The 13th Scandinavian International Conference on Fluid Power, SICFP2013, June 3-5, 2013, Linköping, Sweden

Modeling Hydraulic Accumulators for use in Wind Turbines

Henrik Brun Hansen and Peter Windfeld Rasmussen

R&D/Fritz Schur Energy, Glostrup, DenmarkE-mail: hbh@fsenergy.dk, pwr@fsenergy.dk

Abstract

One of the major challenges in numerical simulation of hydraulic systems, is the long com-putation times of accumulators. In wind power applications, the accumulators must providenecessary hydraulic energy during emergency stops, while the weight of hydraulic equipmentmust be minimized. Therefore, precise and efficient design tools for accumulators are essen-tial.This paper addresses the issue of understanding the dynamic phenomena in piston type ac-cumulators, and how this leads to improved numerical accumulator models. Developing anumerical model has two challenges. First, the unsteady heat transfer between the accumu-lator gas and the wall has to be described. Secondly, a suitable real gas model has to beidentified and employed. To verify the model, a series of experiments were conducted atFritz Schur Energy in Glostrup, Denmark. The experiments were designed to investigategas dynamical properties at various precharge pressures, maximum pressures, and ambienttemperatures. These parameters were varied to obtain parameter independent conclusions.During controlled piston movements, hydraulic and gas pressures were measured togetherwith the piston position and the gas temperature.It was found that the simple thermal time constant approximation by Rotthäuser was suitableand stable for the application. It was also found that the Soave-Redlich-Kwong equation wasoverall best suited with experimental data. The Soave-Redlich-Kwong equation is as muchas six times faster than the widely used Benedict-Webb-Rubin equation, independent of theambient temperature, maximum pressure and precharge pressure.This project concludes that the Soave-Redlich-Kwong equation should be used in simulationof piston type accumulators. It is noted that the pressure available after expansion is some5-10 bar lower than predicted by any model. Therefore, it is suggested that more researchis conducted to obtain an improved model for the heat transfer. Finally, measurement qual-ity was confirmed by comparing measured pressure data by pressure calculations based onmeasured temperature.

Keywords: Hydraulics, Accumulator, Thermodynamics, Modelling, SIMULINK, Matlab,Verification, Thermal time constant

1 Problem IntroductionHydraulic pitch systems in wind turbines require a reliablesafety system to enable emergency stop without the presenceof electrical power. Figure 1 shows the general system layoutfor each blade.

The system must supply volumes in the range 10 to 25 litersof oil in less than 10 seconds for each blade. Hence accur-ate calculation of the accumulator size in a temperature rangefrom −30◦C to +50◦C is essential.

1.1 Motivation

Over the years a number of models have been presented andtested. Otis et al presented the Benedict-Webb-Rubin (BWR)model in 1985 [1], as an expansion of the Beatie-Bridgeman

(BB) model from 1974 [4]. However practical experience hasshown that these models have limited accuracy at low temper-atures. Another aspect in modern computational analysis isthe time required to run the simulation. Debugging the systemmodel shows that the accumulator model is the part requiringthe smallest time steps. Hence reducing the simulation timewill require a faster numerical model of the accumulator.

2 Theory

In order to model the phenomena in the accumulator gas, athermodynamical model is set up using the first law of ther-modynamics:

U = Qcon −W (1)

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Figure 1: The system layout for each wind turbine blade

All symbols are explained in the nonmenclature (table 1 page3). This is a simple energy balance on the gas. Consider thedifferential equation by Otis [2]:

U = mcvdTdt

+m(

T(

∂ p∂T

)v− p)

dVdt

(2)

Since:

W ≡ pdVdt

(3)

and the heat transfer from a system can be written

Qcon = hA(Ta −T ) (4)

Combining equation 1, 2, 3 and 4 yields:

hA(Ta −T )− pdVdt

= mcvdTdt

+

mgas

(T(

∂ pgas

∂T

)v− pgas

)dVdt

(5)

Which is a first order, heterogeneous partial differential equa-tion. It can be simplified to:

dTdt

=hAcvm

(Ta −T )− mTmcv

(∂ pgas

∂T

)v

dvdt

(6)

This equation can further be simplified by introducing thethermal time constant, τ , which can be written in the form:

τ =mgascv

hA(7)

Then equation 6 becomes:

T =(Ta −T )

τ− T

cv

(∂ pgas

∂T

)vdv (8)

This differential equation can be solved when combined withsome equation of state for the accumulator gas. The equa-tion should express gas pressure as a function of temperature,volume and gas specific constants, as:

p = F (T,V,C1,C2, . . . ,Cn) (9)

By taking the partial derivative of the equation of state, it canbe inserted into equation 8:

T =(Ta −T )

τ− T

cv

(∂

∂TF (T,V,C1,C2, . . . ,Cn)

)vdv (10)

The problem of determining the gas temperature has now re-duced significantly. When the change of gas temperature froma small volume change, dv, is known, it can be substitutedback into the chosen equation of state to determine the pres-sure change. By doing this, the pressure and temperature de-velopment in the accumulator gas can be determined accur-ately. However, two questions still remain:

1. How to determine the thermal time constant, τ?

2. Which equation of state should be chosen?

These questions frame the main challenge of this article.

2.1 The Thermal Time Constant

To answer the question regarding the thermal time constant, itis necessary to physically understand it. As seen in equation7, it depends on the heat transfer coefficient, h, gas propertiesand the exposed wall area, A. It therefore expresses some-thing about the time it takes for the gas to transfer heat to theaccumulator wall. It can be expanded by writing out the heattransfer coefficient:

h ≡ QA∆T

(11)

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Symbol Explanation Unit

U Rate of internal energy change Js

QCon Rate of convective heat transfer Js

Q Heat flow Js

W Rate of work done Js

m Gas mass kg

T Gas temperature K

t Time sp Gas pressure Pa (bar when noted)

V Gas volume m3

h Heat transfer coefficient Wm2K

A Exposed wall area m2

Ta Ambient temperature K

cv Specific heat capacity at constant volume JkgK

τ Thermal time constant sC Some constant Some unit

v Specific volume m3

kg

R The gas constant JmolK

Tc Critical temperature K

pc Critical pressure Pa

ω Acentric factor -P0 Preload pressure Bar

T0 Preload (initial) temperature ◦C

A0, a, B0, b, C0, c Benedict-Webb-Rubin equation constants -

Table 1: Nonmenclature

By inserting into equation 7, τ becomes:

τ =mgascv

QA∆T A

=mgascv∆T

Q(12)

In many applications, τ is constant since the heat flux can beassumed constant. Therefore τ is called a thermal time con-stant. However, as indicated by [2], it is not constant whenexamining accumulator dynamics. It must therefore be eval-uated as gas temperature and heat flow changes. Equation 12does not offer easier computations than equation 4. It merelyshows that τ is independent of exposed wall area. It solvesthe problem of finding the heat transfer coefficient betweenthe accumulator gas and the accumulator wall, but instead itintroduces the heat flow. In hydraulics, this property is oftenunknown. However, much effort has been put into approxim-ating τ , for example [2] and [3]. More and less complicatedapproximations have been presented. For the scope of thisresearch, the simple approximation by [3] proved sufficient:

τ ≈ 0.3pV 0.33 +86.2V 0.49 (13)

Note: Only for piston type accumulators.

Even though τ physically varies through compression or ex-pansion, the results obtained when applying the presentedthermodynamical model, equation 8, are not compromisedmuch by assuming a constant τ . This is also noted by [4].

2.2 Equation of State

The subject of choosing the equation of state is the main con-tribution of this article. Parallel models employing equation 8with different equations of state were made and compared toexperimental data.

• The Van Der Waals equation

• The Beatie-Bridgeman equation

• The Benedict-Webb-Rubin equation

• The Redlich Kwong extension to the Van Der Waalsequation

• The Soave extension to the Redlich Kwong extension tothe Van Der Waals equation (The Soave-Redlich-Kwongequation)

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However, since the Benedict-Webb-Rubin equation is an im-provement of the Beatie-Bridgeman equation, and since theSoave-Redlich-Kwong equation is an improvement of theother Van Der Waals based equations, the analysis can be re-duced to the comparison between the two. They are explainedin the following.

2.2.1 The Benedict-Webb-Rubin Equation

This model offers greater complexity than the others. It islike the Beatie-Bridgeman equation based on empirical dataand it is widely recognized for being very accurate. It can bewritten:

p =RTV

+B0RT −A0 − C0

T 2

V 2 +bRT −a

V 3 +aα

V 6 +

c(1+ γ

V 2 )e− γ

V 2

V 3T 2

(14)

2.2.2 The Soave-Redlich-Kwong equation

Soave [5] improved the Redlich-Kwong equation by intro-ducing the acentric factor, ω . In short, the acentric factoris a measure of how non-spherical a molecule is. It therebytakes the shape of the molecules of interest into account. Theequation is especially useful in chemical engineering wherephase transitions are of great importance. Since nitrogen isefficiently described by a cubic equation of state, it is expec-ted to model accumulator gas dynamics accurately. It can bewritten:

p =RT

V −b− a(T )

V (V +b)(15)

Where:

a(T ) = 0.4274R2T 2

c

pc

[1+κ

(1−(

TTc

) 12)]2

(16)

With:κ = 0.480+1.57ω −0.176ω

2 (17)

and:b = 0.08664

RTc

pc(18)

Because the Soave-Redlich-Kwong equation is much simplerthan the Benedict-Webb-Rubin equation, it is as much as sixtimes faster to evaluate numerically.

3 Experimental ApproachThe hydraulic set-up for the experiment is simple. A six literpiston type accumulator is connected to a proportional valvewhich in turn is connected to a pressure source. Pressure andtemperature is measured on both sides of the accumulator. Asketch of the set-up is shown in figure 2.

As the purpose of the research is to compare simulated data,using the thermodynamical model from section 2, to experi-mental data. The data has to span different temperatures andpressures. Furthermore, as the model must predict the dy-namics of accumulators fitted in wind turbines, conclusionsregarding validity of the model must also hold for a wide span

Figure 2: Overview of the experimental set-up

of ambient temperatures. The experiments conducted fall inthe parameter-window illustrated in table 2 .

Maximum pressurePreload Pressure

Ambient Temperature Low Medium HighLow H, M, L H, M, L H, M, LMedium H, M, L H, M, L H, M, LHigh H, M, L H, M, L H, M, L

Table 2: Parameter window. H=High, M=Medium, L=Low.

The table illustrates that all combinations of low, medium andhigh ambient temperatures, preload pressures and maximumpressures. This is a total of 3 · 3 · 3 = 27 combinations. Thelimits of the parameters tested are listed in table 3.

Ambient TemperatureLow -20 C◦

Medium 20 C◦

High 60 C◦

Preload PressureLow 50 barMedium 100 barHigh 150 barMaximum PressureLow 180 barMedium 220 barHigh 250 bar

Table 3: Limits of parameters tested

To obtain this parameter window, the piston was, for eachof the preload pressures and ambient temperatures, movedto different minimum gas volumes. The minimum volumeswere estimated using a steady state model and the ideal gasequation. Relatively small deviation on the maximum pres-sure from the estimated is expected. It is assumed that thesesmall deviations does not influence the validity of the overallconclusions.The desired ambient temperatures were reached by construct-

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ing an insulated chamber. Using solid carbon dioxide, coldambient temperatures was obtained (table 3 gives an estimateof about −20 C◦). A heat source was used for creating thehigh ambient temperature environment. Figure 3 shows howa well insulated heat chamber can be constructed. Pictures ofthe set-up are shown in figure 4.

Figure 3: Sketch of how to insulate an accumulator from theenvironment

In order to stress the numerical models, steep pressure gradi-ents are experimentally obtained. Practical experience hasshown that it is easier to predict the behaviour at small pres-sure and temperature gradients than the opposite. To obtainhigh gradients, the gas was compressed and expanded by arealistic in application rate. An example of the expansion andcompression time series is given by figure 5.

In the following, the measured temperatures and pressuresare plotted versus simulation results with the experimentallyobtained volumes as input. In this way, it is easy to see whichreal gas model best fits experimental results.

4 ResultsA limited portion of the generated results are presented. How-ever, results from both low, medium and high ambient temper-atures are shown. Also, both temperature and pressure meas-urements are compared to calculated results. Please note thata first order filter was used on the temperature data to limitnoise. Of course this results in phase transition which shouldbe ignored.

Figure 4: The setup from two angles.

0 10 20 30 40 500

2

4

6

Time[s]

Vol

ume

[L]

Volume (gas side)

Figure 5: An overview of the experimental compression andexpansion of the accumulator gas.

By driving the real gas model with the measured temperature,that is, using the data from the right hand side plots to drivethe calculations in the left hand side plots, plots like figure 12can be created.

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0 10 20 30 40 50 600

50

100

150

200

Time [s]

Pre

ssure

[bar]

Simulated vs Measured Pressure

BWRVDW(SOA)EXP

Figure 6: p0 = 48 bar, T0 = 30◦C

0 10 20 30 40 50 600

50

100

150

200

250

Time [s]

Pre

ssure

[bar]

Simulated vs Measured Pressure

BWRVDW(SOA)EXP

Figure 7: p0 = 48 bar, T0 =−5◦C

0 10 20 30 40 50 60120

140

160

180

200

220

Time [s]

Pre

ssure

[bar]

Simulated vs Measured Pressure

BWRVDW(SOA)EXP

Figure 8: p0 = 140 bar, T0 = 80◦C

0 10 20 30 40 50 60−50

0

50

100

150

200

Time [s]

Te

mp

era

ture

[C

]

Simulated vs Measured Temperature

BWRVDW(SOA)EXP

Figure 9: p0 = 48 bar, T0 = 30◦C

0 10 20 30 40 50 60−100

−50

0

50

100

150

Time [s]

Te

mp

era

ture

[C

]

Simulated vs Measured Temperature

BWRVDW(SOA)EXP

Figure 10: p0 = 48 bar, T0 =−5◦C

0 10 20 30 40 50 6070

80

90

100

110

120

130

Time [s]

Tem

pera

ture

[C

]

Simulated vs Measured Temperature

BWRVDW(SOA)EXP

Figure 11: p0 = 140 bar, T0 = 80◦C

5 DiscussionThere are two clear observations from the presented results:Firstly, the two real gas models resemble in comparison to

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0 10 20 30 40 50 600

50

100

150

200

250

Time [s]

Pre

ssure

[bar]

Simulated vs Measured Pressure

VDW(SOA)VDW(SOA)

Temp.Con.

EXP

Figure 12: p0 = 48 bar, T0 = −5◦C, pressure calculationswith calculated versus measured temperature input.

the experimental results. They both predict pressures that aresome 5-10 bar lower than measured. At higher compressionand expansion ratios, the calculation error is higher. Secondly,there are some clear deviations between the simulated dataand the measured data. The first observation enlightens thebenefits of the Soave-Redlich-Kwong equation. While main-taining the overall accuracy, it greatly reduces computationtimes.The second observation requires some examination. Clearly,some error sources have to identified. Several such errorsources could be suggested:

• Bad measurements

• Inaccurate real gas model

• Incorrect or inaccurate unsteady model

In the following, each of these suggestions are examined.Great effort has been put into validating and calibrating thetemperature measurement devices in controlled environmentsprior to the research investigations. If the temperature meas-urements were bad despite the validation attempts, bad pres-sure calculations would be a result. However, as seen in figure12, the pressure calculations are much more accurate if thereal gas model is evaluated using the measured temperature.

Figure 12 is data from the same experimental run as figure 7.It can be seen that much more accurate pressure calculationscan be performed if the measured temperature is used as in-put to the real gas model. Similar results are obtained for allother parameters, though not presented here. This serves tovalidate:

• The real gas model

• The pressure measurements

• The temperature measurements

It can therefore be concluded that the primary source oferror is the limited accuracy in the thermodynamical model,equation 8. By analysing the presented data, this work servesto suggest the need for a better thermodynamical model,in order to precisely determine the dynamical behaviour ofaccumulators. The presented thermodynamical model haslimited accuracy, because it does not model the accumulatorwall. It merely suggests that the heat is transferred fromthe accumulator gas to the surroundings. Instead, a moreprecise model would employ a thick wall solution for theaccumulator wall. Also, the heat conduction from thehydraulic oil to the accumulator gas (and vice versa) is notmodelled. Especially when the ambient temperature are verydifferent from the oil temperature, this will have a significantcontribution.Aside from these major issues, minor error sources in thethermodynamical model can be identified. One minor errorsource is the absence of the heat generated by the viscousoil in the model and also of gas flow at the accumulatorwall. Especially the viscous gas flow heat contribution isa challenge to model as a function of volume change. It islikely to have negligible effect at low rate of volume change,since flow phenomena are unimportant to the problemcompared to compressibility effects.The model may be constructed by employing a heat transfermodel for the accumulator wall. The model should includeheat convection from the gas to the wall, heat conductionthrough the wall (such that temperature in the wall changeswith the wall depth), heat conduction from the wall to thesurroundings and also radiation from the accumulator toadjacent surfaces. Also, the model should include heatcontribution from the accumulator oil by conduction thoughthe piston to the gas. This should all serve to enlarge theinsulation properties of the accumulator with respect to thegas, thus increase the peak temperature after compression.As can be seen in the results, the presented model predictstemperature- and pressure changes better after the expansionthan after the compression. As accumulators are loadedduring wind turbine operation at much slower rates thanconducted in this experiment, the need for performancepredictions here vanishes. As the accumulators emptyvery rapidly during emergency stops of the turbines, thecrucial performance phase of the accumulator is during gasexpansion. The need for a better thermodynamical model, asdiscussed above, may therefore have greater importance forareas in hydraulics where accumulators are loaded at highrates of volume change.

6 ConclusionThe presented thermodynamical model has limitations athigh rates of volume changes. This is most likely dueto the inaccurate modelling of the insulation propertiesof the accumulator. The inaccuracies are some 5-10 bardifference between measured and calculated data just aftercompression and expansion. The difference is higher forhigher compression/expansion ratios and the differenceapproaches zero for infinitely slow rates of volume changes.

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A better thermodynamical model can maybe be establishedby implementing a heat balance on the accumulator wall inthe model.The two presented real gas models; The Benedict-Webb-Rubin equation and the Soave-Redlich-Kwong equationare almost similarly accurate. The Soave-Redlich-Kwongmodel suggests a slightly higher pressure after compressioncompared to the Benedict-Webb-Rubin equation. Since theSoave-Redlich-Kwong equation is much more efficient thanthe Benedict -Webb-Rubin equation, this work concludesthat the Soave-Redlich-Kwong equation should replace theBenedict-Webb-Rubin equation in future simulations.The quality of the measurements were confirmed by calcu-lating the pressure using the Soave-Redlich-Kwong equationwith the measured temperature, and then comparing thecalculations to measured pressures. Since the calculated datamatched the measured data extremely well, it is concludedthat the measurements are accurate, dynamic and reliable.Also, it is concluded on this basis that the Soave-Redlich-Kwong real gas model is accurate enough for any applicationwithin the wind turbine technology.

References[1] Pourmovahed A. Otis, D. R. An Algorithm for Com-

puting Nonflow Gas Processes in Gas Springs and Hy-dropneumatic Accumulators. Journal of Dynamic Sys-tems, Measurement, and Control, 107:93–96, 1985.

[2] A Pourmovahed and D. R. Otis. An ExperimentalThermal Time-Constant Correlation for Hydraulic Accu-mulators. Journal of Dynamic Systems, Measurement andControl, 112:116–121, 1990.

[3] S. Rotthäuser. Verfahren zur Berechnung und Unter-suchung hydropneumatischer Speicher. Fakultät fürMaschinenwesen der Rheinisch-Westfälischen Technis-chen Hochschule Aachen, Essen, Germany, 1993.

[4] D. R. Otis. New Delvopment in Predicting and Modify-ing Performance of Hydraulic Accumulators. NationalConference on Fluid Power, 1974.

[5] G. Soave. Equilibrium Constants from a ModifiedRedlichKwong Equation of State. Chem. Eng. Sci.,27:1197–1203, 1972.

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