1、4754 Dynamic Model of a Centrifugal Chiller System-Model Development, Numerical Study, and Validation Satyam Bendapudi, PhD Student Member ASHRAE James E. Braun, PhD, PE Member ASHRAE Eckhard A. Groll, PhD Member ASHRAE ABSTRACT Dynamic models of heat pumps are useful in developing jedback controlle
2、rs and fault-detection-diagnostic (FDD) studies. Several system models have been documented in the literature, but few are for centrijugal chillers. Existing publi- cations focus on model development and validation, providing minimal detail on the numerical aspects of the solution. The solution of t
3、he PDEs that are obtained to model the heat exchangers is critical in terms of accuracy and execution speed. This paper presents the development of a centrifugal chiller system model, using the finite-volume (FV) approach for shell-and-tube heat exchangers und aspects such as mesh dependence, integr
4、ation order, and step size. SufJicient and necessary mesh sizes for accurate steady-state prediction are determined for the heut exchangers. Execution speeds with integration algorithms of thefirst, second, andfourth order are comparedfor equivalent accuracy. The model is based onjrst principles, al
5、lowing it to be used over a wide range of oper- ating conditions and transients. The model is validated using data from a 90-ton R-134a centrijugul chiller. INTRODUCTION Some dynamic models of heat pumps have been devel- oped over the past 25 years for air-to-air systems with recip- rocating compres
6、sors. Liquid chiller modeling has been gaining attention in recent years, but few models exist for centrifugal systems. Liquid chillers using flooded heat exchangers are fundamentally different from other construc- tions due to the large thermal capacitances of the metallic parts, the secondary flui
7、d, and the large quantity of refrigerant. Centrifugal compressors, similarly, are different from recip- rocating machines due to their susceptibility to surge and also their relatively higher tolerance for wet compression. Capacity control, an important dynamic associated with compressor response, c
8、an be effected using inlet-guide vanes, a capability not shared by reciprocating compressors. These issues make the modeling of centrifugal liquid chiller dynamics compli- cated and important. A dynamic chiller system model is required to predict performance during transients associated with start-u
9、p and feedback control since these are the two most significant tran- sients in a chillers operation. If on-off control is used, shut- down transients are equally important. Since large liquid chiller systems do not normally use such control methods, they are not considered here. Shutdown transients
10、 in large chillers can be of interest, though, to determine the time after shut- down when the chiller can be safely restarted. A good dynamic system model can be used within a test bed to evaluate algorithms for feedback controllers and thus minimize the time and expense of experimentation. In the
11、development of automated fault detection and diagnosis (FDD) methods, a dynamic model can be used to generate data with and without faults in the system. Such data can be hard to obtain from real systems for two reasons: first, equipment owners would likely not be willing to risk such experiments on
12、 their systems and, second, it is hard to reproduce faults in areal system with a high level of repeatability. These difficulties can be overcome if a simulation tool is available to generate such data. A significant body of literature exists on the modeling of dynamics of refrigeration equipment of
13、 various configura- tions. In terms of approaches to modeling, the most important differences are in the way the refrigerant in the heat exchang- ers are treated. The two prevailing approaches are the moving boundary and the finite volume. In the moving boundas. Satyam Bendapudi is a graduate studen
14、t, James E. Braun is a professor, and Eckhard A. Groll is an associate professor in the School of Mechanical Engineering, Purdue University, West Lafayette, Indiana. 132 02005 ASHRAE. method, the heat exchanger is divided into control volumes based on the phase of the refrigerant. These volumes are
15、vari- able in time, and the saturated liquid and saturated vapor boundaries move during transients, hence the name. Dhar and Soedel (1978) were among the earliest to model the dynamics of a heat pump using this approach. Their model consists of a system of algebraic ordinary differential equations t
16、hat are integrated using the Euler method. He et al. (1997) also used the moving boundary approach for their model of an air-to-air system. The model is limited to transients associated with load change and the solution is by casting the equations in a linear- ized state-space form. This allowed the
17、 use of a closed-form solution method for small changes near the operating point. The model is used successfully to develop a multi-inPudmulti- output control algorithm. The second prevalent approach used to model heat exchangers is the finite-volume method in which the heat exchanger is divided int
18、o a number of constant control volumes. The transient conservation equations are discretized over these volumes and result in a system of ordinary differ- ential equations. MacArthur and Grald (1 987) were among the first to use this approach. Their model of an air-to-air system uses the sequential,
19、 or iterative, solution method as described by Patankar (1980). The integration algorithm used is the implicit first-order Euler method. Good model speeds are reported under typical start-up and shutdown cycling. Rossi and Braun (1 999) used the finite-volume formulation for a rooftop unit, using a
20、direct inversion method for solving the system of algebraic differential equations. Real-time execu- tion speed is reported using the fourth-order Runge-Kutta inte- gration method. The model uses a dynamic step-sizing algorithm that exploits knowledge of cycling time constants, thereby avoiding the
21、computational overheads of an error- bounded method. Among the literature reviewed, the only dynamic system model found for a centrifugal chiller was the paper by Wang et al. (2000). This system model includes detailed models for single-stage and two-stage centrifugal compressors. The shell-and-tube
22、 heat-exchangers are, however, treated simplis- tically as lumped capacitances. While the preceding is by no means an exhaustive list of work in this area, it is representative in terms of identifying pioneering approaches. A more detailed review of literature on dynamic modeling of refrigeration sy
23、stems was documented by Bendapudi and Braun (2002a, 2002b). Several other researchers have documented dynamic system models that are based essentially on one of the above approaches. Some rele- vant observations can be made based on the above. Very often no justification is provided regarding the ch
24、oice of solution methodology. None of the papers reviewed included a numer- ical study of the models developed in terms of discretization or integration algorithm. While undoubtedly the integration step size would have been chosen after careful numerical study, no information is provided about the m
25、ethodology or the results of such a study. This paper attempts to bridge some part of this discrepancy. TEST SYSTEM Validating model accuracy requires experimental data of system behavior over a wide range of operating conditions and transients. Such data for a centrifugal liquid chiller system were
26、 extensively collected, analyzed, and documented by Comstock and Braun (1999) and are used for the purposes of validating the current model. A detailed description of the experimental test stand is given by Comstock (1999) and Comstock et al. (2001). To aid the discussion on model validation at the
27、end of this paper, a brief description of the system and available data is included here. The chiller is a 90-ton centrifugal system consisting of a shell-and-tube evaporator, a shell-and-tube condenser, a ther- mostatic expansion valve, and a single-stage centrifugal compressor. Capacity control is
28、 achieved by varying the compressors inlet guide vanes. The refrigerant used in the system is R-134a and the secondary fluid is water, which flows in the tubes in both heat exchangers. A schematic of the system and refrigerant flow paths is shown in Figure 1. Parallel to the liquid line that carries
29、 the bulk of high-pressure refrigerant through the expansion device, there is a cooling line tapped at the exit of the condenser. This line carries liquid refrigerant to the motor and transmission housing where it is first expanded across an orifice to the evaporator pressure and then used to cool t
30、he motor and transmission oil. This refrigerant is then returned to the main refrigerant stream at the evaporator inlet. The data collected consist of several sets of tests done with and without different faults introduced into the chiller. Each set consists of running the chiller from start-up thro
31、ugh a sequence of 27 different operating conditions to shutdown. Data were collected at 10-second intervals, capturing all tran- sients associated with start-up, the load-changes triggered by varying return-water temperatures and chilled water setpoint temperature, and the shutdown. MOD EL FORM U LA
32、TI ON A dynamic heat pump system model consists of dynamic component models for the condenser, the evaporator, the compressor, and the expansion device, which are brought together with mutually consistent input-output information, as shown in Figure 1. These component models and the overall system m
33、odel are described in this section. While a good system model should be independent of any specific features of the system, some specificity is unavoidable during valida- tion. In the current model this appears in two aspects. The compressor model is semi-empirical to reduce the computa- tional requ
34、irements. Different empirical models would be employed for different compressor types and depend on the level of detail available in the data. In the heat exchangers, the heat transfer coefficients are tuned using scaling factors to account for surface enhancement effects. These factors would need t
35、o be reestablished when using this model for other chill- ers. In all other respects, the component models are based on first principles. ASHRAE Transactions: Research 133 P, . Evaporator P, * 4,T, h, 4 Qw Tcw 5 RE f. . i * . : 4 : I Condenser l W I W Comp Qcc ,.*. . I I I Figure 1 Schematic of refr
36、igerant paths and system model information flow. Heat Exchangers Formulation. Since the condenser and evaporator are both flooded types, the model development follows a generic approach suitable for either. Refrigerant flowing on the shell side of a shell-and-tube heat exchanger follows a circuitous
37、 path that is a combination of cross-flow and counterflow. It typically enters at one end of the shell and is directed over the tubes several times by baffles before it leaves at the other end. Modeling this flow pattern precisely is difficult and may even be unnecessary. In order to simplifi the fl
38、ow pattern, pure counterflow and pure cross-flow patterns were developed and compared during the early stages of model development. It was found that while both approaches yielded comparable accuracies, the counterflow assumption had the added advan- tage of capturing the temperature profile of the
39、water along the heat exchanger and was also simpler to implement. Therefore, it was used as the flow pattern of choice for approximating the true flow. Physically this can be justified by the facts that the water temperature varies more significantly along the length of the heat exchanger, the refri
40、gerant temperature also varies more signifi- cantly along the length of the heat exchanger in the sin- gle-phase regions, the bulk of the heat transfer occurs in the two-phase region where the refrigerant temperature remains con- stant. Figure 2 shows the flow arrangement used in this model; the out
41、er tube is formed by the shell and has the refrigerant flowing through it, and the inner tubes carry the water. The following simplifiing assumptions are made in the develop- ment. One-dimensionalflow: While the water flow through the tubes is clearly one-dimensional, the refrigerant flow is truly t
42、hree-dimensional. There does, however exist a dominant uniform flow direction between refrigerant inlet and outlet. The consequence of local flow direction variations caused by turbulence is to enhance the heat transfer. This can be accounted for by the selection of an appropriate correlation for th
43、e heat transfer coefficient. This allows the assumption of one-dimensional flow on the refrigerant side also. Negligiblepressure drops: The water and the refrigerant experience some pressure drop in flowing through the heat exchanger. On the water side, this has little impact 134 ASHRAE Transactions
44、: Research I Refrigerante A h Y V w Water _+ z L I Figure 2 Counter-ow assumption and discretization in the heat exchanger: on the heat transfer rate since the properties of water are a very weak function of pressure. On the refrigerant side, for a flooded heat exchanger, the flow path between the i
45、nlet and outlet is fairly short and the sec- tional flow areas available in the shell are large. This results in small pressure drops that can be neglected. Negligible viscous dissipation: As the refrigerant flows through the shell, a small amount of flow energy is irre- versibly converted to heat b
46、ecause of viscous effects between refigerant particles. However, refrigerant vis- cosity even in the liquid phase is very small and the vis- cous dissipation is orders of magnitude smaller than the more dominant energy transfer between the refrigerant and the heat exchanger tube material, and it can
47、 be safely ignored. Negligible axial conduction: The refrigerant state, tube temperature, and water temperature vary along the length of the heat exchanger, and these temperature gra- dients can induce diffusion heat transfer in the axial direction. However, flow rates of both the refigerant and the
48、 water in the heat exchanger are quite high, and the Peclet number (which quantifies the relative effects of convection and diffusion modes of heat transfer) is high enough to make difisive heat transfer a secondary effect that can be neglected. Axial temperature varia- tions in the tube material ca
49、n also be neglected because it is made of a highly conductive material and also because the cross-sectional area available for axial con- duction is very small. Negligible tube material resistance: The conductivity of the tube material is high and offers little resistance to the radial heat transfer between the refrigerant outside and the water inside. The capacitance, however, is more sig- nificant and is accounted for. Negligible shell capacitance: The shell of the heat exchanger is made of steel and is thick enough to consti- tute a large thermal mass. The refrigerant on the shell sid
