Packing Height : The Method of Transfer Units

A newer concept in the analysis of packed column centred on the method of transfer units. This method is more appropriate because the changes in compositions of the liquid and vapour phases occur differentially in a packed column rather than in stepwise fashion as in trayed column.

In this method, height of packings required can be evaluated either based on the gas-phase or the liquid-phase. The packed height (z) is calculated using the following formula:

z = N x H

where

N = number of transfer units (NTU) - dimensionless

H = height of transfer units (HTU) - dimension of length

The number of transfer units (NTU) required is a
measure of the difficulty of the separation. A
single transfer unit gives the change of composition of one of the phases equal
to the average driving force producing the change. The NTU is similar to the number
of theoretical trays required for trayed column. Hence, a larger number of transfer
units will be required for a very high purity product.

The height of a transfer unit (HTU) is a measure of the separation effectiveness of the particular packings for a particular separation process. As such, it incorporates the mass transfer coefficient that we have seen earlier. The more efficient the mass transfer (i.e. larger mass transfer coefficient), the smaller the value of HTU. The values of HTU can be estimated from empirical correlations or pilot plant tests, but the applications are rather restricted.

["Principles of Unit Operations" 2nd Ed., Foust et al, p.391]

The calculation of packing
height follows the same nomenclature as before and this is shown in the Figure
below.

In this Section, we will focus on the applications of the equations rather than any derivation of them. Determination of the packed height can be based on either the gas-phase or the liquid-phase.

For the gas-phase, we have: z = NOG x HOG

and
KY is the overall gas-phase mass transfer coefficient.
"a" is the packing parameter that we had seen earlier (recall the topic
on column pressure drop, e.g. Table 6.3) that characterize the wetting characteristics
of the packing material (area/volume).

Normally, packing manufacturers
report their data with both KY and "a"
combined as a single parameter. Since KY
has a unit of mole/(area.time.driving force), and "a" has a unit of
(area/volume), the combined parameter KY a will
have the unit of mole/(volume.time.driving force), such as kg-mole/(m3.s.mole
fraction). As seen earlier, other than mole fraction, driving force can be expressed
in partial pressure (kPa, psi, mm-Hg), wt%, etc.

y1* is the mole fraction of solute in vapour that is in equilibrium with the liquid of mole fraction x1 and y2* is mole fraction of solute in vapour that is in equilibrium with the liquid of mole fraction x2 .

The values of y1* and y2* can be obtained from the equilibrium line as previously covered (see section on Two-Film Theory). See the Figure below.

(y1 - y1*) is the concentration difference driving force for mass transfer in the gas phase at point 1 (bottom of column) and (y2 - y2*) is the concentration difference driving force for mass transfer in the gas phase at point 2 (top of column).

[ Point P (x, y) as shown is any point in the column. The concentration difference driving force for mass transfer in the gas phase at point P is (y - y*) as shown previously, this time no subscripts are shown. ]

NOTE: Both equilibrium line and operating line are straight lines under dilute conditions.

Alternatively, equilibrium values y1* and y2*
can also be calculated using Henry's
Law ( y = m x, where m is the gradient) which is used to represents the equilibrium
relationship at dilute conditions.

Thus, we have: y1* = m x1 ; y2* = m x2

Similarly for the liquid-phase
we have: z = NOL x HOL

and KX is the
overall liquid-phase mass transfer coefficient, and "a" is the packing
parameter seen earlier. Again, normally both KX
and "a" combined as a single parameter.

Likewise, x1* is the mole fraction of solute in liquid that is in equilibrium with the vapour of mole fraction y1 and x2* is mole fraction of solute in liquid that is in equilibrium with the vapour of mole fraction y2 . Refer to Figure 134 for finding values of x1* and x2* from the equilibrium line.

Alternatively, x1* = y1 /m and x2* = y2 /m.

(x1* - x1) is the
concentration difference driving force for mass transfer in the liquid phase at
point 1 (bottom of column) and (x2* - x2)
is the concentration difference driving force for mass transfer in the liquid
phase at point 2 (top of column).

Using either gas-phase or liquid-phase formula should yield the same required packing height :

[ For more info on packed column design, see Chp. 4, "Process Plant Design",
J.R. Backhurst & J.H. Harker, or for applications of various packed columns,
refer to "Random Packings and Racked Towers", R.F. Strigle Jr. ]