1、4771 Heat Exchanger Dynamic Observer Design Tao Cheng Student Member ASHRAE Harry H. Asada, PhD Xiang-Dong He, PhD Member ASHRAE Shinichi Kasahara ABSTRACT This paper presents the design of model-based nonlinear dynamic observers of an evaporator and a condenser ,for advanced control of air-conditio
2、ning and refrigeration systems. Control-oriented low-order models are derived for the evaporator and the condenser that are spatially distrib- uted. Based on sensor measurements of the evaporating temperature, the nonlinear observer for the evaporator can be used to estimate the heat transfer rate a
3、nd the length of the two- phase section that cannot be measured directly in transient processes. Based on sensor measurements of the condensing temperature, the nonlinear observer for the condenser can be used to estimate the length of the two-phase section and the length of the subcooling section t
4、hat cannot be measured directly. The simulation results and experimental testing are also presented. In air-conditioning and refrigeration applica- tions, the nonlinear observers can be utilized to synthesize feedback linearization nonlinear control andprovide accurate refrigerant charge inventory e
5、stimation and optimization for transient processes. INTRODUCTION Two-phase-flow heat exchangers have been widely used in air-conditioning and refrigeration systems for residential, commercial, and industrial applications. Modeling, estima- tion, and control of two-phase-flow heat exchangers have bee
6、n active research subjects for years in attempts to improve energy efficiency and system reliability. Most ofthese projects are concerned with the steady-state operation of such heat exchangers, despite the fact that steady-state conditions are almost never reached in the presence of dynamic interac
7、tion and varying environmental conditions. Modeling of the dynamic behavior of a complicated and spatially distributed air-conditioning system has been reported in several works (Chi and Didion 1982; MacArthur and Grald 1989; He et al. 1997; He and Asada 2003). With increasing complexity of modern H
8、VAC systems, controlling and optimizing the operation with guaranteed performance, stability, and reliability becomes a challenging issue. In advanced control of HVAC systems, it is necessary to dynamically estimate some immeasurable variables based on available sensor measurements. For example, in
9、nonlinear feedback linearization (He and Asada 2003), the heat transfer rate of evaporators must be estimated because no direct measurement is available. Dynamic estimation of heat transfer rate can also be used for adaptive control of room air temper- ature. To estimate the refrigerant charge inven
10、tory, it is neces- sary to know the length of the two-phase section of a heat exchanger for an accurate estimate of refrigerant in the heat exchanger. Yet, this length is not directly measurable. Due to the development of the modem observer theory and the convergence theory for nonlinear systems (Lo
11、hmiller and Slotine 1998), the design of observers based on nonlinear models of heat exchangers becomes possible and can be used to estimate dynamic variables that cannot be measured. In this paper, a low-order model is developed for the evaporator. The model describes the dynamic relationship betwe
12、en the evap- orating temperature and the compressor side mass flow rate that can be further related to the compressor speed. The model also describes the dynamic relation between the length of the two-phase section of an evaporator and the expansion valve side mass flow rate that can be further rela
13、ted to the expansion valve opening. The evaporator wall temperature is also treated as an independent state variable to improve the model accu- Tao Cheng is a graduate research assistant and Harry H. Asada is a professor and director of the #Arbeloff Laboratory for Information Systems and Technology
14、 at the Massachusetts Institute of Technology. Xiang-Dong He is a senior rescarcher at Daikin US Corporation and visiting scientist at MIT. Shinichi Kasahara is research leader at Daikin Air Conditioning R (2) the refrigerant flowing through the heat exchanger tube can be modeled as one-dimensional
15、fluid flow; and (3) axial heat conduction is negligible. A diagram of a low-order evaporator model is illustrated in Figure 1. Te is the evaporating temperature, Z(i is the length of the two-phase section, Tw(i is the wall temperature of the tube, Ta is the room air temperature, Ini, and Inou, are t
16、he inlet and outlet refrigerant mass flow rates, respectively, q(t) is the heat transfer rate from the tube wall to the two-phase refrigerant, and qa is the heat transfer rate from the room to the tube wall. It is assumed that the two-phase section has invari- ant mean void fraction 7 (Wedekind et a
17、l. 1978). If it can be assumed that the evaporator tube wall temper- ature along the two-phase section is spatially uniform, then the energy balance equation of the tube wall is given by dTW (cppA) - = KD,a,(Tu-T,)-KD,ai(T,-T,), (1) e dt where cp is the specific heat of the copper tube, p is density
18、 of copper, A is the cross-sectional area of the copper tube, Do is the outer diameter, Di is the inner diameter, a, is the heat transfer coefficient between room air and the tube wall, and ai is the heat transfer coefficient between refrigerant and the tube wall. The first term on the right-hand si
19、de of Equation 1 repre- sents the heat transfer rate per unit length from the room to the tube wall. The second term represents the heat transfer rate per unit length from the tube wall to the two-phase refrigerant. Assuming the mean void fraction y is invariant or changes very slowly compared to th
20、e state variables, the liquid mass balance equation in the two-phase section of the evaporator is p,(l-y)Ay = -*+Inin(l-xo), hk where pl is the refrigerant saturated liquid density, q is the heat transfer rate between the tube wall and refrigerant in the two- phase section, xo is the inlet vapor qua
21、lity, and min is the inlet refrigerant mass flow rate, h, = hg - h, (h, and h, are refrig- erant saturated liquid and vapor specific enthalpies). In Equation 2, the left-hand side is the liquid mass time rate of change in the evaporator. On the right-hand side, qlh, represents the rate of liquid eva
22、porating into vapor, and Inin( 1 - xo) is the inlet liquid mass flow rate. The inlet refrigerant mass flow rate In, is dependent on the expansion valve opening A, the low pressure P, and high pressure P, and can be expressed by where a and g, (P,P,) can be identified for a given expansion valve. P,
23、and P, can be measured by two pressure sensors. For the two-phase section, the pressure is an invariant function of the temperature. Therefore, the inlet refrigerant mass flow rate mi, can be expressed as mi, = A:gv(q?, T,). (4) Lets consider the vapor mass balance in an evaporator. The inlet vapor
24、mass flow rate is Ininx, and the outlet vapor mass flow rate is Inour when superheat is present. The rate of vapor generated from liquid during the evaporation process in the two-phase section is qlhg. The time rate change of vapor mass should be equal to the inlet vapor mass flow rate plus the rate
25、 of vapor generated from liquid minus the outlet vapor mass flow rate. Assuming that the vapor volume is much larger than the liquid volume in the low-pressure side, we can obtain the vapor mass balance equation in an evaporator. where A4, is the total vapor mass of the low-pressure side and V is th
26、e total volume of the low-pressure side, and pg is the saturated vapor density of refrigerant and is a function of evaporating temperature Te. The outlet refrigerant mass flow rate is the same as the compressor mass flow rate, which is dependent on the compressor speed, the low pressure P, and high
27、pressure P,. This mass flow rate can be expressed by ASHRAE Transactions: Research 329 where g(Pe,Pd can be identified for a given coinpressor. As said before, the pressure is an invariant function of the temper- ature for the two-phase section. Therefore, the outlet refriger- ant mass flow rate can
28、 be expressed as Therefore, Equation 5 can be written as where Based on Equations 1,2, and 5, the state space represen- tation for the low-order evaporator model is given below, where Te, I, and T, are the three states of the model, and Ta, min, and kaut are the inputs to the system. Equation 9 show
29、s that the evaporator dynamics are nonlinear. Specifically, the nonlinearity comes from the bilin- ear function of 1 and T,-Te. For the nonlinear feedback linearization control of HVAC systems (He and Asada 2003), the heat transfer rate of evap- orators must be estimated because no direct measuremen
30、t or estimation is available. Based on the knowledge of Te, I, and T, the heat transfer rate q can be calculated using q = nDiaiZ(TW- Te). (10) Because only Te can be easily measured using a thermo- couple, our task is to design an observer to estimate the value of the length of two-phase section 1
31、and heat transfer rate q. NONLINEAR OBSERVER DESIGN FOR EVAPORATOR In this section, we will discuss nonlinear observer design for an evaporator to estimate the length of the two-phase section and heat transfer rate q dynamically. The following dynamics of the nonlinear observer are proposed. Figure
32、2 = Observer structure. 7tDiai- x,. 1 . -Z(T, - Te) + -min kk - -mou, 4, - 2 (TepTe) I: where Fe, i, and ?, are dynamic estimations based on the proposed observer, and Te is the actual sensor measurement. L, L2, and L, are the observer gains. The first part of the observer utilizes the nonlinear mod
33、el for evaporator, and the second part is the feedback compensation based on the error between the measured evaporating temperature and the esti- mated value. The question is how we can guarantee that the estimated state variables converge to the actual states of the plant. The contraction theory pr
34、ovides a straightforward procedure for assuring convergence (Lohmiller and Slotine 1998). Accord- ing to the contraction theory, the system i = f(, t) is said to be contracting if f/x is uniformly negative definite. All system trajectories then converge exponentially to a single trajectory, with con
35、vergence rate krnox, where A, is the largest eigenvalue of the symmetric part of /ax. Therefore, if the actual states are particular solutions of the observer and the observer is contracting, then we can conclude that all the trajectories of the observer will converge to the actual states. Figure 2
36、shows the evaporator dynamics and the observer structure. From the proposed observer dynamics, we can see that if Fe is equal to Te, the observer dynamics are the same as the system dynamics. So the actual states that are the solutions of this set of equations are particular solutions of the observe
37、r. If the symmetric part of the Jacobian matrix of the observer dynamics is uniformly negative definite, then the trajectory of 330 ASHRAE Transactions: Research “ , . -O0 I -150 i- 15 0 c ._ z J c 8 10- this means the observed states are the same as the actual states. The Jacobian matrix for the ob
38、server system is as follows: where BI = (C,PA), B2 = nD,a, E3 = 7Di0.i B4 = PlP -!)A The symmetric part of the Jacobian matrix is as follows: According to contraction theory, if the eigenvalues of this matrix are all negative, then the system is contracting. By looking at the characteristic equation
39、 h3 + alh2 + a2h + a, = O, if al O and ala2 - a, 0, then the eigenvalues are all nega- tive. 500 - 400 - 300 2 200 5 I 100- 01 l o- 5 -100 81 t- ni i a -200 + -300 -400 - A -50!OwO0 O 100 200 300 400 500 I-estimate Figure 4 Convergence range for Ll=200, L2=30, L,=l. -5 I O 5 10 15 20 25 30 35 40 45
40、50 I-estimate Figure 5 Zoom plot of the convergence range for L,=200, L2=30, L,=l. By choosing the appropriate values of L, L2, L, we computed the range of I and - ?, such that alO and ala2 - a, O are satisfied as shown in Figure 3. It is concluded that the sufficient condition for alO and ala2 -, a
41、, O is LI = 200, c2 = 30, L3 = 1 by considering the operation space of i and Tw- Te as shown in Figure 4 and Figure 5. Therefore, LI =200, L2 =30, L3=l is a sufficient set of gains to ensure that the Jacobian matrix of the observer model is negative definite. Based on contraction theory, the observe
42、r states will converge to the actual states of the evaporator model because the observer dynamics have been designed in such a way that the original states of the model are the particular solu- tions of the observer model. The criterion of tuning of the observer gains is to make the Jacobian matrix
43、of the observer model negative definite. ASHRAE Transactions: Research 331 93 n 92 0 Figure 6 Diagram of a low-order condenser model. MODEL-BASED NONLINEAR OBSERVERS FOR CONDENSER Figure 6 shows the diagram of a low-order condenser The vapor refrigerant mass balance equation is given by model. where
44、 M, is the vapor mass of condenser and Vc is the vapor volume of condenser, and pg is the saturated vapor density of the refrigerant and is a function of condensing temperature Te. The inlet refrigerant mass flow rate is the same as the compressor mass flow rate. Tc is the condensing temperature, L,
45、 is the two-phase length, and T, is the wall temperature of the condenser. The tube wall energy balance equation for the two-phase section is dTw c icpP4cy = Di, $5, ,iTc - T, cl - nDo, cao,ciTw, c - Ta, c) (15) where Ta.c is the outdoor air temperature. The liquid mass balance equation in the conde
46、nser model is expressed in Equation 16. It is a little bit different from the evaporator model. Two sections have liquid refrigerant. One is the subcooled liquid phase section; the other is the two-phase section. (16) where Lc3 is the subcooled liquid phase length. It can be seen that there are four
47、 unknowns but three equa- tions. We have to get rid of one unknown for the observer design. One assumption is made here, that the length change of the superheated phase is relatively slow dLc 1 -o dt We have 332 Lc3 = L-Lc1-Lc2 Therefore, the liquid mass balance equation can be writ- ten as Equation
48、s 14, 15, and 18 are the condenser model. The model-based observer for the condenser is designed as follows: - Lc2 where LI, L, and L3 are observer gains. As mentioned in the previous section, the criterion of tuning the observer gains is to make the Jacobian matrix of the observer model negative de
49、finite. SIMULATION AND EXPERIMENTAL RESULTS The observer design was evaluated by a matlab simula- tion. For comparison, a standard linear observer is used for the linearized model of the evaporator. Around some operation point, the linearized model of the evaporator is given by where L, L2, and L3 are observer gains. As mentioned previ- ASHRAE Transactions: Research Emprating Temperature _-_ 9 t. Te-linearobserver i L1 O 5 10 15 20 25 30 31 Time) 135- 13 - F 125- pi fi l2 4 115- a * 11- P Two Phase Length _-_ ! I-nonlinearobser
