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Cell[CellGroupData[{
Cell[TextData[StyleBox["Importance of mass transfer fluctuations on reaction \
outcome in multiphase reactors.  ", "Title"]], "Title"],

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  "This notebook has been written in ",
  StyleBox["Mathematica ",
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  "by \n\n",
  StyleBox["Mark J. McCready\nProfessor and Chair of Chemical Engineering\n\
University of Notre Dame\nNotre Dame IN 46556\nUSA",
    FontSize->14],
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  "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \
the World Wide Web and the Internet.\n\nI would hope that as a professional \
courtesy that this notice remain visible to other users. \nThere is no charge \
for copying and dissemination.  \n\nVersion:  5/30/00\nA more recent version \
of this notebook may be available at\n",
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differences in the total conversion but differences in the ",
  StyleBox["selectivity",
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  " of the products if the fluctuation frequency is of the correct order of \
magnitude.  The intended application is gas-liquid reactions involving a \
solid phase catalyst such as trickle bed reactors.   \n\nA CSTR ",
  "(Continuous flow Stirred Tank Reactor) ",
  "model is used.  First the model is demonstrated for a single phase \
reaction.  Next a simple liquid-solid model is used.  When fluctuations in \
the liquid-solid mass transfer coefficient are included, there is a change in \
selectivity with frequency.  Finally the complete gas-liquid-solid model of \
Wu, McCready and Varma (1995) is done.  This shows a maximum in selectivity \
at an intermediate frequency.     "
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is used.  This distinction is very useful in computer algebra and is employed \
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Chemical reaction engineering, reaction selectivity, gas-liquid \
reaction, multiphase reactor, trickle-bed reactor, reactor dynamics     \
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Cell[TextData[{
  "R. Wu, M. J. McCready and A. Varma (1995), ",
  StyleBox["Chemical Engineering Science",
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  ", ",
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  ", pp3333-3334.   T. W. F. Russell and M. M. Denn (1972), ",
  StyleBox["Introduction to Chemical Engineering Analysis",
    FontSlant->"Italic"],
  ", Wiley.  R. W. Felder and R. W. Rousseau (1986) ",
  StyleBox["Elementary Principles of Chemical Processes",
    FontSlant->"Italic"],
  ", 2nd Edition. Wiley.   "
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Cell["Transient and Steady Single Phase CSTR model", "SectionFirst"],

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Cell["Basic problem set-up", "Section"],

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reaction, A + B --> 2 D is carried out.  Second order reaction kinetics are \
assumed given by\n",
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references Russell and Denn or Felder and Rouseau.  It involves writing mass \
balances around the reactor for the chemical species and the total mass.  It \
is critical to realize that it is ",
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Cell["\<\
Consider a tank reactor that has inflows, qa and qb and outflow, q. \
 The overall mass (actually volume) balance is the differential \
equation:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(veq = \(v'\)[t]\  - \ qa\  - qb + \ q\  \[Equal] 0\)], "Input"],

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      TraditionalForm]], "Output"]
}, Open  ]],

Cell["\<\
 The change in volume per time is v'[t] and is nonzero if q does \
not equal qa + qb.\
\>", "Text"],

Cell["\<\
We can do a component balance for \"ca\".  The inflow is: (qa caf), \
the outflow is: q ca[t] and second order reaction that removes \"a\" is: k \
ca[t] cb[t] v[t].  \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(caeq = 
      v[t]\ \(ca'\)[t]\  - qa\ caf + \ q\ ca[t] + \ 
          k\ ca[t]\ cb[t]\ v[t]\  \[Equal] 0\)], "Input"],

Cell[BoxData[
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            RowBox[{
              SuperscriptBox["ca", "\[Prime]",
                MultilineFunction->None], "(", "t", ")"}]}]}], "==", "0"}], 
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Cell["\<\
We also need a similar component balance for \"cb\" with inflow, \
(qb cbf), outflow (q cb[t]) and second order reaction k ca[t] cb[t] v[t], \
terms.  \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(cbeq = 
      v[t]\ \(cb'\)[t]\  - qb\ cbf + \ q\ cb[t] + \ 
          k\ ca[t]\ cb[t]\ v[t]\  \[Equal] 0\)], "Input"],

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            RowBox[{
              SuperscriptBox["cb", "\[Prime]",
                MultilineFunction->None], "(", "t", ")"}]}]}], "==", "0"}], 
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Cell["\<\
We can do a component balance for \"cd\".  There is no inflow,  but \
it does have outflow (q cd[t]) and second order reaction 2 k ca[t] cb[t] \
v[t], terms.  \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(cdeq = 
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          2\ \ k\ ca[t]\ cb[t]\ v[t]\  \[Equal] 0\)], "Input"],

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Cell[TextData[{
  "To recap and make a ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " list the equations are:"
}], "Text"],

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Cell[BoxData[
    \(eqs = {veq, caeq, cbeq, cdeq, v[0] \[Equal] 100, ca[0] \[Equal] 0, 
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                    MultilineFunction->None], "(", "t", ")"}]}]}], "==", 
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Cell["\<\
Note that we have include some initial conditions in the reactor, \
v=100 and the concentrations are initially 0.  \
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Transient reactor example", "Section"],

Cell["\<\
Here we make some substitutions for a transient reaction example.  \
At time =0, the feed compositions of qa and qb are set to caf= .1, and cbf = \
.2.  The flow rates are qa=10 and qb=10 and the exit is set to q = 20 so no \
volume change occurs in the reactor.  We expect that this example will have a \
transient period leading to a steady state. \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(eqsolv1 = 
      eqs /. {\ caf \[Rule]  .1, cbf \[Rule]  .2, qa \[Rule] 10, 
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          q \[Rule] 20, \[IndentingNewLine]k \[Rule]  .1}\)], "Input"],

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            RowBox[{\(0.1`\ \(cb(t)\)\ \(v(t)\)\ \(ca(t)\)\), 
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                RowBox[{
                  SuperscriptBox["ca", "\[Prime]",
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                  SuperscriptBox["cb", "\[Prime]",
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                  SuperscriptBox["cd", "\[Prime]",
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Cell["Here the numerical ODE package solves them.", "Text"],

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}, Open  ]],

Cell["\<\
The command to plot an \"interpolating function\" requires the \
\"Evaluate\" command.  In this figure we are plotting the concentrations  a,  \
b, and  d as a function of time from the solution \"ans1\".

We see that the concentrations reach a steady state.  The highest one is \
\"B\", \"A\" is next and the product is lowest.  \
\>", "Text"],

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Cell[CellGroupData[{

Cell["Additional calculations for this part", "Subsection"],

Cell["\<\
As was mentioned above the behavior is a simple transient leading \
to a steady state.  For your interest you could:

1.  Vary some of the numbers to see how the transient time varies with the \
flow rates, reactor volume and reaction rate.

2. Change the form of the reaction expression to see its effect.

3.  Nondimensionalize the expression to find the dimensionless groups and use \
the values of the groups to note the relative importance of the reaction on \
the time scale of the transient. \
\>", "Text"]
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Cell["Transient and Steady Two-phase CSTR model", "SectionFirst"],

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Cell["Problem set up.", "Section"],

Cell[TextData[{
  "In this section we consider that there is a liquid and a solid phase.  The \
solid phase is a catalyst and it is mass transfer to this catalyst influences \
the reaction.  First this mass transfer will be steady.  Later it will be \
oscillatory. \n\nThe reaction is now ",
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  " + ",
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  " --> ",
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  "+ ",
  StyleBox["e",
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  ".  The reaction to make e occurs only on the surface of the catalyst.  The \
reaction to make d occurs in the bulk.  We can now consider the selectivity \
of the two products by looking at the homogeneous versus heterogeneous \
reaction rates.  Note that mass transfer will play an important role as well \
in the synthesis of ",
  StyleBox["e",
    FontSlant->"Italic"],
  ".    \n\nFirst we have an equation for ",
  StyleBox["a",
    FontSlant->"Italic"],
  ".  The new term is the transfer of a from the liquid phase to the solid \
phase as governed by concentration and a mass transfer coefficient ka.  "
}], "Text"],

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  "Since there are two phases, we need to keep track of the \"a\" on the \
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  "Here are the analogous equations for ",
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Cell[CellGroupData[{

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Cell[CellGroupData[{

Cell[BoxData[
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Cell["Here are the two equations for the product, e.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[TextData[{
  "Finally we need only 1 equation for d as it is formed in the liquid phase. \
 (We could consider d adsorbing onto the catalyst and doing nothing -- which \
it might.   But then again we have not considered an adsorption onto the \
walls of the reactor for ",
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  ", ",
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  ", ",
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Cell[CellGroupData[{

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Cell["\<\
Solution to the transient problem with a steady mass transfer \
coefficient. \
\>", "Section"],

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Cell["\<\
We set the initial volume of the reactor and the feed rates and \
concentrations.  \
\>", "Text"],

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  " can be varied by changing the mass transfer coefficient and the catalyst \
loading.  Do these work exactly together.    \n\n2. Change the relative \
values of the reaction rate constants.\n\n3.  Calculate the average \
concentrations across the transient period.\n\n4.  Look at the effect of \
different initial concentrations in the reactor.  Consider a reactor startup \
in which you wish to minimize the formation of d, how do you set the initial \
concentrations in the reactor?\n\n5.  Make the equations dimensionless and \
see the size of the relative effects by calculating the values of the \
dimensionless groups.  How many groups are there now?  "
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This seems to work well for the mass transfer coefficient.  We can \
now just incorporate it into the model we have been using.\
\>", "Text"],

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Cell["\<\
This example shows the selectivity enhancement with frequency.  \
\
\>", "Section"],

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Cell["\<\
Here is the calculation of the solution for \[Omega] =1.   \
\>", \
"Subsection"],

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            q \[Rule] 2, \[IndentingNewLine]k \[Rule] 3, k2 \[Rule] 10, 
            ka \[Rule] 10, kb \[Rule] 200, 
            vs \[Rule] 100, \[Omega] \[Rule] 1}, {ca[t], cas[t], cb[t], 
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Mistroke
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Mfstroke
[ .06 .02 .02 .02 ] 0 setdash
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mfstroke
[ .06 .02 .02 .02 ] 0 setdash
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Now change the frequency to \[Omega]=0.5,  \
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