Effects of Unsteady Heat Transfer on Behaviour of Commercial Hydro-Pneumatic Accumulators
EEffects of unsteady heat transfer on behaviour ofCommercial Hydro-Pneumatic Accumulators
Jakob Hartig a, ∗ , Benedict Depp a , Manuel Rexer a , Peter F. Pelz a a TU Darmstadt, Chair of Fluid Systems, Otto-Berndt-Straße 2, 64287 Darmstadt
Abstract
Hydraulic accumulators play a central role as energy storage in nearly all fluidpower systems. The accumulators serve as pulsation dampers or energy storagedevices in hydro-pneumatic suspensions. The energy carrying gas is compressedand decompressed, often periodically. Heat transfer to the outside significantlydetermines the transfer behaviour of the accumulator since heat transfer changesthe thermodynamic state of the enclosed gas. The accumulators operating moderanges from isothermal to adiabatic. Simulating fluid power systems adequatelyrequires knowledge of the transfer behaviour of the accumulators and thereforeof the heat transfer. The Engineer’s approach to model heat transfer in tech-nical system is Newton’s law. However, research shows, that in harmonicallyoscillating gas volumes, heat flux and bulk temperature difference change theirphase. Newton’s law is incapable of representing this physical phenomenon. Weperformed measurements on two sizes of commercial membrane accumulators.Experimental data confirm the failure of Newton’s approach. Instead the heattransfer can be modelled with an additional rate dependent term and indepen-dently of the accumulator’s size. Correlation equations for the heat transfer andthe correct accumulator transfer behaviour are given.
Keywords: transfer behaviour, hydraulic accumulator, unsteady heat-transfer,complex Nusselt-number ∗ Corresponding author
Email address: [email protected] (Jakob Hartig)
URL: (Peter F. Pelz) a r X i v : . [ phy s i c s . f l u - dyn ] D ec AS 𝐹 = 𝐹 + ෨𝐹(𝑡) 𝑧 = 𝑙 + ǁ 𝑧 ( 𝑡 ) OIL
𝐹 = 𝐹 + ෨𝐹(𝑡) (a) OILGAS ሶ𝑉 + ෨𝑉(𝑡) ሶ𝑉 (b) HYDRAULIC
ACCUMULATOR
Figure 1: Hydraulic accumulators in a a) Hydropneumatic suspension strut, b) pulsationdamper. In both cases the accumulator is excited with a time dependent volume flow rate.
1. Introduction
Hydro-pneumatic pressure accumulators - in short hydraulic accumulators- are used in oil-hydraulically operated machines and systems. Besides ap-plications like energy storage or temperature compensation, hydraulic accu-mulators are used as potential energy storage in oscillating systems e.g. inhydro-pneumatic suspensions [1] and for pulsation damping [2] c.f. Figure 1.Carrier of potential energy is the periodically compressed working gas, usuallynitrogen, which is undergoing a state change during compression and expan-sion [3]. In analogy to mechanical oscillating systems, hydraulic accumulatorsact as springs. The state-change ranges from being isothermal to being adia-batic, where the nature of the state-change is mainly dependent on excitationfrequency and the accumulator’s size due to heat transport phenomena. Bothdependencies can be captured by the ratio of two time scales: i) the duration ofthe compression cycle 2 π/ Ω and ii) the characteristic time of heat conductionin the accumulator t heat . The ratio of both time scales is a P´eclet-number P e Ω ÉCLET NUMBER 𝑃𝑒 Ω P O L Y T R O P I C E X P O N E N T 𝑛 L O SSA N G L E i n ° -2 -2 Figure 2: Polytropic exponent and lossangle for oscillating gas volume under periodic exci-tation, with (1D-Model) and without (0D-Model) consideration of boundary layer formation.The models are from [4]. which can be seen as a dimensionless frequency.
P e Ω = t heat Ω / π (1)For P e Ω → n = 1), whereas for P e Ω → ∞ thestate change is adiabatic ( n = 1 .
4) [4].Hydro-pneumatic suspensions and vibration absorbers are components oflarger technical systems. For reliable system function and adequate controllerdesign or harshness calculations, these systems have to be modelled adequatelybut concisely [5]. There is ongoing research on adequate dynamic modelling ofhydraulic accumulators in vibration applications regarding state equations [6],operating temperature [7] and heat transfer [8, 9].Lumped parameter models are commonly used to model hydraulic accu-mulators. In lumped parameter models spatial dependency of thermodynamicquantities in the gas is neglected (0D-model). Consequently heat transfer isoften assumed to be proportional to temperature difference between averagedbulk gas temperature ¯ T = 1 /V (cid:82) T d V and ambient temperature T a and there-fore governed by Newton’s law e.g. [8, 10, 6, 11]. However, results from differentfields show (see Sec. 2 and Fig. 2) that, the application of Newton’s law in alumped parameter model is questionable since the heat transfer in harmonically3xcited gas volumes is strongly influenced by the formation of boundary lay-ers [12]. The polytropic exponent in Fig. 2 for the 1D-model with boundarylayers is significantly lower in some frequency ranges compared to models inthe literature [8, 10, 6, 11]. Thus, the assumption of adiabatic state change isunacceptable for a larger frequency range than the models with Newton’s lawlet assume.This is especially relevant for system design, since the transfer behaviour, i.e.the change in pressure p in relation to change in volume V of the accumulatorhighly depends on the polytropic exponent n d p d V = n p p V , (2)where p is the loading pressure, p is the pre-charge pressure and V is theinitial gas volume [4, 13]. Therefore, in this paper we are dealing with thefollowing question: How can the behavior of hydraulic accumulators be modelled adequately butconcisely?
To answer this question a literature overview on heat transfer in periodicallycompressed gas volumes is given first. Then measurements of heat transfer indiaphragm hydraulic accumulators are presented and discussed. Furthermorethe measurements will be used for tuning a semi-empiric lumped parametermodel for heat transfer. Finally a comparison of the model and the standardmodel for hydraulic accumulators, found in the literature, is made.
2. Review of Heat transfer in periodically compressed gas volumes
As seen with Eq. 2, the transfer behaviour of hydraulic accumulators is highlydependent on the nature of the gas’ state change and therefore on the heattransfer between gas and the environment ( ˙ Q ). In the literature on hydraulicaccumulators lumped parameter models with either adiabatic state change ( ˙ Q =0) or Newtonian heat transfer ( ˙ Q ∝ ∆ T ) are used (c.f. [1] or [8, 10, 6, 11]respectively). In the latter case the heat flux ˙ Q is assumed to be proportional4o the temperature difference of the gas bulk temperature ¯ T to the environment∆ T = ( ¯ T − T a ). The proportionality is usually captured with a heat transfercoefficient α , where α is dependent on the fluid flow. The heat ˙ Q transferred tothe surrounding gas with temperature T a is assumed to yield˙ Q = − α ( ¯ T − T a ) A, (3)where A is the heat transfer surface, ¯ T is the gas temperature and α is theheat transfer coefficient. The sign is so that ˙ Q >
N u := αL/λ , where L is a specific length of theproblem an λ is the fluid heat conductivity.Although not considered in hydraulic systems, the dynamic case of heattransfer in harmonically compressed gas volumes, is relevant in different researchfields namely i) cavitation bubbles, ii) air springs, iii) reciprocating piston en-gines. In the latter the heat from compression cannot be entirely separated fromthe combustion heat. Nevertheless the models can aid in understanding heattransfer phenomena at work. A literature review on these three research areasis presented in the following.Pfriem [15] was the first to show with a theoretical analysis of a one dimen-sional model for piston engines that the heat transfer in periodically compressedgas volumes is frequency dependent due to boundary layer formation. In addi-tion to that he proved that heat flux and wall bulk temperature difference areout of phase. Pfriem formulated the energy balance for a differential element ofthe boundary layer near the wall and obtained a convection-diffusion equationfor the transported heat, which was solved by a perturbation approach for smallpressure fluctuations. Similar models have been solved to model diesel engines[16], cavitation bubbles [17, 18] and air springs [19, 4].In his work Pfriem argues that fluid elements in the boundary layer near the5all dissipate the compression heat faster than fluid elements inside the gas.This idea is illustrated in Fig. 3 with a one-dimensional model, where the topmoving wall is adiabatic and the lower one has the constant temperature T a .First, at t = 0 the gas is at its largest volume. The temperature profile of thetemperature difference ∆ T local = T ( z ) − T a is as shown. The temperature T ( z )of fluid elements near the wall is strongly influenced by the wall temperature T a . The local temperature gradient at the wall leads to a local heat flux ˙ Q local in positive z -direction, where heat is transferred from the warmer wall to thecooler fluid. Averaging T ( z ) over the cross-section results in ¯ T that is lowerthan the constant wall temperature T a . With the global temperature difference∆ T global = ¯ T − T a being negative, heat is predicted to flow from the warmerwall into the cooler fluid as well.During compression the moving wall supplies energy to the gas in the formof volume work, which heats the gas over the entire cross-section. At point ii)of the cycle, ¯ T is already greater than T a . However, the temperature of the gasnear the wall does not change as quickly as in the bulk. ∇ T in the zone near thewall still indicates heat flow from the locally warmer wall. However, the globaltemperature difference already predicts an opposite direction of heat flow. Allin all, there is a phase difference in actual heat flux and global temperature dif-ference. The Newtonian approach fails. As the compression progresses further,the temperature profile is monotonous again, so that the directions of globaland local heat flux coincide.Although the authors do not always state explicitly, all models mentionedabove show that the heat transfer coefficient α during periodic compressiononly coincides with the heat transfer coefficient for stationary heat transfer inthe case of small frequencies. For larger frequencies the models for periodicheat transfer predicts a phase difference between the heat flow and the drivingtemperature difference. It can be concluded that local heat flux and globaltemperature difference are transient and show a phase difference in periodicallycompressed gas volumes. So far only flows in one dimension were considered. Inmore than one dimension secondary flows occur and the heat fluxes vary over6 𝑣cos(Ω𝑡) 𝑧 i) ii) 𝑇 a TEMPERATURE DIFFERENCE
𝑇 𝑧 − 𝑇 𝑎 P O S I T I O N I N 𝑧 𝑎 ത𝑇 > 𝑇 𝑎 ii)i) Figure 3: Emergence of boundary layers during compression. Qualitative comparison of localtemperature difference and bulk temperature difference at i) lower turning point and ii) apoint during compression α , heat flux ˙ Q and temperatures T [26]. For harmonic oscillations the real part of the heat transfer then can besimplified to R ( ˙ Q ) = − α (cid:48) ( ¯ T − T a ) A − α (cid:48)(cid:48) Ω d ¯ T d t A. (4)To date,all measurements and nearly all theoretical analyses were performedfor cylindrical components. In cylindrical components, the compression is per-pendicular to the main direction of heat transfer. All measurements were donefor enclosures with one type of material (metal).
3. Method
To answer the research question, measurements of the heat flux in commer-cial hydraulic accumulators have to be done. In this section an overview of themethod for the investigation of heat flux, based on time varying pressure andvolume measurements, is presented, c.f. Fig. 4. Using pressure and volumemeasurements, heat flux and bulk temperature is calculated with the help ofenergy equation and constitutive relations.In Fig. 4, the first step i) are measurements of the pressure response of thehydraulic accumulators. For this a test rig was built so that the accumulatorscould be loaded with different pre-charge and load pressures and excited witha sinusoidal varying flow rate. The measurement data was then smoothed ii)8 ) MEASUREMENT OF PRESSURE RESPONSE ii) SMOOTHING OF MEASUREMENTiii) CALCULATION OF ሶ𝑄(𝑡)
AND
𝑇(𝑡) iv) LEAST-SQUARES-FIT ሶ𝑄 = d𝑈d𝑡 − d𝑊d𝑡 = 𝑓 𝑝, 𝑉𝑇 = 𝑝𝑉𝑚𝑅 CONSTITUTIVE
EQUATIONENERGY BALANCE V O L U M E 𝑉 PRESSURE 𝑝 SPLINE-FIT ሶ𝑄 = 𝐴 𝜆 𝐷 (𝑁𝑢′ 𝑇 𝑎 − ത𝑇 + 𝑁𝑢′′ Ω dത𝑇 d𝑡 ) OILGAS
𝑉 = 𝑉 sin(Ω𝑡)𝑝 = ⋯ V O L U M E 𝑉 PRESSURE 𝑝 Figure 4: Method for measuring and fitting heat flux and bulk temperature in commercialhydraulic accumulators since in step iii) derivatives are calculated. In step iii) constitutive relationsfor the gas and the momentary energy balance are used to calculate heat fluxand bulk temperature. Finally, heat flux and temperature data are used to fitparameters in Eq. 4.
The test-rig for commercial hydraulic accumulators was first presented in[13]. Nevertheless some details on the test-rig are presented below. The basicdesign is similar to a hydro-pneumatic suspension strut (see Fig. 1). A single-acting hydraulic cylinder is displaced in a path-controlled manner. Thus, oilvolume is passed into the accumulator and the resulting pressure in the gasis measured (calibrated pressure transducer with measurement uncertainty of0 . , 𝐹 T ∩ GAS PRESSUREOIL PRESSUREOIL TEMPERATURE
GAS
TEMPERATURELEVEL
ADJUSTMENT
EXCITATION P ∩ P ∩ T ∩ i) ii) mm Figure 5: Schematics and Test Rig
The volume flow is based on the measured displacement of the cylinder z cyl (uncertainty 0 . A cyl = 1963 . . Volume flow in cylinderand accumulator is the same, since pipes and fittings only act as resistanceand inductance between the cylinder and accumulator. The capacity of thepiping is negligible up to 120 bar load pressure since the maximum stiffness ofaccumulators k acc = d p/ d V ≈ N / m is two orders of magnitudes lower thanthe stiffness for oil Shell Tellus S2 HLP 46 k oil = 1 . · N / m and hydraulichoses k hose = 5 . · N / m . [27]The deviation of the excitation from a sinusoidal signal is negligible sincethe form factor X f = RM SAV R (5)where RMS is the root mean square and AVR the mean value of the absolutevalue of the signal is sufficiently small. At the highest excitation (10 mm,10 Hz) and the highest load pressure (120 bar) considered here, the form factoris X f = 1 . . A in 10 − m p in bar 20, 40 20, 40load pressures p in bar 25, 50 30, 60, 90, 120amplitude in ml 4, 8 8, 20, 40 Table 1: Test plan for hydraulic accumulators
95 mm
Figure 6: Hydraulic accumulators used: HYDAC SBO500-0,1 (left) and SBOSBO330-0,75(right)
Both cylinder displacement and pressure signal are recorded in the samedata acquisition device (dSpace DS1103) at 1 kHz. Phase lag between the twosignals is therefore much lower than the measurement frequencies.Two sizes of diaphragm hydraulic accumulators were measured (HYDACSBO500-0,1A6/112U-500AK and SBO330-0,75E1/112U-33AB030) at frequen-cies from 0.002 Hz to 10 Hz, at different amplitudes and pre-charge and loadpressures c.f. Table 1. The gas used was nitrogen and fulfilled the specificationsof DIN EN ISO 14175: N1 (type 5.0 for 0.1 l accumulator with 99 .
999 % purityand type 4.6 for 0.75 l accumulator with 99 .
996 % purity)11 .2. Temperature and Heat Flux Calculation
The pressure and flow rate measurements are used to calculate a momentarybulk temperature (the same temperature that is used in lumped parameter mod-els), a momentary heat flux to the environment and the wall temperature of theaccumulator. Unlike the temperature, there is no spatial pressure dependencyexpected, since the Mach-Number
M a = 0 .
003 and standing waves would leadto pressure drop [28], [4]. This method was first introduced by Kornhauser [26].The method has advantages in comparison to surface temperature and heat-flux measurements since the latter only show local heat-fluxes which may differdepending on location. In contrast to surface temperature measurements, theonly assumption needed is that of constant pressure in the accumulator [12]. Thedisadvantage of the method mentioned by Kornhauser, namely measurementnoise influencing the temperature and heat flux calculations, was taken care ofby cleaning the signals with the help of spline fitting.To obtain the time history of the heat flow ˙ Q over the system boundary,the first law of thermodynamics for a closed system is applied to the hydraulicaccumulator. δQ = dE − δW (6)The variation symbols indicate that W and Q , unlike E , are inexact differentials.The only form of work done is volume work, whose differential can be rewrittenas δW = − pdV with pressure p and volume V . The differential change of theinternal energy of a mass m of a calorically ideal gas is given by dE = mc v dT. (7)Also thermally ideal behavior is assumed. Throughout the experiments, nitro-gen was used. Nitrogen behaves almost ideally at 300 K to 100 bar since thecompressibility factor is very close to 1 [29]. Therefore the ideal gas law in theform pV = mRT (8)12an be used. Combining Eqs. 6 to 8 results in an expression for δQ , which onlydepends on pressure and volume δQ = γγ − p d V + 1 γ − V d p. (9)Additionally the relations γ = c p /c v and R = c p − c v are used. Derivation anddividing by the heat transfer surface A yields ˙ q for the heat flux density1 A δQδt = ˙ q = γγ − p ˙ V + 1 γ − V ˙ p. (10)To calculate ˙ q from Eq. 10 the time signals of p, V and their first time derivatives˙ p, ˙ V are necessary. Small fluctuations in the measurement data with a typicaltime scale much lower than the time scale of the oscillations would lead to largedeviations in the derivatives. Therefore, the measurement data was spline fittedbefore fitting the Nusselt number.Calculation of gas mass m was done with the ideal gas law and temperatureand pressure data from the stationary pre-charge state. The thermodynamic calculation of the previous section provides the timehistory of ˙ q , T ( t ) and ˙ T . These quantities can be linked with each other byEq. 4.However, more common is a dimensionless formulation of α with N u := αD h /λ in Eq. 4 ˙ Q = A λD h ( N u (cid:48) ( T a − ¯ T ) + N u (cid:48)(cid:48)
Ω d ¯ T d t ) . (11)The frequency Ω can be written dimensionless as a P´eclet Number P e := Ω D h λ/ ( c p (cid:37) ) . (12)For one frequency the time series of ˙ q - T - ˙ T defines the unknown real andimaginary parts of the Nusselt number Nu (cid:48) , Nu (cid:48)(cid:48) . Therefore a nonlinear least-square fit is performed with Eq. 11 to determine the unknown parameters N u (cid:48) and
N u (cid:48)(cid:48) averaged over one compression-expansion cycle.13 H AS E D I FF E R E N C E 𝜙 PÉCLET-NUMBER 𝑃𝑒 Ω
25 bar,0.1 l50 bar, 0.1 l30 bar, 0.7 l60 bar, 0.7 l
90 bar, 0.7 l120 bar, 0.7 l 𝜋 Figure 7: Phase difference φ between mean bulk temperature ¯ T and heat flux ˙ q over P´eclet-Number P e Ω .
4. Results and Discussion
The results from the measurements introduced in Sec. 3 are discussed in thissection. First the temperature and heat-flux calculations are discussed. Afterthat the Nusselt-Fit is discussed.
Gas bulk temperature and heat flux were calculated from pressure and vol-ume data. Thermodynamic consistency is given since the residuum of Eq. 6is in the order of one per mille in relation to the calculated heat flux for allfrequencies.Theoretical considerations from Section 2 predict a phase difference betweenthe gas bulk temperature ¯ T and the heat flux ˙ q . Calculating the phase differencebetween these two from the measurement data results in Fig. 7. The phase dif-ference of the different measurements correlates well with P e Ω despite differentaccumulator sizes and load pressures.At low frequencies, in the isothermal region, when temperature rises so slowlythat bulk temperature ¯ T is equal to the local temperature, heat-flux is negativeand proportional to temperature. Therefore, expectations are, that heat-flux14nd temperature are out of phase at low frequencies with a phase difference of π . Then the phase difference decreases until it increases again.In contrast to that, the measurements of φ have one maximum at π and oneminimum at π/
2. Close inspection of the time course of volume and pressuremeasurements at the lowest measurement frequencies (0.002 Hz) for the 0.1 laccumulator shows a non-sinusodial behaviour where the p - V -hysteresis is notclosed. The system was still in a transient state. Two causes may be the reasonfor that: i) Due to constraints in measurement time for the lowest frequenciesonly two instead of five full vibrations were measured. The two oscillations maybe not enough to guarantee that the transient response has decayed. ii) Themeasurements were done in a non-climatised environment. Therefore fluctua-tions in environment temperature or radiant heat cannot be excluded at theselong measurement times. In the following, the measurements at 0.002 Hz wereexcluded.Nevertheless, a phase-difference is visible and failing of the Newtonian ap-proach for heat transfer is evident for spatially averaged models of hydraulicaccumulators. Nusselt Numbers were fitted from the bulk temperature and heat-flux cal-culations for two different accumulator sizes and different pre-charge pressures.The results in Fig. 8 show good correlation with the dimensionless variables
P e Ω and N u for the two different accumulators at different load pressures. Thedifferent volume and heat-transfer between the two accumulators seems to becaptured by the hydraulic diameter D and P e Ω .The trend of the real part of the Nusselt number N u (cid:48) in Fig. 8 can be dividedinto two parts with different slopes. The slope changes at about
P e Ω = 4 · .For small P e Ω the real part N u (cid:48) is larger than the imaginary part
N u (cid:48)(cid:48) .Thus heat flow and temperature difference are exactly out of phase. For high
P e Ω , N u (cid:48) ≈ N u (cid:48)(cid:48) applies and the heat flow precedes the temperature differenceby π/
2. 15
ÉCLET-NUMBER 𝑃𝑒 Ω R E A L P A R T o f N U SS E L T 𝑁 𝑢 ′ I M A G I N A R Y P A R T o f N U SS E L T 𝑁 𝑢 ′′
25 bar,0.1 l
50 bar, 0.1 l
30 bar, 0.7 l60 bar, 0.7 l90 bar, 0.7 l120 bar, 0.7 l
Figure 8: Nusselt Fit for different Pressures and Accumulator sizes
The division in two parts is visible in the model from Pfriem [15] and themeasurements from Kornhauser [12] (see Fig. 9). Compared to the results fromKornhauser [12] however, the real part of Nusselt
N u (cid:48) is one order of magnitudelarger and the P´eclet-number, where the slope change occurs is about 1.5 mag-nitudes higher. We mainly attribute that to the non-similarity of the geometryof the two problems. The heat-transfer area and therefore the surface area tovolume ratio has an enormous influence on the magnitude of
N u and
P e Ω . Thelargest difference between the experiments described here and the measurementsof Kornhauser is the rubber membrane in hydraulic accumulators compared tothe full metal gas enclosure in Kornhauser’s experiments. A Nusselt-fit with thefull accumulators surface (metal + rubber) gave no correlation between the twodifferent accumulator sizes. We therefore concluded, that the rubber surfacemust be excluded from the heat transfer. Comparing thermal diffusivity a formetal (5 · − / s) and NBR-rubber (1 . · − / s) leads to an order ofmagnitude difference between them. Consequently, the Fourier-number is oneorder of magnitude different for the two walls.Another minor difference is the moving element in the gas enclosures: In Ko-16 ÉCLET-NUMBER 𝑃𝑒 Ω R E A L P A R T o f N U SS E L T 𝑁 𝑢 ′ I M A G I N A R Y P A R T o f N U SS E L T 𝑁 𝑢 ′′ FITALL MEASUREMENTS
PFRIEM
KORNHAUSER
Figure 9: All measured Nusselt-Numbers compared with fit from eq. 15 and 16 and resultsfrom Pfriem [15] and Kornhauser [12] rnhauser’s experiment a piston moved up and down, whereas in our experimentsa rubber membrane moves in a complex way in more than one dimension. Incontrast to parallel streamlines in pistons [24] different primary and secondaryflow patterns and therefore mixing of boundary layers at the rubber wall isprobable to occur during movement.The difference in heat transfer area due to the rubber-membrane and thegeneral surface area to volume ratio difference, accounts for about
N u (cid:48) acc ≈ N u (cid:48)
Kornhauser (13)and
P e (cid:48)
Ωacc ≈ P e (cid:48)
ΩKornhauser , (14)compared with Kornhausers results.To use the data in models, a nonlinear least-square fit of the Nusselt-Numberswas done. The data-sets with smallest amplitude (4 ml for 0.1 l, 16 ml and 20ml for 0.75 l) were used at different loading pressures. The data can be fitted17ith log( N u (cid:48) ) = 2 .
011 + 0 . · log( P e Ω ) , (15)for the real part, where R-square = 0.9679 and root mean squared error = 0.1624and log( N u (cid:48)(cid:48) ) = − . . · log( P e Ω ) (16)for the imaginary part, where R-square = 0.9840 and root mean squared error= 0.1411. Thus the fitting equations account for most of the variation in thedata (see for Fig. 9).
5. Extended Lumped Parameter Model for Hydraulic Accumulators
The frequency response of accumulators is highly dependent on heat trans-fer. In this chapter the lumped parameter model for hydraulic accumulatorscommonly found in the literature on fluid power systems is compared to an ex-tended one. In the latter, the influence of periodic compression of gas on heattransfer is integrated.For the state change the two axioms mass conservation (cid:37) d V d t + V d (cid:37) d t = 0 (17)and energy conservation c v V ( d T d t (cid:37) + T d (cid:37) d t ) + T (cid:37)c p d V d t = ˙ Q, (18)need to hold where V is the volume of the accumulator, c v and c p are the specificheat capacities of the gas.As stated above, we assume ideal behaviour for the gas p = (cid:37)RT, (19)where p is the pressure in the accumulator, (cid:37) is the gas density, R is the specificgas constant and T is the temperature. However, all calculations in this papercan be done for non-ideal behaviour. 18n our case the accumulator volume V is changed dynamically, denoted by V = V + ˆ V sin(Ω t ) , (20)where the index 0 denotes the pre-charged average working state of the accu-mulator. In the literature eq. 3 is used for the heat flux ˙ Q .Linearization and transformation of equations 3 and 17 to 20 into frequencyspace yields the p/V -transfer behaviour [4] k + = ˆ p + ˆ V + = − iγN u/P e Ω − γ − iγN u/P e Ω + 1 , (21)where ˆ p + = ˆ p/p and ˆ V + = ˆ V /V . The hat denotes small deviations aroundthe initial, pre-charged state. As used above, N u is the Nusselt number and
P e Ω the dimensionless frequency.Equation 21 is the transfer behaviour of hydraulic accumulators commonlyfound in the literature on fluid power systems (compare Fig. 2).Using eq. 4 instead of 3 for the heat-flux yields k + = ˆ p + ˆ V + = − γ /P e Ω ( N u (cid:48)(cid:48) − iN u (cid:48) ) γ/P e Ω ( N u (cid:48)(cid:48) − iN u (cid:48) ) + 1 (22)Equation 22 is a generalization of equation 21. In general N u is a function of
P e Ω . For this Eq. 15 and 16 were used.In Fig. 10 the two models are compared with measurement data. The newextended model is significantly better than standard model.Furthermore, the model seems to be useful for extrapolation, when comparedwith Fig. 2. The drop in stiffness due to standing waves, as described in [4], isvisible.
6. Conclusion and Outlook
It can be concluded, that the usual assumptions of accumulators being adi-abatic or Newton’s law for heat transfer are wrong for hydro-pneumatic accu-mulators for
P e Ω ≥ .In the measurement data for harmonically excited hydraulic accumulators aphase difference between heat flux and temperature is measurable.19 ÉCLET-NUMBER 𝑃𝑒 Ω i) 0.1 l, 25 bar ii) 0.75 l, 60 bar 𝑁𝑢 = fn(Pe)
𝑁𝑢 = 𝑐𝑜𝑛𝑠𝑡.
MEASUREMENT . D I M E N S I O N L E SS S T I FF N E SS 𝑘 + Figure 10: Comparison of dimensionless stiffness k + of measurement, old model eq. 21 with Nu = const. and new model eq. 22 with Nu = fn( P e Ω ) for i) 0.1 l accumulator and ii) 0.75 laccumulator For Pe-Number
P e Ω ≥ heat transfer in hydraulic accumulators can bemodelled semi-empirically but universally for two different sizes of accumulators References [1] W. Bauer, Hydropneumatic Suspension Systems, Springer -Verlag BerlinHeidelberg, Berlin, Heidelberg, 2011. doi:10.1007/978-3-642-15147-7 .URL http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=371633 [2] Z. Zuti, C. Shuping, W. Huawei, L. Xiaohui, D. Jia, Z. Yuquan, The ap-proach on reducing the pressure pulsation and vibration of seawater pistonpump through integrating a group of accumulators, Ocean Engineering173 (2019) 319–330. doi:10.1016/j.oceaneng.2018.12.078 .URL [3] F. Korkmaz, Hydrospeicher als Energiespeicher, Springer Berlin Heidel-berg, Berlin, Heidelberg, 1982. doi:10.1007/978-3-642-81737-3 .204] P. Pelz, J. Buttenbender, The dynamic stiffness of an air-spring, in:ISMA2004 International Conference on Noise & Vibration Engineering,2004, pp. 20–22.[5] R. Isermann, Mechatronische Systeme: Grundlagen, 2nd Edition, Springer,Berlin, 2008. doi:10.1007/978-3-540-32512-3 .URL http://site.ebrary.com/lib/alltitles/docDetail.action?docID=10218006 [6] S. F. van der Westhuizen, P. Schalk Els, Comparison of different gas modelsto calculate the spring force of a hydropneumatic suspension, Journal ofTerramechanics 57 (2015) 41–59. doi:10.1016/j.jterra.2014.11.002 .[7] P. Els, B. Grobbelaar, Heat transfer effects on hydropneumatic suspensionsystems, Journal of Terramechanics 36 (4) (1999) 197–205. doi:10.1016/S0022-4898(99)00012-9 .[8] A. Pourmovahed, D. R. Otis, Effects of Thermal Damping on the DynamicResponse of a Hydraulic Motor-Accumulator System, Journal of DynamicSystems, Measurement, and Control 106 (1) (1984) 21–26. doi:10.1115/1.3149658 .[9] A. Pourmovahed, D. R. Otis, An Experimental Thermal Time-ConstantCorrelation for Hydraulic Accumulators, Journal of Dynamic Systems,Measurement, and Control 112 (1) (1990) 116–121. doi:10.1115/1.2894128 .[10] S. Rotth¨auser, Verfahren zur Berechnung und Untersuchung hydropneu-matischer Speicher, Ph.D. thesis, RWTH Aachen (15.01.1993).[11] T. H. Ho, K. K. Ahn, Modeling and simulation of hydrostatic transmissionsystem with energy regeneration using hydraulic accumulator, Journal ofMechanical Science and Technology 24 (5) (2010) 1163–1175. doi:10.1007/s12206-010-0313-8 . 2112] A. A. Kornhauser, J. L. Smith, Application of a Complex Nusselt Num-ber to Heat Transfer During Compression and Expansion, Journal of HeatTransfer 116 (3) (1994) 536–542. doi:10.1115/1.2910904 .[13] M. Rexer, P. Kloft, F. Bauer, J. Hartig, P. F. Pelz, Foam accumula-tors: packaging and weight reduction for mobile applications, 12th Interna-tional Fluid Power Conference, Technische Universit¨at Dresden 2020 doi:10.25368/2020.26 .[14] M. Kind, H. Martin, P. Stephan, W. Roetzel, B. Spang, H. M¨uller-Steinhagen, X. Luo, M. Kleiber, R. Joh, W. Wagner, et al., VDI HeatAtlas, Springer, 2010.[15] H. Pfriem, Der periodische w¨arme¨ubergang bei kleinen druckschwankun-gen, Forschung auf dem Gebiet des Ingenieurwesens A 11 (2) (1940) 67–75.[16] K. Elser, Instation¨are W¨arme¨ubertragung bei periodisch adiabater Verdich-tung turbulenter Gase, Forschung auf dem Gebiete des Ingenieurwesens21 (3) (1955) 65–74. doi:10.1007/BF02595010 .[17] M. S. Plesset, Prosperetti A., Bubble Dynamics and Cavitation, AnnualReview of Fluid Mechanics 9 (1977) 145–185.URL [18] P. F. Pelz, A. Ferber, On pressure and temperature waves within a cavi-tation bubble: Ann Arbor, Michigan, USA, 16 - 20 August [2009, in: 7thInternational Symposium on Cavitation 2009 (CAV 2009), Vol. 2.[19] K. Lee, A simplistic model of cyclic heat transfer phenomena in closedspaces, in: iece, Vol. 2, 1983, pp. 720–723.[20] C. Willich, C. N. Markides, A. J. White, An investigation of heat transferlosses in reciprocating devices, Applied Thermal Engineering 111 (2017)903–913. doi:10.1016/j.applthermaleng.2016.09.136 .2221] W. J. D. Annand, D. Pinfold, Heat Transfer in the Cylinder of a MotoredReciprocating Engine, in: SAE Technical Paper Series, SAE Technical Pa-per Series, SAE International400 Commonwealth Drive, Warrendale, PA,United States, 1980. doi:10.4271/800457 .[22] B. Lawton, Effect of compression and expansion on instantaneous heattransfer in reciprocating internal combustion engines, Proceedings of theInstitution of Mechanical Engineers, Part A: Power and Process Engineer-ing 201 (3) (1987) 175–186.[23] A. A. Kornhauser, J. L. Smith, The Effects of Heat Transfer on Gas SpringPerformance, Journal of Energy Resources Technology 115 (1) (1993) 70–75. doi:10.1115/1.2905972 .[24] U. Leki´c, J. B. Kok, Heat transfer and fluid flows in gas springs, The OpenThermodynamics Journal 4 (1) (2010).[25] W. J. Annand, T. H. Ma, instantaneous heat transfer rates to the cylinderhead surface of a small compression-ignition engine, Proceedings of theInstitution of Mechanical Engineers 185 (1) (1970) 976–987.[26] A. A. Kornhauser, Gas-Wall Heat Transfer During Compression and Ex-pansion, Dissertation, Massachusets Institute of Technology (1989).[27] D. Findeisen, S. Helduser, ¨Olhydraulik: Handbuch der hydraulischenAntriebe und Steuerungen, 6th Edition, VDI-Buch, Springer Vieweg,Berlin, 2015. doi:10.1007/978-3-642-54909-0 .[28] J. Spurk, N. Aksel, Fluid mechanics, Springer Science & Business Media,2007.[29] L. Nelson, E. F. Obert, Laws of corresponding states, AIChE Journal 1 (1)(1955) 74–77. doi:10.1002/aic.690010111doi:10.1002/aic.690010111