Dynamic Models of a Home Refrigerator

MACRo 2015-5 th International Conference on Recent Achievements in Mechatronics, Automation, Computer Science and Robotics Dynamic Models of a Home Refrigerator Tamás SCHNÉ 1,2, Szilárd JASKÓ 1,2, Gyula SIMON 1 1 Department of Computer Science and Systems Technology, University of Pannonia, 8200 Veszprém, Egyetem út 10., Hungary, e-mail: schne@dcs.uni-pannon.hu, jasko.szilard@uni-pen.hu, simon@dcs.uni-pannon.hu 2 Kanizsa Felsőoktatásáért Alapítvány, 8800 Nagykanizsa, Zrínyi Miklós utca 33., Hungary Manuscript received January 23, 2015, revised February 9, 2015. Abstract: Home refrigerators produce a substantial part of the annual power consumption in an average household. To further improve the efficiency of these devices, new intelligent control solutions are required. These solutions necessitate the behaviour modeling of the refrigerators. We seek models with as simple structure as possible, since future intelligent controllers may use such models in real time, thus their evaluation must be feasible even on simple microcontrollers. We investigate various dynamic models to describe the behavior of the refrigerator, i.e. the cool-down and warm-up phases. For model parameter identification real data was collected from a real home refrigerator. Data processing, modeling and the parameter identification were performed in MATLAB environment. Keywords: refrigerator modeling, data measurement, identification, simulation 1. Introduction Domestic refrigerators are widely investigated in the literature. Some publications deal with the energy consumption mainly from economical point of view. Hermes and Melo [1] and Marz [2] found that refrigerators require 11%- 16% of the whole energy consumption of a home. This high rate has to be decreased thus different works try to develop mathematical models that can be efficiently used for refrigerator design in the future. They usually concentrate on the four main elements of a refrigerator: the cabinet [3], [4], [5], evaporator [6], [7], [8], condenser [9], [10] and compressor [11], [12]. A common property of these models is that they describe exactly the investigated systems but use too complicated models that cannot be efficiently 103 10.1515/macro-2015-0010

104 T. Schné, Sz. Jaskó, Gy. Simon used in microcontrollers. Thus our purpose is to develop a model applicable for intelligent control, using low-end devices. The structure of the paper is the following: In Section 2 the applied system modeling and evaluation method is defined. Section 3 contains three system models with increasing complexity, Section 4 concludes the paper. 2. Methods The purpose of the modeling is to create a linear, discrete, parametric model of the refrigerator, which can be used for intelligent control of future refrigerators. The model must represent the operation of the refrigerator with enough precision to allow intelligent control, but it does not need to model every little detail of the operation. Note that there exist detailed and precise models [3], [6], [9], [11], but these are too complex to evaluate in small controllers, and the precision they provide is not required. Our modeling concept was to create as simple models as possible, thus we tried to locate those components of a refrigerator that had to be necessarily modeled. We applied incremental modeling with increasing model complexity (i.e. increasing model order). The created models are physical models of the refrigerator, they were created by analyzing the operation of the refrigerator and selecting essential components to be used in the model. The models were built in MATLAB/Simulink, and the parameters of the models were identified using MATLAB. For model identification real measurement data were used: a refrigerator was operated in various conditions to collect data. In Fig. 1 measured data can be seen. On the left hand side there is the cooldown phase of the refrigerator while the warm-up is on the right. They show the whole processes from room temperature to the lowest possible temperature and back. Note that the real measurements contain sudden jumps below zero degrees. The background of this effect is that there is a security valve in the refrigerator that saves the compressor from over charging. This is one of those features which is not necessary to model.

Dynamic models of a home refrigerator 105 Figure 1: Measured data of the refrigerator cool-down (left) and warm-up (right) phase. Thick line - back wall, thin line - cabinet air. For parameter estimation and model evaluation a quadratic error function was used: n e = (s(i) y(i))² i=1 where n is the size of the measurement record, and s and y contain the simulated and the measured back wall data, respectively. The possible simulated models, resulting s will be discussed in Section 3. The parameter estimation was performed by minimizing e, using the "fminunc" function of MATLAB. (1) 3. Models We will introduce three models with increasing complexity in this section. A. Modeling the cabinet air temperature For the first model we chose one main parameter (T c - cabinet temperature) that was thought to be appropriate to characterize the system behavior and to be good for control. This temperature is increased by the ambient temperature (T a 0 C) and decreased by the inlet temperature of the refrigerant of the evaporator (T e<0 C). Heat transfers are bounded by the thermal resistance of the insulation (R i) and the thermal resistance of the wall between the cabinet and the evaporator (R ec). The performance of a refrigerator is also influenced by the cabinet load, i.e. its heat storage capacity (C c).

106 T. Schné, Sz. Jaskó, Gy. Simon The relationship of these parameters is illustrated in the electric circuit shown in Fig. 2. Temperatures, thermal resistances, heat flows and heat storage capacities can be treated as voltages, electric resistances, currents and capacitors (storages) respectively. Figure 2: Refrigerator model with one storage element. If one considers the behavior of a refrigerator then can find that it shows hybrid functionality. It has two discrete states: 1 - compressor off, 2 - compressor on. In each state the system can be described with linear inhomogeneous ordinary differential equations with constant coefficients. State changing is represented with a binary switch (sw) in the circuit, from which the equations for the warm-up (Eq. (2)) and the cool-down (Eq. (3)) phases can be derived: T c(t) = T c (t) + T a (t) (2) C c R i C c R i and T c(t) = T c (t) + T a (t) + T e (t) C c (R i R ec ) C c R i C c R ec (3) Identification results We simulated the system behavior in the two phases and compared the results to the measured data. Both phases show good match (Fig. 3). Note the modeling error below zero, and also note that the trend is adequately modeled.

Dynamic models of a home refrigerator 107 Figure 3: The cool-down and warm-up phases of the one storage model compared to measured data. We can conclude that a single differential equation is enough to describe the behaviour of the inner air of the cabinet. However, the one storage model is not appropriate for control design because it does not model the temperature of the back of the cabinet, a widely used control parameter. B. Modeling the air and evaporator A classical refrigerator control is based on the temperature of the back of the cabinet, which is influenced by the evaporator. The background of this strategy is that impulse-like door openings affect the least in this area. The walls and stored food have a common heat storage capacity, which is great enough to keep the refrigerator compressor from unnecessary operation. Moreover, warm air streaming in cools down quite fast, thus it cannot influence the system temperature considerably. The back of the cabinet, with the evaporator is modeled by a capacitance C e capacity with temperature T e. The capacitance is charged by the condenser (T cond) via the capillary tubes (R cap) as shown in Fig. 4. The binary switch sw has the same role as in Section A. The warm-up and cool-down phases are described by the following equations: Warm-up (sw off): T c(t) = T e (t) T c (t) + T a (t) T c (t) (4) C c R ec C c R i T e(t) = T c (t) T e (t) C e R ec Cool-down (sw on): T c(t) = T e (t) T c (t) + T a (t) T c (t) (5) C c R ec C c R i T e(t) = T c (t) T e (t) T cond (t)+t e (t) C e R ec C e R cap

108 T. Schné, Sz. Jaskó, Gy. Simon Figure 4: Refrigerator model with two storage elements. Identification results The system parameters were identified and the warmup and cool-down phases were simulated, as shown in Fig. 5. While the warmup phase is modeled correctly, there is significant error in the cool-down phase. This modeling error is corrected by the three storage model introduced in Section C. Figure 5: The cool-down and warm-up phases of the two storage model. C. Complex model: air, evaporator, condenser and compressor The fault of the two storage model was that it did not handle the heat storage capacity of the refrigerant container circuit. A significant amount of material can be found in the condenser and the compressor, thus we get a more precise model if the constant voltage source T cond is exchanged with a condenser having C cond capacity and T cond voltage. The compressor supplies refrigerant T comp to the condenser through a narrow tube R cond. The electric circuit representing the relationship between the model elements is depicted in Fig. 6.

Dynamic models of a home refrigerator 109 The warm-up phase of this model is described with Eq. (4), because the new elements influence only the cool-down equations, as follows: T c(t) = T e (t) T c (t) + T a (t) T c (t) C c R ec C c R i T e(t) = T c (t) T e (t) T cond (t)+t e (t) (6) C e R ec C e R cap T cond(t) = T comp (t) T cond (t) T cond (t)+t e (t) C cond R cond C cond R cap Figure 6: Refrigerator model with three storage elements. Identification results The behavior of the identified and simulated system is shown in Fig. 7. In this case both the cool-down and warm-up phases are satisfactorily modeled. The final parameter values can be found in Table 1. Figure 7: The cool-down phase of the three storage model. Solid lines - measured data, dashed lines - simulated data.

110 T. Schné, Sz. Jaskó, Gy. Simon Table 1: Model parameters and values. R i R ec R cap R cond C c C e C cond T a T comp 0.475 0.325 0.01 0.099 3.2 0.64 0.6 20 25 Figure 8: A 37 hour long simulation. We made a long time simulation that took 37 hours and 44 minutes. The result is depicted in Fig. 8. It can be seen that the model follows the real behavior of the refrigerator with small error. 3. Conclusion Three dynamic models of a domestic refrigerator were investigated in this paper. The simplest one describes the system with a single linear inhomogeneous ordinary differential equation with constant coefficients. This model captures the inside air temperature of the refrigerator correctly, but the back of the cabinet (which is important for the control) is not included in the model. The second model improves the first one. Here two equations are defined for both phases. One describes the behavior of the cabinet air and an other one is for the evaporator. This model has significant modeling error in the cool-down phase.

Dynamic models of a home refrigerator 111 The most exact model is the third one, containing three capacitances. It applies two differential equations to describe the warm-up phase and three equations when the compressor is on. It is based on the second model but the cool-down phase is extended with an equation for the condenser of the refrigerator, and models the refrigerator with enough precision to be used in intelligent control applications. Acknowledgements This publication/research has been supported by the European Union and Hungary and co-financed by the European Social Fund through the project TÁMOP-4.2.2.C-11/1/KONV-2012-0004 - National Research Center for Development and Market Introduction of Advanced Information and Communication Technologies and TÁMOP-4.2.2.A-11/1/KONV-2012-0072 project. References [1]. Hermes, C.J.L, Melo, C.: A first-principles simulation model for the start-up and cycling transients of household refrigerators. International Journal of Refrigeration. 31 (8), 1341-1357 (2008) [2]. Marz, M.: The design of intelligent control of a kitchen refrigerator. Mathematics and Computers in Simulation 56 (3), 259-267 (2001) [3]. Hermes, C.J.L.: A first-principles methodology for the transient simulation of household refrigerators. PhD thesis, Federal University of Santa Catarina, Floriano polis-sc, Brazil, 272p (in Portuguese). [4]. Hermes, C.J.L, Melo, C.: How to get the most out from a semi-empirical reciprocating compressor using a minimum set of data. IIR International Conference on Compressors and Coolants, Papiernicka, Slovak Republic. [5]. Azzouz K., Leducq D., Gobin D.: Performance enhancement of a household refrigerator by addition of latent heat storage. International Journal of Refrigeration 31 (5), 892-901 (2008) [6]. He, X.-D., Liu, S., Asada, H.: A moving-interfacemodel of two-phase flow heat exchanger dynamics for control of vapor compression cycle. Heat Pump and Refrigeration Systems Design, Analysis and Applications 32, 69-75 (1994) [7]. Yu, B.F., Wang, Z.G., Yue, B., Han, B.Q., Wang, H.S., Chen, F.X.: Simulation computation and experimental investigation for on-off procedure of refrigerator. IIR International Congress of Refrigeration, The Hague, The Netherlands 3, 546-553 (1994) [8]. Ploug-Sorensen, L., Fredsted, J.P., Willatzen, M.: Improvements in the modelling and simulation of refrigeration systems: aerospace tools applied to a domestic refrigerator. Journal of HVAC& R Research 3 (4), 387-403 (1997) [9]. Hermes, C.J.L., Melo, C.: A heat transfer correlation for natural draft wire-and-tube condensers. IIR Internationa Congress of Refrigeration, Beijing, China. [10]. Hermes, C.J.L., Melo, C., Goncalves, J.M.: A robust mod approach for refrigerant flow through capillary tubes. IIR Internationa Congress of Refrigeration, Beijing, China.

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