Difference between revisions of "User:Kevindunn"
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<rst-options: 'toc' = False/> | <rst-options: 'toc' = False/> | ||
<rst-options: 'reset-figures' = False/> | <rst-options: 'reset-figures' = False/> | ||
.. 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> | </rst> | ||
Revision as of 20:22, 24 October 2010
Contact details
I don't have an office on campus. The best way to contact me is by email:
- dunnkg@mcmaster.ca (preferred)
- kevin.dunn@connectmv.com (alternate)
<rst> <rst-options: 'toc' = False/> <rst-options: 'reset-figures' = False/>
.. 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>