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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):

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The quantities involved are:

All flows and fractions are on molar basis.

Known quantities

If the number of components is $n_c$, the following $5+2\cdot n_c$ quantities are considered known: $P_I, P_R, p, n_0, A, Q_i$ and $z_i$.

Unknowns

The unknowns are $2+2\cdot n_c: n_P, n_R, x_i$ and $y_i$.

Equations

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

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

$n_0\cdot z_i=n_R\cdot x_i+n_P\cdot y_i$ (3.A)

A new variable $P$ 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.

$P=\frac{P_I+P_R} 2$ (4)

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

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

$F_i=Q_i\cdot A \cdot (P\cdot X_i-p\cdot Y_i)$ (5)

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

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

$n_p\cdot y_i=Q_i\cdot A\cdot (P\cdot X_i-p\cdot Y_i)$ (5.A)

You have a total of $2+2\cdot n_c$ equations (1, 2, 3.A and 5.A) that correspond to the $2+2\cdot n_c$ 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:

$n_0=n_P+n_R$ (6.A)

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

$\Theta =\frac{n_P}{n_0}$ (7.A)

Consequently, we obtain:

$\frac{n_R}{n_0}=1-\Theta $ (8)

and:

$\frac{n_R}{n_P}=\frac{1-\Theta }{\Theta }$ (9)

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

$theta_i=\frac{n_P\cdot y_i}{n_0\cdot z_i}=\Theta \cdot \frac{y_i}{z_i}$ (10.A)

Adding up the expression $theta _i\cdot z_i$ upon all the components gives:

$\Sigma \theta _i\cdot z_i=\Theta \cdot \Sigma \frac{y_i}{z_i}\cdot z_i=\Theta \cdot \Sigma y_i=\Theta $ (11.A)

Two additional useful dimensionless quantities are

  1. $\delta$, the ratio of effective pressures across the membrane:

    $\delta =\frac P p$ (12)

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

    $R_i=\frac{Q_i\cdot A\cdot p}{n_0}$ (13.A)

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

$y_i=\frac{R_i}{\Theta }\cdot (\delta \cdot X_i-Y_i)$ (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):

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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:

Again, all flows and fractions are on molar basis.

Known quantities

If the number of components is $n_c$, the following $6+3\cdot n_c$ quantities are considered known: $P_I, P_R, p, n_0, n_{ps}, A, Q_i, w_i$ and $z_i$.

Unknowns

The unknowns are $2+2\cdot n_c: n_P, n_R, x_i$ and $y_i$.

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:

    $n_0\cdot z_i+n_{PS}\cdot w_i=n_R\cdot x_i+n_P\cdot y_i$ (3.B)

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

    $n_P\cdot y_i=Q_i\cdot A\cdot (P\cdot X_i-p\cdot Y_i)+n_{PS}\cdot w_i$ (5.B)

Again, the driving force is expressed as a function of the effective molar fractions at the retentate side ($X_i$) and at the permeate side ($Y_i$), defined in turn as appropriate functions of the molar fractions exiting and entering the two compartments: $Y_i=f(y_i,w_i)$ and $X_i=f(x_i,z_i)$.

We have a total of $2+2\cdot n_c$ equations (1, 2, 3.B and 5.B) corresponding to the $2+2\cdot n_c$ 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:

$n_0+n_{PS}=n_P+n_R$ (6.B)

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

$N=n_0+n_{PS}$ (15)

one can introduce a new dimensionless variable $\Theta°$ defined as:

$\Theta°=\frac{n_P}{n_0+n_{PS}}=\frac{n_P} N$ (7.B)

It is noted that in this case $\Theta°$ 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 $\theta°_i$ (single component separation yield) can be defined:

$\theta_i°=\frac{n_P\cdot y_i}{n_{PS}\cdot w_i+n_0\cdot z_i}=\frac{\frac{n_P\cdot y_i}{N}}{\frac{n_{PS}\cdot w_i+n_0\cdot z_i}{N}}=\frac{\Theta°\cdot y_i}{\frac{n_{PS}\cdot w_i+n_0\cdot z_i}{N}}$

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

$\lambda_i=\frac{n_{PS}{cdot w_i+n_0{cdot z_i}{N}$ (16)

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

$\theta°_i=\frac{\Theta°\cdot y_i}{\lambda_i}$ (10.B)

Adding up the expression $\theta_i\cdot \lambda _i$ upon all the components gives:

$\Sigma \theta_i\cdot \lambda_i=\Theta°\cdot \Sigma \frac{y_i}{\lambda_i}=\Theta°\cdot \Sigma y_i=\Theta°$ (11.B)

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

$R°_i=\frac{Q_i\cdot A\cdot p}{N}$ (13.B)

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

$y_i=\frac{R°_i}{\Theta°}\cdot(\delta\cdot X_i-Y_i)+\frac{n_{PS}\cdot w_i}{\Theta°\cdot N}$ (14.B)

Note that if we place $n_{PS}=0$ we obtain:

$\Theta°=\Theta$, $\theta°_i=\theta_i$, $\lambda_i=z_i$ and $R°_i=R_i$

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

Effective compositions

In the previous paragraphs, the dependency of the retentate side ($X_i$) and permeate side ($Y_i$) effective mole fractions on the molar fractions at the outlet and inlet of the two compartments were left undefined: $Y_i=f(y_i)$ or $Y_i=f(y_i,w_i)$ if the permeate sweep is present, and $X_i=f(x_i,z_i)$.

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

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:

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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 $X_i=x_i$. This leads to an implicit, thus more complex system of equations, which rules out an analytic solution if $n_c > 2$.

“RTSC”

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

The situation can be schematized as in the figure:

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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, $X_i=z_i$ 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:

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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 $X_i=\frac{x_i+z_i}{2}$ . 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 $P_I$ 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 $\delta{CP}$, 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:

$F=Q\cdot A\cdot (P-p)$

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

$\Theta=R\cdot(\delta -1)$ (17)

Substituting the value $\Theta=1$ into the expression (17), we can derive the value of $\delta_{CP}$:

$\delta_{CP}=\frac{1+R}{R}$

The area then corresponding to the complete permeation ($A_{CP}$) will be:

$A_{CP}=\frac{n_0}{(\delta-1)\cdot Q\cdot p}$

Two components

The (14.A) for the case $n_c = 2$ gives rise to the two expressions:

The value of $\delta_{CP}$ can be determined from these expressions, imposing $y_i$ = $z_i$ and $\Theta=1$:

$(1+R_1)\cdot z_1=R_1\cdot \delta\cdot x_1$

$(1+R_2)\cdot(1-z_1)=R_2\cdot \delta \cdot(1-x_1)$

at this point by explicating $x_i$, exploiting normalizations and summing:

$1=\frac{(1+R_1)\cdot z_1}{R_1\cdot \delta}+\frac{(1+R_2)\cdot(1-z_1)}{R_2\cdot \delta}$

from which:

$\delta_cp=\frac{1+R_2}{R_2}+(\frac{1}{R_1}-\frac{1}{R_2})\cdot z_1$

this expression can be compared with that for the case $n_c = 1$ derived above:

$\delta_{CP}=\frac{1+R}{R}$

If $R_2>R_1$, i.e., component 2 is faster, $\frac{1}{R_1}-\frac{1}{R_2}>0$, then the critical pressure ratio $\delta_{CP}$ 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 $(\frac {1}{R_1}-\frac {1}{R_2})\cdot z_1$ due to the slowest component (1). If we express the permeation factor of the faster component $R_2$ based on that of the slower component $R_1$ using the definition of selectivity or separation factor:

$\alpha_{2,1}=\frac{Q_2}{Q_1}=\frac{Q_2}{Q_2}=\cdot A\cdot p}{n_0}=\frac{n_0}{Q_1}=\cdot A\cdot p}=\frac{R_2}{R_1}>1$ and $R_1=\frac{R_2}{\alpha_{2,1}}$

the threshold value for complete permeation becomes:

$\delta_{CP}=\frac{1+R_2}{R_2}+\frac{(\alpha_{2,1}-1)\cdot z_1}{R_2}$

More than two components

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

$(1+R_1)\cdot z_1=R_1\cdot \delta \cdot x_1$

$(1+R_2)\cdot z_2=R_2\cdot \delta \cdot x_2$

…

$(1+R_{n_c})\cdot (1-\sum_{n_c-1} z_i)=R_{n_c} \cdot \delta \cdot (1-\sum_{n_c-1} x_i)$

then making the $x_i$ explicit and summing up the $n_c$ equations gives:

$\delta_{CP}=\frac{1+R_N}{R_{n_c}}+\sum{n_c-1}(\frac{1}{R_i}-\frac{1}{R_{n_c}})\cdot z_i$

this expression can be compared with that for the case $n_c = 2$ derived above:

$\delta_{CP}=\frac{1+R_2}{R_2}+(\frac{1}{R_1}-\frac{1}{R_2})\cdot z_1$

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

$i > j \Leftrightarrow R_i > R_j$

and in particular $R_N$ > {$R_i$}, $\frac{1}{R_i}-\frac{1}{R_{n_c}}>0$ for each i, so $\delta_{CP}$ can be calculated on the basis of the permeation factor of the fastest component ($i = n_c$), increased by the terms $(\frac{1}{R_i}-\frac{1}{R_{n_c}})\cdot z_i$ due to the slowest components (1 . . $n_c-1$). If we express the permeation factor of the fastest component $R_{n_c}$ on the basis of the i-th slowest component $R_i$ using the definition of selectivity or separation factor:

$\alpha_{N,i}=\frac{Q_N}{Q_i}=\frac{Q_N\cdot A\cdot p}{n_0}\cdot \frac{n_0}{Q_i\cdot A\cdot p}=\frac{R_N}{R_i}>1$ and $R_i=\frac{R_N}{\alpha_{N,i}}$ (18)

the critical pressure ratio for complete permeation becomes:

$\delta_{CP}=\frac{1+R_N}{R_N}+\sum_{N-1}\frac{(\alpha_{N,i}-1)\cdot z_i}{R_N}$.

This expression can be interpreted as follows:

Assembly of the complete concentrated parameter model

By substituting expressions of $X_i$ and $Y_i$ 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 $y_i$ can be obtained.

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

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

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

  1. estimation of the overall stage-cut $\Theta$ or separation yield $\Theta°$

  2. calculation of the stage-cuts / separation yield of the components $\theta_i$ / $\theta°_i$.

  3. iterative resolution of the equation $\Sigma z_i\cdot \theta_i=Theta$ i.e. $\Sigma \theta°_i\cdot \lambda_i=\Theta°$ (11.B).

In this way the algebraic problem is greatly simplified, since there is no need to solve an algebraic system in $2+2-n_c$ unknowns, but only one nonlinear equation in one unknown.

One-two models

Global “RTSC” fluidodynamics, two components

We posed $X_i=z_i$ and $Y_i=y_i$.

From equations (2) and (14.A) we obtain a 3-equation system in 3 unknowns ($\Theta$, $y_1$ and $y_2$):

$y_1=\frac{R_1}{\Theta}\cdot (\delta \cdot z_1-y_1)$ (19)

$y_2=\frac{R_2}{\Theta}\cdot (\delta \cdot z_2-y_2)$ (20)

$y_1+y_2=1$ (21)

We substitute (21) into (20):

$(1-y_1)=\frac{R_2}{\Theta}\cdot (\delta \cdot z_2-1+y_1)$ (22)

We derive $\Theta$ from (19):

$\Theta =\frac{R_1\cdot (\delta \cdot z_1-y_1)}{y_1}$ (23)

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

$(1-y_1)=\frac{R_2\cdot y_1}{R_1\cdot (\delta \cdot z_1-y_1)}\cdot (\delta \cdot z_2-1+y_1)$ (24)

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

$y_1^2\cdot (1-\frac{R_2}{R_1})-y_1\cdot (\delta \cdot z_1+1+\frac{R_2}{R_1}\cdot \delta \cdot z_2+\frac{R_2}{R_1})+\delta \cdot z_1=0$ (25)

Obtaining a 2nd degree one-variable equation $y_1$, which is easily solved. Two solutions will be obtained, only one of which is physically acceptable.

Global CSTR fluidodynamics, multi-component

We pose $X_i=z_i$ and $Y_i=y_i$.

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

$y_i=\frac{R_i}{\Theta}\cdot (\delta \cdot x_i-y_i)$

$x_i$ is obtained from equation (3.A):

$x_i=\frac{n_0\cdot z_i-n_P\cdot y_i}{n_R}=\frac{n_0}{n_R}\cdot z_i-\frac{n_P}{n_R}\cdot y_i=\frac{z_i-\Theta \cdot y_i}{1-\Theta}$

We then proceed by substituting the value of $x_i$:

$y_i=\frac{R_i}{\Theta}\cdot (\delta \cdot (\frac{z_i-\Theta \cdot y_i}{1-\Theta})-y_i)$

Solving with respect to $y_i$ will yield that:

$\theta_i=\Theta \cdot \frac{y_i}{z_i}=\frac {1}{1+(\frac{1-\Theta}{\delta})\cdot (\frac{1}{\Theta}+\frac{1}{R_i})}$ (26)

Global “RTSC” fluidodynamics, multi-component

It is posited $X_i=z_i$ and $Y_i=y_i$.

From equation (14.A) we get:

$y_i=\frac{R_i}{\Theta}\cdot (\delta \cdot z_i-y_i)=\frac{R_i}{\Theta}\cdot z_i-\frac{R_i}{\Theta}\cdot y_i$

from which the ratio $y_i$ / $z_i$ can be made explicit:

$\frac{y_i}{z_i}=\frac{\delta}{1+\frac{\Theta}{R_i}}$

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

$\theta_i=\Theta \cdot \frac{y_i}{z_i}=\frac{\delta }{\frac {1}{\Theta }+\frac {1}{R_i}}$ (27)

Global CSTR fluidodynamics, two components

Substituting (26) into (11.A) for 2 components ($CO_2$ and $CH_4$) gives:

$\Theta =\frac{z_{CO_2}}{1+(\frac{1-\Theta}{\delta})\cdot (\frac {1}{\Theta}+\frac {1}{R_{CO_2}})}+\frac{z_{CH_4}}{1+(\frac{1-\Theta}{\delta})\cdot(\frac{1}{\Theta}+\frac {1}{R_{CH_4}})}$

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 $X_i=\frac{z_i+x_i} 2$

Equation (14.A) becomes:

$y_i=\frac{R_i}{\Theta}\cdot (\delta \cdot \frac{(x_i+z_i)}{2}-y_i)$

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

$x_i=\frac{z_i-\Theta \cdot y_i}{1-\Theta}$

and is substituted into the equation:

$y_i=\frac{R_i}{\R_i}{\Theta}\cdot (\delta \cdot \frac{(\frac{z_i-\Theta \cdot y_i}{1-\Theta}+z_i)}{2}-y_i)$

Explicating $y_i$ gives:

$y_i=\frac{z_i\cdot (2-\Theta)}{\frac{2\cdot (1-\Theta)}{\delta}\cdot (\frac{z_i}{R_i}+1)+\Theta}$ (28)

From this we obtain that $\theta_i$ is:

$\theta_i=\frac{(2-\Theta)}{\frac{2\cdot (1-\Theta)}{\delta}\cdot (\frac{1}{R_i}+\frac{1}{\Theta})+1}$ (29)

which is the sought expression of $\theta_i = f(\Theta)$ that allows iterative resolution.

Two-two models

Global “RTSC” fluidodynamics, multi-component

The “RTSC” model leads to defining $Y_i$ and $X_i$ as:

$Y_i=w_i$ and $X_i=z_i$.

Substituting these values into equation (14.A) yields:

$y_i=\frac{R°_i}{\Theta°}\cdot(\delta \cdot z_i-w_i)+\frac{n_{PS} \cdot w_i}{\Theta°\cdot N}$ (30)

Compared with the 2-component case treated above, a more complex system is obtained, for $y_i$, $w_i$ and also $n_{PS}$ are unknowns.

From (30), however, we obtain that $\theta°_i$ is expressible explicitly as a function of the inputs:

$\theta°_i=\frac{R°_i(\delta \cdot z_i-w_i)\cdot N+n_{PS}\cdot w_i}{N\cdot \lambda_i}$ (31)

Thus, iterative resolution is not necessary in this case.

Global CSTR fluidodynamics, multi-component

The CSTR model implies the definition of $Y_i$ and $X_i$ as:

$Y_i=y_i$ and $X_i=x_i$.

Substituting these values into equation (14.A) gives:

$y_i=\frac{R°_i}{\Theta°}\cdot (\delta \cdot x_i-y_i)+\frac{n_{PS} \cdot w_i}{\Theta° \cdot N}$ (32)

From the matter balance for the single component (3.B), the value of $x_i$ is derived:

$x_i=\frac{\lambda_i-\Theta \cdot y_i}{(1-\Theta)}$ (33)

Equation (32) then becomes:

$y_i=\frac{R°_i}{\Theta°}\cdot (\delta \cdot \frac{\lambda_i-\Theta \cdot y_i}{(1-\Theta)}-y_i)+\frac{n_{PS}\cdot w_i}{\Theta ° \cdot N}$

Solving with respect to $y_i$ we get equation (34):

$y_i=\frac{\lambda_i+\frac{n_{PS} \cdot w_i \cdot (1-\Theta°)}{N\cdot R°_i \cdot \delta}}{\Theta°+(\frac{1-\Theta°}{\delta})(\frac{\Theta°}{R°_i}+1)}$ (34)

By substituting equation (34) into (10.B), it is possible to obtain the sought expression of $θ_i$ = $f(\Theta)$ that allows iterative resolution.

Global linear fluidodynamics, multi-component

In this case $Y_i$ and $X_i$ assume the values of:

$Y_i=\frac{y_i+w_i}{2}$ $X_i=\frac{x_i+z_i}{2}$

Equation (14.A), substituting the values of $Y_i$ and $X_i$ and $x_i$ obtained from (33), becomes:

$y_i=\frac{R°_i}{\Theta°}\cdot (\delta \cdot \frac{\frac{\lambda_i-\Theta°\cdot y_i}{(1-\Theta°)}+z_i}{2}-\frac{y_i+w_i}{2})+\frac{n_{PS}}\cdot w_i}{\Theta° \cdot N}$

Solving with respect to $y_i$ we obtain:

$y_i=\frac{\lambda_i+z_i\cdot (1-\Theta°)-\frac{(1-\Theta°)\cdot w_i}{\delta}+2 \frac{n_{PS} \cdot w_i\cdot (1-\Theta°)}{(N\cdot R°_i\cdot \delta)}}{\frac{(1-\Theta°)}{\delta}\cdot (\frac{2\Theta°}{R}°_i+1)+\Theta°}$ (35)

Substituting in (10,B) $\theta°_i=\frac{\Theta°\cdot y_i}{\lambda_i}$ I get:

$\theta°_i=\frac{\lambda_i+z_i\cdot (1-\Theta°)-\frac{(1-\Theta°)\cdot w_i}{\delta}+2\frac{n_{PS}\cdot w_i\cdot (1-\Theta°)}{(N\cdot R°_i\cdot \delta)}}{(\frac{(1-\Theta°)}{\delta}\cdot (\frac {2}{R°_i}+\frac {1}{\Theta°})+1)\cdot \lambda_i}$ (36)

Substituting the values obtained in (36) for each individual component into (11.B), it is possible to obtain the sought expression of $\theta°_i = f(\Theta°)$ that allows iterative resolution.

Distributed parameter models

To create more accurate models, the base concentrated parameters units can be appropriately replicated and interconnected to form distributed parameter models.

Of the four possible effectively distributed global fluid dynamics models (see [Greppi2011b] chapter “Fluidodynamics” page 4), three were considered, excluding the one-side mixing configuration (plug flow for retentate, full mixing for permeate):

It appears evident that Co-current and Counter-current distributed parameter models can be obtained by replicating the fundamental unit with concentrated parameters with two-two connectivity, while Cross-flow is obtained by replicating that with one-two connectivity.

Implementation

Once the models have been derived, their numerical and computer implementation can be carried out.

Detailed information for implementation are at the following links:

Validation

Post-combustion carbon capture is the recovery of $CO_2$ from a stream composed primarily of $N_2$, which is carried out to reduce the carbon dioxide content released into the atmosphere from flue gas from power plants.

This is a similar application to the upgrading of biogas, for which there are many works in the literature, also supported by numerical examples that lend themselves to comparison.

Modeling assumptions

The following chemical species are considered: $CO_2$, $CH_4$, $H_2\O$, $N_2$ and $O_2$.

All fluids are considered to be in the gas phase except in condensers of multi-stage compressors, where vapor-liquid phase equilibrium is calculated using Raoult’s law.

Gas mixtures are considered ideal mixtures of perfect gases, so the volumetric compositions are identical with the molar compositions.

The reference state for the state functions is the pure perfect gas, P=1 atm, T= 25 °C.

The specific heats of ideal gases (in the low-pressure limit) are taken from the Design Institute for Physical Properties (DIPPR) collection in the July 1993 version given in [Perry1999] 2-198 passim.

We also assume STP (Standard Temperature & Pressure) volume conditions of 22.71 ${kmol}/ {Nm^3}$ in accordance with [Iupac1997]:

Finally, the lower heating value of methane LHV[CH4] is assumed to be 802618 kJ/kmol = 35340.8 ${kJ}/{Nm^3s}$.

Conclusions

Based on the review of membrane system models found in the literature [Greppi2011a] concentrated-parameter and distributed-parameter models were formulated and implemented for the simulation of gas separation systems with membranes based on a linear mass transport expression (solution-diffusion model). These models were validated against published data related to carbon dioxide separation with polymeric membranes to verify implementation on applications similar to the one of interest.

Turning to the specific application of biomethane upgrading, a preliminary plant scheme compatible with the reference size was simulated, based on a two-stage process with a single compressor. Permeance values were assumed.

It is noted that the plant meets the specification on carbon dioxide in upgraded biogas, but not the specification on water; also, all the results will still need to be verified against experimental values on permeances. In addition, it will be necessary to complete the plant by providing for other necessary process steps, such as the removal of $H_2S$, which has not been considered here.

The performance parameters for the proposed scheme calculated from the simulation results are:

values that confirm the interest in membrane-based upgrading technology, provided that the assumed permeance values are actually achievable with materials of cost and durability compatible with the application.

The models created are flexible and can be reused in later stages of the project to provide effective modeling support for experimental and design activities:

To further explore the detail of the models, it may also be considered whether it is necessary to include:

References

[Davis2002] R. A. Davis, Simple Gas Permeation and Pervaporation Membrane Unit Operation Models for Process Simulators, Chemical Engineering & Technology, Volume 25, Issue 7 (717–722) July 2002, doi: 10.1002/1521-4125(20020709)25:7<717::AID-CEAT717>3.0.CO;2-N

[Greppi2011a] P. Greppi, E. Arato, B. Bosio, Rapporto PERT1 “Analisi dei modelli di sistemi a membrane reperibili in letteratura”, Marzo 2011

[Greppi2011b] P. Greppi, E. Arato, B. Bosio, Rapporto PERT2 “Proposta di una metodologia di confronto delle tecnologie”, Marzo 2011

[Iupac1997] IUPAC Compendium of Chemical Terminology, 2nd ed., 1997

[Perry1999] Perry’s Chemical Engineer Handbook 7th Edition 1999