Accepted Manuscript Comparison of heat transfer models for reciprocating compressor J. Tuhovcak, J. Hejcik, M. Jicha PII: DOI: Reference:
S1359-4311(16)30608-1 http://dx.doi.org/10.1016/j.applthermaleng.2016.04.120 ATE 8168
To appear in:
Applied Thermal Engineering
Received Date: Accepted Date:
8 January 2016 22 April 2016
Please cite this article as: J. Tuhovcak, J. Hejcik, M. Jicha, Comparison of heat transfer models for reciprocating compressor, Applied Thermal Engineering (2016), doi: http://dx.doi.org/10.1016/j.applthermaleng.2016.04.120
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Comparison of heat transfer models for reciprocating compressor
J Tuhovcak1, J Hejcik and M Jicha Brno University of Technology, FME, Energy Institute, Technická 2, Brno 616 69, CZ E-mail: [email protected]
Abstract. One of the main factors affecting the efficiency of reciprocating compressor is heat transfer inside the cylinder. An analysis of heat transfer could be done using numerical models or integral correlations developed mainly from approaches used in combustion engines; however their accuracy is not completely verified due to the complicated experimental set up. The goal of this paper is to analyse the effect of heat transfer on compressor efficiency. Various integral correlations were compared for different compressor settings and fluids. CoolProp library was used in the code to obtain the properties of common coolants and gases. A comparison was done using the in-house code developed in Matlab, based on 1st Law of Thermodynamics.
Keywords: Reciprocating compressor, heat transfer, integral correlations, volumetric efficiency, isentropic efficiency
Highlights: • Comparison of integral heat transfer models • Influence of heat transfer model on volumetric and isentropic efficiency • Various gases used as working fluid
To whom any correspondence should be addressed.
Thermodynamic efficiency of reciprocating compressors is around 80 %, which is considerably low compared to their electrical or mechanical efficiency. The losses are mainly caused by pressure losses in suction or discharge line, gas leakage and superheating of the working fluid. The last one is the main object of interest in this paper. The main source of heat in the compressor is the compression process itself. Increasing the pressure of gas by reducing its volume causes a necessarily temperature rise. Heat is then transferred to cylinder walls and further to other components. The problem arises when the fresh gas flows through the heated suction line in the cylinder during the suction process. The fresh gas has much lower temperature than the surrounding walls and the heat transfer direction is opposite, from walls to the gas. A high gas temperature demands more power for the compression process than the gas with lower temperature. Almbauer  presented the correlation of compression initial temperature and the COP factor or indicated power. An approximately 1 K rise of gas temperature resulted in the COP reduction of about 0.32 %. Morriesen  also dealt with the same problem. In this case, the increase of temperature from -10 °C to 50 °C resulted in 20 % lower mass flow rate. Superheating of the fluid is present in the whole suction line and also inside the cylinder. The necessity of cylinder cooling in order to enhance higher compressor cycle efficiency was presented also by Balduzzi , who analysed cylinder water cooling. A compressor behaviour, including the heat transfer inside the cylinder, could be expressed by efficiency. Most often authors use isentropic efficiency [equations (1) and (2)], which is defined as the ratio of the specific work consumed by the actual compressor and the ideal one. The ideal compressor is assumed to be identical with the actual compressor regarding its geometry and working conditions; however these processes are reversible. Compression and expansion are adiabatic while the suction and discharge processes are isobaric. An example of difference during the compressor process is shown in Figure 2. This difference could also be found in the working fluid, which behaves as an ideal gas. ηisen =
κ RT1 ⎡⎛ p2 ⎞ ⎢⎜ ⎟ κ − 1 ⎢⎝ p1 ⎠ ⎣
(1) (κ −1) / κ
⎤ − 1⎥ ⎥⎦
Specific work of compressor could also be expressed from the 1st Law of Thermodynamic as in equation (3) (3) w = q − Δh Where wisen is the isentropic work of ideal compressor, wactual is the specific work of actual compressor, κ is the isentropic exponent, R is the gas constant, T is the temperature and p is the pressure. In equation (3) q stands for a heat and h is enthalpy. Castaing-Lasvignottes  used in dynamic simulation of compressor also volumetric efficiency, which is defined as the ratio of volumetric flows ηv =
ሶ is the flow being sucked at low pressure and ܸത௬ the swept volume. where the ܸ௦௨ The same approach was used in the unsteady analysis of Longo , however in volumetric efficiency he used actual mass flows instead of volumetric flows. The difference is just in expressing density though. The work of Pérez-Segarra  offers a detailed analysis of common efficiencies used to describe the compressor behaviour, especially the volumetric efficiency.
A Apaart from m the t aboovee menti m ioneed disccreppancciess beetweenn thhe iddeaal coomppresssorr annd the t acttuall onne, vvoluumeetricc effficiienccy also a o cooverrs leeakaage froom the cyllindder, bacckfllow thrrouggh the t vallve, meechaaniccal ffricction n orr loossees due d to thee unnballancced angular spee s ed of mottor. Hooweeverr thhesee faactors wer w re not n cconnsideeredd inn th he pres p sentt workk duue to t tthe foccus on heaat tran t nsferr. Pérr P rez –Seegarra  [ divvideed tthe vvoluumeetricc efficcienncy intoo thhreee suub-eefficiennciees to t sepa s aratee thhe conntribbution of varriou us proc p cessses iinsiide the com mprresssor; it reeadds (5) ηv = η v, f ⋅η v,cc ⋅η v,v v w wheere ηv,ff staandss foor thhe lossees due d to thee forrcess acctingg onn th he mot m tor dduriing coomppresssionn, ηv,c is the ffirst th heorreticcal subb-voolum mettric effficieency repr r reseentinng a re-exxpaansioon of gass frrom m thhe clea c arannce vvoluumee. A raatio of the t acttual volum metrric fflow w too thee iddeal onee is thee seeconnd ssub--vollum metriic efficciency ηv,v, wh hichh caan be fuurthher split s t intto four f r rattios:: • εvv(irr) – co omppariisonn off maass ente e erinng thhe cylin c ndeer foor th he actu a ual and a ideeal com c mpreessoor • εvv(ssch) – reepreesennts ssuperchharggingg efffects and a wavves in suct s tionn linne • εvv(ssdh) – reepreesennts supe s erdiischharggingg efffectts • εvv(11) – reepreesennts cylin c ndeer leeakaage T Thee main m intteresst iss too evvaluate thee firrst rratioo, εvv(irrr), as a thhe heat h t traansffer in i tthe cyliindeer has h the moost cconnsideerabble inffluencee heere. Eqquattionn (66) expr e resses the rattio of actuual and iddeaal m masss floow rattes tthroough h thhe suct s tionn vaalve. While W e thhe iddeaal prroceess is cons c sideeredd ass addiabaticc, thhe actu a ual proc p cesss usses oonee off thee heeat trannsfeer mod m dels meentiooneed furth f her in the tex xt. Val V ve lossses andd ree-exxpannsioon of o gas g ffrom m th he ccleaarance volu v umee are coonsiiderred in both b h caases.
ε v,v (irrr ) =
H Heaat trranssferr has ann im mpoortannt innfluuencce on o both b h th he issenttrop pic andd voolum metrric effi e icienncyy. Acco A ordinng tto , [ thee moost inefficciennt prroceess in the t com mprressoor is i coomp presssioon. Kee K epinng the t tem t mperratuure of o tthe w worrkin ng fluid f d cons c stannt woul w ld resu r ult in i tthe low westt coomppresssioon wor w rk (Figure 1), buut refr r igerratioon rrequuirees hhighh tem mpeeratuure in the conndeensaationn unnit to t relea r ase heaat. T Theerefoore thee tarrgett is to mak m ke tthe ccom mpreessiion pro ocesss issenttroppic bbecaausee off zerro heat h t floow (Fig ( guree 2).
Fiigurre 1. Proc P essees inssidee the t coomppresssorr cyylind der..
Figgurre 2. 2 Com C mpaarisoon of iseentroopicc andd acctuaal prroceess in the t com c mpreesso or.
T Thee pu urpoose of this t s paaper is to t show s w thhe effe e ect of o heat h traansfe fer on o com c mpreessoor effficienccy, i.e. to shoow tthatt the diiffeerennce betw b weeen tthe actuual com mprresssor and a d thee iddeall onne iss onnly in the t heaat trranssferr. The aactuual com mpressoor uusess vaariouus corr c relattionns too caalcuulatee thhe heeat trannsfeer frrom m thee work w kingg fluuid to t the
ccyliindeer w wallls and a vicce vers v sa, whiile the ideeal com mprresssor hass the heat h traansffer sset to zerro. Man M ny aappproaachees coul c ld be b foounnd inn thhe liter l raturre for f heaat trranssfer moodelllingg; how h weveer thhe corr c rectt on ne has h nnot yett beeen statted. Thhereeforee we w seelecctedd fivve wide w ely useed corre c elattion ns annd ccom mparred thee ressultts.The ccom mpreessoor ggeomettry rem mainns the sam me as weell as the cleearaancee volume annd the t wo orkinng fluiid. O Oveerprresssuree annd uundeerprresssuree caauseed by b the t losses in vallvess arre thhe ssam me for f botth com c mpressorrs. H Heaat trranssfer is con c sideeredd onnly insi i ide the t cyllindder.
The T com mp presssorr mod m el
A sscheemaatic mo odell off a com mpreessoor is i shhow wn in i Figu F ure 3. A cran c nk m mecchannism m conv c vertts thhe rota r ary m mottion n to thee lin nearr mootioon oof thhe pistoon. The T e poositiion of the t pist p ton andd geeom metryy off thee cyylinnderr forrm tthe con ntrool voolum me useed for f ccalcculaationns. Thee mode m el uses u thee ennerggy bala b ancee annd cont c tinuuity equ uatioon tto ccalcculaate tthe fluuid flow w thhroughh thhe cylin c ndeer, the t craank--mootionn eqquaationn too caalcuulatee thhe pisto p on ppossitio on and con c ntroll voolum me, the one-deegreee-oof-ffreedom m eqquaationn to preedicct thhe valv v ve poosittionn, an nd tthe aanaalytiical equatiion forr heeat trannsfeer pred p dictiion. Thhis appproaach hass a bennefiit off faast ccalcculaation n annd ccouuld be b eeasily com c mbinned with ottherr moodeels.
Figgurre 3. Reecipproccatinng ccom mpreessoor 22.1 Flu F uid fflow w T Thee baase of 0D moodell is thee ennerggy bala b ancee ovver a cconttrol volum me calc c culaated usiing thee 1st Laaw of T Theerm modyynam miccs, equ e uatioon (7).. Booth preessuure an nd ttempperaaturre are a connsiddereed as a uunifform m in the cconntroll voolum me, whhichh iss thhe who w ole vollum me of o cyli c indeer. Thee fllow insidee thhe con c ntroll voolum me is nnegglectted.. d + dQ dW d + ∑ dmi ⋅ hi = dU d
w wheere dQ Q staandss foor heeat trannsfeer from fr m/too thee cyylinder, dm mihi is infl i low or outtflow w thhrouughh thee vaalvees, ddU is the t cha c angee of innner ener e rgyy andd dW W iss thhe piisto on work w k.
Solution procedure of this equation is similar to one presented by Farzaneh-Gord . Discrete values of temperature, pressure and the rest of the fluid properties are obtained by solving the differential equation (7) with the finite difference method for a given time-step as follows u t +1 =
1⎧ Δ V Δm ⎫ − ⎨Q + ( m S hS )t − ( m D hD )t − pt ⎬ + ut Δt Δt ⎭ m⎩
Continuity equation is solved similarly
= m in − m out
where min is the mass flow coming into the cylinder, mout is the flow escaping through discharge valve and dmcyl is the change of mass inside the cylinder over a time step. Density of the gas inside the cylinder is calculated knowing volume and mass ρt =
The rest of the thermodynamic properties are obtained employing the CoolProp library . A crankshaft cycle is usually divided into 3600 steps. The first few cycles should be used to balance the flows inside the cylinder. The volume of the cylinder is dependent on the piston position in time, Z(t), which is defined by the crankshaft geometry (r – crank diameter and l – rod length) and the actual angle position of the crank (α(t)), equation (11). Clearance volume affects the overall performance of the compressor and in equation (11) it is represented by clearance length – Z0. Values used in the work are listed in Table 1. 2
⎛r⎞ Z(t ) = r + l + Z0 − r ⋅ cos α(t ) − l 1 − ⎜ ⎟ sin 2 α(t ) ⎝l⎠
Valve behaviour is the main factor affecting the mass flow through coming into the cylinder or out. The valves are usually driven by the pressure force and the spring force of the valve system. Therefore the valve timing could vary for different working conditions. A mathematical description of valve movement was introduced by  as a mass-spring system. This description uses the single degree of freedom equation (12) solved using the 4th order Runge-Kutta method. This approach assumes the valve to be a rigid object. (12) m ⋅ x + d ⋅ x + k ⋅ x = ∑ Fi i
where m is the mass of the valve, k is the spring stiffness and d is the damping constant. For exact values used in the presented work, see the Table 1. Valve position, velocity and acceleration are expressed by x and its first and second derivative. This constant is often neglected in equation (12)  as its value is usually low and hard to obtain. Apart from pressure and spring forces, there is adhesive force acting on the valve; it is caused by the oil present on the valve surface. A flow area, Sflow, is calculated from the valve geometry and substituted in Fliegner’s equation (13) to evaluate flow through a valve: 1
⎛ p ⎞κ m i = S flow ⋅ ζ ⋅ ρ1 ⋅ ⎜ 2 ⎟ ⎝ p1 ⎠
κ −1 ⎛ ⎞ 2κ p1 ⎜ ⎛ p2 ⎞ κ ⎟ 1− ⎜ ⎟ κ − 1 ρ1 ⎜⎜ ⎝ p1 ⎠ ⎟⎟ ⎝ ⎠
The loss coefficient ζ is the function of valve position and must be determined from the experiments or from the CFD simulations. This is one of the most important steps in the procedure, as this value cannot be subtracted from primary parameters of a compressor. There is a dependency on valve geometry, type of fluid used in analysis and geometry of manifold in front of the valve and behind it. Usually the loss coefficient and flow area could be evaluated from steady state CFD analysis or experiments as a single value called “effective flow area” or as two separated values as by Lang . As the valve behavior was not the main interest of this work, the analysis was performed on simplified
geometry, using constant flow area for both valves. The rest of the symbols have a usual meaning, p is the pressure, ρ is the density, and κ is the adiabatic constant. The presented model is commonly used by researchers in compressor industry. It offers reliable results and compared to more sophisticated CFD numerical models, it is much faster. Recently similar model based on the same equations were used with some modifications in the work of ,  or by  2.2
The first model work used ideal air as the working fluid for the compressor. The results were presented at Compressors and their systems 2015 conference in London. The present work includes the library of different fluids: CoolProp . Five basic gases were used for the purposes of the work presented in this paper: Air, Carbon Dioxide and refrigerants R134a, R404a, and R507a. 2.3
An analysis of heat transfer was carried out on the compressor described below in Table 1and Table 2. The compressor was firstly used with air and afterwards with different gases - refrigerants. The temperature of the cylinder wall, piston and cylinder head was firstly set to 90 °C and then to 140 °C to examine the effect of hot wall on the efficiency. The inlet gas temperature was also changed from 50 °C to 80 °C for the same purpose.
Table 1. Compressor specification.
Table 2. Boundary conditions used in the simulation tool.
Boundary condition Value
Cylinder bore Stroke
Suction pressure Discharge pressure
50 / 80
90 / 140
Heat transfer models
Literature offers numerous approaches to heat transfer modelling, although the correct one has not yet been stated. Therefore several correlations were used in this work to verify the differences, which are mainly found in the characteristic velocity determination and in the individual constants in empirical correlations for the Nusselt number. The most significant differences occur during the suction and discharge periods when the gas velocity is highly disordered. Correlations used in this work are taken from Disconzi, Adair, Annand, Woschni and Aigner , – and they all use the nondimensional Nusselt number as the non-dimensional heat transfer coefficient (14) and the Reynolds number as the number that characterizes the gas velocity. Aigner in his correlation equation uses the Stanton number that can be directly related to the friction coefficient. (14) Nu = a Re b Pr c .
ν St = f (C f )
where D is the cylinder diameter, u is characteristic velocity and νis kinematic viscosity of the gas. One of the first semi-empirical correlations for heat transfer in the cylinder was described by Woschni, eq. (17) and Annand, eq. (18) (17) Nu = 0.035 Re 0.7 Nu = 0.7 Re 0.7 Pr 0.7
These equations were mainly developed for internal combustion engines and thus their application for the compressor is limited. A velocity distribution in the compressor cylinder could be different compared to the combustion engine as the latter has the valves operated by camshaft, not only by gas pressure and spring forces. A process inside the cylinder is different. A mixture of air and gasoline is combusted inside the cylinder of the engine, which is not present in the compressor. There are many variations of Woschni and Annand correlation, mainly with different values of the constant a and the exponent b in the general equation (14). Both coefficients in the Annand equation (15) were adjusted according to the previous experience. A difference could also be found in the characteristic velocity in the Reynolds number. Annand uses a directly mean piston speed, whereas Woschni uses two different values: during the gas exchange the velocity is calculated from the equation (16) and when the valves are closed, the equation (17) is used. (19) u = 6.618 ⋅ u p (20) u = 2.28 ⋅ u p where up is the mean piston speed. Very different values of the constant a (0.035 and 0.7 respectively) in both equations (14) and (15) are caused by differently calculated characteristic gas velocities. The Adair correlation (10), [equation(21) ], was developed directly for reciprocating compressor (21) Nu = 0.053 Re 0.8 Pr 0.6 The coefficients, i.e. the constant a (0.053) and the exponent b (0.8) are slightly (in case of the exponent) changed compared to the previous formulations but the constant a is very close to the Woschni equation (14) and very different from that in the Annand equation. This reflects firstly a differently taken characteristic diameter in the Reynolds number and secondly a differently calculated characteristic gas velocity. Adair uses the variable equivalent diameter De, De =
6 ⋅ Cylinder volume Cylinder area
instead of cylinder diameter and the approximated swirl velocity with angular speed of a crankshaft, ω, which has a larger magnitude [equations (23) and (24)]. u = 2ω ⎡⎣1.04 + cos ( 2α )⎤⎦
3 1 π <α < π 2 2
u = ω ⎣⎡1.04 + cos ( 2α )⎦⎤
1 3 π <α < π 2 2
where α is the crank angle. According to Adair, the compressor efficiency could be affected by heat transfer correlation by up to 3 percent; however the volumetric efficiency could decrease by 10 – 20 percent . One of the most crucial problems is the variation of processes inside the reciprocating compressor. Compression and re-expansion processes could be considered as more stable in terms of
velocity fluctuation as the main contribution is done by the piston movement. On the other hand, heat transfer during the suction and discharge process is greatly affected by the flow through the valves. Disconzi  solved this problem by dividing the compressor cycle into four main processes: suction, compression, discharge and expansion. For each of them she proposed a new correlation with different constants a, and exponents b and c in the general equation (11), see Table 3. Characteristic velocity is formed from the mean piston speed and the velocity of exchanging gas, equation(25). Table 3. Reynolds number and constants for processes inside the cylinder according to Disconzi. Process
ρ(t ) Du p μ( t ) ρ (t ) D
0.08 0.8 0.6
+ u p 0.8uc (t )0.2
0.08 0.8 0.6
μ (t )
ρ(t ) Du p μ( t ) ρ (t ) D
0.12 0.8 0.6
+ u p −0.4uc (t )1.4 μ (t )
0.08 0.9 0.6
Velocities in the Reynolds numbers for the individual processes are formed from the mean piston speed up and the velocity of exchanging gas uc, which is calculated from the following equation uc =
m (t )
ρ(t ) ⋅ Ac
where ρ(t) is the density of gas in the cylinder, Ac is the cross-sectional area of the cylinder. The correlation was developed using a numerical CFD simulation of simplified compressor geometry and verified for several working conditions. The last approach uses the Stanton number, which is the ratio of heat flux to the wall and the energy flux in the flow relative to the wall, equation (26). St =
Nu Re Pr
Aigner  used a full three-dimensional CFD simulation to calculate the heat fluxes and after that he proposed Stanton numbers for each process in the cylinder. An advantage of using the Stanton number is that it is directly proportional to the skin friction coefficient Cf, equation (27). There are again several expressions of friction coefficient. Equation (28) is used for compression and expansion process. For the higher Reynolds number, Bejan  suggested a small adjustment for friction coefficient, equation (29), which is used for suction and discharge process. St =
Cf 2 Pr 2/3
C f = 0.078 Re −0.25 C f = 0.046 Re
The heat flux is then expressed for each process inside the cylinder by equation(30)
q = ∑ Sti c p ρi ui ΔT
where cp is the constant pressure heat capacity, ρ is the density of gas, u stands for the reference velocity (velocity of the flow through the valves for suction and discharge and piston speed for compression and expansion), ΔT is the temperature difference between the gas and the wall and i represents the process inside the cylinder.
Results and discussion
Evaluation of heat transfer in the cylinder of reciprocating compressor is not a straightforward task to perform. There are many mathematical approaches which could be used, but not all of them give the same results. Validation with experiments is also complicated as the processes are usually very fast and the temperature sensors have a significant time lag. Therefore it is important to validate and compare these approaches at least numerically and see their effect on the compressor efficiency. Isentropic efficiency [equation (1)] was used for the comparison of models as the research is focused on the refrigerating compressor. Moreover, the volumetric efficiency was evaluated using the ratio of masses [equation (6)]. The wall temperature and the temperature of the gas at the inlet were selected as the main parameters of working conditions to be analyzed. Firstly the wall temperature was set to 90 °C, and then it was increased to 140 °C to enhance the heat transfer and superheating of incoming gas. Both cases were run for all the fluids and both inlet gas temperatures, which were alternatively set to 50 °C and 80 °C. This approach takes into account overheating in the suction line which has a significant effect on the efficiency. In addition, the results for different velocities of the crankshaft are presented. The main reason was to see the effect of piston speed on heat transfer inside the cylinder and also its influence on efficiency. The efficiency of compressor is significantly affected by heat transfer. The higher is the temperature of the gas, the higher is the specific volume of the gas, and thus less mass flowing inside the cylinder at the beginning of compression. For this reason there is a need to decrease the heat flow from the walls to the cylinder, especially in the suction line. However, the first step is to understand the basics of this process and to select a correct approach. In Figure 4 and Figure 5, the heat transfer during the crankshaft cycle was divided into 4 intervals according to the processes inside the cylinder. Interval A limits the compression, B is the discharge process, C is the expansion, and D limits the suction process that is completed almost exactly at BDC. A qualitative behavior of all models is very similar. The gas temperature rises during the compression and after that it overcomes the wall temperature; heat flux changes its direction. This is visible in Figure 4 where the curves in the interval A go from positive numbers to negative ones. The differences between heat transfer models are more visible during the discharge process. Models of Disconzi and Woschni are changing their characteristic velocity during the discharge process; this is displayed by discontinuities in the interval B, see Figure 4. The same occurs during the suction process in the interval C, where the model of Aigner also shows discontinuities because of the same reason. Figure 5 shows a wider solid area for the model of Woschni; this is caused by valve bouncing at the end of the discharge process. This solid area will also be present for the models of Disconzi and Aigner which also change the heat transfer model for open/closed valve conditions. In Figure 5, these areas are overlapped. The model of Disconzi takes into account the velocity of incoming gas causing a rapid
increase in heat transfer rate. The most significant offset from among all the models is observed in the model of Adair which uses the angular crankshaft speed to estimate the characteristic velocity for the Reynolds number. Similar results for the Adair correlation were presented in . Other models show similar tendencies during the compressor cycle; however there are important discrepancies also during the re-expansion and compression processes, when the gas velocity should be mainly affected by piston movement. The reasons are different correlations used to estimate the characteristic gas velocity inside the cylinder and different coefficients used in the Nusselt number. All equations are described in chapter 3. Models predict a different amount of heat transferred between the cylinder walls and the gas during the cycle, see Table 4 and Table 5. Positive values express the flow from the walls to the gas and the negative values the opposite process. A sum of heat fluxes during the cycle is negative for air and CO2, which means for these conditions that the prevailing heat flow is from the walls to the gas. The opposite situation is with refrigerants R134a and R404a; however the sum of transferred heat has no direct consequence on volumetric efficiency. Isentropic efficiency is influenced more directly. The higher is the total amount of transferred heat, the larger is the difference. The influence of different wall temperatures and inlet gas temperatures will be discussed in the following chapters.
Figure 4. Heat flux during the crankshaft cycle for R134 and Twall= 90°C, Tin = 50°C
Figure 5. Heat flux during the crankshaft cycle for CO2 and Twall = 140°C, Tin= 50°C. Table 4. Total heat transferred during the cycle for twall= 90 °C and tin= 50 °C. Transferred heat [J]
Table 5. Total heat transferred during the cycle for twall = 140 °C and tin = 50 °C. Transferred heat [J]
Increasing the wall temperature from 90°C to 140 °C increases the isentropic efficiency; however the volumetric efficiency is lower due to the change of density. Temperature of the gas at the beginning of compression is higher by approximately 10 °C for CO2, 11 °C for the air and 9.4 °C for R134a. Impact of different wall temperatures on isentropic efficiency is smaller for the air (2.1% decrease) and CO2 (4.1% decrease) when compared to refrigerants (16 % decrease) used in this work. A change of volumetric efficiency is very similar for all gases; however it is slightly higher for the air. Differences between the heat transfer models are larger for coolant gases; with increased wall temperature they are even more visible. Setting the wall temperature to 90°C and evaluating the maximum difference between the models, we reach 9.6 % and 3.4 % difference for isentropic and volumetric efficiency of R134a. Changing the wall temperature to 140°C, the differences are much larger, 30.4% and 6.6%. For CO2, the differences are smaller for isentropic efficiency, but slightly larger for volumetric efficiency, 5.7% and 3.8% for 90 °C wall temperature and 8.3% and 7.2% for 140°C. When the air is used as the working fluid, differences in models are even smaller for isentropic efficiency, but higher for volumetric one, similar as for CO2. The model of Adair shows the weakest sensitivity to the change of wall temperature when the isentropic efficiency and volumetric efficiency is evaluated and compared to other models. The reason could be the approximation of velocity by the angular speed of crankshaft, equations (23) and (24) together with the equivalent diameter in equation(22). If the models are mutually compared, it could be seen that those of Adair and Disconzi are the farthest from the average when volumetric efficiency is evaluated. This means that the models of Annand, Aigner and Woschni provide more similar results. However, this is not true for isentropic efficiency, but still the model of Disconzi provides higher numbers compared to the other models. During the suction process, the model of Disconzi predicts the highest amount of heat flux (see Figure 4 and Figure 5) and also the gas temperature at the beginning of the compression is the highest compared to the rest of models; therefore also the isentropic and volumetric efficiencies are influenced most. If the wall temperature is increased, it is even more visible, see Table 6 and Table 7. In this work, refrigerant R404a was also used but the results are almost identical to R134a.
Table 6. Isentropic efficiency, mass ratio and temperature at the beginning of compression for CO2 and inlet temperature Tin = 50 °C. twall = 90 °C
twall = 140 °C
Table 7. Isentropic efficiency, mass ratio and temperature at the beginning of compression for R134a and inlet temperature Tin = 50 °C. twall = 90 °C
twall = 140 °C
The influence of inlet temperatures on compressor efficiency for different heat transfer models is illustrated in
Table 8. The presented values are the ratios of work and mass calculated by the following equations
Wratio = m ratio =
Wtinlet 80 Wtinlet 50 m tinlet 80 m tinlet 50
The work ratio compares the indicated work for a cycle and the mass ratio compares the mass sucked inside the cylinder for one cycle. The wall temperature was set to 90°C. A comparison of different inlet gas temperatures shows that for an increase of 30 °C the mass sucked in the compression chamber differs approximately by 7.5% for all gases and the differences between the heat transfer models are only around 2%. The indicated work difference for these inlet temperatures is represented by work ratio. When evaluating the indicated work of compressor, the results show that coolants are more sensitive to higher inlet temperature. The difference between the models is also higher; for coolants it is 9.5 %, compared to 3.5% for CO2. The temperature at the beginning of the compression is increased by approximately 21 °C for all gases and the differences between the models are below 5%. The sensitivity of Adair model is again very weak when isentropic efficiency is evaluated. However, if the inlet temperature is changed, the volumetric efficiency is influenced most from among all the other models.
Table 8. Ratio of indicated work and mass for different inlet temperatures (Tin_50°C / Tin_80°C). Air
1.2. In the previous section, heat transfer models were only evaluated for one working condition but the dependency on the piston speed in all the models is very high. Changing the RPM is one of the ways how to regulate the compressor; therefore it is important to also evaluate the change of heat transfer rate. Firstly the speed of crankshaft was set to 3500 RPM as the reference value (100 %), and afterwards the speed was changed to 2950 RPM and 2350 RPM, see
Table 9. When CO2 was used as the working fluid, the overall heat transferred between the cylinder and the gas was not changing significantly with the decrease of the crankshaft speed. A decrease from 3500 to 2350 RPM caused less than 10 % increase of heat flux; however the efficiency of compressor remained unchanged. Similarly, for coolant R134a, both isentropic and volumetric efficiencies remain without any important change – less than 2%, but the overall heat transfer rate changes greatly, especially for Woschni and Annand models. These changes are divided among the compression, discharge, expansion and suction processes and the change of piston speed could have a different influence on these processes. Nevertheless, it was expected that the heat transfer rate will increase slightly as the time interval for heat exchange is longer. For better understanding, the overall heat transfer rate should be divided into individual processes.
Table 9.Transferred heat for different crankshaft speed –3500 RPM = 100%. tin = 50 °C, twall = 90 °C CO2
QRPM_3520 [J] 0.00
QRPM_2950 / QRPM_3500 100
QRPM_2350 / QRPM_3500 100
QRPM_3500 [J] 0.00
QRPM_2950 / QRPM_3500 100
QRPM_2350 / QRPM_3500 100
The present study evaluates the influence of heat transfer on the compressor efficiency. Several heat transfer models were examined for the purpose of comparison and to see their effects on the compressor efficiency. Different settings were applied to cover a wider range of application. All the results can be summed up in following points: •
The models predict a different amount of heat transferred during the cycle as a consequence of various correlations of the Nusselt number and the characteristic velocity used to evaluate the Reynolds number. The results show that during all the processes inside the cylinder there are important discrepancies between the models; however the trends are similar.
The isentropic efficiency of compressor is significantly influenced by the type of heat transfer model. When evaluating the volumetric efficiency, the models predict more consistent results.
Both used coolants show a similar behavior throughout the analysis. Evaluating the isentropic efficiency, coolant gases are more sensitive to the variations of wall and inlet gas temperature than CO2 and air.
CO2 and air are slightly more sensitive to temperature changes when the volumetric efficiency is evaluated. These gases have a higher thermal conductivity; however this was not the subject of the present study.
A change of wall or inlet gas temperature has a more significant effect on the efficiency than the heat transfer model, which increases the importance of accurate boundary conditions.
The angular speed of crankshaft influences the piston speed and directly the heat transfer inside the cylinder; however this influence is rather negligible. Efficiencies were almost not influenced by the change of speed.
When the Aigner approach is applied, it is critical to use a proper correlation for the Stanton number. Aigner himself adjusted the Stanton numbers according to the CFD simulation.
Compared to other models, the model of Adair does not significantly react on the change of wall temperature. The influence of inlet gas temperature is more significant, but only when the volumetric efficiency is evaluated.
Further tests and experimental validation are still required to verify the statements for different compressors and conditions. Moreover, dividing the heat transfer through the cycle into particular processes would be beneficial for better understanding of mechanisms inside the compressor.
Nomenclature Ac cp Cf D d F h k l m p Q r R t Sflow T Twall u up uc valve
[m2] [J/kg/K] [-] [m] [-] [N] [J/kg] [N/m] [m] [kg] [Pa] [J/kg] [J] [m] [J/kg/K] [s] [m2] [K] [K] [m/s] [m/s] [m/s]
cross section of the cylinder friction coefficient characteristic dimension damping constant force acting on valve specific enthalpy spring stiffness rod length mass pressure specific heat transferred heat crank diameter gas constant time flow area temperature wall temperature velocity mean piston speed flow velocity through the
U [J] internal energy valve thermalxcapacity[m] for const. pres.position w [J/kg] specific work W [J] work Z [m] piston position Z0 [m] clearance length α κ ηisen ηv ηv,f ηv,c ηv,v εv,v (irr) εv,v (sch) εv,v (sdh) εv,v (1) ρ ς μ ω
[deg] [-] [-] [-] [-] [-] [-] [-] [-] [-] [-] [kg/m3] [-] [m2/s] [rad/s]
crank angle adiabatic exponent isentropic efficiency volumetric efficiency factor of motor forces factor of clearance volume factor of actual vol. flow mass ratio supercharging ratio superdischarging ratio leakage ratio density loss coefficient kinematic viscosity angular speed
Acknowledgement The authors gratefully acknowledge funding from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme I (Project LO1202 NETME CENTRE PLUS)
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Highlights: • Comparison of integral heat transfer models • Influence of heat transfer model on volumetric and isentropic efficiency • Various gases used as working fluid