*To: asloley@distillationgroup.com
Subject: Mass-Transfer Unit Explanation
Date: Wed, 3 Jul 2002 16:18:55 -0400*

I enjoy reading the distillation notes but I have a question concerning the value of a mass transfer unit. What value is a mass transfer unit, as to units ie: lbm, kg, I am confused.

A. K.

North America

*To: A. K.
Subject: Mass-Transfer Unit Explanation
Date:*

A.,

The origin of your confusion is probably a concept mis-match that starts in terminology. When we talk of a unit in general use we mean the dimension attached to a quantity. For a dimension of mass, the unit would be kg or lbm. The meaning of unit in mass-transfer unit is different. The concept involved is the 'unit operation'. A mass-transfer unit is a physical piece of equipment that performs a specific function. What makes this difficult to understand is that no simple, clear-cut physical representation can easily be made as an example of a physical mass transfer unit.

Commonly, two approaches are used to evaluate the performance of a tower or to determine the quantity (height) of mass-transfer equipment needed:

- Theoretical stage approach
- Mass-transfer unit approach

In theory, either approach could be used to evaluate mass-transfer services. However, in practice, each is better suited to specific applications.

The theoretical stage approach is an extension of the Sorel
analysis[1,2]. The McCabe-Thiele [3] diagram simplifies Sorrel
and generates a simple graphical representation. This is especially
suitable for distillation services with significant mass-transfer
in both directions: from the liquid phase to the vapor phase *and*
from the vapor phase to the liquid phase. To move from theoretical
stages to physical equipment we use a tray efficiency or a packing
height equivalent to a theoretical plate (HETP). The single theoretical
stage has a straightforward physical representation, the flash
drum.

Flash drum, one theoretical stage

The mass-transfer unit approach creates a more realistic description of packed towers. Many people refer to this approach as a rate-based model of packed tower performance. The results of this analysis are normally expressed in transfer units (hence the name mass-transfer unit approach). This is especially suitable for services with small quantities of material transferred in essentially one direction only. Classic strippers and absorbers fall into this category. Strippers transfer a component (solute) from the liquid phase (solvent) to the vapor phase. Absorbers transfer a component (solute) from the vapor phase (solvent) to the liquid phase. Additionally, in most of these services the bulk phase rates are essentially constant. The quantity of solute transferred is small compared to both phases in the equipment.

Typical applications of the mass-transfer approach to design, analysis, and troubleshooting include environmental applications (water stripping), humidification (water into dry air), gas treating (amine acid-gas absorbers), dehydration (glycol contactors) and similar services. Many people find the mass-transfer unit more difficult to understand as we have no clear-cut representation of a physical equivalent to a mass-transfer unit. Additionally, graphical solutions to the mass-transfer analysis are rarely presented. This further deepens the common lack of understanding of the method. For graphical methods, rather than thinking of the McCabe-Thiele stage approach, we would more usefully think of the mass-transfer unit approach as being similar to the graphical solution of a heat-exchanger problem with a log-mean temperature being integrated to find the driving force over the exchanger surface.

While many authorities present the analytic approach to the mass-transfer unit, few make the effort explain the underlying concept. Fewer still show the graphical methods that would help understanding. One of the best descriptions of the mass-transfer approach (while lacking the rigor of the design details required) is included in Watson [4]. Strigle [5] also includes a useful description along with detailed examples.

Figure 2 creates our starting point for the graphical approach to the mass-transfer method.

y: concentration in the gas phase

x: concentration in the liquid phase

i: inlet

o: outlet

(i and o should be read as subscripts)

Figure 2 shows an absorber removing a component with concentration y in the gas phase and transferring it to the liquid phase. We will restrict our analysis to one classic case: absorption of a gas contaminant with a small concentration in the gas phase, no heat of mixing affects, steady-state operation, and mass-transfer rate controlled by mass-transfer resistance in the gas phase. Similar methods can be applied to other problems. Those interested should refer to the references [4,5] for application to other cases. This same analysis can be applied to a sub-section of a larger column where the applicable conditions are meet.

With our assumptions, the mass-transfer resistance from the liquid film to the bulk of the liquid is very low. Under this condition, the concentration of the solute in the bulk liquid is essentially equal to the concentration in the liquid film. If we plot the vapor-phase mole concentration against the liquid-phase concentration we construct an equilibrium diagram for the service (Figure 3).

Mass-transfer operating diagram

(Many diagrams are constructed with partial pressures of the vapor phase used instead of mole fractions.)

(Depending upon the system, the interface lines may be curves.)

The concentration at the interface at any point in the tower is located at a lower concentration than the bulk concentration of the component of interest in the gas phase. A series of tie-lines illustrates the join between the equilibrium concentration on the bulk and the concentration at the interface (Figure 4).

Tie-lines showing equilibrium relationships between phases

The rate of mass-transfer per unit volume of transfer device (typically packing) for our case is expressed:

a: interfacial area per unit volume

yg: component concentration in the gas phase

yi: component concentration at the interface

This gives a relation between the number of moles transferred between phases per volume in terms of concentration driving force. Substituting partial pressures in place of gas-phase mole compositions would replace yg with pg and yi with pi. Units must be consistent in this analysis. At steady state and dilute systems we can write the rate of mass-transfer by:

z: vertical height

Integrating across the tower (or section of interest) we get:

The left hand integral is difficult to resolve analytically except in a few cases. Computer programs use numerical methods to solve this integral. However, graphical methods can solve this rapidly as well. By defining the reciprocal of kga/G as a height term G/kga and solving for L we get:

The term being the number of mass-transfer units required. Taking Figure 4, extracting the correct values and plotting yg versus yg-yi we get Figure 5. Integrating the area under the curve from yin to yout gives the graphical solution to the number of transfer units.

Curve for graphical solution to the number of transfer units required for a service.

Analogous techniques can be derived for other cases.

[1] Sorel, E. La Rectification de l'alcohol. Paris, 1893.

[2] Sorel, E. La Distillation, vol. 104 in Encyclopédie scientifique des aide-mémoire. Gauthier-Villars et Fils, Paris, no date.

[3] McCabe, W. L. and Thiele, E. W. *Ind Eng Chem*, **14**,
492 (1922).

[4] Watson, J. S. Separation methods for waste and environmental applications. Marcel Dekker, New York, 1999.

[5] Strigle, R. F., Jr. Packed tower design, Second edition. Gulf Publishing Company, Houston, 1994.

Andrew Sloley

DGI

© 2002 Andrew W. Sloley. All rights reserved.