Difference between revisions of "Oil Shale Pyrolysis"

Oil Shale Pyrolysis
State dimension:	1
Differential states:	2
Continuous control functions:	1
Discrete control functions:	0
Path constraints:	4
Interior point equalities:	2

Revision as of 02:52, 15 March 2019

The following problem is an example from the global optimal control literature and was introduced in [Wen1977]The entry doesn't exist yet.. The process starts with kerogen and is decomposed into pyrolytic bitumen, oil and gas, and residual carbon. The objective is to maximize the fraction of pyrolytic bitumen. There are 5 reactions including:

$A_1 \xrightarrow{k_1} A_2$

$A_2 \xrightarrow{k_2} A_3$

$A_1 + A_2 \xrightarrow{k_3} A_2 + A_2$

$A_1 + A_2 \xrightarrow{k_4} A_3 + A_2$

$A_1 + A_2 \xrightarrow{k_5} A_4 + A_2$

Each reaction is governed by a rate described by:

$k_i = k_{i0} \exp{\left(-E_i/RT\right)}, (i=1,2,3,4,5)$

Mathematical formulation

$\begin{array}{lll} \displaystyle \max_{T} & \displaystyle &x_2(t_N) \\[1.5ex] \mbox{s.t.} & \displaystyle \dot{x}_1 &= -k_1x_1-(k_3+k_4+k_5)x_1x_2\\ & \displaystyle \dot{x}_2 &= k_1x_1-k_2x_2 + k_3x_1x_2\\ & \displaystyle \dot{x}_3 &= k_2x_2 + k_4x_1x_2\\ & \displaystyle \dot{x}_4 &= k_5x_1x_2\\ & \displaystyle k_i &= a_i e^{-\frac{b_i}{RT}},\quad \forall i\in \{1,\dots,5\} \\ [1.5ex] & \displaystyle t &\in \left[t_0,t_N\right] \\ & \displaystyle T(t) &\in \left[698.15K,748.15K\right]\\ & \displaystyle x(t_0) &= (1,0,0,0)^T\\ \end{array}$

Parameters

State variables
Symbol	Initial value ( $t_0$ )
$x_1(t_0)$	$1$
$x_2(t_0)$	$0$
$x_3(t_0)$	$0$
$x_4(t_0)$	$0$

Parameters
Symbol	Value
$a_1$	$20.3$
$a_2$	$37.4$
$a_3$	$33.8$
$a_4$	$28.2$
$a_5$	$31.0$
$b_1$	$\exp{(8.86)}$
$b_2$	$\exp{(24.25)}$
$b_3$	$\exp{(23.67)}$
$b_4$	$\exp{(18.75)}$
$b_5$	$\exp{(20.7)}$

Control variable
Symbol	Interval
$T(t)$	[698.15,748.15]

Reference solution

Coming soon.

Source Code

Model descriptions are not yet available.

References

[Wen1977]

The entry doesn't exist yet.

Difference between revisions of "Oil Shale Pyrolysis"

Revision as of 02:52, 15 March 2019

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Mathematical formulation

Parameters

Reference solution

Source Code

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@@ Line 28: / Line 28: @@
 <math>
 \begin{array}{lll}
-  \displaystyle \min_{T} &  \displaystyle &-x_2(t_N)  \\[1.5ex]
+  \displaystyle \max_{T} &  \displaystyle &x_2(t_N)  \\[1.5ex]
   \mbox{s.t.} &  \displaystyle \dot{x}_1 &= -k_1x_1-(k_3+k_4+k_5)x_1x_2\\
   &  \displaystyle \dot{x}_2 &= k_1x_1-k_2x_2 + k_3x_1x_2\\