## The concept of the transmission unit

The concept of the ideal separation stage lends itself to the design of bottom columns, while packed columns can be calculated with the concept of the transfer unit. This is based on the mass transport equation of film theory.

The concept of the transfer unit assumes that in a differential element a molar current is transferred from the gas phase to the liquid phase without the currents changing $\stackrel{.}{V}$ and $\stackrel{.}{\mathrm{L.}}$ change.

This molar current should be identical to that calculated using film theory and can be derived from it.

- Height of the packed column
- The height of a packed column is calculated using the gas phase as:
- Or using the liquid phase:
- Explanation of symbols for the equation
- Derivation of the equation

- $$H=\underset{0}{\overset{H}{\int}}dH=\frac{\dot{V}}{{\beta}_{G}\text{}a\text{}\mathrm{A.}}\text{}\underset{{y}_{\mathrm{A.}\mathrm{B.}}}{\overset{{y}_{\mathrm{A.}T}}{\int}}\frac{d{y}_{\mathrm{A.}}}{{y}_{\mathrm{A.}i}-{y}_{\mathrm{A.}}}$$

- $$H=\underset{0}{\overset{H}{\int}}dH=\frac{\dot{\mathrm{L.}}}{{\beta}_{\mathrm{L.}}\text{}a\text{}\mathrm{A.}}\text{}\underset{{x}_{\mathrm{A.}T}}{\overset{{x}_{\mathrm{A.}\mathrm{B.}}}{\int}}\frac{d{x}_{\mathrm{A.}}}{{x}_{\mathrm{A.}i}-{x}_{\mathrm{A.}}}$$

Just as the number of theoretical separation stages plays a role in the concept of the ideal separation stage, the number of transmission units (NTU) and the height of a transmission unit (HTU) are taken into account in the concept of the transmission units, whereby a distinction is made between gas and liquid phases.

The amount of a transfer unit decreases when

- the volume flow is reduced,
- the mass transfer coefficient is increased,
- the specific phase interface is increased.

Since the determination of the concentrations at the phase boundary makes it difficult to calculate the packing height, the alternative option chosen is to place the total resistance in one phase. Then you know all the required sizes or can calculate them. One uses the current concentration in one phase and a hypothetical equilibrium concentration at the phase interface which results when there is no mass transport resistance in the other phase.

This results in:

- $${\dot{n}}_{\mathrm{A.}}={\beta}_{O\mathrm{L.}}\text{}a\text{}\left({x}_{\mathrm{A.}}^{*}-{x}_{\mathrm{A.}}\right)\text{}\mathrm{A.}\text{}dH={\beta}_{OG}\text{}a\text{}\left({y}_{\mathrm{A.}}-{y}_{\mathrm{A.}}^{*}\right)\text{}\mathrm{A.}\text{}dH=\dot{V}\text{}d{y}_{\mathrm{A.}}=\dot{\mathrm{L.}}\text{}d{x}_{\mathrm{A.}}$$

Since the concentration gradient is precisely known here, the total mass transfer coefficient (overall coefficient) can be calculated from this.

- $$\frac{1}{{\beta}_{OG}}=\frac{1}{{\beta}_{G}}+\frac{m}{{\beta}_{\mathrm{L.}}}\text{}\text{}\text{}\frac{1}{{\beta}_{O\mathrm{L.}}}=\frac{1}{{\beta}_{\mathrm{L.}}}+\frac{1}{m\text{}{\beta}_{G}}$$

The phase equilibrium has an influence on the mass transport resistance. With very small values of m, the resistance of the liquid phase can be neglected; with very large values, the mass transfer is controlled by the resistance of the liquid phase.

Here, too, the height of the random packing can be estimated

- $$H=\underset{0}{\overset{H}{\int}}dH=\frac{\dot{V}}{{\beta}_{OG}\text{}a\text{}\mathrm{A.}}\text{}\underset{{y}_{\mathrm{A.}\mathrm{B.}}}{\overset{{y}_{\mathrm{A.}T}}{\int}}\frac{d{y}_{\mathrm{A.}}}{{y}_{\mathrm{A.}}^{*}-{y}_{\mathrm{A.}}}$$

or using the liquid phase

- $$H=\underset{0}{\overset{H}{\int}}dH=\frac{\dot{\mathrm{L.}}}{{\beta}_{O\mathrm{L.}}\text{}a\text{}\mathrm{A.}}\text{}\underset{{x}_{\mathrm{A.}T}}{\overset{{x}_{\mathrm{A.}\mathrm{B.}}}{\int}}\frac{d{x}_{\mathrm{A.}}}{{x}_{\mathrm{A.}}^{*}-{x}_{\mathrm{A.}}}$$

The number of transmission units required can be determined by numerical or graphic integration.

A problem with the concept of the transfer units is the estimation of the mass transfer coefficients and the specific phase interface. Both values are strongly influenced by the operating conditions. To facilitate the design of the packing column, the packing manufacturers therefore supply HETP values as a function of the various influencing variables. HETP values indicate the bed height that would be equivalent to an equilibrium level. The total height of a packed column is calculated with:

- $$H={N}_{\mathrm{th}}\cdot \mathrm{HETP}$$