In the 1/12/14 post I mentioned the need to include a temperature survey in your laboratory kinetics program in order to see if multiple states exist in the preferred operating temperature range. This post offers some additional guidance
on where multiple solutions might exist.
The critical Thiele modulus
I have noticed that the multiple solution conditions for n th order reactions are near the condition where the reactant concentration at the center of the catalyst approaches zero. For example, the figure below shows the gap where multiple solutions exist for the conditions shown. This gap occurs where the "inactive" radius begins. (Inactive is not correct, the catalyst is still active but reaction is not significant below this radius.)
In some cases, multiple solutions may not occur at this transition between full and partial catalyst utilization, but the reaction rate can still undergo a steep change as the Thiele modulus increases. This steep change in rate is also undesirable. Thus, by adding a constraint to the model that the reactant concentration at the center is less than 0.02, I was able to solve for the Thiele modulus for this transition. If the Thiele modulus is less than this "critical modulus", then reaction is taking place in the entire catalyst. If the Thiele modulus is greater that the critical value, then reaction is taking place only in an outer shell.
For first order reaction
The plot below shows the critical modulus as a function of heat of reaction and activation energy parameters. The Biot numbers are again assumed to be very high, reducing the resistance of the external film.
Values of Thiele modulus and heat of reaction above a line mean that only the outer shell of the catalyst is being used...below a line, reaction occurs throughout the catalyst. Unfortunately, you need an estimate of the catalyst effective diffusivity and conductivity to use the plot (for Thiele modulus and the dimensionless heat of reaction.)
In order to make the plot, you also need estimates of the particle Biot numbers. Although, the curves change some, the plot below shows that realistic, commercial values for the mass Biot number and delta are not much different from those in the plot above.
Other reaction orders
Similar plots can be made for reaction orders 2 and 3.
Rate expressions with denominator concentration terms can lead to multiple steady states even under isothermal conditions, i.e. Beta = 0. Carberry shows several plots for effectiveness factors for LH rates, both isothermal and nonisothermal. Again, the multiple solutions occur in a narrow band of Thiele modulus. The isothermal step changes are small compared to the nonisothermal step changes.
Operation above the critical Thiele modulus may not be desirable for the commercial system: a high catalyst temperature may harm the catalyst. Also, considerable catalyst volume is not being used. For catalysts made with precious metals, that can be a waste of capital.
The size of the catalyst is usually dictated by pressure drop restrictions. Thus, selection of a smaller catalyst is probably not an option to reduce the Thiele modulus. It may be possible to make the catalyst with active material only in the shell region. This could be more economical, but it doesn't avoid the high temperature. The catalyst activity may be lowered using less active agent and thereby move the operating point vertically down, but then the lower reaction rate may make the reactor too large.
A catalyst with a low activation energy can be used to avoid the step change in reaction rate, but usually the catalyst developer has given you, the reactor engineer, the best catalyst available at that time. "Best catalyst" usually equates to one with a low activation energy.
The best solution is to lower the dimensionless heat of reaction, Beta, which can be accomplished by lowering the reactant concentration in the fluid.
Lowering the reactant concentration moves an operating point horizontally to the left (for a first order reaction). By combining a reduced reactant concentration with a reduced catalyst activity, an intermediate operating point may be found that results in a reasonable reactor size and temperature profiles. Such an intermediate rate would not be possible with the conditions in the multiple solution region.
The above analysis has assumed a first order reaction. For higher order reactions, lowering the reactant concentration also lowers the Thiele modulus, moving an operating point both left and down.
A method was developed that can help locate where multiple steady states might occur for n th order reactions. In addition, the method reveals conditions where the catalyst is not fully utilized. This latter subject is often overlooked in reactor design books.
Students: a challenge
Create a critical Thiele modulus plot for a second order reaction.
To follow: the effect of film resistances on multiple
Carberry, J.J., Chem. and Catalytic Reaction
Eng., Ch. 9, Dover (2001)