This mass balance should applied to equipments provident a pseudo continuos mode of operation. For example: packed towers or bubble towers aimed for liquid-gas contact. The approach would be different if stagewise equipments are being used, such as: tray towers.

CONTENTS

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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In the figure below, a general representation of a global mass balance for a packed tower allowing liquid-gas contact is presented.

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

Being only one soluble gas component in the mixture stream of our interest, only the mole fractions in the liquid and gas phase 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)

$L_2$ $x_2=0$ 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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$L_S$ $x_1$ $G_1$ . One further typical assumption, is that the dilution of the gas component molecules of the gas mixture achieve equilibrium so that diagram of equilibrium become handy at this stage.

Please, read post: Equilibrum 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)

$min$ $max$ $L_1$ $x_1$ $L_{1-min}$ $L_{1-min}'$ can be obtained as,

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

$L_{1-min}$ $x_{1-,max}$ . The maximum fraction $x_{1-max}$ $y_1$ in the gas mixture.

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

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$X$ $Y$ (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)

$p_t$ $p$ the partial pressure of the soluble gas component. As you can see the mole ratio is just the ratio of the fractions of soluble component to that of the carrying media (say, solvent). If the mole ratio $X<1$ $L$ $X>1$ $Y$ $G$ .

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 forseen,

$G_2y_2=G_S\,\dfrac{y_2}{1-y_2}=G_SY_2$ 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 som rearrangement,

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

$x$ $y$ $X$ $Y$ , 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 represention of what occurs in every place of the equipment (from the bottom to the mid). 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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$L_S/G_S$ .

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

$G_1$ ,
$Y_1$ $Y_2$ and
$X_2$ ;

$G_1$ $Y_1$ $Y_2$ $L_S$ $X_2=0$ $L_S$ can be changed. Let us dicuss 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 different 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

$AD$ just because the gradient driving it is reduced to its minimum. Therefore, for the absorption to work at conditions in

$AD$ a larger tower would be required or a larger residence time in it.

$L_S$ $AC$ $AD$ $E$ $AD$ $E$ so that the absorptions process gets interrupted there.

$Y_1$ $X_{1-max}$ $E$ producing a finite height packed tower.

2 PARALLEL FLOW OPERATION

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$-L_S/G_S$ . 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 thw post. I hope you find it useful.

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

Other stuff of interest

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Ildebrando.

Chem Eng stuff

ChemEng stuff followers

Mass balance in a packed tower

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

Other stuff of interest

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