mccabe-thiele Toolbox for GNU-Octave

Installing and Loading mccabe-thiele

```dotnetcli

e.g. this call installs version 0.1.5

pkg install https://github.com/aumpierre-unb/McCabe-Thiele-for-GNU-Octave/archive/refs/tags/v0.1.5.tar.gz pkg load mccabe-thiele ```

Citation of mccabe-thiele

You can cite all versions (both released and pre-released), by using DOI 10.5281/zenodo.7133683. This DOI represents all versions, and will always resolve to the latest one.

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The following is a very short introduction to the mccabe-thiele toolbox for GNU Octave.

This text is divided in two main sections: The Theory and The mccabe-thiele Toolbox.

The Theory

The McCabe-Thiele method is a simplified method to calculate the number of theoretical stages of equilibrium of a distillation column for a two component mixture. The method depends of three premisses:

• components have the same molar heats of vaporization,
• for every mole of liquid vaporized, a mole of vapor is condensed, and
• heat effects are negligible.

Distillation is a unit operation based on the difference of volatility to separate different chemical species of a mixture and on gravity to separate vapor and liquid streams. There are several types of distillation.

Typically, continuous distillation is performed in a vertical column fed at its intermediate section while products at its top and bottom are recovered. As a rule of thumb, the higher the column, the higher the number of stages of equilibrium and the best is the separations of the components of the mixture fed.

Theoretical Stage of Equilibrium

A theoretical stage of equilibrium is an abstract control volume such that the effluent currents are all in thermodynamic equilibrium. All multistage operations are based on the concept of stage of equilibrium.

The Feed Line

Ideally, the feed current separates into a liquid stream q and a vapor stream 1-q, where q is the liquid fraction of the feed. If x<sub>n+1</sub>, y<sub>n</sub> and x<sub>F</sub> are the compositions of the liquid and the vapor streams at the feed inlet and the composition of the feed, respectively, then the material balance is

$$ (1 - q)\ yn + q\ x{n+1} = x_F $$

or

$$ yn = {q \over q-1}x{n+1} - {x_F \over q-1} $$

Operation Lines

The global and specific material balances for any equilibrium stage n are given by

$$ V{n-1} - Ln = Vn - L{n+1} $$

$$ V{n-1}\ y{n-1} - Ln\ xn = Vn\ yn - L{n+1}\ x{n+1} $$

where L<sub>n+1</sub> and V<sub>n</sub> are the liquid and the vapor streams, and x<sub>n+1</sub> and y<sub>n</sub> are compositions of liquid and vapor streams, respectively.

As this is true for any stage in its section, then the material stream is constant in the section. For the last theoretical stage N (the top of the column), the global material balance is given by

$$ D = VN - L{N+1} = Vn - L{n+1} $$

and the specific material balance is given by

$$ D\ xD = Vn\ yn - L{n+1}\ x_{n+1} $$

where D and x<sub>D</sub> are the distillate stream and its composition.

The ratio between the distillate D and the refluxing current L<sub>N+1</sub> is the reflux ratio at the top of the column,

$$ R = {L_{N+1} \over D} $$

Now, introducing the reflux ratio in the specific material balance of the rectifying section, we have

$$ yn = {R \over R+1}x{n+1} + {x_D \over {R+1}} $$

Notice that y<sub>n</sub> = x<sub>D</sub> for x<sub>n+1</sub> = x<sub>D</sub> as well, where x<sub>B</sub> is the column's bottom product composition.

One can apply the very same procedure to find the operation line for the stripping section based on the reflux ratio at the bottom of the column, and find that y<sub>n</sub> = x<sub>B</sub> for x<sub>n+1</sub> = x<sub>B</sub> and that both operation lines and the feed line have a common interception.

It means that given the compositions of the feed and the products, the feed quality and the reflux ratio at the top of the column, the reflux ratio at the bottom of the column can be calculated as

$$ S = {V_0 \over B} $$

where B = F - D is the column's bottom product stream. It is also possible to calculate the stages of equilibrium of a column using the McCabe-Thiele method.

The mccabe-thiele Toolbox

mccabe-thiele provides the following functions:

• stages
• refmin
• qR2S

stages

stages computes the number of theoretical stages of a distillation column using the McCabe-Thiele method given a function y = y(x) that relates the liquid fraction x and the vapor fraction y, or a x-y matrix of the liquid and the vapor fractions, the vector of the fractions of the products and the feed, the feed quality, and the reflux ratio at the top of the column.

If feed is a saturated liquid, feed quality q = 1, feed quality is reset to q = 1 - 1e-10.

By default, theoretical stages are computed from the stripping section to the rectifying section, updown = true.

If updown = false is given, theoretical stages are computed from the rectifying section to the stripping section.

By default, stages plots a schematic diagram of the solution, fig = true.

If fig = false is given, no plot is shown.

Syntax:

dotnetcli [N]=stages(y,X,q,R[,updown[,fig=true]])

Examples:

Compute the number of theoretical stages of a distillation column from the bottom of the column, given a matrix that relates the liquid fraction and the vapor fraction, the composition of the distillate is 88 %, the composition of the feed is 46 %, the composition of the column's bottom product is 11 %, the feed quality is 54 %, and the reflux ratio R at the top of the column is 70 % higher than the minimum reflux ratio:

dotnetcli data=[0. 0.; 0.1 0.212; 0.2 0.384; 0.3 0.529; 0.4 0.651; 0.5 0.752; 0.6 0.833; 0.7 0.895; 0.8 0.942; 0.9 0.974; 1. 1.]; x=[0.88 0.46 0.11]; q=0.54; r=refmin(data,x,q); R=1.70*r; N=stages(data,x,q,R,false,false)

Compute the number of theoretical stages of a distillation column from the top of the column, given the function that compute the vapor fraction given the liquid fraction, the composition of the distillate is 88 %, the composition of the feed is 46 %, the composition of the column's bottom product is 11 %, the feed is saturated liquid, and the reflux ratio R at the top of the column is 70 % higher than the minimum reflux ratio, and plot a schematic diagram of the solution:

dotnetcli y=@(x) (x.^0.9 .* (1-x).^1.2 + x); x=[0.88 0.46 0.11]; q=1; r=refmin(y,x,q); R=1.70*r; N=stages(y,x,q,R)

refmin

refmin computes the minimum value of the reflux ratio of a distillation column using the McCabe-Thiele method given a function y = y(x) that relates the liquid fraction x and the vapor fraction y, or a x-y matrix of the liquid and the vapor fractions, the vector of the fractions of the distillate and the feed, and the feed quality.

If feed is a saturated liquid, feed quality q = 1, feed quality is reset to q = 1 - 1e-10.

Syntax:

dotnetcli [R]=refmin(f,X,q)

Examples:

Compute the minimum value of the reflux ratio of a distillation column, given a matrix that relates the liquid fraction and the vapor fraction, the composition of the distillate is 88 %, the composition of the feed is 46 %, the feed quality is 54 %:

dotnetcli data=[0. 0.; 0.1 0.212; 0.2 0.384; 0.3 0.529; 0.4 0.651; 0.5 0.752; 0.6 0.833; 0.7 0.895; 0.8 0.942; 0.9 0.974; 1. 1.]; x=[0.88 0.46]; q=0.54; r=refmin(data,x,q)

Compute the number of theoretical stages of a distillation column from the top of the column, given the function that compute the vapor fraction given the liquid fraction, the composition of the distillate is 88 %, the composition of the feed is 46 %, the feed is saturated liquid:

dotnetcli y=@(x) (x.^0.9 .* (1-x).^1.2 + x); x=[0.88 0.46]; q=0.54; r=refmin(y,x,q)

qR2S

qR2S computes the reflux ratio at the bottom of a distillation column using the McCabe-Thiele method given the reflux ratio at the top of the column, the vector of the fractions of the products and the feed, and the feed quality.

If feed is a saturated liquid, feed quality q = 1, feed quality is reset to q = 1 - 1e-10.

Syntax:

dotnetcli [S]=qR2S(R,X,q)

Examples:

Compute the reflux ratio at the bottom of the column, given the composition of the distillate is 88 %, the composition of the feed is 46 %, the composition of the column's bottom product is 11 %, the feed quality is 54 %, and the reflux ratio R at the top of the column is 2:

dotnetcli x=[0.88 0.46 0.11]; q=0.54; R=2; S=qR2S(R,x,q)

Compute the reflux ratio at the bottom of the column, given the composition of the distillate is 88 %, the composition of the feed is 46 %, the composition of the column's bottom product is 11 %, the feed is saturated liquid, and the reflux ratio R at the top of the column is 2:

dotnetcli x=[0.88 0.46 0.11]; q=1; R=2; S=qR2S(R,x,q)

See Also

Ponchon-Savarit-for-GNU-Octave, Psychrometrics-for-GNU-Octave, Internal-Fluid-Flow-for-GNU-Octave.

Copyright © 2022 2023 2024 Alexandre Umpierre

email: aumpierre@gmail.com