Difference between revisions of "User:Kevindunn"

From Process Model Formulation and Solution: 3E4
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m (Replaced content with "<html> <div class="math"> \[\begin{split}C^1= \left[ \begin{array}{ccccc} 1 & 1 & 1 & 1 & 0\\ 0 & -3 & -3 & -3 &...")
 
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''' Contact details '''
<html>


I don't have an office on campus.  The best way to contact me is by email:
<div class="math">
 
\[\begin{split}C^1=
* [mailto:dunnkg@mcmaster.ca dunnkg@mcmaster.ca] (preferred)
\left[ \begin{array}{ccccc}
* [mailto:kevin.dunn@connectmv.com kevin.dunn@connectmv.com] (alternate)
  1  &amp;  1  &amp;  1  &amp;  1 &amp;  0\\
 
  0  &amp;  -3  &amp;  -3  &amp; -3  &amp;  4\\
<rst>
  0  &amp;  3  &amp; -5  &amp;  0 &amp;  2\\
<rst-options: 'toc' = False/>
  0  &amp;  -6 &amp;  -6  &amp;  -4  &amp;  -4        \end{array} \right]\end{split}\]</div>
<rst-options: 'reset-figures' = False/>
<script type='text/javascript' src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML-full,local/mwMathJaxConfig'> </script>
 
</html>
.. math::
 
y_F &= 0.5\\
\frac{1}{\beta} y_F - (\delta+1)x_1 + \delta x_2 &= 0 \\
x_1 - (\delta+1)x_2 + \delta x_3 &= 0 \\
x_2 - (\delta+1)x_3 + \delta x_F &= 0 \\
x_F &= 0.1
where :math:`\beta` is a coefficient, assumed constant throughout the absorber tower, that relates the liquid phase composition, :math:`x_n`, to the gas-phase composition, :math:`y_n = \beta x_n`, assuming of course that equilibrium is achieved in each stage.
 
The :math:`\delta` coefficient is a dimensionless number defined as a function of the molar gas and liquid flows in the column, and :math:`\beta`, so that :math:`\delta = \displaystyle \frac{L}{G \beta}`
**5.1** Write the given steady-state equations in the form :math:`A{\bf x} = b` where :math:`{\bf x} = \left[y_F, x_1, x_2, x_3, x_F\right]`.  Report the :math:`A` matrix and :math:`b` vector.
 
**5.2** What condition(s) must be satisfied so that matrix :math:`A` is diagonally dominant?
 
**5.3** You decide to use the Gauss-Seidel method to solve this system of equations. Also, you are given that :math:`L/G = 1.5`, and :math:`\beta = 0.8`.  Is this method guaranteed to converge for these coefficient values?
 
**5.4** Perform one iteration of the Gauss-Seidel method, starting from a suitable initial guess that you believe will converge in fewer iterations than simply using the default guess of :math:`{\bf x} = \left[y_F, x_1, x_2, x_3, x_F\right] = [0, 0, 0, 0, 0]`.  Use :math:`L/G = 1.5`, and :math:`\beta = 0.8` in your matrix, and briefly explain your choice for the initial guess.
 
</rst>

Latest revision as of 07:46, 10 February 2017

\[\begin{split}C^1= \left[ \begin{array}{ccccc} 1 & 1 & 1 & 1 & 0\\ 0 & -3 & -3 & -3 & 4\\ 0 & 3 & -5 & 0 & 2\\ 0 & -6 & -6 & -4 & -4 \end{array} \right]\end{split}\]

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