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Mass balance in a packed tower

This mass balance should be applied to equipment to provide a pseudo-continuous mode of operation. For example, packed or bubble towers are designed for liquid-gas contact. The approach would differ if stagewise equipment is used, such as tray towers.

 

CONTENTS

 1 COUNTERCURRENT FLOW OPERATION

 1.1 A global mass balance on the soluble gas component

 1.1.1 The case of minimum water for an absorption operation

 1.2 A global mass balance for the operation of a packed tower

 1.2.1 Minimum liquid-gas ratio

 2 PARALLEL FLOW OPERATION


1 COUNTERCURRENT FLOW OPERATION

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The figure below presents a general representation of the global mass balance for a packed tower allowing liquid-gas contact.

Fig. 01 Global mass balance for an absorption-packed tower. Notice that only one gas component is of interest. All other components can not be separated under the working conditions.

Since only one soluble gas component in the mixture stream is of our interest, only the mole fractions in the liquid and gas phases are symbolized.

1.1 A global mass balance on the soluble gas component

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This can be easily formulated as,

$G_1 y_1 \,+\, L_2 x_2\,=\, G_2 y_2\,+\, L_1x_1$        Eq. (01)

where of course, if $L_2$ is clean water then, $x_2=0$ and Eq. (01) reduces to,

$G_1 y_1 \,=\, G_2 y_2\,+\, L_1 x_1$        Eq. (02)

1.1.1 The case of minimum water for an absorption operation

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In this case, the minimum amount of water $L_S$ to be used would be that carrying the maximum mole fraction $x_1$ separated from the inlet gas mixture stream $G_1$. One further typical assumption is that the gas component molecules in the gas mixture reach equilibrium, so that the equilibrium diagram becomes helpful at this stage.

Please, read the post: Equilibrium diagrams from gas component solubility data, for more details.

Therefore, Eq. (02) can be rewritten as,

$G_1 y_1\,=\,G_2y_2\,+\, L_{1-min}x_{1-max}$        Eq. (03)

where the subscripts $min$ and $max$ have been introduced to indicate the minimum and maximum features of the enriched liquid stream $L_1$ and the mole fraction $x_1$. From $L_{1-min}$, the stream of minimum required clean water $L_{1-min}'$ can be obtained as,

$L_{1-min}'=L_{1-min}\left( 1-x_{1-max} \right)$        Eq. (04)

Equation (04) has two unknowns $L_{1-min}$ and $x_{1-,max}$. The maximum fraction $x_{1-max}$ corresponds to the equilibrium and, as such, must be searched from an equilibrium diagram at the corresponding mole fraction $y_1$ in the gas mixture.

1.2 A global mass balance for the operation of a packed tower

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The general approach to the mass balance, as presented above, is of interest but provides little insight into the tower's operation. For this purpose, a parameter needs to be introduced: the mole ratio $X$ (in the liquid phase) and $Y$ (in the gas phase), which are defined as,

$X=\dfrac{x}{1-x}$        Eq. (05a)

$Y=\dfrac{y}{1-y}=\dfrac{p}{p_t-p}$        Eq. (05b)

where $p_t$ is the total pressure exerted on the gas mixture and $p$ is the partial pressure of the soluble gas component. As you can see, the mole ratio is just the ratio of the fractions of the soluble component to that of the carrying media (say, solvent). If the mole ratio $X<1$, there is more solvent than dilute gas component molecules in $L$, and vice versa for $X>1$. The same could be said about the $Y$ mole ratio and the stream $G$.

Next, using the idea of the mole ratio, the stream of solvent carrying gas and liquid can be defined as,

$G_S=G\left( 1-y \right)=\dfrac{G}{1+Y}$        Eq. (06a)

$L_S=L\left( 1-x \right)=\dfrac{L}{1+X}$        Eq. (06b)

From Eqs. (6a,6b) It is not hard to see that,

$G_1y_1=G_S\,\dfrac{y_1}{1-y_1}=G_SY_1$        Eq. (07a)

$L_1x_1=L_S\,\dfrac{x_1}{1-x_1}=L_SX_1$        Eq. (07b)

Next, returning to mass balance Eq. (01), another set of formulas can be foreseen,

$G_2y_2=G_S\,\dfrac{y_2}{1-y_2}=G_SY_2$        Eq. (07c)

$L_2x_2=L_S\,\dfrac{x_2}{1-x_2}=L_SX_2$        Eq. (07d)

so that Eq. (01) can now be rewritten as,

$G_S Y_1 \,+\, L_S X_1\,=\, G_S Y_2\,+\, L_SX_1$        Eq. (08)

and after some rearrangement,

$G_S\left(Y_1-Y_2\right)=L_S\left(X_1-X_2\right)$        Eq. (09)

Let us now consider a section inside the tower where the mole fractions and the mole ratios are: $x$, $y$, $X$, and $Y$, as shown in the sketch below.

Fig. 02 Global mass balance for an absorption-packed tower.

Then Eq. (09) can be expressed as,

$G_S\left(Y_1-Y\right)=L_S\left(X_1-X\right)$        Eq. (10)

Eq. (10) is the same mass balance as in Eq. (09), but this is a more general representation of what occurs in every place of the equipment (from the bottom to the middle). Also, Eq. (10) is the equation, in its most general form, of a straight line. Let us rewrite Eq. (10) as follows,

$\dfrac{L_S}{G_S}=\dfrac{Y_1-Y}{X_1-X}$        Eq. (11)

Now, it is clear that $L_S/G_S$ is the slope of a straight line also called the operating line.

Fig. 03 Operating line for a tower of absorption. Notice that the equilibrium curve is given in terms of the mole ratios rather than in terms of the mole fractions.

Notice that Eqs. (09-10) are both represented in the operating line in Fig. (03). One other interesting fact is that the operating line is only a straight line when represented in terms of the mole ratios.

1.2.1 Minimum liquid-gas ratio

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Let us now extend the idea of the minimum solvent requirement to include the gas solvent as well. We may start with the ratio $L_S/G_S$.

The reason for looking for a minimum flow rate of liquid solvent is that:

  • the flow rate of gas mixture $G_1$,
  • the mole ratios $Y_1$ and $Y_2$ and
  • the mole ratio $X_2$;

can be easily set as constants or as design parameters. You can adjust the flow rate $G_1$ with a valve; the mole ratios $Y_1$ and $Y_2$ are set as goal parameters to achieved in the packed tower; and if the liquid solvent $L_S$ is clean $X_2=0$ or you can measure it before the liquid enters the tower. On the other hand, $L_S$ can be changed. Let us discuss a little more on this subject.

Fig. 04 Operating lines for different flow rates of $L_S$.

Consider different operating lines $AB$, $AC$, and $AD$ as presented in Fig. 04, corresponding to various scenarios in the packed tower, corresponding to different $L_S$. As the flow rate $L_S$ is decreased, the slope $L_S/G_S$ decreases too, having in the operating line $AB$ the minimum value for $L_S$, say $L_{S-min}$. For all these operating conditions, the absorption of the soluble gas component in $G_1$ is assured. However, the absorption at $AB$ is easier than at $AD$ simply because the driving gradient is at its minimum. Therefore, for absorption to work under the conditions in $AD$, either a larger tower or a longer residence time would be required.

Also, if you wish a shorter packed tower a larger $L_S$ is required. Then, a convenient operating line $AC$ is desirable. Finally, the operating line $AD$ touches the equilibrium curve at $E$, where no gradient is present, so that no molecules of soluble gas component are absorbed, so that the tower would have infinite height. As you can see, the operating line $AD$ extends beyond $E$, so the absorption process is interrupted there.

How do we overcome the limit of a tower of infinite height? One way is by reducing $Y_1$ so that $X_{1-max}$ ends at $E$, producing a finite height packed tower.

2 PARALLEL FLOW OPERATION

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In a packed tower, with the liquid and gas phases running in parallel, the calculations differ little, but in the end, an operating line of negative slope is obtained: $-L_S/G_S$. This would sound strange, but there are applications for such conditions.

Fig. 05 Operating line for packed tower working with parallel streams $L_1$ and $G_1$.


Again, the operating line may cut the equilibrium curve at the point $X_e, Y_e$ where a packed tower of infinite height would be needed.

This is the end of the post. I hope you find it useful.

Any questions? Write in the comments, and I shall try to help.

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