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Topic: CSI Chemistry (Read 2097 times) |
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HaroldH
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CSI Chemistry
« on: Nov 16th, 2011, 6:04pm » |
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A tank initially contains 50 gallons of chemical A in which there is 10 lbs. of chemical B. Chemical A containing 2 lbs. of dissolved B per gallon flows into the tank at a rate of 5 gallons per minute. The solution is keep uniform by a slurry machine and the slurred mixture simultaneously flows out of the tank at a rate of 3 gallons per minute. How much of B is in the tank at time t=20?
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HaroldH
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Re: CSI Chemistry
« Reply #1 on: Nov 16th, 2011, 7:30pm » |
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Gee. You moved my problem to easy. It is not exactly easy. So, solve it then.
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malchar
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Re: CSI Chemistry
« Reply #2 on: Nov 16th, 2011, 7:44pm » |
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What are the units for t=20?
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HaroldH
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Re: CSI Chemistry
« Reply #3 on: Nov 16th, 2011, 8:00pm » |
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The unit of t is time. It is an initial value problem.
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ThudnBlunder
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Re: CSI Chemistry
« Reply #4 on: Nov 16th, 2011, 8:06pm » |
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on Nov 16th, 2011, 7:30pm, HaroldH wrote:| Gee. You moved my problem to easy. It is not exactly easy. So, solve it then. |
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Sorry, the homework wallah is not here at the moment. 
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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HaroldH
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Re: CSI Chemistry
« Reply #5 on: Nov 16th, 2011, 8:23pm » |
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That is an insult. Solve the problem if you are so experienced to do so.
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Noke Lieu
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pen... paper... let's go! (and bit of plastic)
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Re: CSI Chemistry
« Reply #6 on: Nov 16th, 2011, 8:44pm » |
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Hey HaroldH.
Don't mind T&B- he's just being cautious.
Whilst it's a cute question, it is a matter of crunching through pre-existing formulae.
I have no doubt that Thud can answer this question- even I can feel the right way to approach it- and I'm one of the less competent around here.
Hold tight, and I'll be with you.
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a shade of wit and the art of farce.
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Noke Lieu
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Re: CSI Chemistry
« Reply #7 on: Nov 16th, 2011, 9:16pm » |
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So, does this help?
Or perhaps that?
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| « Last Edit: Nov 16th, 2011, 9:40pm by Noke Lieu » |
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HaroldH
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Re: CSI Chemistry
« Reply #8 on: Nov 16th, 2011, 9:18pm » |
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Thanks. I appreciate your response and UR remarks are well accepted.
But again, I will ask Thundbluder for the soltution since he, well, said that it was easy...
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ThudnBlunder
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Re: CSI Chemistry
« Reply #9 on: Nov 17th, 2011, 2:48am » |
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on Nov 16th, 2011, 9:18pm, HaroldH wrote:
But again, I will ask Thundbluder for the soltution since he, well, said that it was easy... |
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I moved it to Easy because it appears to be solvable using standard methods, in this case ODEs.
However, this does not mean I am in the homework business.
If you need help with homework try posting your progress along with your impersonal, cut-and-pasted question.
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| « Last Edit: Nov 17th, 2011, 7:03am by ThudnBlunder » |
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rmsgrey
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Re: CSI Chemistry
« Reply #11 on: Nov 17th, 2011, 11:54am » |
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on Nov 16th, 2011, 8:00pm, HaroldH wrote:| The unit of t is time. It is an initial value problem. |
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t=20s and t=20 days both have t as 20 time units, and give very different answers...
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| « Last Edit: Nov 17th, 2011, 11:55am by rmsgrey » |
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ThudnBlunder
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Re: CSI Chemistry
« Reply #12 on: Nov 17th, 2011, 6:37pm » |
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on Nov 16th, 2011, 8:23pm, HaroldH wrote:
Handbags at dawn it is then.
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Michael Dagg
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Re: CSI Chemistry
« Reply #14 on: Nov 26th, 2011, 4:06pm » |
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Let A be brine and B salt for better problem clarity.
This is a scalar problem with two initial conditions, one
for both brine and salt which occur before the species are
activated. Time is characterized in minutes, i.e. gal/min ,
so if you wish to use another time division you can simply
scale the problem as given or scale time in the resulting
analytic or numeric solution.
Often what makes these kinds of problems tricky is determining
the concentration factors. In this problem, one is obvious
while the other takes a little reasoning to produce.
The unknown here is the amount of salt at time t . Therefore,
let x be the amount (lb) of salt at time t . Then the model
for this problem is the rate equation of x equal to the sum of
what's flowing into the tank minus what's flowing out of the tank:
x' = (INFLOW - OUTFLOW) lb/min.
The next step requires us to determine the expressions for INFLOW
and OUTFLOW based on what's given.
For INFLOW: Brine with 2 lb of dissolved salt per gallon flows
into the tank at 5 gal/min. Thus,
INFLOW = 2 lb/gal * 5 gal/min = 10 lb/min .
The first factor in the preceding product 2 lb/gal is the
salt concentration coming into the tank while the second
factor is the rate at which the brine arrives. When we take that product
we end up with the amount salt per minute flowing into the tank,
which should have dimensions lb/min, and it does.
For OUTFLOW, the situation is slightly different. Notice that the brine
flows into the tank at slower rate than it leaves, i.e. 5 and 3 gal/min,
and so we have a (net) gain of 5-3 = 2 gal/min. When t=0 the tank
contains 50 gal of brine so the amount of brine in the tank at time t
is 50 + 2t gal.
OUTFLOW must have the same dimensions as INFLOW and also must be
a product whose factors are concentration and rate. This is the
case; again if x is the amount (lb) of salt at time t , then
x/(50 + 2t) lb/gal
is the concentration of salt at time t .
The brine flows out of the thank at 3 gal/min. So,
OUTFLOW = x/(50 + 2t) lb/gal * 3 gal/min = 3x/(50 + 2t) lb/min .
Going back to model and taking the sum gives
x' = INFLOW - OUTFLOW lb/min = 10 - 3x/(50 + 2t) lb/min .
When we put this ODE in standard form
x' + 3x/(50 + 2t) = 10 , x(0) = 10
we can get a better idea of how we can solve it easily. It is a linear
ODE so we have a ton of artillery to solve it. The easiest way I can
suggest is to use an integrating factor since it fits a certain
form.
That is, one integration factor is
u = exp(\int 3/(2t + 50) dt) = (2t + 50)^{3/2} .
That should do it. I will let you take it from here.
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Regards,
Michael Dagg
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Noke Lieu
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pen... paper... let's go! (and bit of plastic)
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Re: CSI Chemistry
« Reply #15 on: Nov 27th, 2011, 4:06pm » |
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That's beautifully written, Michael Dagg.
Clear, concise. Hell- even I can follow it!
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a shade of wit and the art of farce.
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Michael Dagg
Senior Riddler
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Re: CSI Chemistry
« Reply #16 on: Nov 28th, 2011, 5:46pm » |
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Ah, thanks. I intended to be as clear as possible without writing
too much.
I think the problem was disguised somewhat (i.e. chemicals A,B)
perhaps to make it look difficult. In fact, it could give one
the impression that it involves a two dimension system or even
a hidden catalysis; even though it can be wrote as the former
there is no gain in doing so.
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Regards,
Michael Dagg
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guptaharsh18890
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Posts: 7
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Re: CSI Chemistry
« Reply #17 on: Dec 28th, 2011, 10:56pm » |
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That's beautifully written..
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