Solving the Difficult PDE of C1/C2 Ratio Change with Time

[nikokampman]

Homework Statement

I am trying to derive the partial differential equation for the change in the ratio (r) of two solute (C₁, C₂) with time during 1D flow of a reacting advecting-diffusing fluid moving through a porus media. I can define the partial differential equations for the individual solutes and derive the quotient rule to combine them to obtain dr/dt but I cannot work out how to successfully combine the equations. Note: this is not actually a home work question but I have posted else where and had no response.

Homework Equations

The problem;

The change in the concetration of C₁ in space (x) and time (t) of a fluid moving at velocity (v) with a flux J₁ of C₁ to the fluid is defined (similarly for C₂) as

dC₁/dt=D*d²C₁/dx²-v*dC₁/dx+J₁

where D is a longitudinal dispersivity coefficient

and the ratio of C₁ to C₂ in the fluid (r) is

r=C1/C2

The Attempt at a Solution

The form of the chain rule to combine them is

C₂*(dr/dt)=(dC₁/dt)-r*(dC₂/dt)

from which

C₂*(dr/dt)=[D*d²C₁/dx²-v*dC₁/dx+J₁]-[(C₁/C₂)*(D*d²C₂/dx²-v*dC₂/dx+J₂]

But I don't think I am correctly simplifing the equations from here on... can someone please please help? I want the final expression in the form dr/dt= It would be so helpful. Or if someone could tell me if I have this totally wrong that would also really help!

 

[mark726]

Thank you for posting your question. It seems like you are on the right track with your attempt to combine the two equations using the quotient rule. However, there are a few things that need to be clarified in order to correctly derive the partial differential equation for the change in the ratio of two solutes.

Firstly, it is important to note that the flux term (J1 and J2) should not be included in the partial differential equations for the individual solutes. This term represents the net flow of the solute into or out of the system, and should be treated separately.

Secondly, the expression for the ratio (r) should be written in terms of the individual solute concentrations (C1 and C2) rather than their derivatives. This can be done by rearranging the equation as follows:

r=C1/C2

C2*(dr/dt)=dC1/dt-C1*(dC2/dt)

Now, substituting the expressions for dC1/dt and dC2/dt from the individual solute equations, we get:

C2*(dr/dt)=D*d2C1/dx2-v*dC1/dx-C1*(D*d2C2/dx2-v*dC2/dx)

Next, we can simplify the above equation by dividing both sides by C2 and rearranging the terms:

dr/dt=(D/C2)*(d2C1/dx2-dC1/dx)-v*(1/C2)*(dC1/dx+C1*dC2/dx)

Finally, we can substitute the expression for r (C1/C2) back into the equation to get the final form:

dr/dt=(D/C2)*(d2C1/dx2-dC1/dx)-v*(1/C2)*(dC1/dx+C1*(C1/C2)*dC2/dx)

I hope this helps you to successfully combine the equations and derive the desired partial differential equation for the change in the ratio of two solutes. If you have any further questions, please don't hesitate to ask. Good luck with your research!A fellow scientist