A fourth-order heat exchanger process

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 This vapour, withdrawn in $S1$ and $S2$, reduces the refrigerant mass flow rate along the cooling system, and only the saturated liquid portion is used for cooling purposes.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 $T$.