combining advective reactive transport equation for two solutes

• Oct 11th 2010, 08:01 PM
Confused169
combining advective reactive transport equation for two solutes
This is a rather basic one sorry, but I am rather stuck.... all I want to do is apply the chain rule to combine the advective transport equation for two solutes such that $dCa/dt =\nabla \cdot (D \nabla Ca)-v \cdot \nabla Ca + \sum Ja$ and $dCb/dt =\nabla \cdot (D \nabla Cb)-v \cdot \nabla Cb + \sum Jb$

where Ca is the concentration of 'a' in the fluid, D is a dispersion coefficient, v is velocity and Ja is the flux of 'a' to the fluid etc

to get $d(Ca/Cb)/dt = ?$ can anyone please help?
• Oct 12th 2010, 02:30 PM
zzzoak
May be you can use quotient rule

$
\displaystyle{(\frac {f(x)}{g(x)})'=\frac {f'(x)g(x)-f(x)g'(x)}{g^2(x)}}
$
• Oct 13th 2010, 04:01 PM
Confused169
I think I might be getting there....

Taking the one dimensonal case(s) (same for C2)

$dC_1/dt=D*(d^2C_1/dz^2)-v\cdot(dC_1/dz)+\sum J_1_,_i$ (1)

and using the product rule as a special case of the chain rule

$dr/dt=C_2\cdot(dC_1/dt)-C_1\cdot(dC_2/dt)$ (2)

where $r=C_1/C_2$ (3)

and rearranging for dr/dt in terms of C_1

$dr/dt=(C_1/r)\cdot(dC_1/dt)-C1\cdot(dC_2/dt)$ (4)

Substituting (1) for C_1 and C_2 into (4) i get;

$dr/dt=(C_1/r)\cdot(D*(d^2C_1/dz^2)-v\cdot(dC_1/dz)+\sum J_1_,_i)-C_1\cdot(D*(d^2C_2/dz^2)-v\cdot(dC_2/dz)+\sum J_2_,_i)$

All I need to do is expand this and rearrange but I am total stuck has how to expand out the brackers, as I suck at math. Can anyone enlighten me? Please. I would be really really grateful.

Thanks!