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Math Help - combining advective reactive transport equation for two solutes

  1. #1
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    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?
    Last edited by Confused169; October 11th 2010 at 07:14 PM.
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  2. #2
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    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 />
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  3. #3
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    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!
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