energies

Article Transient Thermodynamic Modeling of a Scroll Compressor Using R22 Refrigerant

Jai Pyo Sung 1, Joon Hong Boo 2,* and Eui Guk Jung 1,*

1 Department of Fire & Disaster Prevention Engineering, Changshin University, Gyeongsangnam-do 51352, Korea; [email protected] 2 School of Aerospace and Mechanical Engineering, Korea Aerospace University, Goyang 10540, Korea * Correspondence: [email protected] (J.H.B.); [email protected] (E.G.J.); Tel.: +82-02-300-0107 (J.H.B.); +82-055-250-1307 (E.G.J.)


Received: 18 February 2020; Accepted: 28 July 2020; Published: 31 July 2020


Abstract: In this work, we investigated the transient analysis model for the performance of a scroll compressor. A transient model was developed based on the geometry of the scroll and relevant thermodynamic relations. In particular, the mass and energy conservation equations were transformed to yield pressure and temperature variations over time, respectively. As a result, the transient behavior of the refrigerant was predicted in terms of these two parameters, and the values for the suction and discharge processes had a maximum error of 5% compared to the experimental results. The predicted discharge temperature reliably agreed with the reference values during the entire compression process. The results indicate that the analytical model developed herein is a potentially useful tool for the dynamic analysis of a scroll compressor.

Keywords: scroll compressor; transient behavior; thermodynamic analysis; modeling and simulation; geometry; leakage

1. Introduction The scroll compressor was invented by Leon Creux in 1905 [1]. The compression process of this device is the relative contact between two spiral curves. A major issue in the design is the shape of the scroll, which has a significant impact on compression performance. Numerous researchers contributed to the development of the design technique for the scroll shape [2–10]. Each scroll curve was defined as a circular involute, which is a practical and a classical method to predict its dynamic behavior. Based on the arbitrary initial angle of the involute, the scroll shape is represented by the derivation of each scroll curve from the radius of curvature, parameterized with the angle. Using this process, mathematical expressions for the scroll wrap, operating chamber volume, and leakage area were proposed [11]. If the involute shape of the scroll is determined, the pressure, temperature, and mass flow rate within each chamber volume generated by the contact points of the static and orbiting scrolls can be predicted using the conservation equations. Many geometric design techniques for scroll compressors were introduced in published research articles and patents [10–12]. Although several unique scroll geometries including the square spiral [3] and hybrid wrap [4] were proposed in previous studies, most investigations focused on the interpolation curves using basic circle involutes. In particular, these studies mainly considered geometric design techniques to develop the relationship between volume and leakage as a function of the orbital angle [5]. Blunier et al. [12] presented a mathematical model that described the operation of the scroll compressor. They also proposed a novel method for the design of the shape of a scroll wrap (represented by a circular involute) using the symmetry of the circular involute. Hirano et al. [13] and Lee and Wu [14] conducted studies on the perfect meshing profile (PMP) of a scroll compressor, a design with zero clearance volume. Qiang and Liu [15]

Energies 2020, 13, 3911; doi:10.3390/en13153911 www.mdpi.com/journal/energies Energies 2020, 13, 3911 2 of 21 developed a compression process model based on an integral equation for the scroll mechanism. Wang et al. [16] introduced a general mathematical formulation for scroll wraps, working chamber volume, and leakage areas, based on discretional initial angles of the involute of scroll compressors. Consequently, they derived a general relation that incorporated suction, compression, and discharge. Yanagisawa et al. [17] presented a mathematical model for the chamber volume of a scroll, and they achieved excellent agreement with experimental results. Liu et al. [18] presented a graphical and mathematical method for creating a modified design of the scroll wrap. Bell et al. [19] proposed a mathematical model for geometric shape considering the wall thickness of the scroll as the main variable. To predict the thermodynamic performance of the scroll compressor or expander, a few analytical models were proposed. Blunier et al. [11] used mass and energy conservation equations to express the pressure, temperature, and mass flow changes over time, using the ideal gas state equation. However, their model may not apply to refrigerants under high pressure. In general, the ideal gas model is not suitable for refrigeration systems because refrigerants typically behave non-ideally given that they operate at high pressure and are involved in transcritical processes. Dutta et al. [20] described mass and energy conservation equations using a control volume approach to predict the temperature and pressure inside a scroll and, thus, to investigate the influence of liquid refrigerant injection and heat transfer from the wall. In general, refrigerant injection of scroll compressors is a technique that bypasses a portion of the discharge gas to a suction port to prevent wet compression. These investigators utilized the correlations for convective heat transfer coefficient to estimate the heat transfer terms. Chen et al. [21] combined mass and energy balances for an open control volume to express temperature and mass as functions of the orbital angle, and they utilized the equation of state for a real gas to determine pressure. Since the differential equations for the temperature and mass include the angular velocity, the influence of the compressor frequency could be estimated. Chen et al.’s thermodynamic model [21] was adopted in several studies [22–27]. These thermodynamic models are commonly used; however, a few were modified to account for refrigerant injection [23–25]. Although these models are very useful tools for the prediction of the performance scroll compressors, a few salient points should be noted. Firstly, in the differential equation for temperature variation over the orbiting angle (θ), dT/dθ, a partial derivative term [∂p/∂T V] was included as a parameter, which implies that there is a change of pressure with respect to temperature at a constant specific volume. This term must be initially assumed at θ = 0 to initiate the numerical solution process. For θ > 0, after initiation, the refrigerant temperature inside the reference chamber at the current orbital angle step should be determined by ∂p/∂T V, which was calculated at the previous orbit angle step. Unless the refrigerant is an ideal gas, as in Dutta et al. [20], this term may cause significant deviation because the refrigerant in a compressor usually experiences a wide range of pressures, temperatures, and specific volumes. Secondly, the pressure was determined using the equation of state, considering p = p (T, v). Therefore, the pressure value is solely dependent on the accuracy of this equation, for which the valid range is usually limited. Thirdly, the governing differential equations were obtained as a function of the rotation angle. This is useful for expressing the dynamic behavior of a single compressor, but direct application to predict the transient performance of entire systems is inconvenient. This aspect was studied extensively by many researchers [28]. In this study, a mathematical approach is introduced to address the aforementioned limitations of the thermodynamic model that is widely applied to the scroll compressor. The geometric model of the scroll compressor is considered to be well developed, and a suitable model can be found in the literature. Therefore, emphasis is placed on the derivation of the transient governing equations for the mass, temperature, and pressure of the refrigerant in a scroll compressor, based on a unique approach using generalized thermodynamic relations. In addition, the validation of the prediction results is provided. Energies 2020, 13, x FOR PEER REVIEW 3 of 22 Energies 2020, 13, 3911 3 of 21 2. Geometry and Force Analysis of the Scroll Compressor

2. GeometryEnergiesThe 2020geometric,and 13, x FOR Force PEERdesign Analysis REVIEW of ofa thescroll Scroll is Compressorthe main characteristic that determines3 ofthe 22 scroll compressor’s performance. A scroll is conceptually an independent metal strip machined into an 2.The Geometry geometric and designForce Analysis of a scroll of the is the Scroll main Compressor characteristic that determines the scroll compressor’s involute curve. As shown in Figure 1a, this device consists of a static (or fixed) scroll and an performance. A scroll is conceptually an independent metal strip machined into an involute curve. orbitingThe scroll. geometric On one design side, theof astatic scroll scroll is theis fimainxed on characteristic a flat base, thatand thedetermines orbiting the scroll, scroll which is As showncompressor’s in Figure performance.1a, this device A scroll consists is conceptually of a static an (or independent fixed) scroll metal and strip an orbitingmachined scroll. into an On one placed on a rotating plate, is inserted from the other side. Both scrolls are geometrically identical, side,involute the static curve. scroll As isshown fixed in on Figure a flat 1a, base, this anddevice the consists orbiting of scroll,a static which (or fixed) is placed scroll onand a an rotating but their basic circles are eccentric with an offset distance. Typically, the scroll located at the top is fixed plate,orbiting is inserted scroll. fromOn one the side, other the side.static Bothscroll scrollsis fixed areon a geometrically flat base, and the identical, orbiting but scroll, their which basic is circles to the frame or compressor casing, whereas the scroll located at the bottom moves in an orbital pattern areplaced eccentric on a with rotating an o plate,ffset distance.is inserted Typically,from the other the scrollside. Both located scrolls at theare geometrically top is fixed toidentical, the frame or duebut to their the basiceccentricity circles are of eccentric the motor with shaft, an offset with di stance.a rotational Typically, center the scroll that locatedis aligned at the to top th eis staticfixed scroll. compressor casing, whereas the scroll located at the bottom moves in an orbital pattern due to the Figureto the 1b frame illustrates or compressor the evolution casing, whereas of the the suctio scroll nlocated chamber, at the compressionbottom moves inchamber, an orbital and pattern discharge eccentricity of the motor shaft, with a rotational center that is aligned to the static scroll. Figure1b chamber.due to Thethe eccentricity void space of created the motor by theshaft, two with scrolls a rotational consists center of a thatsuction is aligned cham berto th volumee static inscroll. which the illustrates the evolution of the suction chamber, compression chamber, and discharge chamber. The void refrigerantFigure 1b is drivenillustrates into the the evolution scroll set, of a thecompression suction chamber, chamber compression volume enclosed chamber, by theand twodischarge scroll wraps, space created by the two scrolls consists of a suction chamber volume in which the refrigerant is driven andchamber. a discharge The void chamber space createdvolume by in the which two scrolls the refrigerant consists of ais suction forced cham to exitber volumethe scroll in whichset through the the into the scroll set, a compression chamber volume enclosed by the two scroll wraps, and a discharge centerrefrigerant region. is As driven the intoorbiting the scroll scroll set, eccentrically a compression rotates chamber around volume the enclosed shaft, theby the volume two scroll of the wraps, refrigerant chamberand a discharge volume inchamber which volume the refrigerant in which the is forcedrefrigerant to exitis forced thescroll to exit set the through scroll set thethrough center the region. intercepted by the compression chamber continues to decrease (e.g., V④ > V③ > V② > V①) and As thecenter orbiting region. scrollAs the eccentrically orbiting scroll rotateseccentrically around rotates the around shaft, thethe shaft, volume the ofvolume the refrigerant of the refrigerant intercepted high-pressure compression is achieved until final discharge. intercepted by the compression chamber continues to decrease (e.g., V④ > V③ > V② > V①) and by the compression chamber continues to decrease (e.g., VO4 > VO3 > VO2 > VO1 ) and high-pressure compressionhigh-pressure is achieved compression until is ac finalhieved discharge. until final discharge.

(a) (b) (a) (b) FigureFigure 1. Schematic1. Schematic diagram diagramof of aa scroll structure. structure. (a ()a A) Ascroll scroll set. set. (b) (theb) geometry the geometry of a scroll of a scrollchamber chamber. Figure 1. Schematic diagram of a scroll structure. (a) A scroll set. (b) the geometry of a scroll chamber TheThe shape shape of of the the scrollscroll can be be represented represented by bythe theinvolute involute of a circle, of a circle,as depicted as depicted in Figure in2. By Figure 2. definition,The shape the of distance the scroll between can be points represented on the involute by the involuteand any pointof a circle, on the as tangent depicted of the in Figurebasic 2. By By definition, the distance between points on the involute and any point on the tangent of the basic definition,circle is constant. the distance between points on the involute and any point on the tangent of the basic circle is constant. circle is constant.

Figure 2. Involute of a basic circle.

FigureFigure 2. 2.Involute Involute ofof a a basic basic circle. circle.

Energies 2020, 13, 3911 4 of 21

As such, each of the static and orbiting scrolls is defined by two involutes (inner and outer ones) that are developed around a basic circle with a radius R and are offset by a constant distance. In this study, the geometric modeling of the scroll compressor was based on Reference [6]. This reference considered a scroll expander, for which the geometric modeling method for the scroll is the same as that of the compressor. However, it was considered that the rotating direction of the orbiting scroll of the compressor should be opposite to that of the expander. Geometric models based on suction, compression, and discharge volumes were presented by Ma et al. [6], wherein a method for obtaining the contact points of the static scroll and the orbiting scroll was described in detail. As shown in Figure2, the thickness of the scroll wrap is usually determined by the definition of the initial angles (αo and αi) of the outer and inner involutes. Any axial position of the involutes can be expressed based on the angles (ϕo and ϕi) of the inner and outer involutes, as well as the initial angles of the inner and outer involute functions [16]. Numerous studies on the geometry of scrolls considered new calculations for the volume of the suction, compression, and discharge chambers, including the interaction of static and orbiting arcs based on the initial and involute angles of the involute. To predict the force generated by the gas pressure, a unit normal vector must be defined for the inner and outer involutes of the orbiting scroll. The elements of a unit normal vector for a typical discretized curve can be defined as Equation (1) [29].

   dy  dϕ  nx = r   2  2  dx + dy  dϕ dϕ n =    (1)  dx  − dϕ  ny = r   2  2  dx + dy  dϕ dϕ

As shown in Equation (1), there are unit normal vectors in the x and y directions at arbitrary points on the curve, and these normal vectors are inversely related to each other. For an orbiting scroll, since the direction of the unit normal vector is the direction of the pressure force, the unit normal vector is chosen so that it points toward the scroll wrap. The unit normal vector for the inner and outer involutes of the orbiting scroll is defined by Equation (2) [19].

 _  n = sin(ϕ)iˆ + cos(ϕ) j n =  oi − (2) o  _  noo = sin(ϕ)iˆ cos(ϕ) j − Similarly, the unit normal vector for the static scroll is expressed as follows:

 _  n = sin(ϕ)iˆ cos(ϕ) j n =  si − (3) s  _  nso = sin(ϕ)iˆ + cos(ϕ) j − As shown in Equations (1) and (2), the components of the unit normal vector for a parameterized curve are represented as a general mathematical expression (Equation (1)). There are two unit normal vectors for a given point on the curve (x and y components). As shown in Equation (1), the unit vectors have the same size, but their directions are reversed by the product of 1. As a result, Equation (1) − was defined as a unit vector that points toward the inside and the outside (see Equation (2)). For the involutes of the orbiting scroll, the direction of the unit normal vector was chosen to face the scroll wrap because the force of the pressure acts in the direction of the scroll wrap. Energies 2020, 13, x FOR PEER REVIEW 5 of 22 Energies 2020, 13, 3911 5 of 21 Given that the direction of the unit normal vector points toward the orbiting scroll wrap and the pressure vector and the unit normal vector coincide, the differential force vector acting on the Given that the direction of the unit normal vector points toward the orbiting scroll wrap and the orbiting scroll can be defined as Equation (4). pressure vector and the unit normal vector coincide, the differential force vector acting on the orbiting  scroll can be defined as Equation (4).(−−= )sin ()ϕ ˆ (−+ )cos ()ϕ jhppihppdF =  oi +1 skk +1 skk dFo  _  (4) dF = ()()()()(p ++ −= p )hs sinsin(ϕϕ)iˆˆ+ (p ++ −− p )hs coscos(ϕ)ϕj jhppihppdF dF =  oi − k 11 − k skkoo k 11− k skk (4) o  _  dFoo = (pk+1 pk)hs sin(ϕ)iˆ (pk+1 pk)hs cos(ϕ) j where hs is the height of the scroll, and− Equation (4) can− be defined− as the force per unit length. whereThehs is total the heightforce per of theunit scroll, length and for Equationthe x and (4) y directions can be defined that act as theon the force inner per unitspiral length. curve of the orbitingThe totalscroll force is denoted per unit by length Equation for the(5), xdefineand yddirections as the sum that of the act differential on the inner forces spiral as curve follows: of the orbiting scroll is denoted by Equation (5), defined as the sum of the differential forces as follows:  = Ni  F []()+ −−= hpp ϕ)sin(  ,xoi i=PN 1 skk  i=1 = Foi,x = [ (pk+1 pk)hs sin(ϕ)] Fxy  − − (5) F =  i==1Ni (5) xy  i=N  F P []()()+ −= hpp cos ϕ Foi,y, yoi = [(pk+11 pk)hs cosskk (ϕ)]  i=i=1 − wherewhereN Nis theis the total total number number of discretized of discretized involute involute angles. angles. FromFrom Equatuib Equatuib(5), (5), thethe netnet forceforce actingacting onon thethe wrapwrap ofof the orbiting scroll is defined defined by by Equation Equation (6). (6).

q 2 += 2 oi= 2 , + FFF 2 ,yoixoi (6) Foi Foi,x Foi,y (6)

FigureFigure3 depicts3 depicts the geometricthe geometric model model of the of scrolls the generatedscrolls generated using MATLAB using MATLAB 2015a. The 2015a. positions The ofpositions the static of andthe orbitingstatic and scrolls orbiting are shown scrolls forare fourshown di ffforerent four orbital different angles orbital during angles 360 degreesduring 360 of rotation.degrees Whenof rotation. the scroll When compressor the scroll is in operation,compressor the is orbiting in operation, scroll rotates the counterclockwiseorbiting scroll rotates along thecounterclockwise circular orbit with along a radius the circular equal toorbit the eccentricitywith a radius between equal theto the two eccentricity basic circles between (see Figure the1 twob). Figurebasic 3circles represents (see Figure the operation 1b). Figure cycle 3 of represents the scroll compressorthe operation with cycle the evolutionof the scroll of thecompressor compression with the evolution of the compression chamber (② in the figure) and the discharge chamber (① in the chamber (O2 in the figure) and the discharge chamber (O1 in the figure). The refrigerant is sucked throughfigure). theThe suction refrigerant port is (see sucked also Figurethrough1b) the at thesuctio inletn ofport the (see static also scroll. Figure The 1b) orbital at the angle inlet (ofθ )the is definedstatic scroll. as the The angle orbital between angle the (θ) lineis defined that connects as the angle the centers between of thethe basicline that circles connects of the the static centers and orbitalof the scrollsbasic circles and the of horizontalthe static and axis. orbital scrolls and the horizontal axis.

(a)

Figure 3. Cont. Energies 2020, 13, 3911 6 of 21 Energies 2020, 13, x FOR PEER REVIEW 6 of 22

(b)

(c)

(d)

Figure 3. Scroll positions of a scroll compressor for four different orbital angles during a cycle. (a) θ = Figure 3. Scroll positions of a scroll compressor for four different orbital angles during a cycle. (a) 0, 2π. (b) θ = π/2. (c) θ = π. (d) θ = 3π/2. θ = 0, 2π.(b) θ = π/2. (c) θ = π.(d) θ = 3π/2.

Energies 2020, 13, 3911x FOR PEER REVIEW 7 of 22 21

When θ = 0 or 2π rad, as in Figure 3a, the discharge chamber and two compression chambers When θ = 0 or 2π rad, as in Figure3a, the discharge chamber and two compression chambers are denoted by ① and ②, respectively, for convenience. The suction chamber is defined as the are denoted by O1 and O2 , respectively, for convenience. The suction chamber is defined as the open open volume near the suction port between the static and orbiting scroll wraps. It can be observed volume near the suction port between the static and orbiting scroll wraps. It can be observed that the that the volume of the compression chamber continues to decrease as the orbital angle increases to volume of the compression chamber continues to decrease as the orbital angle increases to π/2, π, and π/2, π, and then 3π/2, as it is driven toward the center of the scrolls. When θ = π/2, as shown in then 3π/2, as it is driven toward the center of the scrolls. When θ = π/2, as shown in Figure3b, a new Figure 3b, a new compression chamber (③) is created. This chamber was previously the suction compression chamber (O3 ) is created. This chamber was previously the suction chamber in Figure3a, chamber in Figure 3a, but the definition was changed because it is completely enclosed by the static but the definition was changed because it is completely enclosed by the static and orbiting scroll and orbiting scroll wraps due to the increase of the orbiting angle. When the orbital angle reaches wraps due to the increase of the orbiting angle. When the orbital angle reaches 2π rad, both scrolls 2π rad, both scrolls resume their initial positions and a geometric cycle is completed. The resume their initial positions and a geometric cycle is completed. The refrigerant achieves the desired refrigerant achieves the desired high pressure after undergoing a few cycles. As the volume of the high pressure after undergoing a few cycles. As the volume of the compression chamber reaches compression chamber reaches its minimum value near the center region, it is driven in the normal its minimum value near the center region, it is driven in the normal direction toward the discharge direction toward the discharge chamber. If the orbital scroll rotates in the opposite direction chamber. If the orbital scroll rotates in the opposite direction (clockwise) and the location of the suction (clockwise) and the location of the suction and discharge chambers are interchanged, the effect of and discharge chambers are interchanged, the effect of expanding the working fluid can be observed, expanding the working fluid can be observed, as in a scroll expander [6]. as in a scroll expander [6].

3. Transient Transient Thermodynamic Model of the Scroll Compressor

3.1. Formulation Formulation of of the the Thermodynamic Thermodynamic Model For aa high-sidehigh-side scroll scroll compressor, compressor, from from the entrance the entrance to discharge, to discharge, the thermodynamic the thermodynamic processes processesassociated associated with the refrigerant with the include refrigerant a pressure include drop a andpressure heat transferdrop and in theheat suction transfer line, in back the pressuresuction line,processes, back heatpressure exchange processes, with theheat scroll exchange walls, heatwith exchangethe scroll with walls, the heat motor, exchange mechanical with components the motor, mechanicaland compressor components shell, leakage and compressor through tolerances, shell, leakage and so on.through The basictolerances, assumptions and so used on. toThe simplify basic assumptionsthe complex problemsused to simplify are as follows the comp [23lex–27 problems]: are as follows [23–27]:

1. TheThe refrigerant refrigerant inside inside the the chamber chamber is is homogeneous; homogeneous; 2. The effects of gravity and kinetic energy due to velocity are negligible; 2. The effects of gravity and kinetic energy due to velocity are negligible; 3. The effect of a lubricant is neglected; 3. The effect of a lubricant is neglected; 4. The heat transfer between the compressor wrap and the working fluid is negligible. 4. The heat transfer between the compressor wrap and the working fluid is negligible. Figure 4 illustrates the control volume for the scroll chamber with energy inflows and outflows. As shownFigure in4 illustratesthe figure, the the control refrigerant volume normally for the flow scrolls in chamber the radial with direction energy inflowsof the scroll and outflows.(denoted byAs shownthe subscript in the figure, r) and the the refrigerant flank (tangential) normally flowsdirection in the of radial the control direction volume of the scroll(denoted (denoted by the by subscriptthe subscript f). Inr) andparticular, the flank if a (tangential) bypass concept direction such of as the vapor control injection volume is (denoted applied, by the the number subscript of entrancesf ). In particular, may increase. if a bypass Since concept the suchcontrol as vaporvolume injection could isexchange applied, theenergy number with of many entrances adjacent may volumesincrease. with Since different the control thermodynamic volume could states, exchange the energyenthalpy with inflow many to adjacent the control volumes volume with may diff erenthave differentthermodynamic values. states,However, the enthalpy the enthalpy inflow outflow to the controlfrom the volume specified may control have di volumefferent values. is considered However, to bethe the enthalpy same at outflow all exits. from the specified control volume is considered to be the same at all exits.

Figure 4. Control volume for the compression chamber with energy balance. Figure 4. Control volume for the compression chamber with energy balance. The conservation of the mass of the refrigerant within the static and orbiting scrolls is expressed by Equation (7) as follows:

Energies 2020, 13, 3911 8 of 21

The conservation of the mass of the refrigerant within the static and orbiting scrolls is expressed by Equation (7) as follows: Xn Xn dm = dm dmout (7) in − l=1 l=1 where m is the mass of the working fluid (refrigerant). The conservation of energy equation is expressed by Equation (8) as follows:

Xn Xn δQ + h dm h dmout = dH VdP (8) in in − − l=1 l=1 where H, h, and Q are the total enthalpy of the homogeneous working refrigerant, the specific enthalpy, and the amount of thermal energy transferred from the compressor surface to the refrigerant, respectively. The subscripts in and out denote the inflow and outflow of the control volume. Since m = ρV, where ρ is the density of the refrigerant, the left side of Equation (7) can be expressed as Equation (9), considering the total derivative. dm = ρdV + Vdρ (9)

Now that ρ is a function of the pressure (p) and temperature (T), in general, the differential form of Equation (7) with respect to time (t) can be written as Equation (10).

  ! ! Xl=n Xl=n dm dV  ∂ρ dp ∂ρ dT  dmin dmout = ρ + V +  = (10) dt dt  ∂p dt ∂T dt  dt − dt T p l=1 l=1

Rearranging Equation (10) for dp/dt, the variation of pressure over time, yields Equation (11).

 l=n l=n  !  dp 1  1 X dmin X dmout  ρ dV ∂ρ dT  =       (11) dt ∂ρ V  dt − dt  − V dt − ∂T dt  = = p p l 1 l 1 ∂ T

As such, mass conservation is rewritten as the pressure change over time (dp/dt). The full derivative of the total enthalpy (H) over time can be written as Equation (12).

dH dh dm = m + h (12) dt dt dt Differentiation of the energy conservation of Equation (8) with respect to time and substituting in Equation (13) yields the following:

l=n l=n dh dm δQ X dm X dm dp m + h = + h in h out + V (13) dt dt dt in dt − dt dt l=1 l=1

Considering h = h (p, T), in general, the time derivative of the specific enthalpy included in Equation (13) can be expressed as follows: ! ! dh ∂h dp ∂h dT = + (14) dt ∂p T dt ∂T p dt

Substitution of Equation (14) into Equation (13) yields

  ! ! Xl=n ! Xl=n !  ∂h dp ∂h dT  dm dQ dmin dmout dp m +  + h = + hin h + V (15)  ∂p dt ∂T dt  dt dt dt − dt dt T p l=1 l=1 Energies 2020, 13, 3911 9 of 21

In the preceding equation, the term dp/dt can be expressed using Equation (11). In addition, the term (∂h/∂p) can be expressed using the thermodynamic equation for generalized enthalpy [28] as   T  ∂h = 1 + T ∂ρ p ρ 2 T . Substituting these terms into Equation (15) and rearranging yields an equation ∂ T ρ ∂ p for the rate of change of temperature with respect to time, dT/dt, as shown in Equation (16).

l=n   " l=n l=n ! # dQ P  dm  ∂ρ P dm P dm ρ + (h h) in T 1 in out + dV dt in dt ρ2 ∂T V dt dt V dt dT l=1 − p l=1 − l=1 = (16)   "    2 # dt  2 −   ∂ρ ∂ρ   ∂h  mT 1 ∂ρ  ∂h T     2 m T 2 ∂ρ T  ∂T p ∂p ρ ∂T  ∂ p − ρ ∂ P T − P ∂p T

In Equation (16), energy conservation is expressed in terms of the rate of temperature change over time (dT/dt). The three parameters in the form of partial derivatives in this equation, i.e.,∂ρ/∂p , T ∂ρ/∂T , and ∂h/∂T , are combinations of the two independent thermodynamic properties of the p |p refrigerant. These values can be obtained using the REFPROP9.1 software. The parameter Q is defined as the thermal energy transferred to the refrigerant from the outside through the scroll wrap. Its value can be ignored if the adiabatic condition is considered. To solve Equations (11) and (16) simultaneously, each control volume change rate over time [=dV/dt] must be entered with the mass flow (dm/dt) for the inlet, outlet, and the leakage of the compressor. The magnitude of each control volume can be regarded as the most important characteristic used in the thermodynamic model of the compressor. The chamber volume (V) can be determined using the scroll curve equation based on the geometric model described in Section2. To calculate dV/dt, however, it is necessary to express dt in terms of the parameter(s) used in the geometrical model. The relationship between the rate of change of the volume over time, dV/dt, and the rate of change of the orbiting angle over time, dθ/dt, can be utilized for this purpose, as in Equation (17). dV dV dθ dV = = ω (17) dt dθ × dt dθ × where ω is the angular velocity of the shaft that can be obtained by converting the compressor input frequency (f ) via ω = 2π f . As shown in Equations (10), (11), (16), and (17), the mass, pressure, and energy conservation equations presented in this study are not based on a fluid dynamic analysis of the shape of the chamber or the configuration of the flow passage. Instead, the analysis is a simplified lumped thermodynamic model that considers only the geometric dimensions of the refrigerant chamber. Using Equation (17), the time interval dt can be obtained using Equation (18).

dθ dθ dt = = (18) ω 2π f

Finally, the compressor work can be expressed as Equation (19).

. dV W = P (19) − dt

3.2. Leakage Mass Flows The mass flow rate that leaks through the radial clearance between the flanks of the wrap and the axial clearance between the tip of the wrap and the endplate can be determined by applying the one-dimensional compressible flow equation at a nozzle, assuming an isentropic process [22–27].

  2/k  (k+1)/k1/2 dm 2k pd pd = CdApu dt r(k 1)Tu pu − pu (20) −     k pd 2 k 1 for − pu ≥ k+1 Energies 2020, 13, 3911 10 of 21

1   (k+1)/(k 1) 2 dm k 2 − = CdApu dt rTu k+1 (21)     k pd 2 k 1 for < − pu k+1 where r and k are defined as the gas constant and the ratio of the specific heat, respectively. In addition, the leakage area A is the sum of the leakage area in the tangential direction (Af) and the radial leakage area (Ar), and their expression was described in detail in Ma et al. [6]. As shown in Equations (20) and (21), if the pressure ratio (pd/pu) is lower than a critical value, the mass flow velocity is not related to the downstream pressure due to choking. In this study, Cd was defined as 0.1 based on the experiment results of R22 in Reference [24].

3.3. Solution Technique The calculation process used in this study is described in detail in Figure5. The basic geometric conditions, as well as the temperature and pressure conditions, were set to be the same as that of a published report [23], as summarized in Table1. Upon applying the input values shown in Table1, the geometric motion equations can then be solved for discretized involute and orbital angles. When the contact condition between the static scroll and the orbiting scroll is met, the volume of each chamber is calculated for all involute angles. The simulation tool is programmed to complete the static and orbiting scroll geometric configuration when the chamber number obtained for the contact condition and the input number of the chamber are equal. When the volume change that occurs during the specified time change (dV/dt) is calculated, this can be applied to Equations (11) and (16), and these governing equations can be solved simultaneously by the fourth-order Runge–Kutta algorithm to obtain the pressure and temperature. Energies 2020, 13, x FOR PEER REVIEW 11 of 22

FigureFigure 5. 5.Flow Flow chart of of the the simulation simulation process. process.

Table 1. Parameters and basic input conditions of the scroll compressor [23]. Items Dimensions Basic circle radius (mm) 2.71 Thickness of the scroll (mm) 3.70 Height of the scroll (mm) 35.7 Number of circles 2.78 Involute start angle (rad) 0.8221 Involute end angle (rad) 19.03 Refrigerant R22 Basic suction pressure (kPa) 324.4 Basic suction temperature (°C) 4.8

In Figure 3, the shape of the scroll set consists of a suction chamber, two compression chambers, and a discharge chamber. By solving for the suction, compression, and discharge chambers in sequence, the discharge pressure and discharge temperature are obtained. The pressure, temperature, and mass flow rate calculated using the temperature and pressure input to the suction port (inlet of the suction chamber in Figure 3) can be specified as the inlet conditions of the neighboring chamber to obtain the temperature, pressure, and mass flow rate of the corresponding chamber outlet. Using this procedure, the exit values of the last chamber (chamber 1 in Figure 3) were calculated and defined as the pressure, temperature, and mass flow rate of the discharge port. If the orbital angle and frequency are given, the corresponding time can be calculated, as shown in Equation (18). Therefore, the algorithm in the red box of Figure 5 was used to perform iterative

Energies 2020, 13, 3911 11 of 21

Table 1. Parameters and basic input conditions of the scroll compressor [23].

Items Dimensions Basic circle radius (mm) 2.71 Thickness of the scroll (mm) 3.70 Height of the scroll (mm) 35.7 Number of circles 2.78 Involute start angle (rad) 0.8221 Involute end angle (rad) 19.03 Refrigerant R22 Basic suction pressure (kPa) 324.4 Basic suction temperature (◦C) 4.8

In Figure3, the shape of the scroll set consists of a suction chamber, two compression chambers, and a discharge chamber. By solving for the suction, compression, and discharge chambers in sequence, the discharge pressure and discharge temperature are obtained. The pressure, temperature, and mass flow rate calculated using the temperature and pressure input to the suction port (inlet of the suction chamber in Figure3) can be specified as the inlet conditions of the neighboring chamber to obtain the temperature, pressure, and mass flow rate of the corresponding chamber outlet. Using this procedure, the exit values of the last chamber (chamber 1 in Figure3) were calculated and defined as the pressure, temperature, and mass flow rate of the discharge port. If the orbital angle and frequency are given, the corresponding time can be calculated, as shown in Equation (18). Therefore, the algorithm in the red box of Figure5 was used to perform iterative calculations based on time. When the governing equations are solved, the mass flow rate is obtained using Equations (20) and (21), and the force acting on the curve of each chamber can be calculated using Equation (9).

4. Results and Discussion To validate the time step size, its influence on the predicted discharge pressure and temperature was investigated as summarized in Figure6. As shown in Equation (18), the time step ( dt) was calculated from the step of the orbital angle (dθ). In Figure6, the calculated discharge pressure and discharge temperature were compared with each other in (a) and (b), respectively, for five different dθ values from 1 10 3 to 12 10 3 degrees. Orbital angle steps larger than 12 10 3 degrees resulted in × − × − × − divergence of the calculation results. It is evident from the figure that, as the orbital angle step (dθ) increases, the initiation time is earlier, and the duration of the compression process is reduced as predicted. As shown in Figure6, a considerable amount of time is required for the calculation as dθ decreases and the discharge pressure is kept constant, regardless of whether dθ is in the steady state (discharge process). Therefore, the orbital angle step (dθ) should be determined by considering the calculation time and experimental results. The orbital angle step of 1 10 3 degrees yielded values that were closest to the experiment × − results, as shown in Figure7. For all calculation results that followed, therefore, dθ assumed a value of 1 10 3 degrees, which corresponded to a time step size of 0.02 ms. In Figure6a, the predicted × − initiation time of the compression process was 5.67 ms when dθ = 1 10 3 degrees. In addition, × − the changes in the predicted discharge pressure were less than 7 kPa among the tested five dθ values, indicating that the discharge pressure was not significantly affected by the step size of the orbital angle. In Figure6b, the discharge temperature was varied from 94 ◦C to 100.2 ◦C, resulting in a difference of 6 ◦C for the entire investigated dθ range. Overall, a larger step size of the orbital angle resulted in a lower discharge temperature and discharge pressure being predicted. Energies 2020, 13, x FOR PEER REVIEW 12 of 22

calculations based on time. When the governing equations are solved, the mass flow rate is obtained using Equations (20) and (21), and the force acting on the curve of each chamber can be calculated using Equation (9).

4. Results and Discussion To validate the time step size, its influence on the predicted discharge pressure and temperature was investigated as summarized in Figure 6. As shown in Equation (18), the time step (dt) was calculated from the step of the orbital angle (dθ). In Figure 6, the calculated discharge Energiespressure2020, 13 ,and 3911 discharge temperature were compared with each other in (a) and (b), respectively, for12 of 21 five different dθ values from 1 × 10−3 to 12 × 10−3 degrees. Orbital angle steps larger than 12 × 10−3 degrees resulted in divergence of the calculation results.

-3 1800 (a) dθ (×10 degrees) 1 3 7 10 12 1600

1400

1200

1000 [kPa]

d

P 800

600 dt (ms) 0.02 0.05 400 0.12 0.17 0.2 200 048121620 t (ms)

120 -3 (b) dθ (×10 degrees) 1 3 7 10 12 100

80

C] 60 ° [

d T Energies 2020, 13, x FOR PEER REVIEW40 13 of 22 dt (ms) assumed a value of 1 × 10−3 degrees,20 which corresponded to 0.02 a time step 0.05 size of 0.02 ms. In Figure 6a, 0.12 0.17 −3 the predicted initiation time of the compression process was 0.2 5.67 ms when dθ = 1 × 10 degrees. In addition, the changes in the predicted0 discharge pressure were less than 7 kPa among the tested 0 4 8 12 16 20 five dθ values, indicating that the discharge pressuret (ms was) not significantly affe cted by the step size of the orbitalFigure angle. 6. Influence In Figure of the time6b, stepthe sizedischarge to provide temperature the validity of wasthe solutions: varied (froma) on the 94 discharge °C to 100.2 °C, Figure 6. Influence of the time step size to provide the validity of the solutions: (a) on the discharge resulting inpressure; a difference (b) on the of discharge 6 °C for temperature. the entire investigated dθ range. Overall, a larger step size of the pressure; (b) on the discharge temperature. orbital angle resulted in a lower discharge temperature and discharge pressure being predicted. It is evident from the figure that, as the orbital angle step (dθ) increases, the initiation time is earlier, and the duration of the compression process is reduced as predicted. As shown in Figure 6, 1600 discharge process a considerable amount of time is required for the calculation as dθ decreases and the discharge Wang et al. [22] pressure is kept constant,1400 regardless of whether dθ is in the steady state (discharge process). Therefore, the orbital angle step (d θpresent) should model be determined by considering the calculation time and experimental results.1200 The orbital angle step of 1 × 10−3 degrees yielded values that were closest to the experiment results, as shown in Figure 7. For all calculation results that followed, therefore, dθ 1000 compression process

[kPa] 800 d P 600

400

200 suction precess

0 0 200 400 600 800 1000 1200 θ [degree]

FigureFigure 7. 7. DISCHARGEDISCHARGE pressure pressure with with respect respect to to the the orbi orbitaltal angle angle predicted predicted by by the the presented presented model model comparedcompared with with experiments. experiments.

Figure 7 compares the prediction results for the discharge pressure according to the orbital angle using the experimental results of Wang et al. [23]. The input conditions for the suction pressure and temperature, in this case, were 324.4 kPa and 4.8 °C, respectively, as shown in Table 1. As denoted in Figure 7, the basic cycle of the scroll compressor consists of a suction process, a compression process, and a discharge process. The discharge pressure remained low and constant during the suction process, increased exponentially during the compression process, and remained constant at a higher pressure during the discharge process. It was observed that the compression process began when the orbital angle reached approximately 323° in the experiment, as well as in the prediction model. The discharge process began when the orbital angle reached 881° in the experiment and 1048° in the prediction, resulting in an error of approximately 167°, a 19% relative error between the experimental and predicted values. Regarding the value of the discharged pressure, the error between the experimental and the prediction results was less than 1% during the suction process. During the compression process, however, the prediction results were higher than the experiment results by up to 17.2% (based on the absolute pressure) at the maximum value. During the discharge process, the pressure obtained based on the experimental and the prediction model was 1517 kPa and 1574 kPa, respectively, indicating that the prediction result was higher by less than 3.14% (based on the absolute pressure). Overall, the pressure predicted using the presented model reliably agreed with the experimental values during the suction and discharge processes. However, large (approximately 17%) errors were observed for the pressure during the compression process as well as the orbital angle at which the discharge process began. Figure 8 represents the discharge temperature according to the orbital angle predicted by the presented model compared to the results obtained by Wang et al. [23]. The figure shows that the presented model exhibits a similar behavior compared to that of Wang et al.

Energies 2020, 13, 3911 13 of 21

Figure7 compares the prediction results for the discharge pressure according to the orbital angle using the experimental results of Wang et al. [23]. The input conditions for the suction pressure and temperature, in this case, were 324.4 kPa and 4.8 ◦C, respectively, as shown in Table1. As denoted in Figure7, the basic cycle of the scroll compressor consists of a suction process, a compression process, and a discharge process. The discharge pressure remained low and constant during the suction process, increased exponentially during the compression process, and remained constant at a higher pressure during the discharge process. It was observed that the compression process began when the orbital angle reached approximately 323◦ in the experiment, as well as in the prediction model. The discharge process began when the orbital angle reached 881◦ in the experiment and 1048◦ in the prediction, resulting in an error of approximately 167◦, a 19% relative error between the experimental and predicted values. Regarding the value of the discharged pressure, the error between the experimental and the prediction results was less than 1% during the suction process. During the compression process, however, the prediction results were higher than the experiment results by up to 17.2% (based on the absolute pressure) at the maximum value. During the discharge process, the pressure obtained based on the experimental and the prediction model was 1517 kPa and 1574 kPa, respectively, indicating that the prediction result was higher by less than 3.14% (based on the absolute pressure). Overall, the pressure predicted using the presented model reliably agreed with the experimental values during the suction and discharge processes. However, large (approximately 17%) errors were observed for the pressure during the compression process as well as the orbital angle at which the discharge process began. Figure8 represents the discharge temperature according to the orbital angle predicted by the presented model compared to the results obtained by Wang et al. [23]. The figure shows that the presented model exhibits a similar behavior compared to that of Wang et al. Energies 2020, 13, x FOR PEER REVIEW 14 of 22

110 discharge process 100 Wang et al. [22] present model 90 80 70 compression process 60 C] °

[ 50 d T 40 30 20 10 suction process 0 0 200 400 600 800 1000 1200 θ [deg.]

FigureFigure 8.8. DischargeDischarge temperaturetemperature withwith respectrespect toto orbitalorbital angleangle predictedpredicted byby thethe presentedpresented modelmodel comparedcompared withwith thosethose byby WangWang etet al. al. [ 23[23].].

SimilarSimilar toto the the discharge discharge pressure, pressure, the dischargethe discharg temperaturee temperature remained remained constant constant during theduring suction the process,suction process, exponentially exponentially increased increased during the during compression the compression process, andprocess, remained and remained constant atconstant a higher at temperaturea higher temperature during the during discharge the process.discharge The process. compression The compression process began process when thebegan orbital when angle the wasorbital approximately angle was approximately 362◦ according 362° to Wangaccording et al.’s to result,Wang whichet al.′s corresponded result, which withcorresponded the prediction with ofthe the prediction presented of model. the presented In addition, model. the dischargeIn addition, process the discharge began when process the orbital began anglewhen was the 913orbital◦ in Wangangle etwas al.’s 913° result, in Wang but 1139 et al.◦ for′s result, the prediction but 1139° based for the on prediction the present based modeled. on the This present resulted modeled. in an errorThis ofresulted approximately in an error 226◦ ,of a 25%approximately relative error 22 between6°, a 25% the relative two values. error Regardingbetween the the valuetwo values. of the Regarding the value of the discharged temperature, the relative error was less than 1% during the suction process. However, the results for the presented model were higher than those reported in the reference by up to 4.1% during the compression and discharge processes. The prediction of the discharged temperature during the entire cycle was in good agreement with Wang et al.’s result. It is noted in Figures 7 and 8 that the presented model produced significantly different predictions, particularly for the orbital angle (and corresponding time) for the termination of the compression process (i.e., the initiation of the discharge process). The presented model predicted an exponential and smooth transition from the compression to discharge processes, whereas the experimental data revealed a sudden transition for the same processes. As shown in Figures 6 and 7, in the discharge process, the predicted results did not exhibit overshoot of pressure and temperature due to over-compression, unlike the experimental results. It was determined that the relative errors for the discharge pressure and orbiting angle were large. As noted in previous studies [30], over-compression depends on the dimensions of the outlet port, shape, and mass flowrate. To solve Equations (11) and (16) simultaneously, the volume change rate over time [= dV dt] must be obtained using the geometric motion equation. As indicated in Figure 3, there were two compression chambers, and they were named compression chambers 1 and 2 for convenience. Compression chamber 1 is adjacent to the suction chamber, whereas compression chamber 2 is close to the discharge chamber. Figure 9a shows the rate of the volume change of compression chambers 1 and 2, whereas Figure 9b shows the volumes of the two compression chambers over time. The input geometric parameters are presented in Table 1. As shown in Figure 9a, the rate of volume change exhibited a 10−3 m3/s order. The volume change rate of compression chambers 1 and 2 ranged from −3.74 × 10−4 m3/s to 2.97 × 10−3 m3/s, and the period was approximately 0.011 ms. The volume change dV/dt in Figure 9a shows the rate of increase of decrease of the volume [6]. The volume changed from positive (the increase of the cavity volume) to negative (the decrease of the cavity volume). The volume of compression chamber 1 ranged from 70 cm3 to 45 cm3 with a period of 0.011 ms.

Energies 2020, 13, 3911 14 of 21 discharged temperature, the relative error was less than 1% during the suction process. However, the results for the presented model were higher than those reported in the reference by up to 4.1% during the compression and discharge processes. The prediction of the discharged temperature during the entire cycle was in good agreement with Wang et al.’s result. It is noted in Figures7 and8 that the presented model produced significantly di fferent predictions, particularly for the orbital angle (and corresponding time) for the termination of the compression process (i.e., the initiation of the discharge process). The presented model predicted an exponential and smooth transition from the compression to discharge processes, whereas the experimental data revealed a sudden transition for the same processes. As shown in Figures6 and7, in the discharge process, the predicted results did not exhibit overshoot of pressure and temperature due to over-compression, unlike the experimental results. It was determined that the relative errors for the discharge pressure and orbiting angle were large. As noted in previous studies [30], over-compression depends on the dimensions of the outlet port, shape, and mass flowrate. To solve Equations (11) and (16) simultaneously, the volume change rate over time [= dV/dt] must be obtained using the geometric motion equation. As indicated in Figure3, there were two compression chambers, and they were named compression chambers 1 and 2 for convenience. Compression chamber 1 is adjacent to the suction chamber, whereas compression chamber 2 is close to the discharge chamber. Figure9a shows the rate of the volume change of compression chambers 1 and 2, whereas Figure9b shows the volumes of the two compression chambers over time. The input geometric parameters 3 3 are presented in Table1. As shown in Figure9a, the rate of volume change exhibited a 10 − m /s order. The volume change rate of compression chambers 1 and 2 ranged from 3.74 10 4 m3/s to − × − 2.97 10 3 m3/s, and the period was approximately 0.011 ms. The volume change dV/dt in Figure9a × − shows the rate of increase of decrease of the volume [6]. The volume changed from positive (the increase of the cavity volume) to negative (the decrease of the cavity volume). The volume of compression chamber 1 ranged from 70 cm3 to 45 cm3 with a period of 0.011 ms. Energies 2020, 13, x FOR PEER REVIEW 15 of 22

0.003 (a) dVc1/dt dVc2/dt 0.002 t 0.001 dV/d 0.000 -0.001 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 80 (b) Vc1 Vc2 70 ]

3 60 cm [ 50 V 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 t [sec]

FigureFigure 9. The variation ofof thethe volumevolume ofof the the two two compression compression chambers chambers with with respect respect to to time: time: (a )(a rate) rate of ofvolume volume change; change; (b) compression(b) compression cavity cavity volumes volumes (subscripts (subscripts c1 and c2c1 denote and c2 compression denote compression chambers 1 chambersand 2, respectively). 1 and 2, respectively).

InIn contrast, contrast, the volume of compression chamber 2 rangedranged fromfrom 7676 cmcm33 toto 4646 cmcm33 with the same period asas compressioncompression chamber chamber 1. A1. detailedA detailed description description of the of volume the volume change change in the scrollin the chamber scroll chamberresulting resulting from the from shaft rotationthe shaft wasrotation described was described in detail inin Referencedetail in Reference [6]. [6]. FigureFigure 10 shows the variation of temperature and and pressure pressure of of compression chambers chambers 1 1 and and 2 2 over over time.time. As As can can be be seen seen in in the the figure, figure, the the compression process process began began at at approximately 6 6 ms. The The pressure pressure andand temperature of of compression compression chamber chamber 1, 1, which which was was adjacent adjacent to to the the suction suction chamber, were calculated to be slightly less than those of compression chamber 2, by approximately 3 °C and 39 kPa, respectively. Since the calculation for the thermodynamic model was performed in the order of the suction chamber, compression chamber 1, compression chamber 2, and discharge chamber, the temperature and pressure were relatively higher for the regions that were closer to the discharge chamber. As such, the volume of a chamber was treated as a control, and an algorithm was utilized such that calculations were performed sequentially from the suction chamber to the discharge chamber.

90 (a) 75 compression chamber temperature Tc1 Tc2 60 C]

° 45 [

T 30 15 0 0 4 8 12 16 20 1500 (b) 1250 compression chamber pressure 1000 Pc1 Pc2 750 [kPa]

P 500 250 0 048121620 t [ms] Figure 10. The pressure and temperature variation against time in the compression chambers (subscripts c1 and c2 denote the compression chambers 1 and 2, respectively).

Figure 11 illustrates the mass flow rate of the refrigerant as a function of time for two different compressor frequencies of 30 Hz and 40 Hz. The mass flow rate was predicted using Equations (20) and (21). In the figure, the mass flow rate increases as the compressor frequency increases, and it

Energies 2020, 13, x FOR PEER REVIEW 15 of 22

0.003 (a) dVc1/dt dVc2/dt 0.002 t 0.001 dV/d 0.000 -0.001 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 80 (b) Vc1 Vc2 70 ]

3 60 cm [ 50 V 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 t [sec] Figure 9. The variation of the volume of the two compression chambers with respect to time: (a) rate of volume change; (b) compression cavity volumes (subscripts c1 and c2 denote compression chambers 1 and 2, respectively).

In contrast, the volume of compression chamber 2 ranged from 76 cm3 to 46 cm3 with the same period as compression chamber 1. A detailed description of the volume change in the scroll chamber resulting from the shaft rotation was described in detail in Reference [6]. EnergiesFigure2020, 13 10, 3911 shows the variation of temperature and pressure of compression chambers 1 and 152 over of 21 time. As can be seen in the figure, the compression process began at approximately 6 ms. The pressure and temperature of compression chamber 1, which was adjacent to the suction chamber, were calculated to be slightly less than those of compression chamber 2, by approximately 3 ◦C and 39 kPa, respectively. Sinceto be slightly the calculation less than for those the thermodynamicof compression chamber model was 2, by performed approximately in the order3 °C and of the39 kPa, suction respectively. chamber, compressionSince the calculation chamber for 1, compressionthe thermodynamic chamber model 2, and was discharge performed chamber, in the theorder temperature of the suction and chamber, pressure werecompression relatively chamber higher 1, for compression the regions chamber that were 2, closer and discharge to the discharge chamber, chamber. the temperature As such, and the pressure volume ofwere a chamber relatively was higher treated for the as regions a control, that andwere an closer algorithm to the discharge was utilized chamber. such As that such, calculations the volume were of a performedchamber was sequentially treated as a from control, the suctionand an algorithm chamber to wa thes utilized discharge such chamber. that calculations were performed sequentially from the suction chamber to the discharge chamber.

90 (a) 75 compression chamber temperature Tc1 Tc2 60 C]

° 45 [

T 30 15 0 0 4 8 12 16 20 1500 (b) 1250 compression chamber pressure 1000 Pc1 Pc2 750 [kPa]

P 500 250 0 048121620 t [ms]

FigureFigure 10.10.The The pressure pressure and and temperature temperature variation variation against ag timeainst in thetime compression in the compression chambers (subscriptschambers c1(subscripts and c2 denote c1 and the c2compression denote the compression chambers 1 andchambers 2, respectively). 1 and 2, respectively).

FigureFigure 1111 illustratesillustrates thethe massmass flowflow raterate ofof thethe refrigerantrefrigerant asas aa functionfunction ofof timetime forfor twotwo didifferentfferent compressorcompressor frequenciesfrequencies ofof 3030 HzHz andand 4040 Hz.Hz. The massmass flowflow raterate waswas predictedpredicted using Equations (20) Energies 2020, 13, x FOR PEER REVIEW 16 of 22 andand (21).(21). InIn the the figure, figure, the the mass mass flow flow rate rate increases increases as the as compressorthe compressor frequency frequency increases, increases, and it variesand it with a constant period. Since the total leakage area, A, in Equations (20) and (21) includes the leakage varies with a constant period. Since the total leakage area, A, in Equations (20) and (21) includes the areas in the flank and radial directions, the mass flow rate exhibited a periodic variation. leakage areas in the flank and radial directions, the mass flow rate exhibited a periodic variation.

0.05

0.04

0.03

0.02

compressor frequency

Massflow rate [kg/s] 30 Hz 40 Hz 0.01

0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 t [sec]

Figure 11. PredictedPredicted refrigerant mass flow flow rate as a functionfunction of time and the compressor frequency.

For a compressor frequency of 30 Hz, the mass flow rate varied between 0.023 kg/s and 0.044 kg/s with a constant period. For a compressor frequency of 40 Hz, the mass flow rate varied periodically between 0.025 kg/s and 0.046 kg/s. Figure 12 represents the discharge temperature over time when the compressor frequency was varied from 30 Hz to 60 Hz, with 10-Hz increments. As the compressor frequency was increased, an earlier initiation time of the compression process was predicted. When a steady state was achieved, the discharge temperature for the compressor frequency of 60 Hz was approximately 0.9 °C higher than that for the compressor frequency of 30 Hz. The compression process was initiated at 5.8, 4.9, 3.9, and 2.4 ms for the input conditions of 30, 40, 50, and 60 Hz respectively.

100

80

60 C] ° [ d 40 T

compressor frequency 20 30 Hz 40 Hz 50 Hz 60 Hz 0 0 2 4 6 8 101214161820 t (ms) Figure 12. Predicted discharge temperature as a function of time for different compressor frequencies.

Figure 13 shows the variation of discharge pressure over time as the compressor frequency was varied from 30 Hz to 60 Hz for the same increments. As the compressor frequency was increased, an earlier initiation time of the compression process was predicted as shown in Figure 13. When the steady state was reached, the discharge pressures at 30 Hz and 60 Hz were calculated to be 1571 k Pa and 1574 kPa respectively, resulting in a difference of only 3 kPa. Based on the results shown in

Energies 2020, 13, x FOR PEER REVIEW 16 of 22

varies with a constant period. Since the total leakage area, A, in Equations (20) and (21) includes the leakage areas in the flank and radial directions, the mass flow rate exhibited a periodic variation.

0.05

0.04

0.03

0.02

compressor frequency

Massflow rate [kg/s] 30 Hz 40 Hz 0.01

Energies 2020, 13, 3911 16 of 21 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 t [sec] For a compressor frequency of 30 Hz, the mass flow rate varied between 0.023 kg/s and 0.044 kg/s Figure 11. Predicted refrigerant mass flow rate as a function of time and the compressor frequency. with a constant period. For a compressor frequency of 40 Hz, the mass flow rate varied periodically between 0.025For kga compressor/s and 0.046 frequency kg/s. of 30 Hz, the mass flow rate varied between 0.023 kg/s and 0.044 Figurekg/s 12with represents a constant the period. discharge For a compressor temperature frequency over time of 40 when Hz, the mass compressor flow rate frequency varied was varied fromperiodically 30 Hz between to 60 Hz,0.025 with kg/s and 10-Hz 0.046 increments. kg/s. As the compressor frequency was increased, Figure 12 represents the discharge temperature over time when the compressor frequency was an earlier initiation time of the compression process was predicted. When a steady state was achieved, varied from 30 Hz to 60 Hz, with 10-Hz increments. As the compressor frequency was increased, an the dischargeearlier initiation temperature time of for the the compression compressor process frequency was predicted. of 60 When Hz was a steady approximately state was achieved, 0.9 ◦C higher than thatthe for discharge the compressor temperature frequency for the compressor of 30 Hz. frequency The compression of 60 Hz was process approximately was initiated 0.9 °C athigher 5.8, 4.9, 3.9, and 2.4than msfor that the for inputthe compressor conditions frequency of 30, 40,of 30 50, Hz. and The 60 compression Hz respectively. process was initiated at 5.8, 4.9, 3.9, and 2.4 ms for the input conditions of 30, 40, 50, and 60 Hz respectively.

100

80

60 C] ° [ d 40 T

compressor frequency 20 30 Hz 40 Hz 50 Hz 60 Hz 0 0 2 4 6 8 101214161820 t (ms)

Figure 12. PredictedFigure 12. dischargePredicted discharge temperature temperature as a function as a func oftion time of time for for diff differenterent compressor compressor frequencies. frequencies. Figure 13 shows the variation of discharge pressure over time as the compressor frequency was Figure 13 shows the variation of discharge pressure over time as the compressor frequency was varied fromvaried 30 from Hz 30 to Hz 60 to Hz 60 Hz for for the the same same increm increments.ents. As Asthe compressor the compressor frequency frequency was increased, was increased,an an earlierearlier initiation initiation time time of of the the compression compression process process was was predicted predicted as shown as shown in Figure in Figure 13. When 13. the When the steady statesteady was state reached, was reached, the the discharge discharge pressures pressures at 30 30 Hz Hz and and 60 60Hz Hzwere were calculated calculated to be 1571 to be k Pa 1571 kPa and 1574 kPa respectively, resulting in a difference of only 3 kPa. Based on the results shown in and 1574Energies kPa 2020, respectively,13, x FOR PEER REVIEW resulting in a difference of only 3 kPa. Based on the results17 of shown22 in Figures 12 and 13, the discharge temperature and pressure of the compressor were not significantly affectedFigures by the 12 frequency.and 13, the discharge This is evident temperature in the and case pressure when the of the compressor compressor is were not installed not significantly in a system but operatedaffected by the itself, frequency. because This the is pressure evident in and the the case temperature when the compressor in the suction is not chamberinstalled in would a system be held at a constantbut operated value by at itself, different because frequencies. the pressure and the temperature in the suction chamber would be held at a constant value at different frequencies.

1750

1500

1250

1000 [kPa] d P 750

compressor frequency 500 30 Hz 40 Hz 50 Hz 60 Hz 250 0 2 4 6 8 101214161820 t (ms) Figure 13. Predicted discharge pressure as a function of time for different compressor frequencies. Figure 13. Predicted discharge pressure as a function of time for different compressor frequencies. Figure 14 shows the variation of the discharge temperature over time when the suction chamber inlet pressure was varied from 3 bar to 5 bar. The suction chamber inlet pressure and temperature were utilized as the input conditions. The suction chamber inlet temperature is defined as the saturation temperature for the input pressure. It can be estimated, based on the figure, that the saturation temperatures for the pressures of 3, 4, and 5 bar are −14 °C, −6 °C, and 0.7 °C respectively, which were calculated using the REFPROP software. The figure shows that the earlier initiation time of the compression process was predicted when a higher pressure was applied at the suction chamber inlet. The initiation times of the compression process were predicted to be 8.52, 4.45, and 2.8 ms for the suction chamber inlet pressures of 3, 4, and 5 bar, respectively. As the suction chamber inlet pressure was increased, the discharge temperature increased due to the increase of the corresponding saturation temperature. For the suction chamber inlet pressures of 3, 4, and 5 bar, the steady-state temperatures of the compression process were calculated to be 75, 82.2, and 95.1 °C, respectively.

100

80

60 C]

° 40 [ d T 20

0 suction inlet pressure 3 bar 4 bar 5 bar 0 2 4 6 8 101214161820 t (ms) Figure 14. Predicted discharge temperature as a function of time for different suction inlet pressures.

Energies 2020, 13, x FOR PEER REVIEW 17 of 22

Figures 12 and 13, the discharge temperature and pressure of the compressor were not significantly affected by the frequency. This is evident in the case when the compressor is not installed in a system but operated by itself, because the pressure and the temperature in the suction chamber would be held at a constant value at different frequencies.

1750

1500

1250

1000 [kPa] d P 750

compressor frequency Energies 2020, 13, 3911 500 30 Hz 40 Hz 17 of 21 50 Hz 60 Hz 250 Figure 14 shows the variation0 of 2 the 4 discharge 6 8 temperature 101214161820 over time when the suction chamber t (ms) inlet pressure was varied from 3 bar to 5 bar. The suction chamber inlet pressure and temperature were Figure 13. Predicted discharge pressure as a function of time for different compressor frequencies. utilized as the input conditions. The suction chamber inlet temperature is defined as the saturation temperature forFigure the 14 input shows pressure. the variation It canof the be discharge estimated, temperature based onover the time figure, when the that suction the saturation chamber inlet pressure was varied from 3 bar to 5 bar. The suction chamber inlet pressure and temperatures for the pressures of 3, 4, and 5 bar are 14 ◦C, 6 ◦C, and 0.7 ◦C respectively, which were temperature were utilized as the input conditions. Th−e suction− chamber inlet temperature is defined as calculatedthe using saturation the REFPROPtemperature for software. the input pressure. The figure It can showsbe estimated, that based the earlier on the figure, initiation that the time of the compressionsaturation process temperatures was predicted for the pressures when a of higher 3, 4, and pressure 5 bar are was−14 °C, applied −6 °C, and at the0.7 °C suction respectively, chamber inlet. The initiationwhich times were calculated of the compressionusing the REFPROP process software. were The predictedfigure shows tothat be the 8.52, earlier 4.45, initiation and time 2.8 of ms for the suction chamberthe compression inlet pressures process wasof predicted 3, 4, and when 5 a bar, higher respectively. pressure was applied As the at suction the suction chamber chamber inlet. inlet pressure The initiation times of the compression process were predicted to be 8.52, 4.45, and 2.8 ms for the suction was increased,chamber the inlet discharge pressures temperature of 3, 4, and 5 increasedbar, respectively. due toAs the the increasesuction chamber of the correspondinginlet pressure was saturation temperature.increased, Forthe the suctiondischarge chambertemperature inlet increased pressures due to of the 3, increase 4, and of 5 bar,the corresponding the steady-state saturation temperatures temperature. For the suction chamber inlet pressures of 3, 4, and 5 bar, the steady-state temperatures of of the compression process were calculated to be 75, 82.2, and 95.1 ◦C, respectively. the compression process were calculated to be 75, 82.2, and 95.1 °C, respectively.

100

80

60 C]

° 40 [ d T 20

0 suction inlet pressure 3 bar 4 bar 5 bar 0 2 4 6 8 101214161820 t (ms) Figure 14. Predicted discharge temperature as a function of time for different suction inlet pressures. Figure 14. Predicted discharge temperature as a function of time for different suction inlet pressures.

As shownEnergies 2020 in, Figures 13, x FOR 8PEER, 10 REVIEW, 12 and 14, the oscillations of the temperature and pressure are18 of predicted,22 and they were evaluated to be dependent on rotational speed and inlet pressure. Figure 15As represents shown in Figures the variation 8, 10, 12, of and the 14, discharge the oscillations pressure of the over temperature time when and the pressure suction are chamber inlet pressurepredicted, was and varied they were from evaluated 3 to 5to bar.be depe Thendent input on rotational conditions speed were and inlet the pressure. same as in Figure 14. Figure 15 represents the variation of the discharge pressure over time when the suction In Figurechamber 15, it is inlet evident pressure that, was as varied the suctionfrom 3 to chamber5 bar. The input inlet conditions pressure were increases, the same the as earlierin Figure initiation time of the14. compressionIn Figure 15, processit is evident is predicted, that, as the and suction the dischargechamber inlet pressure pressure increases increases, accordingly. the earlier For the suction chamberinitiation inlettime pressuresof the compression of 3, 4, and process 5 bar, is the predicted, steady-state and the pressures discharge of pressure the compression increases process were calculatedaccordingly. to be For 1548, the suction 1678, chamber and 1795 inlet kPa, pressures respectively. of 3, 4, and 5 bar, the steady-state pressures of the compression process were calculated to be 1548, 1678, and 1795 kPa, respectively.

2000 suction inlet pressure 1800 3 bar 4 bar 5 bar

1600

1400

1200

[kPa] 1000 d P 800

600

400

200 0 2 4 6 8 101214161820 t (ms) Figure 15. Predicted discharge pressure as a function time for different suction inlet pressures. Figure 15. Predicted discharge pressure as a function time for different suction inlet pressures. Figure 16 illustrates the variation of the force acting on the spiral line of the scroll over time as a function of the suction chamber inlet pressure. This force was predicted using Equation (6) and exhibited periodic variation. As shown in the figure, as the suction chamber inlet pressure increases, the force per unit length acting on the scroll increases. This is because the pressure difference among the different volumes increases relative to the inlet pressure. As can be seen in Figure 14, the differences between the suction chamber inlet pressure and the discharge pressure were 1230, 1255, and 1284 kPa for the inlet pressures of 3, 4, and 5 bar, respectively. This indicates that the pressure difference increased as the suction chamber inlet pressure increased. For this reason, the force increased proportionally to the increase of the inlet pressure. When the suction chamber inlet pressure was 3 bar, the force varied periodically from 5.44 kN/m to 9.05 kN/m. The force periodically varied from 5.17 to 9.15 kN/m for the suction chamber inlet pressure of 4 bar, and it varied from 4.97 to 9.22 kN/m for the inlet pressure of 5 bar.

Energies 2020, 13, 3911 18 of 21

Figure 16 illustrates the variation of the force acting on the spiral line of the scroll over time as a function of the suction chamber inlet pressure. This force was predicted using Equation (6) and exhibited periodic variation. As shown in the figure, as the suction chamber inlet pressure increases, the force per unit length acting on the scroll increases. This is because the pressure difference among the different volumes increases relative to the inlet pressure. As can be seen in Figure 14, the differences between the suction chamber inlet pressure and the discharge pressure were 1230, 1255, and 1284 kPa for the inlet pressures of 3, 4, and 5 bar, respectively. This indicates that the pressure difference increased as the suction chamber inlet pressure increased. For this reason, the force increased proportionally to the increase of the inlet pressure. When the suction chamber inlet pressure was 3 bar, the force varied periodically from 5.44 kN/m to 9.05 kN/m. The force periodically varied from 5.17 to 9.15 kN/m for the suction chamber inlet pressure of 4 bar, and it varied from 4.97 to 9.22 kN/m for the inlet pressure of 5 bar. Energies 2020, 13, x FOR PEER REVIEW 19 of 22

10 suction inlet pressure 3 bar 4 bar 5 bar 9

8

7 [kN/m] F 6

5

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 t [sec]

FigureFigure 16. PredictedPredicted time time variation variation of ofthe the force force acting acting along along the thespiral spiral line linefor different for different suction suction inlet inletpressures. pressures.

5.5. Conclusions AA time-basedtime-based thermodynamicthermodynamic model was derived as a reasonablereasonable tooltool forfor predictingpredicting thethe performanceperformance ofof thethe scroll compressor usingusing thethe R22R22 refrigerant.refrigerant. TheThe validityvalidity ofof thethe analyticalanalytical modelmodel waswas evaluatedevaluated basedbased onon thethe experimentalexperimental resultsresults ofof otherother researchers,researchers, andand thethe dischargedischarge pressurepressure duringduring thethe suctionsuction andand dischargedischarge processesprocesses exhibitedexhibited moderate accuracy with errors less than 3.2% duringduring bothboth processes.processes. However, the pressure yieldedyielded errorserrors ofof upup toto 17.2%17.2% duringduring thethe compressioncompression process.process. In the case of the discharge temperature, excellent accuracy was achieved and the maximum errorerror waswas lessless thanthan 4.5%4.5% for for the the entire entire three three consecutive consecutive processes, processes, compared compared to to the the values values reported reported in thein the literature. literature. In addition,In addition, the etheffects effects of the of simulationthe simulation and inputand input parameters parameters on the on performance the performance of the scrollof the compressorscroll compressor were examined.were examined. In the In analytical the analytical model, model, the size the ofsize the of orbital the orbital angle angle step step and and the compressorthe compressor frequency frequency did notdid significantlynot significantly affect affe thect pressure the pressure and temperature and temperature of the scrollof the outlet,scroll butoutlet, resulted but resulted in differences in differences in the initiation in the initiati time ofon thetime compression of the compression process. Theprocess. suction The chamber suction inletchamber pressure inlet significantlypressure significantly affectedthe affected compressor the compressor outlet pressure outlet pressure and temperature. and temperature. As the suction As the chambersuction chamber inlet pressure inletincreased, pressure theincreased, discharge the pressure discharge and temperaturepressure and increased temperature proportionally. increased proportionally. Author Contributions: Conceptualization, E.G.J.; formal analysis, J.P.S.; investigation, E.G.J. and J.P.S.; methodology, J.H.B.; project administration, J.H.B. and E.G.J.; supervision, J.H.B.; writing—original draft, J.H.B.;Author writing—review Contributions: and Conceptualization, editing, J.H.B. All authors E.G.J.; read investigation, and agreed to E.G.J.; the published methodology, version of theJ.H.B.; manuscript. project administration, J.H.B. and E.G.J.; supervision, J.H.B.; writing—original draft, J.H.B.; writing—review and Funding: This research was funded by Changshin University under grant number [Changshin-2019-50]. editing, J.H.B. All authors read and agreed to the published version of the manuscript.

Funding: This research was funded by Changshin University under grant number [Changshin-2019-50]

Acknowledgments: This work was also supported by the Changshin University Research Fund of 2017 (grant number: Changshin-2019-50).

Conflicts of Interest: The authors declare no conflicts of interest.

Nomenclature

A Leakage Area (m2) Cd Flow (discharge) coefficient dT/dθ Temperature variation over the orbiting angle (°C∙rad−1) dP/dT Pressure variation over the temperature (pa∙°C−1) f Compressor frequency (Hz or s−1) F Force per unit length (N∙m−1) H Enthalpy (J) h Height of scroll (m) or specific enthalpy (J∙kg−1) i Index of the involute angle step

Energies 2020, 13, 3911 19 of 21

Acknowledgments: This work was also supported by the Changshin University Research Fund of 2017 (grant number: Changshin-2019-50). Conflicts of Interest: The authors declare no conflicts of interest.

Nomenclature

A Leakage Area (m2) Cd Flow (discharge) coefficient 1 dT/dθ Temperature variation over the orbiting angle (◦C rad− ) 1 · dP/dT Pressure variation over the temperature (pa ◦C− ) 1 · f Compressor frequency (Hz or s− ) F Force per unit length (N m 1) · − H Enthalpy (J) h Height of scroll (m) or specific enthalpy (J kg 1) · − i Index of the involute angle step iˆ Unit normal vector in x-direction j Index of the orbiting angle step jˆ Unit normal vector in y-direction k Index of the chamber number m Mass (kg) N Maximum index of step n Normal vector p Pressure (pa) Q Heat transfer rate (W, J s 1) · − r Gas constant (J kg 1 K 1) · − · − R Radius of the basic circle (m) S Spiral T Temperature (◦C) t Time (millisecond, ms) u Specific internal energy (J kg 1) · − V Volume (m3) v Specific volume (m3 kg 1) · − x x-Coordinate (m) y y-Coordinate (m)

Greek Symbols

α Initial involute angle (rad) θ Orbiting angle of the scroll (rad) ϕ Involute angle (rad) ρ Density (kg m 3) · − ω Angular velocity of the compressor shaft (rad/s)

Subscripts d Discharge, downstream f Flank i Inner in Inlet l Index of inlet or outlet number n Inlet or outlet number out Outlet o Outer or orbiting involute oi Orbiting inner involute oo Orbiting outer involute r Radial Energies 2020, 13, 3911 20 of 21 s Static or scroll si Static inner involute so Static outer involute u Upstream v Specific volume (m3 kg 1) · −

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