Christina Papadimitriou’s project: Diffusion and Reaction in Catalysts

In this project I will investigate a significant application of differential equations on heterogeneous catalysis. In the first part of the project, and using the reaction rate differential equation of a first order chemical reaction, I will first derive an equation of the concentration of the reactants as a function of time. It is important to mention that this equation varies for chemical reactions of different orders. Then, I will derive a second equation that shows how the concentration of the reactants of a first order chemical reaction, catalyzed by a spherical catalyst, changes as the reactants diffuse in the catalyst. This equation varies for catalysts of different shapes.

In the second part of the project, I will investigate other cases of higher order reactions catalyzed by catalysts of different shapes such as cylinders and slabs.

Part I: Diffusion and Reaction in a spherical Catalyst

What is Catalysis?

A catalyst is a substance that can increase the rate of a chemical reaction without being consumed by the reaction itself. Catalysts can be organic, synthetic or metal.

The process by which the catalyst speeds up a reaction is called catalysis. For any chemical reaction to occur, energy, known as activation energy, is required. When the catalyst is present the activation energy is lowered making the reaction happen more efficiently. Catalysts can be divided into two classes: homogeneous and heterogeneous. Homogeneous catalysts are in the same phase as the reactants, e.g. both are in the liquid phase, while heterogeneous catalysts are in a different phase from the reactants, with most commonly catalysts being solid with gas and liquid reactants.

Why is Catalysis important?

Catalysis underpins many industries and has numerous applications that impact heavily our daily lives. Catalysts are crucial for the chemical industry, as approximately 85-90% of all chemical products are made in catalytic processes. Catalysts are indispensable in the petroleum industry as well, since the process of catalytic cracking, which is the heart of the modern oil refinery, uses a catalyst to convert crude oil to lower fractions such as gasoline and diesel. They also find applications in the industry for treatment of effluents and they are instrumental in abating the discharge of harmful species to the biosphere. One of the most successful applications of catalysis is the catalytic converter, which is found in cars and convert’s almost 98% of the engine’s harmful emissions into non-toxic products. Catalytic converters consist of a honeycomb-shaped ceramic structure, which is coated with metal catalysts, such as Rhodium, Platinum and Palladium, and convert the emissions of polluting gases (CO, NOx etc) into non-polluting gases (N2, H2O, CO2 etc).

First Order Reaction

In a first order, irreversible, chemical reaction with reactant A and product B the reaction rate is:

A \to B \qquad r=k \cdot C_A = - \dfrac{dC_A}{dt}

where k is the rate constant of the reaction and C_A is the concentration of the reactant A.

The rate equation is a differential equation, and it can be integrated to obtain an integrated rate equation that links the concentration of the reactant A with time:

\begin{array}{c} r = - \dfrac{dC_A}{dt} = -k C_A \\ \\ \dfrac{dC_A}{C_A} = -k\, dt \\ \\ \ln C_A = -kt + \ln C_{A_0} \\ \\ C_A = C_{A_0} e^{-kt} \end{array}

where C_{A_0} is the initial concentration of A in the solution and at t = 0, C_A = C_{A_0}.

First Order Reaction in a Spherical Catalyst

During any heterogeneous catalytic reaction, mass transfer of the reactants first takes place from the bulk fluid to the external surface of the catalyst. The reactants then diffuse from the external surface into and through the pores within the catalyst, with reaction taking place only on the catalytic surface of the pores. The chemical reaction is followed by desorption of the products at the catalytic sites and their transport from the catalyst interior to the external surface of the catalyst.

Figure 1: A schematic representation of the diffusion process.

In this project a first order reaction that occurs within a spherical solid catalyst is investigated. The diffusion of the fluid reactant A in the pores of the catalyst takes into account the random distribution of solid and voids as one moves from the exterior to the interior of the catalyst. The diffusion is expressed by the diffusion coefficient D_A, which depends on the physical properties of the reactants and the catalyst.

Figure 2: A spherical solid catalyst with radius r and the pores of its interior.

The concentration of the reactant A through the pores in a spherical catalyst with radius r is expressed by the following differential equation:

D_A \dfrac{1}{r^2} \dfrac{d}{dr} \bigg( r^2 \dfrac{dC_A}{dr} \bigg) - kC_A = 0.

Two boundary conditions are required to solve the equation above:

  1. The concentration remains finite at the center of the catalyst:
    \dfrac{C_A}{dr}\bigg\rvert_{r=0} = 0.
  2. At the external surface of the catalyst (which we assume of maximum radius r=R) the concentration is known, say C_{A_s}.
    C_A\big\rvert_{r=R} = C_{A_s}.
  3. Now we will solve the differential equation above to find an equation that relates the concentration of A to the radius r:

    \begin{array}{c} D_A \dfrac{1}{r^2} \dfrac{d}{dr} \bigg( r^2 \dfrac{dC_A}{dr} \bigg) = k C_A \\ \\ \dfrac{d}{dr} \bigg( r^2 \dfrac{dC_A}{dr} \bigg) = \dfrac{r^2 k C_A}{D_A} \\ \\ r^2 \dfrac{dC_A}{dr} = \dfrac{k C_A}{3D_A}r^3 + C_1 \end{array}

    From the first boundary condition, it must be C_1=0,

    \begin{array}{c} r^2 \dfrac{dC_A}{dr} = \dfrac{k C_A}{3D_A}r^3 \\ \\ \dfrac{dC_A}{C_A} = \dfrac{k}{3D_A}r\, dr \\ \\ \ln C_A = \dfrac{k}{6 D_A}r^2+C_2 \end{array}

    From the second boundary condition:

    \begin{array}{c} C_2 = \ln C_{A_s} - \dfrac{k}{6D_A}R^2 \\ \\ \ln C_A = \dfrac{k}{6D_A}r^2 + \ln C_{A_s} - \dfrac{k}{6D_A}R^2 \\ \\ \ln \dfrac{C_A}{C_{A_s}} = \dfrac{k}{6D_A}(r^2-R^2) \\ \\ C_A = C_{A_s} e^{-\frac{k}{6D_A}(r^2-R^2)}  \end{array}

    Part II: Diffusion and Reaction in Other Geometries

    (in development)

    The reaction-diffusion differential equation for some different geometries are known:

    D_A \dfrac{1}{r^q} \dfrac{d}{dr} \bigg( r^q \dfrac{dC_A}{dr} \bigg) - kC_A = 0

    The differential equations for the cylinder (q=1) and the slab (q=0) can be solved similarly to the equation of the sphere and the concentration of the reactant A can be obtained as a function of r.


    Chemical Reactor Analysis and Design Fundamentals. James B. Rawlings and John G. Ekerdt; Nob Hill Publishing, Madison, WI, 2002.

  1. Anonymous
    December 13, 2012 at 7:28 pm

    In your third step when you integrate in terms of radius you treat Ca on the right side as a constant. Isn’t Ca a function of radius so after integrating in terms of radius Ca would no longer be just Ca?

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