To build an analytical model for a refrigeration system, a deep understanding of the associated physical principles that govern the system behaviour and operational characteristics is needed. The models are usually built from component level before system integration (Koury et al. 2001). For transient analytical models, the key to describe the dynamic behaviour of chillers is to capture the transient characteristics of the heat exchangers (evaporator and condenser), because they hold most of the refrigerant charges of the system. The expansion valve and the compressor can be treated as steady state all the time as their condition changes are much faster compared with the heat exchangers.
For instance, sub-models of evaporator and condenser usually consist of a set of time-space partial differential equations representing heat, mass and momentum balances within the heat exchangers, which can be solved to determine local heat transfer, temperature and pressure at any specific time (Katipamula and Brambley 2005). For compressors and expansion devices, as their thermal inertia are quite small (Bendapudi et al. 2002b), quasi-steady state assumptions are often applied. The component models are then coupled together with the thermodynamic states of refrigerant and mass continuity. The enthalpies, mass flowrates and properties of refrigerant at the outlet of one component become the inlet to the next coupled component.
Detailed mathematical models are difficult to construct and require relatively huge computational time; to reduce the complexity of the modelling and simulation, assumptions are often made. Common assumptions from literatures include using the idealised refrigeration cycle, constant refrigerant charge level (McIntosh et al. 2000), 1-D refrigerant flow in heat exchangers (Nyers and Stoyan 1994; He et al. 1997), etc. Assumptions vary with modelling approaches and they may affect the accuracy of the model and hence the reliability of the FDDs. Many of the assumptions may also be valid for the binary ice system involved in this study. By analysing the application and viability of the common assumptions for various RAC systems, it may help identifying the appropriate ones for the current study.
Many analytical models have been built but they are mainly developed for system or component designs, not many of them have been used for FDDs. Some
38 examples of mathematical models for general vapour compression system simulation are given below. Attention will be paid to the type of system, the selection of modelling parameters and the model’s potential of being employed for FDD purpose.
Browne and Bansal (2002) built a detailed dynamic model for a packaged liquid chiller. For individual system components, correlations of heat and mass transfers were developed. System geometrical parameters, such as size of the heat exchangers, total mass of chilled and cooling water, total amount of refrigerant and the assumed/estimated distribution within individual system components were needed as modelling parameters. The required input variables included chilled water and cooling water temperatures and mass flowrates, wall temperatures of condenser and evaporator, ambient temperature, as well as the estimated building load and the set point temperatures of the evaporator water outlet and condenser water inlet. Empirical regression had been used for the compressor sub-model, to improve the overall model accuracy. The simulation could be applied to calculate the cooling capacity, compressor input work and refrigerant temperatures of condenser and evaporator. The model provided good results for the transient period during system start-up. When the system was operating under steady state, the modelling accuracy was 90%; the errors were due to the omission of the control system in the model, though physically it was incorporated in the system. The model may not be suitable for general FDD applications as the distribution of the refrigerant cannot be easily measured in practical situations, as well as the wall surface temperatures of the heat exchangers.
He et al. (1997) developed a model for describing the dynamics of vapour compression cycles. In particular, the dynamics associated with the evaporator and the condensers were modelled based on a moving-interface approach in which the position of the two-phase/single-phase interface inside a one-dimensional heat exchanger could be properly predicted. Two sets of lumped parameters were applied in this model for the two-phase and the single phase zones separately. However, it is believed the lumped parameter models would not be suitable for binary ice system in which the properties of the binary ice change significantly inside the heat exchanger even under a small temperature variation.
In modelling a variable speed chiller system, Koury et al. (2001) developed a transient distributed model for the condenser and the evaporator, in which the heat
39 exchanger was divided into small elements. Conservation equations and local heat transfer coefficients were applied for individual elements. A steady state assumption was employed for calculating the refrigerant mass flowrate through the compressor and the expansion valve. The model could predict system behaviour during start-up, compressor speed and TEV valve flow area variations. The use of finite element method helped increase the reliability of the model. Moreover, the ability of predicting system behaviours during dynamic change made it possible for dynamic fault detections.
In general, it is difficult to apply and adapt a particular model to other chillers as each has a unique set of heat transfer coefficients depending on the type of heat exchanger employed and flow conditions encountered. However, with the help of calibration variables and/or experimental data, it is possible to modify a detailed physical model, developed for a system, and adapt it for another similar system. McIntosh et al. (2000) modified a detailed model from Braun (1988), which was originally developed for a 5500-ton centrifugal chiller, to be applied to a laboratory 2000-ton centrifugal chiller. This calibration method could in fact be also used to improve modelling accuracies of a given system.
Bendapudi et al. (2002a) developed an analytical model of a centrifugal chiller which they claimed to be potentially suitable for FDD purpose. Unlike all the dynamic models they have reviewed and presented in an earlier report (Bendapudi et al. 2002b), which were not able to predict accurately the dynamic behaviours of centrifugal liquid chillers, this model considered refrigerant re-distribution between components as part of the dynamic features. The influences of the control feedback were also included in the modelling. The model was capable of predicting the compressor start-up and load changes.
Although some previous efforts had been made to applying analytical models to FDDs, full analytical models are still considered not common for FDD applications. The correlations, e.g. He et al. (1997), relating the relevant parameters are difficult to develop and computationally intensive to solve even after simplifying assumptions have been made. Browne and Bansal (2002) showed that the poor accuracy encountered in some of the pure analytical models could be improved by empirical calibrations. In addition, some common essential input parameters such as internal
40 heat exchanger wall temperature and the refrigerant distribution in individual components are difficult to determine in real applications.