State-space modelling for the ejector-based refrigeration system driven by low grade energy
Introduction
With the rapid development of the society, the shortage of energy and pollution of the environment become increasingly serious. Faced with this situation, the potential of exploiting waste heat and low grade energy from industrial process, automobiles, geothermal and solar energies can be a promising alternative in view of ever increasing energy demand and environmental burdens. For these reasons, recent years have witnessed an increasing interest in the study of low grade thermal energy for refrigeration and air conditioning systems.
Among these systems, ejector-based refrigeration systems [1], [2], [3], [4], [5] driven by low-grade waste heat have attracted great attentions from research community, since the ejector has some properties of no moving parts, no electricity power consumption and relatively low cost. However, compared with the conventional refrigeration system, the introduction of the ejector highly complicates the thermo-fluid dynamics of refrigerant in the ejector-based refrigeration system and leads to difficulty in regulating its operating conditions to meet the space cooling requirements. Therefore, it is of necessity to adequately understand the dynamic response by developing a global dynamic model of the system including both the ejector and conventional refrigeration cycle.
Currently, a great number of studies focused on the ejector performance prediction and ejector modelling [1], [2], [3], [4], [5]. Zhu et al. [1] developed a simplified hybrid model to determine the ejector performance, but they did not consider the pressure at outlet of the ejector and give the relationship between the outlet pressure and the primary/secondary flow pressure. He et al. [2] indicated that these models of the ejectors can be classified as thermodynamic model, dynamic model, empirical/semi-empirical model, each of which has respective characters. Lin et al. [3] used the Computational Fluid Dynamics (CFD) technique to develop the ejector model and investigate the influences of cooling load on the Pressure Recovery Ratio (PRR) performance. Different from the method [3], Chou et al. [4] developed empirical model for predicting the maximum flow ratio of the ejector, by using the multi-parameter equation. Compared with the above ejector models, Chen et al. [5] developed a new 1D model for performance prediction of the ejector at critical and subcritical operating condition, and analysed the pressure at the exit of the diffuser. This kind of the 1D model with cheap computation will be adopted in present study, because of easy integration into the model of the conventional refrigeration cycle.
In addition, the dynamic modelling of the conventional refrigeration cycle on the balance between complexity and model accuracy is also a hot research topic [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Two methods for modelling heat exchangers are adopted in system modelling, including finite-volume [6], [7] and moving-boundary lumped-parameter methods [8], [9], [10], [11], [12], [13], [14], [15]. Using finite-volume method, Jahangeer et al. [6] decomposed heat exchanger geometry into finite regions where spatial effects are captured, but complexity is introduced due to spatially varying flow and heat transfer. Bendapudi et al. [7] developed a validated system model for a centrifugal chiller to capture its transient performance using finite-volume method. In contrast to these finite-volume models, lumped-parameter models have many merits such as reducing complexity, less computation and good theoretic background by considering position of refrigerant phase change and considering lumped heat transfer parameters of each fluid phase region. Hence, many scholars employed this method to study simple models of heat exchangers for control system design. He et al. [9]. presented a novel lumped-parameter model to represent dynamics of the vapour compression cycle, based on a moving-interface approach. Following this work, Rasmussen [10] put forward a highly nonlinear state space model based on first principles and experimental results validated its effectiveness. Following the basic ideas of [7], Bendapudi et al. [11] further discussed development and analysis of two methods applied to a flooded evaporator and condenser. Furthermore, a comprehensive understand about the trade-off between finite-volume and moving-boundary methods were given. Li et al. [12] extended their works and introduced some switching schemes between different formulations and pseudo-state variables to derive the transitions of dynamical states from heat exchangers during stop-start transients. Based on this idea, Catano [13] developed a lumped-parameter modelling method for electronics cooling by imposing heat flux boundary condition to evaporator. Furthermore, this proposed model was compared with experimental results at multiple operating conditions. Recently, Yao et al. [14] transformed the ODEs describing dynamics of a chiller into state-space form, by using vector-matrix notation and linearization. Later on, Ding et al. [15] also proposed a hybrid modelling approach for dynamics of condenser, by packing some system variables into less unknown parameters obtained by experimental data using least squares methods.
Although the models of the ejectors and the conventional refrigeration cycles were studied, global dynamic models of the ejector-based refrigeration systems are hardly considered. That is, all the component models are not integrated into a uniform mathematic representation. Furthermore, the above dynamic modelling of the conventional refrigeration cycles considered refrigerant mass flow rates at boundary interfaces, rather than taking into account the variations of the refrigerant mass flow rates in multiple fluid phase regions. This may reduce modelling accuracy due to the loss of some important information on reflecting the dynamics of refrigerant mass flow rates in multiple fluid phase regions.
In this paper, we present a global low-order state-space model of the ejector-based refrigeration system. Much different from the above dynamic models of heat exchangers, reduced-order nonlinear models of evaporator and condenser are respectively developed based on the NTU method by introducing some intermediate variables to describe the variations of refrigerant mass flow rates in multiple fluid phase regions. These two dynamic models are capable of revealing essentially the dynamic characteristics of each of the fluid phase regions and providing much more information to describe the phase changes of the refrigerant, since more physical parameters are fully taken into account. Combining a1D static model of the ejector with two dynamic models of heat exchangers, a global nonlinear state-space model of the ejector-based refrigeration system is obtained. Finally, experimental validation with R134a is carried out based on the real-time experimental test rig. The results show that the proposed model has a good performance in predicting the dynamics of the system and is beneficial for the real-time control and optimization of the ejector-based refrigeration system.
Section snippets
System description
The refrigeration system studied in this paper is a basic vapour compression cycle integrated with an ejector cycle driven by low grade energy. The schematic figures shown in Fig. 1 and Fig. 2 illustrate the change of the refrigerant states during the cycle in pressure-enthalpy chart. Refrigerant vapour with high pressure and high temperature at state ‘1’ is cooled in the condenser by rejecting heat to surrounding air. The condensed refrigerant at state ‘2’ is divided into two parts: one part
System modelling
The ejector-based refrigeration system in study mainly consists of five components, namely evaporator, condenser, ejector, compressor and expansion valve. These component models can be separated into dynamic or static ones relying on their response speeds. Ejector, compressor and expansion valve are usually modelled using static equations involving some empirical formulae [9]. Dynamic modelling of the heat exchangers has been always a hot topic, due to the disadvantages of the existing models
Experimental setup
In order to validate the proposed model, an experimental rig on the ejector refrigeration system shown in Fig. 4 has been built in Process Instrumentation Laboratory of Nanyang Technological University of Singapore. The corresponding schematic diagram is given in Fig. 5 where several sensor locations for study are equipped. The experimental rig consists of the following components:
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A heater with a maximum power of 6 kW is set at the bottom of a heat transfer oil tank as the heat source of the
Results and discussion
The refrigerant R134a can be chosen as working fluid because of its international recognition for no bad effects against ozonosphere and excellent safety performances such as nontoxic, nonexplosive and uninflammable. However, R134a is going to be banned within several countries, due to its high Global Warming Potential (GWP = 1300). Thermodynamics properties of working fluid R134a are obtained from software REFPROP 7.0 developed by the National Institute of Standards and Technology of the
Conclusions
A global low-order state-space model was developed for the ejector-based refrigeration system, based on the first principle of the mass and energy conservation. Compared with the existing methods, the NTU method was applied to evaluate the heat transfer rate between the refrigerant and the tube wall in multiple fluid phase regions of the heat exchangers to obtain the reduced-order models of the evaporator and the condenser respectively. Furthermore, a 1D static model of the ejector proposed by
Acknowledgements
This work was supported by National Research Foundation of Singapore under the grant NRF2011 NRF-CRP001-090.
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