chapter 5. transport in membrane -...

Chang-Han Yun / Ph.D.

National Chungbuk University

October 7, 2015 (Wed)

Chapter 5. Transport in Membrane

2 Chapter 5. Transport in Membrane Chungbuk University

Contents

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


On the point of molecular size alone ⇨ P of large molecule < P of simple small gases(N2)

In real, P of large organic molecule(Ph-CH3 or CH2Cl2) > 104∼105 × P of small molecule(N2)

Difference in interaction ⇨ Difference in solubility ⇨ Difference in permeability

Solubility↑ ⇨ Segmental motion↑ ⇨ Free volume↑

Non-linear relationships between concentration ↔ pressure (<Figure 5-25>)

Strong interaction ⇨ Solubility = non-ideal ⇨ S = f(c) ⇨ not followed by Henry's law

For high solubility in polymers, c↑→ D↑ ⇨ D = f(c)

Flory-Huggins thermodynamics

Convenient method to describe solubility of organic vapor and liquid in polymers

Activity of the penetrant inside the polymer is given by

Flory-Huggins : (5-118)

where χ = interaction parameter, χ > 2(large) : small interaction

0.5 < χ < 2.0(small) : strong interactions ⇨ high permeability

※ Crosslinked polymer : χ < 0.5

5.5.2

interactive Systems 5.5 Transport through Nonporous Membranes


<Figure 5-25> Solubility of CH2Cl2(●),

CHCl3(○) and CCl4(■) in

polydimethylsiloxane(PDMS) as a function of

the vapor pressure.

Component Permeability

(Barrel)

N2 280

O2 600

CH4 940

CO2 3,200

Ethanol(C2H5OH) 53,000

Methylene Chloride(CH2Cl2) 193,000

1.2-Dichloroethane(CH2Cl-CH2Cl) 248,000

Tetrachloride(CCl4) 290,000

Chloroform(CHCl3) 329,000

1,1,2-Trichloroethane(CCl2CHCl) 530,000

Trichloroethene(CCl2=CHCl) 740,000

Toluene(Ph-CH3) 1,106,000

[Table 5-6] Permeability of various components in PDMS at 40°C

5.5.2



Concentration(c) dependence of D

D = f(c) but no unique relationship

(∵ it varies from polymer to polymer and from penetrant to penetrant)

Empirical exponential relationship.

D = D0 exp (γ·ϕ) (5-119)

where D0 = diffusion coefficient at c = 0

γ = plasticising constant(plasticising action of penetrant on segmental motion)

ϕ = volume fraction of the penetrant

D0 dependence of molecular size

Molecular size↓(water) ⇨ D0↑

Molecular size↑(benzene) ⇨ D0↓(see [Table 5-7])

D dependence of γ and ϕ

γ and ϕ appear in the exponent of Eq(5-119)

⇨ highly influence to D ※ Ideal gas ⇨ γ → 0

Component Vm

(cm3/mole)

D0

(cm2/sec)

Water 18 1.2 × 10-7

Ethanol 41 1.5 × 10-9

Propanol 76 2.1 × 10-12

Benzene 91 4.8 × 10-13

[Table 5-7] Effect of penetrant size

on D0 in poly(vinyl acetate)

5.5.2



Concentration(c) dependence of the diffusion coefficient(D)

Describe by free volume theory

<Assume> Penetrant increases the free volume of the polymer.

Relationship between log D and the volume fraction of the penetrant(ϕ) in the polymer

※ similar to Eq(5-119) : D = D0 exp (γ·ϕ)

More quantitative approach than Eq(5-119)

Large difference in permeability of between glassy ↔ rubbery state

Glassy state

mobility of the chain segments is extremely limited

thermal energy too small to allow rotation around the main chain

Rubbery state(above Tg)

Mobility of the chain segments↑

Frozen micro-voids no longer exist

5.5.2.1 Free volume theory

5.5.2



<Figure 5-26> Specific volume of an amorphous

polymer as a function of temperature.

Free volume (Vf) = VT – V0 (5-120)

where VT = observed volume at a temperature T

V0 = volume occupied by the molecules at 0 K

Fractional free volume(vf) = Vf / VT (5-121)

vf ≈ vf,Tg ≈ 0.025 for most of glassy polymers based on viscosity

Vf above Tg ∝ T linearly

vf ≈ vf,Tg + Δα(T - Tg) (5-122)

where Δα = difference (thermal expansion coefficient value above Tg ↔ below Tg)

5.5.2



Basic concept of free volume

Very useful to understand transport of small molecules through polymers

Molecule can only diffuse from one place to another place

if there is sufficient empty space or free volume

Size of penetrant↑ → amount of free volume↑

Probability of finding a 'hole' whose size exceeds a critical value ∝ exp(－B/vf)

where B = local free volume needed for a given penetrant

vf = fractional free volume

Mobility of penetrant ∝ Probability of a hole of sufficient size for displacement

Mobility can be related to thermodynamic diffusion coefficient [see Eq(5-100)]

DT = (mobility coefficient)·RT = m·RT = RT·Af exp(－B/vf) (5 - 123)

where DT = Thermodynamic diffusion coefficient

Af = dependent on the size and the shape of penetrant molecules

B = related to minimum local free volume necessary to allow a displacement

5.5.2



DT = RT·Af exp(－B/vf) ⇨ T↑ & Penetrant size(B↓) ⇨ diffusion coefficient(DT)↑

Non-interacting systems (polymer with inert gases like He, H2, O2, N2, Ar)

• Polymer morphology is not influenced by the presence of these gases

「Meaning」 There is no extra contribution towards the free volume

• By assuming that Af and B ≠ f(polymer)

∙ plot of In(D/(RT·Af)) verse (1/vf) ⇨ Slope = －B from Eq(5-123)

Polyimides deviate from this linear behavior

「Meaning」 Assumptions behind Eq(5-123) are not completely correct

Af and B = f(polymer) ⇨ polymer-dependent parameters need

Interacting systems (e.g organic vapors)

Free volume = f(temperature, penetrant concentration)

• vf = f(ϕ,T) = vf(0,T) + β(T) ϕ (5-124)

where vf(0,T) = vf at temperature T and zero penetrant concentration

ϕ = volume fraction of penetrant

β(T) = constant(extent to which the penetrant contributes to vf)

5.5.2



Diffusion coefficient at zero penetrant concentration : D0 = Dc→0 at Eq(5-123)

(5-125)

Combination of Eq(5-123) and (5-125) gives

(5-126)

(5-127)

「Meaning」

[1/ln (DT/D0)] is related linearly to 1/ϕ

D = D0 exp (γ· ϕ)[Eq(5-119)] and Eq(5-127) are similar when vf(0,T) ≫ β(T)

Plots of ln(D) verse ϕ = linear

5.5.2



Solubility

Gas molecules ⇨ apply Henry's law

(solubility of a gas in a polymer ∝ external partial pressure)

Organic vapor and liquid ⇨ apply Flory-Huggins thermodynamics

(5-118)

where χ = interaction parameter

D(measured diffusion coefficient) ↔ DT(thermodynamic diffusion coefficient)

(5-128)

Penetrant concentration↑ ⇨ difference between the two diffusion coefficients↑

By differentiation of Eq(5-118) with respect to lnϕi

(5-129)

For ideal systems and at low volume fractions(ϕi→0) : dlnai/dlnϕi = 1 and D = DT

5.5.2



Clustering of penetrant molecules ⇨ Cause deviations from free volume approach

Component diffuses not as a single molecule but in its dimeric or trimeric form.

Size of the diffusing components↑ ⇨ Diffusion coefficient↓

(<Ex> water molecules → strong H-bonding ⇨ diffuse by clustered molecules)

Extent of clustering will also depend on

Type of polymer

Other penetrant molecules present

Zimm-Lundberg theory to describe the clustering ability

Cluster function : ability or probability of molecules to cluster inside a membrane

(5-130)

where G11 = cluster integral, V1 = molar volume of penetrant

ϕ1 = volume fraction of penetrant

For ideal system, dlnϕ1/dlna1 = 1 ⇨ G11/V1 = －1 ⇨ no clustering

G11/V1 > －1 ⇨ clustering

5.5.2.2 Clustering

5.5.2



Difference between ternary system and binary system(polymer and liquid) by thermodynamics

Ternary system (a binary liquid mixture and a polymer)

Volume and composition of liquid mixture inside the polymer = important parameters

Composition of liquid mixture inside the polymer ⇨ Sorption selectivity ⇨ Rejection rate

Concentration of a given component i in the binary liquid mixture in the ternary polymeric phase

(5-131)

Preferential sorption is then given by ε = ui – vi i = 1, 2 (5-132)

Δμf,i = Δμm,i + πVi i = 1,2 (5-133)

subscript f (feed) = polymer free phase

subscript m (membrane) = ternary phase

5.5.2.3 Solubility of liquid mixtures

<Figure 5-27> Schematic drawing of a binary

liquid feed mixture in equilibrium with the

polymeric membrane.

5.5.2



Flory-Huggins thermodynamics ⇨ Expressions for the chemical potentials

When V1/V3 ≈V2/V3 ≈ 0 and V1/V2 = m,

Concentration-independent Flory-Huggins interaction parameters and eliminating π gives ;

(5-134)

Composition of the liquid mixture inside the membrane, can be solved numerically

when the interaction parameters and volume fraction of the polymer are known.

In real, Flory-Huggins parameters for these systems = concentration-dependent ⇨ complex

Define where αsorp = sorption (5-135)

「Meaning of Eq(5-134)」

Difference in molar volume

• If only entropy effects ⇨ Selective sorption of smaller molar volume component preferentially

• Polymer concentration↑ ⇨ These effect↑ ※ Maximum at ϕ3→1

5.5.2



Enthalpy of mixing ⇨ Selective sorption of highest affinity component to polymer

By assuming ideal sorption ⇨ this factor only influences the solubility

(the highest affinity leads to the highest solubility)

Influence of mutual interaction with the binary liquid mixture on preferential sorption

depends on the concentration in the binary liquid feed

and on the value of χ12

χ12

For organic liquids, χ12 = f(composition) strongly

For constant interaction parameter, χ12 should be replaced

by a concentration-dependent interaction parameter, gl2(ϕ).

Solvent V1/V2

Methanol 0.44

Ethanol 0.31

Propanol 0.24

Butanol 0.20

Dioxane 0.21

Acetone 0.24

Acetic acid 0.31

DMF 0.23

[Table 5.8] Ratio of molar volumes at 25℃ of various organic solvents

with water (V1 = 18 cm3/mol)

5.5.2



Concentration-dependent systems

Apply Fick's law using concentration dependent diffusion coefficients

Di = D0,i exp(γi·ci) (5-136)

where D0,i = diffusion coefficient at ci→0

γi = plasticising constant(plasticising action influence of liquid on segmental motions)

Substitution of Eq(5-l36) into Fick's law and integration using the BC

BC 1 : ci = ci,lm at x = 0

BC 2 : ci = 0 at x=ℓ

(5-137)

Parameters in Eq(5-137)

∙ D0,i , γ and ℓ = constants

∙ main parameter = concentration inside the membrane (ci,lm)

∙ ci,lm↑ ⇨ permeation rate↑

5.5.2.4 Transport of single liquid

「Meaning of Eq(5-137)」 for single liquid transport

• Interaction between membrane ↔ penetrant

⇨ determine permeation rate

• Affinity between penetrant ↔ polymer↑

⇨ Ji↑ for a given penetrant

5.5.2



5.5.2

interactive Systems

Transport of liquid mixtures through a polymeric membrane

For a binary liquid mixture, Ji = f (solubility, diffusivity)

Strong interaction between Solubility ↔ Diffusivity ⇨ Much more complex

Distinguished phenomena in multi-component transport

Flow coupling ⇨ described via non-equilibrium thermodynamics

Thermodynamic interaction ⇨ preferential sorption

Flux equations for a binary liquid mixture

Ji = Lii dμi/dx + Lij dμj/dx (5-15) • Jj = Lji dμi/dx + Ljj dμj/dx (5-16)

Where Lii dμi/dx = flux of component i due to its own gradient

Lij dμi/dx = flux of component i due to the gradient of component j (coupling effect)

No Coupling (Apply binary system with very low permeability)

Components permeate through the membrane independently of each other

Lij = Lji = 0 ⇨ reduce to simple linear relationships

5.5.2.5 Transport of liquid mixture



5.5.3

Effect of Crystallinity

Crystalline fraction of polymer

Large number of polymers are semi-crystalline(amorphous + crystalline fraction)

However, the crystallinity is quite low in most membranes

Cystallinity < 0.1 → Diffusion resistance by crystalline ⇨ negligible

Effect of crystallinity on the permeation rate is often fairly small

Diffusion coefficient = f (crystallinity)

(5-138)

where Ψcn = fraction of crystalline

B = constant

n = exponential factor (n < 1)

<Figure 5-28> The effect of crystallinity on diffusion resistance



5.6 Transport through Membrane – A unified approach

Classification of model

Based on phenomenological approach

• Black box model

• Provide no information as to how the separation actually occurs

Based on non-equilibrium thermodynamics

Mechanistic models(pore model and solution-diffusion model)

• Relate separation with structural-related membrane parameters

in an attempt to describe mixtures.

• Provide information on how separation actually occurs ⇨ factors are important

Simple model ⇨ starting point

generalized Fick equation

generalized Stefan-Maxwell equation



Flux of component i through a membrane = Velocity × Concentration

Ji = ci(vi + u) (5-139)

Convective flow(u) : main transport through porous membrane

Diffusion flow(vi) : main transport through nonporous

<Figure 5-29> Convective and diffusive flow in membranes.



Comparison of flux contribution in the case of porous membranes (MF)

Given conditions

∙ Membrane with a thickness(ℓ) = 100 μm

∙ Average pore diameter = 0.1 μm

∙ Tortuosity(τ) = 1 (capillary membrane)

∙ Porosity(ε) = 0.6

∙ ΔP for water flow = 1 bar

Convective flux from Poisseuille equation (convective flow)

Diffusion flux

∙ Driving force : difference in chemical potential = f (Δc or Δa, and ΔP)

∙ Δμw = vw∙ΔP = 1.8× 10-5∙105 = 1.8 J/mol



Diffusion flow

Ji = ci∙vi (5-140)

vi = Xi / fi (5-141)

where vi = mean velocity of a component in the membrane

Xi = driving force acting on the component = gradient(dμ/dx)

fi = frictional resistance = RT/DT

DT = thermodynamic diffusion coefficient

If ideal conditions are assumed(DT = Di, observed diffusion coefficient)

Eq(5-140) → (5-142)

Chemical potential : μi = μoi + RT ln ai + Vi∙(P－Po) (5-6)

Eq(5-6) → Eq(5-142) : (5-143)



<Figure 5-30> Process conditions for transport through nonporous membranes.

(superscripts m = membrane, superscripts s = feed/permeate side)

<Assume> Thermodynamic equilibrium exists at the membrane interfaces

μi at the feed/membrane interface is equal in both the feed and the membrane

μi,1m = μi,1

s ⇨ ai,1m = ai,1

s (5-144)

Pressure inside membrane = Pressure on feed side at feed interface (phase 1/membrane)

P1 = Pm ≠ P2



At the permeate interface(membrane/phase 2)

(5-145)

(5-146)

(5-147)

(5-148)

(5-149)



<Assume> diffusion coefficient ≠ f (concentration)

Fick's law[Eq(5-83)] can be integrated across the membrane to give

(5-150)

Eq(5-146), (5-147) and (5-148) → Eq(5-150)

(5-151)

if αi = Ki,2 / Ki,l (i.e. the solubility coefficients are similar at both interfaces) and Pi = Ki ∙ Di ,

then Eq(5-151) converts into

(5-152)

※ Eq(5-152)

Basic equation used to compare various membrane

processes when transport occurs by diffusion.

Process Phase 1 Phase 2

RO L L

Dialysis L L

Gas separation G G

Pervaporation L G

[Table 5-9] Phases involved in diffusion

controlled membrane processes


5.6.1

Reverse Osmosis 5.6 Transport through Membrane – A unified approach

Application

Separate a very low MW solute(salt, very small amount of organic)

Driving force : pressure difference

total flux = water flux(Jw) + solute flux(Js)

Solvent Flux (※ Js = neglected by high selective)

Jtotal = Jw +Js ≈ Jw (5-153)

since Δπ = RT/Vi∙(In cw,2s/cw,1

s) and α1 =1,

(5-154)

or

(5-155)

For small values of x, the term, 1－exp(－x) ≈ －x (5-156)

and since Kw∙cw,1s = cw,1

m (5-157)

Eq(5-154) → (5-158)

Jw = Aw(ΔP - Δπ) with Aw = Dw∙Cw,1m∙Vw / RT∙ℓ (5-159)

where Aw = called the water permeability coefficient(Lp)


5.6.1

Reverse Osmosis 5.6 Transport through Membrane – A unified approach

Solute flux

Reverse osmosis membranes are generally not completely semipermeable

From Eq(5-151) with αj = 1, the solute flux Js can be written as

(5-151)

(5-160)

and since the exponential term is approximately unity (see section 5.6.4),

or Js = B∙Δc (5-161)&(5-162)

where B = permeability coefficient = Ds∙K/ℓ

「Meaning of Eq(5-159) and (5-162)」

Eq(5-159) → Water flux ∝ applied effective pressure difference in reverse osmosis

Eq(5-162) → Solute flux ∝ concentration difference in reverse osmosis


5.6.2

Dialysis 5.6 Transport through Membrane – A unified approach

Dialysis

Liquid phases containing same solvent are present on both sides of membrane

No pressure difference

Flux

Pressure terms = neglected

from Eq(5-152) if αi = 1 → (5-163)

or

(5-164)

「Meaning of Eq(5-159) and (5-164)」

Solute flux is proportional to the concentration difference

Separation arises from differences in permeability coefficients

DT and Distribution coefficients of higher MW < Lower MW species


5.6.3

Gas Permeation 5.6 Transport through Membrane – A unified approach

Gas permeation or Vapor permeation

both the upstream and downstream sides of a membrane consist of gas or vapor

However, Eq(5-152) cannot be used directly for gases.

(5-152)

Concentration of gas in membrane

ci,1m = Pi,1

s∙Ki (5-165)

by combining Eq(5-165) with Eq(5-150)

(5-150)

(5-166)

where Pi = Ki∙Di


Rate of gas permeation ∝ ΔP across the membrane


5.6.4

Pervaporation 5.6 Transport through Membrane – A unified approach

Pervaporation

Feed side = liquid, Permeate side = vapor ⇨ very low pressure in downstream

P2(downside stream) → 0 (or a2s → 0)

exponential term in Eq(5-152) = 1 and can be neglected

(ΔP ≈ 105 N/m2, Vi = 10－4 m3/mol, RT ≈ 2500 J/mol ⇨ exp(－Vi∙ΔP/RT) ≈ 1)

lf partial pressure = activity, then:

γis∙ci

s = Pi (5-167)

Eq(5-165) → (5-168)


Permeate pressure (pi,2s)↑ ⇨ Flux of component i↓

Permeate pressure (pi,2s) = Feed pressure(Pi,l

s) ⇨ Flux of component i = 0



Basic theory on principles : Nernst-Planck equation and Donnan equilibrium

Donnan exclusion(<Figure 5-31>)

Use an ion-exchange membrane in contact with an ionic solution

Same charged ions with the fixed ions in the membrane ⇨ Rejected by membrane

When an ionic solution is in equilibrium with an ionic membrane

※ Activities(= activity coefficient × molar concentration) are used (not concentrations)

※ Electrolyte solutions = generally behave non-ideal (very low concentrations ⇨ ideal behavior)

Chemical potential in ionic solution : μi = μi0 + RT In mi + RT In γi + zi ℱ Ψ (5-169)

Chemical potential in membrane : μim = μo

im + RT ln(mi

m) + RT ln(γim) + zi ℱ Ψm (5-170)

where subscript m = membrane phase

At equilibrium,

• μi = μim at interface (5-171)

• μoi = μo

im in membrane

• Potential difference(Edon) = Ψm－Ψ



(5-172)

(5-173) & (5-174)

For the case of dilute solutions(ai ≈ ci),

(5-175)

<Assume> Swelling pressure(π•Vi) = negligible

※ If not, this term should be added to right-hand side of Eq(5-175).

Calculation example

• Monovalent ionic solute

• Concentration difference = 10

• Edon at the interface

= [(8.314×298)/(96500)] In(1/10) = －59 mV

<Figure 5-31> Schematic drawing of the ionic distribution

at the membrane-solution interface(membrane contains

fixed negatively charged groups) and the corresponding potential



Ion-exchange membrane with a fixed negative charge (R－) with NaCl solution

<Assume> Solution behaves ideal ⇨ ai = ci

At equilibrium, μi = μim

Under ideal conditions (γi → 0)

(5-176)

By electrical neutrality,

∑zi ci = 0 (5-177)

in membrane (5-178)

in solution (5-179)

Combination of Eq(5-176) and (5-178) gives

(5-180)

(5-181)

or (5-182)



For a dilute solution, Eq(5-182) reduces to

(5-183)

⇨ Donnan Equilibrium

Donnan Equilibrium is valid when high concentration of R– is contacted with dilute solution.

Feed concentration↑ ⇨ this exclusion↓

<Example> ∙ Brackish water with 590 ppm NaCl (≈ 0.01 eq/L ≈ 10–5 eq/mL)

∙ Wet-charge density of membrane ≈ 2×10–3 eq/mL

∙ Co-ion (Cl－) concentration in the membrane ≈ 5×10–8 eq/mL

⇨ [Cco-ion]m ∝ [Cco-ion]2 and [fixed charge density in the membrane(R－)]－1

Non-ideal solution ※ Ionic solutions = non-ideal manner in most case

Using activity coefficients(γi)

Mean ionic activity coefficients(γ±)

For a univalent cation and anion, γ± = (γ+•γ–)0.5

where γ+ and γ– = activity coefficients of the cation and anion, respectively



Eq(5-181) → (5-184)

Ion-exchange membranes in combination with an electrical potential difference

Forces act on ionic solutes : ΔC and ΔE

Transport of ion : Fickian diffusion and Ionic conductance

Nernst-Planck equation : (5-185)

NF, RO membranes

Ions are transported across a charged membrane without ΔE

Convective term has to be included

3 contributions for ionic transport : an electrical, a diffusive and a convective term

Ji = Ji,dif + Ji,elec + Ji,conv (5-186)

<Assume> ∙ No coupling phenomena

∙ Ideal conditions

Extended Nernst-Planck equation : (5-187)

chapter 5. transport in membrane -...

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