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Question 3 [1]<br />
==============<br />
<br />
#. Write the Newton-Raphson iteration formula that you would use to solve this nonlinear equation.<br />
#. Apply 3 iterations of this formula, also starting from :math:`T = 380` K, and calculate the error tolerances.<br />
<br />
Solution<br />
--------<br />
<br />
#. The Newton-Raphson algorithm is given on slide 15-17 of the *Section C: Nonlinear Algebraic Equations* slide set.<br />
<br />
To apply the Newton-Raphson method we must first calculate the derivative of our function:<br />
<br />
.. math::<br />
<br />
f'(T) = - 0.26 + 3.38\text{x}10^{-3}T - \frac{1.5\text{x}10^{5}}{T^{2}}<br />
<br />
Therefore the Newton-Raphson iteration formula we use is the following:<br />
<br />
.. math::<br />
<br />
T^{(k+1)} = T^{(k)} - \frac{f(T^{(k)})}{f'(T^{(k)})} = T^{(k)} - \frac{- 24097 - 0.26T + 1.69\text{x}10^{-3}T^{2} + \frac{1.5\text{x}10^{5}}{T} - (-23505)}{- 0.26 + 3.38\text{x}10^{-3}T - \frac{1.5\text{x}10^{5}}{T^{2}}}<br />
<br />
#.<br />
We start with :math:`T^{(0)} = 380 K`:<br />
<br />
.. math::<br />
<br />
\begin{array}{rl}<br />
\Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(T^{(0)}\right) + 1.69\text{x}10^{-3}\left(T^{(0)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(0)}\right)}\\<br />
\Delta H_{r}^{0}\left(T^{(0)}\right) &= - 24097 - 0.26\left(380\right) + 1.69\text{x}10^{-3}\left(380\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(380\right)}\\<br />
\Delta H_{r}^{0}\left(T^{(0)}\right) &= -23557.02715<br />
\end{array}<br />
<br />
*ITERATION 1*<br />
<br />
.. math::<br />
<br />
\begin{array}{rl}<br />
T^{(1)} &= T^{(0)} - \frac{- 24097 - 0.26T^{(0)} + 1.69\text{x}10^{-3}(T^{(0)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(0)}} - (-23505)}{- 0.26 + 3.38\text{x}10^{-3}T^{(0)} - \frac{1.5\text{x}10^{5}}{(T^{(0)})^{2}}}\\<br />
T^{(1)} &= (380) - \frac{- 24097 - 0.26(380) + 1.69\text{x}10^{-3}(380)^{2} + \frac{1.5\text{x}10^{5}}{(380)} - (-23505)}{- 0.26 + 3.38\text{x}10^{-3}(380) - \frac{1.5\text{x}10^{5}}{(380)^{2}}}\\<br />
T^{(1)} &= -3237.72940\\<br />
& \\<br />
\Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(T^{(1)}\right) + 1.69\text{x}10^{-3}\left(T^{(1)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(1)}\right)}\\<br />
\Delta H_{r}^{0}\left(T^{(1)}\right) &= - 24097 - 0.26\left(-3237.72940\right) + 1.69\text{x}10^{-3}\left(-3237.72940\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(-3237.72940\right)}\\<br />
\Delta H_{r}^{0}\left(T^{(1)}\right) &= -5585.4322<br />
\end{array}<br />
<br />
.. math::<br />
<br />
\begin{array}{rl}<br />
\epsilon_{tol,x}^{(1)} &= \left| \frac{T^{(1)}-T^{(0)}}{T^{(0)}} \right| < \epsilon_{tol} ? \\<br />
\epsilon_{tol,x}^{(1)} &= \left| \frac{\left(-3237.72940\right)-\left(380\right)}{\left(380\right)} \right| < \left(10^{-6}\right) ? \\<br />
\epsilon_{tol,x}^{(1)} &= 9.52 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\<br />
& \\<br />
\epsilon_{tol,f}^{(1)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(1)}\right)-\Delta H_{r}^{0}\left(T^{(0)}\right)}{\Delta H_{r}^{0}\left(T^{(0)}\right)} \right| < \epsilon_{tol} ? \\<br />
\epsilon_{tol,f}^{(1)} &= \left| \frac{\left(-5585.4322\right)-\left(-23557.02715\right)}{\left(-23557.02715\right)} \right| < \left(10^{-6}\right) ? \\<br />
\epsilon_{tol,f}^{(1)} &= 0.763 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.<br />
\end{array}<br />
<br />
*ITERATION 2*<br />
<br />
.. math::<br />
<br />
\begin{array}{rl}<br />
T^{(2)} &= T^{(1)} - \frac{- 24097 - 0.26T^{(1)} + 1.69\text{x}10^{-3}(T^{(1)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(1)}} - (-23505)}{- 0.26 + 3.38\text{x}10^{-3}T^{(1)} - \frac{1.5\text{x}10^{5}}{(T^{(1)})^{2}}}\\<br />
T^{(2)} &= (-3237.72940) - \frac{- 24097 - 0.26(-3237.72940) + 1.69\text{x}10^{-3}(-3237.72940)^{2} + \frac{1.5\text{x}10^{5}}{(-3237.72940)} - (-23505)}{- 0.26 + 3.38\text{x}10^{-3}(-3237.72940) - \frac{1.5\text{x}10^{5}}{(-3237.72940)^{2}}}\\<br />
T^{(2)} &= -1640.311\\<br />
& \\<br />
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(T^{(2)}\right) + 1.69\text{x}10^{-3}\left(T^{(2)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(2)}\right)}\\<br />
\Delta H_{r}^{0}\left(T^{(2)}\right) &= - 24097 - 0.26\left(-1640.311\right) + 1.69\text{x}10^{-3}\left(-1640.311\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(-1640.311\right)}\\<br />
\Delta H_{r}^{0}\left(T^{(2)}\right) &= -19214.81711<br />
\end{array}<br />
<br />
.. math::<br />
<br />
\begin{array}{rl}<br />
\epsilon_{tol,x}^{(2)} &= \left| \frac{T^{(2)}-T^{(1)}}{T^{(1)}} \right| < \epsilon_{tol} ? \\<br />
\epsilon_{tol,x}^{(2)} &= \left| \frac{\left(-1640.311\right)-\left(-3237.72940\right)}{\left(-3237.72940\right)} \right| < \left(10^{-6}\right) ? \\<br />
\epsilon_{tol,x}^{(2)} &= 0.493 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\<br />
& \\<br />
\epsilon_{tol,f}^{(2)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(2)}\right)-\Delta H_{r}^{0}\left(T^{(1)}\right)}{\Delta H_{r}^{0}\left(T^{(1)}\right)} \right| < \epsilon_{tol} ? \\<br />
\epsilon_{tol,f}^{(2)} &= \left| \frac{\left(-19214.81711\right)-\left(-5585.4322\right)}{\left(-5585.4322\right)} \right| < \left(10^{-6}\right) ? \\<br />
\epsilon_{tol,f}^{(1)} &= 2.44 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.<br />
\end{array}<br />
<br />
*ITERATION 3*<br />
<br />
.. math::<br />
<br />
\begin{array}{rl}<br />
T^{(3)} &= T^{(2)} - \frac{- 24097 - 0.26T^{(2)} + 1.69\text{x}10^{-3}(T^{(2)})^{2} + \frac{1.5\text{x}10^{5}}{T^{(2)}} - (-23505)}{- 0.26 + 3.38\text{x}10^{-3}T^{(2)} - \frac{1.5\text{x}10^{5}}{(T^{(2)})^{2}}}\\<br />
T^{(3)} &= (-1640.311) - \frac{- 24097 - 0.26(-1640.311) + 1.69\text{x}10^{-3}(-1640.311)^{2} + \frac{1.5\text{x}10^{5}}{(-1640.311)} - (-23505)}{- 0.26 + 3.38\text{x}10^{-3}(-1640.311) - \frac{1.5\text{x}10^{5}}{(-1640.311)^{2}}}\\<br />
T^{(3)} &= -908.1982\\<br />
& \\<br />
\Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(T^{(3)}\right) + 1.69\text{x}10^{-3}\left(T^{(3)}\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(T^{(3)}\right)}\\<br />
\Delta H_{r}^{0}\left(T^{(3)}\right) &= - 24097 - 0.26\left(-908.1982\right) + 1.69\text{x}10^{-3}\left(-908.1982\right)^{2} + \frac{1.5\text{x}10^{5}}{\left(-908.1982\right)}\\<br />
\Delta H_{r}^{0}\left(T^{(3)}\right) &= -22632.0781<br />
\end{array}<br />
<br />
.. math::<br />
<br />
\begin{array}{rl}<br />
\epsilon_{tol,x}^{(3)} &= \left| \frac{T^{(3)}-T^{(2)}}{T^{(2)}} \right| < \epsilon_{tol} ? \\<br />
\epsilon_{tol,x}^{(3)} &= \left| \frac{\left(-908.1982\right)-\left(-1640.311\right)}{\left(-1640.311\right)} \right| < \left(10^{-6}\right) ? \\<br />
\epsilon_{tol,x}^{(3)} &= 0.45 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}. \\<br />
& \\<br />
\epsilon_{tol,f}^{(3)} &= \left| \frac{\Delta H_{r}^{0}\left(T^{(3)}\right)-\Delta H_{r}^{0}\left(T^{(2)}\right)}{\Delta H_{r}^{0}\left(T^{(2)}\right)} \right| < \epsilon_{tol} ? \\<br />
\epsilon_{tol,f}^{(3)} &= \left| \frac{\left(-22632.0781\right)-\left(-19214.81711\right)}{\left(-19214.81711\right)} \right| < \left(10^{-6}\right) ? \\<br />
\epsilon_{tol,f}^{(3)} &= 0.178 < \left(10^{-6}\right) ? \;\; \rightarrow \;\; \text{No, therefore keep going}.<br />
\end{array}<br />
<br />
</rst></div>
Kevindunn