Environmental Engineering Reference
In-Depth Information
Appendix D: Finite difference approach
Equilibrium-stage processes are discrete steps. One approach to the analysis is an evalua-
tion as a finite difference calculation where each stage is an equal and discrete interval in
the process train. Obviously, a process simulator can be used. Finite difference approaches,
including ones shown here, can be implemented on spreadsheets for rapid estimates.
Initially, we can define some finite difference mathematical operations. We will then
demonstrate how the approach is applied to equilibrium-stage separation processes.
D.1
Definitions
Given a function
y
=
f
(
x
), we can define the value at the
n
th interval point as:
y
n
=
f
(
x
0
+
n
x
)
where
x
=
discrete interval which is a constant
x
0
=
initial value of
x
.
The difference in the value of the function
y
between two interval points can be described
as a first forward difference:
y
n
=
f
(
x
0
+
(
n
+
1)
x
)
−
f
(
x
0
+
n
x
)
=
y
n
+
1
−
y
n
.
We can continue this operation:
2
y
n
=
(
y
n
)
=
second forward difference
=
(
y
n
+
1
−
y
n
)
=
(
y
n
+
2
−
y
n
+
1
)
−
(
y
n
+
1
−
y
n
)
=
y
n
+
2
−
2
y
n
+
1
+
y
n
.
This can be generalized to write an expression for the
k
th forward difference:
k
k
!
k
y
n
=
1)
r
(
−
r
)!
y
n
+
k
−
r
.
r
!(
k
−
r
=
0
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