The system under study consists of two sets of single shell heat exchangers filled with water, placed in
parallel and cooled by a liquid saturated refrigerant flowing through a coil system, as it is illustrated in the attached image. The
saturated vapour generated in the coil system is separated from the liquid phase in the stages $S1$ and $S2$, both of
neglected volumes. This vapour, withdrawn in $S1$ and $S2$, reduces the refrigerant mass flow rate along the cooling
system, and only the saturated liquid portion is used for cooling purposes. Table below provides the fluid properties and
equipment dimensions. The temperature of the refrigerant remains constant at $T_C$ as the liquid is saturated, and the
energy exchanged with water is used to vapourise a small portion of the refrigerant fluid. The idividual heat exchanger energy balances can be expressed in terms of deviation variables to define the following LTI system:

$$\begin{align*}
A &= \begin{bmatrix}
-\dfrac{1+\nu_A}{\tau_1} & \dfrac{1}{\tau_1} & 0 & 0\\
0 & -\dfrac{1+\nu_A}{\tau_2} & 0 & 0 \\
0 & 0 & -\dfrac{1+\nu_B}{\tau_3} & \dfrac{1}{\tau_3} \\
0 & 0 & 0 & -\dfrac{1+\nu_B}{\tau_4} \\
\end{bmatrix} \\
B &= \begin{bmatrix}
0 & 0 & 0 & 0\\
\dfrac{k_1}{\tau_1} & 0 & \dfrac{k_3}{\tau_2} & 0\\
0 & 0 & 0 & 0\\
0 & \dfrac{k_2}{\tau_t} & 0 & \dfrac{k_4}{\tau_4}\\
\end{bmatrix},
C = \begin{bmatrix}
\mu & 0 & \mu & 0\\
0 & \mu & 0 & \mu\\
\mu & 0 & \mu & 0\\
0 & \mu & \mu & 0\\
\end{bmatrix},
D = \begin{bmatrix}
0
\end{bmatrix},
\end{align*}$$

where:
$\nu_A \triangleq (hA/C_p(\dot{m}_1 + \dot{m}_3)), \\
\nu_B \triangleq (hA/C_p(\dot{m}_2 + \dot{m}_4)),\\
\tau_1 \triangleq (M_1/(\dot{m}_1 + \dot{m}_3)),\\
\tau_2 \triangleq (M_2/(\dot{m}_1 + \dot{m}_3)),\\
\tau_3 \triangleq (M_3/(\dot{m}_2 + \dot{m}_4)),\\
\tau_4 \triangleq (M_4/(\dot{m}_4 + \dot{m}_3)),\\
k_1 \triangleq(\dot{m}_1/(\dot{m}_1 + \dot{m}_3)),\\
k_2 \triangleq(\dot{m}_2/(\dot{m}_2 + \dot{m}_4)),\\
k_3 \triangleq(\dot{m}_3/(\dot{m}_1 + \dot{m}_3)),\\
k_4 \triangleq(\dot{m}_4/(\dot{m}_1 + \dot{m}_3)),\\
\mu \triangleq(hA/h_{lv}).\\$

This system was discretised with a sampling time of $T = 1 s$ and was discretised while rounding the input and output delays to the closest integer-multiples of $T$.

$C_p$ |
4.217 kJ/kg K |
water specific heat |

$h_{lv}$ |
850 kJ/kg |
refrigerator heat vapourisation |

$T_S(0)$ |
40$^{\circ}$C |
initial temperature in $E_s$ |

$T_{jin}(0)$ |
40$^{\circ}$C |
initial water inlet temperature $j$ |

$T_c$ |
40$^{\circ}$C |
refrigerant temperature |

$\dot{m}_j$ |
1 kg/s |
water mass flow $j$ |

$M_s$ |
50 kg |
mass of water in $E_s$ |

$hA$ |
8 kJ/kg |
overall surface heat transfer |

$V_1$ |
0.5 $\times$ 10$^{-3}$m$^4$ |
inlet water pipe volume 1 |

$V_2$ |
2 $\times$ 10$^{-3}$m$^4$ |
inlet water pipe volume 2 |

$V_4$ |
1.5 $\times$ 10$^{-3}$m$^4$ |
inlet water pipe volume 4 |

$\rho$ |
1000 kg/m$^3$ |
water density |