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Expand file tree Collapse file tree Original file line number Diff line number Diff line change @@ -559,10 +559,8 @@ X_a / (A_a m_u)`, our rate equation is
559559
560560.. math ::
561561
562- \begin {align}
563562 \frac {dX_f}{dt} &= - \frac {r_0 }{m_u} \rho X_f^2 \frac {1 }{A_f} \left (\frac {T}{T_0 }\right )^\nu \equiv \omegadot _f \\
564563 \frac {dX_a}{dt} &= \frac {1 }{2 }\frac {r_0 }{m_u} \rho X_f^2 \frac {A_a}{A_f^2 } \left (\frac {T}{T_0 }\right )^\nu = \frac {r_0 }{m_u} \rho X_f^2 \frac {1 }{A_f} \left (\frac {T}{T_0 }\right )^\nu
565- \end {align}
566564
567565 We define a new rate constant, :math: `\rt ` with units of :math: `[\mathrm {s^{-1 }}]` as
568566
Original file line number Diff line number Diff line change @@ -231,7 +231,7 @@ Some suggestions when setting tolerances:
231231* The VODE integrator has additional logic meant to ensure that
232232 species don't change too much per timestep. This is controlled by
233233 ``integrator.X_reject_buffer ``. If a species $k$, has a mass
234- fraction $X_k > \m box {X_reject_buffer} \c dot \m box {atol_spec}$ then
234+ fraction $X_k > \m athrm {X_reject_buffer} \c dot \m athrm {atol_spec}$ then
235235 we reject a VODE timestep if the mass fraction changes by more than
236236 a factor of 4 in a single VODE timestep and we try again. This is
237237 all done internally to VODE. Making ``X_reject_buffer `` larger will
Original file line number Diff line number Diff line change @@ -58,12 +58,10 @@ $k+1$ appears as:
5858
5959.. math ::
6060
61- \begin {align*}
6261 \Uc ^{m+1 ,(k+1 )} = \Uc ^{m,(k+1 )}
6362 &+ \delta t_m\left [\Advs {\Uc ^{m,(k+1 )}} - \Advs {\Uc ^{m,(k)}}\right ] \\
6463 &+ \delta t_m\left [\Rbs {\Uc ^{m+1 ,(k+1 )}} - \Rbs {\Uc ^{m+1 ,(k)}}\right ]\\
6564 &+ \int _{t^m}^{t^{m+1 }}\Advs {\Uc ^{(k)}} + \Rbs {\Uc ^{(k)}}d\tau .
66- \end {align*}
6765
6866 Solving this requires a nonlinear solve of:
6967
@@ -76,11 +74,9 @@ define ${\bf C}$ for convenience as:
7674
7775.. math ::
7876
79- \begin {align}
8077 {\bf C} &= \left [ {\Advs {\Uc }}^{m,(k+1 )} - {\Advs {\Uc }}^{m,(k)} \right ]
8178 - {\Rbs {\Uc }}^{{m+1 },(k)} \nonumber \\
82- &+ \frac {1 }{\delta t_m} \int _{t^m}^{t^{m+1 }} \left ( {\Advs {\Uc }}^{(k)} + {\Rbs {\Uc }}^{(k)}\right ) d\tau
83- \end {align}
79+ &+ \frac {1 }{\delta t_m} \int _{t^m}^{t^{m+1 }} \left ( {\Advs {\Uc }}^{(k)} + {\Rbs {\Uc }}^{(k)}\right ) d\tau
8480
8581 This can be cast as an ODE system as:
8682
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