orch:Solvers - Revision history

Jaouen: /* Chemical kinetics */

2016-03-07T18:33:33Z

‎Chemical kinetics

Jaouen: /* Lindemann */

2016-03-07T15:59:54Z

‎Lindemann

Jaouen: /* Three-body reactions */

2016-03-07T15:59:00Z

‎Three-body reactions

Jaouen: /* Lindemann */

2016-03-07T15:58:22Z

‎Lindemann

Jaouen: /* Lindemann */

2016-03-07T15:52:43Z

‎Lindemann

Jaouen: /* Lindemann */

2016-03-07T15:52:26Z

‎Lindemann

Jaouen: /* Lindemann */

2016-03-07T15:51:38Z

‎Lindemann

Jaouen: /* Lindemann */

2016-03-07T15:50:39Z

‎Lindemann

Jaouen: /* Lindemann */

2016-03-07T15:49:27Z

‎Lindemann

Jaouen: /* Lindemann */

2016-03-07T15:47:31Z

‎Lindemann

← Older revision		Revision as of 18:33, 7 March 2016
Line 1:		Line 1:
−	~~== Chemical kinetics ==~~
−
−	~~This chapter reports the principles that drive the computation of combustion chemistry in most CFD softwares.~~
−
−	~~=== Chemkin (.scheme .therm .trans), Cantera (xml) ===~~
−
−	~~...~~
−
−
−	~~=== Arrhenius law ===~~
−
−	~~<math>\mathcal{A}_j</math> is the pre-exponential factor, <math>\mathcal{\beta}_j</math> is the temperature exponent and <math>E_{a_j}</math> the activation energy~~
−
−	~~<math>~~
−	~~k_j = \mathcal{A}_j T^{\mathcal{\beta}_j} \exp \left(-\frac{E_{a_j}}{R T}\right)~~
−	~~</math>~~
−
−	~~=== Three-body reactions ===~~
−
−	In the forward direction, three-body reactions involve two species A and B as reactants and yield a single product AB. In that case, the third body <math>\text{M}</math> is used to stabilize the excited product AB*. On the contrary, in the reverse direction, heat provides the energy necessary to break the link between A and B.
−
−	~~<math>......</math>~~
−
−	~~The third body M can be any inert molecule.~~
−
−	~~=== Falloff reactions ===~~
−
−	Under specific conditions, some reaction rate expressions are dependent on pressure and temperature. This is especially true for the rate associated to unimolecular/recombination fall-off reactions which increases with pressure. In such cases, if the chemical process takes place in a high or low pressure limit typical Arrhenius laws are applicable to the reactions that are described. However, if the pressure is in between, an accurate description of the phenomenon requires a more complicated rate expression. In such a case, the reaction is said to be in the ”fall-off” region.
−
−	~~==== Lindemann ====~~
−
−	Several formulas (derived from the Lindemann description) are available to smoothly relate the limiting low and high-pressure rate expressions. With the Lindemann approach, Arrhenius parameters need to be given for both
−
−	* the low pressure limit
−	~~<math>~~
−	~~k_0 = \mathcal{A}_0 T^{\mathcal{\beta}_0} \exp\left(-\frac{E_{a_0}}{R T}\right)~~
−	~~</math>~~
−	* and the high pressure limit
−	~~<math>~~
−	~~k_\infty = \mathcal{A}_\infty T^{\mathcal{\beta}_\infty} \exp\left(-\frac{E_{a_\infty}}{R T}\right)~~
−	~~</math>.~~
−
−	The expression taken at any pressure is based on a combination of both low and high-pressure Arrhenius expressions. The term <math>P_r</math> is here equivalent to a pressure and <math>\text{M}</math> represents the concentration of the mixture, possibly estimated from third-body efficiencies.
−
−	~~==== Troe ====~~
−
−
−
−
−
−	~~<code>~~
−	~~<!-- reaction 0012 -->~~
−	~~<reaction reversible="yes" type="falloff" id="0012">~~
−	~~<equation>O + CO (+ M) [=] CO2 (+ M)</equation>~~
−	~~<rateCoeff>~~
−	~~<Arrhenius>~~
−	~~<A>1.800000E+07</A>~~
−	~~<b>0</b>~~
−	~~<E units="cal/mol">2385.000000</E>~~
−	~~</Arrhenius>~~
−	~~<Arrhenius name="k0">~~
−	~~<A>6.020000E+08</A>~~
−	~~<b>0</b>~~
−	~~<E units="cal/mol">3000.000000</E>~~
−	~~</Arrhenius>~~
−	~~<efficiencies default="1.0">AR:0.5 C2H6:3 CH4:2 CO:1.5 CO2:3.5 H2:2 H2O:6 O2:6 </efficiencies>~~
−	~~<falloff type="Lindemann"/>~~
−	~~</rateCoeff>~~
−	~~<reactants>CO:1 O:1.0</reactants>~~
−	~~<products>CO2:1.0</products>~~
−	~~</reaction>~~
−	~~</code>~~
−
−	~~=== Reaction rates ===~~
−
−	~~The global rate of a reaction j (evolution in concentration per unit of time) varies depending on the proportion of the rates associated to the forward and backward directions.~~
−
−	~~<math>~~
−	~~\mathcal{Q}_j = \mathcal{Q}_{f,j} - \mathcal{Q}_{r,j}~~
−	~~</math>~~
−
−
−	~~=== Species production/consumption source terms ===~~
−
−	~~Species <math>Y_k</math> source terms are deduced from~~
−
−	~~<math>~~
−	~~\dot{\omega}_k = W_k \sum_{j=1}^{N_R} \nu_{k,j} \mathcal{Q}_j~~
−	~~</math>~~
−
	== Solver to build reference trajectories ==		== Solver to build reference trajectories ==

@@ Line 34: / Line 34: @@
 * the low pressure limit
 <math>
-    k_0 = \mathcal{A}_0 T^{\mathcal{\beta}_0} \exp\left(-\frac{E_0}{R T}\right)
+    k_0 = \mathcal{A}_0 T^{\mathcal{\beta}_0} \exp\left(-\frac{E_{a_0}}{R T}\right)
 </math>
 * and the high pressure limit
 <math>
-    k_\infty = \mathcal{A}_\infty T^{\mathcal{\beta}_\infty} \exp\left(-\frac{E_\infty}{R T}\right)
+    k_\infty = \mathcal{A}_\infty T^{\mathcal{\beta}_\infty} \exp\left(-\frac{E_{a_\infty}}{R T}\right)
 </math>.

@@ Line 18: / Line 18: @@
 === Three-body reactions ===
-In the forward direction, three-body reactions involve two species A and B as reactants and yield a single product AB. In that case, the third body M is used to stabilize the excited product AB*. On the contrary, in the reverse direction, heat provides the energy necessary to break the link between A and B.
+In the forward direction, three-body reactions involve two species A and B as reactants and yield a single product AB. In that case, the third body <math>\text{M}</math> is used to stabilize the excited product AB*. On the contrary, in the reverse direction, heat provides the energy necessary to break the link between A and B.
 <math>......</math>
 The third body M can be any inert molecule.
 === Falloff reactions ===

@@ Line 41: / Line 41: @@
     k_\infty = \mathcal{A}_\infty T^{\mathcal{\beta}_\infty} \exp\left(-\frac{E_\infty}{R T}\right)
 </math>.
 ==== Troe ====

@@ Line 37: / Line 37: @@
     k_0 = \mathcal{A}_0 T^{\mathcal{\beta}_0} \exp\left(-\frac{E_0}{R T}\right)
 </math>
-* and the high pressure limit `
+* and the high pressure limit
 <math>
     k_\infty = \mathcal{A}_\infty T^{\mathcal{\beta}_\infty} \exp\left(-\frac{E_\infty}{R T}\right)

@@ Line 33: / Line 33: @@
 Several formulas (derived from the Lindemann description) are available to smoothly relate the limiting low and high-pressure rate expressions. With the Lindemann approach, Arrhenius parameters need to be given for both
-* the low pressure limit <math>k_0 = \mathcal{A}_0 T^{\mathcal{\beta}_0} \exp\left(-\frac{E_0}{R T}\right)</math>
+* the low pressure limit
-* and the high pressure limit <math>k_\infty = \mathcal{A}_\infty T^{\mathcal{\beta}_\infty} \exp\left(-\frac{E_\infty}{R T}\right)</math>.
+<math>
 ==== Troe ====

@@ Line 31: / Line 31: @@
 ==== Lindemann ====
-Several formulas (derived from the Lindemann description) are available to smoothly relate the limiting low and high-pressure rate expressions. With the Lindemann approach, Arrhenius parameters need to be given for both the low pressure limit $k_0$ and the high pressure limit <math>k_\infty</math>.
+Several formulas (derived from the Lindemann description) are available to smoothly relate the limiting low and high-pressure rate expressions. With the Lindemann approach, Arrhenius parameters need to be given for both
 <math>