membranes ii model equations

Separation Processes Laboratory - Prof. Mazzotti - Rate Controlled Separations

Membranes II

Model Equations

Simple well-mixed module, solution-diffusion

model, reverse osmosis, gas permeation,

pervaporation


Outline

1. Simple well-mixed module

2. Solution-diffusion model equations

3. Reverse osmosis & water desalination

4. Gas permeation & process design considerations

5. Pervaporation & process design considerations

6. Comparison


Simple well-mixed module

Total mass balance :

Single species mass balance :

∞

∞

Control volume

Assumptions

• No chemical reactions

• No pressure drop

• Well-mixed in both sides

Nomenclature

• ሶn: molar flow rate

• yi: molar fraction of species i


Simple well-mixed moduleAssumptions




Nomenclature

• ሶn: molar flow rate

• yi: molar fraction of species i

Total mass balance :

Single species mass balance :

∞

∞

Control volume

• J is the total molar flux of permeating species:

• A is the membrane section area where mass transfer occurs


Solution-diffusion model

The solution-diffusion model is applicable.

The different compounds first dissolve in the

membrane matrix and then diffuse through the

membrane under a concentration gradient

=> Separation achieved due to differences in

the solubility of each compound into the

membrane material and their diffusivity

through the membrane

Dense membranes (pore diameter < 10Å)Porous membranes (pore diameter > 10Å)

The pore-flow model is applicable. The

different compounds are transported by a

pressure-driven convective flow through tiny

membrane pores

=> Separation achieved due to differences in

the steric hindrances between the compounds

molecules and the membrane material



Assumptions

• Isothermal process

• Constant pressure within the

membrane, equal to the

retentate pressure

• Components dissolved in

dense membranes act as if

they were a liquid

p

x

pF = pR

pP

Discontinuity in the pressure

δ

The flux happens due to a concentration gradient. Assuming the validity of Fick’s Law within the

membrane, one finds :

Integration over the

membrane thickness

where wi is the molar fraction of species i dissolved in the membrane



Since dissolved components in dense membranes act as if they were liquid, the fugacity of species i inside of

the membranes is given by :

Where :

• is the pure liquid fugacity of species i

• is the Poynting factor of species i

• is the vapor pressure of species i

By replacing wi,R and wi,P in the expression of the permeation flux using the definitions of the fugacities, one

can obtain the final expression of the solution-diffusion model :

Apply the iso-fugacity equilibrium condition at the fluid/membrane interface to obtain the

permeating flux of species i as a function of the retentate and permeate concentrations


Reverse osmosis

Osmosis (reverse or normal) refers to the use of a permselective membrane (membrane that is freely

permeable to water but much less permeable to salt) in order to separate a salt solution from pure water.

More generally, to derive the equations, we have to consider the separation of liquids.

Assumptions

• Local equilibrium at interfaces

• No mass transfer resistance in

the retentate and permeate

streams (bulk concentrations

are equal to membrane

interface concentrations)

Solution-diffusion model :

Fugacities of the gases at the membrane interface and local

equilibrium condition:


Reverse osmosis

The equation is simplified by assuming that the ratio between the activity coefficient is the same at the

retentate or at the permeate side:

: Sorption coefficient of

species i in the membrane

: diffusion coefficient of species i in the

membrane

: permeability of species i

in the membrane (extent

at which a species

dissolves and diffuses

through a membrane)

: permeance of species i in

the membrane


Application: water desalination

At t=0s, we have side (1) that has salt water and side (2) that has only

water. If both p1 and p2 are atmospheric pressure, then we have :

t = 0s

t = ∞s

p1 p2

Salt + H2O

(1)

H2O

(2)

Δπ

Jw

The osmotic pressure Δπ is the pressure difference necessary to

have a null water flux through the membrane, due to the

concentration differences in the two sides of the membrane :

• p1 – p2 < Δπ : Normal osmosis, water flows from the pure water

side to the sea-water side

• p1 – p2 = Δπ : Osmotic equilibrium, no flux since the osmotic

pressure is counterbalanced by the pressure difference

• p1 – p2 > Δπ : Reverse osmosis, water flows from the sea water

side to the pure water side : water desalination


Gas permeation

Gas permeation corresponds to the separation of gases. Typically, a high pressure gas mixture at pR is

fed to one side of the membrane while the permeate is removed on the other side at a lower pressure pP

Assumptions










permeability of species i

in the membrane

: sorption coefficient of species i in the membrane

: diffusion coefficient of species i in the membrane


Diffusion coeff. decreases

with molar volume (larger

molecules interact more

with the polymer chain)

Sorption coeff. increases

with molar volume (larger

molecules are usually

more condensable)

Permeability is proportional to the

product of Di and Ki. For natural rubber

membranes sorption dominates, for

glass membranes diffusion dominates.

Rubbery polymer: Qi

increasing with Vm

Glassy polymer: Qi

decreasing with Vm

In polymeric membranes, Di, Ki(g) and Qi typically depend on the molecular volume:

Diffusion coefficient, sorption coefficient and permeability


Gas permeation

Assuming the equality of sorption coefficients respectively on the retentate and permeate sides, and

considering that the Poynting correction factor is close to 1 for systems at relatively low pressure, we

have:

Where ; : Permeance of species i in the membrane

Selectivity: Pressure ratio:

Permeate concentrations from the permeating fluxes :

For a binary mixture (A+B), the equation becomes quadratic with respect to y1,P and can be solved:


Simple well-mixed module, binary gas mixture

For a binary mixture A-B ,we do not write the subscripts for the compounds anymore, but consider that

yA,P=yP and yA,R=yR. The complete model equations are :

Assumptions




∞

∞Known parameters (3) :

Unknown parameters (8) :

Equations (6)

2 degrees of freedom


Design considerations gas separation

We define characteristic dimensionless parameters for a membrane separation :

Selectivity: Pressure ratio: Stage cut:

Recovery: Purity:

Given the operating conditions (pressures, retentate stream properties, flow patterns), the membrane

technology (permeability, selectivity, thickness) and the upstream conditions (feed stream properties),

the process performance (recovery, purity and required membrane area) can be calculated.

For the simple well-mixed module :

Two degrees of freedom : in the design of a membrane

process, two parameters can be fixed (e.g: Re and Pu)

Separation Processes Laboratory - Prof. Mazzotti - Rate Controlled Separations16

(φ >> α) selectivity limited (α >> φ) pressure-ratio limited

Design considerations : operating conditions

Selectivity limited region

Pressure ratio limited region

Let :


Selectivity and permeability (in Polymeric Membrane Materials) have a pareto-optimum, obtained fromempirical observations.

,log log logi i jP k m

(P is obtained in [barrer] )

Robenson, L. M., The Upper Bound Revisited, 2008.

Design considerations : membrane technology

For materials close to the

Upper Bound, it is not

possible to increase

simultaneously the

selectivity and the

permeance.

=> Trade-off between

increasing the purity or

the stage-cut


Design considerations : process performance

The recovery and purity maps on the A – β shows

the limitations of a one stage membrane process :

• Recovery and purity behave oppositely against

the membrane area. The recovery increases

while the purity decreases due to the

permeation of the least permeant component

along the module.

=> High permeate purity is only achievable at

low recovery rate, which is not economical

• An increase in selectivity (decrease in

permeability) leads to an increase in purity

(decrease in recovery).

• For large selectivity and pressure ratio, the

permeate purity is almost independent from the

membrane area


Applications


Pervaporation

Pervaporation can be considered as an intermediate separation process between gas separation and

reverse osmosis processes : the retentate is in liquid phase and the permeate is in gas phase. The

feed pressure is sufficient to maintain liquid state on the retentate side even as the composition is

changing. The permeate pressure is fixed to a value below the saturation pressure of the permeating

mixture, often applying vacuum.

The isothermal assumption will be assumed for simplicity, but in reality an energy balance equation should

be coupled with material balances to account for the heat of vaporization.

Assumptions











Pervaporation

The solution-diffusion model equation for pervaporation becomes:

: gas sorption coefficient of species i in the membrane

: liquid sorption coefficient of species i in the membrane

: diffusion coefficient of species i in the membrane

: Henry constant solubility for species i (liquid-vapor equilibrium property)

: gas permeability of species i in the membrane

Considering that the Poynting correction factor is close to 1 for systems at relatively low pressure,

we have:


Pervaporation: Separation Factor

In pervaporation, the separation

factor is a product of two terms: One

represents the contribution of

relative volatility, and the other the

contribution of the membrane.

Now we have three factors: the

contribution of relative volatility, the

selectivity of the membrane and the

vapor pressures of feed and

permeate (operating parameters).

We can define the vapor pressures of the liquid retentate (pi,R) and of the permeate (pi,P) :

The separation factor indicates how well was the component i separated compared to component j.

To note that we have :


Species i is preferred and dilute :

23

Separation Factor: Limiting Scenarios

1. High selectivity scenario α >> (pR/pP) :

All these effects are dominated by the

fact that α is not constant, but usually

a strong function of yR:

α = α (yR)

Hence it is possible to have both

α>>(pR/pP) and (pR/pP)>>α for the

same membrane and ratio pR/pP,

simply by varying yR.

2. High pressure ratio scenario (pR/pP) >> α

:

3. Low pressure ratio, low selectivity:

If strong vacuum is applied:


Pervaporation: Combined Effects

Hydrophilic membrane material:

• High Henry‘s constant for water,

low Henry‘s constant for Ethanol

• Strong swelling if yH2O,R >> yEtOH,R

• High permeance and rather low

selectivity if yH2O,R >> y EtOH,R

• No swelling if yEtOH,R >> yH2O,R

• Low permeance and rather high

selectivity if yEtOH,R >> yH2O,R

This leads typically to y(x) curves as

the ones shown in the figure (for

different permeate pressures)


Design considerations for pervaporation processes

• The minor component should permeate

(always favorable in membrane separation)

• To benefit from high selectivities, one should start from highly concentrated feeds

(hence hybrid processes are favorable, e.g. distillation + membrane separation)

• Working at elevated temperatures facilitates the process

(the latent heat of vaporization has to be provided with the feed)

• The pressure ratio is usually increased by applying vacuum on the permeate side

(e.g. by cooling condensation of the permeate stream)

• Mass transfer resistance through the surface boundary layer has to be considered

(especially if high purity of the retentate stream is desired)


Dehydration of Ethanol

• Distillation of 60% EtOH-water feed

• Products are the azeotrope (95.6%

EtOH, top) and water (bottom)

• Pervaporation of the azeotrope yields a

retentate of > 99.5% EtOH (product)

• The permeate (25% EtOH) is condensed

and recycled to the distillation column

• Similar setups for dehydration of other

azeotrope forming organic solvents

Other pervaporation processes :

• Removing VOC‘s from water using hydrophobic membranes (e.g.

silicone rubber)

• Separation of organic solvent mixtures in cases where distillation is

difficult (Methanol / MTBE), (Benzene / Cyclohexane)


Comparison of osmosis, gas permeation and pervaporation

Solution-diffusion model equations

pP,sat


Comparison of osmosis, gas permeation and pervaporation

membranes ii model equations

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