Thiele number

Normally when a small change is made in the condition of a reactor, only a comparatively small change in the response occurs. Such a system is uniquely stable. In some cases, a small positive perturbation can result in an abrupt change to one steady state, and a small negative perturbation to a different steady condition. Such multiplicities occur most commonly in variable temperature CSTRs. Also, there are cases where a process occurring in a porous catalyst may have more than one effectiveness at the same Thiele number and thermal balance. Some isothermal systems likewise can have multiplicities, for instance, CSTRs with rate equations that have a maximum, as in Example (d) following. 

Note that this parameter has the same form as the Thiele number which occurs in the theory of diffusion/reac tion in catalyst pores. 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration,... 

It seems that there is a need to reexamine, some of the basic quantities used in transport processes, like Thiele numbers, attempting to connect them to more chemical quantities. For example, the macroscopic quantity, e the dielectric constant, can be interpreted in terms of dipole moment distribution, and the dipole moment has immediate structural implications. Now to talk of a dielectric constant in the interaction of two atoms would be a rather useless exercise, since the dilectric constant is a continuous matter concept, not a discrete matter concept. In the same... 

The single dimensionless group appearing in the model is referred to as the Thiele number or Thiele modulus in recognition of Thiele s pioneering contribution in this area [20]. The Thiele modulus quantifies the ratio of the reaction rate to the diffusion rate in the pellet. 

In his original paper, Thiele used the term modulus to emphasize that this then unnamed dimensionless group was positive. Later when Thiele s name was assigned to this dimensionless group, the term modulus was retained. Thiele number would seem... 

Effective Area. - The catalyst consists of active metal dispersed on a substrate material. Therefore, there is a difference between the effective area used in mass transfer calculations of chemical reaction and the effective area used in heat transfer culculations which corresponds to the monolith surface area including surface roughness. The ratio of both effective areas must be defined based on experimental results, as they depend on catalyst type and manufacturing processes. The Thiele number is sometimes used for this same purpose. The relationship between effective area ratio and conversion efficiency is shown in Figure 3. This effective area ratio may be one of the characteristic values of the catalyst, which affects catalyst performance and catalyst temperature. The effective area ratio in the present study is estimated to be 0.3 for mass transfer and 1 for heat transfer based on the experimental data. 

There is a plethora of DNs. Some additional DNs are the Grashof number (Gr), Mach number (Ma), Peclet number (Pe), Schmidt number (Sc), Lewis number (Le), Rayleigh number (Ra), Stanton number (Sn), Thiele number (Th), and Weber number (We). 

Thiele proved that the utilization (y>) of the iimer smface is related with the parameter /x for the first order reaction. The equation of the smface reactions in the porous catalysts was proposed for the process with imchanged volrnne of materials. The equation was applied to gasification process of coal and the reduction of porous catalysts. Here, the reduction product layer appearing on the surface should be considered. It must also consider the effect of a variety of solid state processes on controlling all the processes. However, Thiele number does not take the impact into account for solid state process. 

The situation where during the entire process the Thiele number is smaller than 0.3 is easier to model. In that case the effectiveness factor is always 1, and the reaction proceeeds throughout the porous solid with the same rate. The reaction rate per unit internal surface area J (= / ) is assumed to be constant in time. 

This

catalyst particles. For reactions carried out with fluid bed catalysts, the Thiele number is usually low, because of the small particle size, so that often T] 1. The desi of fluid bed reactors is discussed in section 7,22,4 and in the Appendix to section 4.5.1. 

Thiele number for first order reaction or specified with subscripts. Thiele modulus /g for first-order reaction. 

---