# calc

posted by .

I would greatly appreciate your help with a math equation needed for a science experiment I'm researching. The case is the following: there is a pit full of juice and a stream of water is flowing through the pit (so that the amount of water which enters also exits). The goal is for there to be only 3 units of juice in the pit (regardless of the amount of water). As the stream flows through we assume that the stream water mixes evenly with the juice in the pit so that what exits is a proportion of stream water and juice. The question I'm stuck on is, how much stream water needs to pass through the pit in order for there to remain only 3 units of juice water. Since the proportions constantly change, it seems like a calc question to me, although I may be wrong. Could you do me a favor and email me the equation that I could use for any given amount of juice?
Thanks,
Joseph

• calc -

You can set up a first order differential equation to determine the time at which the required concentration remains.

The volume, V of the pit should be known.
The initial amount of juice, j, should be known.

The input concentration c1 of juice is zero.
The input rate r1=r=Q/t (in m³/s) should be known.

Assuming perfect agitation, the output concentration is c2=j/V.
The output rate, r2 equals input rate r1=r.

The equation would be

dj/dt = r1*c1-r2*c2

substituting values,
dj/dt = r*0 - r*(j/V)
dj/dt = -(Q/(Vt))j

Separate variables, to get
dj/j = -(Q/V)*dt/t
Integrate (do not forget the integration constant) and solve for j in terms of t.