Page 269 - Numerical Methods for Chemical Engineering

6 Boundary value problems

Boundary value problems (BVPs) involve the solution of ODEs or partial differential equa-
tions (PDEs) on a spatial domain, subject to boundary conditions that hold on the domain
boundary. Many problems from solid and fluid mechanics, electromagnetics, and heat and
mass transfer are expressed naturally as BVPs. The forms of these differential equations
often resemble each other because they arise from similar conservation principles. Here the
emphasis is upon BVPs that arise from problems in transport phenomena.
This chapter focuses upon real-space methods, in which a computational grid is overlaid
upon the domain. The BVP is then converted into a set of ODEs for a time-dependent
problem or a set of algebraic equations for a steady problem. This technique can be used
evenwhennoanalyticalsolutionexists,andcanbeextendedtoBVPswithmultipleequations
or complex domain geometries. Here, the focus is upon the methods of finite differences,
finite volumes, and finite elements. These methods have many characteristics in common;
therefore, particular attention is paid to the finite difference method, as it is the easiest to
code. The finite volume and finite element methods also are discussed; however, as the
reader is most likely to use these in the context of prewritten software, the emphasis is upon
conceptual understanding as opposed to implementation.

BVPs from conservation principles

Let ϕ(r, t) be some time-varying field, i.e., a function that assigns to each position r and
time t a unique value ϕ(r, t). Common examples of fields in chemical engineering include

ϕ = ρ mass density
ϕ = ρv linear momentum density
(6.1)
1
2
ϕ = ρ|v| + ρ ˆ u total kinetic and internal energy density
2
concentration of species i
ϕ = c i
Each of these fields represents the density of some quantity . Let us consider a closed
domain ,a control volume (CV), with boundary ∂ (Figure 6.1), and write a balance for
the total amount of within ,
 
d ' ' ' '
ϕ(r, t)dr = ϕ[v · (−n)]dS + [J D · (−n)]dS + s(r, t,ϕ)dr (6.2)
 
dt
∂ ∂
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