Page 287 - Numerical Methods for Chemical Engineering

276 6 Boundary value problems

Consider the general form of a second-order differential equation,
2
2
2
∂ ϕ ∂ ϕ ∂ ϕ ∂ϕ ∂ϕ
a + b + c + f t, z,ϕ, , = 0 (6.95)
∂t 2 ∂t∂z ∂z 2 ∂t ∂z
For example, the time-varying 1-D convection/diffusion equation
2
∂ϕ ∂ ϕ ∂ϕ
− + v = 0 (6.96)
∂t ∂z 2 ∂z
in the limit of Pe 1 takes the form above with
a = 1 b = 0 c =− (6.97)

2
In the limit Pe 1, b − 4ac < 0, and the PDE is said to be parabolic. By contrast, in the
limit Pe 1, we have a PDE dominated by convection,
∂ϕ ∂ϕ
+ v = 0 (6.98)
∂t ∂z
Differentiating once with respect to time yields
a = 1
2
2
∂ ϕ ∂ ϕ
+ v = 0 ⇒ b = v (6.99)
∂t 2 ∂t∂z
c = 0
2
Now, b − 4ac > 0, and the equation is said to be hyperbolic. Thus, by changing Pe we
alter the type of the PDE, and as we see below, this changes significantly the way the field
is propagated.
The steady-state diffusion equation
2
∂ ϕ
− = 0 (6.100)
∂z 2
has the general form of a second-order PDE with
a = 0 b = 0 c =− (6.101)

2
Here, b − 4ac = 0 and the equation is said to be elliptic.
Consider some point P at position z p and time t p . What are the set of points in space-time
whose field values influence the field value at (P) and the set of points in space-time whose
field values are influenced in turn by the value at (P)? As a concrete example, consider the
1-D convection equation
∂ϕ ∂ϕ
+ v = 0 (6.102)
∂t ∂z
which describes purely convective transport of ϕ in the direction of increasing z for v> 0.
At future times t > t p , P only influences points (t, z) that are downstream with z p < z < z p +
v(t − t p ). Similarly, only past points (t, z) that are upstream with z p − v(t p − t) < z < z p
influence P. The space-time diagram is shown in Figure 6.9. The two lines that separate the
regions of influence from those of no influence are the characteristic lines for P.
We now relate the characteristic lines to the coefficients a, b, c in (6.95). Let us consider
the point P and a point Q on one of its characteristic lines (Figure 6.9). If P and Q are