moving-boundary heat exchanger ......universityofillinoisaturbana-champaign, 158meb,mc-244,...

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Brian D. EldredgeUniversity of Illinois at Urbana-Champaign,

158 MEB, MC-244,1206 West Green Street,

Urbana, IL 61801e-mail: [email protected]

Bryan P. RasmussenTexas A&M University,

3123 TAMU,College Station, TX 77843-3123e-mail: [email protected]

Andrew G. AlleyneUniversity of Illinois at Urbana-Champaign,

158 MEB, MC-244,1206 West Green Street,

Urbana, IL 61801e-mail: [email protected]

Moving-Boundary Heat ExchangerModels With Variable OutletPhaseVapor compression cycle systems using accumulators and receivers inherently operate ator near a transition point involving changes of phase at the heat exchanger outlets. Thiswork introduces a condenser/receiver model and an evaporator/accumulator model de-veloped in the moving-boundary framework. These models use a novel extension ofphysical variable definitions to account for variations in refrigerant exit phase. System-level model validation results, which demonstrate the validity of the new models, arepresented. The model accuracy is improved by recognizing the sensitivity of the models torefrigerant mass flow rate. The approach developed and the validated models provide avaluable tool for dynamic analysis and control design for vapor compression cyclesystems. �DOI: 10.1115/1.2977466�

Keywords: thermo-fluid systems, dynamic models, vapor compression cycle

IntroductionControl engineers are often faced with the challenging task of

reating simplified dynamic models of complex physical systems.thermofluid system is a good example of a complex physical

ystem that requires an accurate yet simple dynamic model forontrol design purposes. In this work, we present a thermofluidystem dynamic model with specific application to vapor com-ression cycle �VCC� systems. New models are derived for heatxchangers used in conjunction with two common components:eceivers and accumulators. The modeling approach balances therade-offs between complexity and accuracy, arriving at a tool thats useful for both dynamic analysis and control design.

Modeling of VCC systems can be divided into two generalaradigms: finite difference �spatially dependent� models andoving-boundary models �1�. In the finite difference paradigm,

he conservation equations are approximated with a finite differ-nce technique and applied to a number of elements in the heatxchanger �2�. Each element contains its own dynamic states ands independent of fluid phase. As the number of elements in-reases, model accuracy increases as well. However, the increasedumber of elements results in a dynamically complex model thatay be computationally expensive and is unsuitable for model-

ased control design �3�.In the moving-boundary modeling approach, the heat exchang-

rs are divided into regions based on the fluid phase in each re-ion. Model parameters are lumped together in each region. Theocation of the boundary between regions is allowed to be a dy-amic variable, thereby capturing the essential two-phase flowynamics. The resulting models are of low dynamic order, makinghem very well suited for control design. The moving-boundary

odels give the model developer and control designer physicalnsight into the dynamic behavior of the plant. In comparison tohe finite difference models, the compact nature of the moving-oundary models is assumed to reduce accuracy, although no di-ect model validation comparison is available in the literature.rald and MacArthur showed that a spatially dependent model

nd a moving-boundary model for an evaporator have very similarynamic responses �4�.

Contributed by the Dynamic Systems, Measurement, and Control Division ofSME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON-

ROL. Manuscript received June 8, 2006; final manuscript received May 22, 2008;
ublished online September 24, 2008. Assoc. Editor: Huei Peng.
ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

nloaded 28 Oct 2011 to 165.91.141.18. Redistribution subject to ASM

Moving-boundary models have been under development since1979 �5�. A review of the literature shows that they have beenapplied to a variety of VCC systems with many variations in thedetails of the modeling approach �1�. The range of operating con-ditions covered by many system models �5–9� has been limited bythe lack of models for system components frequently used in prac-tice, such as receivers and accumulators. Under normal operatingconditions, heat exchangers used in conjunction with these com-ponents will experience changing fluid phase conditions at theheat exchanger outlet. In the established moving-boundary frame-work, these changing fluid phase conditions would result in thecreation and destruction of dynamic states. Previous works �5,8�used a model switching approach to accommodate multiple mod-eling frameworks in one simulation. A key contribution of Wil-latzen et al. �8� is a presentation of techniques for controllinginactive states. The evaporator model is given in �8� as an ex-ample of the approach. Four model structures are considered: asubcool/two-phase/superheat model, a subcool/two-phase model,a two-phase model, and a two-phase/superheat model. The modelsare implemented with explicit equations for the state derivatives,allowing for direct manipulation of the derivatives. Two guidingprinciples are presented for handling state derivatives when thephysical entities that they represent are not active in the model.First, if a state going to zero will adversely affect numerics, do notallow the state to reach zero. Second, force inactive states to tracka meaningful physical value. The value to be tracked is deter-mined by the desired state value at the time of its activation in themodel. An evaporator switching from a subcool/two-phase/superheat model to a two-phase/superheat model provides an ex-ample of the application of these principles. As the model struc-ture switch occurs, the subcooled region length goes to zero andthe subcooled region wall temperature state becomes inactive. Themodel structure switching is based on the inlet and outlet refrig-erant enthalpies. If, in the case of the evaporator, the inlet enthalpyis less than the saturated liquid enthalpy, the subcooled region isincluded. If the outlet enthalpy is greater than the saturated vaporenthalpy, the superheat region is included. No model validationresults are presented in �8�, but simulation results are presentedfor a number of model outputs.

The current work seeks to address the problem of multiple exitstates without using a switching or multiple model approach. Therest of the paper is organized as follows. The modeling challengeassociated with varying exit phase is addressed in Sec. 2 through
the first-principles derivation of unique integrated condenser/
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eceiver and evaporator/accumulator models. Section 2 expandshe range of operating conditions that may be modeled with the
oving-boundary framework. Section 3 provides model valida-ion results demonstrating the utility of the proposed approach.he accuracy of the model validation in Sec. 3 is enhanced by an

dentified dynamic model of the compressor mass flow. The pro-edure and the model for the compressor are detailed in the Ap-endix. A conclusion summarizes the main points of the paper.

Moving-Boundary Models With Variable Outlethase

2.1 Moving-Boundary Heat Exchanger Model Overview.he moving-boundary approach is based on the assumption ofne-dimensional fluid flow through a pipe with effective diameter,ffective flow length, and effective surface areas �5–9�. The ap-roach also assumes equal pressure throughout the heat ex-hanger. The heat exchanger is divided into regions based on theuid phase, and the effective parameters are lumped in each re-ion. The interface location between fluid phase regions is al-owed to be a dynamic variable. Model parameters in single phaseegions are the averages of the region’s inlet and outlet conditions.

odel parameters in the two-phase region are characterized by aean void fraction �10�. For two-phase refrigerant flowing

hrough a tube, the void fraction gives the ratio of area occupiedy vapor to the total area at a cross section of the tube. Becausewo-phase flow is characterized by saturated conditions, theumped parameters for the heat exchanger’s two-phase region maye found by taking an average of the saturated vapor and theaturated liquid conditions. The mean void fraction, as detailed inec. 2.4.8, is used as the weight in this parameter averaging.The moving-boundary derivation procedure requires the inte-

ration of the governing partial differential equations �PDEs�long the length of the heat exchanger to remove spatial depen-ence. The PDEs for conservation of refrigerant mass and energyre given in Eqs. �1� and �2�. The z-coordinate specifies a locationlong the length of the heat exchanger. The inlet of the heat ex-hanger is described by z=0 and the outlet by z=LT.

��Acs��t

+��m��z

= 0 �1�

��Acsh − AcsP��t

+��mh�

�z= pi�i�Tw − Tr� �2�

Equation �1� is the differential continuity equation for unidirec-ional flow in the z direction. Equation �2� is the differential con-ervation of energy equation for the refrigerant in the heat ex-hanger, which states that the rate of energy change in theefrigerant due to changes in enthalpy, pressure, and position mustqual the rate of heat transfer between the refrigerant and the heatxchanger wall.

The differential conservation of energy equation for the heatxchanger wall energy is given in Eq. �3�. This equation showshat the rate of energy change in the wall due to temperaturehange must equal the rate of heat transfer between the wall andoth the refrigerant and the surrounding air. The PDEs are inte-rated using the Leibniz integration rule given in Eq. �4�. Theonservation of momentum equation is neglected by assuming aniform pressure along the length of the heat exchanger �6–9�.

�Cp�A�w��Tw�

�t= pi�i�Tr − Tw� + po�o�Ta − Tw� �3�

�z1�t�

z2�t��f�z,t�

�tdz =

d

dt��z1�t�

z2�t�

f�z,t�dz� − f�z2�t�,t�d�z2�t��

dt

+ f�z1�t�,t�d�z1�t��

�4�
dt
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This modeling approach has been applied to both evaporatorsand condensers operating in subcritical and transcritical vaporcompression cycle systems �6,11�. The resulting models wereshown to be of low dynamic order and suitable for control designpurposes.

2.2 Receivers and Accumulators in the Moving-BoundaryFramework. A basic vapor compression cycle is composed offour primary components: evaporator, compressor, condenser, andexpansion valve, as shown in Fig. 1. A high side receiver and/orlow side accumulator is/are generally added to the system as ameans of storing excess refrigerant and ensuring safe operationover a large range of operating conditions. From the condenser,the refrigerant flows to a receiver where any excess charge isstored. The refrigerant leaving the evaporator flows to an accumu-lator where any two-phase fluid is captured, thus preventing dam-age to the compressor.

The main challenge presented by the presence of receivers andaccumulators is the possibility of multiple fluid phase conditionsat the heat exchanger outlet. At steady-state conditions, a con-denser with receiver will operate with saturated liquid or two-phase fluid at the outlet, as shown in Fig. 2. During transients, thecondenser outlet fluid condition could deviate to subcooled outletconditions, as shown in Fig. 3. Similarly, if an accumulator con-tains some liquid refrigerant, the evaporator will operate withsaturated vapor or two-phase fluid at the outlet. Transients couldcause the evaporator outlet fluid condition to deviate to super-heated vapor conditions.

The creation of a subcooled region in the condenser or a super-heated region in the evaporator presents the problem of the cre-ation of new dynamic states, as fluid region lengths and wall

Fig. 1 A basic vapor compression cycle with a receiver and anaccumulator

Fig. 2 Two possible steady-state operating conditions for a
condenser with receiver
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emperatures are states in the moving-boundary framework. Whenhe system transients die out, the models face the problem ofisappearing states. In the established moving-boundary frame-ork, multiple model structures would be needed to capture theynamics of heat exchangers operating with the possibility ofultiple fluid phase conditions at the outlet �8�. In the followingodel derivations, we propose an extended definition of quality to

apture the transient behavior of heat exchangers with receivers/ccumulators and avoid the complications of switching modeltructures during simulation. Using this approach, the governingquations of the model do not require direct manipulation duringimulation, regardless of the refrigerant phase at the heat ex-hanger outlet.

2.3 Receiver/Accumulator Model Derivation. High side re-eivers provide a volume for storing excess refrigerant charge andnsuring liquid flow into the expansion device. Low side accumu-ators protect compressors from “slugging” by capturing two-hase refrigerant and allowing only vapor to exit the accumulatornd enter the compressor. For modeling purposes, the two com-onents differ only in their inlet and outlet conditions. Theeceiver/accumulator is modeled as a simple control volume withefrigerant entering and exiting the control volume boundary. Heatransfer may occur between the walls of the receiver/accumulatornd the surroundings. For the receiver, the entering refrigerant isither two-phase, saturated liquid, or subcooled liquid, and thexiting refrigerant is assumed to be saturated liquid, as shown inig. 4. The receiver pressure is assumed to be the condenser pres-ure. For the accumulator, the entering refrigerant is either two-hase, saturated vapor, or superheated vapor, and the exiting fluids assumed to be saturated vapor, as shown in Fig. 5. The accu-

ulator pressure is assumed to be the evaporator pressure.The conservation of mass and energy for the receiver are writ-

en as Eqs. �5� and �6�. Equation �5� shows that the rate of masstorage in the receiver is simply the net mass flow rate into theeceiver. Equation �6� shows that the rate of change of refrigerantnergy stored in the receiver is based on the energy entering andeaving the receiver due to mass flow and the heat transfer withhe surroundings.

ig. 3 A possible transient operating condition for a con-enser with receiver

Fig. 4 Control volume for receiver model

Fig. 5 Control volume for accumulator model

ournal of Dynamic Systems, Measurement, and Control


dmrec

dt= min − mout �5�

d�mrecurec�dt

= minhin − mouthout + UA�Tamb − Trec� �6�

The total refrigerant mass in the receiver is the summation of theliquid and vapor mass, mrec=mg+mf. The total refrigerant energyis the summation of liquid and vapor energy, mrecurec=mgug+mfuf. The total volume of the receiver is the summation of liquidand vapor volumes, Vrec=Vg+Vf. The relationship between massand volume is defined as mg=�gVg and mf =� fVf. Using theserelationships, the volume of each phase may be found in terms ofthe total mass and total volume as Vg= �� fVrec−mrec� / �� f −�g� andVf = ��gVrec−mrec� / ��g−� f�. The net change in volume is zero,

Vg+ Vf =0. By applying the chain rule of differentiation to the lefthand side of Eq. �5�, we find this expression for the receiver massderivative:

dmrec

dt=

d��gVg + � fVf�dt

=d��g�

dtVg +

d�� f�dt

Vf + �gd�Vg�

dt+ � f

d�Vf�dt

= � ��g��Pc

Vg +�� f��Pc

Vf�Pc + ��g − � f�Vg �7�

Taking the same approach for the left hand side of Eq. �6� givesthe receiver energy derivative:

d�mrecurec�dt

=d��gugVg + � fufVf�

dt=

d��gug�dt

Vg +d�� fuf�

dtVf

+ �gugd�Vg�

dt+ � fuf

d�Vf�dt

= � ��gug��Pc

Vg +�� fuf�

�PcVf�Pc + ��gug − � fuf�Vg

�8�

Again, we assume the receiver/accumulator pressure to be equalto that of the upstream heat exchanger. Combining these resultsallows us to write the conservation of refrigerant energy equation

in terms of mrec and Pc, as shown in Eq. �9�. Thus Eqs. �5� and �9�form the governing differential equations for the receiver.

�� d�g

dPcVgug +

d� f

dPcVfuf +

dug

dPc�gVg +

duf

dPc� fVf − ��gug − � fuf

�g − � f

�� d�g

dPcVg +

d� f

dPcVf�Pc + ��gug − � fuf

�g − � f�mrec

= minhin − mouthout + UArec�Tamb − Trec� �9�

The equations for the accumulator are obtained by replacing thereceiver terms with accumulator terms and the condenser termswith evaporator terms.

2.4 Condenser With Receiver Model Derivation. Nomi-nally, the condenser is assumed to have two fluid regions �super-heat and two-phase�, as shown in Fig. 2. The derivation here fol-lows the method presented in �11�. The starting point is thegoverning PDEs for mass and energy, Eqs. �1� and �2�. We firstremove the spatial dependence from the PDEs by integratingalong the length of the pipe. The limits of integration depend onthe fluid region. The conservation of mass and conservation ofenergy equations are first applied to the superheat region and thento the two-phase region. The conservation of energy equation isalso applied to the pipe wall for each region.

2.4.1 Conservation of Mass: Superheat Region. Equation �1�for the conservation of mass is repeated here as the starting point
of the derivation:
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��Acs�
�t+

��m��z

= 0 �10�

ntegrate the first term of Eq. �10� from the inlet of the pipe �z0� to the end of the superheat region �z=L1� and assume a con-

tant cross-sectional area:

�0

L1 ��Acs��t

dz = Acs��0

L1 ��t

dz� �11�

pply Leibniz’s integration rule, Eq. �4�:

�0

L1 ��Acs��t

dz = Acs� d

dt��0

L1 ��t

dz − �gL1� �12�

ssume a lumped density �1 and evaluate the integral. Duringimulation this density will be calculated as the average of theondenser inlet density and the saturated vapor density at the con-enser pressure.

�0

L1 ��Acs��t

dz = Acs� d

dt��1L1� − �gL1� �13�

valuate the time derivative:

�0

L1 ��Acs��t

dz = Acs��1L1 + �1L1 − �gL1� �14�

ressure and enthalpy are chosen as independent variables, ande assume a lumped enthalpy �h1� that is an average of the en-

halpies at the limits of integration. The derivatives of the super-eat region enthalpy and density are given in Eqs. �16� and �17�.

h1 =hin + hg

2�15�

h1 =1

2�hin +

dhg

dPcPc �16�

�1 =��1

�PcPc +

��1

�h1h1 �17�

ubstitute Eq. �16� into Eq. �17� to obtain

�1 =��1

�PcPc +

��1

�h1�1

2�hin +

dhg

dPcPc� �18�

ubstitute Eq. �18� into Eq. �14�:

�0

L1 ��Acs��t

dz = Acs� ��1

�PcPc +

��1

�h1�1

2�hin +

dhg

dPcPc�L1

+ ��1 − �g�L1 �19�

implify to obtain

�0

L1 ��Acs��t

dz = � ��1

�Pc+

1

2

��1

�h1

�hg

�PcAcsL1Pc + �1

2

��1

�h1AcsL1hin

+ ��1 − �g�AcsL1 �20�

ntegrate the second term of Eq. �10�, defining the mass flow ratet the interface between the superheat region and the two-phaseegion as mint:

�0

L1 �m

�zdz = mint − min �21�

ombining Eqs. �20� and �21� gives the desired conservation of
ass equation for the superheat region.
61003-4 / Vol. 130, NOVEMBER 2008


� ��1

�Pc+

1

2

��1

�h1

dhg

dPcAcsL1Pc + �1

2

��1

�h1AcsL1hin + ��1 − �g�AcsL1

= min − mint �22�

2.4.2 Conservation of Energy: Superheat Region. Begin withEq. �2�, repeated here as

��Acsh��t

−��AcsPc�

�t+

��mh��z

= pi�i�Tw − Tr� �23�

Integrating the first term of Eq. �23� along the length of the su-perheat region and applying Eq. �4� yields

�0

L1 ��Acsh��t

dz =d

dt�0

L1

��Acsh�dz − �gAcshgL1 �24�

Again assuming a lumped density and enthalpy, evaluate the inte-gral and the time derivative.

�0


dz = Acs��1h1L1 + �1h1L1 + �1h1L1 − �ghgL1�

�25�

Substitute Eq. �18� and Eq. �16� into Eq. �25� and simplify toobtain the first term of Eq. �23�.

�0


dz = �� 1

�PcPc +

1

2

��1

�h1

�hg

�Pch1 +

1

2

�hg

�Pc�1AcsL1Pc

+ �1

2

��1

�h1h1 +

1

2�1AcsL1hin + ��1h1 − �ghg�AcsL1

�26�

The second term of Eq. �23� is integrated with the same limits.Applying Eq. �4� again and evaluating the integral yields

�0

L1 ��AcsPc��t

dz = Acs� d

dt�PcL1� − PcL1� �27�

Evaluate the time derivative and simplify

�0

L1 ��AcsPc��t

dz = AcsPcL1 �28�

Integrate the third term of Eq. �23�. At the interface between re-gions, the mass flow rate is mint, and the enthalpy is the saturatedvapor enthalpy.

�0

L1 ��mh��z

dz = minthg − minhin �29�

Now combine Eqs. �26�, �28�, and �29� with Eq. �23� and simplifyto find the final conservation of energy equation for the superheatregion.

�� 1

�Pc+

1

2

��1

�h1

dhg

dPch1 + �1

2

dhg

dPc�1 − 1�AcsL1Pc + ��1

2

��1

�h1h1

+ �1

2�1�AcsL1hin + ��1h1 − �ghg�AcsL1

= minhin − minthg + �i1Ai�L1

LT�Tw1 − Tr1� �30�

2.4.3 Conservation of Refrigerant Mass: Two-Phase Region.Integrate the first term of Eq. �10� from the beginning of thetwo-phase region �z=L1� to the end of the pipe �z=LT� and apply
Eq. �4�.


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�L1

LT ��Acs��t

dz = Acs� d

dt�L1

LT

��dz + �gL1� �31�

ssume that the lumped density can be evaluated as a function ofhe void fraction and evaluate the integral.

�L1

LT ��Acs��t

dz = Acs� d

dt�� f�1 − �� + �g��LT − L1�� + �gL1�

�32�

valuate the time derivative, treating enthalpies as functions ofressure.

�L1

LT ��Acs��t

dz = � d� f

dPc�1 − �� +

d�g

dPc��AcsPc�LT − L1� + ��g − � f�

��Acs�LT − L1� + ��g − � f�1 − �� − �g��AcsL1

�33�

ote that LT−L1=L2. Integrate the second term of Eq. �10� andombine the result with Eq. �33� to find the conservation of massquation for the two-phase region.

��g − � f��1 − ��AcsL1 + � d� f

dPc�1 − �� +

d�g

dPc��AcsL2Pc + ��g

− � f�AcsL2� = mint − mout �34�

2.4.4 Conservation of Refrigerant Energy: Two-Phase Region.eginning with Eq. �23�, integrate the first term and apply Eq. �4�.

�L1

LT ��Acsh��t

dz = Acs� d

dt�L1

LT

��h�dz + �ghgL1� �35�

ssume that density and enthalpy can be expressed in terms of theoid fraction and evaluate the integral.

�L1


dz = Acs� d

dt�� fhf�1 − ��L2 + �ghg��L2� + �ghgL1�

�36�

ssume that the enthalpy and density are functions of pressure.he time derivative is then evaluated.

�L1


dz = Acs�� f

�Pchf�1 − ��L2 +

�hf

�Pc� f�1 − ��L2

+��g

�Pchg�L2 +

�hg

�Pc�g�L2Pc + ��ghg − �ghg��

− � fhf�1 − ��L1 + ��ghgL2 − � fhfL2�� 37�

ntegrate the second term of Eq. �23� by applying Eq. �4� andssuming a constant pressure:

�L1

LT ��AcsPc��t

dz = Acs� d

dt�PcLT − PcL1� + PcL1� �38�

valuate the time derivative:

�L1

LT ��AcsPc��t

dz = AcsPc�LT − L1� �39�

valuate the final term of Eq. �23�:



�L1

LT ��mh��z

dz = mouthout − minthint �40�

Combine Eqs. �37�, �39�, and �40� to obtain the conservation ofenergy equation for the two-phase region.

��ghg − � fhf��1 − ��AcsL1 + ��ghg − � fhf�AcsL2�

+ �d�� fhf�dPc

�1 − �� +d��ghg�

dPc�� − 1AcsL2Pc

= minthg − mouthout + �i2Ai�L2

LT�Tw2 − Tr2� �41�

2.4.5 Conservation of Pipe Wall Energy: Superheat Region.Application of Eq. �3� to the superheat region wall gives

�CP�V�w�Tw1 +Tw1 − Tw2

L1L1 = �i1Ai�Tr1 − Tw1� + �oAo�Ta − Tw1�

�42�

2.4.6 Conservation of Pipe Wall Energy: Two-Phase Region.Application of Eq. �3� to the two-phase region wall gives

�CP�V�wTw2 = �i2Ai�Tr2 − Tw2� + �oAo�Ta − Tw2� �43�

2.4.7 Combination of Equations. Equations �22�, �30�, �34�,and �41� together with Eqs. �5� and �9� from the receiver deriva-tion are combined to eliminate the variables mint, mout of the con-denser, and min of the receiver, resulting in a model with sixstates: xc= �L1 Pc mrec � Tw1 Tw2�T. The model inputs are definedas uc= �mc,in mrec,out hc,in Tc,air,in mc,air Tamb�T. The resulting modelis given in Eq. �44� in descriptor form, Zc�xc ,uc� · xc= fc�xc ,uc�,with the elements of the Zc�xc ,uc� matrix found in Table 1.

z11 z12 0 0 0 0

z21 z22 z23 z24 0 0

z31 z32 z33 z34 0 0

0 z42 0 z44 0 0

z51 0 0 0 z55 0

0 0 0 0 0 z66

�L1

Pc

�

mrec

Tw1

Tw2

�=

mc,in�hc,in − hg� + �i1Ai�L1

LT�Tw1 − Tr1�

mrec,out�hg − hc,out� + �i2Ai�L2

LT�Tw2 − Tr2�

mc,in − mrec,out

mrec,out�hc,out − hf� + UArec�Tamb − Trec��i1Ai�Tr1 − Tw1� + �oAo�Ta − Tw1��i2Ai�Tr2 − Tw2� + �oAo�Ta − Tw2�

� �44�

In Eq. �44�, the time derivative of condenser inlet enthalpy issufficiently small so as to neglect it. This was found through nu-merical studies �11�. The model is simulated by calculating xc

=Zc−1fc at each time step and numerically integrating to find the

state vector xc. This model structure can also be linearized locallyto give a state space representation �xc=Acxc+Bcuc� suitable forcontroller design. An actual controller design is beyond the scopeof this paper, which focuses on modeling. The method of switch-ing model structures during simulation �8� requires an explicitequation for each state derivative, so Zc

−1fc must be evaluatedsymbolically. The model presented here is used throughout the
simulation without modification, regardless of the outlet phase of
NOVEMBER 2008, Vol. 130 / 061003-5


ts

Tdtctmorprmv

lqq

�tpc

Tm

z

z

z

z

z

z

z

z

z

z

z

z

z

z

z

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ESL-PA-08-11-01

he condenser. No structural changes take place during simulation,o explicit equations for the state derivatives are not needed.

2.4.8 Condenser Outlet Quality From Mean Void Fraction.he mean void fraction, �, was chosen in the derivation as aynamic state variable. The mean void fraction captures the rela-ionship between vapor mass and liquid mass in the heat ex-hanger and is used in the calculation of lumped parameters in thewo-phase region. We also use the mean void fraction to deter-

ine the outlet quality and hence the fluid properties at the outletf the condenser. A slip-ratio �S� �12� void fraction correlationelates the void fraction to the quality at any point in the twohase region, as shown in Eq. �45� where �s= ��g /� f�S. The slipatio is the ratio of the vapor velocity to the liquid velocity. Nu-erous methods have been introduced for determining suitable

alues of S. The particular method does not impact this derivation.

� =x

x + �1 − x��s�45�

From examination of Eq. �45�, it is readily apparent that theimits on void fraction are the same as the limits on quality. Asuality approaches zero, the void faction also approaches zero. Asuality approaches 1, the void fraction approaches 1 as well.

The mean void fraction, Eq. �46�, results from integrating Eq.45� from xin to xout and dividing by xout−xin. This equation relateshe mean void fraction to the inlet and outlet quality of the two-hase region. Once the outlet quality is obtained, all other outlet

able 1 Entries of Zc „xc ,uc… for the condenser with receiverodel

11 ��1h1−�1hg�Acs

12 �� 1

�Pc+

1

2

��1

�h1

dhg

dPc��h1−hg�+ �1

2

dhg

dPc��1−1�AcsL1

21 �� fhg−� fhf��1− ��Acs

22�d�� fhf�

dPc�1− ��+

d��ghg�

dPc�

−d� f

dPchg�1− ��−

d�g

dPchg�−1�AcsL2

23 �� fhg−� fhf�AcsL2

24 �hc,out−hg�

31 ��1−� f�1− ��−�g��Acs

32 �� d� f

dPc�1− ��+

d�g

dPc��L2+ � ��1

�Pc+

1

2

��1

�h1

dhg

dPc�L1�Acs

33 ��g−� f�AcsL2

34 1

42

d�g

dPcVgug+

d� f

dPcVfuf +mg

dug

dPc+mf

duf

dPc

−��gug−� fuf

�g−� f�� d�g

dPcVg+

d� f

dPcVf�

44 ��gug−� fuf

�g−� f−hc,out�

51 �Cp�V�w�Tw1−Tw2

L1�

55 �Cp�V�w

66 �Cp�V�w

onditions may be calculated.

61003-6 / Vol. 130, NOVEMBER 2008


� =1

�1 − �S�+

�S

�xout − xin��1 − �S�2

�ln� �S + xin�1 − �S��S + xin�1 − �S� + �xout − xin��1 − �S�� 46�

At steady-state, the inlet fluid of the condenser’s two-phaseregion is saturated vapor �xin=1� and the outlet fluid is saturatedliquid �xout=0� or two-phase liquid �0�xout�1�. We would liketo use Eq. �46� to obtain the outlet quality for small deviationsfrom saturated liquid conditions, but the equation is transcenden-tal in outlet quality. Assuming small deviations from saturatedliquid outlet conditions, we set xout=0 outside of the natural log toarrive at Eq. �47�. Solving for xout yields Eq. �48�, where a=�s / �1−�s� and b=1 / �1−�s�.

� �1

1 − �s−

�s

�1 − �s�2 ln� 1

�s + xout�1 − �s�� . �47�

xout = be��−b�/ab� − a �48�During simulation, some transients will cause the condenser’s

outlet fluid to deviate to subcooled conditions. When quality iscalculated from the mean void fraction, Eq. �48�, these deviationsresult in a quality value less than zero. We therefore define apseudoquality that exists outside the normal bounds on quality,x� �0,1�. This pseudoquality is defined as x� �0−� ,0�� 1,1+��, where 0��1. By using the two-phase property relation-ships with the pseudoquality, outlet refrigerant property approxi-mations that capture the gross property behavior are obtained. Theaccuracy of the pseudoquality approximation will be discussed inSec. 2.6.

2.5 Evaporator With Accumulator Model. When connectedto an accumulator containing fluid in the liquid phase, the evapo-rator outlet condition is saturated vapor �xout=1� or two-phasefluid �0�xout�1� at equilibrium. The derivation of the evaporatormodel follows the method presented in �6�, where the evaporatorhas a single two-phase region. This derivation is similar to thederivation of the condenser model and will not be presented indetail. The conservation of mass and conservation of energy equa-tions are

� d� f

dPe�1 − �� +

d�g

dPe��AcsLTPe + ��g − � f�AcsLT� = min − mint

�49�

�d�� fhf�dPe

�1 − �� +d��ghg�

dPe�� − 1AcsLTPe + ��ghg − � fhf�AcsLT�

= minhin − minthint + �iAi�Tw − Tr� �50�

The conservation of energy equation for the evaporator wall isgiven by

�Cp�V�wTw = �iAi�Tr − Tw� + �oAo�Ta − Tw� �51�Combining the evaporator equations with the equations for

the accumulator results in a fourth-order model withstates xe= �Pe macc � Tw�T and inputs ue

= �me,in macc,out he,in Te,air,in me,air Tamb�T. Writing the equations indescriptor form gives Eq. �52�. The entries of Ze�xe ,ue� are found
in Table 2.


AtEfva

Or

ucctdaffp

m

Tt

z

z

z

z

z

z

z

z

z

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ESL-PA-08-11-01

z11 z12 z13 0

z21 z22 z23 0

z31 z32 0 0

0 0 0 z44

�Pe

macc

�

Tw

�=

me,in − macc,out

me,inhe,in − macc,outhg + �iAi�Tw − Tr�macc,out�hg − he,out� − UAacc�Tacc − Tamb�

�iAi�Tr − Tw� + �oAo�Ta − Tw�� 52�

2.5.1 Evaporator Outlet Quality From Mean Void Fraction.s with the condenser with receiver model, the mean void frac-

ion is a dynamic state of the evaporator with accumulator model.quation �46� again provides the means of relating mean void

raction to quality. We assume small deviations from saturatedapor outlet conditions, setting xout=1 inside of the natural log torrive at Eq. �53�. Solving for xout yields Eq. �54�.

� �1

1 − �s+

�s

�xout − xin��1 − �s�2 ln��s + xin�1 − �s�� 53�

xout =�S ln��S + xin�1 − �S��1 − �S�2� − �1 − �S�

+ xin �54�

utlet fluid properties are calculated from the two-phase propertyelationships using either the quality or the pseudo-quality.

2.6 Validity of the Mean Void Fraction Assumptions. Bysing the pseudoquality, the models capture the essential heat ex-hanger outlet refrigerant property behavior without the compli-ations of switching model structures during simulation. This sec-ion addresses the validity of two key assumptions: the smalleviation assumption used to derive Eqs. �47� and �54� and thessumption that approximate refrigerant properties calculatedrom the pseudoquality capture the basic behavior of the true re-rigerant properties with sufficient accuracy for dynamic modelingurposes.

2.6.1 Small Deviation Assumption. Figure 6 compares the

able 2 Entries of Ze„xe ,ue… for the evaporator with accumula-or model

11 �� d� f

dPe��1− ��+ � d�g

dPe��AcsLT

12 1

13 ��g−� f�AcsLT

21 ��d�� fhf�

dPe��1− ��+ �d��ghg�

dPe��−1�AcsLT

22 he,out

23 ��ghg−� fhf�AcsLT

31

� d�g

dPeVgug+

d� f

dPeVfuf +

dug

dPe�gVg+

duf

dPe� fVf�

−��gug−� fuf

�g−� f�� d�g

dPeVg+

d� f

dPeVf�

32 ��gug−� fuf

�g−� f−he,out�

44 �Cp�V�w

ean void fraction of Eq. �46� with the mean void fraction result-



ing from application of the condenser and evaporator small outletquality deviation assumptions developed in Eqs. �47� and �54�. Inthe case of the condenser, deviations over the range xout� �0,0.2� result in errors less than 3.5%. In the case of the evapo-rator, deviations over the range xout� �0.8,1� result in errors lessthan 1%. Therefore, under these normal simulation conditions, thesmall deviation assumptions applied to the mean void fraction donot produce significant errors. The small deviation assumptionsshould not be applied to other operating conditions, such as star-tup and shutdown transients where large deviations in heat ex-changer outlet conditions are probable.

2.6.2 Refrigerant Property Calculation. If a subcooled regiondevelops in a condenser, the outlet enthalpy will decrease with theoutlet temperature. Transitioning from quality to pseudoquality�x�0� mimics the introduction of a subcooled region at the outletof the condenser. Rather than create a new subcooled region andthe associated states, the approximate outlet enthalpy is calculatedas if the refrigerant were two-phase: h=hf�1−x�+hgx. The cre-ation of a subcooled region in the real heat exchanger correspondsto an increase in liquid volume in the heat exchanger and there-fore a lower mean void fraction across the two-phase and sub-cooled regions. As the subcooled region grows longer, thepseudoquality becomes more negative and the outlet enthalpy de-creases. This decreasing pseudoquality results in a decreasing out-let enthalpy approximation, demonstrating that the properties cal-culated from the pseudoquality capture the key behavior of thetrue refrigerant properties. A similar argument could be made forthe case of superheated outlet conditions in the evaporator. Usingthe pseudoquality allows the models to capture the essential tran-sient behavior without using a model switching scheme.

For a numerical example, we consider the case of a condenseroperating with a saturated liquid outlet condition. Assuming a slipratio of 2 and a condenser pressure of 1000 kPa, the mean voidfraction is 0.84 as calculated from Eq. �46�. Using Eq. �48�, wefind that the outlet quality is approximately zero, which agreeswith our initial assumption of a saturated liquid outlet condition.If the model’s mean void fraction state increases above 0.84, theoutlet refrigerant of the condenser will be two-phase. Now weextend our example to the case of a condenser with a subcooledregion at the outlet. The concept of mean void fraction as a rela-tive measure of liquid and vapor in the heat exchanger is appliedto both the two-phase region and the subcooled region. With thesubcooled region included, the mean void fraction will be lessthan 0.84. If the mean void fraction state in our example drops to

Fig. 6 Mean void fraction for two-phase flows operating at900 kPa „condenser… and 320 kPa „evaporator… as a function ofthe outlet quality

0.8, the pseudoquality calculated from Eq. �48� for the condenser

NOVEMBER 2008, Vol. 130 / 061003-7


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utlet is −0.03. The negative pseudoquality value results in anutlet enthalpy value less than the saturated liquid enthalpy.

Model Validation Results

3.1 Model Library. The moving-boundary models describedn this paper are implemented in Thermosys, a MATLAB toolboxrst described in �13�. Thermosys contains linear and nonlinearodels of the basic components of VCC systems. The models of

ndividual components may be used independently or combinedith other components to create a complete system model. Whensystem is modeled, the inputs to each component model are

Table 3 Componen

Component Parameter

Condenser Heat exchanger massHeat exchanger specific heatHeat exchanger frontal areaHydraulic diameter of tubesInternal volumeInternal surface areaExternal surface areaTotal fluid flow lengthFlow cross sectional areaReceiver volume

Evaporator 2 Heat exchanger massHeat exchanger specific heatHeat exchanger frontal areaHydraulic diameter of tubesInternal volumeInternal surface areaExternal surface areaTotal fluid flow lengthFlow cross sectional areaAccumulator volume

EEV Rising slew rateFalling slew rateInput delay

Compressor VolumeRising slew rateFalling slew rate

Pipe Length Compressor to condenserCondenser to valveValve to evaporatorEvaporator to compressorHydraulic diameter

Fig. 7 Photograph of the experimental system

61003-8 / Vol. 130, NOVEMBER 2008


generally the outputs of other component models. For example,the condenser model inputs include the inlet and outlet refrigerantmass flow rates. These mass flow rates are outputs of the com-pressor and valve models. Further details on the Thermosys tool-box are found in �13�.

3.2 Experimental System. The experimental system used forvalidation is located at the Air Conditioning and RefrigerationCenter at the University of Illinois at Urbana-Champaign. Themain components are a 1

2 horsepower variable speed compressor,a tube and fin condenser, two tube and fin evaporators, and fourexpansion devices. The system is instrumented with pressuretransducers, mass flow meters, and thermocouples. For the modelvalidation results presented here, the system is used in the singleevaporator configuration with an electronic expansion valve. Thehigh side receiver and low side accumulator are included in the

hysical parameters

Value Units Comments

4.656 kg Calculated0.467 kJ/kg K Steel

0.0898 m2 From manufacturer0.008103 m Measured0055716 m3 From manufacturer

0.274993 m2 Calculated2.7927 m2 Calculated

10.6895 m Calculated0005156 m2 Calculated

.0007732 m3 Measured

2.7438 kg Calculated0.4877 kJ/kg K Aluminum and copper0.0584 m2 Calculated

.0081026 m From manufacturer0.000591 m3 From manufacturer

0.29166 m2 Calculated3.068 m2 Calculated

11.45794 m Calculated0005156 m2 Calculated

.0028665 m3 Measured

12.5 %/s From manufacturer−12.5 %/s From manufacturer

0.5 s Estimated

.0000304 m3 From manufacturer1000 rpm/s Estimated

−1000 rpm/s Estimated

4.04 m Measured4.79 m Measured1.63 m Measured6.53 m Measured

9.5 mm Estimated

Fig. 8 Schematic of the experimental system

t p

0.0

0.00

0

0.00

0



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ystem, necessitating the use of the new condenser with receiverodel. The low side accumulator contains only vapor, so the

vaporator with accumulator model is not needed. A photographf the system is shown in Fig. 7. A schematic of the system in theodel validation configuration is given in Fig. 8. The physical

arameters of the experimental system components are given inable 3. These parameters were obtained by measurement, bystimation, or from the component manufacturer. Operating con-itions for this model validation scenario are shown in Table 4.ompressor and expansion valve mass flow rates are calculated

rom empirical performance maps. The refrigerant-side heat trans-er coefficients are tuned parameters that remain the samehroughout the simulation. Air-side heat transfer coefficients arealculated from j-factor correlations �14� and are updated duringimulation.

3.3 Model Validation Results. The model validation test sce-ario consists of steps to each of the four controllable inputs:alve opening, compressor speed, evaporator air mass flow rate,nd condenser air mass flow rate, as summarized in Table 5. Thelots in Figs. 9–13 compare experimental data with various modelutputs. It should be noted that the system configuration used hereould not be accurately modeled without the condenser/receiver

Table 4 Model validation operating conditions

vaporator Units Value

ressure kPa 273.1nlet enthalpy kJ/kg 105.9utlet refrigerant temperature °C 18.25efrigerant mass flow rate kg/s 0.00713lip ratio 4.6nlet air temperature °C 23.98utlet air temperature °C 17.04ir mass flow rate kg/s 0.1568wo-phase HTC kW /m2 K 2uperheat HTC kW /m2 K 0.04ir-side HTC 2 �from j factor� kW /m2 K 0.058

ompressor Units Value

nlet pressure kPa 263.5utlet pressure kPa 975

nlet temperature °C 21.91utlet temperature °C 64.42efrigerant mass flow rate kg/s 0.00713ompressor speed rpm 1600

ondenser with receiver Units Value

ressure kPa 970nlet refrigerant temperature °C 59.65efrigerant mass flow rate kg/s 0.00713lip ratio 2.9nlet air temperature °C 25.67utlet air temperature °C 30.44ir mass flow rate kg/s 0.2938uperheat HTC kW /m2 K 0.3879wo-phase HTC kW /m2 K 1ir-side HTC 2 �from j factor� kW /m2 K 0.126eceiver refrigerant mass kg 0.8eceiver ambient temperature °C 27

lectronic expansion valve Units Value

nlet pressure kPa 975utlet pressure kPa 296.5

nlet temperature °C 37.13efrigerant mass flow rate kg/s 0.00713

nput signal % 13

odel presented in this paper. As encircled on Fig. 14, the con-



denser model’s outlet quality briefly deviates to pseudoquality fol-lowing the compressor speed step and the evaporator fan speedstep at approximately 1430 s and 2200 s, respectively. This resultindicates that the model predicted the formation of a subcooled

Table 5 Steps in system inputs

InputSteptime

Beforestep

Afterstep

% ofrange

Valve opencommand

1000 s 13 14.5 9.4%

Compressorrpm

1400 s 1600 rpm 1800 rpm 14.3%

Cond. airmass flow

1800 s 0.2938 kg /s 0.2858 kg /s 2.7%

Evap. airmass flow

2200 s 0.1568 kg /s 0.1363 kg /s 13.1%

Fig. 9 Condenser pressure

Fig. 10 Condenser air outlet temperature

Fig. 11 Evaporator pressure

Fig. 12 Evaporator air outlet temperature

Fig. 13 Evaporator superheat

NOVEMBER 2008, Vol. 130 / 061003-9


rms

4

udetkdmaptmbe

matsfo

A

Com

N

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ESL-PA-08-11-01

egion at the condenser outlet. By using the pseudoquality, theodel was able to capture this condition without interrupting the

imulation to switch model structures.

ConclusionsNew first-principles models are presented for heat exchangers

sed in conjunction with receivers and accumulators, including aetailed derivation of the condenser/receiver model. These modelsxpand the range of operating conditions that may be modeled inhe moving-boundary framework. By extending the definition of aey physical variable �quality�, the models are able to accommo-ate fluctuations in outlet fluid phase conditions without switchingodel structure during simulation, thereby avoiding the creation

nd destruction of dynamic states. Model validation results areresented for a complete system model. The modeled experimen-al system contains a receiver, necessitating the use of the new

odel presented here. The results demonstrate that the moving-oundary models are capable of representing systems with receiv-rs and accumulators.

The validation results indicate that the low-order nonlinearoving-boundary models are potentially a useful tool for dynamic

nalysis of VCC systems. The nonlinear models may be linearizedo provide models that are sufficiently accurate and compact toerve as tools for model-based control design. Future research willocus on further improvements to model accuracy and applicationf the models as a control design tool.

cknowledgmentThe authors wish to thank the sponsoring companies of the Air

onditioning and Refrigeration Center for their financial supportf this work. Mike Keir was instrumental in preparing the experi-ental system used to generate the model validation data.

omenclature

ymbolA area

Cp specific heatL length

Fig. 14 Quality and pseudoqualitystate

P pressure

61003-10 / Vol. 130, NOVEMBER 2008


Q heat transfer rateS slip ratioT temperature

UA product of area and overall heat transfercoefficient

V volumeZ descriptor form matrixe errorh enthalpym masssh superheat

t timeu internal energy, inputs vectorx quality, states vectorx pseudoqualityz spatial coordinate� void fraction� mean void fraction� heat transfer coefficient, weighting factor change� efficiency� density� angular velocity

Subscripts1 fluid region 12 fluid region 2a air

acc accumulatoramb ambient

c condensercs cross-sectionale evaporatorf saturated fluidg saturated vapori inner

in in to control volumeint fluid region interface

k compressork iteration

ulated from the mean void fraction
calc


A

1

vTflsnHpi

rpttdcpw�

Dp

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k–tot compressor, total
o outerout out of control volume

r refrigerantrec receiver

T totalvol volumetric

w wall

ppendix: Dynamic Compressor Mass Flow Models

Static Compressor Mass Flow ModelThe heat exchanger models presented in Sec. 2 are known to be

ery sensitive to inlet and outlet refrigerant mass flow rates �15�.o improve the modeling accuracy, a dynamic compressor massow rate model was incorporated and used for the validation re-ults of Sec. 3. The reader should note that this Appendix is notecessary for the correctness of the models presented in Sec. 2.owever, it was found to greatly impact the accuracy of the com-arison between simulation and data in Sec. 3. Therefore, it isncluded here for completeness.

The modeling of compressors in VCC systems can be catego-ized in two broad groups: simple static models �6,7,9� and com-lex dynamic models �5,16�. The simplified static models rely onime scale separation; the compressor dynamics are much fasterhan the heat exchanger dynamics, and therefore the compressorynamics are treated as if they were instantaneous �11�. A staticompressor model typically uses a semi-empirical modeling ap-roach. The compressor mass flow rate is given by Eq. �A1�,here Vk is the cylinder volume, �k is the compressor speed, and

k is the inlet refrigerant density �6�.

mk = �kVk�k�vol �A1�

uring simulation, the volumetric efficiency, �vol, is found from aerformance map generated through extensive compressor testing.

The performance map is generated by first solving Eq. �A1� forhe volumetric efficiency. By modulating the compressor speednd valve opening settings, the experimental system is run overhe feasible range of inlet and outlet compressor pressures. Thehanges in compressor speed and valve opening are sufficientlylow that each instance of the test can be considered a nearlyteady-state operating condition. Using measurements of mass

Fig. 15 A compressor volumetric efperimentally obtained data „black p
modeling the data


flow rate, compressor speed, and inlet density, the volumetric ef-ficiency is calculated. Performance map indices are chosen ascompressor speed and pressure ratio across the compressor. Anequation with these inputs is fitted to the volumetric efficiencydata with a least-squares linear regression. An example of a per-formance map generated in this fashion is shown in Fig. 15. Theperformance map, together with Eq. �A1�, constitutes a semi-empirical static compressor model.

2 Dynamic Mass Flow ModelThe first step in the identification of the dynamic mass flow rate

model is the isolation of the mass flow dynamics from a completesystem simulation. The static semi-empirical compressor modelpresented in Sec. A.1 provides the base line mass flow model. Aniterative learning control �ILC� algorithm �17,18� is used to iden-tify a mass flow rate correction. The ILC algorithm adjusts themass flow rate to reduce the error in one of the outputs, in thiscase the evaporator pressure. Figure 16 shows a schematic of theisolation procedure.

The base line static compressor model produces a mass flowrate prediction that is used to simulate a particular cycle. After thesimulation has finished, the evaporator pressure model output iscompared to evaporator pressure data to generate an evaporatorpressure error signal. This error signal is passed to the PD-typeILC algorithm given in Eq. �A2�, where P and D are tunable gainson the evaporator pressure error and pressure error derivative, k isthe iteration number, and trial is the data length.

m�t�correction,k = m�t�correction,k−1 + P · e�t�k−1

+ D · e�t�k−1, t � �0,trial� �A2�The ILC algorithm produces a feed-forward mass flow rate cor-rection signal that is applied to the mass flow rate predicted by thestatic compressor model during the next iteration of the simula-tion. Iterations continue until the rms evaporator pressure errorreaches an acceptably small value. At the final iteration, the massflow rate correction signal represents an isolation of the mass flowrate dynamics not captured by the static model.

After isolation of the mass flow dynamics, a model is identifiedto capture the dynamic behavior. First-, second-, and third-ordermodels were identified using standard techniques and compared.The model identification was carried out using the MATLAB Sys-

ency performance map showing ex-ts… and a least-squares fit surface

ficioin

NOVEMBER 2008, Vol. 130 / 061003-11


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Fp

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em Identification Toolbox. The model structure is the linear time-nvariant output error model described in �19�. A first order modelas chosen as a desirable trade-off between accuracy and simplic-

ty since its accuracy was similar to higher order models �15�. Theodel inputs are the deviations from the initial conditions of the

ompressor inlet pressure, outlet pressure, and speed. The massow correction model is of the form given in Eq. �A3�, wherePin

, GPout, and G�k

are the identified transfer functions de-ned in Eqs. �A4�–�A6�

mcorrection = �GPinGPout

G�k� Pin

Pout

�k� �A3�

GPin=

8.698 � 10−7s + 9.014 � 10−7

s + 0.07181�A4�

GPout=

− 4.893 � 10−8s − 4.923 � 10−8

s + 0.01232�A5�

ig. 16 Schematic of the mass flow dynamics isolationrocedure

61003-12 / Vol. 130, NOVEMBER 2008


G�k=

1.591 � 10−7s + 1.659 � 10−7

s + 0.08469�A6�

The total compressor mass flow is then given as

mk�tot = mk + mcorrection �A7�

References�1� Bendapudi, S., and Braun, J. E., 2002, “A Review of Literature on Dynamic

Models of Vapor Compression Equipment,” ASHRAE Report No. 4036-5.�2� MacArthur, J. W., and Grald, E. W., 1989, “Unsteady Compressible Two-

Phase Flow Model for Predicting Cyclic Heat Pump Performance and a Com-parison With Experimental Data,” Int. J. Refrig., 12�1�, pp. 29–41.

�3� Anderson, B. D. O., 1993, “Controller Design: Moving From Theory to Prac-tice,” IEEE Control Syst. Mag., 13�4�, pp. 16–25.

�4� Grald, E. W., and MacArthur, J. W., 1992, “A Moving-Boundary Formulationfor Modeling Time-Dependent Two-Phase Flows,” Int. J. Heat Fluid Flow,13�3�, pp. 266–272.

�5� Dhar, M., and Soedel, W., 1979, “Transient Analysis of a Vapor CompressionRefrigeration System,” Proceedings of the 15th International Congress of Re-frigeration, Venice, Sept. 23–29, Vol. 2, pp. 1031–1067.

�6� Rasmussen, B. P., and Alleyne, A. G., 2004, “Control-Oriented Modeling ofTranscritical Vapor Compression Systems,” ASME J. Dyn. Syst., Meas., Con-trol, 126, pp. 54–64.

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