Two Continuously Stirred-Tank Reactor Process

The process dynamic model consists of six nonlinear ordinary differential equations:

$$\begin{align*} \dot x_{11} &= b_{11} x_{12} \\ \dot x_{12} &= b_{12} u_1 \\ \dot x_{21} &= b_{21} x_{22} + \phi _{21} \left({x_{11},x_{21} } \right) + \Phi x_{31} \\ \dot x_{22} &= b_{22} u_2 + \phi _{22} \left({x_{21},x_{22} } \right) \\ \dot x_{31} &= b_{31} x_{32} + \phi _{31} \left({x_{11},x_{12},x_{21},x_{31} } \right) + \Psi w \\ \dot x_{32} &= b_{32} u_3 + \phi _{32} \left({x_{31},x_{32} } \right) \\ y &= \left[{y_1,y_2,y_3 } \right] = \left[{x_{11},x_{21},x_{31} } \right], \end{align*}$$

where

$\eqalignno{b_{11} &= 1,b_{12} = 1,b_{21} = {{UA} \over {\rho c_p V}},b_{22} = {{F_{j2} } \over {V_j}},b_{31} = {{UA} \over {\rho c_p V}}\cr b_{32} &= {{F_{j1} } \over {V_j}},\Psi = {{F_0 } \over V},\Phi = {{F + F_R } \over V} \cr \phi _{21} &= {{F + F_R } \over V}T_1^d - {{F + F_R } \over V}\left({x_{21} + T_2^d } \right)\cr &\quad - {{\alpha \lambda } \over {\rho c_p}}\left({x_{11} + C_{A2}^d } \right)e^{- \left({{E \over {R\left({x_{21} + T_2^d } \right)}}} \right)}\cr &\quad - {{UA} \over {\rho c_p V}}\left({x_{21} + T_2^d - T_{j2}^d } \right) \cr \phi _{22} &= {{F_{j2} } \over {V_j}}\left({T_{j20}^d - x_{22} - T_{j2}^d } \right)\cr &\quad + {{UA} \over {\rho _j c_j V_j}}\left({x_{21} + T_2^d - x_{22} - T_{j2}^d } \right) \cr \phi _{31} &= {{F_0 } \over V}T_0^d - {{F + F_R } \over V}\left({x_{31} + T_1^d } \right) + {{F_R } \over V}\left({x_{21} + T_2^d } \right)\cr &\quad - {{\alpha \lambda } \over {\rho c_p}}C_A e^{- \left({{E \over {R\left({x_{31} + T_1^d } \right)}}} \right)} - {{UA} \over {\rho c_p V}}\left({x_{31} + T_1^d - T_{j1}^d } \right) \cr \phi _{32} &= {{F_{j1} } \over {V_j}}\left({T_{j10}^d - x_{32} - T_{j1}^d } \right)\cr &\quad + {{UA} \over {\rho _j c_j V_j}}\left({x_{31} + T_1^d - x_{32} - T_{j1}^d } \right) \cr C_A &= {V \over {F + F_R}}\Bigg(x_{12} + {{F + F_R } \over V}({x_{11} + C_{A2}^d })\cr &\quad + \alpha ({x_{11} + C_{A2}^d })e^{- \Big({{E \over {R({x_{21} + T_2^d })}}} \Big)} \Bigg). }$

The values of the process parameters are

$\eqalignno{& \alpha = {\rm 7}{\rm .08} \times {\rm 10}^{{\rm 10}} {\rm h}^{- 1},\quad \rho = 800.9189\,{\rm kg/m}^{\rm 3}\cr & \rho _j = 997.9450\,{\rm kg/m}^3,\quad \lambda = - 3.1644 \times {\rm 10}^{\rm 7} {\rm J/mol}\cr & R = 1679.2\,{\rm J/(mol} {\cdot} {}^{\circ} {\rm C)},\quad E = 3.1644 \times 10^7 {\rm J/mol}\cr & c_\rho = 1395.3\,{\rm J/(kg} {\cdot} {}^{\circ} {\rm C)},\quad c_j = 1860.3\,{\rm J/(kg} {\cdot} {}^{\circ} {\rm C)}\cr & U = 1.3625 \times 10^6{\kern1pt} {\rm J/(h} {\cdot} {\rm m}^{\rm 2} {\cdot} {}^{\circ} {\rm C)},\quad F_0 = F_2 = F = 2.8317\,{\rm m}^{\rm 3}\!{\rm /h}\cr & F_R = 1.4158\,{\rm m}^{\rm 3}\!{\rm /h},\quad F_{j1} = 1.4130\,{\rm m}^{\rm 3}\!{\rm /h}\cr & F_{j2} = 1.4130\,{\rm m}^{\rm 3}\!{\rm /h},\quad T_0^d = 703.7\,{}^{\circ} {\rm C},\quad T_1^d = 750\,{}^{\circ} {\rm C}\cr & T_2^d = 737.5\,{}^{\circ} {\rm C},\quad T_{j1}^d = 740.8\,{}^{\circ} {\rm C},\quad T_{j2}^d = 727.6\,{}^{\circ} {\rm C}\cr & T_{j10}^d \! = \! 629.2\,{}^{\circ} {\rm C},\quad T_{j20}^d \!=\! 608.2\,{}^{\circ} {\rm C},\quad C_{A0}^d \!=\! 18.3728\,{\rm mol/m}^{\rm 3}\cr & C_{A1}^d = 12.3061\,{\rm mol/m}^{\rm 3},\quad C_{A2}^d = 10.4178\,{\rm mol/m}^{\rm 3}\cr & V_1 = V_2 = V = 1.3592\,{\rm m}^{\rm 3},\quad V_{j1} = V_{j2} = V_j = 0.1090\,{\rm m}^{\rm 3}\cr & A = 23.2\,{\rm m}^{\rm 3} . }$