Solutions to the Diffusion Equation

In Chapter 4 we described many of the general features of the diffusion equation and several methods of solving it when D varies in different ways. We now address in more detail methods to solve the diffusion equation for a variety of initial and boundary conditions when D is constant and therefore has the relatively simple form of Eq. 4.18 that is, 

This equation is a second-order linear partial-differential equation with a rich mathematical literature [1]. For a large class of initial and boundary conditions, the solution has theorems of uniqueness and existence as well as theorems for its maximum and minimum values.1 

Many texts, such as Crank s treatise on diffusion [2], contain solutions in terms of simple functions for a variety of conditions—indeed, the number of worked problems is enormous. As demonstrated in Section 4.1, the differential equation for the diffusion of heat by thermal conduction has the same form as the mass diffusion equation, with the concentration replaced by the temperature and the mass diffusivity replaced by the thermal diffusivity, k. Solutions to many heat-flow 

1If the diffusivity is imaginary, the diffusion equation has the same form as the time-dependent Schrodinger s equation at zero potential. Also, Eq. 4.18 implies that the velocity of the diffusant can be infinite. Schrodinger s equation violates this relativistic principle. 

Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 99 

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The theory of shape selection has been examined by many investigators concerned with solidification from the melt, and its status has recently been reviewed by Caroli and Muller-Krumbauer [63], The problem is to find stable, quasi-stationary solutions to the diffusion equation where a propagating branch maintains a constant shape and velocity. If the interface is assumed to have a uniform concentration, a family of such solutions exists, but there is no unique solution owing to the lack of a characteristic length. The solutions fix the peclet number. 

The solution to the diffusion equation for the experimental situation in which the coated surface is uncovered is... 

The next several sections discuss initial and boundary conditions, and methods to solve Equation 3-10 given such conditions. In the process of learning the methods, typical solutions to the diffusion equation will be presented. These solutions will be encountered in later discussions. More solutions are presented in Appendix 3. 

Some simple solutions to the diffusion equation at steady state... 

Since it is assumed that f and g are solutions to the diffusion equation, then... 

This result is known as the principle of superposition. The principle is useful in solving diffusion equations with the same boundary conditions, but different initial conditions, or with the same initial conditions but different boundary conditions, or other more general cases. Suppose we want to find the solution to the diffusion equation for the following initial condition ... 

The above derivation has not made use of the initial and boundary conditions yet, and shows only that A may take any constant value. The value of A can be constrained by boundary conditions to be discrete Ai, A2,..., as can be seen in the specific problem below. Because each function corresponding to given A is a solution to the diffusion equation, based on the principle of superposition, any linear combination of these functions is also a solution. Hence, the general solution for the given boundary conditions is... 

Simple component exchange between solid phases is accomplished by diffusion. If only two components (such as Fe " and Mg) are exchanging, the diffusion is binary. The boundary condition is often such that the exchange coefficient between the surfaces of two phases is constant at constant temperature and pressure. The concentrations of the components on the adjacent surfaces may be constant assuming interface equilibrium. The solution to the diffusion equation... 

Solid sphere Next we consider the solution to the diffusion equation in a solid sphere of radius a with constant D, uniform initial concentration, zero surface concentration, and a constant production rate of p. The diffusion equation is... 

Chapter 2 The Diffusion Equation. The diffusion equation provides the mathematical foundation for chemical transport and fate. There are analytical solutions to the diffusion equation that have been developed over the years that we will use to our advantage. The applications in this chapter are to groundwater, sediment, and biofihn transport and fate of chemicals. This chapter, however, is very important to the remainder of the applications in the text, because the foundation for solving the diffusion equation in environmental systems will be built. 

To repeat equation (E2.2.3) is a solution to the diffusion equation, and we have shown that it meets the boundary conditions, ft is therefore a solution to our problem as we have formulated it. This may seem like a fairly extensive example for one solution. However, we can use equation (E2.2.3) as a basis for an entire set of Dirac delta solutions that can model instantaneous spills. Thus, equation (E2.2.3) is a building block for many of the solutions we will model. 

Solution to the Diffusion Equation with a Slow Release of Chemical... 

E. SOLUTION TO THE DIFFUSION EQUATION WITH A STEP IN CONCENTRATION... 

Equation (2.38) has a first-order sink and a zero-order source, which meets our criteria for an analytical solution to the diffusion equation. Ce is the concentration of C at equilibrium for the reaction. This technique of assuming that multiple reactions are zero-order and first-order reactions will be utilized in Example 2.9. 

EXAMPLE 4.2 Post hole depth to avoid frost heave (solution to the diffusion equation with oscillating boundary conditions)... 

A number of other special solutions to the diffusion equations may be found in the literature. A few of these are mentioned here ... 

The diffusion equation is the partial-differential equation that governs the evolution of the concentration field produced by a given flux. With appropriate boundary and initial conditions, the solution to this equation gives the time- and spatial-dependence of the concentration. In this chapter we examine various forms assumed by the diffusion equation when Fick s law is obeyed for the flux. Cases where the diffusivity is constant, a function of concentration, a function of time, or a function of direction are included. In Chapter 5 we discuss mathematical methods of obtaining solutions to the diffusion equation for various boundary-value problems. 

For problems with relatively simple boundary and initial conditions, solutions can probably be found in a library. However, it can be difficult to find a closed-form solution for problems with highly specific and complicated boundary conditions. In such cases, numerical methods could be employed. For simple boundary conditions, solutions to the diffusion equation in the form of Eq. 4.18 have a few standard forms, which may be summarized briefly. 

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