Mixing problems using DE's

Woopydalan

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''?

miglo

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

Woopydalan

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?

chiro

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.

HallsofIvy

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].

Woopydalan

I was saying water is flowing in and a mixture is flowing out

Chestermiller

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 (x_in -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 x_in 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.