Linear rate of reaction

When the reaction rate is linear in concentration, the temperature is constant, and the geometry is planar, the problem reduces to

The exact solution to this problem is easily found by assuming a solution of the form exp(kr). Then, substitution into the differential equation gives

Thus,

We then write the solution as the sum of two terms

This is

where

Application of the boundary conditions shows that A = 0 and B = 1/cosh f, and

The effectiveness factor is then

The effectiveness factor is plotted as a function of Thiele modulus f in the figure.

At small f the effectiveness factor is one, meaning that the rate of reaction is relatively uninfluenced by diffusion. For large f the effectiveness factor is smaller than one, meaning that the average reaction rate is reduced below what it would be without diffusion limitations. The reaction rate as a function of f is shown in the next figure.

We see that the rate is proportional to f² for small f but is proportional to f for larger f values. Since f² is proportional to the reaction rate constant, this means that the actual reaction rate is lowered due to the influence of diffusion. This effect must be correctly modeled by the chemical engineer in the design and operation of catalytic chemical reactors. The concentration profiles inside the pellet shown in the next figure illustrate the same phenomenon.

For small f the concentration remains at the boundary value and diffusion effects are minimal. For larger f, the concentration decreases away from the pellet surface due to diffusion, and since the reaction rate is less when the concentration is less, the inner part of the pellet contributes less to the overall reaction rate. For the larger f's shown the mass is confined to a narrow layer near the boundary.

Continue on to: Nonlinear Problem.

Back up: Formulation, Non-dimensional problem, Combination of Variables, Effectiveness Factor.