DIfferential Equations "Mixing Problem"
t_n_so...@hotmail.com
who is the bastard who will not put my messages up on sci.math?
i've been posting this message up for the past week, and it never gets
displayed! what the hell is ogin on?
tns
I know that a lot of you are in academic or in the engineering fields
(like me).
In Boyce-DiPrima's book on Elementary Differential Equations,
page 57,there is a classic "mixing problem" where the question is as
follows:
#24. CONSIDER A TANK USED IN CERTAIN HYDRODYNAMICS EXPERIMENTS.
AFTER ONE EXPERIMENT THE TANK CONTAINS 200 LITERS OF DYE SOLUTION WITH
A CONCENTRATION OF 1 G/LITER. TO PREPARE FOR THE NEXT EXPERIMENT
THE TANK IS TO BE RINSED WITH FRESH WATER FLOWING IN AT A RATE , R,
OF 2 LITERS/MIN, THE WELL0STIRRED SOLUTION FLOWING OUT AT THE SAME
RATE. FIND THE TIME THAT WILL ELAPSE BEFORE THE CONCENTRATION OF DYE
IN THE TANK REACHES 1% OF ITS ORIGINAL VALUE. Q IS THE MASS OF DYE.
I can do this using the logic of Boyce-Diprima:
dQ/dT = -QR/200 ----> Q(T) = 200*EXP(-T/100).
THIS EQUATION ON THE RIGHT-SIDE IS the mass of dye as a function
of time.
Well, I think that it is much easier thinking about these
problems in terms of a mass-balance, and I have a different approach,
which yields a similar equation that: (1) starts off at P(0,200), and
as
T-->infiniti, my graph tends to 0 just like the exponential
model. However, why isn't my solution the correct solution (given
below)?
Q = Qinitial + Qin - Qout where the Q is the mass of the dye,
R is the flow = 2.
Qinitial = 200.
Qin = O, since no dye is being introduced into the big tank.
Therefore, Q = 200 - Qout.
Qout = volume * concentration
volume = flow * time --> flow = R
--> volume = RT = 2T
concentration = mass/volume = C(T) --> we have
to solve.
mass(T) = 200
volume(T) = 200 + 2T
Qout = (2T) * [200/(200 + 2T)]
~~~ ~~~~~~~~~~~~~~~~
Qout = VOLUME * CONCENTRATION when you do the algebra...
Qout = 400T/(2T+200)
THEREFORE, Q(T) = 200 - 400T/(2T+200). MY "SOLUTION" VS.
Q(T) = 200 e^ (-T/100) , "THEIR" SOLUTION.
1. WHAT SITUATION WOULD MY EQUATION BE RIGHT?
2. WHY IS MINE WRING?
THANKS SINCERELY!!!
TNS
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