Tutorial 7 - 2010

Due date(s):	11 November 2010
(PDF)	Tutorial questions
Other instructions	Hand-in at class.

<rst> <rst-options: 'toc' = False/> <rst-options: 'reset-figures' = False/>

.. |m3| replace:: m\ :sup:`3` .. highlight:: python

.. rubric:: Tutorial objectives: Numerical differentiation and plotting.

Question 1 [2]

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This question uses the experimental data from the class notes. A batch system was charged with reactant :math:`{\sf A}` at time :math:`t=0` at a concentration of 1.0 mol/L. As the assumed first-order reaction evolved, the concentrations were recorded:

.. csv-table:: :header: "Time (minutes)", "Concentration (mol/L)"

0, 1.00 5, 0.84 10, 0.72 20, 0.57 30, 0.47 45, 0.37 60, 0.30

The dynamic balance for this system can be shown to be :math:`\displaystyle \frac{dC_{\sf A}(t)}{dt} = -k C_{\sf A}(t)`.

1. . Reproduce the above table and add a column that provides an approximation of :math:`\displaystyle \frac{dC_{\sf A}(t)}{dt}` at each time step.
2. . Add a fourth column that lists the order of your approximation listed in part 1.
3. . Use a 2nd order Lagrange polynomial to approximate the :math:`C_{\sf A}(t)` data, and use this polynomial to estimate :math:`C_{\sf A}\,'(t)` at time :math:`t=15` minutes.
4. . Can you provide an :math:`O(h^2)` estimate of :math:`\displaystyle \frac{dC_{\sf A}(t)}{dt}` at :math:`t=0`? If so, please calculate it.

Question 2 [1.5]

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A second order reaction in a constant-volume batch reactor has the following time-dependent behaviour:

.. math::

C_{\sf A}(t) = \frac{C_{\sf A}(0)}{C_{\sf A}(0)kt + 1} \qquad\,\text{with}\qquad\, k = 0.002 \frac{\text{m}^3}{\text{mol.s}} \qquad\,\text{and}\qquad\,C_{\sf A}(0) = 2.5 \frac{\text{mol}}{\text{m}^3}

Starting with :math:`h = 1.0` and decreasing in powers of 10, which step size, :math:`h=\Delta t`, provides the lowest error estimate of :math:`\displaystyle \left.\frac{dC_{\sf A}}{dt}\right|_{t=100}` at :math:`t=100` seconds?

Present your results in tabular form with these 5 columns:

* Step size, :math:`h = \Delta t` * Forward difference approximation * Error in the forward difference approximation * Central difference approximation * Error in the central difference approximation

Bonus question [0.5]

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Show the two error columns from question 2 in graphical form, using log-axes for both the x- and y-axis. Search for help over the internet if you don't know how to plot log-plots in MATLAB, or with Python's ``matplotlib`` library. Add a legend, grid lines and axis labels to the plot.

.. raw:: latex

\vspace{0.25cm} \hrule \begin{center}END\end{center} </rst>