Chapter 18. Introduction to Membrane Separation Processes

This chapter presents an introduction to the four membrane separation methods most commonly used in industry: gas permeation, reverse osmosis (RO), ultrafiltration (UF), and pervaporation (pervap). These membrane separation processes are based on the rate at which solutes transfer though a semipermeable membrane. They are not operated as equilibrium-staged processes. The key to understanding these membrane processes is the rate of mass transfer, not equilibrium. Yet, despite this difference, we will see many similarities in the solution methods for different flow patterns with the solution methods developed for equilibrium-staged separations. Because the analyses of these processes are often analogous to the methods used for equilibrium processes, we can use our understanding of equilibrium processes to help understand membrane separators.

Because membrane processes are usually either complementary or competitive with distillation, absorption, and extraction, some knowledge of the membrane separations will prove very helpful even if the engineer usually uses equilibrium-based processes. In gas permeation components selectively transfer through the membrane. Gas permeation competes with cryogenic distillation as a method to produce nitrogen. Absorption and gas permeation are competitive methods for removing carbon dioxide from natural gas streams. In RO a tight membrane that rejects essentially all dissolved components is used. Water dissolves in the membrane and passes through under a pressure difference up to 6000 kPa (800 psi) (Li and Kulkarni, 1997). RO has in many cases displaced distillation as a method for desalinating seawater and is extensively used to make wastewater potable (Reisch, 2007). UF membranes are fabricated to pass low molecular weight molecules and to retain high molecular weight molecules and particulates. The pressure difference, 70 to 1400 kPa (10–100 psi) is more modest than in RO. UF is a useful method for separating proteins and other large molecules that essentially have no vapor pressure and, thus, cannot be distilled. UF competes with extraction, crystallization, and chromatography as a separation method for biochemicals. In pervap the feed is a liquid, while the permeate product is removed as a vapor. Pervap is used as a method to break azeotropes and is often coupled with distillation columns. Since membrane separations are based on different physicochemical properties than equilibrium-staged separations, membrane separators can often perform separations such as separation of azeotropic mixtures and separation of nonvolatile components, which cannot be done by distillation or other equilibrium-based separations. Descriptions of these membrane separation processes are found in Baker (2012), Baker et al. (1990), Eykamp (1997), Geankoplis (2003), Kucera (2010), Noble and Stern (1995), Ho and Sirkar (1992), Strathmann (2011), Wang and Zhou (2013), and Wankat (1990).

There are several other membrane processes that are in commercial use but are not covered in this chapter. In dialysis, small molecules in a liquid diffuse through a membrane because of a concentration driving force (Strathmann, 2011). The major applications are hemodialysis and the artificial kidney developed by Kolff and Beck in 1944 for treatment of people whose kidneys do not function properly (Lonsdale, 1982). Electrodialysis (ED) uses an electrical field to force cations through cation exchange membranes and anions through anion exchange membranes (Lonsdale, 1982; Strathmann, 2011). The membranes are alternated in a stack, and every alternate region becomes concentrated or diluted. ED is used for desalination of brackish water and in the food industry. Vapor permeation is similar to gas permeation except vapors that are easily condensed are processed (Huang, 1991). This process has not met its potential partly because of difficulties with condensation of liquid. Liquid membranes use a layer of liquid instead of a solid polymer to achieve the separation (Wankat, 1990). Liquid membrane systems can be operated as countercurrent processes, and, to some extent, compete with extraction. Microfiltration is similar to UF but is used for particles between the sizes processed by UF and normal filtration (Noble and Terry, 2004). Nanofiltration removes molecules and particles that are in-between the sizes removed by RO and UF (Strathmann, 2011). The design procedures developed for RO and UF can be applied to microfiltration and nanofiltration. The entire spectrum of membrane separations is summarized in Table 18-1. Nanofiltration is not listed separately but is similar to RO except that Δp is from 0.3 to 3.0 MPa, and the approximate size retained is 8-50Å.

At the level of this introduction the mathematical sophistication needed to understand membrane processes is approximately the same as that needed for the equilibrium-staged processes. A background in mass transfer (Chapter 15) is helpful but not essential.

Note: A nomenclature list for this chapter is included in the front matter of this book.

1. Explain differences and similarities between the different membrane separation systems

2. For a perfectly mixed system with no concentration polarization, predict performance of an existing membrane separation system, and design a new membrane separation system using both analytical and graphical analysis procedures

3. Explain and analyze the effect of concentration polarization, and include it in the design of perfectly mixed membrane separators

4. Explain and analyze the effect of gel formation, and include it in the design of UF systems

5. Determine appropriate operating conditions for pervaporation systems

6. Explain and analyze effects of flow patterns on separation achieved in membrane systems

Common commercial geometries are shown in Figure 18-1 (Leeper et al., 1984). More extensive construction details are shown by Baker et al. (1990) and Eykamp (1997). The plate-and-frame system (Figure 18-1A) is similar to a parallel-plate heat exchanger or a plate-and-frame filter press, except that the filter cloth is replaced by flat sheets of membranes. This design is used for food-processing applications in which rigorous cleaning by disassembly may be required, for electrodialysis (which involves passing a current through the membrane), and for membrane materials that are difficult to form into more complicated shapes.

The tube-in-shell system (Figure 18-1B) is occasionally used. This configuration is very similar to a shell-and-tube heat exchanger. The membrane is coated on a porous support. The main advantage of these systems is that they can be cleaned by passing sponge balls through the separators. The surface area per unit volume is larger than plate and frame but smaller than spiral wound.

The spiral-wound configuration (Figure 18-1C) is more complicated but has a significantly larger surface area per unit volume. With proper design of the channels there is significant turbulence at the membrane surface that promotes mass transfer. The spiral-wound configuration is the most common module for RO because it has a high surface area/volume and does not become easily clogged with particulates. These systems have also been used for carbon dioxide recovery and UF of relatively clean solutions.

The hollow-fiber configuration (Figure 18-1D) looks schematically very similar to the tube-in-shell system, except the tubes are replaced by a very large number of hollow fibers made from the membrane. This configuration has the largest area-to-volume ratio. The hollow fibers can be optimized for a particular separation. For RO the inner diameter is about 42 microns, and the outer diameter is about 85 microns. The separation is done by a 0.1 to 1 micron skin on the outer surface. The remainder of the membrane is a structural support. For gas permeation the requirement of a small pressure drop inside the tubes dictates a larger inner-fiber diameter. For UF, in which feed can be dirty, the fibers are 500 to 1100 microns inside diameter. Typically the feed is inside the tubes, and the thin membrane skin is on the inside of the fibers. Care must be taken that particulates do not clog the fibers. Hollow-fiber membranes are technologically the most difficult to make.

The most common systems are spiral wound and hollow fiber. Systems are usually purchased as off-the-shelf modules in a limited number of sizes. Since a standard size is unlikely to provide the required separation and flow rate for a given problem, a series of modules is cascaded to obtain the desired separation (Figure 18-2). The parallel configuration (Figure 18-2A) allows one to increase the feed flow rate and is used with all of the membrane separations. Parallel operation is roughly analogous to increasing the diameter of a distillation or absorption column, except there is very little economy of scale [the exponent in Eq. (11-3) is ∼1].

The retentate-in-series system (Figure 18-2B) increases the purity of the more strongly retained components and simultaneously increases the recovery of the more permeable species. Unfortunately, purity of permeate and recovery of retained components both decrease. This configuration is used for production of less-permeable nitrogen from air at relatively high pressure since the nitrogen has not permeated through the membrane. The system is roughly analogous to a crossflow extractor or stripper. Parallel and retentate-in-series systems are often combined (Figure 18-2C).

The permeate-in-series system (Figure 18-2D) is used when the permeate product is not of high enough purity and one or more additional stages of separation are required. Designers try to avoid this configuration if possible. The major cost of operating most membrane separators is the energy required to create the pressure difference to force the permeate through the membrane. This configuration requires an additional compressor or pump to repressurize the permeate for the next membrane separator. One of the major disadvantages of membrane separators is that additional stages for the permeate do not reuse the energy-separating agent (pressure). The equilibrium-staged separations have the advantage that energy-separating agents (heat and cooling) can easily be reused. Ideally, a commercial membrane will be available that produces permeate of the desired purity in one stage.

A final common membrane cascade is the retentate-recycle or feed-and-bleed mode (Figure 18-2E). The recycle allows for a very high flow rate in the membrane module to increase mass transfer rates and minimize fouling. The same high flow rates can be obtained without recycle, but a very long membrane system is required to have a sufficiently long residence time. Recycle is used extensively with continuous and batch UF systems. Recycle in Figure 18-2E is similar to recycle in Figure 18-2D since high-pressure retentate is recycled in both cases. Thus, the compressor or pump needs only to boost pressure (e.g., because of friction losses for flow inside a hollow-fiber membrane), not overcome the large pressure difference that results when fluid permeates through the membrane.

The flow patterns on the retentate (feed) and permeate sides of the membrane have major effects on the mass balances and the separation. These flow patterns can be perfectly mixed on both sides, plug flow on one side and perfectly mixed on the other side, plug flow in the same direction on both sides (cocurrent), plug flow in opposite directions on the two sides (countercurrent), mixed on one side and crossflow on the other, and somewhere in between these ideal regimes. For example, if we consider the hollow-fiber module shown in Figure 18-1D, the flow inside the hollow fibers is very close to plug flow. Depending on the shell-side design, the flow on the shell side could be an approximately well-mixed, cocurrent plug flow or a countercurrent plug flow. Countercurrent plug flow usually gives the most separation.

To simplify the mathematics, in Sections 18.2 to 18.6 we assume that both sides of the membrane are perfectly mixed. The resulting designs are conservative in that the actual apparatus results in the same or better separation than predicted. The effect of other flow patterns is explored in Section 18.7.

A large number of polymers (Figure 18-3) have been used to make membranes for membrane separators. Cellulose, ethyl cellulose, and cellulose acetate (actually a polymer blend of cellulose, cellulose acetate, and cellulose triacetate [Kesting and Fritzsche, 1993]) were the first commercially successful membranes and are still used. These materials have relatively poor chemical resistance, but their low cost makes them attractive when they can be used. The most common commercial membrane material is polysulfone, which has excellent chemical and thermal resistance. Various polyamide polymers are also commonly used in RO. Silicone rubber, which has very high fluxes but low selectivity, is often used in pervaporation to preferentially permeate organics and as a coating on composite membranes for gas permeation. A large variety of grafted polymers, specialty polymers, and composite membranes has been developed to optimize flux and selectivity, particularly for pervaporation (Huang, 1991; Strathmann, 2011) but also other membrane separations. Kesting and Fritzsche (1993) are an excellent source for information on the relationship between polymer structure and membrane properties for the polymers used for gas separation.

The basic equation for flux of permeate through the membrane is

Although this equation is derived from experiments and is not a fundamental law, it is so basic and powerful that you need to memorize it and explore its implications in as many ways as possible. The exact form of the equation depends on the type of flux and the type of separation.

The flux can be written as a volumetric flux,

or as a mass flux,

or as a molar flux,

These fluxes are obviously related to each other:

where ρ_m and ρ are molar and mass densities, respectively. Once flux is known, it is used in conjunction with feed rate and feed concentration to determine the membrane area required.

Equation (18-1a) can be used to determine flux once appropriate terms on the right-hand side of the equation are known. Permeability, P, is a transport coefficient that can be determined directly from experiment. In some cases permeability can be estimated from more fundamental variables such as solubility and diffusivity (see next section). The membrane separation thickness, t_ms, is the thickness of the portion of the membrane that is actually doing the separation. This is often a thin skin with a thickness less than 1 micron (10^–6 m) and, in polymeric hollow-fiber gas permeation membranes, can be as low as 0.1 micron. Since t_ms can be difficult to measure, the ratio (P/t_ms), the permeance, is often reported from experimental data. The units of P and (P/t_ms) depend on the flux and driving force employed.

The driving force depends on the type of membrane separation. For gas permeation the driving force is the difference in partial pressure of the transferring species across the membrane. For RO the driving force is the pressure difference minus the osmotic pressure difference across the membrane. For UF the driving force is the same as for RO but usually simplifies to pressure difference across the membrane since osmotic pressures are small. These driving force effects are specific to each membrane separation and are discussed in more detail in later sections.

Since flux is flow rate per membrane area, increasing flux (while maintaining the desired separation) decreases membrane area. The result is a more compact, less expensive device. Equation (18-1a) shows that flux can be increased by increasing permeability or driving force or by decreasing the separation thickness. Permeability of the membrane depends on interactions between the molecular structure of the membrane material and the solutes. These effects are briefly discussed in the sections on each membrane separation. More details on molecular structure-permeability effects are in the books by Baker (2012), Kesting and Fritzsche (1993), Ho and Sirkar (1992), Noble and Stern (1995), and Strathmann (2011).

For many years attempts were made to make very thin membranes, but there was always difficulty in making the membranes mechanically strong enough. Sidney Loeb and Srinivasa Sourirajan (1960, 1963) made the breakthrough discovery. They found that anisotropic (asymmetric) membranes with a very thin skin (0.1 to 1 µm) on a much thicker porous support have the necessary mechanical strength while having a very small separation thickness. Schematics of isotropic and anisotropic membranes are shown in Figure 18-4. Another method for producing an asymmetric membrane is to cast a thin layer of one polymer onto a porous supporting layer of another polymer to form a composite membrane. Although not all membrane separators use asymmetric membranes, they are the most common.

Membranes must be essentially totally hole free. If there is a pinhole high-pressure fluid will be pushed through in convective flow. This fluid has not undergone the separation process and will contaminate the product. Example 18-4 calculates the magnitude of this hole effect.

Gas permeation systems typically use hollow-fiber or spiral-wound membranes, although hollow-fiber systems are more common (Baker, 2004). Cellulose acetate membranes are used for carbon dioxide recovery, polysulfone coated with silicone rubber is used for hydrogen purification, and composite membranes are used for air separation. Feed gas is forced into the membrane module under pressure. Retentate, which does not go through membrane, becomes concentrated in less permeable gas. Retentate exits at a pressure that is close to inlet pressure. More permeable species are concentrated in permeate. Permeate, which has passed through the membrane, exits at low pressure. Operating cost for a gas permeator is the cost of the compression of feed gas and the irreversible pressure difference that occurs for gas that permeates the membrane. A typical hollow-fiber unit contains 5000 m² membrane area per m³ at a cost of approximately $200/m².

For gas permeation steady-state overall flux Eq. (18-1a) written for volumetric flux of gas A becomes

where J_A is volumetric flux of gas A, F_m,p is molar transfer rate of permeate (gas that has permeated through membrane), y_p,A is mole fraction A in the permeate, ρ_m,p,A is molar density of solute A in permeate, A is membrane area available for mass transfer, P_A is permeability of species A, driving force for separation Δp_A is difference in partial pressure of A across membrane, and t_ms is thickness of membrane skin which actually does the separation. Ratio (P_A/t_ms) is called permanence and is often a variable that is measured experimentally. Since partial pressure = (mole fraction)(total pressure) or p_A = y_Ap, Eq. (18-2a) can be expanded to

where p_r is the total pressure on the retentate (high pressure) side, y_r,w,A is mole fraction A on the retentate side at the membrane wall, and p_p and y_p,w,A are pressure and mole fractions at the wall on the permeate (low pressure) side. The flux equation for other components will look exactly the same as for component A, except subscript A will be changed to B, C, . . . as appropriate. Mole fractions at the membrane wall depend on flow patterns and mass transfer rates in the membrane module (Sections 18.3.2 and 18.7).

where y without subscript A or B refers to the mole fraction of faster permeating species A. In a binary system, flux of slower moving component B is

Since pressure and mole fractions can vary along the membrane, in principle Eqs. (18-2a) to (18-2c) are applied point by point along the membrane. In almost all systems there is no concentration gradient on the permeate side perpendicular to the membrane; thus, we normally replace y_p,w,A with y_p,A, which can vary along the membrane. In addition, if mass transfer on the retentate side is not rapid, the mole fractions and partial pressures at the membrane surface will not be the same as in bulk gas or feed gas (see concentration polarization in Section 18.4). Usually mass transfer rates in gas systems are high enough that the mole fraction at the membrane surface is almost equal to the mole fraction in bulk gas.

Most gas permeation membranes work by a solution-diffusion mechanism. On the high-pressure side of the membrane, gas first dissolves into the membrane. It then diffuses through the membrane’s thin skin to the low-pressure side where the permeate reenters the gas phase. For this type of membrane, permeability is the product of gas solubility H_A in the membrane times diffusivity D_A,M in the membrane:

Solubility parameter H_A is very similar to a Henry’s law coefficient. Representative permeabilities for several polymer membranes are shown in Figure 18-5 and in Table 18-2. Small molecules such as helium have high diffusivities but low solubilities, while large gas molecules such as carbon dioxide have high solubilities but low diffusivities. The resulting product, permeability P, is relatively large for both small and large molecules but has a minimum for molecules around the size of nitrogen.

If driving forces are equal, gases with higher permeabilities transfer through membranes at higher rates. It is useful to define selectivity α_AB as

which can be used as a short-hand method of comparing ease of separation of gases. Selectivity of two solutes can be estimated from Figure 18-5 and Table 18-2. For example, from Figure 18-5, α_CO2∇N2 = P_CO2 / P_N2 = 24/0.3 = 80 in a CA membrane. Selectivity is a function of concentration, pressure, and temperature since individual permeabilities depend on these variables. However, α_AB tends to be less dependent on these factors than individual permeabilities. Note that α_AB is analogous to relative volatility defined in Chapter 2. Since gas permeation is usually operated as one-stage systems, α_AB values of 20.0 and higher are preferred instead of the much more modest values commonly employed in distillation.

Robeson (1993) found an upper limit to performance of polymer membranes in the commercially important separation of oxygen and nitrogen from air. On a log-log plot of selectivity versus oxygen permeability (a Robeson plot), the upper bound plots as a straight line (see Problem 18.D17 for details). Although theoretical reasons for this limit have not been found, very few new membranes have been developed that are able to perform better than Robeson’s limit. Membrane research has focused on ways to do better than Robeson’s upper limit.

while from Eq. (18-3b) for less permeable component B in a binary separation

Because permeate is typically at low pressures where gas molecules are far apart, transfer of A is usually independent of transfer of B and vice versa. Thus, P_A and P_B are constant. Solve both equations for F_m,p, and set them equal to each other.

This equation can be solved for either y_r or y_p, although solving for y_r is easier. After some algebra, the result is

Equation (18-6a), the rate transfer (RT) equation, relates mole fractions in retentate and permeate based on membrane parameters and driving force. Although Eq. (18-6a) was derived for perfectly mixed gas permeators, it is applicable to other flow configurations if applied point by point on the membrane. Since it was derived based on transfer rates, Eq. (18-6a) is not an equilibrium expression; however, it can replace equilibrium expressions for binary systems.

Completely mixed membrane permeators (Figure 18-6) are rate equivalent of flash distillation (Hoffman, 2003). The geometries are identical—a single feed goes into a well-mixed chamber, and two products are withdrawn. The more permeable species concentrates in the permeate product, which is analogous to the more volatile component concentrating in vapor. However, the two products are not in equilibrium but are related by RT Eq. (18-6a). Because of geometric similarity, analysis is surprisingly similar to binary flash distillation analysis.

The overall mole balance for Figure 18-6 is

and the mole balance on the more permeable component is

Equations (18-7) can be combined and solved for either y_r,out or for y_p. Since solving for y_p keeps the analogy with flash distillation intact, we will solve for y_p. In molar units

Equation (18-8) is an operating equation for well-mixed gas permeators. The ratio θ = (F_m,p/F_m,in), called cut, is an important operating parameter. RT Eq. (18-6a) and operating Eq. (18-8) are the equations needed to solve problems with perfectly mixed, binary gas permeators. Solution is illustrated in Example 18-1 when y_p is known and in Example 18-2 when θ is specified.

Once permeate y_p and retentate y_r mole fractions are known, volumetric fluxes can be determined from Eqs. (18-2b) and (18-3b), and the total volumetric flux is

If the area is known, the molar transfer rate can be determined from

or from

If the molar feed rate is known, then F_m,p = θ F_m,in, and area A is

or from

Gas permeators often operate at temperatures and pressures at which gases are very close to ideal. Ideal gas behavior simplifies calculations. At constant pressure and temperature molar densities of all ideal gases are equal, ρ_m,A = ρ_m,B = p/(RT), and densities cancel in Eq. (18-6a). The resulting RT equation is

The volumetric flow rate units of permeabilities listed in Table 18-2 are cm³ (STP)/s. For ideal gases this is equivalent to a molar flow rate since 1.0 mol occupies 22,400 cm³ at standard temperature and pressure (STP). These simplifications are illustrated in Example 18-1.

A graphical solution to perfectly mixed gas permeators is straightforward. On a graph of y_p vs. y_out, Eq. (18-8a) plots as a straight line with a slope of –(1 – θ)/θ. Note that cut θ is analogous to f = V/F in a flash distillation system. Intersection with y_p = y_out line occurs at y_p = y_out = y_in, which is analogous to y = x = z in a flash system. Intersections with axes can also be determined (see Example 18-2 and Figure 18-7). Simultaneous solution is obtained at the point of intersection of the straight line representing Eq. (18-8a) and the curve representing Eq. (18-6a). This method is illustrated in Figure 18-7 for a number of values of θ (see Example 18-2).

Membrane separations are often most effective at low concentrations. This is exactly where distillation is most expensive. Thus, hybrid systems that combine a membrane separator with distillation are often used commercially.

There is one other significant difference between the membrane separators and the equilibrium-staged separations we have studied. Companies are much more likely to buy off-the-shelf or turnkey membrane units. Off-the-shelf systems are modules in standard sizes that are connected to more or less perform the desired separation. The engineer needs to determine performance of these units since they will not be exactly the same as the design desired. With a turnkey unit companies buy from a manufacturer that guarantees a given performance level. This situation arose because only a limited number of companies have the technical expertise necessary to make membrane separators. A large number of companies are capable of making distillation and absorption systems. If your company decides to buy a turnkey unit, knowledge of membrane separations will enable you to help a company lawyer include all pertinent items in the contract and to negotiate a contract that is more favorable for your company. After delivery, you will be able to perform appropriate tests to determine if the membrane system meets contract specifications. If specifications are not met, your company can demand that the vendor fix the problems.

The operating equation is essentially Eq. (18-8a) written for each component:

For ideal gases since molar ratios are equal to volume ratios, θ is the same in molar and volume units. In addition, for ideal gases mole fraction equals volume fraction; thus, Eq. (18-8c) can be used with either molar or volumetric flow rates. The rate expression, Eq. (18-4), for molar flows is

or in volumetric flow rates is

Solving Eq. (18-11b) for y_p,i, we obtain

where, after some algebraic rearrangement,

If Eqs. (18-8c) and (18-11c) undergo exactly the same algebraic steps used for flash distillation [from Eqs. (2-36) to (2-37)], the resulting equation for y_r,i (equivalent to x_i in flash) is

and the summation equation ∑ y_r,i = 1.0, (equivalent to ∑ x_i = 1.0 in flash) is

which is essentially same as flash distillation Eq. (2-40a) (see Problem 18.C5). If we multiply Eq. (18-11e) by K_M,i and note y_p,i = K_M,iy_r,i we obtain an equation equivalent to ∑ y_i = 1.0 in flash.

Thus, K_M,i is the membrane equivalent of the equilibrium K_i value in flash distillation.

In a simulation problem all P_i and y_i,in values, t_ms, p_r, p_p, F_in, and area A are known, but permeate volumetric flow rate F_p or equivalently cut θ are unknown. This problem is equivalent to the classic multicomponent flash distillation problem in which V/F is unknown. In flash distillation the best convergence scheme was the Rachford-Rice equation derived by starting with ∑ y_i – ∑ x_i = 0. For membranes the equivalent convergence method, ∑y_p,i – ∑y_r,i = 0, also appears to be best. The procedure mimics the procedure used for flash distillation:

1. Pick a value of θ between 0.0 and 1.0.

2. Calculate F_p = θ F_in.

3. Calculate each K_M,i from Eq. (18-11d).

4. Calculate ∑y_p,i from Eq. (18-11h) and ∑y_r,i from Eq. (18-11f).

5. Calculate Check = [∑y_p,i – ∑y_r,i].

6. If Check ≠ 0, pick a new value of θ and repeat. A convenient method for obtaining convergence is to use Goal Seek in Excel.

This procedure is illustrated in Example 18-3. Although shown only for gas permeation, this approach can be used with other perfectly mixed membrane separators.

We can do an approximate calculation of flow rate through a pinhole and estimate the magnitude of the effect of the pinholes. Assume the hole is a tube of radius R and length L = t_ms, flow is laminar, change in potential energy (ρgΔz) is negligible, there are no entrance or exit effects, and flow is at steady state. The equation for volumetric flow rate Q is (Bird et al., 2006).

where µ is viscosity, and Δp = p_r – p_p. Although Eq. (18-12) is not strictly applicable because there are certainly entrance and exit effects, it will give a reasonable estimate of magnitude of the effect of the holes on purity.

RO is the most commonly used process to purify and desalinate water (Reisch, 2007). RO is an integral part of industrial water management plans to reach zero-liquid discharge goals (Kucera, 2010). In RO liquid water is forced under pressure through a nonporous membrane in the opposite direction of osmosis (osmosis is defined shortly). Most salts and uncharged molecules are retained by the membrane. Thus, permeate is much purer water and retentate becomes significantly more concentrated. A typical seawater desalination system recovers between 50% and 55% of feed water as potable water.

Commonly used membranes shown in Figure 18-3 are: 1) a blend of cellulose acetate and cellulose triacetate, 2) aromatic polyamides (aramids), and 3) cross-linked aromatic polyamides (Eykamp, 1997). Currently, the best membranes are thin-film interfacial composite membranes (Baker, 2004; Uemura and Henmi, 2008). Both hollow-fiber and spiral-wound modules are used for RO (Figure 18-1), but about 85% of applications use spiral-wound membranes (Baker, 2004). Spiral-wound membranes do not clog as easily as hollow fibers; thus, less pretreatment is needed.

Feed to an RO system usually requires pretreatment to remove particulates that would clog the membrane. If there are ions or solutes in solution that have limited solubility, design must include a solubility calculation to determine if they will precipitate onto the membrane when retentate is concentrated. If precipitation is likely, these ions or solutes must either be removed or made more soluble to prevent them from precipitating. A schematic of a simple RO system including the most important auxiliary equipment is shown in Figure 18-8. In practice, large-scale systems may have hundreds of membrane modules arranged both in series and in parallel (Figure 18-2C) in what is often called a “Christmas-tree” pattern (Baker, 2004). Large-scale systems have pressure exchangers on high-pressure retentate waste lines that recover over 90% of the energy used to pressurize fluid that has not passed through the membrane. More details on equipment are available in Baker et al. (1990), Eykamp (1997), Ho and Sirkar (1992), and Noble and Stern (1995). Kucera (2010) discusses details of operation and maintenance of RO systems.

There are two reasons for studying osmosis in a textbook on separations. First, we need to understand osmosis to design RO systems. The driving force for solvent flux in RO is the difference between the pressure drop across the membrane and the osmotic pressure difference across the membrane (Δp – Δπ). Mass flux of solvent is

In these equations K′_solv is solvent (usually water) permeability through the membrane with an effective membrane skin thickness of t_ms. Pressure drop across the membrane, Δp = p_r – p_p. Δπ is the difference in osmotic pressure across the membrane.

Osmotic pressure is a thermodynamic property of the solution. Thus, π is a state variable that depends on temperature, pressure, and concentration but does not depend on the membrane as long as the membrane is semipermeable. Osmotic equilibrium requires that chemical potentials of solvent on the two sides of the membrane be equal. Note that solutes are not in equilibrium since they cannot pass through the membrane. Although osmotic pressure can be measured directly, it is usually estimated from other measurements (e.g., Reid, 1966). For an incompressible liquid, osmotic pressure can be estimated from vapor pressure measurements:

where V_solvent is partial molar volume of solvent. Another common method is to relate osmotic pressure to freezing-point depression (Reid, 1966).

For dilute systems osmotic pressure is often a linear function of concentration:

As solutions becomes more concentrated, osmotic pressure increases more rapidly than predicted by a linear relationship. For some dilute systems the linear constant can be estimated from the van’t Hoff equation:

Since derivation of the van’t Hoff equation assumes that Raoult’s law is valid, Eq. (18-14d) will be incorrect if solute associates or dissociates even though empirical Eq. (18-14b) may still be accurate. Osmotic pressure, in Pa, for natural waters containing salts can be determined from the following empirical correlations with T in °C and salt concentration C_s in kg/m³ (Geraldes et al., 2005):

Since C_s = xρ_solution with ρ_solution in kg/m³, Eq. (18-14e) can be written as

However, solution density increases as x increases, which means the equation is no longer linear. Solution densities at 25.0°C (Green and Perry, Table 2-110, Table 2-90, 2008) are: 0.0 wt% NaCl, ρ = 0.997 g/cm³; 1.0 wt% NaCl, ρ_solution = 1.00409; and 2.0 wt% NaCl, ρ_solution = 1.01112. If we take an average value of ρ_solution between x = 0.0 and 0.02 [Eq. (18-14e) is valid up to a concentration C_s = 20.0 kg/m³ or x = 0.0197], we obtain ρ_avg = 1004.1 kg/m³. If we convert Eq. (18-14g) to π in atm with a′ in atm/weight fraction, the resulting equation is very close to linear in the range of validity.

At T = 25.0°C, a′ = 705.9 atm/weight fraction.

The second reason for studying osmosis is a novel separation system called forward osmosis (FO), which may become important in the future. FO is osmosis but with the addition of chemicals to form a draw solution that is more concentrated than feed water. The result is that pure water is drawn out of the feed water and into the draw solution (Cath et al., 2006; Chung et al., 2012). The process has also been studied for pharmaceutical manufacturing and food processing. There are several attractive features of the process, the main one being that since the FO step is the natural direction of water movement, energy requirements are very low. In addition, the draw solution can be tailored for specific separations such as difficult and high-salinity waters. The major hurdle is one that all mass separating–agent processes (e.g., extraction) must overcome—how to economically recover the pure product from the draw solution and recycle the draw solution. If it is possible to use low-grade, waste energy (e.g., heat from a diesel generator), there may be applications of FO in small desalination plants.

The mass solute flux across the membrane, J′_A, can be written as

where K′_A is solute permeability, and x_w and x_p are mass fractions of solute at the membrane wall on the retentate side and in the permeate, respectively. For a membrane with perfect solute retention, K′_A = 0. Typical units for terms in Eqs. (18-13) and (18-15) are pressure p and osmotic pressure π in atm, solvent permeance (K′_solv/t_ms) in g/(atm·s·m²), solvent flux J′_solv in g/(m²s), salt permeance (K′_A/t_ms) in g/(m²s·mass fraction), and salt flux J′_A in g/(m²s). If a large amount of water is removed as product, in addition to increasing osmotic pressure because x_r and x_w increase, salt flux into permeate also increases.

The analysis procedure developed in Section 18.3 for gas permeation is the starting point for analyzing RO. However, RO analysis is more complicated because of 1) osmotic pressure, which is included in Eq. (18-13), and 2) mass transfer rates are much lower in liquid systems. Since mass transfer rates are relatively low, the weight fraction of solute at the membrane wall, x_w, is greater than the weight fraction of solute in retentate bulk, x_r. Buildup of solute at the membrane surface occurs because solvent movement from the bulk fluid to the membrane carries solute with it. Since solute does not pass through the semipermeable membrane, its concentration builds up at the wall, and it must back diffuse from the wall to the bulk solution. This concentration buildup, concentration polarization, is illustrated in Figure 18-10. Concentration polarization has a major effect on RO and UF separations (see Section 18.5 for UF). Since concentration polarization causes x_w > x_r, osmotic pressure becomes higher on the retentate side and, following Eq. (18-13), flux declines. Concentration polarization also increases (x_w – x_p) in Eq. (18-15) and the flux of solute may increase, which is also undesirable. In addition, since concentration polarization increases solute concentration at the wall, precipitation and fouling become more likely.

Solvent flux Eq. (18-13) can be expanded by noting F′_solv = F′_p(1 – x_p) [x is normally mass fraction in RO]:

Osmotic pressure on the retentate side depends on the concentration of solute at the membrane wall. To simplify the analysis, we temporarily assume osmotic pressure is a linear function of weight fraction, Eqs. (18-14c), (18-14g), and (18-14h). Then flux is

To relate wall concentration to retentate concentration, we define the concentration polarization modulus, M, in terms of weight fractions:

Methods to measure or predict M are developed shortly. Substituting the definition for M in Eq. (18-16b),

Solute flux Eq. (18-15) can also be expanded and written in terms of concentration polarization modulus.

Essentially the same procedure used to develop the RT equation for gas permeators will be used. That is, after assuming that is not zero and is independent of solvent transfer rate, and that is independent of solute transfer rate, we will solve Eqs. (18-16c) and (18-18) for F′_p, set the two equations equal to each other, and solve for the desired concentration. It is convenient to define membrane selectivity α′ as

It is easier to solve for x_r:

This RT equation represents the transfer rate of solvent and solute through the membrane in mass fraction units. If , α′ will be infinite, x_p will be zero, and x_r must be determined from a mass balance. If selectivity is not constant, α′ will depend on x_w = Mx_r. It will then be convenient to solve for x_p as a function of x_r.

Since RO often operates with high retention of solute and thus very low values of x_p, it is useful to simplify RT Eq. (18-20) for small x_p. As x_p → 0 [(x_p α′ a′) << 1.0], Eq. (18-20) becomes

If α′(p_r – p_p)>>1,

The simplified linear equations are easier to use but are not accurate if (x_p α′ a′) is not small enough. Use of Eq. (18-21b) with the mass balance is illustrated in Example 18-6.

Solving for x_p, we obtain the operating equation

or the alternative

θ′ = F′_p/F′_in is the cut in mass units. Equation (18-23a) is analogous to Eq. (18-8a) for a well-mixed gas permeator.

In well-mixed modules x_r is constant and equal to x_out. Thus, Eqs. (18-20), (18-21a), and (18-21b) are valid with x_r replaced by x_out; however, since there is usually concentration polarization, x_w = Mx_out, and M must be determined. Once M is known, simultaneous solution of Eqs. (18-20) and (18-23) or of Eqs. (18-21) and (18-23) can be obtained either analytically (see Examples 18-5 and 18-6) or by plotting both equations on a graph of x_p vs. x_out. Equation (18-23) is a straight line that is identical to the operating line obtained for gas permeation. The RT curve representing Eq. (18-20) or (18-21) can be plotted by calculating values in the same way as in Example 18-2. Note that this curve is below the x_p = x_out line on a graph of x_p vs. x_out.

RO data are often reported as the rejection coefficient R.

The rejection coefficient measured with no concentration polarization (M = 1.0) is called the inherent rejection coefficient, R^o, since x_out = x_r = x_w when M = 1.0. The inherent rejection coefficient can be used with concentration polarization if we know the wall concentration or M.

From Eqs. (18-24a) and (18-24b) we can relate R to R^o for any value of M.

If R^o is known and mass fractions x_p and x_out are specified, the maximum value of M is

If R is known, we can solve Eq. (18-24a) simultaneously with operating Eq. (18-23).

Alternatively, we can solve for x_p:

These are very convenient and simple solutions, but R has to be known for the current operating conditions that include concentration polarization effects. Use of these equations is illustrated in Example 18-5.

Solving Eq. (18-20) for selectivity, we obtain

If we do an experiment in a well-mixed membrane module with rapid stirring so that there is no concentration polarization, M = 1.0, Eq. (18-26) simplifies to

Since all terms on the right-hand side are known, we can calculate α′. If linear Eq. (18-21b) is used with Eq. (18-24) for R and we solve for α′, we obtain

Both methods are illustrated in Example 18-5.

By manipulating Eqs. (18-21b) and (18-24), we can find a relationship between retention, R, at one set of conditions and R at another set of conditions for dilute, well-mixed systems. A convenient base case (Case A) condition is an experiment with no concentration polarization (M_{Case A} = 1.0 and R_{Case A} = R^o). Substitute Eq. (18-21b) into the definition of R, Eq. (18-24), for both Cases A and B. Solve for 1.0 – R in both equations and then divide Case B equation by Case A equation. The ratio is

This equation is valid for linearized RT equation (18-21b), which requires

Equation (18-29a) allows calculation of R for different values of M and (p_r – p_p) for the same membrane if R is known at Case A conditions and the inequalities in Eq. (18-29b) are satisfied. Unfortunately, the restrictions in Eq. (18-29b) are quite severe for salts because a′ is large.

If pressure difference (p_r – p_p) is the same in the two runs, then

Remarkably, Eq. (18-29c) can be obtained exactly without linearization (except for osmotic pressure) by starting with Eq. (18-20) and requiring constant (p_r – p_p) and constant permeate mass fraction x_p. Since Eq. (18-20) required that osmotic pressure be a linear function of weight fraction (a term), Eq. (18-29c) is restricted to relatively low concentrations [less than 2.0% by weight salt according to Eq. (18-14h)], but this restriction is much less confining than Eq. (18-29b).

Since all terms on the right-hand side are known or measured, M can be calculated from experimental data.

However, estimation of M will allow us to avoid doing expensive and time-consuming experiments. Concentration polarization was shown schematically in Figure 18-9. This figure applies to the simplest situation–steady-state, one-dimensional back diffusion of one solute into a bulk retentate stream with a perfectly rejecting membrane (R = 1.0). (If rejection is not almost complete, more detailed theories are required [e.g., Ho and Sirkar, 1992; Noble and Stern, 1995; Wankat, 1990].) The differential mass balance for this simple situation using a Fickian analysis is (Problem 18.C3):

where D is Fickian solute diffusivity in liquid solution in m²/s, and ρ_solv is in g/m³. Boundary conditions are solute concentration equals solute wall concentration at z = 0:

and concentration becomes bulk concentration x_r when z is greater than or equal to boundary layer thickness δ.

The value of δ depends on operating conditions (geometry, velocity, T). Defining the mass transfer coefficient in the usual form,

the solution is

Since δ depends on operating conditions, mass transfer coefficient k also depends on operating conditions. k has units of m/s. If there are multiple solutes, a Maxwell-Stefan analysis (Section 15.7) is recommended.

This short, and overly simplified, development is useful to determine what affects concentration polarization. If solvent flux J′_solv increases, M increases. If mass transfer coefficient k increases, concentration polarization decreases. Increasing diffusivity increases k. Thus, operating at a higher temperature decreases M, although there are obvious limits based on membrane thermal stability and thermal stability of solutes (e.g., most proteins are not thermally stable). Decreasing boundary layer thickness δ by promoting turbulence or operating at very high shear rates in thin channels or narrow tubes also increases k.

The quantitative use of Eq. (18-34a) requires either experimentally determined values of mass transfer coefficient k or a correlation for k (which is ultimately based on experimental data). If experimental data are available that allow calculation of M from Eq. (18-31), then k can be determined by solving Eq. (18-34a) for k.

A large number of mass transfer correlations are available for a variety of geometries and flow conditions (Wankat and Knaebel, 2008). Five that are useful for membrane separators are correlations for turbulent flow in tubes, for turbulent flow in spiral wound membrane modules, for laminar flow in tubes and between parallel plates, and for well-mixed tanks (Blatt et al., 1970; Schock and Miquel, 1987; Wankat, 1990; Wankat and Knaebel, 2008). For turbulent flow in tubes the mass transfer coefficient can be estimated from

where Sh is the Sherwood number, and the Reynolds number, Re, and Schmidt number, Sc, are defined as

d is tube diameter, and u_b is bulk velocity in the tube. Fully developed turbulent flow certainly occurs for Re > 20,000, and usually appears in UF devices for Re > 2000.0.

Spiral-wound membranes (Figure 18-1C) are most commonly used for water treatment. Spacers in the feed channel are designed to promote turbulence and increase mass transfer rates. Schock and Miquel (1987) experimentally determined a mass transfer correlation for typical spiral-wound modules.

Here D_s is salt diffusivity, and Re is defined by Eq. (18-35b) with d = the height of feed channel. This equation is similar to Eq. (18-35a) but predicts a higher mass transfer coefficient because of the turbulence-promoting spacers.

For laminar flow in tubes of length L and radius R with a bulk velocity u_b, the average mass transfer coefficient is

For laminar flow between parallel plates with a spacing of 2 h, the average mass transfer coefficient is

For flat membranes in a well-stirred turbulent tank, the mass transfer coefficient can be estimated from

In Eqs. (18-37a) and (18-37b) k is in cm/s, ω is stirrer speed in radians/s, d_tank is tank diameter in cm, D is diffusivity in cm²/s, and kinematic viscosity ν = µ/ρ is in cm²/s. Stirred tanks are a convenient laboratory configuration.

Experiment c required a number of steps to eventually find mass transfer coefficient k. This is invariably the case, since k is not a directly measured variable but depends on interpretation of data using a model. Knowing the value of k, we have a single data point to compare to correlation Eq. (18-37a). They disagreed. You may be tempted to use the correlation and ignore the data point. However, mass transfer correlations are not very accurate. They usually predict trends well (such as the effect of Reynolds and Schmidt numbers), but absolute values predicted can be significantly off. If it is reliable, a single data point can be used to adjust the constant in the correlation for application to this particular system. If more data were available, we could check the entire correlation.

Equipment for UF systems often looks very similar to RO systems, although they operate at lower pressures. However, this similarity does not extend to the molecular level. Remember that RO membranes are nonporous and separate based on a solution-diffusion mechanism. UF membranes are porous and separate based on size exclusion. Large molecules are excluded from pores in the thin membrane skin, and thus, large molecules are retained in retentate. Small molecules fit into pores and pass through to permeate. Since there is usually a distribution of pore sizes, molecules within the range of pore sizes partially permeate and are partially retained. In a somewhat oversimplified picture, UF is crossflow filtration at the molecular level.

Because of their porous structures, UF membranes have significantly higher fluxes than RO membranes. Thus, concentration polarization is usually worse in UF than in RO because there is a much greater solvent flow from bulk fluid through the wall. Concentration polarization can cause membrane fouling, which not only decreases flux but also can drastically decrease membrane life. Hydrophilic membranes tend to foul less rapidly but have shorter lives than more stable hydrophobic membranes. The best membrane depends on the operating conditions. Cellulose acetate (Figure 18-3) membranes were the first commercial membranes and are still used where their low level of interaction with proteins is more important than their relatively short life. Polymeric membranes are used where more basic conditions are encountered. The most common polymeric membrane is polysulfone (Figure 18-3). Membranes are tailor-made to sieve molecules in different size ranges depending on the purpose of the separation (Figure 18-11). Nominal molecular weight exclusion (shown by x in Figure 18-11) is often reported by manufacturers, but it is not nearly as useful as a complete retention curve.

With low concentration polarization UF can be analyzed by procedures used for RO; thus, Eqs. (18-12) and (18-16) are valid. However, in UF retained molecules are often very large, and osmotic pressure difference is very low. For most UF applications osmotic pressure difference can be ignored, which simplifies solvent flux equations. In mass units solvent flux is

where F′_p is mass transfer rate of permeate (e.g., kg/s), and x_p is weight fraction of solute in permeate. As we will see, ignoring the osmotic pressure difference is a more important simplification than it looks at first.

For sieve-type membranes if a pore does not exclude solute, it will pass solute at the wall weight fraction, x_w = Mx_r. In sieve-type membranes inherent rejection R^o (M = 1.0) can be interpreted as fraction of flux carried by pores, which exclude solute. Then 1.0 – R^o is fraction of flux carried by pores that do not exclude solute. Ideally, R^o is independent of (p_r – p_p), and by definition R^o is independent of M.

The permeate weight fraction, x_p, is then

This equation can also be solved for retentate weight fraction:

Equations (18-39) and (18-40) are the RT equations for UF. They are particularly simple because inherent rejection R^o is based on experimental data. The retention definition for UF is microscopic, which is different than the macroscopic definition used for RO [Eq. (18-24)]. RT equations for UF depend only on inherent solute rejection and M. For a perfectly mixed membrane module we assume that x_r = x_out. Then either Eq. (18-39) or (18-40) written in terms of x_out can be solved simultaneously with mass balances, Eqs. (18-22) or operating Eq. (18-23). Equations (18-23) and (18-39) or (18-40) can also be solved simultaneously to obtain

Note the differences between UF Eqs. (18-39) to (18-41) and RO Eqs. (18-25) and (18-26). Since R^o in UF is approximately constant, the polarization modulus appears in Eqs. (18-39) to (18-41) from the concentration of material transferred through pores. In RO the value of R depends on both M and (p_r – p_p), and M does not appear explicitly in Eqs. (18-25) and (18-26).

Experimental results with low-concentration feeds or under conditions where M is close to 1.0 are in good agreement with theoretical predictions. However, when wall concentration becomes high, solvent flux J_solv often cannot be controlled by adjusting the pressure difference. Thus, Eq. (18-38) no longer holds! Some other phenomenon must be controlling solvent flux. Careful examination of membrane surfaces after these experiments shows a gellike layer covering the membrane. This gel layer alters the flux-pressure drop relationship and controls solvent flow rate.

The effect of gel formation at the wall can be studied using the diffusion equation. Return to Figure 18-10. We implicitly assumed that wall concentration was a variable that could increase without bound. In gelling systems once wall concentration equals the gel concentration, x_g, concentration at the wall becomes constant at x_w = x_g. As additional solute builds up at the wall, the gel concentration is unchanged, but the gel layer thickness increases; thus, x_w becomes a constant set by solute gelling behavior. This is illustrated in Figure 18-12. The values of x_g vary from less than 1 wt% for polysaccharides to 50 vol% for polymer latex suspensions (Blatt et al., 1970).

To analyze gelling systems we can again use Eq. (18-32a) when R = 1.0. (Once a gel forms it is usually quite immobile, and 100% retention is reasonable.) With the coordinate system redefined as in Figure 18-12, boundary conditions are the same as Eqs. (18-32b) and (18-32c). The solution has same form as in Eq. (18-34); however, since x_w and x_r are fixed, solvent flux is the variable. If we solve for solvent flux, we obtain

The consequence of this equation is that once a gel has formed solvent flux is set by rate of back diffusion of solute. We no longer control solvent flux!

What happens if we increase pressure drop across the membrane for a system with a gel present? Solvent flux temporarily increases, but more solute is carried to the membrane than can be removed by back diffusion. The extra solute is deposited on the gel layer, which increases the gel layer’s thickness. This increases flow resistance, which reduce flux through the membrane until the steady-state solvent flux given in Eq. (18-42) is again obtained. This effect is usually not reversible. Reduction of pressure results in a flow rate less than that predicted by Eq. (18-42) because the thicker gel layer remains on the membrane. A gel layer can often be removed by shutting down the system and back flushing (running pure solvent at a higher pressure on the permeate side) or by mechanical scrubbing of the membrane. Unfortunately, if gel fouls the membrane, it may be very difficult to return the membrane to its original flux behavior. Fouling-resistant membranes should be used under gelling conditions.

Designers have some control since decreasing boundary layer thickness, δ, with thin channels or turbulence increases mass transfer coefficient k and hence flux. To a limited extent D can be increased by raising temperature. Effect of these variables on mass transfer coefficients can be explored using correlation Eqs. (18-35) to (18-37). If fouling is severe after gel formation, it may be necessary to operate so that gels never form: x_w = Mx_r < x_g at all times. This can be done by operating under conditions that reduce M (high values of k) and keeping x_r low (low feed concentrations and low cuts). The feed-and-bleed system in Figure 18-2 is frequently used because very high velocities and hence larger k can be achieved. When a gel does not form solvent flux in UF is controlled by Eqs. (18-38), the concentration polarization modulus can be calculated from Eq. (18-34), and solution is straightforward. When feed-and-bleed systems are used to concentrate solute, the purpose is to retain all solute (often a polymer), and operation is often at the highest flux rate, which is given by Eq. (18-42) when gel just starts to form. Foley (2016) illustrates solution methods for these steady-state systems.

Most feeds processed by UF contain a number of different solutes. How should we analyze these separations? An almost overwhelming temptation is to study each solute individually and then assume that superposition is valid. That is, we assume that each solute in a mixture behaves in the same way it does alone. After all, this is what we did for gas permeation and RO. Thus, we would predict that large molecules will be retained and small molecules will pass through the membrane. If no gel forms, this behavior is often observed; however, if a gel forms, the gel is usually much tighter (less porous) than the membrane, and the gel layer usually captures small molecules. Thus, separation behaviors with and without gels can be quite different. If the purpose of UF is to separate large and small molecules, then we must operate under conditions where a gel will not form. Gel formation and fouling have prevented UF from achieving its full potential because they often limit both separation and flux.

Although fluxes are lower, RO membranes can also cause gelling and foul if they are operated with feeds that can gel or that contain particulates. To prevent fouling it is common to put an UF system before a RO system. Then the UF system removes particulates and large molecules that could foul the RO membrane. This procedure is followed in whey processing in which UF retains proteins and RO retains sugars and salts.

Warning! Experimental results and equations for both UF and RO are reported in many different units. It is easy to make a unit mistake if you do not carefully carry units in the equations. It is especially easy to make a subtle mistake in the appropriate solution density to use when converting from concentration c in g/L or mole/L to weight fraction x. The correct conversion from g/L to weight fraction is

Unfortunately, the solution density ρ_solution is a function of concentration. For a relatively pure permeate (R close to 1.0) the permeate solution density is approximately the solvent density. For more concentrated streams, such as the feed or the retentate, the density needs to be known or estimated as a function of concentration.

When concentrations are given in units of g/L, approximations are often made to calculate fluxes. For example, the correct equation for the solute mass flux is

This is often approximated as

For relatively high rejection coefficients permeate is quite pure, J_solv >> J_A, and the approximation is quite good. In other cases the approximation may not be as accurate.

The inherent rejection and concentration polarization modulus are often defined in concentration units:

With these definitions the solute mass flux is

Equations (18-46) and (18-47) are valid ways to formulate the problem. Unfortunately, it may be assumed that and M = M_c when the exact equations are

If ρ_solution,out and ρ_solution,p are very different, confusing R with R_c and M with M_c can result in significant error. To be exact, check how all terms are defined, use the appropriate solution densities of permeate and retentate to convert to weight fraction units, and then solve in weight fraction units.

while local partial pressure on the downstream side, labeled 2, is

and local driving force is

Because x_i,1, y_i,2, and p_tot,2 vary, local values of partial pressures and driving force depend on flow patterns and pressure drop in the membrane module.

Driving force can be increased by lowering p_tot,2 either by drawing a vacuum as shown in Figure 18-13 or, less frequently, by reducing y_i,2 using a sweep gas on the permeate side. Driving force will also be increased if upstream partial pressure of component i is increased. This occurs for more concentrated feeds (higher x_i,1) and at higher upstream temperatures (larger VP_i,1). Use of higher upstream temperatures may require a higher upstream pressure to prevent vaporization of liquid on the upstream side. Higher upstream pressure does not increase the permeation rate significantly (Neel, 1991). Although feed concentration is not usually a variable the designer can control in a stand-alone, single-pass system (Figure 18-13), it is a design variable in hybrid systems (Figure 18-14).

As usual with membrane separations, the membrane is critical for success. Currently, two different classes of membranes are used commercially for pervap. To remove traces of organics from water a hydrophobic membrane, most commonly silicone rubber coated on a microporous support, is used. To remove traces of water from organic solvents a hydrophilic membrane such as cellulose acetate, ion exchange membranes, polyacrylic acid, chitosan or Nafion^® is used. Both types of membranes are nonporous and operate by a solution-diffusion mechanism. Selecting a membrane that will preferentially permeate the more dilute component will usually reduce the membrane area required. Membrane life is typically about four years (Baker, 2004).

Because evaporation increases the driving force, pervap can use a highly selective membrane and retain a reasonable flux if the more permeable component is also more volatile. However, evaporation complicates both equipment and analysis. Typical permeate pressures are quite low (0.1–100 Pa) (Leeper, 1992). Because of this low pressure, the permeate needs a large cross-sectional area for flow or the pressure drop caused by flow of permeate vapor will be large. Thus, plate and frame, spiral wound, and hollow fibers with feed inside the fibers are used commercially. Figure 18-13 shows that a vacuum pump and a condenser may both be required to recover dilute low-pressure vapor. Unless the cut is small, an additional energy source is required to supply heat of evaporation. If the cut is small, energy in the hot liquid can supply this energy. For larger cuts a portion of retentate can be heated and then recycled. Huang (1991) discusses equipment and equipment design, and Baker (2012) outlines several pervap processes.

Although selectivity in pervap units can be quite high (e.g., Leeper [1992] reports values from 1.0–28,000), the values are not infinite. There will always be some less permeable species in the permeate and some more permeable species in the retentate. If feed concentration and selectivity are high enough, a single-pass system (Figure 18-13) can produce high-purity permeate or high-purity retentate. However, the other stream will contain a significant amount of valuable product. Single-pass systems can produce high purity or high recovery but not both simultaneously.

If both high purity and high recovery are desirable or a single-pass system cannot meet product specifications, a hybrid system with recycle is often used (Huang, 1991; Suk and Matsura, 2006; Wankat, 1990). Hybrid systems use two different types of separation. The most common pervap hybrid is to combine it with distillation with either two columns (Figure 18-14A) or a single column (Figure 18-14B). In Figure 18-14A feed to distillation column 1 forms a minimum-boiling azeotrope. Bottoms product is essentially pure component A. Distillate product from column 1, which approaches the azeotrope concentration, is fed to the pervap unit. If component A preferentially permeates the membrane, the permeate will be more concentrated in A than the distillate. The permeate is then recycled to column 1 to recover the A product. Component B is concentrated in the retentate, which is fed to distillation column 2. Distillate from column 2 also approaches the azeotrope concentration and is part of the feed to the pervap unit. Bottoms product from column 2 is essentially pure B. The two distillation columns in Figure 18-14A are similar to the columns in Figure 8-4A. Since the pervap unit replaces the liquid-liquid separator in Figure 8-4A, the hybrid system does not require a heterogeneous azeotrope; thus, Figure 18-14A is more generally applicable.

If the selectivity is high enough, the retentate may meet purity specifications. Then distillation column 2 is not needed, which results in obvious savings in capital and operating costs. This is illustrated with a simplified one-column flowchart used for breaking the ethanol-water azeotrope (Figure 18-14B). Open steam heating may be used instead of a reboiler (Leeper, 1992). A hydrophilic membrane that selectively permeates water is used. This figure is similar to Figure 8-3A with the pervap system replacing the liquid-liquid separator.

Hybrid system pervap units are usually designed with a low cut per pass to prevent a large temperature drop. Total cut for the entire unit can be any desired value. Typically, if required, a vacuum pump is the highest operating expense. Vacuum pump load can be decreased by refrigerating the permeate condenser’s final stage. Appropriate design of heat exchangers and thermally integrating pervap and distillation systems can significantly reduce energy costs.

Because separation factor data are often reported in weight units, in this section we define x and y as weight fractions in liquid and vapor, respectively. Molar units can also be used and are preferred for theoretical analysis (Baker et al., 2010). Equation (18-50) is superficially similar to the definition of relative volatility in distillation. Of course, here the separation factor is for a rate process and represents the RT curve, not equilibrium. Separation factor data are usually obtained under conditions where concentration polarization is negligible. Experimentally determined separation factors are functions of temperature and liquid mole fraction (e.g., see Figure 18-15).

Separation factor data can be converted to y versus x format by solving for y_A in Eq. (18-50b):

Since separation factor usually depends on liquid concentration, Eqs. (18-50a) to (18-50c) are valid locally.

We again use membrane modules that are well mixed on both the retentate and permeate sides. In this situation Eq. (18-50c), with y replaced by y_p and x replaced by x_out, becomes the RT equation

where y_p and x_out refer to weight fractions of the more permeable component in the permeate and the retentate, respectively. Selectivity β in Eq. (18-51) must be determined at operating temperature T and liquid weight fraction x_out. If data are available in the form of Figure 18-15, it must be at operating temperature, T, of the pervap system.

The overall mass balance for a single-pass system (Figure 18-13) is

and the mass balance for the more permeable species is

These equations are identical to the balances for gas permeation, Eqs. (18-7a) and (18-7b), except that x has replaced y for feed and retentate. Solving Eqs. (18-52a) and (18-52b) simultaneously for y_p

where cut θ′ = F′_p/F′_in. This result is essentially the same as Eq. (18-8). If we want to use a graphical procedure, the operating equation will plot as a straight line on a graph of y_p vs. x_out.

If x_out and T are specified (which means β_AB is known), Eqs. (18-51) and (18-53) can be solved simultaneously. The resulting equation is quadratic in y_p and linear in θ′ (Wankat, 1990, pp. 707–709). Unfortunately, this procedure is more complicated and less useful in real situations than it appears at first glance. Since neither x_out nor T is usually known, selectivity is not known, and the calculation becomes complicated. We will use a simultaneous graphical solution to develop a somewhat simpler procedure that is illustrated in Example 18-10.

An energy balance is needed to estimate pervap system temperature. We assume that the pervap system is adiabatic and is operating at steady state. Then the energy balance for the system shown in Figure 18-13 is

We choose pure liquid component A at temperature T_ref as the reference. Assuming heat of mixing is negligible, enthalpies of liquid streams are

and vapor enthalpy is

where λ_p is mass latent heat of vaporization of the permeate determined at T_ref, and heat capacities are also in mass units. Combining Eqs. (18-54a) to (18-54c), we obtain

Since the membrane module is well mixed, it is reasonable to assume the system is in thermal equilibrium, T_out = T_p. If we arbitrarily set reference temperature equal to outlet temperatures, T_ref = T_out = T_p, we obtain simplified forms of the energy balance.

From Eq. (18-55b) we can determine the cut necessary to obtain a specified outlet temperature, and Eq. (18-55c) allows us to determine outlet temperature for any specified cut. The relationship between cut and temperature drop of the liquid stream is linear. Examining Eq. (18-55c) is instructive. Since latent heat is significantly greater than heat capacity, outlet temperature drops rapidly as cut is increased. To prevent this drop in temperature a recycle system with a low cut per pass is often used.

We are now ready to develop the solution procedure for completely mixed pervap systems. This procedure is straightforward if selectivity data are available and either outlet temperature or cut are specified. If cut is specified, calculate T_out from Eq. (18-55c). To plot the RT curve, pick arbitrary values for x_out, calculate β_AB from appropriate data such as Figure 18-15 or 18-16, calculate y_p from RT Eq. (18-51), and plot the point on the curve. If experimental y vs. x data at operating temperature T are available, plot it directly as y_p vs. x_out. The simultaneous solution to the selectivity equation and mass balances is at the point of intersection of the RT curve and the straight operating line, Eq. (18-53) (see Figure 18-16 in Example 18-10).

Of course, there are many different feasible designs. If a higher permeate concentration is desired, the ethanol-water feed mixture in Example 18-10 would probably be concentrated in an ordinary distillation column to a concentration much closer to the azeotrope concentration (see Figure 18-14B). Retentate is typically recycled to the distillation column. Pervaporation, azeotropic distillation (Chapter 8), and adsorption (Chapter 19) are all used commercially to break the ethanol-water azeotrope.

The corresponding molar flux equations are

There are two important differences between Eq. (18-58a) and Eq. (18-2c). First, p_A,1 is calculated from Eq. (18-49a), p_A,1 = x_A,1(VP)_A,1, in pervap, while retentate partial pressure in vapor permeation is calculated from p_A,r = y_A,r p_total,r. Vapor pressure of A in a binary mixture can be calculated from

Equation (18-59) requires vapor pressure of pure component and activity coefficient γ_A, which depends on the binary mixture, mole fraction x_A, and temperature, T. Pure component vapor pressures and correlations of these vapor pressures in terms of Antoine equation constants are readily available in handbooks. If component A is water, steam tables provide very detailed tables of saturation pressure (called vapor pressure). Activity coefficient γ_A (x_A, T) can be determined with an appropriate equation of state. Since process simulators calculate activity coefficients while calculating VLE, activity coefficients can be conveniently determined with a process simulator (see Example 18-11).

Based on Equations (18-58) pervaporation flux data should be reported as permeabilities, or if membrane thickness is not known, as permeances, P/t_ms (Baker et al., 2010). Ability of the system to separate the two components should be reported as selectivity, α_AB, defined for gas permeation in Eq. (18-4b).

The second major difference between the pervap and vapor permeation equations is permeability in pervap depends on both evaporation (vapor-liquid equilibrium) and permeation through the membrane, whereas in vapor permeation, permeation occurs only through the membrane. The separation factor for pervap can be calculated as the product of separation factors for evaporation and for membrane permeation (Baker, 2012).

It is convenient to use staged models (Figure 18-19) to analyze these flow patterns. Essentially, use of staged models is a numerical integration of the differential equations. Coker et al. (1998) developed programs to solve the simulation problem (area is known) to find cut, outlet flow rates, and mole fractions.

For a design problem (cut is specified and membrane area is unknown) solution is easiest if we make permeate flow rate, F_p,j, for each small, well-mixed stage the same F_p,j = F_p; thus, each stage has different areas. Then flow rates from the crossflow system are

Since stages are well-mixed and rapid mass transfer is assumed, y_r = y_r,w. We can write Eqs. (18-6a) and (18-8a) for each small, well-mixed stage j using variables y_p,j, y_r,j (both for more permeable component), F_m,p, and F_m,r,j. For constant selectivity these equations can be solved simultaneously by substituting Eq. (18-8a) in (18-6a), rearranging, and solving with the quadratic equation. The resulting solutions for each stage are identical to Eqs. (18-10) (in Example 18-2), except y_in is replaced with y_r,j–1, and θ is replaced by

We can substitute volumetric rate, F_p, for molar rate, F_m,p, because volumetric flow rates in cm³(STP)/s are equivalent to molar flow rates. Retentate flow rates and retentate mole fractions for each stage can be found from external mass balances, Eq. (18-7), which can also be written in terms of volumetric flow rates in cm³(STP)/s. (Remember: For ideal gases, mole fraction = volume fraction.) Then outlet retentate volumetric flow rate and mole fraction are

Starting with stage j = 1, the solution procedure is as follows: Quadratic Eq. (18-10), which results from simultaneous solution of RT and operating equations, is solved on stage j for y_p,j. Since F_p and F_r,j–1, feed flow rate to stage j, are known, and y_p,j was just calculated, Eqs. (18-65a) and (18-65b) can be solved to determine F_r,j and y_r,j. The area A_j for each stage can be determined by rearranging Eq. (18-5a) and substituting F_p = F_m,p/ρ_m,p:

The counter j is increased by 1, which makes the previously calculated values the values for j – 1, and the process is repeated.

Once every stage has been calculated, the mole fractions for the crossflow system are

and the total area is

This set of equations is easy to program within a mathematical package. If N is large enough (N = 1000 is more than sufficient and requires little time) very good agreement will be obtained with the analytical solution, with less effort and less time. Numerical solution methods can easily include a variable selectivity. A spreadsheet program and example are in this chapter’s appendix.

Crossflow systems with unmixed permeate have more separation than completely mixed systems. Crossflow works better than well mixed because average driving force, [∑(p_ry_r,j – p_py_p,j)]/N, is considerably larger in crossflow than the single driving force, (p_ry_r,out – p_py_p), in well-mixed systems. A real system probably has some but not complete mixing on both the permeate and retentate sides; thus, the permeate mole fraction is probably between the values calculated for completely mixed and crossflow.

The analysis for crossflow can be extended to multicomponent systems. Now each stage is a perfectly mixed multicomponent system with a feed from the previous stage. The results for each stage can be calculated using the procedure in Section 18.3.3. Thus, a trial-and-error procedure is needed on each stage, but the entire cascade is calculated only once. Coker et al. (1998) illustrate a different procedure for multicomponent crossflow.

Tubular and hollow-fiber modules can be arranged to have approximately countercurrent flow patterns (Figure 18-18C). Since countercurrent flow is an advantage in equilibrium-staged systems, we would expect that it might be advantageous in membrane separations also. Example 18-12 shows that this is true. A staged model for countercurrent flow is shown in Figure 18-19C. Note that permeate streams from all stages right of stage j are mixed together, and the resulting mixture is permeate for stage j. This flow pattern is used to relate permeate mole fraction y_p,j to the permeate mole fraction transferring through the membrane plus permeate from later stages, y_p,j+1. Since later stages have not been calculated yet, the analysis requires trial and error. Coker et al. (1998) present a solution method for a staged model for countercurrent, multicomponent permeation when the membrane area is known but cut is unknown.

Students interested in computer programs to solve cocurrent and/or countercurrent gas permeation are referred to Coker et al. (1998) or the third edition of Separation Process Engineering (Wankat, 2012).

Baker, R. W., “Membrane Technology,” in A. Seidel (Ed.), Kirk-Othmer Encyclopedia of Chemical Technology, 5th ed., Vol. 15, Wiley-Interscience, New York, 2004, pp. 796–852.

Baker, R. W., and I. Blume, “Permselective Membranes Separate Gases,” Chem. Tech., 16, 232 (1986).

Baker, R. W., E. L. Cussler, W. Eykamp, W. J. Koros, R. L. Riley, and H. Strathmann, Membrane Separation Systems, U.S. Department of Energy, DOE/ER30133-H1, April 1990, Reprinted by Noyes Data Corp., Park Ridge, NJ, 1991.

Baker, R. W., J. G. Wijmans, and Y. Huang, “Permeability, Permeance and Selectivity: A Preferred Way of Reporting Pervaporation Performance Data,” J. Membrane Sci., 348 (1–2), 346 (2010).

Binous, H., “Gas Permeation Computations with Mathematica,” Chem. Eng. Educ., 140 (Spring 2006).

Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Revised 2nd ed., Wiley, New York, 2006.

Blatt, W. F., A. Dravid, A. S. Michaels, and L. Nelson, “Solute Polarization and Cake Formation in Membrane Ultrafiltration: Causes, Consequences and Control Techniques,” in J. E. Flinn (Ed.), Membrane Science and Technology, Plenum Press, New York, 1970, pp. 47–97.

Cath, T. Y., A. E. Childress, and M. Elimelech, “Forward Osmosis: Principles, Applications, and Recent Developments,” J. Membrane Sci., 281, 70–87 (2006).

Chern, R. T., W. J. Koros, H. B. Hopfenberg, and V. T. Stannett, “Material Selection for Membrane-Based Gas Separations,” in D. R. Lloyd (Ed.), Materials Science of Synthetic Membranes, Am. Chem. Soc., Washington, D. C., Chapter 2, 1985.

Chung, T.-S., S. Zhang, K. Y. Wang, J. Su, M. M. Ling, “Forward Osmosis Processes: Yesterday, Today and Tomorrow,” Desalination, 287, 78–81 (2012).

Coker, D. T., B. D. Freeman, and G. K. Fleming, “Modeling Multicomponent Gas Separation Using Hollow-Fiber Membrane Contactors,” AChE Journal, 44, 1289 (1998).

Conlee, T. D., H. C. Hollein, C. H. Gooding, and C. S. Slater, “Ultrafiltration of Dairy Products in a ChE Laboratory Experiment,” Chem. Eng. Educ., 32 (4), 318 (Fall 1998).

Drioli, E., and M. Romano, “Progress and New Perspectives on Integrated Membrane Operations for Sustainable Industrial Growth,” Ind. Engr. Chem. Research, 40, 1277 (2001).

Eykamp, W., “Membrane Separation Processes,” R. H. Perry and D. H. Green (Eds.), Perry’s Chemical Engineers’ Handbook, 7th ed., McGraw-Hill, New York, pp. 22-37 to 22-69, 1997.

Foley, G., “The Lambert W Function in Ultrafiltration and Diafiltration,” Chem. Engr. Educ., 50 (2), 107 (2016).

Geankoplis, C. J., Transport Processes and Separation Process Principles (Includes Unit Operations), 4th ed., Prentice Hall, Upper Saddle River, NJ, 2003, Chapter 13.

Geraldes, V., N. E. Pereira, and M. Norberta de Pinho, “Simulation and Optimization of Medium-Sized Seawater Reverse Osmosis Processes with Spiral-Wound Modules,” Ind. Engr. Chem. Research, 44, 1897 (2005).

Green, D. W., and R. H. Perry (Eds.), Perry’s Chemical Engineers’ Handbook, 8th ed., McGraw-Hill, New York, 2008.

Henis, J. M. S., and M. K. Tripodi, “The Developing Technology of Gas Separating Membranes,” Science, 220 (4592), 11 (April 1, 1983).

Ho, W. S. W., and K. K. Sirkar (Eds.), Membrane Handbook, Van Nostrand Reinhold, New York, 1992.

Hoffman, E. J., Membrane Separations Technology: Single-Stage, Multistage, and Differential Permeation, Gulf Professional Publishing (Elsevier Science), Amsterdam, 2003.

Huang, R. Y. M. (Ed.), Pervaporation Membrane Separation Processes, Elsevier, Amsterdam, 1991.

Kesting, R. E., and A. K. Fritzsche, Polymeric Gas Separation Membranes, Wiley, New York, 1993.

Kucera, J., Reverse Osmosis: Industrial Applications and Processes, Scrivener Publishing, Salem, MA, 2010.

Leeper, S. A., “Membrane Separations in the Recovery of Biofuels and Biochemicals: An Update Review,” in N. N. Li and J. M. Calo (Eds.), Separation and Purification Technology, Marcel Dekker, New York, 1992, pp. 99–194.

Leeper, S. A., D. H. Stevenson, P. Y.-C. Chiu, S. J. Priebe, H. F. Sanchez, and P. M. Wikoff, Membrane Technology and Applications: An Assessment, U.S. Department of Energy, February 1984.

Li, N. N., and S. S. Kulkarni, “Membrane Separations,” in McGraw-Hill Encyclopedia of Science and Technology, 8th ed., Vol. 10, McGraw-Hill, New York, 1997, pp. 670–671.

Loeb, S., and S. Sourirajan, “Seawater Demineralization by Means of a Semipermeable Membrane,” University of California at Los Angeles Engineering Report No. 60-60, 1960.

Loeb, S., and S. Sourirajan, “Seawater Demineralization by Means of an Osmotic Membrane,” Advan. Chem. Ser., 38, 117 (1963).

Lonsdale, H. K., “The Growth of Membrane Technology,” J. Membrane Sci., 10, 81 (1982).

Nakagawa, T., “Gas Separation and Pervaporation,” in Y. Osada and T. Nakagawa (Eds.), Membrane Science and Technology, Marcel Dekker, New York, 1992, Chapter 7.

National Academy of Engineering, Grand Challenges, www.engineeringchallenges.org/9142.aspx (2015).

Neel, J., “Introduction to Pervaporation,” in R. Y. M. Huang (Ed.), Pervaporation Membrane Separation Processes, Elsevier, Amsterdam, 1991, pp. 1–109.

Noble, R. D., and A. Stern (Eds.), Membrane Separations Technology, Elsevier, Amsterdam, 1995.

Noble, R. D., and P. A. Terry, Principles of Chemical Separations with Environmental Applications, Cambridge University Press, Cambridge, UK, 2004.

Perry, R. H., and D. W. Green (Eds.), Perry’s Chemical Engineers’ Handbook, 7th ed., McGraw-Hill, New York, 1997.

Porter, M. C., “Membrane Filtration,” in P. A. Schweitzer (Ed.), Handbook of Separation Techniques for Chemical Engineers, 3rd ed., McGraw-Hill, New York, 1997, Section 2.1.

Reid, C. E., “Principles of Reverse Osmosis,” in U. Merten (Ed.), Desalination by Reverse Osmosis, MIT Press, Cambridge, MA, 1996, Chapter 1.

Reisch, M. C., “Filtering Out the Bad Stuff,” Chem. & Eng. News, 21 (April 23, 2007).

Robeson, L. M., “Correlation of Separation Factor versus Permeability of Polymeric Membranes,” J. Membrane Sci., 62, 165 (1993).

Schock, G., and A. Miquel, “Mass Transfer and Pressure Loss in Spiral-Wound Modules,” Desalination, 64, 339 (1987).

Strathmann, H., Introduction to Membrane Science and Technology, Wiley-VCH, Weinheim, Germany, 2011.

Suk, D. E., and T. Matsura, “Membrane-based Hybrid Processes: A Review,” Separ. Sci. Technol., 41, 595 (2006).

Uemura, T., and M. Henmi, “Thin-Film Composite Membranes for Reverse Osmosis,” in N. H. Li, A. G. Fane, W. W. Ho, and T. Matsuura (Eds.), Advanced Membrane Technology and Applications, Wiley, New York, 2008, Chapter 1.

Wang, H., and H. Zhou, “Understand the Basics of Membrane Filtration,” Chem. Engr. Progress, 109 (4), 33–40 (April, 2013).

Wankat, P. C., Rate-Controlled Separations, Kluwer, Amsterdam, 1990, Chapters 12 and 13.

Wankat, P. C., Separation Process Engineering, Includes Mass Transfer Analysis, 3rd ed., Prentice Hall, Upper Saddle River, NJ, 2012, Chapter 17.

Wankat, P. C., and K. S. Knaebel, “Mass Transfer,” in D. W. Green and R. H. Perry (Eds.), Perry’s Chemical Engineers’ Handbook, 8th ed., McGraw-Hill, New York, 2008, pp. 5-43 to 5-83.

Woods, D. R., Rules of Thumb in Engineering Practice, Wiley-VCH, New York, 2007.

A1. Membrane systems are rate processes, and flash distillation is an equilibrium process. Explain why solution methods are so similar for well-mixed membrane separators and flash distillation.

A2. How would you use the crossflow spreadsheet program in this chapter’s appendix as a simulator instead of as a design program?

A3. What are the advantages of asymmetric membranes compared to symmetric membranes?

A4. What are the advantages of hollow-fiber membranes compared to other geometries? When would you use other geometries?

A5. In large membrane systems it is common to have membrane cascades with membranes arranged both in parallel and in series. What are advantages of this arrangement? The arrangements after the first set of membranes in series usually have fewer membranes in parallel. This has been called a Christmas-tree pattern. Draw this pattern.

A6. For multicomponent gas permeation in perfectly mixed systems use of ∑y_r,i = 1.0 or ∑y_p,i = 1.0 may lead to incorrect solution, but use of ∑y_r,i – ∑y_p,i = 0 for convergence does not lead to incorrect solutions. Explain what can go wrong with the use of ∑y_r,i = 1.0 or ∑y_p,i = 1.0.

A7. We are operating a stirred-tank UF system to concentrate an intermediate molecular weight polypeptide. Unfortunately, for the membrane available, the retention is significantly lower than desired. Two options for improving the retention have been proposed. Explain the conditions for which each option will or will not improve retention.

a. Increase the stirrer speed.

b. Decrease the stirrer speed.

A8. We try decreasing the stirrer speed to increase the retention of the intermediate molecular weight polypeptide and find it works. Unfortunately, the separation between the intermediate molecular weight polypeptide and a low molecular weight compound becomes much worse. Explain what happened.

A9. The Fly-by-Night Membrane Separation Company claims to have the solution for the osmotic pressure limitation that occurs in RO of aqueous solutions of concentrated salts. The company’s proposal is basically to use UF instead of RO. Your boss wants to know if your company should invest in the Fly-by-Night Membrane Separation Company. What do you tell your boss, and what is the reason?

A10. Some azeotropic mixtures can be separated by sending the vapor mixture to a gas permeation system—designated as vapor permeation if the mixture is easily condensed (Huang, 1991; Neel, 1991)—and some (probably different) azeotropic mixtures can be separated by sending a liquid mixture to an RO system. Why is pervaporation a much more popular method of separating azeotropic mixtures? Note: In all cases a hybrid membrane distillation system will probably be used.

A11. Someone attached the feed line to the permeate outlet fitting on the hollow fiber RO unit. The permeate product line was attached to the feed fitting. If salty feed water at 50.0 bar is sent to the unit while connected this way, what is likely to happen?

B. Generation of Alternatives

B1. There is a number of ways distillation and membrane separators can be combined as hybrid systems. Brainstorm as many methods as you can.

B2. Devise schemes that will increase UF separation of intermediate molecular weight polypeptide from a low molecular weight compound if retention of intermediate molecular weight polypeptide is initially too low.

B3. FO has been proposed as a method to increase concentration of retentate after RO and produce more water. Propose a method for doing this, including recycle of draw solution. Note: The main economic driver is reducing amount of waste that has to be disposed of, not increasing amount of water produced.

C. Derivations

C1. Derive Eqs. (18-10) for a gas permeator.

C2. Solve Eq. (18-20) for x_p as a function of x_r.

C3. Derive Eq. (18-32a) from shell balances. You should obtain a second-order equation. Do a first integration, and apply boundary condition R = 1.0 at z = 0.

C4. For an RO system in which osmotic pressure is a linear function of weight fraction of solute, π = a′x, show that at the same values of x_p, the relationship between R, R^o, and M is

R = 1 – M(1 – R^o)

What assumptions have to be made to use this relationship for a different RO module?

C5. Derive Eq. (18-11e).

C6. Since analytical solutions are easier for some problems, graphical solutions are easier for other problems, and numerical spreadsheet solutions are easier for others, it is important to become comfortable with all three methods.

a. Solve Example 18-1 graphically, and compare your answer with the example.

b. Solve Examples 18-1 and 18-2 numerically with a spreadsheet, and compare your answers with the examples.

D. Problems

*Answers to problems with an asterisk are at the back of the book.

D1.* A gas permeation system with a cellulose acetate membrane will be used to purify a carbon dioxide–methane stream. Permeabilities are given in Example 18-2. The effective membrane thickness t_ms = 1.0 µm (a micron = 1.0 × 10^-6 m). Operation is at 35.0°C. p_H = 12.0 atm and p_L = 0.2 atm (a vacuum). Feed is 15.0 mol% carbon dioxide. Assume ideal gases.

a.* A single-stage, well-mixed membrane separator will be used. Operate with a cut, θ = F_p/F_in = 0.32. Find y_p, y_out, and flux of carbon dioxide. If feed rate = 1.0 kmol/h, calculate membrane area, F_p and F_out.

b. Design the two-step gas permeation system shown in the following figure. Each stage is well mixed. Use same feed to first stage as in part a. Pressures and feed mole fractions are same as in part a. Set F_p1 = F_p2 = 0.5 F_p,parta (then F_out,2 = F_out,parta). Find y_p1, y_p2, y_out1 = y_in2, y_out2, and F_out1. Find carbon dioxide fluxes in both stages and membrane areas required in both stages.

D2. We have a mixture that is 40.0 mol% carbon dioxide and 60.0 mol% oxygen being processed in a perfectly mixed membrane gas permeation separator. A silicone rubber membrane is used at 25.0°C. Permeabilities are P_CO2 = 2700 × 10^–10 cm³(STP)cm/[cm²·s·cm Hg] and P_O2 = 500 × 10^–10 cm³(STP)cm/[cm²·s·cm Hg]. Thickness of active layer of membrane is t_ms = 0.0003 cm. Permeate pressure is p_p = 65.0 cm Hg, and retentate pressure is p_r = 365 cm Hg. Assume gases are ideal.

a. If we want the highest possible value of permeate mole fraction carbon dioxide, y_p, what value of cut θ should be used, what is the value of y_p, and what is the value of the retentate mole fraction of carbon dioxide, y_r?

b. If we want the lowest possible value of retentate mole fraction carbon dioxide, y_r, what value of cut θ should be used, what is the value of y_r, and what is the value of permeate mole fraction of carbon dioxide, y_p?

D3. We are doing RO experiments with a completely mixed laboratory unit.

Experiment A. With a very high stirrer speed so that there is no concentration polarization, we do an experiment with a feed that is 0.77 wt% sodium chloride in water. Cut θ′ (in mass units) is 0.22, and measured inherent retention is R^o = 0.9804. Permeate pressure is 1.1 atm, T = 40.0°C, and solvent flux is 415.4 g/(m²⋅s). K′_solv/t_ms = 33.29 g/(atm m²⋅s). Data: Assume solutions have the same density as water = 997 kg/m³. Osmotic pressure correlations are given in Eq. (18-14h).

For experiment A, retentate pressure was not recorded. Calculate the value of retentate pressure that was used for this experiment.

D4.* A cellulose acetate membrane is being used for RO of aqueous sucrose solutions at 25.0°C.

Data: Density of solvent (water) is ρ = 0.997 kg/L.

Density (kg/L) of dilute aqueous sucrose solutions is ρ = 0.997 + 0.4x where x is weight fraction sucrose.

At low sucrose weight fraction osmotic pressure (in atm) can be estimated as π = 59.895x where x is the weight fraction sucrose.

Molecular weight of water is 18.016. Molecular weight of sucrose is 342.3.

(Note: Some of this data may not be needed to solve this problem.)

Experiment A. This experiment is done in a well-mixed stirred tank. At 1000.0 rpm with a 3.0 wt% solution of sucrose in water using p_r = 75.0 atm and p_p = 2.0 atm, we obtain J′_solv = 4.625 g/(m²s). Mass transfer coefficient k = 6.94 × 10^–5 m/s. We measure x_r = x_out = 0.054 and x_p = 3.6 × 10^–4.

a. Calculate concentration polarization modulus M.

b. Calculate selectivity α′, K′_solv/t_ms, and K′_A/t_ms.

c. Estimate mass transfer coefficient k if rpm is increased to 2000.0.

D5. We are using a cellulose acetate RO membrane to concentrate a dilute sucrose mixture.

Operation is at 25.0°C.

a. We measure pure water flux (no sucrose present), and find at a pressure drop of 102.0 atm across the membrane J′_solv = 1.5 × 10^–3 [g/(cm² s)]. An experiment in a highly stirred system (M = 1.0) with a dilute sucrose solution gives a rejection R^o = 0.997 for an inlet weight fraction of 0.050, and cut θ′ = 0.45 (in weight units). Find K′_solv/t_ms and selectivity α′.

b. We will design a perfectly mixed RO system to operate at conditions where M = 3.0. Feed is 2.0 wt% sucrose. We want a cut of θ′ = 2/3. Feed rate will be 5.0 kg/s. Pressure drop across the membrane is 78.0 atm. Plot the RT curve, Eq. (18-20), and find the intersection with the operating line, Eq. (18-23a), to find x_p and x_r = x_out. This is a complete nonlinear (except for linearizing osmotic pressure dependence on sucrose concentration) solution. Then find J_solv and the membrane area.

c. Resolve part b for x_p and x_out using linearized solution Eq. (18-29a) with Eqs. (18-25a) and (18-25b).

d. Compare your answer in part c with your answer in part b. Which answer is more accurate?

Data are in Problem 18.D4.

D6. UF of a dextran solution in a well-mixed system with gel formation gives following data:

Determine k and the weight fraction at which dextran gels.

D7. An UF membrane is first tested in a stirred cell in which there is no concentration polarization. Experimental values obtained are listed in the following table. Then the same membrane is used in a spiral-wound module in which there is concentration polarization. Values listed are obtained. Assume membrane thickness and permeabilities are the same. Estimate R^o_c from stirred-cell data, and assume inherent membrane rejection is unchanged. Calculate expected solvent flux and concentration polarization modulus M_c in the spiral-wound membrane system. No gel layer forms. Use concentration units, g/L. Assume density = 0.997 kg/L.

D8. A perfectly mixed membrane module is used to concentrate helium in retentate. Feed is 5.0 vol% helium and 95.0 vol% hydrogen at 5.0 atm. Membrane is natural rubber of the type used by Nakagawa (Table 18-2), and operation is at 25.0°C. The active portion of membrane is 1.0 micron thick. Permeate pressure is 1.0 atm. Assume gases are ideal.

a. If y_pHe = 0.025, what are cut θ and y_rHe?

b. If θ = 0.75, what are y_pHe and y_rHe?

c. If y_rHe = 0.06, what are cut θ and y_pHe?

d. For part a, what membrane area is required for a feed of 100.0 m³(STP)/h?

D9. The following data were obtained for UF of skim milk in a spiral-wound system (Conlee et al., 1998).

a. Estimate gelling weight fraction, x_g, and mass transfer coefficient, k.

b. If you were the technician’s supervisor and one more run could be done, what data would you want the technician to obtain to improve estimate of x_g?

D10. We are separating a feed that is 40.0 mol% nitrogen and 60.0 mol% hydrogen in a completely mixed membrane module containing an ethyl cellulose membrane at 30.0°C. Feed rate is 2.0 kmol/min. Permeate cut is θ = 0.20. Membrane thickness is 0.8 µm (0.8 × 10^–4 cm). Permeabilities are in Table 18-2 (Geankoplis data). High pressure is p_r = p_out = 5.5 atm, and permeate pressure is p_p = 1.12 atm. Find:

a. Mole fraction of hydrogen in permeate, y_p, and in retentate, y_r.

b. Volumetric total flux and volumetric fluxes of hydrogen and nitrogen.

c. Membrane area required, m².

D11. We are testing a UF membrane in a perfectly stirred system. With pure distilled water we obtain a pure water flux of J = 5000.0 L/(m² day) at a permeate pressure of 1.0 bar and a retentate pressure of 4.0 bar.

Next we do experiments in the completely mixed system with a constant stirrer speed to separate an aqueous feed that is 0.1 wt% polysaccharide. Previous experiments showed that with this stirred speed, mass transfer coefficient k = 2.89 × 10^–5 m/s. Since the system is quite dilute, solvent density is the same as pure water. Inherent rejection coefficient is R^o = 1.0. Permeate pressure is 1.0 bar, and a cut θ′ (in weight units) of 0.40 is used. Retentate pressure was increased in steps, and the following data were collected:

Density of water is 997 kg/m³. Find the weight fraction of polysaccharide at which gelling occurs.

D12. In Example 18-3 we want to increase the nitrogen mole fraction in retentate. To do this we increase retentate pressure to p_r = 780.0 mm Hg. Find the value of θ and retentate and permeate mole fractions.

D13. We are running an RO system for an aqueous feed with 3.5 wt% NaCl. This approximates concentration of seawater, but seawater is much more complex. Typically, highest cut for RO of seawater is approximately 0.50 to 0.55. In the United States, EPA lists preferred salt weight fraction in drinking water as 0.00050. In other countries, a higher drinking water maximum salt concentration is used. Operation is at 25.0°C.

a. If x_feed = 0.035, x_p = 0.00050, and θ′ = 0.55, what value of retention, R, (including concentration polarization effects) is required?

b. If x_feed = 0.035, x_p = 0.00100, and θ′ = 0.55, what value of retention, R, (including concentration polarization effects) is required?

Data: For dilute systems osmotic pressure of sodium chloride is π = a′x where x is weight fraction, π is in atm, and a′ = 705.9 atm/(weight fraction) at 25.0°C. Although this is not a dilute system, to simplify the analysis, assume that the linear equation for osmotic pressure is valid.

D14. A pervaporation system will be used to remove water from n-butanol using a cellulose 2.5-acetate membrane in a perfectly mixed module. Feed is 90.0 mol% n-butanol. Feed rate is 100.0 lb/h. Flux is 0.2 lb/ft²h. Pervaporation unit operates at T_p = T_out = 30.0°C where selectivity of water compared to butanol is 43.0 (in mole fraction units. WATCH UNITS!).

Data: Latent heats: butanol = 141.6 cal/g; water = 9.72 kcal/gmol.

C_p,B (45.0°C) = 0.635 cal/g °C, C_p,w (45.0°C) = 1.0 cal/g °C, MW_B = 74.12, MW_w = 18.016. Assume heat capacities and latent heats are independent of temperature.

a. If feed is at 60.0°C, find cut, permeate mole fraction, outlet liquid mole fraction, and membrane area.

b. If cut θ = 0.08 is used, find permeate mole fraction, outlet liquid mole fraction, and feed inlet temperature.

c. If outlet water liquid mole fraction is 0.05, find permeate mole fraction, cut, and inlet feed temperature.

D15.* An UF system is concentrating latex particles in an aqueous suspension. System is perfectly mixed on the retentate side and operates with p_p = 1.0 bar and p_r = 2.2 bar. Because of extensive stirring, concentration polarization modulus is M = 1.2, and no gel forms (x_wall < x_gel = 0.5). Osmotic pressure of latex particles is negligible. Density of solutions can be assumed to equal density of pure water = 0.997 kg/L. All latex particles are retained by the membrane (R^o = 1.0). Feed to UF module is x_r,in = 0.10 weight fraction latex (for all parts). We operate with a cut θ = 0.20. Feed rate of suspension, F′_r,in, is 100.0 kg/h. Flux rate of the membrane with pure water is 2500.0 L/(m² day).

a.* Find F′_r,out, x_p, and x_r,out.

b. What membrane area is required?

c. If we decrease stirring and M increases, at what value of M will gel formation occur (θ′ = 0.20)?

d. At M = 1.2, what value of the cut θ′ will cause gel formation to occur?

e. If M = 1.2 and θ′ = 0.20, at what feed weight fraction will gel formation occur? What is x_r,out?

f. We are doing an experiment of slowly decreasing the amount of stirring. If gel formation first occurs with x_F = 0.20 and θ′ = 0.25, what are the values of M and mass transfer coefficient k with this amount of stirring? J_solv = 2500.0, but the area has changed.

g. With the same amount of stirring as in part f but otherwise new conditions, we do an experiment with x_F = 0.20, p_r = 3.4 bar, p_p = 1.0 bar, and θ′ = 0.20. Does a gel form? What is the solvent flux in L/(m²day)?

h. We do an experiment with the same conditions as part g, except θ′ = 0.26. Does a gel form? What is the solvent flux in L/(m²day)?

D16. We are doing RO experiments with a completely mixed laboratory unit. K′_solv/t_ms = 1.387 g/(atm m²s). Data: Assume solutions have same density as water = 997 kg/m³. Osmotic pressure of sodium chloride is π = a′x where x is in weight fraction, π is in atm, and at 25.0°C, a′ = 705.9 atm/(weight fraction).

Experiment B. With a very high stirrer speed so that there is no concentration polarization, we do an experiment with a feed that is 0.077 wt% sodium chloride in water. Cut θ′ (in weight units) is 0.22, and measured inherent retention is R^o = 0.996. p_p = 1.1 atm, and p_r = 12.06 atm.

Experiment C. A feed with 0.093 wt% sodium chloride was separated with a permeate pressure of 1.1 atm and a retentate pressure of 15.2 atm. Cut (in weight units) θ′ = 0.26. Stirring speed was lower, and concentration polarization is expected. R = 0.992 was measured.

a. What is the value of concentration polarization modulus M in Experiment C?

b. What are the values of x_out, x_p, and solvent flux J′_solv in Experiment C?

D17. On a Robeson (1993) plot (log-log plot of selectivity versus oxygen permeability in Barrer) the upper bound for separation of oxygen from nitrogen plots as a straight line. Approximate values of end points are P_O2 = 0.0001 Barrer, α_O2–N2 = 42.0; and P_O2 = 10,000 Barrer, α_O2–N2 = 1.9. Barrer = 10^–10 [cc STP cm]/[cm² s cm Hg].

a. Find the equation for the straight line.

b. If we need an α_O2–N2 = 8.0, what is the maximum value of P_O2 that can be obtained?

D18. Repeat Example 18-3 except with unknown area and θ = 0.3. Find the value of area, A, and the retentate and permeate mole fractions. Suggestion: Solve with a spreadsheet.

D19. An experiment is being done to determine permanence (permeability/thickness of active layer of membrane), cm³(STP)/[cm²·s·cm Hg], for helium and hydrogen. A perfectly mixed gas permeation unit with 0.20 m² of area is operated with p_r = 13.0 bar and p_p = 1.26 bar. With pure He we collect 126.8 cm³(STP)/s. With pure hydrogen we collect 180.1 cm³(STP)/s. Find:

a. Permanence of helium and permanence of hydrogen.

b. Selectivity of H₂/He.

c. If t_ms = 0.0001 cm, calculate P_He and P_H2 in cm³(STP)cm/(cm²·s·cm Hg).

D20. A mixture of carbon dioxide, methane, and nitrogen is fed to a perfectly mixed gas permeation system with a natural rubber membrane. Membrane properties are given in Geankoplis data in Table 18-2. Membrane active layer thickness is 1.0 µm. Feed pressure is 17.0 bar = retentate pressure. Permeate pressure is 1.1 bar. Feed is 15.0 mol% carbon dioxide, 75.0 mol% methane, and 10.0 mol% nitrogen. For θ = 0.40 find retentate mole fractions, permeate mole fractions, and membrane area (in m²) needed for 1.0 m³/s of feed gas.

Suggestion: Set up on a spreadsheet or other computer solution method and converge ∑y_p – ∑y_r = 0 using Goal Seek to change the guessed value of area A.

D21. We are purifying nitrogen by removing oxygen in a gas permeation system. The membrane is a silicone rubber. Thickness of the active layer of the membrane is 0.00007 cm. Feed F_m,1,in = 10.0 kmol/minute and is y_1,in = 0.209 mole fraction oxygen. The device has two well-mixed permeation modules coupled in series with retentate from the first module sent as feed to the second. p_p = 1.05 atm, and p_r = 8.4 atm for both modules. Cut θ₁ = 0.4, and we desire y_r2,O2 = 0.12 mole fraction O₂. Permeability data are in Table 18-2.

Find y_r1,O2, y_p,1,O2, y_p2,O2, θ₂, membrane selectivity α, and A₂ in m².

D22. In Example 18-4 we saw that small holes in a membrane are a major problem in gas permeation if high-purity permeate is desired. If we want to produce high-purity water, are holes a problem for RO membranes? One would think answer has to be yes, but let’s calculate and see.

a. We have an RO system producing very pure water (1.0 ppm salt) from a feed that contains 50.0 ppm salt (already purer than normal tap water). If we have one 10.0 µm–diameter pinhole per m², how much does this affect purity? Conditions: p_r = 15.0 bar, p_p = 1.1 bar, A = 1.0 m², K′_solv/t_ms = 0.50 g/(bar·s·m²), cut = θ′ = 0.20, very well mixed: M = 1.0, low concentration: can ignore osmotic pressure.

b. If product water is at 250.0 ppm and the feed is brackish water at 1.0 wt% (10,000 ppm), what is the effect of one 10.0-µm pinhole per m² on purity? Conditions: p_r = 25.0 bar, p_p = 1.1 bar, A = 1.0 m², K′_solv/t_ms = 1.0 g/(bar·s·m²), cut = θ′ = 0.2, very well mixed: M = 1.5, include osmotic pressure. Assume the same density and viscosity as pure water.

Data: Densities of saltwater solution are listed between Eqs. (18-14g) and (18-14h).

Use Eq. (18-14e), (18-14f), or (18-14h) for osmotic pressure. Assume viscosity is 1.0 cp.

D23. UF systems are commonly used for changing solvent or buffer systems when solute is a large molecule such as a protein. Usually the process is operated in a batch mode either in a stirred tank or feed and bleed (Figure 18-2E) but with no bleed of retentate. This process is doing the same function (changing solvents) as batch systems in Section 9.3 but by a totally different mechanism. We have a large solute molecule that is excluded from pores, R^o_solute = 1.0. Initial solvent, A, and final solvent, B, are not excluded from pores, R^o_A = R^o_B = 0, and have equal fluxes. We want same solute weight fraction in both solvents; thus, if there is initially F kg of solution, at the end of the process we want F kg of solution. Two operating methods are proposed: 1. Dilute a sample with the appropriate amount of solvent B and then do UF to remove excess solvents until you obtain x_A,final. 2. Add solvent at the same mass flow rate as the solvents are transferred through the UF membrane.

a. In terms of symbols, how many kg of solvent B are needed to replace F(x_A,initial – x_A,final) kg of solvent A if on a solute-free basis initial weight fraction of solvent A is x_A,initial and final weight fraction of solvent A is x_A,final using process 1?

b. In terms of symbols, how many kg of pure solvent B are needed to replace F(x_A,initial – x_A,final) kg of solvent A if on a solute-free basis initial weight fraction of solvent A is x_A,initial and final weight fraction of solvent A is x_A,final using process 2?

c. If F = 100.0 kg of A, x_r,A,initial = 1.0, and x_r,A,final = 0.10, how much solvent is used for each process?

d. If F = 100.0 kg of A, x_r,A,initial = 1.0, and x_r,A,final = 0.01, how much solvent is used for each process?

D24. For the hybrid system shown in Problem 4.D21, what value of selectivity α is required in a membrane separator to achieve the desired values of y_p and y_r. Note: This problem does not require solution of Problem 4.D21.

D25. A completely mixed gas permeator is using a 10.0 µm–thick silicone rubber membrane to separate a feed that is 30.0 mol% CO₂ and 70.0 mol% CH₄ at 25.0°C. High pressure is 20.0 atm, and low pressure is 1.2 atm. Permeabilities are in Table 18-2. Assume gases are ideal.

a. What is the highest possible mole fraction of CO₂ in retentate, what is the corresponding CO₂ mole fraction in permeate, and what is the value of θ?

b. What is the lowest possible mole fraction of CO₂ in retentate, what is the corresponding CO₂ mole fraction in permeate, and what is the value of θ?

c. If feed rate is 100.0 m³ STP/s and cut = 0.597, find y_r,CO2, y_p,CO2, and the membrane area in m².

D26. A poly (4-methyl pentene) membrane will be used to separate oxygen from nitrogen. Both permeabilities are given in Table 18-2. Thickness of active layer is 0.0002 cm. Feed (and retentate) pressure is 6.1 atm, and permeate pressure is 1.15 atm. Inlet gas is 20.9 mol% oxygen. We desire a permeate mole fraction of y_p = 0.42 mole fraction oxygen. Feed rate is F_m,in = 1.2 kmol/min. Assume ideal gases and a well-mixed device. Find the mole fraction of oxygen in retentate = y_r, cut θ = F_m,p/F_m,in, and the membrane area, A, in m².

D27. A mixture of helium, methane, and nitrogen is fed to a perfectly mixed gas permeation system with a silicone rubber membrane. Membrane properties are given in Table 18-2. Membrane active layer thickness is 1.2 µm. Feed pressure is 12.4 bar = retentate pressure. Permeate pressure is 1.1 bar. Feed is 1.6 mol% helium, 87.5 mol% methane, and 10.9 mol% nitrogen. Feed rate is 1.13 m³/s STP.

a. Find retentate mole fractions, permeate mole fractions, and membrane area (in m²) for θ = 0.26.

b. Find retentate mole fractions, permeate mole fractions, and θ for membrane area = 10.0 m².

D28. We are purifying nitrogen by removing oxygen in a gas permeation system. The membrane is a silicone rubber. Thickness of the active layer of membrane is 0.00011 cm. Feed F_m,1,in = 10.0 kmol/min and is y_1,in = 0.21 mole fraction oxygen. The device has two well-mixed permeation modules coupled in series with retentate from the first module sent as feed to the second. p_p = 0.8 atm and p_r = 6.4 atm for both modules. F_m,p,1 = F_m,p,2 = 2.0 kmol/min. Permeability data are in Table 18-2.

a. Find y_r1,O2, y_p,1,O2, y_r2,O2, y_p2,O2, θ₂, membrane selectivity α, and areas A₁ and A₂ in m².

b. If a single perfectly mixed membrane is separating the same feed with the same conditions except F_m,p = 4.0 kmol/min, calculate y_p,O2, y_r,O2, y_p,O2, θ, and area in m².

E. More Complex Problems

E1. Your company’s well water, which contains 190.0 ppm (weight) of salt, is fine for most purposes but not for a particular manufacturing process that needs purer water. A vendor is trying to sell you a spiral-wound RO unit that has an inherent salt rejection coefficient of R^o = 0.98, a high pure water flux of J′_water = 102.0 g/(m²s) at Δp = 35.0 bar, and approximately 600.0 m² of area per m³ volume. Modules are available with volumes of 0.5 (300.0 m²), 1.0 (600.0 m²) and 2.0 m³ (1200.0 m²). Modules can be put in parallel or in series. Your company wants to produce 20,000 m³/day of water that contains 12.0 ppm of salt. The retentate, at 480.0 ppm and 1.4 bar, will be used elsewhere in the plant. Inlet pressure of the water is 28.0 bar and cannot be safely increased without a major capital expense. Operation will be at 25.0°C. The sales person wants to do the design for the company (at a price) and will not release his other data. However, he estimates that two large and one medium module will provide the water your company wants.

a. Using available data, estimate the area of the membrane and the operating conditions required to produce the desired permeate and retentate products. Use mass transfer data for spiral-wound systems, but do calculations for a perfectly mixed system.

b. How did the sales person calculate his number?

Data: Typical spiral-wound unit: channels 0.03 to 0.1 cm, velocity = 0.5 to 1.5 m/s, Re = 500.0 to 1000.0 (Wang and Zhou, 2013).

E2. We are doing RO of dilute aqueous sucrose solutions at 25.0°C. A feed that is 2.20 wt% sucrose is separated in a very well-stirred system (M = 1.0) with p_r = 60.0 atm and p_p = 1.1 atm. The flux J′_water = 3.923 g/(m²s) when x_p = 0.00032 and x_r,out = 0.056. Data are in Problem 18.D4.

a. Calculate cut θ′, inherent rejection R^o, and sucrose flux J′_sucrose.

b. Calculate selectivity α′_{water–sucrose}, K′_water/t_ms, K′_sucrose/t_ms, and J′_water.

c. If we now repeat the experiment with less mixing so that M = 2.1 with x_feed = 0.0220, p_r = 60.0 atm, p_p = 1.1 atm, and θ′ = 0.61, calculate x_r, x_p, J′_water, and J′_sucrose.

E3. Feed from a distillation column to a pervaporation unit is 90.0 wt% ethanol. After pressurizing and heating to 85.0°C, it is fed to a system similar to Figure 18-14, except retentate can be reheated to 85.0°C and sent to a second pervap stage. Additional stages can be added if needed. A 20.0 micron–thick polyacrylonitrile film membrane is used (see Figure 18-15). Assume each stage of pervap is perfectly mixed and operates at 25.0°C. Final retentate (product) needs to be at 98.5 wt% ethanol or higher. Permeate streams are pressurized and recycled to the distillation column. F = 100.0 kg/h of 90.0 wt% ethanol distillate. Find:

a. Number of stages needed in pervap system.

b. For each stage: cut, permeate flow rate, and product flow rate.

c. Weight fraction of combined permeate streams and retentate weight fraction.

d. Membrane area required for each stage.

F. Problems Requiring Other Resources

F1. A perfectly mixed pervap unit is separating a benzene–isopropyl alcohol mixture. Figure 12-23 in Wankat (1990) shows the separation factor of benzene with respect to isopropyl alcohol versus the weight fraction of benzene in liquid for pervap. If operation is at 50.0°C = T_p = T_out, x_in = 30.0 wt% benzene and the θ = 0.10, find y_p, x_out, and T_in. Assume heat capacities and latent heats are independent of temperature.

Note: Use lines on the selectivity diagram, not data points.

H. Spreadsheet Problems

H1. A spreadsheet or other computer method (e.g., MATLAB, Mathmatica) is required. Repeat Problem 18.D10 except for a crossflow system. Report the average y_p, y_r, membrane area, volumetric total flux, and volumetric fluxes of hydrogen and nitrogen. Turn in a copy of your spreadsheet or other program with numbers for both parts a and b.

a. Find the solution for N = 1, which is the perfectly mixed answer.

b. Find the solution for N = 1000, which is the crossflow answer.

H2.* In Section 18.3.3 the following statement appears, “Geankoplis (2003) solves the multicomponent permeator system by a different method, but the results are identical.” Solve Example 13.5-1 in Geankoplis (2003) using the method in Section 18.3.3 with a spreadsheet, and show that the results agree with Geankoplis’ solution.

H3. Pervaporation problems are trial and error but can be conveniently solved on a spreadsheet if selectivity is constant. Solve Problem 18.D14a with a spreadsheet, and check your answer with graphical solution. Graphical and spreadsheet solutions should agree.

H4. Repeat Problem 18.D14a but with the feed at 80.0°C.

H5. RO experiments are done in a laboratory stirred-tank membrane system at 45.0°C. With pure water we measure the pure water flux = 17.89 g/(m² s) when p_r = 15.2 atm and p_p = 1.1 atm. We then do experiment A with a highly stirred system (M = 1.0) with p_r = 13.4 atm and p_p = 1.1 atm and find R^o = 0.991. We now plan on doing an experiment with p_r = 11.3 atm and p_p = 1.1 atm, θ′ = 0.55, x_in =0.03 wt% salt, and a mass transfer coefficient of k = 4.63 × 10^–5 m/s. Density of water is 0.997 kg/L. Predict M, R, J′_solv, x_p, and x_r for the new experiment. Use of a spreadsheet or other computer solver is highly recommended.

H6. Repeat Problem 18.H5 except with x_in = 0.0006, and find the value of k that gives M = 1.1. Report k, R, J′_solv, x_p, and x_r. Use of a spreadsheet or other computer solver is highly recommended.

H7. Repeat Problem 18.H5 (x_in = 0.003) but with p_r = 21.4 atm and p_p = 1.1 atm, θ′ = 0.45, and a mass transfer coefficient of k = 4.63 × 10^–5 m/s. Predict M, R, J′_solv, x_p, and x_r for the new experiment.

Introduction to Design Problems 18.H8 and 18.H9

Find the configuration with the lowest total annual cost for the separation assigned to your team.

Estimate the total annual cost (TAC) as

Although there are other costs (e.g., capital cost for heat exchangers to cool gas after compression and energy cost for cooling water), include only the following three costs:

1. Capital cost for membrane systems

2. Capital cost for compressors

3. Electricity cost for operation of compressors

Use electricity cost as $0.06/kWh. 8760 h/yr.

Compressor hp requirements use the Quick chart from www.agcsi.com/estimate/chart1A1.htm, which is based on compressing natural gas. The chart gives the horsepower required to compress 1.0 million cubic feet per day at the inlet condition (not STP).

Centrifugal compressor costs with motor including installation factor and cost index (Woods, 2007).

Base 1000.0 kW. $1.592 × 10⁶, Cost at x kW = cost 1000 kW(x kW/1000 kW)ⁿ, n = 0.9 for 2000 to 4000 kW.

Base 10,000 kW. $12.2 × 10⁶, Cost x kW = cost 10,000 kW(x kW/10,000 kW)ⁿ, n = 0.71 for 8000 to 25,000 kW.

Membrane module, not including compressor: cost including installation factor and cost index (Woods, 2007): Base 560.0 m², $153,000. Cost at x m² = cost at 560.0 m²(x m²/560.0)ⁿ, n = 1.0 for 560.0 to 10,000 m².

If required areas are larger, put modules in parallel. Assume ideal gases.

Design group report requirements are listed in Chapter 11 in the introduction to Problems 11.G6 to 11.G9.

H8. Helium is in short supply worldwide. There are natural gas sources that contain large amounts of low-concentration helium. Your job is to develop a membrane cascade using spiral-wound membrane permeators, which are essentially crossflow systems, to produce 1.0 m³STP product/min of a 90.0 mol% or slightly higher helium product from a feed that is 1.0 mol% He and 99.0 mol% CH₄. Use membrane properties in Table 18-2. Feed gas is at 25.0°C and 25.0 atm. In a real design you can pick any retentate pressures you want, but you have to pay for compression. You could also pick any permeate pressure you want for each module. To simplify the problem use a retentate pressure of 25.0 atm and a permeate pressure of 1.1 atm for all membranes. Active layer is t_ms = 3.0 µm. Assume the supply of natural gas is larger than you need. Methane depleted of helium will be sent back to the pipeline at 25.0 atm. Assume costs of compressing retentate to pipeline pressure is negligible. Helium product needs to be compressed to 25.0 atm. Report feed rate, membrane area, y_p,avg, y_r, and θ for each membrane module in your cascade. Include a sketch of your cascade.

H9. A natural gas field produces a gas that is 30.0 mol% carbon dioxide and 70.0 mol% methane. Design crossflow separators that will process 100.0 m³ STP/s of fresh feed. Use Table 18-2 for permeability data. Membrane is in a spiral-wound system and is 10.0 µm thick. Feed pressure is 20.0 atm, and permeate pressure is 1.2 atm. Use a constant permeate pressure of 1.2 atm. You can vary retentate pressure. The methane and carbon dioxide products will be delivered at 20.0 atm.

a. If we use a cross-flow system with a silicone rubber membrane at 25.0°C find the value of cut θ that produces a retentate with CO₂ mole fraction = 0.10 or slightly less. Report membrane area, y_p,avg, y_r, and θ. Then, set N = 1, find y_r at same value of θ (y_r will be > 0.1), and check your answer for a perfectly mixed membrane system with a hand solution of problem 18.D25c. The purpose is to benchmark the spreadsheet program.

b. If we use a cross-flow system with a ethyl cellulose membrane at 30.0°C find the value of cut that produces a retentate with CO₂ mole fraction = 0.10 or slightly less. Report membrane area, y_p,avg, y_r, and θ.

c. The membrane in part a is high flux but relatively low selectivity. In part b the membrane is higher selectivity but lower flux. What are the advantages and the disadvantages of each membrane?

d. Determine the most economical cross-flow system (based on lowest value of TAC) using one or more modules to obtain a methane product that has CO₂ mole fraction = 0.10 or slightly less with a methane yield of 80.0% (or more). Include a sketch of your membrane cascade. Report membrane type, membrane area, y_p,avg, y_r, and θ for each membrane module in the cascade.

Crossflow calculation requires no trial and error and is fast even with N = 1000, which is the recommended value. Data is input on the spreadsheet, which also calculates y_{p,stage j} from Eq. (18-10e) and A_{stage j} from Eq. (18-66). These values are transferred to the VBA program, which calculates y_r,j, F_r,j, y_p,total, and A_total and increments the stage number. The spreadsheet with equations is shown in Figure 18-A1 and with results for Example 18-12 in Figure 18-A2. The VBA program is shown in Table 18-A1.

Although splitting calculations between a spreadsheet and the VBA program has advantages, it is important to ensure correct values of parameters that change as the stage number is incremented are used to calculate y_p and area. Equations (18-10c) and (18-10d) show that y_p is calculated with the inlet value of y_r (y_r,j–1 labeled yrinj in VBA) and the value of θ from Eq. (18-64c). Area is calculated with y_p,j and x_r,j. Since spreadsheets update values immediately, whereas VBA programs update values sequentially in the For loop, if you are not careful, there can be a mismatch in update timing.