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


5.1 Introduction

Definition of membrane : permselective barrier between two homogeneous phases.

Driving force(F) = gradient in potential = ∂X/∂x ≒ ΔX/Δx = ΔX/ℓ [N/mol] (5-1)

where ΔX : Potential difference across the membrane

ℓ : membrane thickness

Main potential differences in membrane processes

Chemical potential difference (Δμ)

Electrical potential difference (ΔF)

※ Electrochemical potential = chemical potential(Δμ) + electrical potential(ΔF)

※ Other possible forces(※ not considered here)

• Magnetic fields

• Centrifugal fields

• Gravity

Passive membrane transport of

components from a phase with a high potential to

one with a low potential.


5.1 Introduction

Average driving force (Fave) = －ΔX /ℓ (5-2)

Flux (J) = proportionality factor (A) × driving force (X) (5-3)

Proportionality factor(A)

determines transport speed of components through membrane

measure of the resistance exerted by the membrane

Passive transport : Transfer from a high potential to a low potential

Facilitated transport(or Carrier-mediated transport)

Enhanced transport by a (mobile) carrier

Carrier

• Interacting with one or more specific components in feed

• Increasing transport by additional mechanism

Transport proceeds in co-current or counter-current fashion

• Simultaneous transport of another component

• Increasing chemical potential gradient by 2nd component

Schematic drawing of basic form of passive transport

(C is carrier and AC is carrier-solute complex).


5.1 Introduction

Active transport

Transport against their chemical potential gradient

Possible only by adding the energy to the system

※ Living cell membranes providing the energy by ATP

Coupling of driving force with fluxes

Described by non-linear phenomenological equations

『Meaning』 explained by non-equilibrium thermodynamics

Permeating individual component dependently from each other

• Δp across the membrane

→ solvent flux + mass flux and the development of a solute concentration gradient

• Δc across the membrane

→ diffusive mass transfer

→ buildup of hydrostatic pressure

Osmosis : coupling between Δc ↔ Δp

Electro-osmosis : coupling between ΔF ↔ Δp


5.2 Driving Forces

Potential difference

Difference in pressure(p), concentration(c), temperature(T), electrical potential(F)

μi = μio + RT ln ai + ViP at constant T(isothermal conditions) (5-4)

μio = constant

ai = activity = concentration or composition in order to express non-ideality

ai = γi xi (5-5)

where γi = activity coefficient

xi = mole fraction

For ideal solutions

activity coefficient(γi) → 1

activity(ai) = mole fraction(xi)

Δμi = RT ln Δai + ViΔP from Eq(5-4) (5-6)


5.2 Driving Forces

Contribution of T and P to the Δμi

Temperature contribution at room temperature : RT = 2,500 J/mole

Pressure contribution(ViΔP)

• Molar volume of liquids = small

molar volume of water = 1.8×10－5 m3/mol (18 cm3/mol)

molar volume of ordinary organic solvent(MW=100 g/mol, ρ=1 g/mL) = 10－4 m3/mol

• ViΔP = 100 J/mol for water, 500 J/mol for the solvent at ΔP = 50 bar(5×106 N/m2)

Dimensionless comparison

Driving force = Electrochemical potential = Chemical potential + Electrical potential

Ideal conditions (ai = xi and Δln xi = Δxi/xi)

From Eq(5-2) and Eq(5-6) (5-7)

By Eq(5-7) × ℓ/RT, then dimensionless driving force is

(5-8) & (5-9)

where P* = RT/Vi and E* = RT/(ziℱ)


5.2 Driving Forces

For liquid

Concentration term(Δxi/xi) ≒ 1

Pressure term :

strongly dependent on components involved

(i.e. on the molar volume)

For gases

P* = P (assuming ideal gas)

Electrical potential depends on the valence(zi)

(5-10)

Electrical potential : very strong driving force(Pressure potential : very weak)

For same water transport

1 unit of concentration term = 1/40 volt of electrical potential difference(for zi = 1)

= pressure of 1,200 bar

[Table 5-1] Estimated value of P*

Component P*

Gas P

Macromolecule 0.003∼0.3 MPa

Liquid 15∼40 MPa

Water 140 Mpa



Flux equations derived from irreversible thermodynamics

Real description of transport through membranes

Consider membrane = black box

No information about the structure of the membrane

No physico-chemical view

(No information about permeation of molecules or particles through the membrane)

Advantage of black box concept

Very clearly describe existence of coupling of driving forces and/or fluxes

Transport processes through membranes

Non-equilibrium processes

Thermodynamics of the irreversible processes can be used.

In irreversible processes (and thus in membrane transport)

Free energy(G) is dissipated continuously. (if a constant driving force is maintained)

Entropy(S) is continuously produced and Entropy(S) production is irreversible energy loss.



Dissipation function (ϕ) : Entropy increase

Summation of all irreversible processes

Each can be described as the produce of conjugated flows (J) and forces (X).

ϕ = T(dS/dt) = ∑Ji • Xi (5-11)

where ϕ = dissipation function

J = conjugated flows

X = forces

Transport of mass, transfer of heat and of electric current → flow

At close to equilibrium

Each force is linearly related to the fluxes ⇨ Xi = ∑Rij • Jj (5-12)

Each flux is linearly related to the forces ⇨ Ji = ∑Lij • Xj (5-13)

For single component

J1 = L1ㆍX1 = －L1(dμ1/dx) (5-14)



In the case of the transport of two components l and 2

two flux equations with four coefficients (L11, L22, L12, L21)

If no electrical potential, Driving force = Chemical potential gradient

(5-15)

(5-16)

The 1st term of Eq(5-15) ⇨ flux of comp. 1 under its own gradient

The 2nd term of Eq(5-15) ⇨ contribution of comp. 2 gradient(dμ2/dx) to comp. 1 flux(J1)

L12 : coupling coefficient(represents the coupling effect)

L11 : main coefficient

Simplification of coefficients

L12 = L21 ⇨ three phenomenological coefficients (5-17)

L11 (and L22) ≧ 0 (5-18)

L11 • L22 ≧ L122 (5-19)



Positive coupling ⇨ Coupling coefficients(L12 or L21) : positive

Flux of one component↑ → Flux of the 2nd component↑ ⇨ Selectivity↓

For all kinds of membrane processes

Appling non-equilibrium thermodynamics

Dilute solutions consisting of a solvent (usually water) and a solute

⇨ The characteristics of a membrane : 3 coefficients or transport parameters

• Solvent permeability(L)

• Solute permeability(ω)

• Reflection coefficient(σ)

※ index w : water as the solvent / index s : solute

Dissipation function in dilute solution

ϕ = Jw•Δμw + Js•Δμs (5-20)

Δμ of water, Δμw = μw,2 - Δμw,1 = Vw(P2 - P1) +RT[ln(a2) - ln(a1)] (5-21)

where subscript 2 : phase 2(permeate side) and subscript 1 : phase 1(feed side)



Define osmotic pressure, (5-22)

By Eq(5-22) → Eq(5-21) : Δμw = Vw(ΔP - Δπ) (5-23)

Writing Δμs for the solute as : (5-24)

By Eq(5-23) & Eq(5-24) → Eq(5-20)

(5-25)

where the 1st term : total volume flux (Jv) = Jw•Vw + Js•Vs (5-26)

the 2nd term : diffusive flux (Jd), (5-27)

Dissipation function, ϕ = Jv•ΔP + Jd•Δπ (5-28)

Corresponding phenomenological equations

Jv = L11•ΔP + L12•Δπ (5-29)

Jd = L21•ΔP + L22•Δπ (5-30)



For no osmotic pressure(Δπ = 0 → c1=c2 → Δc=0)

(Jv)Δπ=0 = L11•ΔP (5-31)

(5-32)

※ L11 : hydrodynamic permeability or water permeability of the membrane (Lp)

For no hydrodynamic pressure difference across the membrane (ΔP = 0)

Eq(5-30) → (Jd)ΔP=0 = L22•Δπ (5-33)

or (5-34)

L22 : osmotic permeability or solute permeability(ω)

Reflection coefficient(σ)

No volume flux(Jv = 0) under steady state

Eq(5-29) → L11•ΔP + L12•Δπ = 0 (5-35)

or (5-36)

Process Lp, L/(m2•hr•atm)

RO

UF

MF

< 50

50 ∼ 500 > 500



Eq(5-35) → L11 = L12 at ΔP = Δπ

There is no solute transport across the membrane

Membrane is completely semipermeable

※ Membranes are not usually completely semipermeable

※ L12/L11 : reflection coefficient(σ) → measure of selectivity

σ = － L12/L11 ≦1 (5-37)

σ = 1 : ideal membrane, no solute transport (5-38)

σ < 1 : not a completely semipermeable membrane: solute transport (5-39)

σ = 0 : no selectivity. (5-40)

By Eq(5-37) → Eq(5-29) and Eq(5-30)

Jv = Lp (ΔP - σΔπ) (5-41)

Js = ĉs (1 - σ)Jv + ωΔπ (5-42)

『Meaning』

3 transport parameters characterize

transport across a membrane

Water (solvent) permeability (LP)

Solute permeability (ω)

Reflection coefficient (σ)



Solute is not completely retained ⇨ σ ≠1 ⇨ Osmotic pressure difference = σ·Δπ ≠ Δπ

Freely permeable to the solute (σ = 0)

Osmotic pressure difference approaches zero ( σ·Δπ → 0)

Volume flux(Jv) = Lp·ΔP (5-43)

This is a typical equation for porous membranes (Jv ∝ ΔP)

(Ex, Kozeny-Carman and Hagen-Poiseuille equations for porous membranes)

Experiments with pure water at Δπ=0 → Lp via Eq(5-43)

Schematic representation of pure water

flux as a function of the applied pressure.

▶ high Lp : more open membrane ▶ low Lp : more dense membrane



Eq(5-42) → Solute permeability (ω) ※ Js = ĉs (1 - σ)Jv + ωΔπ (5-42)

(5-44)

Solute permeability (ω) and Reflection coefficient (σ)

Eq(5-42) → (5-45)

where Δc = concentration difference between the feed and the permeate

ĉ = logarithmic mean concentration [ĉ = (cf – cp) ln(cf / cp)]

By plotting Js/Δc verse Jv· ĉ /Δc

• Intercept of ⇨ Solute permeability(ω)

• Slope of ⇨ Reflection coefficient(σ)

Schematic drawing to obtain solute permeability

coefficient(ω) and reflection coefficient(σ) according to Eq(5-45)



When pores size increased(from RO to NF or UF)

Major retention contribution : Pore size → Solute molecular size

Stokes-Einstein equation [Eq(5-46)] ⇨ Solute size

(5-46)

※ Valid only for spherical and quite large particles

Solute size ↑ → Reflection coefficient(σ) ↑ (Selectivity ↑)

※ No information about the transport mechanism by thermodynamics

※ Coefficients in multi-component transport ⇨ not easy to determine

[Table 5-3] Some characteristic data for low

molecular weight solutes.

Solute MW Stockes radius(Å) σ polyethylene glycol

vitamin B12

raffinose

sucrose

glucose

glycerine

3000

1355

504

342

180

92

163

74

58

47

36

26

0.93

0.81

0.66

0.63

0.30

0.18



Coupling in electro-osmosis(electrical potential difference and hydrostatic pressure)

ΔE without ΔP ⇨ Occurring solvent transport

Porous membrane separating two (aqueous) salt solutions by electro-osmosis

Ion transport by ΔE (electrical potential difference)

Solvent transport by ΔP

Entropy production = sum of conjugated fluxes and forces

(5-47)

or I = L11 ΔE + L12 ΔP (5-48)

I = L21 ΔE + L22 ΔP (5-49)



By Assuming that Onsager's relationship applies (L12 = L21)

1) In the absence of an electric current (I = 0)

• Developing ΔE by ΔP (Streaming Potential)

(5-50)

2) When the pressure difference is zero (ΔP = 0)

• Electric current ⇨ Occurring solvent transport (Electro-osmosis)

(5-51)

3) When the solvent flux across the membrane is zero (J = 0)

• Electro-osmotic pressure is built up by an electrical potential difference.

(5-52)

4) In the absence of an electrical potential difference (ΔE = 0)

• Solvent flow across the membrane ⇨ generate Electrical current(I)

(5-53)



Structure-related membrane models

More useful than irreversible thermodynamic approach

Partly based on the principles of the thermodynamics of irreversible processes

Types of structure

Porous membranes : MF, UF (※ Pore size : 2 nm ∼ 10 μm)

Non-porous membranes : Pervaporation

Porous membrane

Transport occurs through the pores

Structure parameters : pore size, pore size distribution, porosity

Selectivity : based mainly on differences between particle and pore size



Dense membranes(Non-porous membrane)

The only dissolved molecule can permeate via membrane.

Affinity between membrane ↔ low MW component ⇨ determine solubility

Transported from one side to the other via diffusion

Selectivity : determined by differences in solubility and/or differences in diffusivity

Transport parameters

• Thermodynamic interaction or affinity between the membrane and the permeant

Interaction between polymers and gases is low

Interaction between polymer and liquids is strong

Affinity ↑ → Swelling of polymer network ↑(considerable effect on transport)

chapter 5. transport in membrane -...

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