Exact Analytical Solution Steady-State Approximation

3 Exact Analytical Solution (Steady-State Approximation) 

Steady-state approximations are useful and thus are used extensively in the development of noathematical models of kinetic processes. Take, for example, the reaction A B — C (Fig. 1.15). If the rate at which A is converted to B equals the rate at which B is converted to C, the concentration of B remains constant, or in a steady state. It is important to remember that molecules of B are constantly being created and destroyed, but since these processes are occurring at the same rate, the net effect is that the concentration of B remains unchanged (rf[B]/rft = 0), thus  

Decreases in [A] as a function of time are modeled as a first-order decay process  

The value of k can be determined as discussed previously. From Eqs. (1.137) and (1.138) we can deduce that 

If the steady state concentration of B [Bss], the value of k, and the time at which that steady state was reached (tss) are known, k2 can be determined from 

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The rotating ring—disc electrode (RRDE) is probably the most well-known and widely used double electrode. It was invented by Frumkin and Nekrasov [26] in 1959. The ring is concentric with the disc with an insulating gap between them. An approximate solution for the steady-state collection efficiency N0 was derived by Ivanov and Levich [27]. An exact analytical solution, making the assumption that radial diffusion can be neglected with respect to radial convection, was obtained by Albery and Bruckenstein [28, 29]. We follow a similar, but simplified, argument below. 

Although in the vast majority of this book we consider approximate solutions, there are a few exceptional classes of problems for which an exact analytic solution is possible, and we focus on them in this chapter. This will give us a chance to review analytic solution techniques for linear PDEs. More importantly, it will give us an opportunity to introduce a number of important concepts about scaling and nondimensionalization in a framework of relatively straightforward physical problems and to explore some aspects of the time evolution of steady flows from a state of rest. 

Once the differential equations and mass balance have been written down, three approaches can be followed in order to model complex reaction schemes. These are (1) numerical integration of differential equations, (2) steady-state approximations to solve differential equations analytically, and (3) exact analytical solutions of the differential equations without using approximations. 

Analytical methods provide exact solutions that allow for direct analysis of the influence of experimental variables and the determination of the conditions for particular behaviours such as the achievement of a steady-state signal. Nevertheless the use of analytical methods is not always feasible due to the complexity of the problems. In such cases numerical methods offer a very accurate approximation to the true solution once the conditions of the simulation are optimised. 

Coupling may also occur via boundary conditions, e.g. the reaction rate in a catalyst pellet depends on the concentration and the temperature of the fluid surrounding the pellet. At steady state, when coupling between equations occnrs throngh boundary conditions, an exact or approximate analytical solution can be calculated with boundary conditions as variables, e.g. the effectiveness factor for a catalyst particle can be formulated as an algebraic fimction of surface concentrations and temperature. The reaction rate in the catalyst can then be calculated using the effectiveness factor when solving the reactor model. However, this is not possible for transient problems. The transport in and out of the catalyst also depends on the accumulation within the catalyst, and the actual reaction rate depends on the previous history of the particle. 

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