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Hello All,
I am in the process of making a large scale composter for my college, and need some help modeling the residence times I would get from stringing multiple CSTR's in series in order to ensure sufficient process time. So far, I've tried working through with very limited sucess, can anyone point me in the right direction?
Thanks!
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Cstrs In Series
Started by Guest_prometheus_*, Jun 11 2010 01:37 PM
3 replies to this topic
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#2
siretb
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Posted 14 June 2010 - 01:54 AM
The easiest way to model is to use the Laplace transform
Google CSTR + Laplace. The following link gives you the general approach.
http://www.chemeng.q...ments/mod2c.PPT
Google CSTR + Laplace. The following link gives you the general approach.
http://www.chemeng.q...ments/mod2c.PPT
#3
lopez8
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Posted 16 June 2010 - 09:45 PM
Hey there,
Modeling CSTR's in series using Laplace properties, is done by taking the Laplace transform of the mass balance, acc=gen+input-output. For CSTR there is no accumulation. Therefor, acc=0, then 0=gen+input-output, in other word it's at steady state. 0=Co-C+V/Qr--->C=Co/(1-V/Qr)
"Assuming that you know what all the variables mean."
Steady state eq. C=Co/(1+tau k) Eq.1
+ The sign in the denominator depends on the type of reaction!!! This will also determine what type of reactor you should use!!
+ I used a first-order reaction going to from reactants to products.
Eq.1 is a function of flow and volume. If the flow is constant then the design variable is the volume of the reactor (V).
For the series part.
+ Recognize that C is the concentration going into the following reactor. Therefore, C1=Co/(1+tau1k)..C2=C1/(1+tau2k0... then,
Cn=Co/{(1+tau1k)*(1+tau2k)*(1+tau"n"k)}. If all of the reactor are the same (volume and flow ) then Cn=Co/(1+tauk)^n
To elaborate an equation for series of CSTR as a function time (transient), set acc=gen+input-output, and use the Laplace transform.
" Assume Co is a step function (constant)"
VdC/dt=-kVC+QCo-QC ----> V/Q dC/dt+(1+kV/q)C=Co Eq.2
After taking the Laplace transfrom and simplifying Eq. 2
C(s)/C0(s)=K/(Ts+1) Where K=Q/(Q+Vk) T=V/(Q+Vk) Eq.3
Eq.3 is in time-constant from; which make's it easier analyse..." find the poles and gain."
There is no need to invert the Eq.3 from (s) space to (t) space, if you have access to MathLab, if not then invert it and use good old excel.
Every thing would change if Co is Sinusoidal, ramp, exponential, rectangular, an time-delay function.
The same analogy applies for the series part...
Note:
The more reactor you have in series, the more time it will take for the last reactor to see the change. Also the responses becomes
sluggish.
I think this is a good start for your project. Don't hesitate to ask if you need more help. Please read more about CSTR
A.L
Modeling CSTR's in series using Laplace properties, is done by taking the Laplace transform of the mass balance, acc=gen+input-output. For CSTR there is no accumulation. Therefor, acc=0, then 0=gen+input-output, in other word it's at steady state. 0=Co-C+V/Qr--->C=Co/(1-V/Qr)
"Assuming that you know what all the variables mean."
Steady state eq. C=Co/(1+tau k) Eq.1
+ The sign in the denominator depends on the type of reaction!!! This will also determine what type of reactor you should use!!
+ I used a first-order reaction going to from reactants to products.
Eq.1 is a function of flow and volume. If the flow is constant then the design variable is the volume of the reactor (V).
For the series part.
+ Recognize that C is the concentration going into the following reactor. Therefore, C1=Co/(1+tau1k)..C2=C1/(1+tau2k0... then,
Cn=Co/{(1+tau1k)*(1+tau2k)*(1+tau"n"k)}. If all of the reactor are the same (volume and flow ) then Cn=Co/(1+tauk)^n
To elaborate an equation for series of CSTR as a function time (transient), set acc=gen+input-output, and use the Laplace transform.
" Assume Co is a step function (constant)"
VdC/dt=-kVC+QCo-QC ----> V/Q dC/dt+(1+kV/q)C=Co Eq.2
After taking the Laplace transfrom and simplifying Eq. 2
C(s)/C0(s)=K/(Ts+1) Where K=Q/(Q+Vk) T=V/(Q+Vk) Eq.3
Eq.3 is in time-constant from; which make's it easier analyse..." find the poles and gain."
There is no need to invert the Eq.3 from (s) space to (t) space, if you have access to MathLab, if not then invert it and use good old excel.
Every thing would change if Co is Sinusoidal, ramp, exponential, rectangular, an time-delay function.
The same analogy applies for the series part...
Note:
The more reactor you have in series, the more time it will take for the last reactor to see the change. Also the responses becomes
sluggish.
I think this is a good start for your project. Don't hesitate to ask if you need more help. Please read more about CSTR
A.L
#4
Guest_prometheus_*
Guest_prometheus_*
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- guestGuests
Posted 17 June 2010 - 09:07 AM
Wow! Lopez that was VERY helpful. I appreciate the guidance, I'm a biochemist, so this is a little out of my field. I'm not terribly intimidated by it, though, as decomposition is a pretty amiable reaction.
Quickly, I'm not terribly familiar with those variables, am I correct to believe:
Cn = concentration leaving reactor 'n'
Co = concentration initial
Tau = residence time
k = rate constant for the reaction
Literature I've read indicates that the rate of this kind of thing follows first order kinetics, so in that case i can just use
Cn=Co/(1+tauk)^n
to find the exit concentration of the material left undigested, right?
If thats the case, I think I'm mostly done! I don't think I really need to model the system with respect to time, although given the variability of the waste I'll be using, that might be useful for trouble shooting (also, considering I'm planning on having automated pH adjustment systems at each step, it would be useful to check the effect on reactors downstream).
Thanks again for your help! If you guys want, I can try and post some pictures as this thing comes together.
Quickly, I'm not terribly familiar with those variables, am I correct to believe:
Cn = concentration leaving reactor 'n'
Co = concentration initial
Tau = residence time
k = rate constant for the reaction
Literature I've read indicates that the rate of this kind of thing follows first order kinetics, so in that case i can just use
Cn=Co/(1+tauk)^n
to find the exit concentration of the material left undigested, right?
If thats the case, I think I'm mostly done! I don't think I really need to model the system with respect to time, although given the variability of the waste I'll be using, that might be useful for trouble shooting (also, considering I'm planning on having automated pH adjustment systems at each step, it would be useful to check the effect on reactors downstream).
Thanks again for your help! If you guys want, I can try and post some pictures as this thing comes together.
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