Mixture problem. How to solve for C?


by Jeff12341234
Tags: mixture, solve
Jeff12341234
Jeff12341234 is offline
#1
Feb22-13, 09:13 PM
P: 179
I need to solve for C. I know it's probably simple but i don't remember how to. This is what I have so far:
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Chestermiller
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#2
Feb22-13, 10:33 PM
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PF Gold
P: 4,508
Because the volume flow rate entering is different from the volume flow rate leaving, you need to write down two differential equations, rather than 1:

Volume Input - Volume Output = accumulation for the volume of fluid in the tank

Chemical X Input - chemical X Output = accumulation for chemical X in the tank

If V(t) is the volume of fluid in the tank at time t, fin is the volumetric flow rate of fluid in, and f_out is the volumetric flow rate of fluid out, what is the differential equation for V?

If C(t) is the concentration of chemical X within the tank at time t, and C_in is the concentration of chemical X in the feed to the tank, what is the differential equation for the rate of change of total chemical X in the tank?

The next step is to multiply the differential equation for V by C, and subtract the resulting relationship from the mass balance on chemical X.
Jeff12341234
Jeff12341234 is offline
#3
Feb22-13, 10:46 PM
P: 179
I don't follow.. The way I did it is the way the professor instructed us and the steps match the steps in his example. To solve for C, I now realize from an example in the book that A(0)=35. With that information, I can solve for C.


Chestermiller
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#4
Feb23-13, 07:41 AM
Sci Advisor
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Thanks
PF Gold
P: 4,508

Mixture problem. How to solve for C?


OK. I see what you did, and, of course, it is right. But, here's my alternate version to consider:

[tex]\frac{dV}{dt}=f_{in}-f_{out}[/tex]
[tex]\frac{d(VC)}{dt}=f_{in}C_{in}-f_{out}C[/tex]
Multiply the first equation by C and subtract it from the second equation:

[tex]V\frac{dC}{dt}=f_{in}(C_{in}-C)[/tex]

where [itex]V=V_0+(f_{in}-f_{out})t[/itex]
So,

[tex]\frac{dC}{(C_{in}-C)}=f_{in}\frac{dt}{V_0+(f_{in}-f_{out})t}[/tex]


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