Chapter 7. Polarization Phenomena & Membrane Fouling (Part...

Chang-Han Yun / Ph.D.

National Chungbuk University

November 18, 2015 (Wed)

Chapter 7. Polarization Phenomena & Membrane Fouling

(Part I)

2 Chapter 7. Polarization Phenomena & Membrane Fouling Chungbuk University

Contents

Contents Contents

7.5 Flux Characteristics in Pressure Driven Membrane Operation

7.4 Pressure Drop

7.3 Turbulence promoters

7.2 Concentration Polarization in Pressure Driven Processes

7.1 Introduction

7.6 ∼ 7.8 Concentration Polarization Model


7.1 Introduction

Concentration polarization and Fouling ⇨ flux↓over time

In MF/UF : very severe

In gas separation and pervaporation : less severe

<Figure 7-2> Overview of various types of resistance

towards mass transport across a membrane in

pressure driven processes.

<Figure 7-1> Flux behavior as a function of time.

Resistance

Rp : Pore blocking

Ra : Adsorption

Rm : Membrane

Rg : Gel layer forming

Rcp : Concentration polarization


7.1 Introduction

Additional resistances on the feed side to the transport ⇨ cause of flux decline

Concentration polarization(Rcp)

Adsorption(Ra)

Gel layer formation(Rg)

Plugging of the pores(Rp)

※ Feed quality and process ⇨ determine the extent of these phenomena

Convective flux through the membrane

(7-1)

For pressure driven processes(MF, UF, NF, RO)

(7-2)

※ In the ideal case, Rtot = Rm (the membrane resistance)


7.1 Introduction

Concentration polarization resistance(Rcp)

Retained solutes ⇨ accumulated at near membrane surface

⇨ concentration of retained solutes near membrane↑ ⇨ form resistance

Gel layer resistance(Rg)

Concentration of the accumulated solute molecules = so high

This mainly happens when the solution contains proteins.

Pore-blocking resistance(Rp) and Adsorption resistance(Ra) in porous membrane

Some solutes to penetrate into membrane and block pores ⇨ Rp

Finally leading to adsorption phenomena

Adsorption : on membrane surface as well as within pores themselves



Solvent : permeate through the membrane

Solutes : (partly) retained by the membrane

Solute balance at steady state

Convective transport of solute

(bulk → membrane surface)

= permeation through membrane

+ diffusive back flow(membrane → bulk)

(7-3)

cp < cb ⇨ Concentration polarization

<Figure 7-3> Membrane separation; the basic concept.

<Figure 7-4> Concentration polarization.



Integrate Eq(7-3) with boundary conditions

BC 1 : x = 0 → c = cm

BC 2 : x = δ → c = cb

(7-4)

(7-5)

Mass transfer coefficient(k), (7-6)

Define intrinsic retention(Rint), (7-7)

Eq(7-5) → (7-8)

Concentration polarization modulus : cm/cb

J↑ ⇨ cm/cb ↑

Rint↑ ⇨ cm/cb ↓

k↓ ⇨ cm/cb ↑



When the solute is completely retained by the membrane (Rint = 1.0 and cp = 0) :

Eq(7-5) and Eq(7-8) ⇨ (7-9)

※ Basic equation for concentration polarization

• Membrane part ⇨ Flux (J)

• Hydrodynamics ⇨ Mass transfer coefficient (k)

Mass transfer coefficient(k) dependency on hydrodynamics of the system

Sherwood number (Sh)

(7-10)

Where Re = Reynolds number, (7-11)

Sc = Schmidt number, (7-12)

a, b, c and d = constants

ν = kinematic viscosity, η = dynamic viscosity, dh = hydraulic diameter

v = flow velocity, L = length of the tube or channel, D = diffusion coefficient



Hydraulic diameter(dh)

For a pipe (hollow fibers, capillary membranes or tubular membranes)

dh = 4 A/S = 4 (π/4)∙d2/π∙d = d

For a rectangular slit (plate-and-frame) of height(h) and width(w)

dh = 4 w∙h/2(w+h) = 2 w∙h /(w+h)

Eq(7-10) → k = f(v, D, η, ρ, module configuration(dh, L))

Important parameters : v, D

if module configuration = constant ⇨ k = f (v, D)

MF and UF when comparison with RO or gas separation and pervaporation

D of macromolecules(or SS) = low

fluxes = relatively high

Appearance Laminar Turbulent

Tube Sh = k∙dh/D = 1.62 (Re∙Sc∙dh/L)0.33 Sh = 0.04 Re0.75 Sc0.33

Channel Sh = 1.85 (Re∙Sc∙dh/L)0.33 Sh = 0.04 Re0.75 Sc0.33

[Table 7-1] Mass transfer coefficients in various flow regimes

Concentration Polarization of MF = severe



To minimize concentration polarization

Flux (J)↓

Mass transfer coefficient (k)↑

To enhance Mass transfer coefficient(k)

Diffusion coefficient(D)

※ increased only by increasing temperature

Flow(solvent) velocity(v)↑

※ Flux of solvent↑ ⇨ Polarization↑

Module configuration to increase turbulence

※ Reynolds number(Re) : Turbulent flow > Re = 2,000 > Laminar flow

Concentration Polarization↓

k↑

<Figure 7-5> Fully developed laminar and

turbulent velocity profiles in a pipe or slit.



To increase of turbulence

velocity↑

using turbulence promoters to break the boundary layer

• adapting corrugated membranes

• pulsating flow

feed temperature↑ ⇨ D of solutes↑, η of solvent↓ ⇨ k↑ ⇨ Polarization↓



Concentration profile

Pressure driven process

• Convective process

• Solute retained

Concentration driven process

• Diffusive process

• Fast solute permeating

7.2.1 Concentration

Profile

Transport Mechanism Process Resistance Concentration Profile

Convective transport

(Solute retained)

MF

UF

NF

RO

rm > rbl <Figure 7-4> or

<Figure 7-6a>

Diffusive transport

Gas separation

Pervaporation

Dialysis

Carrier mediated transport

rm < rbl <Figure 7-6b>

rm > rbl ⇨ developing concentration profile

rm < rbl ⇨ developing concentration profile


7.2 Concentration Polarization in Pressure Driven Processes 7.2.1 Concentration

Profile

<Figure 7-6> Concentration profiles

in membrane processes

a) convective transport

b) diffusive transport

Membrane operation Influence Origin

RO

UF

MF

gas separation

pervaporation

electrodialysis

dialysis

diffusion dialysis

carrier mediated transport

moderate

strong

strong

(very) low

low

strong

low

low

moderate

k large

k small / J large

k small / J large

k large / J small

k large / J small

-

J small

J small / k large

J large※ / k large

[Table 7-2] Consequences of concentration polarization


7.2 Concentration Polarization in Pressure Driven Processes 7.2.1 Concentration

Profile

Process Concentration

Polarization

Flux

(J)

Permeating

Species

Diffusivity

(m2/sec)

k

(= D/δ)

MF or UF Severe High Macro

solutes 10－10 ∼ 10－11 Low

RO Less severe Lower Micro

solutes 10－9 Higher

Gas separation

Pervaporation Low or negligible Low

Micro

solutes 10－4 ∼ 10－5 High

Dialysis

Diffusion dialysis Not generally severe Low

Micro

solutes 10－9 Higher

Concentration polarization of pervaporation compared with gas separation

J : Pervaporation < Gas separation

k : Pervaporation < Gas separation

※ VOC removal from water : c = very low & selectivity = very high ⇨ show more severe effect

Plate-and-frame or Spiral wound module

Spacer materials ⇨ turbulence↑ ⇨ k↑

Concentration polarization :

Pervaporation > Gas separation

※ Facilitated membrane and Membrane contactors : Moderate ※ Electrodialysis : very severe


7.3 Turbulence promoters

Hydrodynamic performance ⇨ characterize k

Flow conditions (v, η, ρ, D) and module geometry ⇨ determine k by correlations

Turbulence promoters

Mass transfer↑

Correlation through Sherwood Number(Sh) when k ≠ f(spacer)

Sh = k∙dh/D = 0.0096 Re0.5 Sc0.6 (7-14)

<Figure 7-7> Schematic drawing of a spacer (upper figure) and a spacer filled channel (lower figure).

k↑(ex, feed spacer in spiral wound module)

Δp & Energy↑ k↑

concentration polarization↓(wall concentration↓) J↑


7.3 Turbulence Promoters

Mass transfer coefficient(k) when spacer is adapted for laminar flow region(Re < 2000)

(7-15)

where Δl is the distance between successive corrugations

<Figure 7-7> Schematic drawing of a flow channel with turbulence promoters


7.4 Pressure Drop

Well developed flow (7-16)

Fanning Equation (relation between Δp ↔ v) for a channel and a straight pipe

Δp = f (S∙L / A) 0.5 ρv2 (7-17)

where L = length of the tube, ρ = density of the liquid (fluid), v = flow velocity

f = friction factor, S = circumference, A = cross-sectional area

Pressure drop by introducing an spacer or turbulence promoter

(7-18)

where Δl = distance between successive turbulence promoters

[Table 7-3] Friction factors in various systems

Shape

Flow Region Channel Tube

Laminar f = 24 Re－1 f = 16 Re－1

Turbulent f = 0.133 Re－0.25 f = 0.079 Re－0.25

Fiction factor = f(Re) weakly

in turbulent region

⇨ Δp = f(v) strongly in turbulent region



Water flux for pure water through a porous membrane in pressure driven processes

(7-19)

where Rm = hydrodynamic resistance of the membrane

Hydrodynamic permeability, Lp = 1 / η∙Rm ⇨ J = LpΔp

Hydrodynamic resistance Rm = membrane constant ≠ f(feed composition or pressure)

Water flux for solution through a porous membrane in pressure driven processes

J ∝ Δp before finite Δp

J = constant after finite Δp

J∞ = Limiting flux = f(cb, k)

where cb = concentration in the bulk of the feed

k = mass transfer coefficient

(7-20)

from (7-9)

J ∝ Δp for pure water

<Figure 7-9> Flux as a function of Δp both for

pure water and for a solution.



(7-20)

Slope = - k

<Figure 7-11> Limiting flux (J∞) plotted as a function of the logarithm of the bulk concentration.

<Figure 7-10> The flux as a function of the applied

pressure for different cb and different k.

Feed concentration(cf)↑ at constant k ⇨ J∞↓

Mass transfer(k)↑ at constant cf ⇨ J∞↑

Typical for UF

Lesser extent for MF

Not for RO

Description of concentration polarization

Same for both UF and RO formally

Difficulty : UF > RO

(∵ properties of concentrated

macromolecular solutions)


7.6 Gel Layer Model

Concentration polarization in UF : very severe

flux through the membrane = high

diffusivity of the macromolecules = rather low

retention is normally very high

Gel concentration

depends on

independent of the bulk concentration

Gel formation

reversible or irreversible

very important factor in membrane cleaning

solute concentration at surface

= very high

<Figure 7-12> Concentration polarization and gel layer formation.

• Size

• Shape

• Chemical structure

• Degree of solvation


7.6 Gel Layer Model

Gel layer model

Good model for theory of concentration polarization and limiting flux behavior in UF

Describing the occurrence of limiting flux

<Assume> Solute = completely retained by the membrane

Δp↑ ⇨ Solvent flux(Jw↑) until a critical concentration ⇨ reach at gel concentration(cg)

Δp↑ further ⇨ gel layer = become thicker and/or compacter

• Resistance of the gel layer (Rg)↑ to solvent transport ⇨ constant flux

• Gel layer becomes the limiting factor in determining flow

Total resistance (see <Figure 7-12>)

Gel layer resistance(Rg) + Membrane resistance(Rm)

(7-21)

<Assume> Gel concentration = constant across gel layer

Slope = - k

Intercept on the abscissa (J∞ = 0) = ln(cg) <Figure 7-13> J∞ verse ln(cb)


7.6 Gel Layer Model

Drawbacks of gel layer model

cg = not constant

= f(cb, cross flow velocity)

cg for a given solute = varying widely

k = assumed to be constant

D of the macromolecular solute = f(concentration)

Gel formation phenomena = Different from characteristics of macromolecule

proteins = form a gel easily

dextranes = do not gel so easily even at very high concentrations


7.7 Osmotic Pressure Model

J = high

Rejection = high

k =low

the flux equation

(7-22)

Here, ΔP = hydraulic pressure difference

Δπ = osmotic pressure difference

cm (not by cb) ⇨ determine Δπ

Relationship between concentration ↔ π

For dilute low MW solutions ⇨ linear relationship(van't Hoff relationship)

In general, exponential rather than linear

π = a∙cn (7-23)

where a = constant

n = exponential factor > 1

concentration of macromolecule

at surface = very high

osmotic pressure cannot

be neglected anymore



For semi-dilute or concentrated polymer solutions : n > 2

a and n = f(MW, type of polymer)

Another expression for non-ideality : virial expansion

(7-24)

Non- linear relationship between flux ↔ ΔP

By combining Eq(7-23), Eq(7-22) and Eq(7-9)

(7-25)

(7-26)

<Figure 7-14> Schematic drawing of π as a function

of the concentration for various values of n.



Combining Eq(7-23) and (7-25), and substituting the result into Eq(7-26) leads to

(7-27)

or (7-28)

Two extreme cases

• Δπ = very high ⇨ ∂J/∂ΔP → 0

ΔP ↑ ⇨ J = not increase : J∞ region

• Δπ → 0 ⇨ ∂J/∂ΔP = 1/ (η Rm)

Multiplying with η Rm to Eq(7-28) ⇨ two dimensionless numbers :

(7-29)

and (7-30)

where superscript o : pure solvent



ηRm(∂J/∂ΔP)

Ratio between slope of the plot of J versus ΔP and that of pure solvent flux(Jo) versus ΔP

Measurement of effectiveness of ΔP

Maximum slope = (∂J/∂ΔP)o

At high ΔP

• ∂J/∂ΔP ↓ ⇨ Rm (∂J/∂ΔP) ↓ : effectiveness of an increase in ΔP↓ (Rm ↑ or η Rm ↑ in fact)

• Δπ↑

(Δπ n)/( η Rm k)

Ratio between osmotic pressure resistance

and membrane resistance

<Figure 7-15> Effectiveness of pressure increase as a function of the

ratio between the osmotic resistance and the membrane resistance.

Rm↑

Rm (∂J/∂ΔP) ↓



Influence of Δπ as a function of ΔP

Eq(7-25) ⇨ J as a function of ΔP(<Figure 7-16>)

J∞ can be attained at even lower pressures at low Rm than that in <Figure 7-16>

<Figure 7-16> Calculated values of J plotted as a function of the applied ΔP

at varying cb and the following parameters:

a=100; n=2; Rm=5×105 bar∙sec/m; k=2×10－6

cb



Relationship between Gel layer model ↔ Osmotic pressure model

Gel layer model

• Plot of J versus In(cb) ⇨ straight line (slope = －k)

Osmotic pressure mode

• Eq(7-25) ⇨ similar J versus In(cb) relationship, (7-31)

• Rm↓ ⇨ (Δπ n)/(η Rm k) ≫ 1 ⇨ the right-hand side of Eq(7-31), ∂J/∂ln(cb) = slope = －k

J∞ = 0 ⇨ ΔP = Δπ, High values of (Δπ n) / (η Rm k) ⇨ J↓ (∵ osmotic pressure effects)

Factors leading to high (Δπ n) / (η Rm k)

• High ΔP or Low Rm ⇨ High J

• High bulk concentration(cb)

• Low mass transfer coefficient(k)

• High value of n(macromolecular solute)

<Figure 7-17> Flux J∞ as a function of the

concentration in the bulk, cb.


7.8 Boundary Layer Resistance Model

<Assume> 100 % rejection of solutes and steady-state

Convective flow of solute towards membrane surface = Diffusive flow back to bulk of feed

∴ 100% rejection ⇨ average velocity of the solute molecules in the boundary layer = zero

Boundary layer resistance model(<Figure 7-18>)

Concentration↑ ⇨ hydrodynamic resistance in boundary layer (Rbl)↑

Rbl & Rm ⇨ solute flux(J) with assuming no gelation

(7-32)

Boundary layer = considered concentrated solution

Permeability(P) of stagnant layer = f(c, MW of solute)

• Rbl for macromolecular (UF) : much greater

• Rbl for low MW (RO) : lower relatively

P of the solvent = f[distance(x)] where 0<x<δ

(∵ concentration profile in boundary layer)

<Figure 7-18> Schematics of boundary layer resistance model.



phenomenological Darcy equation

osmotic gradient

(driving force for solvent flow in boundary layer)

Integration over the boundary layer leads to

(7-34) where (7-35)

Simplifying Eq(7-34) ⇨ (7-36)

Sedimentation measurement ⇨ Permeability (P) ⇨ Rbl

permeability ↔ sedimentation coefficient

(7-37)

Where v1 = partial molar volume of the solvent

v2 = partial molar volume of the solute

c = solute concentration.

volume flux, (7-33)

<Figure 7-19> Correlation between the sedimentation of

a solute and the permeation of a solvent.



Ultracentrifugation ⇨ Sedimentation coefficient(s)

(7-38)

where dr/dt = sedimentation velocity of that particle

ω2r = acceleration in centrifugal field

Concentration dependence of sedimentation coefficient

(7-39)

Substitution of Eq(7-37) → Eq(7-35)

(7-40)

(7-41)

<Assume> 100 % rejection ⇨ c(x) = cb exp (J∙x / D) (7-9)



Since integration of P(x)－1 over the boundary layer gives

<Assume> diffusion coefficient D = constant ≠ f(c)

(7-42)

Substitution of Eq(7-9) → Eq(7-41) and integration over the boundary layer gives:

(7-43)

⇨ ΔP, J, Rm, cb, k, s and D ⇨ calculate Rbl

Error in k ⇨ large effect on the calculated Rbl ⇨ need exact k ⇨ difficult for application

(∵ since cm is related to k via an exponential function)

※ Boundary layer resistance model = osmotic pressure model on the point of concept

(7-44)

※ Difficulties in practical use : boundary model > osmotic model

(Independent measurements are essential for both models.)

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