mass transfer in multiphase systems - Greenleaf University

MASS TRANSFER IN MULTIPHASE SYSTEMS: VOLATILE ORGANIC COMPOUND
REMOVAL IN THREE-PHASE SYSTEMS
This appendix shows how the transport equations (conservation of mass used for illustration) are the same
regardless of the observer. The basic development (Bird 1960) is that there are three types of
concentration derivatives:
As a fixed observer of flow quantifying the concentration of some quantity of mass in a stream.
For this, it is simply C/t, the partial of C with respect to t holding x, y, and z constant.
As a random moving observer in the stream, the derivatives must include the motion:
dC C C dx C dy C dz
= + + +
dt t x dt y dt z dt
(83)
As an observer flowing with the stream, the substantial derivative is as follows:
DC C C C C
= + v + v + v
x y z
Dt t x y z
(84)
The substantial derivative for a moving body with the flow is explained in reference to the relations for a
fixed position in the following. Extensive development and analysis is used from the masterful work by
Anderson in computational fluid dynamics (CFD). Similar analysis below and many other mathematical
tools are available in (Anderson 1995).
Conservation of mass
For a fluid particle moving between 2 points, a Taylor series provides
2 1 ( x2 x1) ............
x
t
(85)
Dividing by (t 2 -t 1)
D
t t v
......
t t x t Dt
lim 2 1
2 1
2 1
(86)
The substantial derivative is shown below in operator form:
D
Dt
v
t
(87)
f
f ( x, y, z, t)
(88)
Any function f can be shown using calculus of several variables, e.g.,
df f f dx f dy f dz
dt t x dt y dt z dt
(89)
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