Page 308 - Numerical Methods for Chemical Engineering

The finite volume method 297

The second derivative with respect to y then takes a simple form
2
2
∂ ϕ ∂ ∂ϕ 1 ∂ ϕ
= = (6.181)
2
∂y 2 ∂y ∂y [h(ξ L)] ∂η 2
By contrast, the second derivative with respect to x is much more complex,
2
∂ ϕ ∂ ∂ϕ 1 ∂ ηh (ξ L) ∂ 1 1 ∂ϕ ηh (ξ L) ∂ϕ 1

= = − − (6.182)
∂x 2 ∂x ∂x L ∂ξ h(ξ L) ∂η L ∂ξ h(ξ L) ∂η
Through this approach, we can employ a finite difference discretization on a regular grid
in (ξ, η) space; however, the differential equation now involves more complex derivatives.
The finite element method, described below, allows us to solve BVPs in complex geome-
tries without performing such coordinate transformations (which are not always possible
anyway).

The finite volume method

Above, our focus has been on the finite difference method, which is easy to implement in
domains of rather simple geometry. In complex domains, it is difficult to place a grid and
keep track of neighbors when the grid points are required to lie along the coordinate axes.
Here, we discuss another method that is not subject to this condition. We again consider the
2-D Poisson equation but now instead of the microscopic equation
2
2
∂ ϕ ∂ ϕ
2
−∇ ϕ =− 2 − 2 = f (x, y) (6.183)
∂x ∂y
we consider the corresponding macroscopic balance
' '
0 = [(−n) · (−∇ϕ)]dS + f (x, y)dV (6.184)
∂
where we have used the equivalence between (6.2) and (6.7). In the finite volume method,
we apply this macroscopic balance to each of a number of control volumes, or cells, that
partition the computational domain (Figure 6.23). In the center of each cell, we place a
computational node, at which we would like to compute the solution. To compare the
resulting finite volume method to that of finite differences, let us consider a regular grid of
uniformly-spaced points, with each cell having a “2-D volume” V c = ( x)( y). The nodal
coordinates are (x i , y j ), x i = i( x), y j = j( y) and each cell is surrounded by neighbors
to the “north,” “south,” “east,” and “west.”
To obtain a set of algebraic equations for the node field values, we apply the macroscopic
balance to each cell, denoted as the control volume (i,j) , that is centered on the nodal
position (x i , y j ),
' '
0 = [n · ∇ϕ]dS + f (x, y)dV (6.185)
∂ (i, j) (i, j)