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# Rate Of Change In Concentration With Time

4 replies to this topic
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### #1 jordan1111

jordan1111

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Posted 17 September 2021 - 09:31 AM

Hello everyone,

I have a question regarding the rate of change in concentration with time. Let's say we have a stream of unsaturated brine with a concentration of 50 g/l NaCl and a flow rate of 20 l/min going into a tank, the tank has 700 l of 120 g/l NaCl brine. The outflow from the tank is also 20 l/min. My question is how long does it take to reach a new equilibrium and what is the concentration in the tank? I think I need to use derivative function to solve the problem, but i dont remember how to set the equation.

Greatly appreciated.

### #2 latexman

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Posted 17 September 2021 - 11:25 AM

Yes, you want to integrate the differential mass balance.  There is no reaction going on, so you start with:

Accumulation = input - output

d(VCNaCl)/dt = qinCin - qoutCNaCl

Does that help?

### #3 jordan1111

jordan1111

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Posted 17 September 2021 - 04:20 PM

Thank you so much! I solved the problem.

### #4 Pilesar

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Posted 17 September 2021 - 04:40 PM

You have two questions. 1) How long to reach equilibrium and 2) What is the final concentration in the tank? These two questions can be answered without doing any calculations whatsoever!!! Think more on that and do not jump into the math details before trying logic to solve the riddle.

To better visualize what is happening in the tank, I put together a spreadsheet for the fun of it. 1) Calculate the amount of NaCl in the tank. 2) Choose a time step (smaller steps are more accurate) and increment the time and calculate the tank NaCl again. 3) Repeat Step 2 and Step 3. The linked picture shows my approach: https://tinyurl.com/rybf6nhc

The formula I used to calculate the tank NaCl in cell B14 is:

=+B13+_TimeStepSize*(_InflowFlowRate*_InflowConcentration-_OutflowFlowRate*B13/_TankVolume)

### #5 Pilesar

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Posted 18 September 2021 - 12:16 AM

My initial response above was that both questions could be answered without doing calculations. I still think that is true of the second question. I have been thinking more about the problem as presented and now believe calcs are necessary to answer 'How long to reach equilibrium.' I had overlooked the consequences of Avogadro's number. While large, Avogadro's number is not infinite. The time to equilibrium for a specific tank cannot be predicted precisely. But for a large number of tanks, there is a statistical probability distribution near the 38 hour mark by my calcs. This is a mathematical academic solution using zero tolerance. A real world engineering problems might look similar, but it would likely have a much looser tolerance than zero.