## Determining Mass Transfer Coefficients

It isn't reasonable to expect mass transfer coefficients to be readily available for any and all systems.

The "best" solution is to experimentally measure coefficients on a bench scale (using a wetted-wall column, etc.) and then use the results to design a full scale separation column. When this isn't feasible, more approximate arrangements must be made.

### Correlations

Dimensional analysis of mass transfer suggests correlations of the form:

A number of correlations matching this form are presented in your textbook, pp. 533-40.

Treybal (1987) suggest the following correlations for use with beds packed with Raschig rings or Berl saddles:

subject to the following definitions

We will be using these in a class example.

### Analogies

Since the basic mechanisms of heat, mass, and momentum transport are essentially the same, it is sometimes possible to directly relate heat transfer coefficients, mass transfer coefficients, and friction factors by means of analogies.

Analogies involving momentum transfer are only valid if there is no form drag, hence they are pretty much limited to flow over flat plates and inside (but not across) conduits. Also recognize that if there is much heat or mass transfer, it may change fluid and flow characteristics enough to make analogy worthless; in some cases, a viscosity correction may be used to compensate.

A simple, crude analogy recognizes that turbulent eddies transport heat and mass as well as momentum, thus one can argue that the eddie diffusivities are the same for all modes of transport, that is: ET = EH = EM. These values are seldom at hand, though.

Another analogy -- probably the oldest -- is the "Reynolds Analogy", which relates the Fanning friction factor for fluid flow to heat transport:

where the right hand side is the "Stanton Number". The Stanton number is a dimensionless group made up of other, more familiar groups. It can be defined for heat transfer or for mass transfer.
The Reynolds analogy gives reasonable values for gases where the Prandtl number is roughly one.

Note that one-half the friction factor is the ratio of the overall momentum transported to the wall to the inertial effects in the mainstream. The Stanton number represents the ratio of the overall heat transport to the wall to the convective effects in the mainstream. The Reynolds analogy says that these ratios are equal for mass and momentum transport.

The Reynolds analogy postulates direct interaction between the turbulent core of the flow and the walls. If a laminar sublayer is included between these, the Prandtl-Taylor analogy applies:

This form includes the ratio of the mean velocities in the sublayer and core as well as the Prandtl number for heat transfer. Note that when the Prandtl number is equal to one, this equation reduces to the Reynolds analogy.

Probably the most widely used is the Colburn (or Colburn-Chilton) analogy. It is based on correlations and data rather than on assumptions about transport mechanisms. The Colburn "j-factor" for heat transfer and the Colburn-Chilton j-factor for mass transfer are:

The heat transfer factor may be modified with the Seider-Tate viscosity correction
although this does not seem to be universally done.

When the j-factors are used, the fluid properties in the Stanton number are evaluated at the mean bulk average temperature and those for the Prandtl number at the film temperature (this means two heat capacities!).

The Colburn-Chilton analogy is simply

valid for turbulent flow in conduits with NRe > 10000, 0.7 < NPr < 160, and tubes where L/d > 60 (the same constraints as Seider-Tate).

A wider range of data is correlated by the Friend-Metzner analogy:

which is valid when NRe > 10000, 0.5 < NPr < 600, 0.5 < NSc < 3000.

### Coefficients from Reference Conditions

Another possibility is to estimate mass transfer coefficients by comparison with measured values for reference systems.

For instance, the overall mass transfer coefficients for the oxygen-water system has been measured (see MSH Fig 18.21, p. 581) and can be used to predict overall coefficients for other systems using

MSH suggest a typical value of n=0.3, so new values can be obtained using

For gas-film coefficients, MSH provide data for ammonia-water, and recommend estimation using

References:

1. Brodkey, R.S. and H.C. Hershey, Transport Phenomena: A Unified Approach, McGraw-Hill, 1988, pp. 516-20.
2. McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering (5th Edition), McGraw-Hill, 1993, pp. 348-52.
3. McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering (6th Edition), McGraw-Hill, 2001, pp. 532-40, 580-88.

R.M. Price
Original: 12/99
Modified: 1/27/2003