Multiple steady states for catalyst: III

Film effect on steady states
Carberry (2001), Section 5-14, discusses the possibility of the fluid film causing multiple steady states even for a reaction that occurs only on the outer surface of a solid catalyst. In my investigations, it appears that multiple solutions to the heat balance for the catalyst are not likely. [See also Froment(1972)]. In order for multiple solutions to exist, the values of five parameters need to be "just right". The parameters are shown below.

In my trials using various values for the above parameters, a condition that would lead to multiple solutions has a small "window" for the parameter values. For example, the Stanton number has to be just right in order for the heat removal line to line up with the heat generation curve's transition portion. Another requirement is that the heat generation curve has an "S" shape. Not all combinations of Da, Beta_bar, and gamma lead to that shape: many result in a shape that allows only one intersection by the heat removal line.

The parameters also have relationships not shown by their definitions. The mass transfer and heat transfer parameters in Da and St are related by the Nusselt and Sherwood correlations. These correlations also relate the velocity, u, to the mass transfer parameter. Thus, K and Da are linked. These additional constraints need to be included when determining the possibility of multiple solutions.

Carberry also develops a nonisothermal external effectiveness factor in Section 5-4. By solving the heat and mass balances, he obtains the following formula:

There are only three parameters in the formula for the external effectiveness factor instead of five. That is due to the inclusion of the additional constraints mentioned above. The heat of reaction parameter has been redefined to include the Lewis number which relates the thermal diffusivity to the mass diffusivity. In addition, the 2/3 power for the Lewis number indicates that a high Reynolds number has been assumed in calculating the heat transfer and mass transfer parameters from the Nusselt and Sherwood correlations (see below). With these changes, Carberry's Fig 5-5 through 5-7 show no indication of multiple steady solutions.

The above analysis is for n th order reaction rates. Langmuir-Hinshelwood rates can lead to isothermal multiplicity from external film resistance (Carberry Sec. 5-15).

Laboratory experimental design
Even if there are no multiple states, a high Beta can result in a very high catalyst temperature. Such a condition may be mistaken for a second, high temperature solution. To avoid the high temperature, lower the nominal concentration of the reactant to reduce Beta. High temperatures may also be caused by low velocity, which reduces the mass transfer parameter. Try to avoid extremely low velocities in laboratory experiments.

Conclusions:
This series of posts has shown that kinetic analysis using the commercial size catalyst should not be confounded by multiple steady states. The multiple solutions that do exist are due to the intraparticle resistances or adsorption phenomena and these multiple solutions occur over a narrow range of parameter values. The interphase film resistance does not appear to be a cause for multiple solutions.

Conducting kinetic studies on the commercial catalyst avoids severe errors that are due to the range of possibilities for the catalyst effective transport parameters. For example, the ratio of the mass and energy Biot numbers can vary over four orders of magnitude. In contrast, the transport parameters for the external film can be determined from more precise molecular theory for gases and from empirical correlations for liquids. Also, measured values of thermal conductivity and viscosity are available for many compounds. These values may be available in the Prode Properties database.  

Carberry, J.J., Chem. and Catalytic Reaction Eng., Dover
(2001)
Froment, G.F., Adv. Chem. Ser., 109, 1 (1972)