A major premise for the use of kinetic models based on the commercial catalyst is that there is a unique reaction rate for each set of concentrations and temperature on the catalyst surface. Fortunately, the conditions that lead to multiple steady states are limited. Let's examine the limitations.
A model for n th order reactions
Before discussing the limitations, we need to define the model and the parameters. For reactions that are a function of only one reactant, the heat and mass balances may be expressed as shown below.
I have not defined all of the parameters in the dimensionless groups, but they should not be difficult to identify. For further details, see Carberry, Finlayson, or Smith.
Finlayson shows how the two balances can be solved to obtain the temperature profile as a function of the internal concentration. This reduces the model to the mass balance as shown below.
Intrapellet multiple steady states
Our main concern is the existence of multiple steady states when there is very little film resistance. A low film resistance is the desired condition for the laboratory kinetic experiments using the commercial catalyst. We will first examine the possibility of multiple steady states due to the reactions within the pellet, given the surface conditions. In a later post, we will examine the possibility of the film causing multiple surface conditions.
To eliminate the film resistance, high values may be used for the two Biot numbers.
Results for first order reaction
Smith, Fig 11-13, p 502 and Carberry, Fig9-1, p 458 show effectiveness factors for Biot mass and Biot heat numbers of infinity. Both refer also to Weisz and Hicks.
Both plots were made with gamma = 20. They both indicate that multiple solutions occur when beta = 0.4 or greater. This is a fairly high heat of reaction, so most systems will not have multiple steady states.
The second observation is that the multiple steady states exist only over a small range of the Thiele modulus. This too limits the presence of multiple steady states.
A final opinion
You certainly don't want to operate a reactor in the multiple steady state region because it could present control problems. However, if the reactor is designed to operate with Thiele modulus values above the multiple steady state region, you need to be aware of that region and plan your startup procedure accordingly as the Thiele modulus may pass through the region during heatup. Thus, your laboratory program should include a temperature survey to see if the reaction rate has a step change at some temperature. Develop your model in the temperature region above or below this step temperature (if found) depending upon the needed reaction rate.
The figures in Smith and Carberry assume that the derivative of temperature and concentration are zero at r = 0. In practice, these derivatives may be zero at a r > 0. If that fact is recognized in the model, the effectiveness factor plots look quite different. The above conclusions regarding the multiple states are not affected, but it appears that the effectiveness factor for high Thiele modulus values may be different than depicted. Stay tuned for more on this subject.
Carberry, J.J., Chem. and Catalytic Reaction Eng., Dover (2001)
Finlayson, B.A., Nonlinear Analysis in Chem. Eng., McGraw-Hill (1980)
Smith, J.M., Chemical Eng. Kinetics, McGraw-Hill (1981)
Weisz, P.B. and J.S. Hicks, Chem. Eng. Sci., 17, p265 (1962)