Gas Membrane separation modeling - additional theoretical information

Model specification

Concentrated parameters models

Units with one-two connectivity

The simpliest unit for membrane separations of a gas stream can be represented as a black-box divided into two compartments, with one inlet (supply) and two outlets (retentate and permeate):

The quantities involved are:

• Input to the high-pressure compartment (gas mixture to be separated): .

• Transfer of matter from the high-pressure compartment to the low-pressure compartment: F, .

• Output from the high-pressure compartment (retentate): .

• Output from the low-pressure compartment (permeate): .

All flows and fractions are on molar basis.

Known quantities

If the number of components is , the following quantities are considered known: and .

Unknowns

The unknowns are and .

Equations

Molar fractions must meet the closure conditions (2 equations):

• retentate molar fractions: (1)

• molar fractions of the permeate: (2)

The molar mass balances for each component can be expressed as ( equations):

(3.A)

A new variable is introduced to represent the effective pressure in the retentate compartment, assumed to be equal to the arithmetic mean between the inlet and outlet pressures.

(4)

The effective pressure in the permeate compartment is equal to the permeate outlet pressure .

The equations for the transport through the membrane of each component are:

(5)

here the driving force is expressed as a function of the effective mole fractions on the retentate side and permeate side and , which will be expressed later as appropriate functions of the inlet and outlet mole fractions of the two compartments: and (see the chapter “Effective compositions”).

The equations for the transport of each component can be rewritten as:

(5.A)

You have a total of equations (1, 2, 3.A and 5.A) that correspond to the unknowns.

Dimensionless formulation

It was found convenient to reformulate the problem by introducing new variables in order to make it dimensionless.

Based on the overall mass balance:

(6.A)

one can introduce a new dimensionless variable (stage-cut), defined as the fraction of the feed that permeates through the membrane:

(7.A)

Consequently, we obtain:

(8)

and:

(9)

Similarly, based on the partition of the i-th component, the single component stage-cut () can be defined:

(10.A)

Adding up the expression upon all the components gives:

(11.A)

Two additional useful dimensionless quantities are

1. , the ratio of effective pressures across the membrane:

   (12)

2. and , the permeation factor of the i-th component:

   (13.A)

The transport equation for the single component (5) can then be rewritten as:

(14.A)

Units with two-two connectivity

In some cases, gas separation units operate with a second inlet to the low-pressure compartment (permeate sweep), in which case the fundamental unit can be represented as a black box with two inlets (feed and permeate sweep) and two outlets (permeate and retentate):

This representation is also more convenient for discretizing units with axial flow on the permeate side; note that in the case of concentrated parameter modeling it is not important whether the permeate and retentate side flow have equal or opposite directions.

The quantities involved are:

• Input to the high-pressure compartment (gas mixture to be separated):

• Input to the low-pressure compartment (permeate sweep):

• Transfer of matter from the high-pressure compartment to the low-pressure compartment: ,

• Output from the high-pressure compartment (retentate):

• Output from the low-pressure compartment (permeate):

Again, all flows and fractions are on molar basis.

Known quantities

If the number of components is , the following quantities are considered known: and .

Unknowns

The unknowns are and .

Equations

In this case the equations to be considered will be:

1. the closure conditions on the mole fractions of retentate (1) and permeate (2)

2. the molar mass balances for each component:

    (3.B)

3. the transport equations across the membrane for each component:

   (5.B)

Again, the driving force is expressed as a function of the effective molar fractions at the retentate side () and at the permeate side (), defined in turn as appropriate functions of the molar fractions exiting and entering the two compartments: and .

We have a total of equations (1, 2, 3.B and 5.B) corresponding to the unknowns.

Dimensionless formulation

Again it seems convenient to reformulate the problem by introducing new variables in order to make it dimensionless.

Based on the overall matter balance:

(6.B)

and defining a new variable to represent the combined flow rate entering the subunit:

(15)

one can introduce a new dimensionless variable defined as:

(7.B)

It is noted that in this case does not represent the stage-cut, but rather a separation yield.

Similarly, based on the balance of the individual i-th component, a new variable (single component separation yield) can be defined:

By defining equal to the average molar fraction in the combined feeds:

(16)

the previous equation can be expressed more compactly in the form:

(10.B)

Adding up the expression upon all the components gives:

(11.B)

Redefining the permeation factor of the i-th component to account for the permeate sweep:

(13.B)

the transport equation for the single component (5.B) can be rewritten as:

(14.B)

Note that if we place we obtain:

, , and

thus returning to the equations related to one-two connectivity.

Effective compositions

In the previous paragraphs, the dependency of the retentate side () and permeate side () effective mole fractions on the molar fractions at the outlet and inlet of the two compartments were left undefined: or if the permeate sweep is present, and .

These effective molar fractions depend on the fluid dynamic assumption being made, and can be imposed according to three approaches:

• CSTR;

• “RTSC”;

• Linear average.

CSTR

The Continuously Stirred Tank Reactor (CSTR) model, from a fluid-dynamic point of view, corresponds to perfect mixing and leads to setting the average concentrations equal to those at the outlet.

The situation can be schematized as in the figure:

where the horizontal lines in the subunits represent the molar fraction of a component, equal to the outlet fraction from each subunit.

For example, for the retentate subunit with the CSTR assumption, one posits . This leads to an implicit, thus more complex system of equations, which rules out an analytic solution if .

“RTSC”

We denote by “RTSC”, a kind of reverse CSTR.

The situation can be schematized as in the figure:

where the horizontal lines in the subunits represent the mole fraction of a component, equal to the entry fraction in each subunit; the exit fraction from each subunit will be different.

For example, with the “RTSC” hypothesis for the retentate compartment, is posed.

This is a little-used assumption in process modeling, but it leads to an interesting simplification of the equations.

Linear average

In the CSTR and “RTSC” models, the concentration profile along the membrane is stepped, that is, for each subunit a constant mole fraction is considered, (step function).

A more realistic model is obtained by considering a linear profile:

The approximation of linear profiles leads to assuming arithmetic averages between input and output as effective molar fractions.

For example, for the retentate compartment we pose . This model also leads to an implicit system of equations that does not allow for an analytic solution.

Analysis of the degrees of freedom

It should be noted that the operating pressure of the retentate compartment is not always an independent variable. In fact, if there is excess permeant area or for low flow rates, complete permeation of the entire inlet flow may occur.

In such a case, a distinction must be made between the inlet pressure of the feed and the pressure at which the retentate compartment equilibrates, and if necessary a localized inlet pressure drop can be assumed, to bring down the pressure from that at the inlet to that determined by the flow through the P membrane.

Although this is a highly unlikely circumstance under real operating conditions (in which one rather operates with insufficient area and maximum flow), it must be taken into account when writing the equations and defining the dimensionless variables.

Once the conditions of complete permeation are exceeded, for even lower inlet flow rates a reverse flow phenomenon may come about whereby one of the fluid will tend to enter from the outlet of the retentate. To prevent this reverse flow, a check valve can be assumed present at the outlet to the retentate.

Excluding flow reversal, and considering the one-two configuration, the conditions for complete permeation will be discussed below, in terms of the dimensionless parameter critical pressure ratio , for which all the incoming flow is transferred to the permeate compartment.

Single component

In the case of a pure component, the mole fraction will be unity on both sides of the membrane along its entire length, so equation (5) becomes:

Applying the dimensionless equation (11.A) we get:

(17)

Substituting the value into the expression (17), we can derive the value of :

The area then corresponding to the complete permeation () will be:

Two components

The (14.A) for the case gives rise to the two expressions:

The value of can be determined from these expressions, imposing = and :

at this point by explicating , exploiting normalizations and summing:

from which:

this expression can be compared with that for the case derived above:

If , i.e., component 2 is faster, , then the critical pressure ratio can be calculated by an equation similar to that valid for a single component, based on the permeation factor of the fastest component (2), increased by the term due to the slowest component (1). If we express the permeation factor of the faster component based on that of the slower component using the definition of selectivity or separation factor:

and

the threshold value for complete permeation becomes:

More than two components

In this case, again starting from (14.A) and taking advantage of normalization, we obtain:

…

then making the explicit and summing up the equations gives:

this expression can be compared with that for the case derived above:

Again assuming that the components are in order of decreasing permeation rate:

and in particular > {}, for each i, so can be calculated on the basis of the permeation factor of the fastest component (), increased by the terms due to the slowest components (1 . . ). If we express the permeation factor of the fastest component on the basis of the i-th slowest component using the definition of selectivity or separation factor:

and  (18)

the critical pressure ratio for complete permeation becomes:

.

This expression can be interpreted as follows:

• If all components permeate at the same rate () the critical pressure ratio that determines complete permeation is controlled solely by the first term;

• The sense of the first term is that, for high permeation factors, the critical pressure ratio tends to 1 indicating that conditions of complete permeation are easy to achieve;

• The sense of the second term is that when very slow components are present ( >> 1), they can become controlling and prevent complete permeation, provided that the product between selectivity and the inlet mole fraction is comparable to the permeation factor of the fastest component: .

Assembly of the complete concentrated parameter model

By substituting expressions of and according to the type of average assumed for the concentrations in (14.A) for one-two connectivity or in (14.B) for two-two connectivity, an expression of can be obtained.

Only in certain cases is analytical resolution possible. In other cases, one proceeds numerically by estimating a value for the variable or alone, which substituted into equation (10.A) (i.e., into (10.B)) will give the value of or ).

Through such an expression it is possible to compute the partitions of all components if the overall partition (or ) is known.

This allows us to simplify the iterative procedure proposed in [Davis2002] as follows:

1. estimation of the overall stage-cut or separation yield

2. calculation of the stage-cuts / separation yield of the components / .

3. iterative resolution of the equation i.e. (11.B).

In this way the algebraic problem is greatly simplified, since there is no need to solve an algebraic system in unknowns, but only one nonlinear equation in one unknown.

One-two models

Global “RTSC” fluidodynamics, two components

We posed and .

From equations (2) and (14.A) we obtain a 3-equation system in 3 unknowns (, and ):

 (19)

(20)

 (21)

We substitute (21) into (20):

(22)

We derive from (19):

(23)

We substitute the expression of from (23) into (22):

(24)

The latter is an equation in the y~1~ unknown only; we then proceed with its explication:

 (25)

Obtaining a 2nd degree one-variable equation , which is easily solved. Two solutions will be obtained, only one of which is physically acceptable.

Global CSTR fluidodynamics, multi-component

We pose and .

To determine we proceed from equation (14.A) in which we substitute values for the pattern of and :

is obtained from equation (3.A):

We then proceed by substituting the value of :

Solving with respect to will yield that:

(26)

Global “RTSC” fluidodynamics, multi-component

It is posited and .

From equation (14.A) we get:

from which the ratio / can be made explicit:

Starting from the definition of the stage cut for the single component () (10.A), it is possible to obtain the sought expression of = f() which allows iterative resolution:

(27)

Global CSTR fluidodynamics, two components

Substituting (26) into (11.A) for 2 components ( and ) gives:

which is the equation in Θ to be solved.

With respect to the one-two “RTSC” two-component case, we obtain a third-degree polynomial instead of a second-degree polynomial, which is theoretically (but not practically) analytically solvable.

Global linear fluidodynamics, multi-component

The assumption underlying this model implies that

Equation (14.A) becomes:

The value of is obtained from (3.A):

and is substituted into the equation:

Explicating gives:

(28)

From this we obtain that is:

(29)