Transport&reaction.doc - Transport and Reaction Equation...

Transport and Reaction Equation
IPOLA Equation
The fundamental mass balance equation is
    
    A L O P I
( 0 )
where:
 I = inputs
 P = production
 O = outputs
 L = losses
 A = accumulation
Consider a 3-D ‘cell’ like this
where J
x
|
x
indicates the flux density in the x direction at the point x. Fluxes into the
volume represent Inputs (I), while fluxes out of the volume are Outputs (O). Production
(P), Losses (L), and Accumulation (A) inside the volume are also possible.
The cell can represent a small piece of many different things; it might represent a volume
of water in a pond, part of the atmosphere, a portion of an aquifer, or a block of steel.
 x
 z
J
x
|
x
J
x
|
x+  x
J
z
|
z
J
z
|
z+  z
 y
J
y
|
y
J
y
|
y+  y
P L
A

Similarly, the flux density J might be a liquid flow, a heat (energy) flux, or a chemical
flux. Since our primary interest is in the movement of chemicals in the environment,
we’ll consider only the flux of chemical mass here. The approach has broad applicability
though.
Advective Flux Density
The mass flux (ML -2 T -1 ) can consist of a number of components. First, there may be a
fluid (liquid or gas) flow with velocity v that carries along a mass of chemical. Let’s
consider just the x direction. In time  t, a velocity v (at x) will sweep in a volume V = v|
x
 t  y  z. The volume swept out may be different because it depends on the velocity at x
+  x.
The chemical mass going in or out in time  t depends on the fluid volume and the
concentration C in the fluid:
Mass/Time = Concentration x Volume/Time.
If we divide everything by the area of the inflow (or outflow) faces, we get the mass flux
density, which ends up being very simple:
J
x
|
x
= C v
where C is concentration and v is the velocity at point x.