used in the calculation, the hypothetical vapor is represented as a lowercase x to indicate it is a feed-side concentration.

Given this information about partial pressures, Equation 5 can be rewritten for V·P and PV operations as:

$$\text{Vapor\hspace{0.17em}Permeation:\hspace{0.17em}\hspace{0.17em}}{j}_i=\frac{P_i}{\mathcal{l}}\left(x_i{p}^F-y_i{p}^P\right)$$ Equation 9

$$\text{Pervaporation:\hspace{0.17em}\hspace{0.17em}}{j}_i=\frac{P_i}{\mathcal{l}}\left(x_i{\gamma }_i{p}_i^{\mathit{\text{sat}}}-y_i{p}^P\right)=\frac{P_i}{\mathcal{l}}\left(x_i^{*}{p}^{F*}-y_i{p}^P\right)$$ Equation 10

The driving force for V·P is then controlled by altering the applied feed-side and permeate pressures. In pervaporation, the applied feed pressure plays little role, other than to maintain the fluid in a liquid state. Instead, the feed-side temperature has the leading role in PV driving force, primarily through the near-exponential effect on saturated vapor pressure. The effects of composition and temperature on the partial vapor pressures of water and ethanol in equilibrium with a liquid mixture in the range of 0–20 wt% water are shown in log-log plots in Figure 6. Both water and ethanol partial pressures depend heavily on temperature at all concentrations, increasing by a factor of 2 to 2.5 for every 20 °C temperature rise. Operating at the highest practicable temperature will yield the highest feed-side partial pressures and, therefore, the highest partial pressure driving force, resulting in the highest flux. Also evident from Figure 6 is that the water partial pressure is strongly dependent on composition while the ethanol partial pressure varies only modestly with water concentration in this range. Consequently, as the water content in the feed-side fluid decreases as the fluid traverses the membrane grid or as a batch is processed, the feed-side vapor pressure of water will decrease. According to Equation 5, For there to be a positive flux of water through the membrane, the permeate partial pressure of water must be below the feed-side partial pressure of water. As indicated in the figure, the feed-side partial pressure of water may be quite small relative to that of the solvent and relative to the initial water partial pressure, depending on the initial water content. This may necessitate a lower absolute permeate pressure for some, or all, of the system or operating time in order to motivate water transport through the membrane.

Figure 6.

Effect of temperature on the partial vapor pressures of (a) water and (b) ethanol in equilibrium with a water-ethanol liquid mixture as a function of the water weight fraction in the liquid phase ($w_w^L$) over the 0–20 wt% range for temperatures ranging from 30 to 130 °C. Values calculated with NRTL model.

The solvent in the mixture greatly impacts the partial vapor pressure of water, even for the same water composition. First, the type of solvent has an effect through the activity coefficient as shown in Figure 5. The y-axis intercept of each curve in Figure 5a represents the infinite dilution activity coefficient for water (i.e. ${\gamma }_w\to {\gamma }_w^{\infty }\text{\hspace{0.17em}as\hspace{0.17em}}{x}_w\to 0$), which varies from just over 1 in DMF to almost 28 in MtBE. For most of the solvents, γ_w decreases markedly as water content increases. On the other hand, the activity coefficients for the solvents in the 0–20 wt% water range are close to 1, never exceeding 1.6. The effect of solvent type and composition on the partial vapor pressures of water and the solvent at fixed temperature of 100 °C are shown in Figure 7. The same temperature was selected for all the mixtures to hold $p_w^{\mathit{\text{sat}}}$ constant in this comparison and 100 °C was selected because it is the normal boiling point of water. In the 0–20 wt% water range, the partial pressure of a solvent is nearly that of the pure solvent and, as such, the solvent partial pressures at 100 °C are ordered in the figure based primarily on their normal boiling point. However, water partial pressure varies with the solvent type based primarily on the effect of the solvent on the activity coefficient of water. For mixtures containing 1 wt% water, the water partial pressure at 100 °C changes by over a factor of 20, from 3.1 kPa in methanol to 72.6 kPa in MtBE. Even within the four alcohols in the targeted solvent list, the water partial pressure at 1 wt% water ranges from the 3.1 kPa in methanol to 17.2 kPa in 1-butanol. Thus, as solvent hydrophobicity increases, so does the water partial pressure in the feed, increasing the PV driving force for permeation in Equation 10. This will increase the driving force and flux for water removal from the more hydrophobic compounds.

Figure 7.

Partial vapor pressure of (a) water and (b) solvent in equilibrium with a water-solvent liquid mixture at 100 °C as a function of the water weight fraction in the liquid phase ($w_w^L$) over the 0–20 wt% range as calculated with NRTL model. Values for MIBK and MtBE shown in the range of solubility of each with water^{53, 54}.

Effect of non-negligible permeate pressure on PV and V·P observations

Equation 9 and Equation 10 can be used to convert the flux and permeate composition results reported for a membrane under specified operating conditions to membrane permeance and selectivity values. These membrane performance parameters can be used to better estimate performance under other conditions. In many studies reported in the literature, permeate partial pressures are negligible compared to the feed partial pressures. This is due to the study of moderate feed water concentrations (5–15 wt%) and collection of the permeate using liquid nitrogen-cooled traps in combination with vacuum pumps capable of delivering low absolute pressures (< 0.1 kPa). Such permeate collection conditions are not usually practical in larger industrial systems. In cases where the permeate pressures are non-negligible, the ratio of total feed vapor pressure (p^F for V·P or p^F* for PV) to total permeate pressure p^P, termed “the pressure ratio” ϕ, is an important design and operating factor, which can result in an actual observed separation quite different from the ideal membrane separation.

The separation attributable to the membrane alone is the ratio of the permeabilities (or permeances) of two compounds in the membrane material, termed the molar permselectivity, α, or simply “selectivity”:

$${\alpha }^{\mathit{\text{mem}}}=\frac{P_1}{P_2}=\frac{{\Pi }_1}{{\Pi }_2}$$ Equation 11

Combining Equation 11 with Equation 6, Equation 9, and Equation 10 results in the following expressions for V·P and PV, relating the separation factors calculated from observed values of x_i and y_i to the membrane selectivity:

$${\text{Vapor\hspace{0.17em}Permeation:\hspace{0.17em}\hspace{0.17em}β}}_{12}^{\text{VP}}=\left(\frac{P_1}{P_2}\right)\left[\frac{1-\frac{1}{\varphi }\frac{y_1}{x_1}}{1-\frac{1}{\varphi }\frac{y_2}{x_2}}\right]={\alpha }^{\mathit{\text{mem\hspace{0.17em}}}}\left[\frac{1-\frac{1}{\varphi }\frac{y_1}{x_1}}{1-\frac{1}{\varphi }\frac{y_2}{x_2}}\right]$$ Equation 12

$$\text{Pervaporation}:{\beta }_{12}^{\text{PV}}=\left(\frac{P_1}{P_2}\right)\left[\frac{\left(x_1^{*}/x_1\right)}{\left(x_2^{*}/x_2\right)}\right]\left[\frac{1-\frac{1}{\varphi }\frac{y_1}{x_1^{*}}}{1-\frac{1}{\varphi }\frac{y_2}{x_2^{*}}}\right]{=\alpha }^{\mathit{\text{mem\hspace{0.17em}}}}{\alpha }_{12}^{\mathit{\text{VLE}}}\left[\frac{1-\frac{1}{\varphi }\frac{y_1}{x_1^{*}}}{1-\frac{1}{\varphi }\frac{y_2}{x_2^{*}}}\right]$$ Equation 13

According to these expressions, as ϕ increases, the right hand bracketed terms become 1 and the separation factor for V·P approaches the selectivity of the membrane (${\beta }_{12}^{\text{V·P}}\Rightarrow {\alpha }^{\mathit{\text{mem}}}$), while the separation factor for PV approaches the product of the membrane selectivity and the relative volatility of the feed-side liquid (${\beta }_{12}^{\mathit{\text{PV}}}\Rightarrow {\alpha }^{\mathit{\text{mem}}}·{\alpha }_{12}^{\mathit{\text{VLE}}}$). Thus, in PV, favorable VLE augments membrane selectivity while unfavorable VLE detracts from membrane selectivity. Fortunately, unfavorable VLE behavior can be overcome with a high membrane selectivity to yield a separation. In this way, a water-selective membrane can remove water even from the water/solvent systems in Figure 4 exhibiting azeotropes or low water/solvent relative volatilities. Ideally, the inherent properties of both the membrane and VLE work in the same enrichment direction. Since the separation in a V·P unit is not affected by VLE behavior, there may be unfavorable VLE situations that would benefit from employing V·P instead of PV.

Calculation of permeate composition

With regard to the permeate composition, Baker provides a very useful relationship for calculating the permeate concentration of the higher permeance species in a binary gas or vapor mixture from a specific feed-side concentration (x_i) and pressure ratio (Equation 8.19 in the reference)¹⁶:

$$y_i=\left(\frac{\varphi }{2}\right)\left[x_i+\frac{1}{\varphi }+\frac{1}{{\alpha }^{\mathit{\text{mem}}}-1}-\sqrt{{\left(x_i+\frac{1}{\varphi }+\frac{1}{{\alpha }^{\mathit{\text{mem}}}-1}\right)}^2-\frac{4{\alpha }^{\mathit{\text{mem}}}{x}_i}{\left({\alpha }^{\mathit{\text{mem}}}-1\right)\varphi }}\right]$$ Equation 14

Although developed for gas or vapor separation, Equation 14 can be used for pervaporation by replacing x_i with the hypothetical feed-side vapor mole fraction ($x_i^{*}$) and basing the pressure ratio on the total hypothetical feed-side pressure (p^F*). Clearly, both membrane selectivity and pressure ratio play a significant role in determining the permeate composition. In the extremes of ϕ relative to α^mem, Equation 14 simplifies to:

$$\text{if}\varphi \gg {\alpha }^{\mathit{\text{mem\hspace{0.17em}}}}\text{("Selectivity-limited")}:y_i=\frac{{\alpha }^{\mathit{\text{mem}}}{x}_i}{1+x_i\left({\alpha }^{\mathit{\text{mem}}}-1\right)}$$ Equation 15

$$\text{if}{\alpha }^{\mathit{\text{mem}}}\gg \varphi \text{("Pressure\hspace{0.17em}Ratio-limited")}:y_i=\varphi x_i$$ Equation 16

Thus, under a pressure ratio-limited situation (Equation 16), the permeate composition of the preferentially permeating compound is independent of the membrane selectivity, dictated only by the applied pressure ratio and feed-side mole fraction – a concept that may be difficult to appreciate for those focused only on selectivity. In such a case, raising the flux of the solvent will raise the water flux since the ratio of fluxes is a constant (because y_i is fixed). For example, under pressure ratio-limited conditions, switching to a membrane with lower selectivity but higher solvent permeance will raise water flux while delivering the same permeate water composition.

PV and V·P Process Design Considerations

PV and V·P processes are primarily designed in one of two modes of operation: single-pass or recirculating batch, as depicted in Figure 2 and Figure 3. In single-pass mode, the membrane dehydration system reduces the water content in a stream of constant flow rate from the inlet/feed mole fraction, $x_w^F$, to the desired product/retentate solvent product mole fraction, $x_w^R$. In recirculating batch mode, the concentration of water in a discrete charge of water-contaminated solvent is reduced over time, τ, from an initial mole fraction, $x_{w_0}^F$, to the desired final value for the product solvent, $x_{w_{\tau }}^F$. The membrane area required for the dehydration task, A, is a function of the membrane material and module, feed mixture, feed conditions, permeate conditions, and either the flow rate, if continuous, or the charge volume and cycle time (τ) if a recirculating batch operation. Unless the membrane area required can be met with a single membrane module, the membrane system will consist of a grid of membrane modules and housings arranged in parallel and/or in series to accomplish the separation.

Minimizing membrane area is a general PV/V·P process design objective because area represents both an initial capital cost and an operating expense related to periodic replacement of the membrane elements. However, minimizing area may require feed-side and permeate conditions that either result in large increases in other process costs or are just too extreme for the membrane or module materials. The effect of membrane performance characteristics and several process parameters on membrane area, permeate purity, and solvent recovery will be examined in this section. To do so, estimations of membrane performance will need to be established.

Single-Pass Systems

The most straightforward dehydration system to model is a single-pass operation in which a constant molar feed flow rate of a solvent/water mixture (${\dot{N}}^F$) is treated with a membrane system to reduce the water content from the inlet feed mole fraction ($x_w^F$) to a final retentate water content ($x_w^R$). If the illustrations in Figure 1 are taken as representing a differential membrane element (dA), then the differential loss of component i from the feed-side fluid stream ($d{\dot{N}}_i$) as the fluid traverses the membrane element is equal to the flux through the membrane multiplied by the membrane area as:

$$\begin{gathered} V·P:d{\dot{N}}_i=d\left(\dot{N}{x}_i\right)=-j_i\mathit{\text{\hspace{0.17em}dA}}=-\frac{P_i}{\mathcal{l}}\left(x_i{p}^F-y_i{p}^P\right)\mathit{\text{\hspace{0.17em}dA}} \\ PV:d{\dot{N}}_i=d\left(\dot{N}{x}_i\right)=-j_i\mathit{\text{\hspace{0.17em}dA}}=-\frac{P_i}{\mathcal{l}}\left(x_i^{*}{p}^{F*}-y_i{p}^P\right)\mathit{\text{\hspace{0.17em}dA}}=-\frac{P_i}{\mathcal{l}}\left(x_i{\gamma }_i{p}_i^{\mathit{\text{sat}}}-y_i{p}^P\right)\mathit{\text{\hspace{0.17em}dA}} \end{gathered}$$ Equation 17

Spreadsheets were created to estimate performance of single-pass V·P and PV systems, operating in a crossflow configuration as depicted in Figure 1, by dividing the membrane unit into 100 sub-units linked in series, describing the transport of each species in a sub-unit according to Equation 17 (detailed in Supporting Information). The soundness of using 100 sub-units was confirmed by determining that an increase to 1000 sub-units resulted in relative changes in calculated values of less than 1.4%. The advantage of fewer sub-units is faster spreadsheet refreshing and iterative solving. The spreadsheets enable the study of different process variables to assess their effect on system performance. A similar spreadsheet was previously employed by the author’s group to estimate performance of a V·P system with multi-component vapor feed streams^{17, 18}. A rougher estimate of performance can be made by applying several simplifying assumptions to Equation 17. For a PV system, the assumptions are:

• Temperature is constant (therefore $p_i^{\mathit{\text{sat}}}$ values are constant)

• Activity coefficients are constant

• Total solvent flow rate is constant at the feed solvent flow rate, ${\dot{N}}_s^F$ (i.e. negligible solvent permeates the membrane)

• Permeances are constant

• Permeate pressure is negligible

With these assumptions and substituting $\dot{N}={\dot{N}}_s^F/\left(1-x_w\right)$, Equation 17 (for PV) can be integrated to yield the following relationship between the retentate/outlet water composition ($x_w^R$), feed/inlet water composition ($x_w^F$), area, flow rate, membrane permeance, activity coefficient, and saturated vapor pressure:

$$\text{ln\hspace{0.17em}}\left[\frac{\raisebox{1ex}{$x_w^R$}\!\left/ \!\raisebox{-1ex}{$\left(1-x_w^R\right)$}\right.}{\raisebox{1ex}{$x_w^F$}\!\left/ \!\raisebox{-1ex}{$\left(1-x_w^F\right)$}\right.}\right]+\frac{1}{1-x_w^R}-\frac{1}{1-x_w^F}\cong -\frac{A}{{\dot{N}}_s^F}\left(\frac{P_w}{\mathcal{l}}\right){\gamma }_w{p}_w^{\mathit{\text{sat}}}$$ Equation 18

For small water concentrations, Equation 18 simplifies further to:

$$\text{ln\hspace{0.17em}}\left(\frac{x_w^R}{x_w^F}\right)\cong -\frac{A}{{\dot{N}}_S^F}\left(\frac{P_w}{\mathcal{l}}\right){\gamma }_w{p}_w^{\mathit{\text{sat}}}$$ Equation 19

For a single-pass V·P system with complete evaporation (see Option 2 in Figure 3), Equation 18 and Equation 19 can be applied after substituting the total feed pressure p^F for (${\gamma }_w{p}_w^{\mathit{\text{sat}}}$).

Recirculating Batch Systems

In a recirculating batch membrane process, the amount of each species in the feed source tank ($N_i^t$) changes with time as does the concentration in the tank ($x_i^t$). In the membrane system, the concentration of each species changes with both time and position along the membrane system. The tank is charged initially with amount of mixture N⁰ at water concentration $x_w^0$. A stream is removed from the source tank and processed through the membrane system, lowering the water content from the feed concentration at a given time ($x_w^{F@t}$) to the retentate concentration at that time ($x_w^{R@t}$), corresponding to the separation achieved by the membrane at that time. The retentate is returned to the feed source tank. The source tank is considered well stirred such that the feed concentration to the membrane is the same as the tank concentration at a given time ($x_w^{F@t}=x_w^t$). The incremental change in the amount of a compound in the source tank ($d{N}_i^t$) over a time increment (dt) is equal to the permeation through the membrane system of area A at a particular time, represented as:

$$\begin{gathered} V·P:d{N}_i^t=d\left(N^t{x}_i^t\right)=-{\overline{j}}_{i}A\mathit{\text{dt}}=-{\left[{\int }_0^A\frac{P_i}{\mathcal{l}}\left(x_i{p}^F-y_i{p}^P\right)\mathit{\text{\hspace{0.17em}dA}}\right]}_{@t}\mathit{\text{dt}} \\ PV:d{N}_i^t=d\left(N^t{x}_i^t\right)=-{\overline{j}}_{i}A\mathit{\text{\hspace{0.17em}dt}}=-{\left[{\int }_0^A\frac{P_i}{\mathcal{l}}\left(x_i{\gamma }_i{p}_i^{\mathit{\text{sat}}}-y_i{p}^P\right)\mathit{\text{\hspace{0.17em}dA}}\right]}_{@t}\mathit{\text{dt}} \end{gathered}$$ Equation 20

Where the left side of each relationship represents the change in moles of species i in the feed tank, ${\overline{J}}_i$ is the area-averaged flux at time t, and the integration on the right-hand side is over the area of the membrane system at the conditions of time t. A spreadsheet was created to estimate recirculating batch PV system performance by dividing the batch cycle into 100 time intervals, describing the transport of each species in a time interval according to Equation 20 (as described in Supporting Information). A spreadsheet model of a recirculating batch V·P operation was not developed because this type of operation is not particularly practical due to the large amount of energy required to evaporate and condense the recirculation stream.

A rougher but simpler estimate of recirculating batch system performance can be established by making the same assumptions as above for the single-pass system estimate, except assuming the amount of solvent in the feed source tank is constant and equal to the amount of solvent in the initial tank charge (i.e. $N_s^t=N_s^0$) rather than that the flow rate of solvent is constant. In addition, if it is assumed that the single-pass removal of water in the membrane unit is small such that $x_w^{R@t}\cong x_w^{F@t}$, then the right-side integral in Equation 20 simplifies to $\left(P_w/\mathcal{l}\right)x_w^t{p}^{F}A$ for V·P and $\left(P_w/\mathcal{l}\right)x_w^t{\gamma }_w{p}_w^{\mathit{\text{sat}}}A$ for PV. With these assumptions and substituting $N^t=N_s^0/\left(1-x_w^t\right)$, Equation 20 for PV can be integrated with respect to time to yield the following relationship between the tank water composition at the cycle time τ ($x_w^{\tau }$), initial tank water composition ($x_w^0$), area, initial tank charge of solvent, membrane permeance, activity coefficient, saturated vapor pressure, and processing time:

$$\text{ln\hspace{0.17em}}\left[\frac{\raisebox{1ex}{$x_w^{\tau }$}\!\left/ \!\raisebox{-1ex}{$\left(1-x_w^{\tau }\right)$}\right.}{\raisebox{1ex}{$x_w^0$}\!\left/ \!\raisebox{-1ex}{$\left(1-x_w^0\right)$}\right.}\right]+\frac{1}{1-x_w^{\tau }}-\frac{1}{1-x_w^0}\cong -\frac{A\tau }{N_s^0}\left(\frac{P_w}{\mathcal{l}}\right){\gamma }_w{p}_w^{\mathit{\text{sat}}}$$ Equation 21

Note that this equation is of the same form as Equation 18 for a single-pass PV system, with the molar solvent flow rate ${\dot{N}}_s^F$ replaced by the total moles of solvent in the initial tank charge divided by the processing time ($N_s^0/\tau$). For small water concentrations, Equation 21 further simplifies to:

$$\text{ln\hspace{0.17em}}\left(\frac{x_w^{\tau }}{x_w^0}\right)\cong -\frac{A\tau }{N_s^0}\left(\frac{P_w}{\mathcal{l}}\right){\gamma }_w{p}_w^{\mathit{\text{sat}}}$$ Equation 22

which is of the same form as Equation 19 for a single-pass system with the molar solvent flow rate ${\dot{N}}_s^F$ replaced by the total moles of solvent in the initial tank charge divided by the processing time ($N_s^0/\tau$).

If a recirculating batch V·P system with complete evaporation (Option 1 in Figure 3.) were being considered, Equation 21 and Equation 22 could be used to estimate performance after substituting the total feed pressure p^F for (${\gamma }_w{p}_w^{\mathit{\text{sat}}}$), mirroring the situation for single-pass V·P with complete evaporation. If, however, the vapor for a recirculating batch V·P system is generated in a batch evaporator as in Option 1 of Figure S-3 (where the liquid in a source vessel is heated to generate a vapor through VLE in the vessel rather than a complete evaporator), then Equation 21 and Equation 22 apply without alteration because of the VLE established in the batch evaporator.

Implications of simplified process equations

While Equations 18/19 and 21/22 are rough estimates of how a single-pass or recirculating batch process will reduce water in a solvent, they can be used to identify important factors that will affect performance of these systems. First, according to these equations, the area required to reduce water by a certain extent in a PV system is proportional to the flow rate in a single-pass system or to the amount of material to be processed in a given time for a recirculating batch operation and is inversely proportional to water permeance, water activity coefficient, and the saturated vapor pressure of water. Because $p_w^{\mathit{\text{sat}}}$ increases rapidly with increasing temperature, operating at the maximum temperature will greatly reduce required membrane area or processing time. The corollary for V·P systems is that operating at maximum p^F will minimize area. Second, for PV systems, the specific solvent in the mixture impacts water removal mainly through γ_w. Since γ_w can vary significantly from one solvent to another (as noted earlier, ${\gamma }_w^{\infty }$ ranges from about 1 to near 30 among the target solvents), the area or time required to accomplish a desired decrease in water concentration can vary significantly between solvents. As indicated in Figure 5 the differences in γ_w between solvents become less pronounced as water concentration increases. Thirdly, the ratio of final and initial water concentrations is roughly an exponential function of the other parameters but is relatively independent of the initial concentration. Thus, reducing water from 10 wt% to 1 wt% requires approximately the same area as reducing water from 1 wt% to 0.1 wt%, despite a ten-fold difference in the amount of water removed, based on the assumptions made.

Just as illustrative are the conditions under which the simplifying assumptions used in establishing the simplified equations fail:

Assumption: Temperature is constant

Since PV involves a phase change from the feed liquid to the permeate vapor, the feed liquid cools as it traverses the membrane system, thus temperature is not constant in PV and this will have a dramatic effect on local flux due to a drop in partial vapor pressure. For example, based on the heat of evaporation of water and ethanol/water mixture heat capacities, reducing water content in ethanol from 10 wt% to 9 wt% requires 24.6 kJ of heat per kg of the feed mixture and results in a 7.4 °C temperature drop. If the feed were at 100 °C, this ΔT would result in a drop in the saturated vapor pressure of water of 24%. As a result, heat may need to be added between, or in, membrane modules to counteract the heat lost due to evaporation. V·P does not involve a phase change in the membrane module, so this assumption is generally reasonable. The corollary assumption for V·P is that the total feed-side pressure is constant. However, in real systems, p^F will fall due to viscous flow pressure drop as the vapor moves through the membrane module(s).

Assumption: Activity coefficients are constant

As illustrated in Figure 5 and discussed above, γ_w can vary significantly with water content and this variability is solvent-dependent. Thus, this assumption is invalid when: γ_w varies significantly with water content, the initial water content is high, and the change in water content is large. As shown in Figure 5, reducing water from 20 to 0.1 wt% results in γ_w increasing by at least 50% for all of the solvents except methanol and DMF. However, as water is reduced from only 1 to 0.1 wt%, γ_w increases by 50% or more for just one solvent, MtBE.

Assumption: Solvent molar flow rate or amount of solvent in the feed tank is constant

This assumption is reasonable when the amount of solvent in the permeate is low relative to that in the feed, typically when membrane selectivity is high and permeate pressure is low – or if membrane area is low as might be the case for a small (e.g. <50%) relative change in water concentration. For all of the solvents, a selectivity of 100 or more will result in less than 10% transfer of solvent to the permeate for a reduction in water content from 10 to 0.1 wt%, when permeate pressure is negligible. The effect of selectivity and permeate pressure on solvent transfer to the permeate will be explored later.

Assumption: Permeances are constant

In theory, permeances are less dependent on temperature and concentration than fluxes because they have been normalized by partial pressures – which are highly dependent on temperature and concentration. However, permeance values reported in the literature or in vendor datasheets were obtained under experimental conditions that might not relate directly to the conditions of the solvent drying process being considered. Permeability is the product of solubility and diffusivity. As a result, parameters that alter solubility and diffusivity will affect permeability and permeance. For example, increasing temperature generally decreases solubility but increases diffusivity. These changes partially offset each other, resulting in relatively small changes in permeance, particularly over small temperature changes (e.g. ΔT of 5 to 10 °C). For some materials, such as inorganic and amorphous perfluoro polymeric materials, permeances are reasonably independent of water concentration. However, hydrophilic polymers that plasticize upon water sorption can exhibit large changes in permeance with water concentration. For example, the PV water permeance of a commercial poly(vinyl alcohol) membrane increased by 140% as water in the water/ethanol feed increased from 0.7 to 15 wt%¹⁹. The permeance of such a material will depend on the water activity on both the feed and permeate sides of the selective layer. A variety of models and methods of accommodating such fluctuations have been developed^20–22. An additional factor for V·P is superheat, in which the feed-side vapor is heated above the dew point temperature to avoid condensation in the membrane module. Superheat reduces the activity of water and solvent experienced by the membrane that may alter the performance of the membrane.

Assumption: Permeate pressure is negligible.

This assumption is often the first to be made because it greatly simplifies flux equations but is the hardest to justify in commercial PV or V·P processes due to the cost of supplying low absolute vacuum pressures. In addition, if low retentate water concentrations are required, low feed-side partial pressures of water will result – making it difficult to achieve permeate pressures that are negligible by comparison. Further complicating this assumption is the permeate pressure drop contributed by the porous support layers that provide mechanical stability to the thin, permselective layer as well as the pressure drop due to vapor flow through the membrane element.

Some of these non-idealities can be partially accounted for in the simplified equations. The effect of temperature on $p_w^{\mathit{\text{sat}}}$ and of concentration on γ_w can be approximated in the PV performance estimations using the geometric mean of the two parameters between the feed and retentate conditions, akin to the use of geometric mean of relative volatility (${\overline{\alpha }}_{12}^{\text{VLE}}$) in determining the minimum number of distillation stages in Equation 3. Similarly, an average permeance could be employed to account for the effect of concentration on permeance for some membranes. Of the above assumptions, the spreadsheet calculations used herein only require constant permeances within a membrane unit. Inclusion of permeances as functions of temperature and component activity are being considered for future work.

Impact of membrane properties and process variables on system design

In this section, the effect of process parameters on system performance will be evaluated using the estimating equations developed above and the more in-depth spreadsheet calculations developed from Equation 17 and Equation 20 for single-pass PC, recirculating batch PV, and single-pass V·P (see Supporting Information for descriptions of the spreadsheet calculations). The effect of membrane properties on ethanol/water separation performance will be studied based on the permeance and selectivity ranges outlined in the materials review paper for commercially available membrane materials. The benchmark separation discussed below will involve reducing water in ethanol from 10 to 0.5 wt% (22.1 to 1.27 mol%) using a membrane with a constant water permeance of 2000 GPU and a constant water/ethanol selectivity of 2000. This set of conditions was selected to be representative of the middle ground of the seven common PV/V·P membrane materials removing water from ethanol/water mixtures by pervaporation for feeds containing 5–15 wt% water at temperatures of 50–80 °C that was detailed in the previous paper⁹. The 10 to 0.5 wt% water benchmark separation was selected to illustrate the removal of water from a concentration above the azeotrope (4 wt% water in the ethanol azeotrope) to a product concentration that is consistent with fuel ethanol specifications (0.3 wt% in the European Union²³ to 1.3 wt% in the United States²⁴). The benchmark product water level of 0.5 wt% is also consistent with generally acceptable water impurity levels in recovered solvents to retain solvency (1% water) or to avoid affecting reaction equilibria (0.1% water)²⁵. Product water levels of 100 ppm (0.01%) or lower might be necessary if water deactivates a catalyst or reagent in the intended application of the reclaimed solvent. For other solvents, the target product water concentrations will likely be similar to those in ethanol, although the initial water concentration will depend on a number of factors, including the azeotrope for the specific solvent/water system. The effect of varying the feed and product water concentrations on ethanol/water separation system design will be explored.

The benchmark membrane system is a 70 °C isothermal, single-pass PV process with a solvent feed rate of 0.05 kg s⁻¹ (about 2 million liters per year) operating with negligible permeate pressure (p^P = 0). For a benchmark recirculating batch PV system, an initial solvent charge of 1440 kg, equal to 8 h of a 0.05 kg s⁻¹ solvent flow, is assumed to be processed in an 8 h cycle, all other parameters being the same as the single-pass benchmark conditions.

Retentate concentration and permeate pressure

As calculated with the single-pass PV spreadsheet, the membrane area required to perform the benchmark separation is 75.5 m². The estimate of area for this separation from simplified Equation 18 with a geometric mean activity coefficient is 78.2 m², which is only 3.5% higher than that from the spreadsheet. The impact of varying retentate concentration and permeate pressure on the membrane area required for the separation, keeping all other parameters the same as the benchmark scenario, is displayed in Figure 8. Each curve represents a different retentate concentration of water, ranging from 0.1 to 5 wt%, with total permeate pressure as the x-axis. The intercept of each curve with the y-axis is the membrane area needed when the permeate pressure is negligible (e.g. 75.5 m² for 0.5 wt% retentate). The closed symbols shown on the y-axis are the rough area estimates calculated from Equation 18, for which p^P =0 had been assumed in the derivation. Each open symbol on the curves represents the point at which the area has increased 50% from the area calculated at p^P =0 for a specific retentate concentration. A 50% change in membrane area will be used herein as a subjective indicator that a variable has reached a value that causes a notable change in membrane area.

Figure 8.

Membrane area of a single-pass PV system required to reduce the water concentration of an ethanol/water solution from 10 wt% in the feed stream to 0.1, 0.5, 1, or 5 wt% as a function of the total permeate pressure. Values calculated with single-pass PV spreadsheet under benchmark conditions: isothermal operation at 70 °C, 0.5 kg s⁻¹ ethanol feed rate, 2000 GPU water permeance, and water/ethanol α = 2000. The closed symbols on the y-axis are the areas calculated from simplified Equation 18 with geometric mean of ${\gamma }_w{p}_w^{\mathit{\text{sat}}}$. Open symbols represent the permeate pressure at which the area has increased 50% from the area calculated at p^P=0 for a specific retentate concentration. Arrows along the x-axis are estimates of the maximum practical permeate pressure as described in the text.

It is evident from the proximity of the solid symbols to the y-axis intercepts of the curves that the areas calculated from simplified Equation 18 are reasonable estimates of the spreadsheet calculations for p^P =0, with relative differences ranging from only 0.2 to 5.4%. For non-negligible permeate pressures, as expected, the required area increases as p^P increases due to the reduced driving force for flux through the membrane. However, the magnitude of this impact depends on the specified retentate water concentration - as indicated in Figure 8 by the pressure ranges where the areas rapidly increase and by the relative positions of the open symbols at a 50% area increase. For instance, the benchmark system of 0.5 wt% retentate needs 50% more area at p^P =0.86 kPa than at p^P =0, while a system treating down to a higher retentate concentration of 5 wt% water would see a 50% increase at a higher permeate pressure of p^P =3.6 kPa. As mentioned earlier, the permeate vacuum is usually generated with a condenser in combination with a vacuum pump. Lower permeate pressures require lower condenser temperatures or booster vacuum pumps, each adding capital and operating costs. Condensing pure water at absolute pressures of 0.86 and 3.6 kPa requires temperatures of 5 and 27 °C, respectively. Simply considering this 22 °C difference in condenser temperature makes the impact of permeate pressure on permeate condensation conditions apparent. The presence of solvent in the permeate reduces condenser temperatures further due to the impact of solvent on the bubble point of the permeate mixture. Figure S-8 in Supporting Information displays the effect of varying both feed and retentate water concentration in the range of 0.1 to 20 wt% on membrane area, permeate solvent concentration, and evaporation energy of the, otherwise, benchmark single-pass PV system.

The benchmark recirculating batch PV system should, in theory, require the same area as the benchmark single-pass system because the amount of the batch solvent charge and cycle time were chosen to parallel the single-pass solvent flow rate. In fact, at 78.2 m², the estimate of area for the recirculating batch PV operation from Equation 21 is exactly the same as the estimate for the single-pass PV operation from Equation 18. The fuller spreadsheet calculation for the recirculating batch system yields an area of 74.4 m², which is only slightly lower than the single-pass spreadsheet result of 75.5 m². Thus, the recirculating batch and single-pass benchmark systems yield analogous results for equivalent operating parameters. In addition, the permeate pressure that yields a 50% increase in recirculating batch system area is 0.87 kPa, comparable to the 0.86 kPa observed for the single-pass system. Thus, the spreadsheet calculations yield similar area results under the benchmark parameters even when permeate pressure is varied.

A single-pass V·P system in which the 10 wt% water feed stream is fully evaporated at 70 °C, generating a total feed pressure of 71.6 kPa (based on a dew point calculation with CHEMCAD), would require 76.3 m² of membrane, very similar to that of the PV systems. A 50% increase in V·P membrane area is estimated to occur at a permeate pressure of 0.79 kPa, also similar to that of the PV systems. The V·P area is slightly different than that of the PV systems because of the difference between the mole fraction and partial pressure of water from the complete evaporator in the V·P system and those corresponding to the hypothetical vapor in equilibrium with the liquid and the membrane in a PV system.

Individual pervaporation membrane modules range in active membrane area from 1 to 50 m², depending on the type of membrane. Plate-and-frame modules, in which flat sheets of membrane are layered with feed and permeate spacers, have been used most extensively for pervaporation dehydration and have reported active areas of up to 50 m^{216, 26, 27}. By comparison, a 1 meter long, 4 inch (10 cm) diameter spiral wound membrane module contains 3 to 6 m² of active membrane and an 8 inch diameter spiral wound module contains from 20 to 40 m²¹⁶. Individual zeolite membrane tubes of 12 mm outer diameter and 0.8 m length have an active area of 0.03 m² (based on outer active surface). Multi-tube zeolite modules with areas up to 7.3 m² have been described for pervaporation dehydration^{28, 29}. Multi-element hybrid silica membrane modules with areas of up to 3.8 m² are also available³⁰. Similarly, multiple spiral wound or plate-and-frame modules can be plumbed together in a single pressure vessel, allowing for much larger membrane areas in a single assembly. For example, an early industrial pervaporation plant drying ethanol from 7 wt% water to 0.2 wt% water consisted of 42 plate-and-frame modules, divided between three pressure vessel assemblies. Each assembly contained about 50 m² of membrane, for an average assembly area of 700 m² and total system area of about 2100 m^{227, 31}. Thus, the calculated membrane areas presented in Figure 8 are well within the practical range of commercial membrane module/assembly designs. The exact design of the membrane system will be determined by process specifics, such as reheating requirements, and membrane/module type.

Estimation of maximum practical permeate pressure

Permeate pressure clearly plays a significant role in determining many PV and V·P system characteristics. It would be useful to have an estimate of the maximum practical permeate pressure ($p_{\mathit{\text{max}}}^P$) and maximum permeate condenser temperature for a given set of feed conditions. One such estimate can be derived from the target concentration of water in the solvent product and the operating temperature. According to the flux equations, the partial pressure of water in the permeate must be below that on the feed side of the membrane to maintain a flux of water (i.e. $p_w^P<p_w^F$ in Equation 5). The lowest feed-side partial pressure of water occurs at the lowest water concentration – either in the retentate of a single-pass system or in the recirculation tank at the end of a batch cycle. A rough estimate of the minimum pressure ratio can be calculated by dividing the maximum permeate composition from Equation 15 by the specified retentate water concentration, yielding:

$${\varphi }_{\mathit{\text{min}}}\cong \frac{{\alpha }^{\mathit{\text{mem}}}}{1+x_w\left({\alpha }^{\mathit{\text{mem}}}-1\right)}$$ Equation 23

where x_w for a single-pass system is either $x_w^R$ for V·P or $x_w^{R*}$ for PV. For recirculating batch systems, the end-of-cycle tank concentrations of $x_w^{\tau }$ (V·P) or $x_w^{\tau *}$ (PV) would be used. The maximum total permeate pressure would then be the total feed-side vapor pressure under the retentate, or final, conditions (p^R) divided by ϕ_min as:

$$p_{\mathit{\text{max}}}^P\cong \raisebox{1ex}{$p^R$}\!\left/ \!\raisebox{-1ex}{${\varphi }_{\mathit{\text{min}}}$}\right.$$ Equation 24

For processes involving only small reductions in water concentration, the feed and retentate partial pressures of water ($p_w^F$ and $p_w^R$) may not be very far apart. In such cases, $p_{\mathit{\text{max}}}^P$ based only on retentate conditions using the above calculation would appreciably reduce the water vapor pressure driving force not just at the end of the system or cycle, but throughout the membrane system, leading to a large increase in membrane area. In those situations, an alternate $p_{\mathit{\text{max}}}^P$ could be calculated based on a specified allowable relative increase in membrane area (Ψ). For example, the 50% increase in membrane area used as a reference in Figure 8 would correspond to Ψ=1.5. Using the geometric mean of the feed-side water partial pressure, $p_{\mathit{\text{max}}}^P$ would be estimated in this way as:

$$p_{\mathit{\text{max}}}^P\cong \left(1-\frac{1}{\Psi }\right)\frac{\sqrt{p_w^F{p}_w^R}}{y_w^{\mathit{\text{gm}}}}$$ Equation 25

where $y_w^{\mathit{\text{gm}}}$ is the maximum possible water concentration in the permeate associated with the geometric mean of the feed and retentate hypothetical feed concentrations (i.e. $\sqrt{x_w^{F*}{x}_w^{R*}}$), as calculated with Equation 15 for the “selectivity limited” condition.

The lesser of the permeate pressure estimates from Equations 24 and 25 would then be used as $p_{\mathit{\text{max}}}^P$. The lesser values are presented as arrows along the x-axis of Figure 8 in order of increasing water in the retentate (color coded to match the respective curve). Calculations of $p_{\mathit{\text{max}}}^P$ are provided in Table S-5 of the Supporting Information. For example, the feed and retentate partial pressures of water in the benchmark single-pass PV scenario are 14.0 and 0.98 kPa, respectively, with $x_w^{F*}$=0.193, $x_w^{R*}$=0.0134, and p^R=73.2 kPa. From Equation 15, the maximum permeate composition associated with the geometric mean feed-side concentration ($y_w^{\mathit{\text{gm}}}$) is 0.991. Based on Equation 23 and Equation 24, ϕ_min=72 and $p_{\mathit{\text{max}}}^P$=1.0 kPa. Alternatively, with Ψ=1.5, Equation 25 yields $p_{\mathit{\text{max}}}^P$=1.2 kPa. Thus, the lesser of the two is 1.0 kPa (the value indicated in Figure 8), which is similar to the 0.87 kPa permeate pressure associated with a 50% area increase based on the spreadsheet calculations. For the 0.1 and 0.5 wt% retentate examples, Equation 24 yielded the lesser value of $p_{\mathit{\text{max}}}^P$, while $p_{\mathit{\text{max}}}^P$ from Equation 25 was the lesser value for the 1 wt% and 5 wt% retentate examples. For each curve, $p_{\mathit{\text{max}}}^P$ is situated in the region where the required area begins to increase sharply with increasing total permeate pressure. As a result, the estimate of $p_{\mathit{\text{max}}}^P$ is useful to assess when total permeate pressure will impact process performance and the area required. In addition, the maximum permeate condenser temperature can be estimated from the bubble point temperature of the permeate at $p_{\mathit{\text{max}}}^P$.

An additional limitation to achieving low permeate pressures via condensation is the freezing temperature of the permeate, which limits both the minimum temperature achievable in a simple liquid condenser and the associated vapor pressure at that temperature^{32, 33}. For example, permeates with low solvent concentrations will freeze near the melting point of water. Pure water ice at 0 °C has a water vapor pressure of 0.6 kPa. Even ice subliming at −20 C exerts a vapor pressure of 0.1 kPa³⁴.

Water permeance and selectivity

The review of membrane materials noted the wide range of water permeances and water/ethanol permselectivities reported for commercially available solvent dehydration membranes⁹. To investigate the effect of changing water permeance and/or selectivity on the performance of the benchmark single-pass PV system, the spreadsheet PV calculations were carried out for water permeances and selectivities ranging from 500 to 20,000 GPU and 20 to 20,000, respectively. Varying water permeance while holding selectivity constant resulted only in a change in the membrane area required to reach the specified retentate water concentration. Area was proportional to the inverse of the water permeance, as is also predicted by the simplified performance equations (Equations 18/19 and 21/22). Modeling the system with a non-negligible permeate pressure did not alter this inverse relationship. The product $A\left(P_w/\mathcal{l}\right)$ was constant for a fixed permeate pressure. All other calculated process values, such as the concentration of ethanol in the permeate or the permeate pressure that resulted in a 50% increase in required area compared to a negligible permeate pressure, were unaffected by a change in water permeance at constant selectivity.

Altering water/ethanol selectivity has the opposite effect, it changes almost all calculated process values. The effects of selectivity values of 20, 200, 2000, and 20,000 on (a) membrane area; (b) bubble point temperature of the permeate (i.e. permeate condenser temperature); (c) concentration of ethanol in the permeate; and (d) fraction of ethanol recovered in the retentate product are shown in separate graphs in Figure 9. The x-axis variable in all graphs in Figure 9 is the total permeate pressure. All other benchmark conditions are retained. The benchmark case, with a selectivity of 2000, is shown as a solid blue line in the graphs. Figure 9a confirms that the membrane area required when the permeate pressure is negligible is independent of selectivity – as indicated by the convergence of the curves at the y-axis. However, as total permeate pressure increases, the impact of selectivity becomes significant as indicated by the diverging curves. The higher the selectivity, the more sensitive membrane area is to permeate pressure. The reason for this behavior is that a lower selectivity results in ethanol diluting the water in the permeate, effectively reducing the partial pressure of water in the permeate. As shown in Figure 9c, there is a dramatic difference in the concentration of ethanol in the permeate for each decade change in selectivity. When p^P=0, the overall permeate ethanol concentration covers the range of 0.15 wt% for α=20,000 to 58.8 wt% when α=20. More importantly, the local permeate concentration of ethanol at the retentate end of the system with p^P=0 is 90.9 wt% (80 mol%) for α=20 but only 0.99 wt% (0.39 mol%) for α=20,000. As a result, the low selectivity membrane inherently dilutes the permeating water with ethanol. Whereas the benchmark system (α=2000) requires 50% more area at a permeate pressure of 0.86 kPa, the system with α=20 does not increase in area by 50% until a much higher permeate pressure, 3.4 kPa. Higher permeate pressures also result in more favorable (i.e., higher) condensation temperatures, presented in Figure 9b as the bubble point of the permeate compositions in Figure 9c for the range of membrane selectivities (bubble point calculated with CHEMCAD). For a given permeate pressure, a higher permeate ethanol concentration necessitates a lower temperature to completely condense the vapor. Higher permeate pressure results in a higher permeate ethanol concentration, but the bubble point temperature is higher due to the higher partial pressures.

Figure 9.

The effect of water/ethanol selectivity of membrane (α = 20, 200, 2000, 20000) in single-pass PV system on (a) membrane area; (b) bubble point temperature of the permeate; (c) concentration of ethanol in the permeate; and (d) fraction of ethanol recovered in the retentate product. All are shown as a function of the total permeate pressure. Curves in (b) only shown above the freezing point temperature. Values calculated with single-pass PV spreadsheet under the following benchmark conditions: 10 wt% water in feed, 0.5 wt% water in retentate, isothermal operation at 70 °C, 0.5 kg s⁻¹ ethanol feed rate, and 2000 GPU water permeance.

The bubble point curves are only shown for temperatures above the freezing points of the corresponding permeate compositions shown in Figure 9c^{32, 33, 35, 36}. Except for the lowest membrane selectivity, permeate pressures achievable by liquid condensation will be above 0.7 kPa. As illustrated in Figure 8 and Figure 9a, this permeate pressure may result in a sizeable increase in membrane area. In order to achieve lower permeate pressures, a more complex permeate processing system is required, such as inserting a booster vacuum pump before the condenser, designing a permeate freezing/thawing system, or adding an absorption/desorption unit. For the ethanol/water system, a lithium bromide absorber has been proposed, utilizing the ability of this salt to reduce the vapor pressure of both ethanol and water^{32, 33, 37, 38}.

Although the higher permeate pressures possible with lower selectivity membranes have the potential to lower the capital and operating costs associated with the vacuum system, they also result in an unwelcome reduction in the fraction of ethanol recovered in the retentate product, as shown in Figure 9d. The maximum ethanol recovery for α=20 is 85% whereas the benchmark α=2000 system has a maximum solvent recovery of 99.8%. Further, the resulting higher level of solvent in the permeate may necessitate more advanced and costly processing, or disposal options, for the permeate. A fractional condenser could be added to the permeate condenser system to purify the permeate vapor. Such a “dephlegmator” condenser has been demonstrated for separating ethanol/water permeate vapors from a PV system^{39, 40}. Alternatively, the permeate condenser/vacuum system could be divided into two or more zones, each capturing the permeate vapor from a section of the membrane system. The pressure and condenser temperature of each zone could then be optimized with the potential to process the permeate condensate from each membrane section separately⁴¹. A more complex system might involve utilizing membranes with different properties in each section to further optimize the separation and permeate processing⁴². Finally, if the solvent recovery system already consists of a batch distillation or evaporation unit, then that unit could be used to strip the solvent from batches of condensed permeate containing unacceptably high concentrations of solvent.

These same permeate management considerations apply to single-pass V·P systems. For recirculating batch PV systems, the permeate condensate from different time intervals of the batch cycle could be collected and managed separately, according to the solvent concentration. Additionally, a booster vacuum pump upstream of the permeate condenser could be brought on-line at a certain time in the batch cycle to reduce the permeate pressure and boost water partial pressure driving force to reduce membrane area, cycle time, and permeate solvent concentration.

Effect of feed temperature on PV system performance

Because the partial pressures of water and ethanol increase dramatically with increasing temperature (see Figure 6), the membrane area required to achieve a given reduction in water concentration with a PV system will vary significantly with feed temperature. This effect on membrane area is exhibited in Figure 10 for a single-pass PV system operated at temperatures ranging from 30 to 110 °C, all other properties being those of the hypothetical benchmark single-pass PV system. Each curve represents areas calculated for a fixed feed temperature with the area plotted as a function of total permeate pressure to illustrate the interplay between feed temperature and permeate pressure. As in Figure 8, the intercept of each curve with the y-axis is the area corresponding to negligible permeate pressure (e.g. A=75.5 m² for the benchmark 70 °C feed) and the closed symbols shown on the y-axis are the areas calculated from Equation 18. As with the effect of varying retentate concentration on membrane area, the Equation 18 calculation provides a reasonable estimate of the spreadsheet calculations for membrane area at p^P=0. Each open symbol represents the permeate pressure at which the calculated area has increased 50% from the area calculated at p^P=0 for a specific feed temperature.

Figure 10.

Effect of PV temperature on the membrane area required in a single-pass PV system to reduce the water concentration of an ethanol/water stream from 10 wt% to 0.5 wt% as a function of the total permeate pressure. Values calculated with single-pass PV spreadsheet under the following benchmark conditions: isothermal operation, 0.5 kg s⁻¹ ethanol feed rate, 2000 GPU water permeance, and water/ethanol α = 2000. The closed symbols on the y-axis are the areas calculated from simplified Equation 18 with geometric mean of ${\gamma }_w{p}_w^{\mathit{\text{sat}}}$. Open symbols represent the permeate pressure at which the area has increased 50% from the area calculated at p^P=0 for a specific temperature.

The vertical spacing of the curves and the log scale of the area axis in Figure 10 indicates that temperature does have a marked effect on the area required for a PV separation. The minimum area at 110 °C is calculated to be 17 m² whereas it is 534 m² at 30 °C. This 31-fold increase explains the motivation to operate PV systems at the highest practical temperature. Despite this dramatic change in required membrane area, other process parameters are unchanged, or only modestly altered, with temperature. For example, the concentration of ethanol in the permeate (p^P=0) for the 30 °C feed is calculated to be 1.61 wt%, but is calculated to change only to 1.37 wt% for a 110 °C feed.

As p^P increases at each temperature in Figure 10, the predicted membrane area increases due to the reduced driving force for flux through the membrane. However, the magnitude of this impact depends on the specified feed temperature as indicated in the figure by the relative positions of the open symbols that note a 50% area increase for each curve. Systems operating at higher temperatures are able to accommodate higher permeate pressures before system area is impacted. As before, the benchmark system operating at 70 °C needs 50% more area at p^P=0.86 kPa than at p^P=0, while systems operating at 30 and 110 °C are predicted to need 50% more area at p^P of 0.12 and 3.8 kPa, respectively. The high membrane area and low permeate pressure projected for a 30 °C PV operation make such low feed temperatures impractical unless thermally-sensitive materials are present.

Increasing temperature affects the performance of recirculating batch PV systems in the same manner as single-pass PV systems, dramatically reducing the membrane area required to reduce the water in a given initial charge of water/ethanol mixture over a fixed cycle time. For single-pass V·P operations, the effect of temperature is through the partial pressures and total feed pressure generated in a complete evaporator operated at the designated temperature. Since those pressures are determined by the evaporation temperature, the effect of temperature on V·P system performance and membrane area will mirror that of the single-pass PV system.

Effect of temperature drop on PV system performance

The performance calculations described above assumed isothermal operation of the PV systems. Unfortunately, as noted in the discussion of assumptions, PV involves a phase change from the feed-side liquid to the permeate vapor resulting in a cooling of the feed-side liquid. This temperature change is small if a small amount of permeate is generated per unit feed liquid, as might be the case for a small change in water concentration. However, the enthalpy of vaporization ($\Delta H_i^{\mathit{\text{vap}}}$) of each species considered herein is large compared to the heat required to alter the temperature of a solvent/water liquid mixture. As a result, it does not require much water evaporation to cause a noticeable temperature drop and, as noted above, system performance is strongly dependent on temperature.

If the temperature drop is physically impractical, or simply more than desired, then heat must be added. In a single-pass system, this usually involves inter-stage heat exchangers located so the temperature change does not exceed a target threshold⁴³. This involves a tradeoff as a smaller reheat threshold will necessitate more, but lower power reheaters whereas a larger reheat threshold will reduce the number of reheaters but cause portions of the membrane system to operate at lower temperatures, depressing system performance and increasing membrane area due to a lower average operating temperature⁴⁴. In a recirculating batch PV system, inter-stage reheaters could be used, but it could also be operated with just a feed heater and a higher recirculation rate – in which case the maximum temperature drop (ΔT_max) for the system over the cycle time could be specified, which would then specify a minimum recirculation rate given the separation required in that time interval. The maximum temperature drop in a recirculating batch cycle is expected to be observed at the outset of the batch cycle due to the relatively high water flux at the initial water concentration in the cycle.

The effects of single-pass inter-stage reheat threshold and batch ΔT_max on membrane area and system performance, as calculated with the spreadsheets based on Equation 17 and Equation 20, are shown in Figure 11 for two permeate pressure conditions: the benchmark case of p^P=0 and the permeate pressure resulting in a 50% increase in area for isothermal operation (0.86 or 0.87 kPa). For the curves representing p^P=0 (the lower two curves in the figure), the required membrane area increases relatively linearly and modestly relative to both reheat threshold and ΔT_max. A 50% increase in required membrane area for p^P=0 is predicted to result from a 16 °C inter-stage reheat threshold or batch ΔT_max of 21 °C. Consequently, allowing both single-pass and recirculating batch systems to operate in a temperature window between the feed temperature of 70 °C down to about 50 °C would only increase area by 50%.

Figure 11.

Effect of the reheat temperature drop threshold of a single-pass PV system (solid lines) and the maximum temperature drop in a batch PV system (dashed lines) on the membrane area required to perform the benchmark separation for systems operating with negligible permeate pressure and with a permeate pressure that is predicted to cause a 50% increase in area for an isothermal system. Values calculated with single-pass and batch PV spreadsheets under the following benchmark conditions: 10 wt% water in feed, 0.5 wt% water in retentate, feed temperature 70 °C, 0.5 kg s⁻¹ ethanol feed rate or 1440 kg ethanol batch charge, 2000 GPU water permeance, and water/ethanol α = 2000.

Introducing a non-negligible permeate pressure changes the dynamic of temperature drop in a single-pass system, but not for a recirculating batch system. As indicated in Figure 11, the area required for a single-pass system rises rapidly with reheat threshold when p^P=0.86 kPa, with a 50% increase above the base 113 m² predicted to occur at a reheat threshold of only 7 °C, noticeably lower than the 16 °C threshold that caused a 50% area increase when p^P=0 was assumed. However, for a recirculating batch system operated with p^P=0.87 kPa, a 50% increase above the base 112 m² is predicted to occur when ΔT_max is 21 °C, the same as for p^P=0. The reason for the difference between recirculating batch and single-pass systems at non-negligible permeate pressures is that the fixed reheat threshold for single-pass PV results in the lowest temperature coinciding with the lowest water concentration, severely lowering the feed-side vapor pressure of water. Conversely, with the recirculating batch system, ΔT_max is observed in the initial time interval and diminishes as the cycle proceeds due to less water evaporating per unit circulating fluid near the target final water concentration. The impact of non-negligible permeate pressure on single-pass reheat threshold could be moderated by allowing a higher reheat threshold in the first membrane sections when the water is highest while reducing the reheat threshold in the last membrane sections, or by simply adding an inter-stage heater later in the flow path.

Effect of feed-side pressure drop on V·P system performance

As noted previously, the controlling parameter for V·P system performance is the applied feed-side pressure. Because there is no phase change in V·P, temperature drop due to permeation is not a decisive issue for V·P as it is in PV. However, feed-side pressure drop is a critical issue. The simple act of the feed vapor flowing through the membrane module causes a viscous flow pressure drop, the result of which is similar to that of temperature drop in PV – a decline in the feed-side partial pressure of both species and a drop in the driving force. The net result is an increase in required membrane area and/or a need to operate with a lower permeate pressure. The single-pass V·P spreadsheet allows for consideration of a feed-side pressure drop as a fixed pressure drop for each sub-unit of the system.

Effect of solvent on general system performance

The previous sections investigated how changes in process variables and membrane performance characteristics are predicted to affect membrane area requirements, permeate purity, solvent recovery, and permeate condenser/vacuum demands for the removal of water from ethanol. In this section, the impact of changing the solvent in the mixture from ethanol to one of the other targeted solvents will be assessed. In Figure 7, the impact of solvent type on the partial vapor pressures of both water and of the solvent at 100 °C were presented for water content up to 20 wt%. The more hydrophobic solvents exhibited higher water partial pressures. How these partial pressure differences translate to PV system performance was assessed first by calculating the areas required to reduce the water content from 10 to 0.5 wt% for each solvent at 100 °C. Areas calculated with the single-pass PV prediction spreadsheet are presented in Figure 12, normalized by the area required when ethanol is the solvent and p^P=0 (A_min=23.9 m²). The membrane was assumed to deliver the same water permeance (2000 GPU) and selectivity (2000) for all solvents. The areas associated with four permeate pressure conditions are shown as individual vertical bars for each solvent, p^P=0, 1, 2.7, and 5 kPa. The permeate pressure of 2.7 kPa is the pressure that causes a 50% increase in area for ethanol at this temperature and is included as a reference for the other solvents. The other two non-negligible pressures of 1 and 5 kPa were selected to provide a range of permeate conditions for comparison. MIBK and MtBE are not included in the figure because 10 wt% water exceeds the solubility limit of each at 100 °C. It should be noted that 100 °C is above the normal boiling point of most of the solvents considered here, requiring operation of the PV system above atmospheric pressure to maintain the feed as a liquid. If operating the system at atmospheric pressure, then temperature would be limited to the minimum boiling point of the mixture over a given concentration range.

Figure 12.

Effect of solvent in a binary water/solvent mixture on the area required for water reduction for single-pass PV operating at 100 °C as a function of total permeate pressure. All areas are relative to the area required when ethanol is the solvent and p^P=0 (A_min = 23.9 m²). Values calculated with single-pass PV spreadsheet under the following benchmark conditions: 10 wt% water in feed, 0.5 wt% water in retentate, isothermal operation, 0.5 kg s⁻¹ solvent feed rate, 2000 GPU water permeance, and water/solvent α = 2000. MIBK and MtBE not shown because water solubility is less than the 10 wt% water assumed for the feed.

For negligible permeate pressure (left-most bar for each solvent), the membrane area required to reduce water from 10 to 0.5 wt% decreases in the order: methanol > DMF > ethanol > 2-propanol > acetone > acetonitrile > 1-butanol > THF. This order is the reverse of the order of decreasing water partial pressures displayed in Figure 7. Thus, only when the solvent was DMF or methanol did the calculations predict more area was required than when ethanol was the solvent. Although not shown, the area calculated from Equation 18 using the geometric mean feed-side water partial pressure provided a reasonable estimate of the spreadsheet calculations for membrane area of each solvent at p^P=0 (the average difference between the area from Equation 18 and the area from the spreadsheet calculation was 4.4%). Solvent/water systems requiring the least amount of membrane area when p^P=0 showed the lowest relative increase in area due to increasing p^P. THF is predicted to require the least membrane area for this separation. A 50% increase in area when THF is the solvent is predicted to occur when p^P rises to 12.1 kPa, markedly higher than 50% increase caused by p^P=2.7 kPa for ethanol. At the other end of the range, a 50% area increase is predicted for methanol when p^P is only 1.5 kPa. This trend is disrupted only for DMF as the area to remove water from DMF rises above that of even methanol at higher permeate pressures. Similar trends in area were calculated for an equivalent recirculating batch PV system, also operated at 100 °C.

A change in solvent will also impact the concentration of solvent in the permeate. The permeate solvent concentrations corresponding to the areas and conditions used to generate Figure 12 are displayed in Figure 13. Here the trends are not as obvious as for area because the average solvent permeate concentration depends primarily on the product of the solvent partial pressure and membrane area. In this way, although DMF has the lowest partial pressure of all the solvents, 1-butanol is predicted to yield the lowest permeate concentration because the membrane area required for the water/DMF separation is about 3 times that required when 1-butanol is the solvent. When p^P=0 (solid black bars in Figure 13), the permeate concentrations for all solvents are predicted to be in the vicinity of 1 wt%, ranging from 0.21 wt% 1-butanol to 2.9 wt% methanol. The effect of operating with non-negligible permeate pressures on permeate solvent follows the trend for membrane area with methanol predicted to reach 16 wt% in the permeate if p^P=5 kPa, further illustrating the impact of p^P on many aspects of system performance.

Figure 13.

Effect of solvent in a binary water/solvent mixture on the concentration of solvent in the permeate stream of a single-pass PV operating at 100 °C as a function of total permeate pressure as calculated with single-pass PV spreadsheet. Same conditions as described for Figure 12.

Effect of Solvent on Membrane Performance

The above predictions for the effect of solvent on PV system performance assumed water permeance and selectivity were the same for all solvents. These assumptions are useful to scrutinize the impact of solvent vapor pressure properties on system parameters, but it is more probable that the solvent will impact membrane performance, particularly selectivity. The effect of selectivity on permeate solvent concentration for selectivities of 20, 200, 2000, and 20000 is shown in Figure 14 for the same separation conditions as applied in the previous figure, including a fixed water permeance of 2000 GPU, and with p^P=0. The solid black bars designate α=2000 and are the same as the black bars in Figure 13. As indicated in Figure 14 for each solvent, a 10-fold decrease in α results in approximately a 10-fold increase in permeate concentration - until a 10-fold increase in concentration is not physically possible. For the lowest selectivity of 20, each solvent would be present at over 10 wt% in the permeate and most of the solvents would be nearly 10 wt% in the permeate even for α=200. The average permeate concentration falls to about 0.1 wt%, about 1,000 ppm solvent in water, when selectivity is 20,000.

Figure 14.

Effect of water/solvent selectivity (α) on the concentration of solvent in the permeate stream of a single-pass PV operating at 100 °C. Values calculated with single-pass PV spreadsheet with the same conditions as described for Figure 12 except variable selectivity and p^P=0.

For selective layer materials that minimally swell in water and in the solvents, ($P_w/\mathcal{l}$) should be relatively constant between solvents. However, due to the differences in molecular size of the solvents and membrane-solvent interactions, solvent permeance will depend on the solvent and, therefore, so will α. For example, Tang et al. and Huang et al. reported the strong effect of molecular size on permeability through amorphous perfluoro polymers^45–47. In one study, the permeability of water in the amorphous perfluoro polymer Hyflon AD 60 was about 10 times that of methanol, 100 times that of 2-propanol, and 200 times that of 1-butanol⁴⁵. Although zeolite membranes operate, in part, on a size sieving mechanism and the NaA, T-type, and CHA zeolites possess pore sizes that should reject most solvents, the permeance of organic compounds with actual zeolite membranes does not always track with molecular size. The presence of non-zeolitic defects and non-selective grain boundary defects may result in transport behavior that does match that of zeolite pores⁴⁸. Sommer and Melin tested NaA and T-type zeolite membranes, as well as amorphous silica membranes, for the dehydration of a variety of organic solvents^{49, 50}. For the solvents in Figure 12, water/solvent permselectivities for a NaA zeolite membrane varied from 400 for acetonitrile up to 31,000 for methanol⁵⁰. Each of the four alcohols studied (methanol, ethanol, 2-propanol, and 1-butanol) was reported to have a water/solvent selectivity of over 10,000, as did THF. The selectivity fell to 2,700 for acetone and feel further to 520 and 400 for DMF and acetonitrile, respectively. Based on those values, the permeate concentrations for the alcohols and THF with that NaA membrane would be better represented by the right-most (red) bars in Figure 14, whereas acetone would be the black bar, and the permeate for DMF or acetonitrile would be closer to the α=200 gray bars. Thus, although DMF would have nearly the lowest permeate concentration of the solvents if selectivity was independent of solvent type, it is likely to produce one of the highest for the selectivities reported by Sommer and Melin for a NaA zeolite membrane. While there was a 75-fold range in selectivity for the NaA membrane, the ratio of highest to lowest water permeance for the eight solvents in Figure 12 was only 3.3⁵⁰. Thus, the solvent that elicits the lowest water permeance would require 3.3 times as much membrane area as that with the highest water permeance, all other factors being equal. The magnitude of this range in required area is on the order of the ratios of membrane areas between solvents simply due to the solvent properties as shown in Figure 12.

For membrane materials that sorb water and swell as water activity increases, a change in solvent could impact performance of water-swelling materials by impacting the water activity experienced by the membrane, thereby affecting both ($P_i/\mathcal{l}$) and α. For example, Yave recently reported on the effect of feed water concentration on two commercial poly(vinyl alcohol)membranes fabricated with different degrees of crosslinking¹⁹. A higher degree of poly(vinyl alcohol)crosslinking resulted in a lower dependence of ($P_i/\mathcal{l}$) and α values on water content and higher selectivity and lower permeances than that of the less crosslinked poly(vinyl alcohol). If the effect of water content on membrane performance is known, then that could be factored into the estimates, possibly as a geometric mean permeance in the rough calculations or as variable permeances in the spreadsheet calculation^20–22. Similarly, if the effect of temperature on permeances was parameterized, this relationship could be added to the estimation spreadsheets.

Designing a PV system for drying multiple water/solvent mixtures

If a facility generates mutiple water/solvent mixtures that can be accumulated in batches, then a single recirculating batch PV system, or a single-pass PV system that processes separate batches of water-contaminated solvent, could be contemplated to process the mixtures⁵¹. Batch systems provide more flexibility since time is an additional design degree freedom⁵². The processing limitations and specifications of the range of mixtures would need to be considered in designing the PV unit. Before engaging in a detailed process design, the rough design (membrane area requirements and cycle times) for each mixture could be estimated with the rough estimation equation and evaluated more thoroughly with a spreadsheet estimator or process simulator. The maximum permeate pressures could then be predicted along with the condenser temperatures based on bubble points of the permeate compositions at the estimated pressures.

Conclusions

The performance of PV and V·P membranes is often boiled down to flux and separation factor for a given set of process conditions. In some cases, a single parameter has been used to compare membranes. Unfortunately, test and literature data reported for membranes may not be directly useful for membrane system end-users because such systems rarely operate at the water concentration, temperature, or permeate pressure reported. In addition, test data may not be available for the solvent being considered for drying. While coupon/module testing is recommended for identifying complications and preparing a final system design, rough calculations can be performed with limited data to arrive at an estimate of membrane area requirements and process performance.

Even for the single binary system of water/ethanol, when operated under a non-negligible permeate pressure, a change in one process condition resulted in changes in all process outcomes including: membrane area, permeate concentration, solvent recovery, permeate condenser temperatures, or heating requirements. In other words, when describing PV or V·P process performance: “everything is connected to everything else” – an expression of the Haida people to describe the interdependency and interconnectivity in the natural world. While the interdependency in a PV/V·P process pales in comparison to that of nature, the response of process performance to the parameters studied herein indicates that PV/V·P processes are more complex than generally portrayed.

The simplified equations developed herein proved useful in estimating the minimum membrane area requirements when the system is operated under negligible permeate pressure. The spreadsheet calculations enabled predictions for non-negligible permeate pressures in a cross-flow configuration. The permeate pressure threshold for raising required membrane area by 50% was lowest for the lowest feed temperature, lowest retentate water concentration, and highest selectivity. A method of estimating the maximum practical permeate pressure was developed with values that mirror those identified with the spreadsheet based on the 50% area increase threshold.

While each of the membrane materials is expected to be capable of removing water from almost all the targeted solvents, the specific solvent was shown to greatly affect process outcomes. As a result, although a single PV or V·P system could be contemplated to dry batches of different solvent/water mixtures, process parameters would need to be tailored to each batch to account for the effect of the solvent and any other processing difference, such as initial water content and required final water content for reuse, on the process. In addition to being more flexible, recirculating batch PV processes were predicted to be less sensitive to permeate pressure increases. However, the overarching observation from this work is that the process outcomes depend on the full range of process parameters under non-ideal conditions.