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Generic network modeling of reciprocating compressors
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 5 ( 2 0 1 4 ) 1 0 7 e1 1 9
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Generic network modeling of reciprocating compressors Jian Hu a,b, Liang Yang a, Liang-Liang Shao a, Chun-Lu Zhang a,* a b
School of Mechanical Engineering, Tongji University, Shanghai 201804, China China R&D Center, Carrier Corporation, No.3239 Shen Jiang Road, Shanghai 201206, China
article info
abstract
Article history:
With the increasing applications of CO2 trans-critical cycles, the design of reciprocating
Received 24 March 2014
compressors returns to the center stage. Quick design and optimization of a compressor
Received in revised form
with arbitrary configuration is always a big challenge. This paper presents a new generic
8 June 2014
modeling approach to reciprocating compressors design. The reciprocating compressors
Accepted 10 June 2014
were firstly torn down to components, e.g. compression chamber, valve, shaft, motor,
Available online 16 June 2014
crankcase, etc. Then the component models were developed to feature the sub-processes inside the components. Refrigerant flow, heat flow, power flow, and air flow (for inter-
Keywords:
mediate cooler) between components were described on a network basis. Finally, the
Reciprocating compressor
object-oriented programming method was applied to develop a graphical user interface for
Model
generic drag-and-drop modeling of reciprocating compressors with arbitrary configuration.
CO2
Experimental data of a CO2 two-stage compressor and a R410A single-stage compressor
R410A
were used to validate the generic modeling tool. The deviations in the mass flow rate and power consumption of R410A compressor are mostly within ±3% and ±5%, respectively, while the deviations in the mass flow rate and power consumption of CO2 compressor are mostly within ±8% and ±5%, respectively. © 2014 Elsevier Ltd and IIR. All rights reserved.
lisation par re seau ge ne rique de compresseurs a piston Mode piston ; Mode le ; CO2 ; R410A Mots cles : Compresseur a
1.
Introduction
Reciprocating compressors are widely used in various refrigerating units covering a large range of capacity. Due to relatively lower volumetric efficiency and larger dimension, reciprocating compressor is nowadays replaced
* Corresponding author. Tel.: þ86 136 71825 133. E-mail address: [email protected] (C.-L. Zhang). http://dx.doi.org/10.1016/j.ijrefrig.2014.06.007 0140-7007/© 2014 Elsevier Ltd and IIR. All rights reserved.
by rotary compressors (e.g. rolling-piston, scroll, screw compressors) in most applications. However, with the increasing applications of carbon dioxide (CO 2) transcritical cycles (Austin and Sumathy, 2011; Bansal, 2012; Pearson, 2005), reciprocating compressor is returning to the center stage because of its advantages in high pressure
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Nomenclature a A C d F Fd Fs Fi Fp Frod g h k m p pc pd Pinput Fb,x Fb,y Q S
acceleration, m s2 area, m2 flow coefficient diameter, m force between two solid bodies, N force acting on the discharge valve, N force acting on the suction valve, N inertial force of piston, N pressure force on the piston, N total force acting on the rod, N gravity acceleration, m s2 enthalpy, J kg1 spring factor of the suction valve mass flow rate, kg s1 pressure, Pa pressure in the crank chamber, Pa discharge pressure, Pa input power, W Bearing force along x direction, N Bearing force along y direction, N heat flow, W valve displacement, m
operation and good efficiency when running at lower pressure ratio. To well design reciprocating compressor, numerical simulation has become a powerful approach and different simulation models are found in the literature. Some CFD simulations have been carried out for the compressor components (e.g. mufflers and valves) (Nakano and Kinjo, 2008; Pereira et al., 2008b) and the whole compressor (Birari et al., 2006). Despite of the recent advances in numerical methodologies, the computational cost of a full three-dimensional simulation of a reciprocating compressor is still impracticable for optimization purposes (Pereira et al., 2008a). Therefore, simpler methodologies which can offer satisfactory results for a preliminary design are still very important and rez-Segarra et al. (2003) and worth further development. Pe Rigola et al. (2003) developed a detailed numerical model of the thermal and fluid dynamic behavior of small reciprocating compressors which are commonly used in household refrigerators and freezers. Later, to simplify the process on compressor performance evaluation, they developed a detailed model for the thermodynamic efficiencies to characrez-Segarra terize the hermetic reciprocating compressors (Pe et al., 2005). They focused on the volumetric efficiency, isentropic efficiency and combined mechanical-electrical efficiency and detached them into several partial efficiencies so as to denote effects of different physical sub-processes. More recently, they presented a more generic object-oriented unstructured modular modeling methodology of reciprocating compressors (Damle et al., 2011). The new approach offers advantages of handling complex circuitry (e.g. parallel paths, multiple compressor chambers, etc.), coupling different simulation models for each element and adaptability to different configurations without changing the source code. Yang et al. (2013) found there was no comprehensive models
t T V X vs vd M
time temperature, K volume, m3 unknown variable set suction valve velocity, m s1 discharge valve velocity, m s1 Mass kg
Greek symbols ε convergence tolerance h efficiency r density, kg m3 t time, s G torque, N m q crank angle k Specific heat ratio Subscripts i inflow I inertial o outflow d discharge s suction l leakage
for CO2 reciprocating compressors in the literature. They therefore presented a comprehensive model to predict the CO2 reciprocating compressor performance, which included both the frictional losses at piston ring-cylinder liner interface and at the journal bearings. For more information, state-of-the-art reviews of numerical methodologies applied to reciprocating compressors were made available by Rasmussen and Jakobsen (2000) and Ribas et al. (2008). Most of the compressor models mentioned above can only cover a specific compressor or a series of compressors with fixed or similar configuration. In addition, the programming methods used were typically the ‘functional-programming’ approach which is of poor extension ability. Therefore, quick design and optimization of a compressor with arbitrary configuration is still a big challenge for both modeling and implementation methods. Different from the existing ones, we apply a generic network based modeling methodology with the objectoriented programming method to carry out a graphical dragand-drop modeling and simulation platform for reciprocating compressor design. The network model involves refrigerant flow, heat flow, power flow, and air flow between compressor components. Different configurations and complex circuitry of reciprocating compressors can be handled by an easy-of-use graphical drag-and-drop style. At last, the method is validated with different compressors.
2.
Reciprocating compressor model
Fig. 1 is the typical schematic of a reciprocating compressor. Generally, the reciprocating compressor consists of a set of components. The low pressure refrigerant vapor from the evaporator enters the crankcase and is heated by the
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dvc 1 dVc Vc dmc ¼ 2 mc dt dt mc dt
(4)
Substituting Equation (2) into Equation (4) yields dvc Ap dx Ap x þ V0 dmc ¼ dt mc dt m2c dt
(5)
The gas temperature inside the cylinder can be calculated as dTc dQ ZRT dVc ¼ mcv dt cv V dt dt
Fig. 1 e Schematic of reciprocating compressor. motor and the wall of shell. After that, it goes across the suction stub to the cylinder where it is compressed by the piston and the pressure is lifted. Some part of the refrigerant mass will be leaked back to the crankcase during the whole compression process. At last, the high-pressure gas is discharged from the cylinder to the discharge plenum, namely the cylinder header and go through the manifold towards the condenser.
2.1.
Component models
To simulate the compressor in a generic way, we propose a network modeling approach. Firstly we divide the whole compressor system into individual component parts. Then the components can be connected with each other through different mass and energy flows, e.g. refrigerant flow, heat flow, and power flow. Inside each component model, fundamental governing equations (e.g. conservation equations) and empirical correlations (e.g. correlations for heat transfer coefficients) are applied to describe the different physical sub-processes. The major components in our compressor component library are described as follows.
2.1.1.
Compression chamber and valve
The model calculates the pressure, temperature and volume as a function of the crank angle for an entire revolution. The RungeeKutta fourtheorder method is applied to calculate the mass flow rate and discharge temperature of the refrigerant and power consumption. For the compression process inside cylinder, we have Specific volume of refrigerant gas inside cylinder: vc ¼
Vc mc
where dQ is the heat transfer rate between the gas and cylinder dt c is the cylinder volume wall, Z is the compression factor and dV dt change rate. Upon Equations (3) and (6), the refrigerant mass and temperature inside the cylinder can be calculated. Therefore the corresponding pressure can be obtained.
2.1.2.
Valve model
A simple valve model for the flow through the discharge or suction port is developed. A schematic of the model is shown in Fig. 2. The valve is modeled with single degree of freedom object. For brevity, we only take the discharge valve as an example. If the pressure in the discharge plenum is larger than the discharge pressure, the valve opens. Otherwise it will close. The distance y, which is the valve open distance, is calculated by the function d2 1 p y ¼ p pd 4 k
(7)
where d is the diameter of discharge port, p is the pressure in the discharge plenum and k is the spring constant of the valve. Apparently, the maximum distance that the valve can reach is determined by the valve stop. The flow area is then calculated by Ad ¼ yp
(8)
The mass flow rate through the discharge valve is determined using the equation for isentropic compression flow (Fox and McDonald, 1992). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k kþ1 dmd 2k pd pd k ¼ Cflow Ad p ðk 1ÞRT dt p p
(9)
Valve Stop
(1)
The volume inside the cylinder: Vc ¼ Ap x þ V0
(6)
Valve Spring
y Valve Seat
(2)
where Vc is the cylinder clearance volume. Considering the changes with respect to time, we have
y P
dmc dms dmd dml ¼ dt dt dt dt
Discharge Opening
(3) Fig. 2 e Schematic of the discharge valve model (single degree of freedom system).
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where Ad is the area of the suction plenum opening. k is the specific heat ratio. Cflow is the correction factor, which is 0.58 and 0.6 for suction and discharge process, respectively.
2.1.3.
Leakage model
FP is the pressure force on the piston (N), pc is the cylinder gas pressure (Pa), pb is the pressure (Pa) in the crankcase. The acceleration can be calculated using the following equation (Lin and Sun, 1987): 2 3
The gas leakage through the piston ring gap is modeled as an isentropic, compressible fluid flowing through an orifice. The mass flow rate can be determined as follows (Span, 1996).
2 6 l cos 2 q 1 l3 sin 2q 7 6 7 a ¼ ru2 6cos q þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 3 7 2 4 1 l2 sin q 4 1 l2 sin2 q 2 5
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 u u 2k kþ1 u k 7 6 dml 2k u 7; pd > 0:54 6 pd pd ¼ Cflow Agap pu u 5 pu tZRTu ðk 1Þ 4 pu dt pu
To simplify the calculation, the following equation is used to calculate the acceleration. (15) a ¼ ru2 ðcos q þ l cos 2 qÞ (10)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 u u kþ1 u k1 7 pd dml 2 u k 6 6 7; < 0:54 choked ¼ Cflow Agap pu u 5 pu tZRTu 4 k þ 1 dt
(11)
Crankshaft
This model calculates the torque and bearing load, the contacting force for any schematic of reciprocating compressor with arbitrary cylinder configuration. In each time step, it uses the pressure data from the cylinder chamber to calculate the mechanical parameters, such as the acceleration of the piston, the torque load on the main bearing and motor end bearing. The results will be further used for the bearing component.
2.1.5.
The total force acting on the rod 1 Frod ¼ F cos 4 ¼ F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 l2 sin q
(16)
This force will cause a torque on the crankshaft system
where the discharge coefficient Cflow is assumed to be 0.86 (Span, 1996) and the compressible factor Z is calculated by the equation of state (Span, 1996; Stachowiak and Batchelor, 2001).
2.1.4.
(14)
Physical shaft loss system
In order to determine the exact power consumption of a reciprocating compressor, various losses in the compressor need to be considered. Firstly, a local coordinate is built on each cylinder as shown in Fig. 3. The force on the cylinder, which is denoted as F, is divided into two parts, the inertia force and the gas pressure force. F ¼ Fp þ FI
(12)
where,
Τ ¼ Frod r sinðq þ 4Þ ¼ Frsinðq þ 4Þ=cos 4
Then the force acting on the crank bearing is also divided into two orthogonal directions. X direction : Fcb;x ¼
Fp cos b ¼ FP cos b
(18)
Y direction : Fcb;y ¼
Fp sin b ¼ FP tan b cos b
(19)
Here we have gotten the force and torque acting on each single crankshaft system. To get the force acting on the main bearing and pump end bearing on the whole crankshaft system, a four cylinder example of which is displayed in Fig. 4. We need to convert the local coordinate to the global coordinate. The force analysis method is the same, but the crank angle needed to be converted based on the following equation: qi ¼ q þ 4i þ fi
(13)
(20)
Here q is the crank angle, while 4i is the bank angle between the ith cylinder to the 1st cylinder, and fi is the shaft angle between the ith cylinder to the 1st cylinder. At each direction, we have X
p Fp ¼ pc pb D2 4
(17)
Fcb;x þ Fmb;x þ Fpb;x ¼ 0
Suction and compression process
B
F φ A
Fp
Fig. 3 e Schematic of crankshaft system.
θ
(21)
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Y
E2
X1
0
2
1
1
Y1
2 3
a
4
0
PUMP END BEARING
X
MAIN BEARING
a
ω
E1 3
4
E3 E4 L Fig. 4 e Schematic of crankshaft system built on the global coordinate.
X X X
Fcb;y þ Fmb;y þ Fpb;y ¼ 0
(22)
Gshaft;xi ¼ Fmb;x L
(23) (24)
For the whole crankshaft system, the torque generated from the force acting on each of crank bearing can be calculated as: Gshaft;xi ¼ Fcb;x Ei
(25)
Gshaft;yi ¼ Fcb;y Ei
(26)
where Ei is the distance from ith cylinder to the pump end bearing as shown in Fig. 4 Substituting Equations (25) and (26) into (23) and (24), we get !, n X Ei Fcb;x L (27) Fmb;x ¼ i¼1
n X
!, Ei Fcb;y
Wt c2
W* ¼
Gshaft;yi ¼ Fmb;y L
Fmb;y ¼
where v1;oil , v2;oil are the kinematic viscosities of the oil at 37.8 C and 93.3 C, respectively. The dimensionless load capacity is calculated by
L
mUcir Lbearing R2journal
Meanwhile the total force can be calculated by Wt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F2x þ F2y þ F2y
(31)
After we finish calculating the force on each cylinder, some conversion work from the local coordinate to the global coordinate is still needed. The global coordinate is built on the whole crankshaft system, as shown in Fig. 4. After these two conversion steps are done, we can calculate the torque for each cylinder. There are two unknowns, the force acting on main bearing and the pump end bearing, meanwhile we have two Equations (27) and (28) to solve them. The results will be further used to calculate the shaft efficiency using equation (32). The power loss on the shaft can be calculated with a motor performance correlation as follows.
(28)
i¼1
Now we have four Equations (21), (22), (27) and (28) with four unknown parameters, namely the force acting on the main bearing,Fmb , which is divided into Fmb;x , Fmb;y , and force acting on the pump end bearing, Fpb;x , Fpb;y , therefore the equations can be solved. Then we use a regression method to predict the frictional power losses at the crankshaft bearing and the crank journal bearing (Stachowiak and Batchelor, 2001). 1:577 0:477 2:240 1:278 Pbearing ¼ 3:9307$103 $v0:706 1;oil $v2;oil $Lbearing $Djournal $Nj
1:324 1 þ ln W* $c0:249 $T0:204 sup
(30)
(29)
hmechanical ¼
Wcompression ¼ f Gshaft Wshaft
(32)
Here hmechanical is the mechanical efficiency and Wcompression is the compression work rate. A correlation f ðGshaft Þ can be curve-fitted from experimental data to calculate the shaft efficiency.
2.1.6.
Motor
This component model calculates the motor performance based on the motor efficiency curve. The shaft work of compressor can be therefore determined by Wshaft ¼ hmotor P
(33)
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where the P is the overall power input to the motor or compressor. The motor efficiency, hmotor can be assumed to be 95% or specified in terms of engineering best practice. To sum up, with the motor-mechanical efficiency and the calculated compression power, the overall motor consumption can be calculated. Meanwhile, the isentropic efficiency of the compression process can be figured out as well. ho;is ¼
mðh2s h1 Þ Pinput
(34)
where h1 is the enthalpy at the suction state 1 while h2s is the one at the discharge state 2 assuming the refrigerant is experiencing an isentropic process when compressed from state 1 to state 2.
2.1.7.
Crankcase
This model computes the heat transfer effect to the suction fluid based on the following parameters: the fraction of motor and bearing power losses added as heat, the area and coefficient for heat transfer from ambient to the suction fluid. Energy equation: hcyl;suc ¼ ðmtube;suc htube;suc þ ml hl Þ ðmsuc;tube þ ml Þ
(36)
Continuity equation msuc;cyl ¼ msuc;tube þ ml
(37)
Thermal Parameter -ParaName -ParaValue +GetParaName() +GetParaValue()
Thermal ParameterList +ReadVarList() +WriteVarList() +FindVar()
2.2.
Compressor network model
As we mentioned previously, component models solve the individual physical sub-processes inside the components. To develop a generic modeling platform for reciprocating compressors, we should know how to generally describe the connections among components in an arbitrary reciprocating compressor system. Between the components, there are different “flows”: refrigerant mass flow, power flow or power transmission, heat flow or heat transfer, and air flow (taking place mainly in intermediate cooler). A generic network model is therefore proposed.
2.2.1. (35)
Momentum equation psuc;out ¼ psuc;tube
Here, m1;tube is the mass flow rate from the suction tube entering the crankcase, ml is the mass flow rate leakage from cylinder to the crankcase during the compression process, msuc;cyl is the mass flow rate entering the cylinder. It should be noted that all these force and moment balances are assumed under quasi-static conditions, namely keeping static balances at each time step of the crank angle.
Port and network
The port is used to represent the inlet or outlet of a component and the key performance parameters are defined on vertices of the network. Typically, the vertices are categorized into four types of port as shown in Fig. 5 in terms of the physical principles. Port is an abstract class and inherited by the refrigerant, air, heat and mechanical ports. Each type of port is a set of thermal parameters regarding the refrigerant, air, heat and mechanical parts, respectively. Then each type of port
Refrigerant Port Port -ID +ReadFluidNode() +FindVar() +AddVar() +DeleteVar()
Air Port
Heat Port
Port Network +ReadArray() +WriteArray() +SetNodeInfo() +GetNodeInfo() +SetVarValue() +AddNode() +GetVar()
Heat Network Mechanical Port
Mechanical Network Air Network
Refrigerant Network
Fig. 5 e Structure of the port and network.
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will constitute a network describing a complete process occurring inside compressor. Each type of port has its attribute parameters, such as the mass flow rate, enthalpy and pressure on the refrigerant type vertex, the shaft speed and power on the mechanical vertex, and the air temperature and relative humidity on the air vertex. As long as the unknown parameters on the node are solved, the compressor system performance can be determined.
2.2.2.
Refrigerant port
This type of port is defined to represents refrigerant entering or leaving state of the component model. In each refrigerant port, the mass flow rate, pressure, and enthalpy are defined as the independent variables so that the refrigerant state can be determined. Then the set of refrigerant ports, which we call the refrigerant linked list, can be used to determine the refrigerant flow from the suction stub to the discharge cylinder head.
2.2.3.
Mechanical port
This type of port is designed for the calculation of the mechanical friction losses and shaft load torque. A set of mechanical ports constitute a mechanical linked list so that the shaft power transfer path can be represented and calculated.
end, the air port linked list is able to represent the whole air cooling process. In order to determine the heat transfer effect from the compressor internal elements (e.g. bearing and motor), to the suction state of the refrigerant before compression, we need to know the temperature of each element and the whole temperature distribution inside the compressor. For simplicity, the compressor is divided into the following lumped element: the cylinder, crankcase, shaft (including the bearing) and the ambient. Then the temperature heat network inside the compressor is established as shown in Fig. 6. For each of the lumped element, an energy balance of the form 0 ¼ Qin Qout Qgen
2.2.5.
Tcyl Tgas þ Qloss ¼ 0 Rgas cyl
(39)
Tgas Tmotor þ Qmotor ¼ 0 Rgas motor
(40)
Tcylinder :
Heat port
This type of port is defined to determine the surface temperature of a component and heat flow rate between this specific component and the attached refrigerant. With a set of heat ports, a thermal network will be established to determine the temperatures as well as the heat flow rates between component and refrigerant.
(38)
can be established, where Qin and Qout are the heat flow into and out of the element, respectively. Qgen is the heat generated inside the compressor component, for instance, the heat generated due to the friction loss. The application of Equation (38) to the components are as follows.
Tmotor :
2.2.4.
Tshaft :
Tgas Tshaft þ Qfriction_loss ¼ 0 Rgas_shaft
Tcrankcase :
Tgas Tcrankcase Tambinet Tcrankcase þ ¼0 Rgas_cc Rambinet
Tgas : msuc;pipe ðhsuc hin Þ ¼ Qloss þ Qmotor þ Qfriction_loss
Air port
In the multiple-stage compressor, sometimes the air convection method is applied to cool the cylinder in the intermediate stage so as to decrease the discharge temperature and improve efficiency. Therefore air ports are designed to represent the air state entering or leaving the cylinder. In the
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(41)
(42)
(43)
Note that heat flow rate for each component can be obtained by solving the related component model. Here we just take cylinder for instance. It is a function of the refrigerant temperature, shell temperature and the geometry of the cylinder.
TCylinder TMotor TShaft Rgas_Shaft Rgas_Motor
Rgas_Cylinder
TGas Rgas_crankcase
TCrank_case
Rcase_ambient
TAmbient
Fig. 6 e Equivalent electrical circuit for thermal resistance between elements.
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Qloss ¼ f Tgas ðqÞ; Tcyl
(44)
Now we have six Equations (39)e(44) and six unknowns which are the temperature of cylinder, motor, shaft, crankcase, the average temperature and enthalpy (Tcylinder, Tmotor, Tcrankcase, Tgas, Tshaft, hgas). Therefore the problem can be determined and numerically solved.
after another and transfer the results to their connected vertices. All equations on the vertices of network will be solved simultaneously using the NewtoneRaphson method, which provides a generic approach for modeling a reciprocating compressor with arbitrary configuration. A flow chart for the entire model solving process is detailed in Fig. 7.
3.2.
3.
Implementation
3.1.
Numerical algorithm
After compressor system network model is established, we need to find a way to solve the model efficiently. As mentioned above, the conservative equations are all invoked in each component. To improve the robustness and make it easy to debug the model, all those equations are not solved simultaneously. Instead, the component models can be solved one
Object-oriented Programming (OOP)
Object-oriented programming (OOP) has roots that can be traced back to the 1960s. Researchers studied ways to maintain software quality and developed OOP methodology in part to address common problems by strongly emphasizing discrete, reusable units of programming logic (Eckel, 2002). In OOP, each object is capable of receiving messages, processing data and sending message to other objects. ‘Methods’ on these objects are closely associated with the object. A programming usually consists of different types of objects, each corresponding to a particular kind of complex data to manage. Each
Fig. 7 e Global algorithm of solver.
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Compressor Component +Simulate() +GetRes()
Motor
Suction tube Valve
Compression chamber
+Simulate() +GetRes()
Crankcase
+Simulate() +GetRes()
+Simulate() +GetRes()
Discharge tube +Simulate() +GetRes()
+Simulate() +GetRes()
+Simulate() +GetRes()
Suction muffer
Journal Bearing
+Simulate() +GetRes()
+Simulate() +GetRes()
Discharge Muffer +Simulate() +GetRes()
Fig. 8 e Schematic of component library structure based on OOP.
object will have its own standardized methods for performing particular operations on its data. The major two features of OOP are ‘Inheritance’ and ‘Polymorphism’. The inheritance enables the son class to get the data and method from its parents class so that code can be reused. The polymorphism enable to offer standardized methods across different types of objects, provided they all derivate from one parent class. Same code will invoke different functions. For example, solver is able to send same ‘Simulate’ command to each component class, and the compiler is able to invoke different ‘Simulate’ functions according to the type of son class. A ‘cylinder’ object will invoke the compression procedure while the ‘valve’ invokes a ‘Fanno flow’ procedure. The benefit of this feature is high extensibility. New component model can be added into the existing system without any modification to the whole solver framework, so
long as the new component implements its own ‘Simulate’ procedure. Because this new code can be invoked by the solver without any modification, the framework is closed for modification and open for the function extensibility. The ‘Compressor Component’ just provides a pure virtual function ‘Simulate’ and its derived class ‘Compression chamber’, ‘Valve’, ‘Motor’, ‘Crankcase’ will implement ‘Simulate’ function to provide concrete implementations. The detailed UML class relationship is shown in Fig. 8. In conclusion, due to the advanced feature of OOP, we have implemented a solver structure that is open for extension and closed for modification. Meanwhile the system model is constructed by joining component from the standard library. Therefore, a reciprocating compressor platform which can handle any complex compressor configuration is established.
Fig. 9 e Compressor system schematic in design tool interface.
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Fig. 10 e Object-Oriented single-stage compressor system model. The generic reciprocating compressor modeling platform is developed as a graphical drag-and-drop modeling tool with friendly user interface, as shown in Fig. 9. In this tool, the component models are represented as icons on the component library panel. The user can use mouse to drag, move, and drop different icons in the model editor window. After connecting the icons (components) in a logical way, a
reciprocating compressor with desired configuration is then ready for simulation.
3.3.
Examples
After those four types of port vectors being established, arbitrary compressor configuration can be easily set up. Engineers
Fig. 11 e Object-Oriented two-stage compressor system model.
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between the first and second stages, the schematic of which is shown in Fig. 11.
are able to do a trial numerical simulation to evaluate different design concepts. Figs 10 and 11 are two typical compressor examples implemented according to the principles we defined above. Fig. 10 demonstrates a single-stage reciprocating compressor, while Fig. 11 represents a twostage reciprocating compressor.
4.
4.1.
Compressor testing rig
A schematic of a hot gas bypass load stand is shown in Fig. 12. At point 1, the refrigerant is suctioned into compressor, compressed to discharge pressure and temperature at point 2. Note that a hot gas bypass line is setup at the discharge port, and then the refrigerant gas is divided into two streams. One stream goes through the bypass tube and expanded to the pressure of suction, point 5. Another stream is cooled down in a condenser or gas cooler, then expanded through an expansion valve and evaporator, finally joins the previous stream at the suction chamber. The discharge pressure is tuned by an
Model validation
Two compressors and their lab testing data are used for model validation. One is a single-stage R410A compressor, the schematic of which is shown in Fig. 10. Another is a two-stage trans-critical CO2 compressor with an intermediate cooler
Water chilling unit
Water flow meter
3
Gas cooler
5 Flow meter
Oil Separator
2 Pdis Control
Bypass EEV
Discharge EEV
P
Psuc Control
EEV
T
Pressure Sensor
6
Temperature Sensor
CO2 compressor
P
T Evaporator
1 7
5
Heater
T
Tret_wat control
Heater
Water flow meter
Fig. 12 e Schematic of compressor testing rig.
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350 +3%
1700
Predicted mass flow rate (kg/h)
Predicted mass flow rate (kg/h)
2000
-3% 1400
1100
800
300
+8% -8%
250
200
150
100
500 500
800 1100 1400 1700 Measured mass flow rate (kg/h)
2000
50 50
Fig. 13 e Numerical and Experiment mass flow rate comparison of R410A compressor.
100 150 200 250 300 Measured mass flow rate (kg/h)
350
Fig. 15 e Numerical and Experiment mass flow rate comparison of CO2 compressor.
expansion valve located at the discharge line, while the suction pressure can be adjusted by changing the opening rate of EEV (electronic expansion valve) located at the bypass line. The specified superheat is obtained by controlling the additional heater installed at the suction chamber. The major testing instrumentation of the test rig is the same for R410A and CO2. We made adjustment on the specific model number of pressure transducers and mass flow meters, since the pressure and mass flow rate varies dramatically for those two refrigerants. We also ensured the plastic sealed parts are compatible for both R410A and CO2. The testing rig utilizes thermocouples (Omega KMQSS-125G-6) and pressure transducers (Omega PX32B1-2.5KGV for CO2 and PX32B11KGV for R410A, respectively) to measure and adjust the suction pressure, suction temperature, discharge pressure, mass flow measurements with a mass flow meter (MicroMotion
DH25 for CO2 and DH100 for R410A, respectively), a volumetric flow meter (Sponsler SP717) and electric power analyzer with accuracies of 0.05K, 0.25%, 0.5%, 0.5% and 1% respectively.
4.2.
Model validation
The comparison between the model predictions and experimental data are illustrated in Figs. 13e16 . For the single-stage R410A compressor, the deviations of mass flow rate and power consumption between predictions and experiments are mostly within ±3% and ±5%, respectively. For the two-stage CO2 compressor, the deviations of mass flow rate and power consumption between predictions and experiments are mostly within ±8% and ±5%, respectively. The present model
7
Predicted power consumption (kW)
Predicted power consumption (kW)
22 +5%
20
-5%
18 16 14 12 10 8
+5% 6
-5%
5
4
3 8
10
12
14
16
18
20
22
Measured power consumption (kW) Fig. 14 e Numerical and Experiment power consumption comparison R410A compressor.
3
4 5 6 Measured power consumption (kW)
7
Fig. 16 e Numerical and Experiment power consumption comparison of CO2 compressor.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 5 ( 2 0 1 4 ) 1 0 7 e1 1 9
accuracy is very competitive in comparison with the reciprocating compressor models in the open literature. However, there are still some room for improving the model accuracy. The deviation of mass flow rate for the CO2 compressor is fairly larger than that of the R410A compressor. One reason is that there is a discharge plenum in the R410A compressor to decrease the pressure pulsation and the corresponding pressure loss effect. The CO2 compressor does not have such design. Since the pressure pulsation is very difficult to model, it wasn't taken into account in the simulation, which introduces additional deviation in simulating the CO2 compressor. Another reason is the assumption of isentropic compression for calculating the leakage mass flow rate for CO2 may not be very accurate.
5.
Conclusions
In this paper, a new generic network model of reciprocating compressors was developed. The whole compressor is divided into individual components. Inside each component model, the conservative equations were used to describe physical sub-processes. Between components, a network model involving refrigerant flow, power flow, heat flow, and air flow was developed to describe the connections in a generic way so that arbitrary configuration of reciprocating compressors can be easily modeled. Based on the OOP method, a graphical drag-and-drop modeling platform was developed. Same methodology might be extended to other types of refrigerant compressors modeling. A single-stage R410A compressor and a two-stage transcritical CO2 compressor were modeled and validated with experimental data. For the single-stage R410A compressor, the deviations of mass flow rate and power consumption between predictions and experiments are mostly within ±3% and ±5%, respectively. For the two-stage CO2 compressor, the deviations of mass flow rate and power consumption between predictions and experiments are mostly within ±8% and ±5%, respectively.
Acknowledgments This work is partially supported by the National Natural Science Foundation of China (Grant No. 51206123) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 11ZZ30).
references
Austin, B.T., Sumathy, K., 2011. Transcritical carbon dioxide heat pump systems: a review. Renew. Sustain. Energy Rev. 15, 4013e4029.
119
Bansal, P., 2012. A revieweStatus of CO2 as a low temperature refrigerant: fundamentals and R&D opportunities. Appl. Therm. Eng. 41, 18e29. Birari, Y.V., Gosavi, S.S., Jorwekar, P.P., 2006. Use of CFD in design and development of R404A reciprocating compressor. In: International Compressor Engineering Conference. Purdue University, West Lafayette, IN, USA. Paper 1727. rez-Segarra, C.D., Castro, J., Oliva, A., 2011. Damle, R., Rigola, J., Pe Object-oriented simulation of reciprocating compressors: numerical verification and experimental comparison. Int. J. Refrigeration 34, 1989e1998. Eckel, B., 2002. Thinking in Cþþ. In: Introduction to Standard Cþþ, second ed., vol. 1. Prentice Hall PTR, New Jersey, USA. Fox, R.W., McDonald, A.T., 1992. Introduction to Fluid Mechanics. John Wiley & Sons, New York. Lin, M., Sun, S., 1987. Principles of Reciprocate Compressor. Mechanical Industry Press, Beijing, China. Nakano, A., Kinjo, K., 2008. CFD applications for development of reciprocating compressor. In: International Compressor Engineering Conference. Purdue University, West Lafayette, IN, USA. Paper 1842. rez-Segarra, C., Rigola, J., Oliva, A., 2003. Modeling and Pe numerical simulation of the thermal and fluid dynamic behavior of hermetic reciprocating compressorsdpart 1: theoretical basis. HVAC&R Res. 9, 215e235. rez-Segarra, C., Rigola, J., Soria, M., Oliva, A., 2005. Detailed Pe thermodynamic characterization of hermetic reciprocating compressors. Int. J. Refrigeration 28, 579e593. Pearson, A., 2005. Carbon dioxidednew uses for an old refrigerant. Int. J. Refrigeration 28, 1140e1148. Pereira, E.L., Deschamps, C.J., Ribas, F.A., 2008a. A comparative analysis of numerical simulation approaches for reciprocating Compressors. In: International Compressor Engineering Conference. Purdue University, West Lafayette, IN, USA. Paper 1879. Pereira, E.L., Deschamps, C.J., Ribas, F.A., 2008b. Performance analysis of reciprocating compressors through computational fluid dynamics. Proc. Inst. Mech. Eng. Part J. Process Mech. Eng. 222, 183e192. Rasmussen, B.D., Jakobsen, A., 2000. Review of compressor models and performance characterizing variables. In: International Compressor Engineering Conference. Purdue University, West Lafayette, IN, USA. Paper 1429. Ribas, F.A., Deschamps, C.J., Fagotti, F., Morriesen, A., Dutra, T., 2008. Thermal analysis of Reciprocating Compressors-A Critical Review. In: International Compressor Engineering Conference. Purdue University, West Lafayette, IN, USA. Paper 1907. rez-Segarra, C., Oliva, A., 2003. Modeling Rigola, J., Pe and numerical simulation of the Thermal and fluid dynamic behavior of hermetic reciprocating compressorsdPart 2: experimental investigation. HVAC&R Res. 9, 237e249. Span, W., 1996. A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressure up to 800 MPa. J. Phys. Chem. Ref. Data 6 (25), 1509e1596. Stachowiak, G.W., Batchelor, A.W., 2001. Engineering Tribology, second ed. Butterworth-Heinemann, Boston. Yang, B., Bradshaw, C.R., Groll, E.A., 2013. Modeling of a Semihermetic CO2 reciprocating compressor including lubrication Submodels for piston rings and bearings. Int. J. Refrigeration 36, 1925e1937.
Available online at www.sciencedirect.com
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journal homepage: www.elsevier.com/locate/ijrefrig
Generic network modeling of reciprocating compressors Jian Hu a,b, Liang Yang a, Liang-Liang Shao a, Chun-Lu Zhang a,* a b
School of Mechanical Engineering, Tongji University, Shanghai 201804, China China R&D Center, Carrier Corporation, No.3239 Shen Jiang Road, Shanghai 201206, China
article info
abstract
Article history:
With the increasing applications of CO2 trans-critical cycles, the design of reciprocating
Received 24 March 2014
compressors returns to the center stage. Quick design and optimization of a compressor
Received in revised form
with arbitrary configuration is always a big challenge. This paper presents a new generic
8 June 2014
modeling approach to reciprocating compressors design. The reciprocating compressors
Accepted 10 June 2014
were firstly torn down to components, e.g. compression chamber, valve, shaft, motor,
Available online 16 June 2014
crankcase, etc. Then the component models were developed to feature the sub-processes inside the components. Refrigerant flow, heat flow, power flow, and air flow (for inter-
Keywords:
mediate cooler) between components were described on a network basis. Finally, the
Reciprocating compressor
object-oriented programming method was applied to develop a graphical user interface for
Model
generic drag-and-drop modeling of reciprocating compressors with arbitrary configuration.
CO2
Experimental data of a CO2 two-stage compressor and a R410A single-stage compressor
R410A
were used to validate the generic modeling tool. The deviations in the mass flow rate and power consumption of R410A compressor are mostly within ±3% and ±5%, respectively, while the deviations in the mass flow rate and power consumption of CO2 compressor are mostly within ±8% and ±5%, respectively. © 2014 Elsevier Ltd and IIR. All rights reserved.
lisation par re seau ge ne rique de compresseurs a piston Mode piston ; Mode le ; CO2 ; R410A Mots cles : Compresseur a
1.
Introduction
Reciprocating compressors are widely used in various refrigerating units covering a large range of capacity. Due to relatively lower volumetric efficiency and larger dimension, reciprocating compressor is nowadays replaced
* Corresponding author. Tel.: þ86 136 71825 133. E-mail address: [email protected] (C.-L. Zhang). http://dx.doi.org/10.1016/j.ijrefrig.2014.06.007 0140-7007/© 2014 Elsevier Ltd and IIR. All rights reserved.
by rotary compressors (e.g. rolling-piston, scroll, screw compressors) in most applications. However, with the increasing applications of carbon dioxide (CO 2) transcritical cycles (Austin and Sumathy, 2011; Bansal, 2012; Pearson, 2005), reciprocating compressor is returning to the center stage because of its advantages in high pressure
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Nomenclature a A C d F Fd Fs Fi Fp Frod g h k m p pc pd Pinput Fb,x Fb,y Q S
acceleration, m s2 area, m2 flow coefficient diameter, m force between two solid bodies, N force acting on the discharge valve, N force acting on the suction valve, N inertial force of piston, N pressure force on the piston, N total force acting on the rod, N gravity acceleration, m s2 enthalpy, J kg1 spring factor of the suction valve mass flow rate, kg s1 pressure, Pa pressure in the crank chamber, Pa discharge pressure, Pa input power, W Bearing force along x direction, N Bearing force along y direction, N heat flow, W valve displacement, m
operation and good efficiency when running at lower pressure ratio. To well design reciprocating compressor, numerical simulation has become a powerful approach and different simulation models are found in the literature. Some CFD simulations have been carried out for the compressor components (e.g. mufflers and valves) (Nakano and Kinjo, 2008; Pereira et al., 2008b) and the whole compressor (Birari et al., 2006). Despite of the recent advances in numerical methodologies, the computational cost of a full three-dimensional simulation of a reciprocating compressor is still impracticable for optimization purposes (Pereira et al., 2008a). Therefore, simpler methodologies which can offer satisfactory results for a preliminary design are still very important and rez-Segarra et al. (2003) and worth further development. Pe Rigola et al. (2003) developed a detailed numerical model of the thermal and fluid dynamic behavior of small reciprocating compressors which are commonly used in household refrigerators and freezers. Later, to simplify the process on compressor performance evaluation, they developed a detailed model for the thermodynamic efficiencies to characrez-Segarra terize the hermetic reciprocating compressors (Pe et al., 2005). They focused on the volumetric efficiency, isentropic efficiency and combined mechanical-electrical efficiency and detached them into several partial efficiencies so as to denote effects of different physical sub-processes. More recently, they presented a more generic object-oriented unstructured modular modeling methodology of reciprocating compressors (Damle et al., 2011). The new approach offers advantages of handling complex circuitry (e.g. parallel paths, multiple compressor chambers, etc.), coupling different simulation models for each element and adaptability to different configurations without changing the source code. Yang et al. (2013) found there was no comprehensive models
t T V X vs vd M
time temperature, K volume, m3 unknown variable set suction valve velocity, m s1 discharge valve velocity, m s1 Mass kg
Greek symbols ε convergence tolerance h efficiency r density, kg m3 t time, s G torque, N m q crank angle k Specific heat ratio Subscripts i inflow I inertial o outflow d discharge s suction l leakage
for CO2 reciprocating compressors in the literature. They therefore presented a comprehensive model to predict the CO2 reciprocating compressor performance, which included both the frictional losses at piston ring-cylinder liner interface and at the journal bearings. For more information, state-of-the-art reviews of numerical methodologies applied to reciprocating compressors were made available by Rasmussen and Jakobsen (2000) and Ribas et al. (2008). Most of the compressor models mentioned above can only cover a specific compressor or a series of compressors with fixed or similar configuration. In addition, the programming methods used were typically the ‘functional-programming’ approach which is of poor extension ability. Therefore, quick design and optimization of a compressor with arbitrary configuration is still a big challenge for both modeling and implementation methods. Different from the existing ones, we apply a generic network based modeling methodology with the objectoriented programming method to carry out a graphical dragand-drop modeling and simulation platform for reciprocating compressor design. The network model involves refrigerant flow, heat flow, power flow, and air flow between compressor components. Different configurations and complex circuitry of reciprocating compressors can be handled by an easy-of-use graphical drag-and-drop style. At last, the method is validated with different compressors.
2.
Reciprocating compressor model
Fig. 1 is the typical schematic of a reciprocating compressor. Generally, the reciprocating compressor consists of a set of components. The low pressure refrigerant vapor from the evaporator enters the crankcase and is heated by the
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dvc 1 dVc Vc dmc ¼ 2 mc dt dt mc dt
(4)
Substituting Equation (2) into Equation (4) yields dvc Ap dx Ap x þ V0 dmc ¼ dt mc dt m2c dt
(5)
The gas temperature inside the cylinder can be calculated as dTc dQ ZRT dVc ¼ mcv dt cv V dt dt
Fig. 1 e Schematic of reciprocating compressor. motor and the wall of shell. After that, it goes across the suction stub to the cylinder where it is compressed by the piston and the pressure is lifted. Some part of the refrigerant mass will be leaked back to the crankcase during the whole compression process. At last, the high-pressure gas is discharged from the cylinder to the discharge plenum, namely the cylinder header and go through the manifold towards the condenser.
2.1.
Component models
To simulate the compressor in a generic way, we propose a network modeling approach. Firstly we divide the whole compressor system into individual component parts. Then the components can be connected with each other through different mass and energy flows, e.g. refrigerant flow, heat flow, and power flow. Inside each component model, fundamental governing equations (e.g. conservation equations) and empirical correlations (e.g. correlations for heat transfer coefficients) are applied to describe the different physical sub-processes. The major components in our compressor component library are described as follows.
2.1.1.
Compression chamber and valve
The model calculates the pressure, temperature and volume as a function of the crank angle for an entire revolution. The RungeeKutta fourtheorder method is applied to calculate the mass flow rate and discharge temperature of the refrigerant and power consumption. For the compression process inside cylinder, we have Specific volume of refrigerant gas inside cylinder: vc ¼
Vc mc
where dQ is the heat transfer rate between the gas and cylinder dt c is the cylinder volume wall, Z is the compression factor and dV dt change rate. Upon Equations (3) and (6), the refrigerant mass and temperature inside the cylinder can be calculated. Therefore the corresponding pressure can be obtained.
2.1.2.
Valve model
A simple valve model for the flow through the discharge or suction port is developed. A schematic of the model is shown in Fig. 2. The valve is modeled with single degree of freedom object. For brevity, we only take the discharge valve as an example. If the pressure in the discharge plenum is larger than the discharge pressure, the valve opens. Otherwise it will close. The distance y, which is the valve open distance, is calculated by the function d2 1 p y ¼ p pd 4 k
(7)
where d is the diameter of discharge port, p is the pressure in the discharge plenum and k is the spring constant of the valve. Apparently, the maximum distance that the valve can reach is determined by the valve stop. The flow area is then calculated by Ad ¼ yp
(8)
The mass flow rate through the discharge valve is determined using the equation for isentropic compression flow (Fox and McDonald, 1992). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k kþ1 dmd 2k pd pd k ¼ Cflow Ad p ðk 1ÞRT dt p p
(9)
Valve Stop
(1)
The volume inside the cylinder: Vc ¼ Ap x þ V0
(6)
Valve Spring
y Valve Seat
(2)
where Vc is the cylinder clearance volume. Considering the changes with respect to time, we have
y P
dmc dms dmd dml ¼ dt dt dt dt
Discharge Opening
(3) Fig. 2 e Schematic of the discharge valve model (single degree of freedom system).
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where Ad is the area of the suction plenum opening. k is the specific heat ratio. Cflow is the correction factor, which is 0.58 and 0.6 for suction and discharge process, respectively.
2.1.3.
Leakage model
FP is the pressure force on the piston (N), pc is the cylinder gas pressure (Pa), pb is the pressure (Pa) in the crankcase. The acceleration can be calculated using the following equation (Lin and Sun, 1987): 2 3
The gas leakage through the piston ring gap is modeled as an isentropic, compressible fluid flowing through an orifice. The mass flow rate can be determined as follows (Span, 1996).
2 6 l cos 2 q 1 l3 sin 2q 7 6 7 a ¼ ru2 6cos q þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 3 7 2 4 1 l2 sin q 4 1 l2 sin2 q 2 5
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 u u 2k kþ1 u k 7 6 dml 2k u 7; pd > 0:54 6 pd pd ¼ Cflow Agap pu u 5 pu tZRTu ðk 1Þ 4 pu dt pu
To simplify the calculation, the following equation is used to calculate the acceleration. (15) a ¼ ru2 ðcos q þ l cos 2 qÞ (10)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 u u kþ1 u k1 7 pd dml 2 u k 6 6 7; < 0:54 choked ¼ Cflow Agap pu u 5 pu tZRTu 4 k þ 1 dt
(11)
Crankshaft
This model calculates the torque and bearing load, the contacting force for any schematic of reciprocating compressor with arbitrary cylinder configuration. In each time step, it uses the pressure data from the cylinder chamber to calculate the mechanical parameters, such as the acceleration of the piston, the torque load on the main bearing and motor end bearing. The results will be further used for the bearing component.
2.1.5.
The total force acting on the rod 1 Frod ¼ F cos 4 ¼ F pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 l2 sin q
(16)
This force will cause a torque on the crankshaft system
where the discharge coefficient Cflow is assumed to be 0.86 (Span, 1996) and the compressible factor Z is calculated by the equation of state (Span, 1996; Stachowiak and Batchelor, 2001).
2.1.4.
(14)
Physical shaft loss system
In order to determine the exact power consumption of a reciprocating compressor, various losses in the compressor need to be considered. Firstly, a local coordinate is built on each cylinder as shown in Fig. 3. The force on the cylinder, which is denoted as F, is divided into two parts, the inertia force and the gas pressure force. F ¼ Fp þ FI
(12)
where,
Τ ¼ Frod r sinðq þ 4Þ ¼ Frsinðq þ 4Þ=cos 4
Then the force acting on the crank bearing is also divided into two orthogonal directions. X direction : Fcb;x ¼
Fp cos b ¼ FP cos b
(18)
Y direction : Fcb;y ¼
Fp sin b ¼ FP tan b cos b
(19)
Here we have gotten the force and torque acting on each single crankshaft system. To get the force acting on the main bearing and pump end bearing on the whole crankshaft system, a four cylinder example of which is displayed in Fig. 4. We need to convert the local coordinate to the global coordinate. The force analysis method is the same, but the crank angle needed to be converted based on the following equation: qi ¼ q þ 4i þ fi
(13)
(20)
Here q is the crank angle, while 4i is the bank angle between the ith cylinder to the 1st cylinder, and fi is the shaft angle between the ith cylinder to the 1st cylinder. At each direction, we have X
p Fp ¼ pc pb D2 4
(17)
Fcb;x þ Fmb;x þ Fpb;x ¼ 0
Suction and compression process
B
F φ A
Fp
Fig. 3 e Schematic of crankshaft system.
θ
(21)
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Y
E2
X1
0
2
1
1
Y1
2 3
a
4
0
PUMP END BEARING
X
MAIN BEARING
a
ω
E1 3
4
E3 E4 L Fig. 4 e Schematic of crankshaft system built on the global coordinate.
X X X
Fcb;y þ Fmb;y þ Fpb;y ¼ 0
(22)
Gshaft;xi ¼ Fmb;x L
(23) (24)
For the whole crankshaft system, the torque generated from the force acting on each of crank bearing can be calculated as: Gshaft;xi ¼ Fcb;x Ei
(25)
Gshaft;yi ¼ Fcb;y Ei
(26)
where Ei is the distance from ith cylinder to the pump end bearing as shown in Fig. 4 Substituting Equations (25) and (26) into (23) and (24), we get !, n X Ei Fcb;x L (27) Fmb;x ¼ i¼1
n X
!, Ei Fcb;y
Wt c2
W* ¼
Gshaft;yi ¼ Fmb;y L
Fmb;y ¼
where v1;oil , v2;oil are the kinematic viscosities of the oil at 37.8 C and 93.3 C, respectively. The dimensionless load capacity is calculated by
L
mUcir Lbearing R2journal
Meanwhile the total force can be calculated by Wt ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F2x þ F2y þ F2y
(31)
After we finish calculating the force on each cylinder, some conversion work from the local coordinate to the global coordinate is still needed. The global coordinate is built on the whole crankshaft system, as shown in Fig. 4. After these two conversion steps are done, we can calculate the torque for each cylinder. There are two unknowns, the force acting on main bearing and the pump end bearing, meanwhile we have two Equations (27) and (28) to solve them. The results will be further used to calculate the shaft efficiency using equation (32). The power loss on the shaft can be calculated with a motor performance correlation as follows.
(28)
i¼1
Now we have four Equations (21), (22), (27) and (28) with four unknown parameters, namely the force acting on the main bearing,Fmb , which is divided into Fmb;x , Fmb;y , and force acting on the pump end bearing, Fpb;x , Fpb;y , therefore the equations can be solved. Then we use a regression method to predict the frictional power losses at the crankshaft bearing and the crank journal bearing (Stachowiak and Batchelor, 2001). 1:577 0:477 2:240 1:278 Pbearing ¼ 3:9307$103 $v0:706 1;oil $v2;oil $Lbearing $Djournal $Nj
1:324 1 þ ln W* $c0:249 $T0:204 sup
(30)
(29)
hmechanical ¼
Wcompression ¼ f Gshaft Wshaft
(32)
Here hmechanical is the mechanical efficiency and Wcompression is the compression work rate. A correlation f ðGshaft Þ can be curve-fitted from experimental data to calculate the shaft efficiency.
2.1.6.
Motor
This component model calculates the motor performance based on the motor efficiency curve. The shaft work of compressor can be therefore determined by Wshaft ¼ hmotor P
(33)
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where the P is the overall power input to the motor or compressor. The motor efficiency, hmotor can be assumed to be 95% or specified in terms of engineering best practice. To sum up, with the motor-mechanical efficiency and the calculated compression power, the overall motor consumption can be calculated. Meanwhile, the isentropic efficiency of the compression process can be figured out as well. ho;is ¼
mðh2s h1 Þ Pinput
(34)
where h1 is the enthalpy at the suction state 1 while h2s is the one at the discharge state 2 assuming the refrigerant is experiencing an isentropic process when compressed from state 1 to state 2.
2.1.7.
Crankcase
This model computes the heat transfer effect to the suction fluid based on the following parameters: the fraction of motor and bearing power losses added as heat, the area and coefficient for heat transfer from ambient to the suction fluid. Energy equation: hcyl;suc ¼ ðmtube;suc htube;suc þ ml hl Þ ðmsuc;tube þ ml Þ
(36)
Continuity equation msuc;cyl ¼ msuc;tube þ ml
(37)
Thermal Parameter -ParaName -ParaValue +GetParaName() +GetParaValue()
Thermal ParameterList +ReadVarList() +WriteVarList() +FindVar()
2.2.
Compressor network model
As we mentioned previously, component models solve the individual physical sub-processes inside the components. To develop a generic modeling platform for reciprocating compressors, we should know how to generally describe the connections among components in an arbitrary reciprocating compressor system. Between the components, there are different “flows”: refrigerant mass flow, power flow or power transmission, heat flow or heat transfer, and air flow (taking place mainly in intermediate cooler). A generic network model is therefore proposed.
2.2.1. (35)
Momentum equation psuc;out ¼ psuc;tube
Here, m1;tube is the mass flow rate from the suction tube entering the crankcase, ml is the mass flow rate leakage from cylinder to the crankcase during the compression process, msuc;cyl is the mass flow rate entering the cylinder. It should be noted that all these force and moment balances are assumed under quasi-static conditions, namely keeping static balances at each time step of the crank angle.
Port and network
The port is used to represent the inlet or outlet of a component and the key performance parameters are defined on vertices of the network. Typically, the vertices are categorized into four types of port as shown in Fig. 5 in terms of the physical principles. Port is an abstract class and inherited by the refrigerant, air, heat and mechanical ports. Each type of port is a set of thermal parameters regarding the refrigerant, air, heat and mechanical parts, respectively. Then each type of port
Refrigerant Port Port -ID +ReadFluidNode() +FindVar() +AddVar() +DeleteVar()
Air Port
Heat Port
Port Network +ReadArray() +WriteArray() +SetNodeInfo() +GetNodeInfo() +SetVarValue() +AddNode() +GetVar()
Heat Network Mechanical Port
Mechanical Network Air Network
Refrigerant Network
Fig. 5 e Structure of the port and network.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 5 ( 2 0 1 4 ) 1 0 7 e1 1 9
will constitute a network describing a complete process occurring inside compressor. Each type of port has its attribute parameters, such as the mass flow rate, enthalpy and pressure on the refrigerant type vertex, the shaft speed and power on the mechanical vertex, and the air temperature and relative humidity on the air vertex. As long as the unknown parameters on the node are solved, the compressor system performance can be determined.
2.2.2.
Refrigerant port
This type of port is defined to represents refrigerant entering or leaving state of the component model. In each refrigerant port, the mass flow rate, pressure, and enthalpy are defined as the independent variables so that the refrigerant state can be determined. Then the set of refrigerant ports, which we call the refrigerant linked list, can be used to determine the refrigerant flow from the suction stub to the discharge cylinder head.
2.2.3.
Mechanical port
This type of port is designed for the calculation of the mechanical friction losses and shaft load torque. A set of mechanical ports constitute a mechanical linked list so that the shaft power transfer path can be represented and calculated.
end, the air port linked list is able to represent the whole air cooling process. In order to determine the heat transfer effect from the compressor internal elements (e.g. bearing and motor), to the suction state of the refrigerant before compression, we need to know the temperature of each element and the whole temperature distribution inside the compressor. For simplicity, the compressor is divided into the following lumped element: the cylinder, crankcase, shaft (including the bearing) and the ambient. Then the temperature heat network inside the compressor is established as shown in Fig. 6. For each of the lumped element, an energy balance of the form 0 ¼ Qin Qout Qgen
2.2.5.
Tcyl Tgas þ Qloss ¼ 0 Rgas cyl
(39)
Tgas Tmotor þ Qmotor ¼ 0 Rgas motor
(40)
Tcylinder :
Heat port
This type of port is defined to determine the surface temperature of a component and heat flow rate between this specific component and the attached refrigerant. With a set of heat ports, a thermal network will be established to determine the temperatures as well as the heat flow rates between component and refrigerant.
(38)
can be established, where Qin and Qout are the heat flow into and out of the element, respectively. Qgen is the heat generated inside the compressor component, for instance, the heat generated due to the friction loss. The application of Equation (38) to the components are as follows.
Tmotor :
2.2.4.
Tshaft :
Tgas Tshaft þ Qfriction_loss ¼ 0 Rgas_shaft
Tcrankcase :
Tgas Tcrankcase Tambinet Tcrankcase þ ¼0 Rgas_cc Rambinet
Tgas : msuc;pipe ðhsuc hin Þ ¼ Qloss þ Qmotor þ Qfriction_loss
Air port
In the multiple-stage compressor, sometimes the air convection method is applied to cool the cylinder in the intermediate stage so as to decrease the discharge temperature and improve efficiency. Therefore air ports are designed to represent the air state entering or leaving the cylinder. In the
113
(41)
(42)
(43)
Note that heat flow rate for each component can be obtained by solving the related component model. Here we just take cylinder for instance. It is a function of the refrigerant temperature, shell temperature and the geometry of the cylinder.
TCylinder TMotor TShaft Rgas_Shaft Rgas_Motor
Rgas_Cylinder
TGas Rgas_crankcase
TCrank_case
Rcase_ambient
TAmbient
Fig. 6 e Equivalent electrical circuit for thermal resistance between elements.
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Qloss ¼ f Tgas ðqÞ; Tcyl
(44)
Now we have six Equations (39)e(44) and six unknowns which are the temperature of cylinder, motor, shaft, crankcase, the average temperature and enthalpy (Tcylinder, Tmotor, Tcrankcase, Tgas, Tshaft, hgas). Therefore the problem can be determined and numerically solved.
after another and transfer the results to their connected vertices. All equations on the vertices of network will be solved simultaneously using the NewtoneRaphson method, which provides a generic approach for modeling a reciprocating compressor with arbitrary configuration. A flow chart for the entire model solving process is detailed in Fig. 7.
3.2.
3.
Implementation
3.1.
Numerical algorithm
After compressor system network model is established, we need to find a way to solve the model efficiently. As mentioned above, the conservative equations are all invoked in each component. To improve the robustness and make it easy to debug the model, all those equations are not solved simultaneously. Instead, the component models can be solved one
Object-oriented Programming (OOP)
Object-oriented programming (OOP) has roots that can be traced back to the 1960s. Researchers studied ways to maintain software quality and developed OOP methodology in part to address common problems by strongly emphasizing discrete, reusable units of programming logic (Eckel, 2002). In OOP, each object is capable of receiving messages, processing data and sending message to other objects. ‘Methods’ on these objects are closely associated with the object. A programming usually consists of different types of objects, each corresponding to a particular kind of complex data to manage. Each
Fig. 7 e Global algorithm of solver.
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Compressor Component +Simulate() +GetRes()
Motor
Suction tube Valve
Compression chamber
+Simulate() +GetRes()
Crankcase
+Simulate() +GetRes()
+Simulate() +GetRes()
Discharge tube +Simulate() +GetRes()
+Simulate() +GetRes()
+Simulate() +GetRes()
Suction muffer
Journal Bearing
+Simulate() +GetRes()
+Simulate() +GetRes()
Discharge Muffer +Simulate() +GetRes()
Fig. 8 e Schematic of component library structure based on OOP.
object will have its own standardized methods for performing particular operations on its data. The major two features of OOP are ‘Inheritance’ and ‘Polymorphism’. The inheritance enables the son class to get the data and method from its parents class so that code can be reused. The polymorphism enable to offer standardized methods across different types of objects, provided they all derivate from one parent class. Same code will invoke different functions. For example, solver is able to send same ‘Simulate’ command to each component class, and the compiler is able to invoke different ‘Simulate’ functions according to the type of son class. A ‘cylinder’ object will invoke the compression procedure while the ‘valve’ invokes a ‘Fanno flow’ procedure. The benefit of this feature is high extensibility. New component model can be added into the existing system without any modification to the whole solver framework, so
long as the new component implements its own ‘Simulate’ procedure. Because this new code can be invoked by the solver without any modification, the framework is closed for modification and open for the function extensibility. The ‘Compressor Component’ just provides a pure virtual function ‘Simulate’ and its derived class ‘Compression chamber’, ‘Valve’, ‘Motor’, ‘Crankcase’ will implement ‘Simulate’ function to provide concrete implementations. The detailed UML class relationship is shown in Fig. 8. In conclusion, due to the advanced feature of OOP, we have implemented a solver structure that is open for extension and closed for modification. Meanwhile the system model is constructed by joining component from the standard library. Therefore, a reciprocating compressor platform which can handle any complex compressor configuration is established.
Fig. 9 e Compressor system schematic in design tool interface.
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Fig. 10 e Object-Oriented single-stage compressor system model. The generic reciprocating compressor modeling platform is developed as a graphical drag-and-drop modeling tool with friendly user interface, as shown in Fig. 9. In this tool, the component models are represented as icons on the component library panel. The user can use mouse to drag, move, and drop different icons in the model editor window. After connecting the icons (components) in a logical way, a
reciprocating compressor with desired configuration is then ready for simulation.
3.3.
Examples
After those four types of port vectors being established, arbitrary compressor configuration can be easily set up. Engineers
Fig. 11 e Object-Oriented two-stage compressor system model.
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between the first and second stages, the schematic of which is shown in Fig. 11.
are able to do a trial numerical simulation to evaluate different design concepts. Figs 10 and 11 are two typical compressor examples implemented according to the principles we defined above. Fig. 10 demonstrates a single-stage reciprocating compressor, while Fig. 11 represents a twostage reciprocating compressor.
4.
4.1.
Compressor testing rig
A schematic of a hot gas bypass load stand is shown in Fig. 12. At point 1, the refrigerant is suctioned into compressor, compressed to discharge pressure and temperature at point 2. Note that a hot gas bypass line is setup at the discharge port, and then the refrigerant gas is divided into two streams. One stream goes through the bypass tube and expanded to the pressure of suction, point 5. Another stream is cooled down in a condenser or gas cooler, then expanded through an expansion valve and evaporator, finally joins the previous stream at the suction chamber. The discharge pressure is tuned by an
Model validation
Two compressors and their lab testing data are used for model validation. One is a single-stage R410A compressor, the schematic of which is shown in Fig. 10. Another is a two-stage trans-critical CO2 compressor with an intermediate cooler
Water chilling unit
Water flow meter
3
Gas cooler
5 Flow meter
Oil Separator
2 Pdis Control
Bypass EEV
Discharge EEV
P
Psuc Control
EEV
T
Pressure Sensor
6
Temperature Sensor
CO2 compressor
P
T Evaporator
1 7
5
Heater
T
Tret_wat control
Heater
Water flow meter
Fig. 12 e Schematic of compressor testing rig.
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350 +3%
1700
Predicted mass flow rate (kg/h)
Predicted mass flow rate (kg/h)
2000
-3% 1400
1100
800
300
+8% -8%
250
200
150
100
500 500
800 1100 1400 1700 Measured mass flow rate (kg/h)
2000
50 50
Fig. 13 e Numerical and Experiment mass flow rate comparison of R410A compressor.
100 150 200 250 300 Measured mass flow rate (kg/h)
350
Fig. 15 e Numerical and Experiment mass flow rate comparison of CO2 compressor.
expansion valve located at the discharge line, while the suction pressure can be adjusted by changing the opening rate of EEV (electronic expansion valve) located at the bypass line. The specified superheat is obtained by controlling the additional heater installed at the suction chamber. The major testing instrumentation of the test rig is the same for R410A and CO2. We made adjustment on the specific model number of pressure transducers and mass flow meters, since the pressure and mass flow rate varies dramatically for those two refrigerants. We also ensured the plastic sealed parts are compatible for both R410A and CO2. The testing rig utilizes thermocouples (Omega KMQSS-125G-6) and pressure transducers (Omega PX32B1-2.5KGV for CO2 and PX32B11KGV for R410A, respectively) to measure and adjust the suction pressure, suction temperature, discharge pressure, mass flow measurements with a mass flow meter (MicroMotion
DH25 for CO2 and DH100 for R410A, respectively), a volumetric flow meter (Sponsler SP717) and electric power analyzer with accuracies of 0.05K, 0.25%, 0.5%, 0.5% and 1% respectively.
4.2.
Model validation
The comparison between the model predictions and experimental data are illustrated in Figs. 13e16 . For the single-stage R410A compressor, the deviations of mass flow rate and power consumption between predictions and experiments are mostly within ±3% and ±5%, respectively. For the two-stage CO2 compressor, the deviations of mass flow rate and power consumption between predictions and experiments are mostly within ±8% and ±5%, respectively. The present model
7
Predicted power consumption (kW)
Predicted power consumption (kW)
22 +5%
20
-5%
18 16 14 12 10 8
+5% 6
-5%
5
4
3 8
10
12
14
16
18
20
22
Measured power consumption (kW) Fig. 14 e Numerical and Experiment power consumption comparison R410A compressor.
3
4 5 6 Measured power consumption (kW)
7
Fig. 16 e Numerical and Experiment power consumption comparison of CO2 compressor.
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accuracy is very competitive in comparison with the reciprocating compressor models in the open literature. However, there are still some room for improving the model accuracy. The deviation of mass flow rate for the CO2 compressor is fairly larger than that of the R410A compressor. One reason is that there is a discharge plenum in the R410A compressor to decrease the pressure pulsation and the corresponding pressure loss effect. The CO2 compressor does not have such design. Since the pressure pulsation is very difficult to model, it wasn't taken into account in the simulation, which introduces additional deviation in simulating the CO2 compressor. Another reason is the assumption of isentropic compression for calculating the leakage mass flow rate for CO2 may not be very accurate.
5.
Conclusions
In this paper, a new generic network model of reciprocating compressors was developed. The whole compressor is divided into individual components. Inside each component model, the conservative equations were used to describe physical sub-processes. Between components, a network model involving refrigerant flow, power flow, heat flow, and air flow was developed to describe the connections in a generic way so that arbitrary configuration of reciprocating compressors can be easily modeled. Based on the OOP method, a graphical drag-and-drop modeling platform was developed. Same methodology might be extended to other types of refrigerant compressors modeling. A single-stage R410A compressor and a two-stage transcritical CO2 compressor were modeled and validated with experimental data. For the single-stage R410A compressor, the deviations of mass flow rate and power consumption between predictions and experiments are mostly within ±3% and ±5%, respectively. For the two-stage CO2 compressor, the deviations of mass flow rate and power consumption between predictions and experiments are mostly within ±8% and ±5%, respectively.
Acknowledgments This work is partially supported by the National Natural Science Foundation of China (Grant No. 51206123) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 11ZZ30).
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