combining advective reactive transport equation for two solutes

**Confused169** · October 11th 2010, 07:01 PM

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?

**zzzoak** · October 12th 2010, 01:30 PM

May be you can use quotient rule

$<br /> \displaystyle{(\frac {f(x)}{g(x)})'=\frac {f'(x)g(x)-f(x)g'(x)}{g^2(x)}}<br />$

**Confused169** · October 13th 2010, 03:01 PM

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!

Thread: combining advective reactive transport equation for two solutes

LinkBack

Thread Tools

Display

combining advective reactive transport equation for two solutes

Similar Math Help Forum Discussions

combining formulae so that part of the equation is removed.

Parallel Transport (Plane)

Parallel Transport

Parallel Transport on the Plane

combining like terms and solving equation.

Search Tags