## Model description:

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 This vapour, withdrawn in $S1$ and $S2$, reduces the refrigerant mass flow rate along the cooling system, and only the saturated liquid portion is used for cooling purposes.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

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## Publication details:

Title | Discretisation of continuous-time dynamic multi-input multi-output systems with non-uniform delays |

Publication Type | Journal Article |

Year of Publication | 2011 |

Authors | Kassas, Z.M. |

Journal | Control Theory & Applications, IET |

Volume | 5 |

Issue | 14 |

Start Page | 1637 |

Pagination | 1637-1647 |

Date Published | 09/2011 |

ISSN | 1751-8644 |

Accession Number | 12228423 |

Keywords | continuous time systems, delays, discrete systems, MIMO systems |

Abstract | Input and output time delays in continuous-time (CT) dynamic systems impact such systems differently as their effects are encountered before and after the state dynamics. Given a fixed sampling time, input and output signals in multiple-input multiple-output (MIMO) systems may exhibit any combination of the following four cases: no delays, integer-multiple delays, fractional delays and integer-multiple plus fractional delays. A common pitfall in the digital control of delayed systems literature is to only consider the system timing diagram to derive the discrete-time (DT) equivalent model; hence, effectively `lump` the delays across the system as one total delay. DT equivalent models for systems with input delays are radically different than those with output delays. Existing discretisation techniques for delayed systems usually consider the delays to be integer-multiples of the sampling time. This study is intended to serve as a reference for systematically deriving DT equivalent models of MIMO systems exhibiting any combination of the four delay cases. This algorithm is applied towards discretising an MIMO heat exchanger process with non-uniform input and output delays. A significant improvement towards the CT response was noted when applying this algorithm as opposed to rounding the delays to the closest integer-multiple of the sampling time. |

DOI | 10.1049/iet-cta.2010.0467 |