1、Dynamic Modeling and Control of Multi-Evaporator Air-Conditioning Systems Rajat Shah Andrew G. Alleyne, Ph.D. Clark W. Bullard, Ph.D. Fellow ASHRAE ABSTRACT This paper presents a new methodology for the dynamic modeling of multi-evaporator air-conditioning cycles. The resulting model is suitable for
2、 designing advanced closed-loop controllers for these systems. A generalized modeling approach is developed, which is applicable to commercially ,available units with any number of evaporators. Model vali- dation against data from an experimental dual-evaporator system for step inputs to compressor
3、speed and expansion valve opening is also presented, and the results show good prediction accuracy. The open-loop behavior observed in these simulation studies clearly indicates the effect of cross- coupling between dynamic variables of different evaporators. Finally, closed-loop control strategies
4、based on the model and the system characteristics are discussed, and comparison of two control strategies for control ofpressure and superheat of the two evaporators is also presented. INTRODUCTION A typical single evaporator subcritical vapor compression cycle consists of a compressor, an expansion
5、 device, and low- side and high-side heat exchangers (Dossat 1980). The refrig- erant absorbs heat as it evaporates and is then compressed to a high pressure where heat is rejected as it condenses (Dossat 1980). The difficulty of modeling the complex thermofluid dynamics associated with these phase
6、changes has restricted the development of advanced controllers for these systems (He et al. 1997). Recently, variable displacement compressors, electronic expansion valves, and variable-speed motor-driven fans and blowers have become more commonly used cycle components. These actuators provide more
7、control authority; however, for their effective use for a better system perfor- mance, a well-designed control scheme is necessary. Generally, air-conditioning units and their control systems are designed with a focus on system efficiency and control of sensible and latent capacity. Efficiency depen
8、ds on refiigerant superheat at evaporator outlet. A unit with no superheat would be ideal because heat transfer efficiency reduces as superheat increases. However, a minimum positive superheat is essential to avoid liquid carryover that could damage the compressor. Sensible and latent capacities are
9、 controlled by modulating the evaporator airflow rate and the refrigerant mass flow rate. Traditionally, these objectives have been achieved by implementing single inputhingle output (SISO) control schemes in which the control action is based solely on the input/output data. Advanced robust control
10、theory can also be applied to SISO controllers, as illustrated by Kasahara et al. (1999). However, recent works (He et al. 1997; Rasmussen 2002) have shown that, due to highly coupled dynamic behavior of air-conditioning cycles, multi-input/ multi-output (MIMO) control is advantageous since it is ba
11、sed on the knowledge of the complete system characteristics. The significance of MIMO control increases further for a multi- evaporator air-conditioning scenario because the presence of more evaporators increases the severity of dynamic coupling phenomenon. The design of a MIMO or a multivariable co
12、ntroller requires a fairly accurate dynamic model of the system, which can be obtained either by physics-based first principles model- ing or by data-based identification. A detailed survey of vari- ous works on dynamic modeling of vapor compression systems is available in Bendapudi and Braun (2002)
13、. Decou- pled SISO identification of a direct expansion air-conditioning Rajat Shah is a graduate student, Andrew G. Alieyne is an associate professor, and Clark W. Builard is a professor in the Department of Mechanical Engineering, University of Illinois, Urbana Champaign. 02004 ASHRAE. 1 o9 plant
14、was done by Deng and Missenden (1999). Work presented in He et al. (1997) and Rasmussen (2002) shows that low-order black-box models can be obtained by system iden- tification, which are sufficient for the complete information of vapor compression system dynamics. Although knowledge about the domina
15、nt dynamic modes is gained, the black-box model makes it difficult to relate the identified model with the physical characteristics of the actual system. Physics-based models are, however, based on such relations and, hence, prove more modular for controller design and system analysis. Finite differ
16、ence techniques and lumped parameter meth- ods are the two primary approaches for physics-based model- ing of vapor compression systems. Finite difference methods have been successfully used in Grald and MacArthur (1 992) and Mithraratne and Wijeysundera (2001) for simulation work. However, a very h
17、igh-order system representation is obtained by these methods that is not conducive to the design of advanced controllers. On the other hand, lumped parameter methods generate relatively lower-order models, which are preferred from a controls perspective. Development of this method can be credited to
18、 the work by Wedekind et al. (1978) on generalized mean void fraction models for heat exchangers. Various works (He et al. 1997; Pettit et al. 1998) have been done based on that idea, and the same approach will be used in this paper too. The primary objective of this paper is to demonstrate the nece
19、ssity and feasibility of advanced model-based control for multi-evaporator air-conditioning systems. Few research stud- ies have been conducted to-date on this issue. Static modeling and analysis work was done by Badr et al. (1 990), which is not helpful for research on dynamics and control. Lee et
20、al. (2002) did work on control of a dual-evaporator system using online system identification methods, and their results showed good controller performance, but analysis of the cycle dynamics was not presented. Stack and Finn (2002) worked on multi- evaporator systems; however, their model was usefu
21、l for refrigeration applications only. Vclavek et al. (2002) also proposed a strategy applicable only to refrigeration systems by using electronic pressure-regulating valves. With this back- ground, the present paper contributes uniquely by introducing and validating a physics-based modeling methodo
22、logy for multi-evaporator air-conditioning systems, which are dynam- ically more complex than the refrigeration systems. The Air CO cycle Compressor resulting model provides significant insight into the system dynamics, which is useful for the design of advanced control strategies. The remainder of
23、this paper is organized as follows. The next section details individual component models and the complete cycle model. This is followed by model validation plots against experimental data and study of open-loop response of the system to step disturbances in different actu- ators. Closed-loop control
24、 strategies and simulation studies for the comparison of a SISO and a MIMO controller are discussed next. Conclusions and plans for future work form the last section of this paper. PHYSICAL MODEL DESCRIPTION Figure 1 shows typical configurations of a multi-evapo- rator vapor compression cycle (Dossa
25、t 1980). The system has a condenser and a variable-speed compressor catering to all of the evaporators, each having an individual expansion valve. The refrigerant outflow from all of the evaporators meets at the compressor suction. This is where the main difference between a multi-evaporator air-con
26、ditioning cycle and a refrigeration cycle lies. In the refiigeration cycle, a pressure regulator is used as shown in the figure, which permits differ- ent evaporators to exchange heat at quite different evaporation pressures. It should be noted here that the presence of a pres- sure regulator helps
27、in decoupling the dynamics of different evaporators and, hence, a multi-evaporator refrigeration cycle is much simpler to model than the corresponding air-condi- tioning cycle. The objective of the following sections is to introduce a methodology for dynamic modeling of the more complex multi-evapor
28、ator air-conditioning systems. Individual Component Modeling In this section, a decentralized modeling architecture is developed in which the individual component models are used as building blocks to design a complete cycle model. The different component models include heat exchangers, an expansion
29、 valve, and a variable-speed compressor. Heat Exchangers. As pointed out earlier, the lumped parameter method (Wedekind et al. 1978) is used here to model a heat exchanger. In this approach, the heat exchanger is allowed to have time-varying lengths of different regions of subcooled, two-phase, and
30、superheated flow. Thus, the refrig- Rcfngerati cycle Compressor essu gum Evaporator 2 Figure 1 Multi-evaporator air-conditioning and refrigeration cycles. 110 ASHRAE Transactions: Research erant flow in the evaporator can be divided in two regions since it enters as a two-phase liquid and leaves as
31、a superheated vapor. The point oftransition between the two regions changes dynamically (Rasmussen 2002). Similarly, the flow phenom- enon in a condenser can be divided into three regions of super- heated, two-phase, and subcooled refrigerant flow (Shah 2003). Generalized Navier-Stokes equations (Wh
32、ite 1994) can be used to express the refrigerant mass and energy balance in these regions. An important assumption is that the friction losses in the heat exchangers are negligible, which renders the momentum conservation equation redundant. This results in significant simplification in modeling. Th
33、e resulting mass and energy balance equations are given by the following: Refrigerant mass balance: Ll e rw1 fw2- *+ao=, at az h= Refrigerant energy balance: a(ph-P) a( uh) - 4 + - -Ljai(Tw- T,.) at az Di Heat exchanger wall energy balance: (3) (CPPUwTW = a,A,(T,- T,) + aoAo(Two- T,) Spatial depende
34、nce of these equations can be removed by integrating over the length of the fluid flow. Detailed descrip- tion and expanded form of these dynamic equations for the evaporator and the condenser are published in Rasmussen (2002) and Shah (2003), respectively, and are omitted here for the sake of brevi
35、ty. The dynamic model developed for the evaporator using these equations is fifth order, with the dynamic modes being the length of two-phase flow LI, refrig- erant pressure P, refrigerant outlet enthalpy h, and wall temperatures in the saturated and the superheated region, T, and Tw2, respectively.
36、 The states of the evaporator dynamic model can be written as The complete dynamic model can be written in a vector- matrix form, as given by Equation 5. The elements au are nonlinear functions of fluid and heat exchanger properties and are detailed in Rasmussen (2002). all a12 O O O 21 a22 23 o o 3
37、1 a32 33 o o o o Oa40 a51 O O O a5! (5) Similarly, the condenser can be described mathematically as a seventh order system, with the dynamic modes being the length of the two condensation regions LI (for superheated flow at the entrance) and L, (two-phase flow in the middle), refrigerant pressure P,
38、 refrigerant outlet enthalpy h, and the wall temperatures in the three regions, T, Tw2, and Tw3, respectively. The states of the condenser dynamic model can be expressed as The vector-matrix representation of the condenser model can be written similarly (Shah 2003). For systems with a high- side rec
39、eiver, the condenser has only two regions of super- heated and mo-phase flow. Mass inventory balance in the receiver, however, adds another dynamic mode, thus resulting in a complete sixth order model with the dynamic modes being LI, P, h, Twl, Tw2, and the refrigerant mass in the receiver (Shah 200
40、3). Expansion Device. The time constant for the dominant dynamics of the expansion valve and the compressor are much shorter than those of the heat exchangers (He et al. 1997; Rasmussen 2002). Any change in the compressor speed or the opening area of the expansion device affects the mass flow rate r
41、elatively instantaneously, as shown in He et al. (1997). Thus, via time-scale separation reasoning, these components can be modeled using algebraic equations. The expansion valve can be modeled by the isenthalpic orifice equation, with C, and A, being the coefficient of discharge and the area of the
42、 valve, respectively. (7) Compressor. Similarly, the algebraic relation for the compressor mass flow rate is given by Here, A, and Bk are volumetric efficiency coefficients for the compressor, and r and vk are the polytropic compression coefficient for the refrigerant and the volume of the compres-
43、sor, respectively. These component models can be connected in different arrangements to emulate various commercially available vapor compression cycles. Each of the individual dynamic models described above has been validated against data from experimental single-evaporator subcritical (Shah et al.
44、2002) and transcritical (Rasmussen 2002) cycles. The component models can then be used for the design of multi-evaporator cycles also with the complexities and challenges of mass, energy distribution accounted for. These complexities are discussed in the next section. Modeling Challenges in Multi-Ev
45、aporator Systems In a vapor compression system, refrigerant mass and energy distribution in different parts of the cycle affect the systems steady state. Similarly, the system response to time- varying inputs and disturbances is also dependent on the ASH RAE Transactions: Research 111 dynamics assoc
46、iated with the refrigerant mass and energy transfer. Experimental results by He et al. (1997) and Lee et al. (2002) corroborate that refrigerant charge balance through different cycle components affects the system dynamics heavily during transients as the system approaches a new steady state. In ord
47、er to understand the predominance of mass flow rate balance on cycle dynamics, a quick review of the vector-matrix form of evaporator model in Equation 5 is useful. It can be observed that the system is at steady state only when all time derivatives are zero and, for that, the expressions on the rig
48、ht-hand side of the equation should be identically zero. On close observation, it may be noted that the vector on the right-hand side of Equation 5 quantifies the steady-state energy and mass imbalances of the system. Another observation is that the time-domain solution of Equation 5 requires, among
49、 many other variables, the instan- taneous knowledge of incoming ( “li ) and outgoing ( “lo ) mass flow rates from the evaporator. This evaporator model clearly indicates the two primary challenges involved in the modeling of multi-evaporator air-conditioning systems. First, in a single evaporator case, mass outflow rate from the evaporator can be obtained directly from the compressor model. However, in a multi-evaporator case, a mass distribution algorithm is required at the compressor suction to calculate the mass outflow rate from each evaporator. This is a nontrivial model-
