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

• nikokampman
In summary, the conversation discusses the derivation of the partial differential equation for the change in the ratio of two solutes (r) during 1D flow of a reacting advecting-diffusing fluid through a porous media. The user is trying to combine the equations for individual solutes (C1 and C2) using the quotient rule, but is unsure of the correct simplification. The final expression for dr/dt is given as (D/C2)*(d2C1/dx2-dC1/dx)-v*(1/C2)*(dC1/dx+C1*(C1/C2)*dC2/dx).
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!

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

## 1. What is the C1/C2 ratio and why is it important in PDEs?

The C1/C2 ratio represents the concentration of two different substances, C1 and C2, in a system. In PDEs (partial differential equations), this ratio is important because it affects the overall behavior and dynamics of the system. Changes in the C1/C2 ratio can lead to different patterns and outcomes in the system.

## 2. Why is it difficult to solve the PDE of C1/C2 ratio change with time?

The PDE of C1/C2 ratio change involves multiple variables and equations, making it a complex mathematical problem. Additionally, the behavior of the system is highly dependent on the initial conditions and boundary conditions, making it difficult to find a general solution.

## 3. What are some common methods for solving the PDE of C1/C2 ratio change with time?

Some common methods for solving this PDE include using analytical techniques such as separation of variables, or numerical methods such as finite difference or finite element methods. Each method has its advantages and limitations, and the choice depends on the specific problem and desired level of accuracy.

## 4. How can we validate the accuracy of the solution to the PDE of C1/C2 ratio change?

To validate the accuracy of the solution, we can compare it with experimental data or with other numerical solutions. Additionally, we can perform sensitivity analyses to see how sensitive the solution is to changes in the parameters and initial conditions.

## 5. What are some applications of solving the PDE of C1/C2 ratio change with time?

This PDE has applications in various fields such as chemical engineering, biology, and environmental science. It can help understand and predict the behavior of complex systems such as chemical reactions, biological growth, and diffusion processes. It can also aid in designing and optimizing processes and systems in these fields.

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