A control oriental model for combined compression-ejector refrigeration system

Energy Conversion and Management 138 (2017) 538–546

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A control oriental model for combined compression-ejector refrigeration system Jiapeng Liu a, Lei Wang a,⇑, Lei Jia a, Zhen Li a, Hongxia Zhao b a b

School of Control Science and Engineering, Shandong University, Jinan 250061, China School of Energy and Power Engineering, Shandong University, Jinan 250061, China

a r t i c l e

i n f o

Article history: Received 8 October 2016 Received in revised form 18 January 2017 Accepted 12 February 2017

Keywords: Ejector refrigeration system Compressor Control model Parameter identification Pressure pulsating phenomenon

a b s t r a c t Combined compression-ejector refrigeration systems have attracted lots of attention in recent years. In order to improve the running stability of the complex refrigeration system, it is necessary to obtain a simple and accuracy mathematical model for system control. In this paper, a control oriental model for combined compression ejector system is proposed. By analyzing the inner relationship between compressor and ejector, a hybrid model is built based on thermodynamic principles and lumped parameter method. Comparing with traditional theoretical models, the model is more suitable for system control due to its simpler structure and less parameters. Then the pressure pulsating phenomenon inside the piping system between compressor and ejector is investigated based on the model. The effectiveness of the proposed model is validated by experimental data. It is shown that the model can reflect the system performance under variable operating conditions. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The ejector refrigeration technology is one of the active research areas in energy utilization for decades due to its usage of the low grade thermal energy such as solar energy, automobiles’ waste heat and industrial exhaust heat [1,2]. However, there are two key factors that hold back the wide application of the ejector refrigeration technology [3]. One factor is the coefficient of performance (COP) of the ejector refrigeration system is still lower compared with the traditional compressor refrigeration system. And the other factor is that the system performance is greatly affected by the instability of the low grade thermal energy. A great deal of research has been done to solve the above problems in recent years such as combined compression-ejector refrigeration system(CERS) [4], ejector-adsorption refrigeration system (EAdRS) [5], transcritical ejector refrigeration system(TERS) [6] and multi components ejector refrigeration system(MERS) [7,8]. CERS is a hybrid refrigeration system combined compressor and ejector which was proposed by Sokolov [9]. The use of a compressor changes the operating condition of the ejector, thereby the performance of CERS can be improved. Zhu [10] introduced a novel hybrid compression refrigeration system which could be improved by 9.1% comparing with the traditional

⇑ Corresponding author. E-mail address: [email protected] (L. Wang). http://dx.doi.org/10.1016/j.enconman.2017.02.033 0196-8904/Ó 2017 Elsevier Ltd. All rights reserved.

compression refrigeration system. Xue [11] performed a dynamic model based on state space for CRES, the experimental data and simulation result both shown that the compressor operating condition has influence on the ejector performance. Wang [12] conducted an experimental study on a hybrid refrigeration system with R134a, the experimental result shown that there are optimal values for evaporator and generator pressure which could improve the COP by 34%. Zhao [13] analyzed the exergy destruction in a combined compressor ejector refrigeration system by building the thermodynamic model, results shown that about a half of the exergy destruction occurs in the ejector comparing with the whole refrigeration system. CRES can overcome the effect of the instable low grade thermal energy with the help of the compressor driven by electronic power. Wang [14] carried out an investigation on the CRES driven by automobile exhaust waste heat, the result shown that the hybrid system can run steadily under variable operating conditions. Yan [15] performed an experimental study of a hybrid ejector compressor system with R134a. The experimental results shown that the performance of refrigeration system is improved by adjusting the EEV expansion based on compressor frequency. CERS is more effective and stable compared with traditional ejector refrigeration system, however the complexity of the system also increases due to complex dynamic characteristics in the compressor. Especially the reciprocating compressor which is widely used in medium and small scale refrigeration system has complex

J. Liu et al. / Energy Conversion and Management 138 (2017) 538–546

539

Nomenclature

c A D N n P R Rg T

specific heat ratio of gas area (m2 ) diameter (m) compressor motor speed (r/min) exponent of velocity distribution pressure (K Pa) radius (m) gas constant (J=ðkg  KÞ) temperature (K)

thermodynamic behaviors because of its periodic working mode. Even though the ejector structure is very simple, the thermodynamic phenomenon inside ejector chamber is too complex to be expressed as a simple model. These factors increase the difficulty of system modeling and real time control. Theoretical analysis based on conservation laws and auxiliary relationships is the main method in previous modeling processes [16]. Keenan and Newman [17] proposed a one-dimensional model to predict constant area ejector performance, the model is built on the basis of ideal gas dynamics, ignoring heat and friction losses and the conservation laws for mass, momentum and energy. Then Keenan [18] modified this model by introducing the concepts of constant pressure mixing and constant area mixing during mixing progress. Munday [19] developed the constant pressure model by introducing the concept of ‘‘hypothetical throat”. The primary flow doesn’t mix with the secondary flow until it reaches the ‘‘hypothetical throat” located downstream of the primary nozzle exit. Huang [20] presented a critical-mode model by assuming that the constant pressure mixing occurs in the constant area chamber. This model could explain the choking phenomenon of secondary fluid comparing with previous ejector models. Zhu [21] introduced the shock wave model by considering the nonuniform distribution of the secondary flow inside the mixing chamber. Calculation results shown that the shock wave model has a stronger ability to predict the ejector performance comparing with other 1D models. Cardemil [22] proposed a general model for the ejector used in the refrigeration system. Both dry and wet working fluids are considered based on the real gas equations. The theoretical model can reflect the system thermodynamic performance accurately, however theoretical models contain many parameters which would need more time to be identified. And most theoretical models need a iterate solution procedure which are not suitable for the real time control of refrigeration system. In hybrid modeling progress, the complexity of theoretical model can be decreased by using parameter classification and identification. Zhu [23] proposed a hybrid model for ejector by simplifying the shock wave model. This model only contains two or three parameters and is much simpler comparing with traditional 1D ejector models. Ding [24] proposed a hybrid model for condenser in the ejector refrigeration system, nonlinear least squares method is used to identify parameters in this hybrid model. The experimental data shown that this model could predict the condenser performance accurately and the maximum error is less than
10%. Although these hybrid models can be used to control the separate component successfully, the interaction between those components are not considering in the modeling process. Considering the complex thermodynamic phenomenon in the piping system between ejector and compressor, a simple model which can predict the relationship between ejector and compressor performance is a key factor in the control of CRES. Moreover, the pressure pulsating phenomenon in the reciprocating compressor piping system,

V

volume (m3 )

Subscripts e evaporator g generator p primary flow s suction flow

which is neglected at most of previous work in the open literature during the modeling process [11,12], may influence the effectiveness of the control strategy and even lead to the change in operational mode of ejector. The prediction of the pressure pulsating phenomenon is necessary to obtain a better system control effect. In this paper, a control oriental model for CRES is proposed. By combining the ejector model and compressor model, a uniform model is built to predict the ejector and compressor performance based on thermodynamic principles. Subsequently, the fundamental equations are simplified with the help of the lumped parameter method. The model is suitable for refrigeration system control due to only six parameters need to be determined. Considering the pressure pulsating phenomenon in the reciprocating compressor piping systems, the transient responses on the ejector suction pressure is analyzed based on the hybrid model.

2. Mathematical model A combined compression-ejector refrigeration system is shown schematically in Fig. 1. The whole system mainly consists of a generator, an ejector, a condenser, an evaporator, a compressor, a pump and an electronic expansion valve (EEV). The system is very similar to the conventional ejector refrigeration system except that the steam flow through the evaporator is inhaled by a compression firstly. The steam is compressed by the compressor from Pe to Ps , and then, through the ejector, to pressure Pc . Thus, the ejector can have a higher suction pressure, which can lead to the improvement of ejector performance. It should be noted that the pressure pulsating phenomenon is existed in the outlet of the reciprocating compressor due to the

Fig. 1. Schematic diagram of the combined compression-ejector refrigeration system.

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J. Liu et al. / Energy Conversion and Management 138 (2017) 538–546

periodic operate mode, the pulsating intensity is dependent on the reciprocating compressor frequency and the structure of the piping system. The modeling process would be difficult if the pressure pulsating phenomenon is considered. In this study, the main purpose of the model is used for real time control, so the average value in steady state is used to identify system parameters. Then based on several assumptions, this model could be used to analyze the transient response of the system under pressure pulsating phenomenon later. This analysis will instruct the control strategy to eliminate the influence of pulsating phenomenon. At first, the ejector and compressor models are described in the following section. 2.1. Ejector

Fig. 3. Operational modes of ejector.

Fig. 2 depicts a typical modern ejector. It consists of a primary nozzle and a converging-diverging outlet nozzle. The primary flow under high pressure is accelerated to the supersonic speed through the primary nozzle. The low pressure caused by the supersonic fluid draws the secondary fluid into the mixing chamber. After passing through the throat of the ejector, the mixed fluid expands to convert the velocity energy back into the pressure energy. Ejector operational condition can be categorized into three modes, namely critical mode, subcritical mode and back flow mode. The relationship between ejector entrainment ratio and operational mode is shown in Fig. 3. In this study, the shock wave model is used to analyze the dynamic performance of ejector. In order to build ejector model, several assumptions are made as follow: 1. The working fluid is an ideal gas with constant thermodynamic property. 2. The primary and secondary flow are both saturated steam. It should be noted that the vapor comes from the compressor may have a degree of superheat. But according to the previous studies [20,23,25], the superheat levels in both primary and secondary flow only have a litter influence on the ejector performance. Hence, in this study this assumption can be used to simplify the ejector model. 3. The primary flow are uniformly distributed in the radial direction, and it will be mixing with secondary flow at constant area mixing chamber when secondary flow reaches the choking condition. 4. the pressure and temperature of the secondary flow are uniformly distributed in the radial direction. 5. An equivalent coefficient is used in calculating the energy loss inside the ejector. 6. The operating conditions of the ejector is under critical model, which means both primary flow and secondary flow are at choking conditions.

According to the shock model by Zhu [21], the main governing equations can be expressed as follows: The primary flow rate follows

m_ p ¼ Pp At




cwp

Rg T p

12


2ðccþ1 1Þ 2 1þc

ð1Þ

where P p and T p is the pressure and temperature of the primary flow at the nozzle inlet, wp is the isentropic efficiency, At is the area of the primary nozzle throat, c is the specific heat ratio of gas. The secondary flow rate follows

_s¼ m

"
nþ1
2nþ1 # 2pPS V pA nR22 RpA n nR22 RpA n  1 1 Rg T sA n þ 1 R2 nþ1 R2

ð2Þ

where Ps is the secondary flow pressure in the suction chamber inlet, RpA ; P pA and T pA are the radius, pressure and temperature of the primary flow in the ‘‘hypothetical throat” respectively, RsA ; P sA and T sA are the radius, pressure and temperature of the secondary flow in the ‘‘hypothetical throat”, R2 is the constant area mixing chamber radius, n is the exponent of velocity distribution. The entrainment ratio (ER) of the ejector can be obtained as

_ m

x¼ _s mp

ð3Þ

the relationship between RpA and R2 is expressed as

#12
c1
1 1 "
c1
nþ1
RpA n c  1 2 Pp c Pp 2c T sA 2 1 ¼ 1 2 R2 Ps Ps Tp 8 " #14 9 c þ1 c þ1 c 1
1


 < = c  1 4 c þ 1 4ð1cÞ Rt Pp 4c Pp c 12 1  wexp 1 : ; 2 2 R2 P s Ps where wexp is the isentropic efficiency.

Fig. 2. Schematic view of a ejector.

ð4Þ

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J. Liu et al. / Energy Conversion and Management 138 (2017) 538–546

Combining Eqs. (2) and (4), we obtain

m_ s ¼


12

2p P s

c

1

Rg

T 2sA

Eq. (8) can be simplified to




 n n RpA  1 R22 n þ 1 2n þ 1 R2

8 #14 9
1
 cþ1
c4þ1c "
c1 < = Pp Pp c 12 c  1 4 c þ 1 4ð1cÞ Rt 1 1  wexp : ; 2 2 R2 P s Ps




14 Rt c4þ1c c1 ða1  a2 ba3 ÞN ¼ b1 f 1 1  b2 a ða c  1Þ R2 ð5Þ

2.2. Compressor

Note that function f 1 still contains parameters T sA and RpA , which are hard to measured. Hence Eq. (10) is not suitable for system control and the further simplification is necessary. Using the lumped parameter analysis, the function f 1 can be redefined as

f 1 ¼ b3

For a combined compression-ejector system, as well as for traditional compression system, the compressor plays an important role in improving system performance. The accurate dynamic model of reciprocating compressor is hard to build due to the complex mass, pressure and thermal fluctuation inside the compressor. In this study, the hybrid model proposed by Ding [26] is used to predict the compressor performance under steady state, then the pressure fluctuation in outlet of the compressor is investigated based on several assumptions. The mass flow rate can be obtained by

h i 1 VN _ c ¼ 1 þ C  C ðPdisc =Psuct Þn m

vb

ð6Þ

where v b is the specific volume, N is the compressor motor speed, C is the value of clearance volume. Pdisc is the pressure in outlet of the compressor which is equal to the ejector suction pressure Ps ; Psuct is the compressor inlet pressure which is same with the evaporator pressure Pe .


b4 R2 ab5 Rt

f 1 ¼ b6 ab5

b6 ¼ b3


b4 R2 Rt




14 Rt c4þ1c c1 ða1  a2 ba3 ÞN ¼ b1 b6 ab5 1  b2 a ða c  1Þ R2

cwp

12 

Rg

2ðccþ1 1Þ

2 1þc

. In practice, as the refrigerant and ejec-

tor geometries are fixed, parameters c; Rg and At are constant. As the outlet of the compressor is connected with the inlet of sec_ s for steady state. _c¼m ondary flow of the ejector, we can obtain m
12 
 h i 1 VN 2pP s c n n RpA 1 þ C  C ðP disc =Psuct Þn ¼ 1 R22  1 v b T 2 Rg R2 n þ 1 2n þ 1 sA 8 ð8Þ #14 9 cþ1 cþ1 "
c1 1

 
  < = 4c Pp c 12 c  1 4 c þ 1 4ð1cÞ Rt P p 1  wexp 1 : ; R2 P s Ps 2 2

ð14Þ

Finally, Eq. (8) is simplified as



cþ1



1c

14 

ac k1  k1 k2 a 4c a c  1

k1 ¼ b1 b6 Rt k2 ¼ b2 R2



ð13Þ

Then Eq. (10) is expressed as

Even though Eqs. (6) and (3) can reflect the ejector and compressor performance, there are still many parameters in those equations such as R2 and RpA , which are difficult to be measured. These formulas need to be further simplified. The mass rate of the primary flow can be redefined as

where b ¼ At

ð12Þ

where

where

ð7Þ

ð11Þ

The ejector geometrical parameters can be seemed as constant for the fixed ejector, so parameters R2 and Rt are constant. Hence f 1 can be constructed as

2.3. model simplify

Pp _ p ¼ pffiffiffiffiffi b m Tp

ð10Þ

¼ ða1  a2 ba3 ÞN

ð15Þ

ð16Þ ð17Þ

Based on the above analysis, Eqs. (7) and (15) can be characterized by only three measurable variables a; b; N, and six unknown parameters c; k1 ; k2 ; b; a1 ; a2 . Furthermore, k2 is only governed by wexp and c. The value of c can be calculated by the refrigerant thermodynamic parameters, however the values of wp is not easy to determined. According to the CFD simulation by Varga [27], the isentropic efficiency is influenced by the geometrical parameters and operate conditions. So in this study the efficiency is assumed as an unknown value. Compared with previous thermodynamic models, this hybrid model is more suitable for control refrigeration system due to its simpler structure and less parameters. The entrainment ratio is expressed as

_ m

P b

p pffiffiffiffiffi x¼ _s ¼ mp ða1  a2 ba3 ÞN T p

ð18Þ

set

b1 ¼ 2p


12

12

c

Rg


b2 ¼ wexp f1 ¼

Ps R22 1

c1



2.4. the pressure pulsating phenomenon in the piping system

14


cþ1

2 2
 n n RpA  1 n þ 1 2n þ 1 R2

T 2sA a1 ¼ ð1 þ C ÞV=v b a2 ¼ CV=v b a3 ¼ 1=n P a¼ s Pp Ps b¼ Pe

In order to analyze the pressure pulsating phenomenon in the piping system between reciprocating compressor and ejector, the following additional assumptions are introduced.

þ1 4ðc1 cÞ

ð9Þ

1. Thermodynamic performance inside the piping system between ejector and compressor is uniformly distributed. 2. There is no fluctuation in the compressor outlet mass flow rate during the compressor suction or discharge process, which means the mass flow ratio in outlet is constant during the suction or discharge process. 3. As the change ratio of the temperature is much slower comparing with the compressor motor speed, the temperature can be seen as constant during one working period.

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J. Liu et al. / Energy Conversion and Management 138 (2017) 538–546

Then the mass flow rate in the outlet of the compressor during one working period can be expressed as

mc

; 0 < t < eL

e

_ c;out ¼ m

ð19Þ

eL < t < L

0;

where L is the running time period, e is the ratio between the discharge period and the whole time. During the compressor discharge period, the ejector suction pressure is changing from Ps;min to P s;max . The mass flow rate in the outlet of the compressor is expressed as

_ c;out ¼ m

ða1  a2 ba3 ÞN

ð20Þ

e

the mass flow rate in the suction chamber is expressed as

 14  cþ1  1c _ s ¼ ac k1  k1 k2 a 4c a c  1 m

The relationship between pressure and temperature can be expressed as

ð23Þ

The temperature can be assume constant during on operating period, so the ratio of the pressure can be expressed as

_ ¼ P_ s Dm

Vm Rg T

ð24Þ

Z

eT




a k1  k1 k2 a1 "

cþ1




 4c

a k1  k1 k2 a2 c 2

"

cþ1

 4c

acj k1  k1 k2 aj

1c

1 # 4  1  a1  a2 ba13 N1 ¼ 0

1c

1 # 4   a1  a2 ba23 N2 ¼ 0    1

a1

c

a2

c




1 #

1c

   a1  a2 baj 3 Nj ¼ 0

4

aj c  1

ð31Þ

Then, define an objective function: j  X

cþ1

 4c

aci k1  k2 k2 ai

ðai

1c

c

 1Þ

14



2  ða1  a2 bai 3 ÞRi

i¼1

ð32Þ where f ðXÞ is the sum of the squares of the residuals between the calculation result and the experimental data. The LevenbergMarquardt method is implemented to solve the optimization problem [28]. The solution procedure is implemented by an iterative program. In the iterative cycle, X iþ1 is defined as

X iþ1 ¼ X i þ PðiÞ ðXÞ

ð33Þ

where the downhill iteration direction PðiÞ ðXÞ can be obtained by solving the following equation: T

MðiÞ PðiÞ ðX Þ ¼ J ðJÞ ðX Þr ðiÞ ðX Þ

P s;max can be obtained as

Ps;max ¼

cþ1

 4c

ð21Þ

ð22Þ

PV ¼ MRg T

" c 1

f ðXÞ ¼

Then the mass flow is obtained as

_s _ ¼m _ c;out  m Dm

For j experimental tests, Eq. (30) can be written into j equations as

P_ s dt þ Ps;min

ð25Þ

0

During other period, the ejector suction pressure is changing from Ps;max to Ps;T . The mass flow rate can be expressed as

_ c;out ¼ 0 m

ð26Þ

The mass flow in the outlet of the compressor can be expressed as



cþ1 

1c

_ s ¼ ac k1  k1 k2 a 4c a m

c

1

14 

ð27Þ

ð34Þ T

The matrix M ðiÞ is defined as MðiÞ ¼ J ðjÞ ðX ÞJ ðjÞ ðX Þ þ kðjÞ I, I is the identity matrix and the Jacobian matrix J is defined as

2 @r1

@k1

6 6 .. 6 . 6 6 @ri 6 @k1 6 6 . 6 . 4 . @r j @k1

@r1 @k2

.. . @ri @k2

.. .

@r j @k2

@r 1 @km

3

7 .. 7 . 7 7 @r i 7 @km 7 7 .. 7 7 . 5

ð35Þ

@r j @km

Then

_ ¼ m _ s ¼ P_ s Dm

Vm Rg T

ð28Þ

Ps;T ¼

T

eT

P_ s dt þ Ps;max

Then PðjÞ is in a descent direction. Hence the error between X iþ1 and X i can be converge in a sufficiently small neighborhood.

At the end of the one cycle, the pressure can be obtained as

Z

By choosing the value of J, the matrix M ðiÞ can be positive definite.

ð29Þ

An iterative procedure is used to solve those below equations, calculation flowchart is shown in Fig. 4.

a a k k CN Initical Value

Ps

Equaon(20)(21)(24)(25) Ps

3. Parameter identification Equaon(27)(28)(29)

To establish the model, values of variables Pp ; P s ; P e and N are required. The measurement of pressures is relatively easy, which can be obtained by the readings of pressure transducers installed at the ejector entrance and evaporator outlet. Compressor speed can be estimated by the frequency of the motor. Eq. (7) is a linear equation, the value of parameter b can be easily obtained with the help of the standard least squares method. To determine the empirical parameters in Eq. (15), rearranging Eq. (15) as



cþ1



1c

14 

ac k1  k1 k2 a 4c a c  1

 ða1  a2 ba3 ÞN ¼ 0

ð30Þ

Ps

Ps

Ps Ps T

Ps

Ps

error

no

yes end Fig. 4. Calculation flowchart in pressure pulsating phenomenon analysis.

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J. Liu et al. / Energy Conversion and Management 138 (2017) 538–546

4. In the condenser, the resulting compressed steam is cooled by ambient air. The condensed working fluid is led back to feed the evaporator and generator to complete the cycle.

4. Model validation 4.1. Experimental setup Experimental studies are conducted on CRES which is shown in Fig. 5. The system has seven main components such as an ejector, an evaporator, a condenser, an expansion valve, a variable speed pump, a receiver and a generator. The main geometrical parameters of the ejector are shown in Table 1. The working principle of the system is as follows:

The uncertainties of experimental results are calculated based on Moffat analysis [29]. And the calculations are shown in Table 2. The performance of the model is evaluated through the relative error ER and the root mean square error ERMS which are defined as follow

1. The working fluid (R134a) is heated by a 6 kW electrical heater installed in the generator to generate the high pressure steam. 2. A prototype reciprocating compressor is used to generate the low pressure steam for evaporator, the ratio between the discharge period and the whole time is 0.25. An inverter is applied to adjust compressor motor speed. 3. The electronic expansion valve is driven by a step motor which can be controlled by host computer.

ER ¼



v alueexp  v aluemod

ð36Þ

v alueexp

ERMS ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi uP v alue v alue 2 u j exp mod t i¼1 v alueexp

ð37Þ

j

where j is the number of fitted points, v alueexp is the value obtained from experimental data and v aluemod is the value calculated by hybrid model.

Table 3 Model parameters and performance. Parameters

Value

b k1 k2 a1 a2 c

0.0007508 0.01689 0.002486 0.02664 0.01052 0.0517

ERMS

3.89 %

0.2

Calcalated entrainment ratio

0.19

Fig. 5. Photograph of the experimental rig.

Table 1 Main geometry parameters of the ejector Parameters

value (mm)

Primary nozzle throat diameter Primary nozzle exit diameter Ejector throat diameter Mixing chamber length Diffuser length NXP

2.5 2.8 5.5 22 66 0

0.18 +6%

0.17 0.16

-6%

0.15 0.14 0.13 0.12 0.11

0.1 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

Measured entrainment ratio Fig. 6. Experiment fittings by hybrid model.

Table 2 The uncertainties of experimental results Parameters Primary flow pressure P p (Bar) Secondary flow pressure P s (Bar) Evaporator pressure P e (Bar) Primary flow temperature T p (°C) Secondary flow temperature T s (°C) Primary flow mass rate mp (L/h) Secondary flow mass rate mp (L/h)

Accuracy

Absolute uncertainty

Minimal measured value

Uncertainty

0.02% 0.01% 0.01% 0.3% 0.3% 1.92% 0.32%

0.008 0.0015 0.0015 0.45 0.45 2.304 0.064

20.23 5.86 2.46 107.86 37.91 80.06 8.12

0.0003954 0.000256 0.000609 0.00417 0.01187 0.02878 0.00788

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J. Liu et al. / Energy Conversion and Management 138 (2017) 538–546

4.2. Validation results

10 9

Relative error (%)

8 7 6 5 4 3 2 1 0

0

5

10

15

20

25

Fitting points

Entrianment ratio

Fig. 7. Relative error for experiment fittings by hybrid model.

0.2

4.3. simulation and discussion

0.18 0.16 0.14 750

Exp Mod

800

850

900

950

Compressor motor speed(r/min)

(a) Entrianment ratio

During the validation progress, the system performance is adjusted by changing the compressor motor speed and EEV expansion. The compressor motor speed is changed from 750 r/min to 950 r/min and the value of the EEV expansion is changed from 0.01 to 0.04. The parameters in the hybrid model are identified by the standard least squares method and Levenberg-Marquardt method form the experimental data. The parameters and relative error are shown in Table 3. Using the method to predict the system performance under variable operating conditions, Fig. 6 shows the comparing result between hybrid model and experimental data for the entrainment ratio and Fig. 7 shows the relative errors for experimental fittings. These results show that this hybrid model can predict the system performance. The calculate errors are less than
6% under most operating conditions.

0.19 0.18 0.17 0.16 750

Exp Mod

800

850

900

950

Compressor motor speed(r/min)

(b)

Pressure (Pa)

Fig. 8. Effect of the compressor motor speed on entrainment ratio. (a) EEV expansion is 0.028. (b) EEV expansion is 0.035.

The hybrid model is applied to predict the performance of the ejector and a iterate solution is used to obtain the dynamic pressure response in the piping system between compressor and ejector. The value of EEV is set to 0.028 and 0.035, the compressor speed is selected as 750 r/min, 800 r/min and 850 r/min, the iterate error is set to 1  106 . The relationships between ejector entrainment ratio and compressor motor speed at different values of EEV expansion are given in Fig. 8. Both experimental date and simulation result show that ejector performance increases with the rising of the compressor motor speed and EEV expansion. Figs. 9 and 10 show the simulation result and experimental data for the pressure pulsating phenomenon at different compressor motor speed and EEV expansion. The relative error between simulation result and experimental data is shown in Fig. 11. It is observed that the oscillation period of the suction pressure is consistent with the compressor working cycle. The result shows that

× 105

6.4 6.2

Mod Exp

6 0

0.02

0.04

0.06

0.08

0.1

0.12

Time (sec)

Pressure (Pa)

(a) × 105

6.6 6.45

Mod Exp

6.3 0

0.02

0.04

0.06

0.08

0.1

0.12

Time (sec)

Pressure (Pa)

(b) × 10

7

5

6.7 Mod Exp

6.4 0

0.02

0.04

0.06

0.08

0.1

0.12

Time (sec)

(c) Fig. 9. The pressure pulsating phenomenon with EEV expansion 0.028. (a) Compressor motor speed is 750 r/min. (b) Compressor motor speed is 800 r/min. (c) Compressor motor speed is 850 r/min.

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Pressure (Pa)

J. Liu et al. / Energy Conversion and Management 138 (2017) 538–546

× 105

7 6.5

Mod Exp

6 0

0.02

0.04

0.06

0.08

0.1

0.08

0.1

0.12

Time (sec)

Pressure (Pa)

(a) × 10

6.9

5

6.7 Mod Exp

6.5 0

0.02

0.04

0.06

0.12

Time (sec)

Pressure (Pa)

(b) × 10

7.4 7.3 7.2 7.1 7

5

Mod Exp

0

0.02

0.04

0.06

0.08

0.1

0.12

Time (sec)

(c) Fig. 10. The pressure pulsating phenomenon with EEV expansion 0.035. (a) Compressor motor speed is 750 r/min. (b) Compressor motor speed is 800 r/min. (c) Compressor motor speed is 850 r/min.

model can achieve accurate performance prediction, the error is less than
6% under most operating conditions. Then the pressure pulsating phenomenon inside the piping system between compressor and ejector is investigated under different operating conditions. The results show that the hybrid model can predict the pressure pulsating phenomenon in the piping system. The intensity of the pressure pulsating phenomenon is about 3%. In order to improve the control accuracy, advanced control strategies can be used to eliminate effects of the pressure pulsating phenomenon based on the proposed model.

2 Fig.8(a) Fig.8(b) Fig.8(c) Fig.9(a) Fig.9(b) Fig.9(c)

1.8

Relative error (%)

1.6 1.4 1.2 1 0.8 0.6 0.4

Acknowledgements

0.2 0 2

4

6

8

10

12

Fitting points Fig. 11. Relative error for experiment fittings by the suction pressure.

the hybrid model can predict the pressure pulsating phenomenon in the piping system between compressor and ejector. The intensity of the pressure pulsating phenomenon is quite low, with the maximum value is about 3%. That means the pressure pulsating phenomenon may cause the ejector to work at subcritical mode when the ejector works at critical point. Then the performance of refrigeration system is reduced. Hence, in order to improve the system performance, advanced control strategies can be used to eliminate effects of the pressure pulsating phenomenon based on the intensity value. 5. Conclusion A control oriental model for CRES is proposed based on hybrid modeling method. Comparing with traditional theoretical models, the model only contains six parameters which can be determined by the experimental data easily. Furthermore, the model is suitable for system control due to its simple structure. The experimental data is used to validate the hybrid model. The results show the

The work was funded by the Natural Science Foundation of Shandong Province of China under the Grant No. ZR2016FM24 and the Fundamental Research funds of Shandong University under the Grant No. 2014JC022.

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