Membrane processes Separation of liquid or gaseous mixtures by mass transport through membrane (= permeation). Membrane is selective, i.e. it has different permeability for different components. Conditions for separation on membrane are either hydromechanical (similar to filtration) or diffusion-based. Membranes can be: porous transport is via diffusion or by pressure difference nonporous dissolution + diffusion Membrane are made of: natural products: acetyl-cellulose synthetic: cross-linked polymers other materials: ceramics, metals Asymmetric membranes: thin skin (actual membrane) is attached to a thick, porous and sturdy support resistance to large pressure differences Some types of membrane processes micro + ultra filtration (UF): Analogous to filtration driving force is pressure difference (hydrodynamical process) up to 1 MPa. Separation of suspensions and colloids with particles of 10-1 10-3 μm. reverse osmosis (RO): Also hydromechanical process with pressure differences up to 10 MPa. Separation of components from true solutions (desalination of water) gas permeation: Separation of gaseous mixtures by diffusion driven by partial pressure difference. Air separation, gas cleaning. pervaporation: Separation of liquid solutions by diffusion; mixture evaporates after passing through the membrane dialysis: Diffusion driven transport in solution. Used in medicine artificial kidney electrodialysis: Driving force is electric potential difference (voltage); desalination of water Membrane modules tubular a tube or capillary of diameter 10-1 10 1 mm, easy to clean, low performance per volume of the module; use for UF plate-and-frame like filter press, i.e. a series of flat membrane sheets stacked one above another and separated by spacers; advantage can be cleaned spiral wound plate-and-frame rolled into a cylinder high surface to volume ratio; use for all membrane processes hollow fibers thin fibers placed in a shell high surface to volume ratio; used for all membrane processes Balance of membrane modules We assume two components: - A solute (or dispersed component) - B solvent (or dispergant)
Balance on moles - total: n F = n R + n P - component A: n Fx AF = n Rx A + n Py A x AF = (1 θ)x A + θy A, where θ = n P n F - component B: x BF = (1 θ)x B + θy B Enthalpy balance The area of the membrane is then calculated from Concentration polarization h F + Q = (1 θ)h n R + θh P F h molar enthalpy n Py k = Φ k A where k = A or B Φ k is the flux which can be express (see later) is called cut Occurs in membrane separation of liquids when using pressure difference (RO, UF). Component A (solute) tends to accumulate at the membrane, its concentration there is high A leaks through the membrane by diffusion because of and increased driving force c AW c Ap could be counteracted by stirring. A measure of quality of the membrane is rejection coefficient: R = c AR c Ap c AR M = c AW polarization modulus c AR
Kinetics of some membrane processes Generally, kinetics of transport through membrane is formulated as follows: Kinetics for RO flux = permeability driving force thickness flux = flow area ; flow can be [m3 s -1 ] or [mol s -1 ] Membrane is permeable for B only a) osmosis flow of B into the solution if p < π b) equilibrium is reached after osmosis increases p 2 so that p = π... osmotic pressure c) reverse osmosis if p 2 is increased (by pump or hydrostatic pressure) so that p > π In a real membrane (rejection coefficient R < 1) A occurs in both sides of the membrane. Each side of the membrane has its own osmotic pressure (with respect to pure solvent) Flux for RO is then Φ VB = P B δ [(p 2 p 1 ) (π 2 π 1 )] = P B ( p π) δ Φ VB volumetric flux [m 3 m -2 s -1 ] P B δ permeability of membrane for B thickness Osmotic pressure can be calculated from van t Hoff equation: π = βc A RT = ac A
Kinetics for UF A is dispersed in B, typically A is a large molecule c A is small π is small π = 0. Therefore Φ VB = P B δ p However, in UF a gel layer often forms close to the membrane on the retentate side. Additional explanation to RO If the membrane is non-ideal, i.e. partly permeable for A (rejection coefficient < 1) then the flux of A due to diffusion is Also from definition Φ A = J A = P A δ (Mc AR c AP ) Φ A = c AP Φ V c AP Φ VB i.e. the total volumetric flux is approximately equal to flux of B. Using expression for Φ VB, Φ A and definition of R = 1 c Ap car we obtain: R = 1 c Ap M = 1 c AR 1 + α BA ( p a(mc AR c AP )) where M is polarization modulus and α BA = P B P A selectivity, α BA has a unit [Pa -1 ] or [atm -1 ] Therefore there are two resistances for the transfer of B through the membrane and the gel (and in addition there can be concentration polarization). Extended flux equation is then
where P g is the permeability of the gel for B. Φ VB = p δ M PB + δ g P g sum of resistances Carman-Kozeny equation (hydrodynamic flow through porous material) determines the permeability of the gel: d 2 p ε 3 P g = 150η(1 ε) 2 where Remarks: d p diameter of the particles forming the gel ε porosity of the gel = η viscosity of the fluid volume of free space volume of gel a) Symbol J V is also used instead of Φ V b) Flux is sometimes expressed by explicitly using viscosity η or where Φ VB = J VB = P B p no gel layer ηδ Φ VB = J VB = ηδ m P B p + ηδ g P g P B = P B η and P g = P g η SI unit of P B and P g is [m 2 s -1 Pa -1 ]; unit of P B, P g is [m 2 ] c) When a gel is formed J VB is constant (balance between convective transport of A toward the membrane and back diffusion). When p is suddenly increased J VB instantaneously increases, however, the gel layer thickness δ g starts growing until the original value of J VB is restored due to increased resistance. d) If the membrane is non-ideal the molar flux of a through the membrane is J na = Mc AR (1 R V )J VB R V = 1 c AP c AR apparent rejection coefficient < 1
M = c AW c AR polarization modulus Gas permeation Gases are dissolved in and diffuse through the membrane. Modules with hollow fibers and spiral-wound membranes are used purification of hydrogen, CO 2, production of N 2 and O 2 from air. Driving force is partial pressure. The flux is for A: for B: n P molar flux of permeate J Vi = P i δ m (p ir p ip ) = P i δ m (p R x i p P y i ) J Vi = V ip A = n Py i Ac i c i molar density of pure component i; for ideal gases c i = p P does not depend on RT component n P(1 y A ) Ac B n Py A Ac A = P A δ m (p R x A p P y A ) = P B δ m (p R (1 x A ) p P (1 y A )) From mass balance: x A = x AF θy A 1 θ We assume ideal gas, c A = c B and divide first two equations: y A 1 y A = α BA p R x A p P y A p R (1 x A ) p P (1 y A ) α BA = P B P A selectivity (should be > 40)
By using mass balance we determine y A : ay 2 A + by A + c = 0 a = ( θ 1 θ + p P ) (α p BA 1) R b = (1 α BA ) ( p P + θ p R 1 θ + x AF 1 θ ) 1 1 θ x AF c = α BA 1 θ Remark: P i can be approximately expressed as P i = K i 1 D i = S i D i K i D i S i = K i 1 Henry constant absorption to membrane diffusivity diffusion through membrane solubility Pervaporation (PV) Retentate is a liquid. Components diffuse through the membrane and evaporate on the permeate side due to lowered pressure. Membrane is swollen in the retentate side and shrunk on the permeate side. Also temperature drops across the membrane complex diffusive process. PV is used to separate azeotropic mixture obtained in rectification. A simplified description of the transport through the membrane makes use of empirically determined selectivity α BA. It is assumed that (analogy to distillation) Mass balance α AB = y A xa y = y A(1 x A ) B xb x A (1 y A ) by eliminating x A : x A = x AF θy A 1 θ Enthalpy balance for PV (α AB 1) θy A 2 [(1 θ) + (α AB 1)x AF + α AB θ]y A + α AB x AF Feed is hot, retentate is colder and permeate is coolest due to evaporation. There is no external heat supply or removal. For simplicity we will assume that t R = t P (not true but for enthalpy balance is OK)
n Fc pf (t F t ref ) = n Rc pr (t R t ref ) + n P(c pp (t P t ref ) + h vap ) where h vap = y A h vap,a + (1 y A ) h vap,b and similarly c pf = x AF c pa + (1 x AF )c pb c p is evaluated at t = t F+t ref ; c 2 pr and c pp are analogous but not needed Assuming t ref = t P = t R Then the cut n Fc pf (t F t ref ) = n P h vap θ = n P = c pf(t F t ref ) n F h vap