# Tutorial 2 - 2013

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 Due date(s): 21 January 2013 (PDF) Tutorial questions

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• Assignment objectives**: math refresher; mol balances; working with conversion

* Reminder: always state assumptions in this tutorial, in assignments, midterms and exams.

Make sure you can do these in a test/exam (i.e. without internet access). Let :math:X be conversion; find:

#. :math:\displaystyle \int{ \frac{1}{(1-X)^2} \,dX} = #. :math:\displaystyle \int_{X_1}^{X_2}{ \frac{1}{(1-X)^2} \,dX} = #. And in general, what is: :math:\displaystyle \int{ \frac{a + x}{bx} \,dx} =

For the question we covered in the end of class last week (see last page of this tutorial), we showed the volume of a PFR required is the area under the curve. The volume required was :math:2.16\,\text{m}^3 to obtain 80% conversion.

If we used 4 CSTRs in series:

#. What would be the size of each reactor if we wanted 20% conversion in each reactor? #. What is the total volume of these 4 reactors? #. How does this total CSTR volume compare with (a) the single CSTR volume and (b) the single PFR volume? #. What is the reaction rate in each reactor?

The following reaction rate, :math:-r_A measured in units of :math:\left[\dfrac{\text{kmol}}{\text{hr.m}^3}\right] is observed at a particular conversion, :math:X:

.. csv-table:: :header: "Reaction rate", "Conversion" :widths: 15, 15

78, 0.0 106, 0.2 120, 0.4 70, 0.6

We showed in class that the area under this curve is related to volume of the plug flow reactor.

#. Start from the general mol balance and derive the equation that shows the area is equal to the plug flow reactor's volume; clearly state all assumptions used in your derivation. #. Assuming these assumptions are all met, calculate the plug flow reactor's volume to achieve a 60% conversion given a feed rate of :math:15\,\text{mol.s}^{-1} to the reactor. #. If there is zero conversion at the entry to the PFR and 60% at the exit; what is the conversion half-way along the reactor? #. What is the conversion at 25% of the way along the reactor? #. Now plot a graph a graph of conversion throughout the reactor, from start to end. The :math:x-axis on your plot should be the volume co-ordinate, :math:V. #. What is the reaction rate at the entry of the reactor? #. And at the midpoint? #. And at the exit? #. Plot a curve that shows the reaction rate throughout the reactor, from start to end. The :math:x-axis on your plot should be the volume co-ordinate, :math:V.

A new drug is being prototyped in a batch reactor; as is becoming common-place now, this drug is grown *inside* a cell as a by-product of the regular cellular processes. So far, experiments have shown the rate of consumption of the starting material, an animal-derived cell :math:A, is the only concentration in the rate expression.

.. math::

-r_A = \frac{5.5C_A}{20+C_A}

where :math:-r_A has units of :math:\left[\dfrac{\text{mol}}{\text{day.m}^3}\right]

#. Why is a batch reactor suitable for this type of testing? #. 30 mols of cellular material are added to a batch reactor of :math:0.5\,\text{m}^3; the liquid food source is added at the same time to the reactor, in excess. Calculate the amount of cellular material remaining in the tank after 10 days. #. How many days are required to convert 80% of the starting cellular material. #. Show a plot of the concentration in the tank over time until there is essentially 100% conversion.

*Note*: in tutorial 1 you solved a similar problem, but for a CSTR and PFR.

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