How do I determine the 'other stuff' in DE mixing problems?

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In summary, the conversation discusses setting up a differential equation for a chemical mixing problem involving water and a certain rate of flow. The equation takes the form dx/dt = in - out, where x is the amount of chemical and in is the rate of flow coming in. The out part is usually the number multiplied by some other factors, such as A/V where A is the amount of chemical and V is the volume of water in the tank. When setting up the equation, A is left alone since it is the quantity being solved for, while V is calculated using the initial amount of volume and the rate of flow. The conversation also explores the scenario of pouring in pure water when there is already a mixture of water and chemical in the tank
  • #1
member 392791
I am having major problems understanding these types of questions, where you will have water and in will be chemicals mixing at a certain rate, and coming out of another tube at a rate, and then the question is to find out certain things, concentration at a time or whatever.

My question, the equation takes the form

dx/dt = in - out

x is the amount of chemical

The in is the rate coming in, however it seems the out part is usually the number multiplied by some other stuff. The question is, how do I know what to put in ''the other stuff''?
 
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  • #2
the other stuff will be A/V where A is the amount of chemical and V is the volume of water in the tank
when setting up the differential equation you leave A alone since that is what were trying to solve for
for V, you take initial amount of volume+(rate in-rate out)*t
where rate in and rate out are in liters/min or whatever units you are using
 
  • #3
how about when a chemical and water mix is already in the tank, and you are now pouring in pure water? How do I express that?
 
  • #4
Woopydalan said:
how about when a chemical and water mix is already in the tank, and you are now pouring in pure water? How do I express that?

Then if your rate of change depends on that it will be some function of what's already in the tank which is some function of x.
 
  • #5
If the amount of chemical is A and the volume is V, then the amount of chemical per unit volume is A/V as miglo said. If the water is flowing out at rate "r" then the chemical is flowing out at rate (A/V)r= (Ar/V). Since that is flowing out the rate is "out"= -(Ar/V). Saying that there is only water flowing out means that "in= 0". So the differential equation is
[tex]\frac{dA}{dt}= -Ar/V[/tex].
 
  • #6
I was saying water is flowing in and a mixture is flowing out
 
  • #7
If the volumetric flow rate out is equal to the volumetric flow rate in, and, if the tank is well-mixed so the concentration of the chemical coming out is equal to the concentration within the tank, then the mass balance for the chemical in the tank goes:

Vdx/dt = F (xin -x)

where F is the volumetric flow rate, V is the volume of fluid in the tank, x is the species concentration within the tank (and in the exit stream), and xin is the species concentration in the feed. This reduces to HallsofIvy's equation for the case in which the concentration in the feed is zero (pure water). I think HallsofIvy meant to say that only water is flowing in.
 

1. What are mixing problems in the context of differential equations?

Mixing problems involve the study of how substances are mixed or dispersed in a given medium, such as a liquid or gas. Differential equations are used to model and solve these types of problems.

2. How are differential equations used to solve mixing problems?

Differential equations are used to describe the rate of change of a substance in a mixture. By setting up and solving these equations, we can determine how the concentration of a substance changes over time and how it is affected by factors such as diffusion and convection.

3. What are some common types of mixing problems that can be solved using differential equations?

Some common types of mixing problems include tank mixing, chemical reactions in a solution, and air pollution dispersion. These problems can involve one or multiple substances and can be modeled using different types of differential equations, such as ordinary or partial differential equations.

4. What are the challenges associated with solving mixing problems using differential equations?

One of the main challenges is accurately describing the system and its behavior with a suitable differential equation. This often requires knowledge of the physical and chemical properties of the substances involved and the underlying mechanisms of mixing. Additionally, the solutions to these equations can be complex and may require numerical methods for approximation.

5. What are some real-world applications of mixing problems and their solutions using differential equations?

Mixing problems and their solutions have a wide range of practical applications, including in chemical and biochemical engineering, environmental science, and pharmaceutical research. For example, understanding how pollutants disperse in the atmosphere can help improve air quality, and modeling drug diffusion in the body can aid in the development of more effective medications.

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