Chapter 5. Transport in Membrane

Chapter 5. Transport in Membrane

Contents Contents

5.5 Transport through Nonporous Membranes

5.7 Transport in Ion-exchange Membrane 5.6 Transport through Membrane

5.4 Transport through Porous Membranes 5.3 Non-equilibrium Thermodynamics

5.2 Driving Forces 5.1 Introduction

 Parallel cylindrical pores perpendicular or oblique to the membrane surface

 Length of each of the cylindrical pores = Membrane thickness

 Hagen-Poiseuille equation with assumption of same pore size ⇨ Volume flux

(5-54)

where ε = surface porosity = fractional pore area = n_p·π·r²/A_m n_p = total number of pore

A_m = membrane area r = radius of pore

τ = pore tortuosity(for cylindrical perpendicular pores, the tortuosity = 1) η = viscosity

<Figure 5-5> Geometries of pores porous membranes (a) cylindrical

(b) sintered membrane

(c) phase inversion membrane

 By comparing Eq(5-54) with the phenomenological Eq(5-43) : Volume flux(J_v) = L_p·ΔP

(5-55)

 Non-cylindrical and non-parallel pore

 Kozeny-Carman relationship, (5-56)

where ε = volume fraction of the pores S = internal surface area

K = Kozeny-Carman constant (depends on shape of pores and tortuosity)

 Asymmetric membrane or composite membrane : used in gas separation

 Gas molecules : diffuse from the high-pressure to the low-pressure side

 Rate determining step

 For gas membrane or RO : transport through the dense nonporous top layer

 For UF or composite membrane : Sub-layer

• Surface porosity < 1%

• Effective thickness of sub-layer ≫ actual top layer thickness

<Figure 5-6> Transport in an asymmetric Membrane as a result of various mechanisms.

• transport through a dense (nonporous) layer

• Knudsen flow in narrow pores

• viscous flow in wide pores

• surface diffusion along the pore wall top layer(bulk diffusion)

narrow pores (Knudsen diffusion)

wide pores (viscous flow)

 The average diffusion length can be given by

(5-57) where ℓ₀ : actual top-layer thickness

ℓ_eff : effective thickness(strongly dependent on ε of the sub-layer)

<Figure 5-7> Schematic drawing of various diffusion paths in a composite membrane.

 Viscous flow

 Large pore sizes (r > 10μm)

 Gas molecules collide exclusively with each other

(in fact they seem to ignore the existence of the membrane)

 No separation between the various gaseous components

 Flux is proportional to r² from Hagen-Poiseuille equation, 5.4.1.1 Knudsen flow

<Figure 5-8> Schematic drawings depicting Poisseuille(or viscous flow) and Knudsen flow.

 Knudsen diffusion

 Pores are smaller and/or pressure of the gas is reduced

 Mean free path(λ) of the diffusing molecules > pore size of the membrane

 Collisions between the gas molecules < collisions with the pore wall

 Mean free path (λ) for liquid

 λ = average distance traversed by a molecule between collisions

 λ = few Å for liquid ⇨ Knudsen diffusion = neglect

 Mean free path (λ) for gas

 λ of gas : depend on the pressure and temperature

 For gas, λ = kT / (π·d_gas²·P·√2) where d_gas = diameter of the molecule (6-58)

 Pressure ↓ ⇨ Mean free path ↑

 Mean free path = f(T) at constant P 5.4.1.1 Knudsen flow

※ At 25°C, λ of O₂ = 70Å at 10 bar ⇨ 70μm at 10 mbar

※ Pore size of UF = 20∼200 nm(0.02∼0.2 μm)

 Knudsen flow

(6-59) where D_k : Knudsen diffusion coefficient,

T : temperature

M_w : molecular weight r : pore radius

 Flux ∝ (M_w)^－0.5

 Separation : inversely proportional to the ratio of the square root of MW 5.4.1.1 Knudsen flow

 Friction model

 Passage through pores by viscous flow and diffusion

 Pore sizes = very small

• Solute molecules cannot pass freely through the pore

• Friction occurs between

▸ solute ↔ pore wall ▸ solvent ↔ pore wall ▸ solvent ↔ solute

 Frictional force(F) per mole

 related linearly to the velocity difference or relative velocity

 Proportionality factor : Friction coefficient(f)

 Frictional force(F) per mole

• F_sm =－f_sm(v_s－v_m) =－f_sm v_s (5-60)

• F_wm =－f_wm(v_w－v_m) =－f_wm v_w (5-61)

• F_sw =－f_sw(v_s－v_w) (5-62)

• F_ws =－f_ws(v_w－v_s) (5-63)

subscripts s = solute

w = water(solvent) m = membrane

 Linear relationships between the fluxes and forces at isothermal conditions

 Force, (5-64)

 When other(external) forces acting on component i, such as the frictional force(F_i) (5-65)

 Diffusive solute flux = (Mobility) × (Concentration) × (Driving force)

(5-67) Diffusion Viscous Flow

where m = Mobility = D / RT, c_sm = concentration of solute in pore (5-66)

 By assuming an ideal solution(volume ≠ f(concentration)),

(5-68)

 For dilute solutions,

(5-69)

 Frictional force per mole of solute, (5-70)

 By relating the mobility of the solute in water to f_sm,

Mobility of the solute in water, m_sw = 1 / f_sw (5-71)

 Define a parameter b = relating the frictional coefficient f_sm to f_sw, then

(5-72)

 By combining Eq(5-67), (5-68), (5-69), (5-71) and (5-72), Solute flux(J_s) is

(5-73)

 Distribution coefficient of solute between bulk and pore(K) = c_sm / c (5-74)

 D_sw = RT / f_sw where D_sw = diffusion coefficient of solute in dilute solutions (5-75)

 With J_v = ε•v, J_i = J_s•ε and ξ = τ•x,

Eq(5-73) ⇨ since c_p = J_s / v_s (5-76) & (5-77) where ε = volume fraction of the pores τ = pore tortuosity

ξ = pore length c_p = solute concentrations in permeate

 Integration of Eq(5-76) with the boundary conditions BC 1 : x = 0 → c_l.sm = K•c_f

BC 2 : x = ℓ → c_2.sm = K•c_p

Where K = solute distribution coefficient between solute ↔ membrane

c_f = solute concentrations in feed, c_p = solute concentrations in permeate (5-78)

『Meaning』

 Maximum value of c_f/c_p : asymptotic value at b/K

 J_v ↑ ⇨ c_f/c_p↑

 K↓ ⇨ Solute uptake by membrane ⇨ c_f/c_p↑

 f_sm > f_sw ⇨ friction factor(b) ↑

 b ↑ and K ↓ ⇨ b/K ↑ ⇨ c_f/c_p↑

<Figure 5-9> Schematic drawing of

concentration reduction (c_f/c_p) versus flux

 Solute rejection, (5-79)

 Maximum rejection, R_max(J_v→∞) from Eq(5-77) and (5-78)

(5-80)

 Spiegler and Kedem equation (5-81)

 For a highly selective membrane

• (Exclusion +Kinetic)terms↓

• Solubility of the solute in the membrane↓

 Solution-diffusion mechanism

 Solution ↔ exclusion term

determine selectivity Distribution coefficient(K)

(equilibrium thermodynamic parameter) Frictional forces(F_sm) (kinetic parameter)

 Diffusion ↔ kinetic term

 Permeate concentration = c_p = J_s/J_v , then the rejection R can be written as (5-82)

 Eq(5-42) → Eq(5-82) gives

※ J_s = c_s (1 - σ)J_v + ωΔπ (5-42)

where ω = solute permeability σ = Reflection coefficient

(5-83) or

(5-84)

 Eq(5-44) → Eq(5-84) gives [※ Eq(5-44) : ω = (L₂₂/L₁₁－ σ²) ĉ L₁₁]

(5-85)

 When J_v → ∞ ⇨ Maximum rejection(R^∞)

(5-86) where ĉ^∞ = average solute concentration at J_v → ∞

 Assuming that ĉ^∞ ≈ c_f then R^∞ = σ

 Furthermore if R = Δπ/π_f and substitution of Eq(5-86) into (5-85) then (5-87)

 Plot 1/R versus the 1/J_v → straight line • abscissa ⇨ 1/ R^∞

• Slope ⇨ [L₁₁/L₂₂－(R^∞)²]

 Pore of non-porous membranes : molecular level described in terms of free volume

 Difference between liquids and gases

 Affinity of liquids ↔ polymers ≫ Affinity gases ↔ polymers (Solubility of a liquid in a polymer ≫ Solubility of a gas)

 Flow through a dense membrane

• Gases : flow in a quite independent manner

• Liquids : influenced by flow coupling and thermodynamic interaction

 Solution-diffusion mechanism

• Basic transport mechanism of non-porous membrane

• Permeability (P) = Solubility (S) × Diffusivity (D) (5-88)

 Solubility

 Thermodynamic parameter

 Measure of the amount of penetrant sorbed by the membrane under equilibrium conditions.

 Solubility of gases in elastomer polymers

• very low

• described by Henry's law

 Solubility of organic vapors or liquids(which cannot be considered as ideal)

• Can not be described by Henry's law

 Diffusivity

 Kinetic parameter(indicates how fast a penetrant is transported through membrane)

 dependent on penetrant geometry(molecular size ↑ ⇨ diffusion coefficient ↓)

 Diffusion coefficient is concentration-dependent with interacting systems ※ large(organic) molecules ⇨ swell the polymer ⇨ large diffusion coefficients

 Quite low(< 0.2% by volume)

 <Assume> Ideal system ⇨ Diffusion coefficient = constant ≠ f(concentration) ⇨ Applying Fick's law

 Solubility of organic liquids (and vapors) in polymers

 Relatively high (depending on the specific interaction)

 <Assume> Diffusion coefficient = f(concentration) ⇨ concentration ↑ → diffusivities ↑

 Two separate cases

 Ideal systems : Diffusivity and Solubility = constant

 Concentration-dependent systems : Diffusivity and Solubility = f(concentration)

 Solubility ≠ f(concentration)

 Sorption isotherm = linear (Henry's law)

 Concentration inside the polymer ∝ applied pressure (<Figure 5-10a>)

 This behavior is normally observed with gases in elastomers.

 Non-ideal systems

 Strong interactions between organic vapor / liquid ↔ polymer

 With glassy polymers the sorption isotherm is generally curved (<Figure 5-10b>)

 Sorption isotherms = highly non-linear, especially at high vapor P (<Figure 5-10c>)

 Described by free volume models and Flory-Huggins thermodynamics

<Figure 5-10> Schematics of sorption isotherms for ideal and non-ideal systems.

 Obtained from equilibrium measurements

 Measure gas volume taken up by polymer sample at a known applied pressure

 Dual sorption theory

 Solubility of a gas often deviates in the manner shown in <Figure 5-10b> for glassy polymers

 Assume that two sorption mechanisms occur simultaneously

• Sorption according to Henry's law

• Sorption according to Langmuir type

 Gas concentration in polymer = Sorption by Henry's law + Sorption by Langmuir

c = c_d + c_h (5-89)

(5-90)

where k_d = Henry's raw constant (cm³(STP).cm^－3.bar^－1) = Solubility coefficient (S) b = Hole affinity constant (bar^－1)

c'_h = Saturation constant (cm³(STP).cm^－3)

 The dual sorption model : three parameter model

 Good description of observed phenomena

• Very frequently used to describe sorption in glassy polymers.

• Imply the existence of two different types of sorbed gas molecules

⇨ Two different sorption modes for one membrane ⇨ Difficult to understand

<Figure 5-11> The two contributions in the dual sorption theory.

 Permeability

 Permeability(P) = Solubility(S) × Diffusivity(D) (5-88)

 The simplest way to describe the transport of gases

Fick’s 1^st law : (5-91)

 Diffusion (<Figure 5-12>)

 Statistical molecular transport as a result of the random motion of the molecules

 Concentration difference ⇨ occur macroscopic mass flux

<Figure 5-12> Schematic drawing of diffusion as a result of random molecular motions.

 Modeling (<Figure 5-13>)

 More molecules on left side than on right side ⇨ Net mass flux : left to right

 Two planes (a thin part of a membrane) at x and x+δx

 Quantity of penetrant at x = J•δt at time δt

 Quantity of penetrant leaving at x+δx = [J + (dJ/dx) δx] δt

 The change in concentration (dc) in the volume between x and x+δx

(5-92) & (5-93)

 By δx → 0 and δt → 0, (5-94)

 By Eq(5-91) → Eq(5-94), adapting Fick’s 1^st law to Eq(5-94),

 Fick’s 2^nd law : (5-95)

<Figure 5-13> Diffusion across two planes situated at points x and x+δx in the cross-section of a membrane

 Fick’s 2

(5-96)

 D of gases in gases at room temperature = 0.05 ∼ 1 cm²/sec

 D of low molecular weight liquids and gases in liquids = 10^-4 ∼ 10^-5 cm²/sec

 Diffusivity coefficient (D)

 D depends on diffusing particles size and on nature of diffusion medium

 In general, particle size ↑ ⇨ diffusion coefficients ↓ ※ Stokes-Einstein equation [Eq(5-46)],

Noble Gas MW Diameter(Å) Diffusion Coefficient(cm²/sec)

He 4 2.6 ≈ 0.5 × 10^-4

Ne 20 2.75 ≈ 1.0 × 10^-6

Ar 40 3.4 ≈ 1.0 × 10^-8

Kr 84 ≈ 0.5 × 10^-8

[Table 5-4] Diffusion coefficients of noble gases in polyethylmethacrylate at 25℃

<Figure 5-14> Diffusion coefficients of components in water and in an elastomer membrane as a function of MW(left figure) and in a polymer as a function of the degree of swelling for a given low MW penetrant.

 Diffusion in non-interacting systems

 In water, MW ↑ → D ↓(only slightly)

 In rubber, MW ↑ → D ↓(strongly)

 Diffusion in concentration-dependent systems

 Swell membrane considerably ⇨ change diffusing medium significantly

 Strong interactions ⇨ Large impact on diffusion phenomena

 Swelling ⇨ Penetrant concentration inside the polymer ↑ ⇨ D ↑ ⇨ Effect of the particle size on D ↓

 In general, D ↓ at lower swelling values ⇨ effect of concentration ↑

 Diffusion processes in terms of friction (Another way of describing diffusion processes)

 Chemical potential gradient(dμ/dx) ⇨ Force on penetrant molecules ⇨ move through the membrane with a velocity v

where f = frictional resistance coefficient (5-97)

 Since mobility coefficient m = 1/f

(5-98)

 Quantity of molecules passing through the cross-sectional area per unit time

(5-99)

 The thermodynamic diffusion coefficient(D_T) is related to the mobility by the relation

D_T = m•RT (5-100)

 and chemical potential μ = μ⁰ + RT ln(a) (5-101)

 Eq(5-99) → (5-102)

 By comparison with Fick’s 1st law,

(5-103)

 For ideal systems

 Activity(a) = Concentration(c)

 D = D_T

 Eq(5-102) ⇨ Fick's 1^st law

 For non-ideal systems (organic vapors and liquids)

 Must use activities instead of concentrations.

 D_T = f(concentration or activity)

⇨ Presence of the penetrant modifies the properties of the membrane.

 Henry's law

 c = S•p (5- 104)

where p = external pressure

c = concentration of solute inside the membrane

 Eq(5-104) → Fick's law [Eq(5-91)] and integrating across the membrane leads to:

(5-105) where p₁ = pressure on the feed side (x=0) p₂ = pressure on the permeate side (x=ℓ)

c₁ = penetrant concentration in polymer c₂ = penetrant concentration on permeate side

and since the permeability coefficient P = D·S (5-106)

(5-107)

 Friction verse radius and diffusivity of solute

 Friction resistance, f = 6π·η·r (5-108)

 Diffusion coefficient, (5-109)

 Solubility verse radius

 radius ↑ ⇨ solubility ↑, since interaction of a gas with a polymer = very small

 Non-interacting gas : He, H₂, N₂, O₂ and Ar

 Some interaction : CO₂, C₂H₄, C₃H₆, etc.

<Figure 5-15> Solubility and diffusivity of various gases in natural rubber.

 Main parameter that determines the solubility = ease of condensation

 Radius ↑ ⇨ molecules becoming more condensable

 Critical temperature T_c = measure of the ease of condensation

 Radius of molecule ↑ ⇨ T_c and S of the gas in the polymer ↑

Gas T_c(K) S(cm³·cm^-3·cmHg^-1)

H₂ 33.3 0.0005

N₂ 126.1 0.0010

O₂ 154.4 0.0015

CH₄ 190.7 0.0035

CO₂ 304.2 0.0120

[Table 5-5] Critical temperature(T_c) and Solubility coefficient(S) of various gases in natural rubber

<Figure 5-16> The P-V isotherms for a gas at various temperatures.

 Implication of <Figure 5-17> and <5-18> : Permeability of various gases

 Smaller molecules do not automatically permeate faster than larger molecules.

 High diffusivity ⇨ High permeability of smaller molecules(H₂, He)

 High solubility ⇨ High Permeability of larger molecule(CO₂)

 Low diffusivity and solubility of N ⇨ low permeability

<Figure 5-17> Solubility of various inert gases in silicone rubber (PDMS) as a function of critical temperature (T_c) and Lennard-Jones potential (ε/k)

<Figure 5-18> Permeability of various gases in natural rubber

 Permeability of various gases according to polymer nature

 Higher permeable in rubbery polymers

 Lower permeable in glassy polymers.

<Figure 5-19> The permeability of various gases in different polymers

 Diffusion coefficient = constant for ideal systems

 Determined by a permeation method(Time-lag method)

 Very suitable for ideal systems with a constant diffusion coefficient

 Determination of diffusion coefficient by measuring flow rate

 Membrane is free of penetrant at the start of the experiment

 Amount of penetrant (Q_t) passing through the membrane in the time t (5-110) where c_i = concentration on the feed side and n = integer

 When t →∞, the exponential term in Eq(5-110) = neglected and it simplifies to:

(5-111)

 Linear plot of Q_t versus t ⇨ intercept on time axis = θ (time lag)

(5-112) 5.5.1.1 Determination of the diffusion coefficient

 Determination of diffusivity by measuring permeate pressure(p₂)

 p₂ versus t plot ⇨ Time-lag

 Permeability coefficient(P)

 obtained from Eq(5-107), J = P(p₁-p₂)/ℓ, at steady-state

 Concentration dependent systems : use more complex relationships for the time-leg

<Figure 5-20> Time-lag measurement of gas permeation.

 D & P → Solubility coefficient(S) from the ratio P over D [Eq(5-106), P = D•S]

 Direct method to determine solubility coefficient

 Gravimetric method using a microbalance

 Quartz spring

 Pressure decay method

 Pressure decay method

 high accuracy

 employed in a single and dual volume concept 5.5.1.2 Determination of the solubility coefficient

<Figure 5-21> Schematic drawing of a single and a dual volume pressure decay set-up.

 Experimental method

• place polymer sample in a closed and constant volume

• evacuated for a certain period to remove present interfering molecules

• gas is applied at a certain pressure

• pressure will be decreased by sorption of gas in polymer until equilibrium

• calculate amount of penetrant inside polymer

 Plot M_t/M_∞ versus √time ⇨ D = slope according to Eq(5-113) where M_t = mass uptake at time t

M_∞ = mass uptake at infinite time t_∞ (5-113)

(5-114)

 Temperature dependence of the permeability coefficient(P)

 Arrhenius type of equation : P = P_o exp (- E_P/RT) (5 - 115) where E_P = activation energy (35∼45 kJ/mol for various gases in PE) 5.5.1.3 Effect of temperature on the permeability coefficient

<Figure 5-23> Temperature dependence of P of non-interactive gases in PE

 Temperature(T) dependence of solubility(S) of non-interactive gases in polymers

 S = S_o exp (- ΔH_S/RT) (5-116)

where ΔH_S = heat of solution S_o ≠ f(T)

 ΔH_S = heat of mixing term + heat of condensation(+ : endothermic, ─ : exothermic)

 Heat of solution(ΔH_S)

 For small non-interactive gases(N₂, He, CH₄, H₂)

∙ ΔH_S = small positive value ⇨ Endothermic ⇨ T ↑ → solubility ↑

 For large molecules(organic vapor) : situation is much more complex ∙ ΔH_S = negative value ⇨ Exothermic ⇨ T ↑ → solubility ↓

 Temperature(T) dependence of diffusion(D) of non-interactive gases in a polymer

 Diffusion coefficient(D)

Arrhenius behavior : D = D₀ exp (- E_d/RT) (5-117)

where E_d = activation energy for diffusion D₀ ≠ f(T), pre-exponential factor

「Caution」 Don't confuse above D₀ with D₀ of diffusion coefficient at c=0.

 Combination of Eq(5-88) with (5-116) and (5-117) ⇨ Eq(5-115)

(5-115)

※ Eq(5-88) : Permeability(P) = Solubility(S) × Diffusivity(D), Eq(5-116) : S = S_o exp (- ΔH_S/RT)

 For small non-interactive gases

 T effect on S = weak

 T effect on D = strong

∴ P and D dependence on T = almost same

 For the larger molecules

 Effect of T on D Effect of T on S

 D and S = f(c)

Major effect of T on P coming from D

opposite

More complex

 Comparison of activation energy of permeation(E_P) in elastomeric and glassy polymers (Slope of <Figure 5-24>) ※ P = P_o exp (- E_P/RT)

 Polyvinyl acetate with a glass transition temperature(T_g) = 29℃

 E_p in elastomeric region > E_p in glassy region

 Rubbery polymer : much more segmental mobility and rotational freedom, but E_p = high

「Meaning」 E_p can not be related explicitly to the ease of permeation.

<Figure 5-24> Temperature dependence of the permeability of Ne in polyvinylacetate.