Page 271 - Numerical Methods for Chemical Engineering

260 6 Boundary value problems

As source terms often arise from chemical reaction, (6.9) is also known as a convec-
tion/diffusion/reaction equation.
In this chapter, we consider the numerical solution of PDEs of the form (6.9). We focus
first upon stationary problems, for which the steady-state field ϕ(r) satisfies the PDE
2
0 =−∇ · (ϕv) + ∇ ϕ + s(r, t,ϕ) (6.10)
and then treat time-dependent problems. Here, we consider general transport-type PDEs,
but do not address the particular numerical issues that arise in computational fluid dynamics
(CFD).Whilethemethodsoffinitedifferences,finitevolumes,andfiniteelements,described
here, are used in CFD, a few additional concerns arise, particularly involving the coupling
of the pressure and velocity fields. To do justice to this subject would require a dedicated
text; therefore, for further details the reader is referred to Ferziger & Peric (2001).

Real-space vs. function-space BVP methods

There are two general approaches to solving BVPs. In a function-space method, we write
the solution as a linear combination of basis functions {χ m (r)}, each satisfying all boundary
conditions,

ϕ(r, t) = c m (t)χ m (r) (6.11)
m
ϕ(r, t) then automatically satisfies all boundary conditions (assumed to be linear in ϕ), and
we have merely to find the {c m (r)} that best satisfy (6.9).
Alternatively, in a real-space method, we specify a number of grid points r [j] ∈ and
compute numerically the field values ϕ(r [j] , t) at these points. While function-space methods
can yield analytical solutions for some linear PDEs in simple domains, real-space methods
require numerical solution yet are more generally applicable, especially for problems with
nonlinear source terms or complex domain geometries. We thus restrict our focus to real-
space methods, although the finite element method will be seen to mix both approaches. For
further discussion of function-space approaches, consult Bird et al. (2002), Deen (1998),
and Stakgold (1979).

The finite difference method applied to a 2-D BVP

Let us consider a steady 2-D BVP on a rectangular domain involving only diffusion and
2
a position-dependent source term, −∇ ϕ = f (r), the Poisson equation. We require the
solution to be zero on all boundaries, a Dirichlet-type boundary condition. Thus, the BVP
is
2
2
∂ ϕ ∂ ϕ
2
−∇ ϕ =− − = f (x, y) 0 ≤ x ≤ L 0 ≤ y ≤ H
∂x 2 ∂y 2
BC 1 ϕ(0, y) = 0 0 ≤ y ≤ H
BC 2 ϕ(L, y) = 0 0 ≤ y ≤ H (6.12)
BC 3 ϕ(x, 0) = 0 0 ≤ x ≤ L
BC 4 ϕ(x, H) = 0 0 ≤ x ≤ L