Differential equations for series variable volume reactions

[maistral]

How do you make a differential equation for such?

Say for example. I have two reactions in series, A → R and R → S going in a gas-phase reaction. If I'm correct, the ODE for the conversion of A is

dXA/dt = kA*[(NAo/vo)*(1-XA)/(1+δA*YAo*XA)]*[(vo/NAo)*(1+δA*YAo*XA)].

I don't know now how to get the differential equation for R since it's a series reaction - or is there another method for me to get the values of A, R, and S simultaneously? Please keep it simple. Thanks.

 

[SteamKing]

There may not be a single differential equation to describe such processes. However, a system of several differential equations may do the trick.

 

[maistral]

Yep, it's what I'm after. I may be able to derive at least the constant-volume version but I am beginning to have issues upon the determination of δ for R and S and the setting up of the differential equation itself. I need something to pattern upon.

 

[Chestermiller]

It would be very helpful if you could define the parameters in your equation. I've had a lot of experience solving problems like this, but I don't recognize your parameters.

Chet

 

[maistral]

Thanks for expressing interest to help me.

It's an isothermal variable volume batch reaction that happens to move as a series reaction type. I have no issues with setting up an equation for A, but I am at a total loss when setting up the equations for R and S.

A → 2R → S

δA = from Fogler, d/a+c/a-b/a-a/a, at least in this equation it's r/a-a/a. These small letters are the coefficients.
NAo = incoming moles
vo = initial volume
XA = percent conversion of A.




While I was researching for more possible clues, I found this thing:

http://www.umich.edu/~elements/course/lectures/six/

And at the example C, it's approximately close to what I need (except it's a parallel reaction in a PFR, what I need is a batch reactor - but hey I can manage to make series rate equations, at least for constant volume ones). I just have particular trouble in trying to understand the stoichiometry part and with that, I have some important questions to ask:

1) I don't understand why did these terms exist:

Is it somehow similar to what Fogler's ECRE which is δ=d/a+c/a-b/a-1? I could not comprehend what made it this way. Further isn't F for molar flowrates? If I use this for batch reactors instead, should I use N? Lastly, what is y there?

Sorry for asking too many questions, I'm trying to study on my own before I try to implement it as a code (with no other sources other than these books that would not give series nor parallel gas-phase reaction examples). Thanks a lot guys.

 

[Chestermiller]

I can't follow Fogler's notation. Let me say the problem back to you to test my understanding.

You have a batch reactor in which the volume is varying with time. Since you don't say anything about how this comes about, I will assume that the pressure of the reactor is constant, and the volume is varying in proportion to the total number of moles of the three species. The initial volume is v₀. Initially, only species A is present, and the initial number of moles of this species is NA₀. I assume that you are dealing with an ideal gas system. You want to determine the number of moles of A, R, and S, and the volume of the reactor as functions of time. Is this a correct statement of the problem?

Incidentally, I know Scott Fogler. About 50 years ago, he was a young assistant professor when I was a graduate student at the U of M.

Chet

 

[maistral]

Sorry for the reaction, but really, oh my gosh You hit the head of the nail perfectly
I was toying around with it and this is what came out:

dNA/dt = -k1A * [(NA/V)^α] * V
dNR/dt = [k1R * [(NA/V)^α] - k2R* [(NR/V)^γ]] * V

Then I realized, dV/dt = ...?

 

[Chestermiller]

Sorry for the reaction, but really, oh my gosh You hit the head of the nail perfectly
I was toying around with it and this is what came out:

dNA/dt = -k1A * [(NA/V)^α] * V
dNR/dt = [k1R * [(NA/V)^α] - k2R* [(NR/V)^γ]] * V

Then I realized, dV/dt = ...?

This is much closer. I assume that the first reaction is α order and the second reaction is γ order, correct? Also, in the second version you gave, there were 2 moles or R produced for every mole of A that reacted. Also, in your second equation above, k1R should be the same as k1A, unless k1R includes the stoichiometric factor of 2, in which case k1R = 2k1A, correct?

You don't need a differential equation for V. The volume increases in proportion to the total number of moles. So...

V = v₀(NA + NR + NS)/(NA₀)

I don't like writing the reactions as a linear sequence, because it produces ambiguity. Are the reactions

A -> 2R
R + R -> S

or are they

A ->2R
R ->S

Depending on which you mean, the mass balance expressions for the 2nd reaction will be different, as will the rate constants. As long as you know what you mean, this won't be a problem, but I prefer writing them as explicitly separate reactions.

 

[maistral]

Yes, just to safeguard them if they would turn out to be non-elementary (I mean, the α and γ).

So what you mean is, I just plug in the entire V = vo*(NA + NR + NS)/(NAo) on each V term?

What about the differential equation for R? Also, what if I had a combination reaction instead? Say for example, this reaction:

aA + bB → rR
rR + sS → tT

If I'm correct, the differential equations would be:
dNA/dt = - k1A * [(NA/V)^α] * [(NB/V)^β] * V
dNB/dt = - k1B * [(NA/V)^α] * [(NB/V)^β] * V
dNR/dt = {k1R * [(NA/V)^α] * [(NB/V)^β] - k2R * [(NR/V)^γ] * [(NS/V)^ω]} * V
dNS/dt = - k2S * [(NS/V)^ω]} * V
dNT/dt = k2T * [(NT/V)^τ]} * V

what would be the equation for V in this case? Actually I'm trying to evade a material balance solution since that would defeat the purpose of the program I'm trying to implement - I really don't mind a massive Runge-Kutta table for systems of DE, it's just this V term is troubling.

Sorry for the barrage of questions, I thank you very much for your kind attention.EDIT: What I meant above there is, actually when I resolved my constant-volume program, what I did was since they were all in terms of concentration, I solved them all numerically without the aid of a material balance (which is what I'm trying to pull off this time as well)

I mean this one:
aA + bB → rR
rR + sS → tT in a constant volume setup,

dCA/dt = - k1A * [(CA)^α] * [(CB)^β]
dCB/dt = - k1B * [(CA)^α] * [(CB)^β]
dCR/dt = {k1R * [(CA)^α] * [(CB)^β] - k2R * [(CR)^γ] * [(CS)^ω]}
dCS/dt = - k2S * [(CS)^ω]}
dCT/dt = k2T * [(CT)^τ]}

Then given inital values for each component, I would solve the differential equations simultaneously using Runge-Kutta.

 

[Chestermiller]

OK. To get V in the constant pressure case,

V=v₀(NA+NB+NR+NS+NT)/(NA₀+NB₀+NR₀+NS₀+NT₀)

Please be careful in precisely specifying the reaction rate and rate constant for each of the reactions. For example, if the reaction rate is in moles/(volume-sec), you need to specify which species this number of moles references to. Be careful with the stoichiometric coefficients in the mass balances also.

Chet

 

[maistral]

OK. To get V in the constant pressure case,

V=v₀(NA+NB+NR+NS+NT)/(NA₀+NB₀+NR₀+NS₀+NT₀)

Please be careful in precisely specifying the reaction rate and rate constant for each of the reactions. For example, if the reaction rate is in moles/(volume-sec), you need to specify which species this number of moles references to. Be careful with the stoichiometric coefficients in the mass balances also.

Chet

I'm really really sorry but my brain just won't get what you said I tried reading it and understanding it well but I could not get it precisely. I'm the type of guy who builds off from an example, I can't grasp text concepts well.

By any chance, do you mean that I should watch out at how I construct my differential equations - for example if it is 2A → R, do you mean that I am to set kA = 2k1?

As my final question, given your prescribed equation for V, if the stoichiometric coefficients for the reaction are not 1, will they have certain coefficients beside them as well? I mean say

A + 2B → 2R
R → 2S

will it still be V = vo*(NA+NB+NR+NS)/(NAo+NBo+NRo+NSo) ?

In advance, I'd like to thank you very much for your time and effort for answering my series of questions. If possible (and hoping you wouldn't find it very rude), I have a certain message sent to you which would reveal the reason of the series of questions. Once again, thank you very much.

 

[Chestermiller]

I'm really really sorry but my brain just won't get what you said I tried reading it and understanding it well but I could not get it precisely. I'm the type of guy who builds off from an example, I can't grasp text concepts well.

By any chance, do you mean that I should watch out at how I construct my differential equations - for example if it is 2A → R, do you mean that I am to set kA = 2k1?

Yes. If the reaction is aA+bB=cD+dD, and you calculate the reaction rate as r = kA^aB^b, then the rate that A disappears is ar, the rate that B disappears is br, the rate that C appears is cr, and the rate that D appears is dr.

As my final question, given your prescribed equation for V, if the stoichiometric coefficients for the reaction are not 1, will they have certain coefficients beside them as well? I mean say

A + 2B → 2R
R → 2S

will it still be V = vo*(NA+NB+NR+NS)/(NAo+NBo+NRo+NSo) ?

Yes. You accounted for the numbers of moles involved in the individual species mass balances. If you included the stoichiometric coefficients again here, this would be doing it twice.

 

[maistral]

Thanks a lot! I'll try practicing with the equations for a while then I would begin to implement it. :D

 

[maistral]

Errr, I forgot to ask this question.

If there are inerts in the incoming feed, do I just ignore them? I mean the V equation still holds true even, say for example:

aA + bB → rR
rR + sS → tT

The volume equation is still V=v0(NA+NB+NR+NS+NT)/(NA0+NB0+NR0+NS0+NT0)? Or do I add NIo at the denominator and NI at the numerator (I = the inert component), since I won't be setting the differential equations for NI anyway (and thus they will be held constant) - such that the volume equation is now:

V=v0(NA+NB+NR+NS+NT+NI)/(NA0+NB0+NR0+NS0+NT0+NI0) where NI would probably be the same numbers since they are inerts? I have a tingly feeling that they should be added because they are in the batch reactor chamber and therefore they take space (and must be added). Thanks again.