Equations

Variables and parameters

Let’s consider a substance \(S\). Let’s call:

  • \(S_b(t)\): Concentration of the substance on the bulk (liquid) phase, outside all particles. The substance could be a substract or a product of the reaction. It is usually measured in mols per liter.
  • \(V_b\): Total volume of the bulk (liquid) phase. Usually measured in liters.
  • \(S(t,r,R_i)\): Concentration at time \(t\) and radial position \(r\), inside a particle of radius \(R_i\). Measured in the same units of \(S_b(t)\).
  • \(f(R)\): Particle size distribution. Typically, this is a discrete approximation of the real (measurable but ultimately unknown) particle size distribution. For practical purposes we will consider a finite discrete distribution with \(N_R\) different particle sizes, where the probability \(p_i\) for a particle having radius \(R_i\) for \(i \in \{1, 2, \cdots, N_R \}\) with \(\sum_{i=1}^{N_R} p_i = 1\).
  • \(V_R\): total volume of particles, experimentally obtained with the total weight and density of the catalyst particles. Measured in the same units as \(V_b\).
  • \(D_S\): Effective diffusion coefficient of substance \(S\) inside the (porous) particle. It has the units meters squared / second.
  • \(v_e\): Effective reaction rate at which the amount of substance \(S\) changes without considering diffusional restrictions. If \(v_e>0\) it is ussually called a product, while \(v_e<0\) is called a substract. This is usually measured in the units of \(S_b\) per second.

Impact of a particle distribution

We define \(N_R\) the number of different particle radii. A discrete particle size distribution has probability \(p_i\) for a particle having radius \(R_i\), for \(i \in \{1, 2, \cdots, N_R \}\) with \(\sum_{i=1}^{N_R} p_i = 1\). The probability \(p_i\) is interpreted in a frequentist approach: it is simply the fraction of particles of the size \(R_i\), given by \(p_i = n_i / n\) with \(n = \sum_{i=1}^{N_R} n_i\) being the total number of particles.

We can then work out explicitely the total number of particles from the total volume of the particles:

\[\begin{split}V_R & = \sum_{i=1}^{N_R} n_i \frac{4 \pi}{3} R_i^3 = \sum_{i=1}^{N_R} p_i \ n \ \frac{4 \pi}{3} R_i^3 \\ & = n \ \frac{4 \pi}{3} \sum_{i=1}^{N_R} p_i R_i^3\end{split}\]

That is, the total number of particles is given by the total volume and the expected volume of a single particle.

\[n_i = p_i n = p_i \frac{V_R}{\frac{4 \pi}{3} E \left[ R^3 \right]}\]

Let’s consider a numerical example. Let’s imagine we have a total volume of \(V_R = 10 \ [m l] = 1.0\ E-8 \ [m^3]\) Let’s consider the following distribution \(p_1 = 0.4\) and \(p_2 = 0.6\), for \(R_1 = 0.9 R_0\) and \(R_2 = 1.1 R_0\), where \(R_0=6.5 \ E-9 \ [m]\) respectively.

We can compute the following values:

  • The expected radius is \(E[R] = p_1 R_1 + p_2 R_2 = XxX\)
  • The volume for a expected radius is \(\frac{4}{3} \pi (E[R])^3 = XxX\)
  • The number of particles is \(V_R / \frac{4}{3} \pi (E[R])^3 = XxX\).
  • The number of particles of size \(R_1\) and \(R_1\) are \(n_1 = p_1 n = XxX\) y \(n_1 = p_1 n = XxX\), respectively.
  • The total surface is

How to model the effective reaction rate

The effective reaction rate \(v_e\) is a function of several terms, and occurs only inside the particles, where the catalist is attached to the surface of the porous structure. In a particle of radius \(R_i\), it would be:

\[\begin{split}v_e(t,r,R_i) &= v_e \left( S(t,r,R_i), E(t,r, R_i), \textrm{other relevant parameters} \right) \\ & \approx v \left( S(t,r,R_i), E_{max}, \textrm{other relevant parameters} \right) \times I(t) \times Z(r, R_i)\end{split}\]

Where:

  • \(v(S, E, \textrm{other relevant parameters})\): the reaction rate, measured in the units of \(S_b\) per second.
  • \(I(t)\): Enzime Inactivation. It only has time dependance, being bounded between 0 and 1 and decreasing: \(0 \leq I(t) \leq 1\). It has no units. It models tha catalyst inhibition growing over time.
  • \(Z(r, R_i)\): Enzime radial distribution, that only has space dependance and being non-negative, \(0 < Z(r, R_i)\) and such that the total enzime applied to all particles is a known value \(E_0\):
\[\begin{split}E_0 &= \sum_{i=1}^{N_R} n_i \int_0^{R_i} E_{max} Z(r, R_i) 4 \pi r^2 dr \\ &= n \sum_{i=1}^{N_R} \frac{n_i}{n} \int_0^{R_i} E_{max} Z(r, R_i) 4 \pi r^2 dr \\ &= n E_{max} 4 \pi \sum_{i=1}^{N_R} p_i \int_0^{R_i} Z(r, R_i) r^2 dr\end{split}\]

Here we have used \(n_i\) the number of particles of size \(R_i\), and \(n\) the total number of particles, and the relationship between volume and particle size distibution:

\[n = \frac{V_R}{\frac{4 \pi}{3} E \left[ R^3 \right]} = \frac{V_R}{\sum_{i=1}^{N_R} p_i \frac{4}{3} \pi R_i^3}\]

The equations

The equations, boundary conditions and initial conditions are given for \(S_b(t)\) and \(S(t,r,R_i)\).

The reaction diffusion equation, for \(t>0\) and \(0<r<R_i\):

\[\frac{\partial S}{\partial t}(t,r,R_i) = D_S \left(\frac{\partial^2 S}{\partial r^2}(t,r,R_i) + \frac{2}{r}\frac{\partial S}{\partial r}(t,r,R_i)\right) - v_e\left(S(t,r,R_i)\right) I(t) Z(r, R_i)\]

The boundary condition at the center of the particle for \(t>0\):

\[\frac{\partial S}{\partial r}(t, 0, R_i) = 0\]

The boundary conditions at the surface of the particles are

\[S_b(t) = S(t, R_i,R_i)\]

and

\[\begin{split}\frac{d S}{d t}(t, R_i, R_i) &= - 3 D_S \frac{V_c}{V_R E \left[ R^3 \right] } E \left[ R^2 \left. \frac{\partial S}{\partial r} \right|_{r=R} \right] \\ &= - 3 D_S \frac{V_c}{V_R \sum_{i=1} ^{N_R} R_i^3} \sum_{i=1} ^{N_R} R_i^2 \frac{\partial S(t,R_i, R_i)}{\partial R} \\\end{split}\]

The initial conditions are

\[\begin{split}S_b(0) &= S_0 \\ S(0,r, R_i) &= 0 \textrm{ for } 0 \leq r < R_i \textrm{ and } i \in \{ 1, 2, \cdots, N_R \}\end{split}\]