Assignment 4 - 2010

Due date(s):	09 December 2010
(PDF)	Assignment questions
Other instructions	Hand-in at class.

<rst> <rst-options: 'toc' = False/> <rst-options: 'reset-figures' = False/> .. |m3| replace:: m\ :sup:`3`

Question 1 [2]

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A new type of `thermocouple <http://en.wikipedia.org/wiki/Thermocouple>`_ is being investigated by your group. These devices produce an *almost* linear voltage (millivolt) response at different temperatures. In practice though it is used the other way around: use the millivolt reading to predict the temperature. The process of fitting this linear model is called *calibration*.

1. . Use the following data to calibrate a linear model:

================= ==== ==== ==== ==== ==== ==== ==== ==== ==== ==== Temperature [K] 273 293 313 333 353 373 393 413 433 453 ----------------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- Reading [mV] 0.01 0.12 0.24 0.38 0.51 0.67 0.84 1.01 1.15 1.31 ================= ==== ==== ==== ==== ==== ==== ==== ==== ==== ====

Show the linear model and provide the predicted temperature when reading 1.00 mV.

1. . Are you satisfied with this model, based on the coefficient of determination (:math:`R^2`) value?

1. . What is the model's standard error? Now, are you satisfied with the model's prediction ability, given that temperatures can usually be recorded to an accuracy of :math:`\pm 0.1` K with most thermocouples.

Question 2 [2]

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Batch reactors are often used to calculate reaction rate data, since the dynamic balances are a direct function of the reaction rate:

.. math::

\frac{dC_{\sf A}(t)}{dt} = r_{\sf A}

assuming constant volume and assuming all energy-related terms are negligible. We are unsure about the order of our reaction rate, :math:`\alpha`, and we also don't know the reaction rate constant, :math:`k` in the expression :math:`r_A = - k C_{\sf A}^{\alpha}`.

Integrating the above differential equation from time :math:`t=0` to time :math:`t=t` gives:

.. math::

\int_{C_{\sf A,0}}^{C_{\sf A}}{ \frac{1}{C_{\sf A}^\alpha} dC_{\sf A}} &= -k \int_{0}^{t}{dt} \\

                          \left.          \frac{C_{\sf A}^{1-\alpha}}{1-\alpha} \right|_{C_{\sf A,0}}^{C_{\sf A}} &= -kt  \qquad \text{where}\,\, \alpha \neq 1

The following concentrations are collected as a function of time

================== ==== ==== ==== ==== ==== ==== ==== ====

Time [hours] 0 0.5 1 2 3 5 7.5 10 17

---

---- ---- ---- ---- ---- ---- ---- ---- ----

Concentration [mol/L] 3.5 1.5 0.92 0.47 0.29 0.16 0.09 0.06 0.03

================== ==== ==== ==== ==== ==== ==== ==== ====

1. . Use these concentration-time data to determine the values of :math:`\alpha` and :math:`k`, using a least squares model.

1. . Once you have the rate constants, simulate the batch reactor (use the ``ode45`` or Python integrator) and superimpose the data from the above table on your concentration-time trajectory. Do they agree? Which regions of the plot have the worst agreement (this is where you would collect more data in the future).

Question 3 [3]

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The yield from your lab-scale bioreactor, :math:`y`, is a function of reactor temperature, impeller speed and reactor type (one with with baffles and one without). You have collected these data from various experiments.

.. csv-table:: :header: Temp = :math:`T` [°C], Speed = :math:`S` [RPM], Baffles = :math:`B` [Yes/No], Yield = :math:`y` [g] :widths: 30, 30, 30, 30

82, 4300, No, 51 90, 3700, Yes, 30 88, 4200, Yes, 40 86, 3300, Yes, 28 80, 4300, No, 49 78, 4300, Yes, 49 82, 3900, Yes, 44 83, 4300, No, 59 65, 4200, No, 61 73, 4400, No, 59 60, 4400, No, 57 60, 4400, No, 62 101, 4400, No, 42 92, 4900, Yes, 38

1. . Use a built-in software function in MATLAB or Python (please don't use Excel) to fit a linear model that predicts the bioreactor yield from the above variables. Your model should be of the form:

.. math::

\hat{y} = a_0 + a_T T + a_S S + a_B B

*Hint*: convert the No/Yes variables to integers with ``0=No`` and ``1=Yes``.

1. . Interpret the meaning of each coefficient in the model and comment on the standard error.

1. . Calculate and show the :math:`\mathbf{X}^T\mathbf{X}` and :math:`\mathbf{X}^T\mathbf{y}` matrices for this linear model in order to calculate the 4 coefficients in :math:`{\bf a} = \left[a_0, a_T, A_S, A_B\right]` yourself. Your answer must include the MATLAB or Python code that computes these matrices and the :math:`{\bf a}` solution vector.

Question 4 [2]

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.. Similar to Tutorial 8, 2009

The viscosity of sulphuric acid, :math:`\nu`, varies with purity, :math:`p` in the following manner:

======================== ===== =====

math:`p` [%] 20 60 80

---

----- ----- -----

math:`\nu` [millipascal] 1.40 5.37 17.4

======================== ===== =====

1. . Express :math:`\nu(p)` as a quadratic function using Lagrange interpolating polynomials. Do not simplify the polynomial.
2. . Express :math:`\nu(p)` as a quadratic function using Newton interpolating polynomials. Do not simplify the polynomial.
3. . Fit a cubic spline through the data points: clearly show your :math:`{\bf Xa = y}` linear system of equations, then solve them using computer software; finally report your spline coefficients.
4. . Use computer software to plot:

	*	the Newton interpolating polynomial 

* the cubic spline, * and the 3 data points on the same graph.

1. . What is the estimated viscosity at :math:`p` = 40% purity using linear interpolation?
2. . Which of the estimation procedures that you used above has the closest estimate to the true value of 2.51 millipascal?

Question 5 [2]

===

The following data are collected from a bioreactor experiment, during the growth phase.

============= ===== ===== ===== =====

Time [hours] 0 1.0 2.0 4.0 6.0

---

----- ----- ----- ----- -----

math:`C` [g/L] 0.1 0.341 1.102 4.95 11.24

============= ===== ===== ===== =====

Fit a natural cubic spline for these data and use it to estimate the number of cells at time 3, 5, and 7 hours.

Show your matrix derivation for the linear system of equations, and solve it using computer software. Plot the cubic spline over the time range 0 to 8 hours.

Bonus question [0.5]

========

Use the cubic spline from the previous question and find the time where the cell count was approximately 10.0 g/L. Do not solve the equation by hand, but investigate `MATLAB's <http://www.mathworks.com/help/techdoc/ref/roots.html>`_ or `Python's <http://docs.scipy.org/doc/numpy/reference/generated/numpy.roots.html>`_ polynomial root finding function: ``roots(...)``

.. raw:: latex

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