Difference between Thiele modulus and Damköhler number

[MexChemE]

Hi, PF! In the study of mass transfer involving chemical reactions, I have seen the use of two different non-dimensional parameters which apparently quantify the same ratio. These are the second Damköhler number and the Thiele modulus, which are defined as
$$\textrm{Da}^{\textrm{II}} = \frac{\textrm{reaction rate}}{\textrm{diffusion rate}} = \frac{\textrm{diffusion time}}{\textrm{reaction time}}$$
$$\phi = \frac{\textrm{reaction rate}}{\textrm{diffusion rate}} = \frac{\textrm{diffusion time}}{\textrm{reaction time}}$$
I would like to know why is a distinction made between both parameters, i.e. why do we need both dimensionless numbers if they quantify the same physical ratio. Right now, the only difference I found between the two is that Da^II is used only for heterogeneous reactions, whereas the Thiele modulus is used for both homogeneous and heterogeneous reactions.

My guess right now is that, even if they physically represent the same, the physical quantities needed to define each of them appear in different kinds of systems and equations. So in one system, the Damköhler number may appear when scaling its governing equation, and in a different one, the Thiele modulus appears when scaling the equation. But I would like to think there's more to it than that.

Thanks in advance for any input!

 

[gfd43tg]

Good question. I'm doing a reaction engineering laboratory right now and I have to analyze the Thiele Modulus of my catalyzed reaction. I will bring this up with my professor and let you know what he says

 

[MexChemE]

Good question. I'm doing a reaction engineering laboratory right now and I have to analyze the Thiele Modulus of my catalyzed reaction. I will bring this up with my professor and let you know what he says

That will be very appreciated!

A further question has popped up in my head. Whenever there's a heterogeneous reaction going on at a surface, the boundary condition we normally use is
$$N_{Az} |_{z = z_0} = k_1'' C_A |_{z = z_0}$$
If the reaction going on at the surface happens to be instantaneous, the boundary condition becomes
$$C_A |_{z = z_0} = \frac{N_{Az} |_{z = z_0} }{k_1''} = 0$$
Because $k_1'' \rightarrow \infty$. Now, for homogeneous reactions happening in the whole volume of the system, we don't use a boundary condition. We include a reaction rate term in the mass balance
$$R_A = k_1''' C_A$$
If we consider the reaction to be instantaneous, then, using the same reasoning as above, we have
$$C_A = \frac{R_A}{k_1'''} = 0$$
Because we again consider $k_1''' \rightarrow \infty$. But in this case $C_A$ is the concentration profile of A in the whole system, so this result means that as soon as A enters the system, it is consumed instantly by the homogeneous reaction, so there's no concentration profile to analyze for A. Is my reasoning right?

 

[BvU]

That is indeed the idea of an ideally stirred checmical reactor. Hence the I in CISTR

 

[MexChemE]

That is indeed the idea of an ideally stirred checmical reactor. Hence the I in CISTR

That's right! Wherein concentration is just a function of time and not of position. It feels nice when the dots connect.

So, can we say that one of the assumptions used to arrive at the ideal CSTR model is that the reaction is instantaneous? Does this apply to ideal batch reactors too?

 

[BvU]

So, can we say that one of the assumptions used to arrive at the ideal CSTR model is that the reaction is instantaneous?

Certainly not ! For a finite k you can also solve the equations and find a value for C_A.

 

[MexChemE]

Certainly not ! For a finite k you can also solve the equations and find a value for C_A.

Indeed, we can do an unsteady state mass balance and find C_A as a function of time. Macroscopically, in this model concentration is constant with respect to position, but not with respect to time. That being said, perhaps I should reword my question: When studying an ideal CSTR reactor from the microscopic (geometric) mass transfer viewpoint, do we neglect the spatial concentration gradients because we assume a high rate of reaction (and maybe high rate of diffusion too), or is it just because we imagine the impeller is doing a great job with the mixing? Or maybe a little bit of both.

 

[BvU]

The I in the acronym applies to the mixing: there is no spatial dependence - the reaction rate is the same everywhere in the reactor. And it can be any value.

 

[gfd43tg]

The dahmkohler number included transport from bulk to the surface of the pellet, whereas thiele modulus accounts for transport from surface to inside the pellet

 

[MexChemE]

The dahmkohler number included transport from bulk to the surface of the pellet, whereas thiele modulus accounts for transport from surface to inside the pellet

That works for me. It reminds me of the difference between the Biot and Nusselt numbers in heat transfer.