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
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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.
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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).
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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
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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)
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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)
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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
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5.3 Non-equilibrium Thermodynamics
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.
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5.3 Non-equilibrium Thermodynamics
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)
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5.3 Non-equilibrium Thermodynamics
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)
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5.3 Non-equilibrium Thermodynamics
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)
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5.3 Non-equilibrium Thermodynamics
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)
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5.3 Non-equilibrium Thermodynamics
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
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5.3 Non-equilibrium Thermodynamics
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)
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5.3 Non-equilibrium Thermodynamics
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
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5.3 Non-equilibrium Thermodynamics
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)
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5.3 Non-equilibrium Thermodynamics
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
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5.3 Non-equilibrium Thermodynamics
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)
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5.3 Non-equilibrium Thermodynamics
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)
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5.3 Non-equilibrium Thermodynamics
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
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5.3 Non-equilibrium Thermodynamics
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)
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