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State Space Based Modeling andPerformance Evaluation of an Air-Conditioning SystemMahendra Kumar a , I.N. Kar b & Anjan Ray ba Department of Electrical Engineering, Indian Institute ofTechnology Delhi, Indiab Departments of Electrical Engineering and MechanicalEngineering, Indian Institute of Technology DelhiPublished online: 25 Feb 2011.

To cite this article: Mahendra Kumar , I.N. Kar & Anjan Ray (2008) State Space Based Modelingand Performance Evaluation of an Air-Conditioning System, HVAC&R Research, 14:5, 797-816

To link to this article: http://dx.doi.org/10.1080/10789669.2008.10391040

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VOLUME 14, NUMBER 5 HVAC&R RESEARCH SEPTEMBER 2008

797

State Space Based Modeling and Performance Evaluation of an Air-Conditioning System

Mahendra Kumar I.N. Kar, PhD Anjan Ray, PhD

Received October 15, 2007; accepted April 20, 2008

This paper presents a state space based multi-input multi-output model for a direct expansionair-conditioning system. The dynamic behavior of both refrigeration and air circuits was takeninto account in this model, and the condensation of moisture in air was considered in the modelto obtain a better representation of an actual system. The model can serve as a platform fordesigning and implementing different modern control strategies. A performance evaluationfunction in the form of internal variables was also designed and computed for ON-OFF control.Experimental validation was done on an air-conditioning system with ON-OFF control on anIndian Railway passenger coach. The transient and steady-state responses for dry-bulb temper-ature and specific humidity of inside air were obtained from the model and an experiment. Thesewere compared and found to be in good agreement.

INTRODUCTIONThe control of an air-conditioning system plays a major role in its performance and is one of

the most challenging problems in the field of process control. The challenge in controllingair-conditioning systems lies in achieving maximum thermal comfort at minimum energy con-sumption. The design of an efficient air-conditioning controller that maintains a good thermalenvironment in a given space at minimum energy consumption largely depends on the availabil-ity of an accurate dynamic model of the system. Modeling of refrigeration and air-conditioning systems has been the subject of many studies over the last few decades (Wu andShiming 2006; Wang and Jin 2000; Shiming 2000; Yiu and Wang 2007). Several researchershave contributed to the development of modeling the dynamics of air-conditioning systems forcontrol purposes. He et al. (1995, 1998) presented a state space multi-input multi-output(MIMO) dynamic model for a vapor-compression system. In their research, only the refrigera-tion circuit was considered and the effect of humidity in air on a dynamic model was not consid-ered. Browne and Bansal (2002) gave a dynamic component-based model forvapor-compression liquid chillers for predicting performance and on-line fault detection anddiagnosis. Huang et al. (2006) and Arguello-Serrano and Velez-Reyes (1999) derived a dynamicmodel for an air circuit, but the effect of condensation of moisture in air was not considered intheir studies. Most of the work in the literature presented dynamic models for refrigeration cir-cuits or air circuits separately. Wu and Shiming (2006) and Shiming (2000) presented a compo-nent-based dynamic mathematical model that takes into account the behaviors of a directexpansion (DX) refrigeration plant, i.e., the refrigeration circuit and the variable-air-volume(VAV) air distribution subsystem simultaneously. The model is component-based, and all theinternal variables of the system that contribute to the performance of an air-conditioning systemwere not analytically expressed. The variables (length of two-phase section in the evaporator

Mahendra Kumar is a research student in the Department of Electrical Engineering, Indian Institute of Technology Del-hi, India, on study leave from Indian Railway. I.N. Kar and Anjan Ray are associate professors in the Departments ofElectrical Engineering and Mechanical Engineering, respectively, at Indian Institute of Technology Delhi.

©2008, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC&R Research,Vol. 14, No. 5 (September 2008). For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is notpermitted without ASHRAE's prior written permission.

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and condenser, evaporator and condenser pressure, evaporator temperature, duct air temperatureand humidity, etc.) were not modeled in their work. Hence, it is difficult to use the Wu andShiming (2006) and Shiming (2000) models for feedback control design.

In this paper, a state space based lumped parameter dynamic model is proposed for represent-ing a DX air-conditioning system for control purposes. This model represents a full air-conditioning system, i.e., the refrigeration subsystem or circuit as well as the air subsystem orcircuit simultaneously. The cross-coupling between the refrigeration circuit and the air circuitwas also considered. The proposed state space model considers twelve state variables, four reg-ulated variables, four output variables, and five disturbance input variables. The state variables,such as temperature and humidity of air inside thermal space, have a direct effect on thermalcomfort. Thus, all the state variables that contribute to energy consumption and maintaininggood thermal comfort are considered. Some of these state variables cannot be measured directly,but with the help of this model the behavior of these variables can be estimated by designing anappropriate observer (like extended Kalman filter-based prediction). For example, the length ofthe two-phase section in the evaporator cannot be measured, but its behavior can be obtainedusing the proposed model.

The effect of condensation of moisture in air was considered in this model for calculatinghumidity of duct air and the temperature of the evaporator wall and duct air. If the temperatureof the air passing through the evaporator goes below the dew-point temperature, then condensa-tion of moisture in air takes place. The heat energy released due to condensation of moisture inair, specifically for hot, humid climates like most of India, is very substantial. Hence, the con-densation of moisture in air must be considered in dynamic modeling of an air-conditioning sys-tem for humid climates to obtain better mathematical representation of an actual system.

In this paper, modeling has been done also for an air-conditioning system with ON-OFF con-trol. Here, ON-OFF control implies the air-conditioning system remains on (blower, condenser,and compressor motor are on) until the temperature of the conditioned space is greater than thepreset temperature and the system turns off (blower motor on and condenser and compressormotor off) when the temperature of the conditioned space is less than the preset temperature.When the air conditioner is off, then there is fast change in the air temperature and air humidityof a conditioned space. Feedback control is implemented on the basis of air temperature andhumidity measurements. The timing of switching on and off an air-conditioning system can beoptimized in terms of energy consumption and thermal comfort with knowledge of an ON andOFF model.

The air-conditioning system of a passenger coach of Indian Railway was used as a case studyand for experimental validation of the proposed model. The design of a controller for mobile airconditioning, like air conditioning of a passenger railway coach, is more challenging comparedto air conditioning of a stationary installation due to the complicating influences of highly vary-ing thermal load, non-uniform air distribution, and reliability of air-conditioning equipment. Thethermal load on mobile air-conditioning systems varies with changes in direction and location ofthe moving vehicle due to the change of exposure to the sun and due to varying wind effects.The ambient conditions also change more frequently in mobile air-conditioning systems than instationary ones, which also leads to varying thermal load. The conditioned air speed distributionis not uniform due to partitions and different air speed obstruction materials inside the railwaycoach. The non-uniform air distribution gives rise to variations in temperatures in the thermalspace. The efficiency of an air-conditioning controller and energy consumption are directlyrelated, i.e., if the efficiency of the controller is good then energy consumption will be less andthermal comfort will be better.

The performance measure of any air-conditioning system depends on two factors: thermalcomfort and energy consumption. So, the performance evaluation function in this paper was

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designed by taking into account the thermal comfort and energy consumption. Thermal comfortwas calculated in terms of a well-known index, Fanger’s (1970) predicted mean vote (PMV).The second part of performance evaluation function is energy consumption. In this part, energyconsumed by a compressor motor, blower motor, and condenser motor were considered. Pres-ently, the most widely used control in the industry is ON-OFF control, so its performance was ana-lyzed by computing the proposed evaluation function evaluated for ON-OFF control of anair-conditioning system on a railway coach. This function will characterize thermal comfort andenergy consumption of the air-conditioning system. This ON-OFF control can be taken as a refer-ence for designing better control strategies, thereby leading to improved performance indices.

The main contributions of this paper are as follows:

1. A MIMO state space based dynamic model is proposed for a full air-conditioning system,i.e., the refrigeration circuit as well as the air circuit.

2. The proposed state space model includes the dynamic effect of moisture condensation in air,which has not previously been reported in the literature.

3. The proposed model was validated through simulation and experimental studies. The air-conditioning unit of a railway coach was taken as a case study for simulation and experimen-tal validation.

4. A performance evaluation function composed of parts representing thermal comfort andenergy consumption is proposed.

SYSTEM DESCRIPTION AND OPERATION The air-conditioning system can be divided into two main subsystems or circuits:

a. a refrigeration circuit and b. an air circuit or a VAV air distribution subsystem.

The refrigeration circuit, or DX refrigeration plant, is composed of an evaporator coil, a con-denser coil, a compressor, and an expansion valve. The air circuit is composed of a circulating orblower fan, thermal space (passenger area), connecting ductwork, dampers, and a mixing aircomponent.

The refrigeration circuit shown in Figure 1 is a vapor-compression refrigeration system andconsists of a cooling coil (evaporator), a compressor, a condenser, and an expansion valve.

The basic operation for the air circuit is as follows: fresh air is allowed in from outsidethrough a filter and gets mixed with the return air. The ratio of fresh air to return air, which isgenerally 1:3, can be varied using dampers, as shown in Figure 1. The mixed air enters the evap-orator coil, where it loses heat and gets conditioned. Conditioned air enters the duct; this air isknown as supply air. Supply air enters the thermal space through air distributors or grilles to off-set sensible and latent heat. Finally, air from the thermal space is drawn through a fan, where75% gets recirculated and the rest is exhausted from the system. The cooling of air is done by anevaporator coil, which is a part of the refrigeration circuit.

Most industrial air-conditioning systems in operation are controlled using ON-OFF controllers,which switch the compressor on or off according to the air temperature of the thermal space.

DYNAMIC MATHEMATICAL MODELINGA dynamic model was developed based on principles of mass and energy conservation. The

physical phenomena occurring during the lowering and increasing of temperature of air andrefrigerant were also considered. This model is of lumped parameter type and consists of a set offirst-order ordinary differential equations derived for all internal variables of the system. These

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variables contribute directly or indirectly to the thermal comfort and energy consumption of thesystem. A basic energy-conservation equation for a system can be written as: energy stored insystem (Δs) = energy supplied to system (Hsu) + energy generated within system (Hg) – energydissipated from system (Hd):

where m is the mass of fluid, Cp is the specific heat of fluid, and dT/dt is the change of tempera-ture with respect to time. The following assumptions were made to derive this model:

a. Only ON and OFF conditions of the plant are considered.b. All heat exchangers are considered to be long, thin, horizontal tubes.c. Refrigerant flow through the heat exchanger tube can be modeled as a one-dimensional fluid

flow.d. Axial heat conduction is negligible.e. Refrigerant is distributed uniformly between pipes.f. Thermal resistance of pipe walls can be neglected.g. The fluid consists of a liquid and a gas phase, which are ideally mixed and of equal velocity

in every cross section.h. The two phases are in thermodynamic equilibrium, and isentropic process only is assumed. i. Coefficients of heat transfer and of pressure drop correspond to the homogeneous two-phase

case considered.j. System time delay has not been considered.k. Ideal gas behavior and perfect gas mixing have been assumed.l. Negligible transient effects in the flow splitter and mixer have been assumed.m. Thermal load and moisture load of the system are assumed constant.n. The temperature of air at the exit of the evaporator is assumed to be spatially uniform.

Figure 1. Air-conditioning system.

Δs Hsu Hg Hd–+ mCpdT dt⁄= =

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o. The thermal storage due to furniture, curtains, carpet, etc. inside the conditioned space isassumed to be negligible. The change in moisture level inside the conditioned space due toroom surface material is also assumed to be negligible.

Refrigeration Circuit ModelingIn this section, a model for a vapor-compression based refrigeration circuit is presented for ON

and OFF processes. When the compressor is turned off, the presented model assumes that therefrigerant mass flow rate drops to zero instantaneously. There are significant changes in proper-ties of air-conditioning systems, such as heat transfer coefficients, the length of the liquid phase,and the pressure in the evaporator and condenser, when the compressor is switched off.

The refrigeration circuit consists of two components: the evaporator and the condenser, whichare described in the following sections.

Evaporator (Cooling Coil). The evaporator is of the louver-fin type, and counter-flow heatexchange between refrigerant and air is assumed. In this section, a differential equation involv-ing the internal variables of the evaporator has been derived. Figure 2 shows a schematic of anevaporator with four state variables: liquid-phase length in evaporator le, evaporator wall tem-perature Twe, evaporator refrigerant temperature Te, and evaporator pressure Pe.

Liquid-phase length in evaporator. The evaporator refrigerant side is divided into tworegions: a two-phase region and a superheated region, as shown in Figure 2. The two-phaseregion can be further divided into liquid refrigerant and vapor refrigerant. During the ON pro-cess, the rate of change of the liquid phase length in the evaporator is given by He et al. (1998):

(1)

During the OFF period of the plant, the following changes are affected in Equation 1:

a. = 0.b. The values ρle, γe, hlge, and αie are taken for the plant OFF condition.

Figure 2. Schematic of evaporator model.

ρlehlgeAe 1 γe–( )dle dt⁄ m· v hie hge–( )– αieπDiele Twe Te–( )–=

m· v

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802 HVAC&R RESEARCH

The length of the liquid phase in the evaporator during the plant OFF condition in steady statewill become zero.

Evaporator wall temperature. To derive an expression for evaporator wall temperature, con-densation of moisture in air was considered. If the temperature of the evaporator wall becomeslower than the dew-point temperature of air, then condensation of moisture in air takes place.The latent heat of vaporization of water is released due to formation of liquid water from watervapor. The heat generated due to latent heat of vaporization is absorbed by the refrigerantthrough evaporator tubes. The amount of heat released will be directly proportional to the massof moisture condensed. Thus, there will be two different expressions for evaporator wall temper-ature depending on the following two possibilities:

a. Evaporator wall temperature is less than the dew-point temperature of air.b. Evaporator wall temperature is greater than the dew-point temperature of air.

The rate of change of the evaporator wall temperature during the plant ON state and when thesupply air temperature is less than the dew-point temperature is given by

. (2)

The first term on the right-hand side of Equation 2 corresponds to cooling input by refrigerantor sensible heat absorbed by the system, which results in an increase in temperature from Te toTwe. The second term corresponds to the heat transfer rate from supply air to tube wall. The thirdterm corresponds to condensation of water vapor on the evaporator coil. This term is the productof mass of water vapor condensed and latent heat of vaporization released, sensible heatabsorbed by condensed water being subtracted from the above product. The bypass factor of theevaporator was taken into account by multiplying the second and third terms by (1 – b) becausebypassed air will not contribute to the change of the evaporator wall temperature.

The mass of condensed water vapor is calculated by taking the difference between the abso-lute humidity of incoming air and the outgoing air to the evaporator. The absolute humidity ofoutgoing air from the cooling coil is calculated from a psychrometric chart with the help of thecurve-fitting method. The term on the left-hand side of Equation 2 corresponds to the net heatsupply rate in the evaporator wall section.

During plant OFF condition, Equation 2 is modified by changing the value of αie and αoe toOFF condition values without any change in the rest of the quantities.

The rate of change of the evaporator wall temperature during plant ON and supply air temper-ature is more than dew-point temperature and is given by

. (3)

During plant OFF condition, Equation 3 is modified by changing the value of αie and αoe toOFF condition values; the rest of the expression will remain the same.

Evaporator refrigerant temperature. The rate of change in evaporator refrigerant tempera-ture with time for constant evaporator refrigerant mass during the ON process can be given byBrowne and Bansal (2002):

(4)

MeCe( )dTwe dt⁄ αieAei Te Twe–( ) 1 b–( ) 1 ξ–( )αoeAeo T3 Twe–( ) ξαoeAeoT0 Twe–+( )+=

1 b–( )vρa hfg CpwTwe–( ) ξW0 1 ξ–( )W3 0.001 0.0214Twe2 0.1177Twe 4.15502+ +( )–+( )+

MeCe( )dTwe dt⁄ αieAei Te Twe–( ) 1 b–( ) 1 ξ–( )αoeAeo T3 Twe–( ) ξαoeAeo T0 Twe–( )+( )+=

MeCe( )dTe dt⁄ m· r– hge hie–( ) αieAei Twe Te–( )+=

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During the OFF process, the following changes are affected in Equation 4:

a. = 0.b. The value of αie is taken for plant OFF condition.

Evaporator pressure. The rate of change of evaporator pressure during the ON process can begiven as (He et al. 1998):

(5)

(6)

During the OFF process, the following changes are to be made in Equation 5:

a. and = 0. b. The values of hlge, αie, and (dρge/dPe) are taken for plant OFF condition.

Condenser. The condenser is air cooled and assumed to be of plate-fin tube type. Accordingto the state of the refrigerant, the condenser is divided into three regions: a desuperheatingregion, a two-phase region, and a subcooling region. The two-phase region is further dividedinto a liquid phase and a vapor phase. In this section, state variables of the condenser arederived. Figure 3 shows a schematic of a condenser with four state variables: liquid-phase lengthin condenser lc, condenser wall temperature Twc, condenser refrigerant temperature Tc, and con-denser pressure Pc.

Liquid-phase length in condenser. The rate of change of the liquid-phase length during theON process in the condenser is given by He et al. (1998):

(7)

m· r

AeLe dρge dPe⁄( ) dPe dt⁄( ) m· v hie hge–( ) hlge αieπDiele Twe Te–( ) hlge m· com–⁄+⁄=

m· com ωcρcVc 1 c c Pd Ps⁄( )1 n⁄–+( )=

m· v m· com

Figure 3. Schematic of condenser model.

ρlchlgcAc 1 γc–( )dlc dt( )⁄ m· com hlc hic–( ) αicπDiclc Tc Twc–( )+=

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804 HVAC&R RESEARCH

During the OFF process, the following changes are to be made in Equation 7:

a. = 0.b. The values ρlc, γc, hlgc, and αic are taken for plant OFF condition.

Condenser wall temperature. The condenser wall temperature when the plant is on is givenby Browne and Bansal (2001):

(8)

During plant OFF condition, Equation 8 is modified by changing the value of αic and αoc toOFF condition values; the remainder of the expression remains unaltered.

Condenser refrigerant temperature. The condenser refrigerant temperature when the plant ison is given by Browne and Bansal (2001):

(9)

During the OFF process, the following changes are required in Equation 9:

a. = 0. b. The value of αic is taken for plant OFF condition.

Condenser pressure. The rate of change of the condenser pressure during the ON process isgiven by He et al. (1998):

(10)

During the OFF process, the following changes are affected in Equation 10:

a. = 0. b. The values of hlgc, αic, and (dρgc/dPc) are taken for plant OFF condition.

Air-Handling Circuit It is assumed that both indoor temperature and moisture content are the same throughout the

whole conditioned space, i.e., that these quantities are spatially uniform.Supply Air Temperature. The rate of change of supply air temperature with respect to time

has two expressions analogous to evaporator wall temperature. The first expression is when theevaporator wall temperature is less than the dew-point temperature. In this case, condensation ofwater vapor takes place and the expression is given by

.

(11)

The first term on the right-hand side of Equation 11 corresponds to the sensible heat transferrate from the evaporator wall at temperature Twe to the duct air at temperature Tae. The multiply-ing factor (1 – b) corresponds to the bypass factor of the evaporator. The second term corre-sponds to the heat transfer rate due to air bypassed through the evaporator coil; this term is thesum of fresh air sensible heat gain and return air sensible heat gain. The third term is due to the

m· com

McCc( )dTwc dt⁄ α icAic Tc Twc–( ) αocAoc T0 Twc–( )+=

McCc( )dTc dt⁄ m· r hic hoc–( ) αicAic Tc Twc–( )–=

m· r

AcLc dρgc dPc⁄( ) dPc dt⁄( ) m· com αicπDiclc Tc Twc–( ) hlgc⁄–=

m· com

CpρaVd( )dTae dt⁄ 1 b–( )vρaCp Twe Tae–( ) b 1 ξ–( )vρaCp T3 Tae–( ) ξvρaCp T0 Tae–( )+( )+=

1 b–( )vρahfg 0.001 0.0214Twe2 0.1177Twe 4.15502+ +( ) Ws–( )– bvρahfg ξW0 1 ξ–( )W3 Ws–+( )–

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difference in latent heat of incoming air and outgoing air; if there is an increase in latent heat,then correspondingly sensible heat gain decreases, and again the multiplying factor (1 – b) cor-responds to the bypass factor of the evaporator. The humidity of the incoming air is calculatedfrom a psychrometric chart with the help of the curve-fitting method. The fourth term corre-sponds to the difference in the latent heat of the bypassed incoming air and the latent heat of theoutgoing air. The bypassed incoming air has two components: one corresponding to fresh airand the second corresponding to return air. If there is an increase in latent heat, then correspond-ingly sensible heat gain decreases, and the multiplying factor b corresponds to the bypass factorof the evaporator. The left-hand side of Equation 11 gives the rate of change of supply air tem-perature with time and is equal to the net cooling available for reducing the supply air tempera-ture. The supply air temperature changes with respect to time when the evaporator walltemperature is more than the dew-point temperature and is given by

. (12)

Coach or Thermal Space Temperature. The rate of change of the coach temperature isgiven by

. (13)

The first term on the right-hand side of Equation 13 corresponds to the cooling input rate orthe heat absorption rate by the thermal space from temperature Tae to T3. The second term cor-responds to the sensible heat load, which has two terms corresponding to the thermal load fromthe inside as well as the moisture load. Here, qi is the sensible heat load generated inside theconditioned space. The sources of qi are electrical appliances, occupants, etc. This load is notdependent upon ambient conditions. M0 is the moisture load due to occupants. This load is nota strong function of the room conditions for the range of room conditions considered in thissimulation. The third term, qo, corresponds to the sensible heat gain through sealed windows,the roof, side walls, end partitions, and the floor. It is variable and depends upon many factorssuch as the difference between outdoor air temperature and indoor air temperature, solar radi-ant intensity, heat transfer coefficient, surface area, etc. Obtaining the correct estimation of qois a tedious task. Some researchers (Yao et al. 2006) have tried to solve this problem using softcomputing techniques such as neural networks. In the research reported in this paper, an esti-mated representative value of qo was used. The fourth term corresponds to the difference inlatent heat associated with incoming air and outgoing air; if there is an increase in latent heatthen, correspondingly, sensible heat gain decreases. The term on the left-hand side gives thenet heat rate to the thermal space.

Specific Humidity of Supply Air. The rate of change of specific humidity of the supply airwith respect to time has two expressions similar to evaporator wall temperature and supply airtemperature, depending upon whether condensation of water vapor takes place or not. When theevaporator wall temperature is less than the dew-point temperature, then condensation of watervapor takes place. The expression for supply air humidity using the conservation of mass princi-ple in this case is given by

. (14)

CpρaVd( )dTae dt⁄ 1 b–( )vρaCp Twe Tae–( )=

b 1 ξ–( )vρaCp T3 Tae–( ) ξvρaCp T0 Tae–( )+( ) vρahfg ξW0 1 ξ–( )W3 Ws–+( )–+

CpρaVs( )dT3 dt⁄ vρaCp Tae T3–( ) 1 ξ⁄( ) qi hfgM0–( ) qo vhfgρa Ws W3–( )–+ +[ ]=

ρaVd( )dWs dt⁄ 1 b–( )vρa 0.001 0.0214Twe2 0.1177Twe 4.15502+ +( ) Ws–( )( )=

bvρa 1 ξ–( )W3 ξW0 Ws–+( )+

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The first term on the right-hand side of Equation 14 indicates the mass of moisture in theincoming air, which comes in contact with the evaporator wall, and the mass of moisture in theoutgoing air. The humidity of the incoming air is calculated from a psychrometric chart with thehelp of the second-order curve-fitting method. The multiplying factor, (1 – b), corresponds tothe bypass factor of the evaporator. The second term amounts to mass of moisture in the incom-ing air bypassed through the evaporator and the mass of moisture in the outgoing air. The multi-plying factor b corresponds to the bypass factor of the evaporator. The rate of change of specifichumidity of supply air when the evaporator wall temperature is more than the dew-point temper-ature is given by

. (15)

The first two terms on the right-hand side of Equation 15 correspond to absolute or specifichumidity of incoming air, and the third term corresponds to outgoing supply air humidity.

Specific Humidity of Thermal Space. The rate of change of the specific humidity of thethermal space is given by

. (16)

The first term on the right-hand side of Equation 16 corresponds to the change in humiditydue to the supply air humidity Ws, and the second term on the right-hand side corresponds tomoisture load.

The superheat of refrigerant at the outlet of the evaporator can be expressed as He et al.(1998):

(17)

PERFORMANCE EVALUATION FUNCTION

The performance of any air-conditioning system depends on two factors:

a. thermal comfort b. energy consumption

Thermal Comfort

The most important variables that influence the condition of thermal comfort are air tempera-ture, water vapor pressure in ambient air, relative air velocity, mean radiant temperature, activitylevel (heat production in the body), thermal resistance of the clothing, and purity of the air.Fanger (1970) defined a thermal sensation index by considering air temperature, water vaporpressure in ambient air, relative air velocity, mean radiant temperature, activity level, and ther-mal resistance of the clothing. PMV is the thermal sensation index given by Fanger (1970) andis internationally standardized. It is easy to use PMV when controlling an air-conditioning sys-tem because PMV is an index of human thermal sensation with one value for all seasons and iseffective for all temperatures that are neither extremely high nor extremely low. PMV rangesfrom –3 (cold) to +3 (hot). When PMV is zero, then thermal sensation will be neutral, i.e., nei-ther cold nor hot. Broadly speaking, when PMV approaches zero then thermal comfortincreases.

ρaVd( )dWs dt⁄ vρa 1 ξ–( )W3 ξW0 Ws–+( )=

ρaVs( )dW3 dt⁄ vρa Ws W3–( ) M0+=

SH Tae Te–( ) 1 exp αoeπDie– Le le–( ) Cpm· com⁄( )–[ ]=

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The performance evaluation function corresponding to thermal comfort is defined as

. (18)

PMV can be computed from the equations suggested by Fanger (1970).Energy Consumption. There are three equipment components that consume electrical

energy in an air-conditioning system: the compressor motor, the blower motor or evaporatormotor, and the condenser fan motor.

The performance evaluation function corresponding to energy consumption is defined as

. (19)

The power consumed by the compressor is , the power consumed by the evaporatormotor is , the power consumed by the condenser fan motor is , and is a designedvalue of total power consumed by the compressor, evaporator and condenser fan motors. Theperformance evaluation function J is given by

. (20)

The variables λ1 and λ2 are weighting factors; they lie between 0 and 1. When thermal com-fort and energy consumption are given equal weights in a performance function, λ1 and λ2 haveequal values. When thermal comfort is given more weight than energy consumption, λ2 < λ1,and when thermal comfort is given less weight than energy consumption, λ2 > λ1. In case onlythermal comfort has to be optimized without considering energy consumption of an air-condi-tioning system, then λ2 = 0 and λ1 = 1, and when only energy consumption is optimized withoutconsidering thermal comfort, λ2 = 1 and λ1 = 0.

STATE SPACE MODELThis dynamic mathematical model can be represented in state space form. The state space

model represents the internal behavior of the system in addition to the input-output behavior. Itcan be utilized to design a feedback controller based on nonlinear control theory. The nonlineardifferential equations (Equations 1–18) can be represented in state space form as

(21)

and

, (22)

where

and

.

J1 PMV( )2 td0

tf

∫=

J2 E· com E· e E· cf+ +( ) E· dgn÷[ ] td0

tf

∫=

E· comE· e E· cf E· dgn

J λ1 PMV( )2 td λ2 E· com E· e E· cf+ +( ) E· dgn÷[ ] td0

tf

∫+0

tf

∫=

x· g x u d, ,( )=

y h x u,( )=

x x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12[ ]'=

x1 le x2 Twe x3 Te x4 Tae x5 T3 x6 Ws x7 W3 x8 Pe x9 Pc ;=;=;=;=;=;=;=;=;=

x10 Tc x11 Twc x12 lc=;=;=

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808 HVAC&R RESEARCH

The control input vector is given by u and the disturbance vector is given by d:

where

The output vector is given by , where and.

It must be noted that some state variables cannot be measured directly; those variables can beestimated by designing an appropriate observer.

Rewriting Equations 1–18 gives the following state equations.State equation for length of two-phase section in the evaporator:

State equation for the evaporator wall temperature:Case 1, when the evaporator wall temperature is more than the dew-point temperature of air

Case 2, when the evaporator wall temperature is less than the dew-point temperature of air

State equation for the evaporator temperature:

State equation for the supply air temperature:Case 1, when the evaporator wall temperature is more than the dew-point temperature of air

Case 2, when the evaporator wall temperature is less than the dew-point temperature of air

State equation for the coach air temperature:

State equation for specific humidity of supply air:Case 1, when the evaporator wall temperature is more than the dew-point temperature

u u1 u2 u3 u4[ ]'= d d1 d2 d3 d4 d5[ ]'=

u1 av u2 m· com u3 v u4 ξ d1 T0 d2 W0 d3 qi d4 M0 d5 qo=;=;=;=;=;=;=;=;=

y y1 y2 y3 y4[ ]'= y1 T3 y2 Te y3 SH=;=;=y4 W3=

x· 1 Cvu1 ρv x9 x8–( ) hge hie–( ) αieπDiex1 x2 x3–( )–[ ] ρlehlgeAe 1 γe–( )÷=

x·2 αieAei x3 x2–( ) 1 b–( ) 1 u4–( )αoeAoe x5 x2–( ) u4αoeAoe d1 x2–( )+( )+[ ] MeCe÷=

x·2αieAei x3 x2–( ) 1 b–( ) 1 u4–( )αoeAoex x5 x2–( ) u4αoeAoe d1 x2–( )+( )+

0.5 1 b–( )u3ρa hfg Cpwx2–( ) u4d2 1 u4–( )x7 0.001 0.0214x22 0.1177x2 4.15502+ +( )–+( )+ MeCe÷

=

x· 3 Cvu1 hie hge–( ) ρv x9 x8–( ) αieAei x2 x3–( )+[ ] MeCe÷=

x·4

1 b–( )u3ρaCp x2 x4–( ) b 1 u4–( )u3ρaCp x5 x4–( ) u4u3ρaCp d1 x4–( )+( )+

u3hfgρa u4d2 1 u4–( )x7 x6–+( )–CpρaVd÷=

x·4

1 b–( )u3ρaCp x2 x4–( ) b 1 u4–( )u3ρaCp x5 x4–( ) u4u3ρaCp d1 x4–( ) 1 b–( )u3hfgρa–+( )+

0.001 0.0214x22 0.1177x2 4.15502+ +⎝ ⎠

⎛ ⎞ x6–⎝ ⎠⎛ ⎞ bu3ρahfg u4d2 1 u4–( )x7 x6–+( )–

CpρaVd÷=

x·5 u3ρaCp x4 x5–( ) hfgu3ρa x6 x7–( ) 1 u4⁄( ) d3 hfgd4–( ) d5+ +–[ ] VsρaCp÷=

x· 6 u3 Vd⁄( ) u4d2 1 u4–( )x7 x6–+[ ]=

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VOLUME 14, NUMBER 5, SEPTEMBER 2008 809

Case 2, when the evaporator wall temperature less than dew-point temperature

State equation for the specific humidity of the thermal space:

State equation for the evaporator pressure:

where

State equation for the condenser pressure:

where

State equation for the condenser temperature:

State equation for the condenser wall temperature:

State equation for the length of the two-phase section in the condenser:

Output equations:

CASE STUDIES

In this work, the air-conditioning system of a railway passenger coach that runs on theShatabdi trains of Indian Railways is considered. The physical parameters for the mathematicalmodel are listed in Table1. The physical parameters, such as mass, length, diameter, and speed,were taken from the manufacturer data sheet, and other physical parameters like the heat transfercoefficients and thermal load are computed from relationships given in Stoecker and Jones(1982) and McQuiston et al. (2001). The values of the internal variables at the time of start-upwere as follows:

x·6 u3 Vd⁄( ) 1 b–( ) 0.001 0.0214x22 0.1177x2 4.15502+ +( ) x6–[ ] u3 Vd⁄( )b 1 u4–( )x7 u4d2 x6–+( )+=

x· 7 1 Vs⁄( ) u3 x6 x7–( ) d4+[ ]=

x· 8 Cvu1 hie hle–( ) ρv x9 x8–( ) hlge u2–÷ αieπDiex1 x2 x3–( ) hlge÷+[ ] AeLe der( )÷=

der dρge dPe⁄=

x· 9 u2 αicπDicx12 x10 x11–( ) hlgc÷–[ ] AcLc der2( )÷=

der2 dρgc dPc⁄=

x· 10 Cvu1 hic hoc–( ) ρv x9 x8–( ) αicAic x10 x11–( )–[ ] McCc÷=

x· 11 αicAic x10 x11–( ) αocAoc d1 x11–( )+[ ] McCc÷=

x· 12 u2 hlc hic–( ) α icπDicx12 x10 x11–( )+[ ] ρ lchlgcAc 1 γc–( )÷=

y1 x5= y2 x3= y3 x4 x3–( ) 1 exp αoeπDie Le x1–( ) Cpu2÷–( )–[ ]= y4 x7=;;;

le 0 m= Twe 36°C= Te 36°C= Tae 36°C= T3 37.2°C= Ws 26.5 g/kg (dry air)=;;;;;

W3 26.5 g/kg (dry air)= Pe 0.6 MPa= Pc 0.6 MPa= Tc 38°C= Twc 39°C= lc 0 m=;;;;;

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810 HVAC&R RESEARCH

Solution MethodThe set of differential equations provided by the mathematical model was integrated numeri-

cally in time using MATLAB’s (2007) ode45 function, which is an implementation of theRunge-Kutta fourth-fifth order method. An initial condition has to be given to solve these differ-ential equations.

Experimental PlantAn experimental reading was taken in the air-conditioned railway coach. There are two

roof-mounted package units in one railway coach, and one package unit has a cooling capacityof 24 kW. Temperatures were recorded with the help of digital and mercury thermometers in themiddle of the coach. Humidity was measured with the help of a relative humidity meter andwet-bulb and dry-bulb thermometers. All curtains and doors remained closed when the experi-ment was conducted. One package unit was switched on. The technical specifications of theplant are listed in Table 2.

Results and Discussions for Plant ON Condition Here, plant ON condition implies that the evaporator fan, condenser fan, and compressor

motor are working at full capacity. The test results for plant ON condition are depicted inFigures 4 and 5. The plant was run for 3600 s, uninterrupted, in manual cooling mode, and

Table 1. Physical Parameters of Case Studies

Ae 3.982 × 10–5 m2 W0 19 g/kg (dry air) Mc 33.5 kg

γe 0.85 Le 0.665 m Vd 5 m3

0.0706 kg/s ωc 47.5 rev/s Aic 2.05 m2

αie 566.5 W·m–2·K–1 c 0.04 αoc 40 W·m–2·K–1

Die 0.00712 m Vc 117.65 × 10–6 m3 Lc 1.13 m

Me 17.2 kg Ac 3.982 × 10–5 m2 qo 5750 W

Aie 1.189985 m2 0.125 Aoc 7.26842 × 10–5 m2

αoe 124 W·m–2·K–1 qi 900 W Vs 150 m3

Aeo7.26842 × 10–5

m2 T0 42°CM0 0.000094 kg (dry air)/s

Parameters During OFF Process

ξ 0.25 αic 1764 W/m2·K αie 20 W·m–2·K–1 αic 20 W·m–2·K–1

v 1.1111 m3/s Dic 0.00712 m αoe 10 W·m–2·K–1 αoc 4 W·m–2·K–1

Table 2. Technical Specifications of Plant

Rated cooling capacity of one package unit 12 × 2 = 24 kW

Compressor Reciprocating type

Evaporator (height × width) Φ9.62 mm, 4 row 16 tubes (406 × 114)

Condenser Φ9.62 mm, 3 row 27 tubes (673 × 63)

Expansion device Capillary

Indoor fan Centrifugal type (Φ280 mm)

Outdoor fan Two propeller fans (Φ533 mm)

Airflow rate 1.1111 m3/s

m· r

γc

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VOLUME 14, NUMBER 5, SEPTEMBER 2008 811

experimental readings were recorded after each 120 s. There were four persons inside the rail-way coach at the time of the experiment. Figure 4a shows the dry-bulb temperature of the pas-senger area of the coach. One curve shows the simulated result and the other shows theexperimental results. The readings were taken at the same ambient conditions. It is clear fromthe figure that the maximum difference between the experimental and simulated results is lessthan 10% at t = 120 s and that the simulated results closely follow the experimental results. As tincreases, the deviation between the two results approaches zero. As mentioned previously, inthis simulation, the thermal load was assumed to be constant.

Figure 4b shows the simulated and the experimental absolute humidities in the passenger areaof the coach. It is clear from the figure that the maximum difference between the experimentaland simulated results is 10% of the experimental result, and the simulated output is in closeagreement with the experimental output.

Figure 4c shows the behavior of the duct air absolute humidity obtained through simulation.The duct air humidity is less than the passenger area humidity by 0.65%. This result is expecteddue to the humidification of air due to occupants.

Figure 4d shows the behavior of the duct air temperature, evaporator temperature, and evapo-rator wall temperature with time. These results were obtained from the mathematical modelthrough simulation. Here, the evaporator temperature is the temperature of the refrigerant in the

Figure 4. Temporal variation of different state variables when the plant is continuouslyon: (a) coach air temperature in °C, (b) specific humidity of coach air in kg/kg (dry air),(c) specific humidity of duct air in kg/kg (dry air), and (d) duct air, evaporator wall, andevaporator refrigerant temperature in °C.

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812 HVAC&R RESEARCH

evaporator. The duct air temperature is more than the temperature of the evaporator wall, and thetemperature of the evaporator wall is more than the evaporator refrigerant temperature. Thus, thebehavior of these parameters is as expected.

Figure 5 shows the behavior of the condenser wall temperature, condenser refrigerant temper-ature, liquid-phase length in the evaporator, liquid-phase length in the condenser, evaporatorpressure, and condenser pressure with respect to time for the model. The results are taken for theinitial 100 s, as there is no change in behavior after this, and these results are as expected.

Results and Discussion for ON/OFF Control This section presents the results and discussion for the ON and OFF conditions of the air-condi-

tioning plant. A comparison is made between the experimental and simulation results. In thiscase, ON-OFF control, which is the most widely used control strategy in the industry for air-con-ditioning applications, was analyzed by evaluating the proposed performance evaluation func-tion. This performance function can serve as a reference for comparison between differentcontrol strategies. In this case, the air-conditioning system of a railway coach was set in auto-matic mode with the thermostat at 25.1°C. Here, automatic mode implies that compressors willbe switched off when the temperature of the thermal space becomes less than or equal to 24.8°Cand that compressors will be switched on when the temperature of the thermal space becomesgreater than or equal to 25.1°C. The air-conditioning system was run without interruption for

Figure 5. Variation of internal variables of plant with time during ON operation: (a) con-denser wall and condenser refrigerant temperatures in °C, (b) liquid-phase length in theevaporator and condenser in m, and (c) evaporator and condenser pressure in MPa.

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VOLUME 14, NUMBER 5, SEPTEMBER 2008 813

1800 s, and experimental readings were recorded after each 120 s. There were four personsinside the railway coach at the time of the experiment. The test results for ON/OFF control areshown in Figure 6. The values of different state variables at start time were as follows:

Figure 6a shows the time variation of the dry-bulb temperature of the passenger area of thecoach. Both simulated and experimental results are shown. It is clear from the figure that themaximum difference between the experimental and simulated results is around 2% and that thesimulation results are in close agreement with the experimental results. Figure 6b shows thevariation of absolute humidity in the passenger area of the coach. These curves are obtainedthrough simulation and experimental results. The absolute humidity was measured with a rela-tive humidity meter and wet-bulb and dry-bulb thermometers. It is clear from the figure that themaximum difference between the experimental and simulated results is around 3% and that thesimulated results closely follow the experimental results.

Figure 6c shows the behavior of the duct air absolute humidity of the model obtained through sim-ulation. Figure 6d shows the simulated behavior of the duct air temperature, evaporator temperature,

Figure 6. Temporal variation of different state variables during plant ON-OFF operation:(a) dry-bulb temperature of conditioned space in °C, (b) specific humidity of conditionedspace in kg/kg (dry air), (c) specific humidity of duct air in kg/kg (dry air), and (d) evapo-rator wall, evaporator refrigerant, and duct air temperature in °C.

le 0 m= Twe 24.5°C= Te 25.0°C= Tae 25.6°C= T3 26.8°C= Ws 14.75 g/kg (dry air)=;;;;;

W3 14.75 g/kg (dry air)= Pe 0.6 MPa= Pc 0.6 MPa= Tc 36°C= Twc 37°C= lc 0 m=;;;;;

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814 HVAC&R RESEARCH

and evaporator wall temperature with time. Here, the evaporator temperature is the temperature of therefrigerant in the evaporator. The duct air temperature will be more than the evaporator wall tempera-ture, and the evaporator wall temperature will be more than the evaporator temperature. Thus, trendsof these parameters are as expected.

Evaluation of Performance Function

Determination of Performance Function J1. Performance function J1 was evaluated for thevariation in air temperature and specific air humidity, keeping the other variables influencingthermal comfort constant. This performance function was evaluated for ON-OFF control with thetemperature setpoint at 25°C. The values of the thermal comfort variables are as follows:

PMV is determined by solving Fanger’s (1970) equation. J1 is calculated using Equation 18.Figure 7a shows variation of J1 with time.

MADu--------- 58.2 W/m= η 0= fc1 1.1= Ic1 0.5= μ 0.3 m/s= tmrt T3=;;;;;

Figure 7. Variation of performance function with time: (a) thermal comfort variation and(b) variation of normalized energy.

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VOLUME 14, NUMBER 5, SEPTEMBER 2008 815

Determination of Performance Function J2. The performance function J2 in Equation 19corresponds to energy consumption. There are three power-consuming equipment components,and their corresponding values are as follows:

J2 is determined using Equation 19. Figure 7b shows the variation of J2 with time. Perfor-mance evaluation function J is the sum of J1 and J2 and can be determined by usingEquation 20.

CONCLUSIONSThe conclusions regarding state space based modeling and performance evaluation of a

mobile air-conditioning system are as follows:

1. A multi-input multi-output (MIMO) state space based complete dynamic model for a directexpansion air-conditioning system was presented by using physical laws. In this model, allinternal variables that affect the performance of the system were considered.

2. The condensation of moisture in air was considered in this model since energy released dur-ing condensation is substantial in hot and humid climates.

3. This model will be useful in predicting the internal behavior of an air-conditioning system inaddition to the measured inputs and outputs.

4. The proposed model is a lumped parameter model and, consequently, the accuracy of thismodel depends upon the accuracy of estimating the thermal load, heat transfer coefficient,bypass factor, etc. This is a limitation of this model that exists in general for mathematicalapproximation of any physical plant. The modeling of an air-conditioning unit of a passengerrailway coach was considered as a case study in this paper, and the simulated outputs are inclose agreement to the experimental outputs.

5. Based on the knowledge of system parameters and initial conditions, this model is easy touse, as it requires the solution of a set of first-order ordinary nonlinear differential equa-tions. This model also can be exploited to design a feedback controller based on nonlinearcontrol theory.

NOMENCLATURE

a = valve opening A = cross-sectional area α = heat transfer coefficientb = bypass factor of cooling coil c = clearance factor C = specific heatCp = constant pressure specific heat of airCpw = specific heat of liquid water Cv = specific heat of air at constant volumeD = heat exchanger tube diameter

= power consumed by motorfc1 = clothing area factor

= mean void fractionh = refrigerant enthalpyhc = convective heat transfer coefficienthfg = latent heat of vaporizationhw = enthalpy of liquid water

η = mechanical efficiency Ic1 = thermal resistance from skin to outer sur-

face of clothed bodyJ = cost functionl = length of the two-phase sectionL = length of heat exchanger

= mass flow rateM = mass, kg Mo = moisture loadM/ADu= metabolic rateμ = relative air velocity n = compression index N = number of tubes P = pressure q = heat transfer rate from tube wall to refrig-

erant qi = sensible heat load from inside

E· com 5 kW= E· e 1 kW= E· cf 1 kW= E· dgn 7 kW=;;;

E·

γ

m·

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816 HVAC&R RESEARCH

qo = sensible heat load from outside ρ = densitySH = superheatt = thickness of copper tube in evaporator

and condenser tc1 = mean temperature of outer surface of

clothed bodytmrt = mean radiant temperature T = temperatureT3 = dry-bulb temperature of thermal space

T2 = dry-bulb temperature of supply airv = volumetric flow rate of airVc = volumetric displacement of compressorVd = volume of ductVs = volume of thermal spaceW3 = humidity ratio of thermal spaceWo = humidity ratio of outdoor airWs = humidity ratio of supply air ω = compressor speedξ = ratio of fresh air to total intake air

Subscripts

a = air c = condenser cf = condenser fan d = discharge side e = evaporatorg = gas i = inlet or inside

l = liquid

o = outlet or outside

r = refrigerant

s = suction-side compressor

v = valve

w = tube wall

REFERENCESArguello-Serrano, B., and M. Velez-Reyes. 1999. Nonlinear control of a heating, ventilating and air condi-

tioning system with thermal load estimation. IEEE Transactions on Control Systems Technology7(1):56–63.

Browne, M.W., and P.K. Bansal. 2001. Different modelling strategies of in-situ liquid chillers. Proc. Inst.Mech. Engrs (UK) A:357–74.

Browne, M.W., and P.K. Bansal. 2002. Transient simulation of vapour-compression packaged liquid chill-ers. International Journal of Refrigeration 2:597–610.

Fanger, P.O. 1970. Thermal Comfort Analysis and Application in Environmental Engineering. New York:McGraw-Hill.

He, X.-D., S. Liu, H.H. Asada, and H. Itoh. 1995. Modelling of vapor compression cycles for advancedcontrols in HVAC systems. Proceedings of the American Control Conference, pp. 3664–68.

He, X.-D., S. Liu, H.H. Asada, and H. Itoh. 1998. Multivariable control of vapor compression systems.HVAC&R Research 4(3):205–30.

Huang, W.Z., M. Zaheeruddin, and S.H. Cho. 2006. Dynamic simulation of energy management controlfunctions for HVAC systems in buildings. Energy Conversion and Management 47:926–43.

McQuiston, F.C., J.D. Parker, and J.D. Spitler. 2001. Heating, Ventilating, and Air Conditioning Analysisand Design. Singapore: John Wiley and Sons.

Shiming, D. 2000. A dynamic mathematical model of a direct expansion (DX) water cooled air condition-ing plant. Building and Environment 35:603–13.

Stoecker, W.F., and J.W. Jones. 1982. Refrigeration & Air Conditioning. New York: McGraw-Hill.The MathWorks, Inc. 2007. MATLAB. The MathWorks, Inc., Natick, Massachusetts.Wang, S., and X. Jin. 2000. Model-based optimal control of VAV air-conditioning system using genetic

algorithm. Building and Environment 35:471–87.Wu, C., and D. Shiming. 2006. Development of a dynamic model for a DX VAV air conditioning system,

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Yiu, J.C.-M., and S. Wang. 2007. Multiple ARMAX modeling scheme for forecasting air conditioning sys-tem performance. Energy Conversion & Management 48:2276–85.

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