HYDRAULIC ACCUMULATOR. Principles DIMENSIONING OSP 050. The ideal and real gas. Boyle-Mariotte's Law. Equation for an real gas

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1 HYDRAULIC ACCUMULATOR DIMENSIONING OSP 050 Priniles The aumulor dimensioning method is based on the stus hange of the gas ontained in the aumulor. The same hanges our with oil. Equion for an real gas At onstant temerure and isothermal ste hange: While dimensioning hydrauli aumulors, the following two elements have to be aounted for: the rinile wants th the aumulor filling gas (nitrogen) behaves as an ideal gas, whih in rie is not the ase high ressures and low temerures as the temerure exhange roess is unknown, isothermal or adiabi hanges an only be assumed. The ideal and real gas Ideal gas: Real gas: this gas does not exist all known gases are real. The more they move away from their ondension oint (the oint where a gas hanges to liquid), the more the gas feures get loser to those of an ideal gas. Nitrogen ondension oint: -196 C It is obvious, th the ste equion = onstant does not desribe the real behaviour of a gas, seially high ressures and low temerures. The omressibility for "k" makes it ossible to take aount of and desribe the behavioural differenes between the real and ideal gas. Assuming th we have to deal with an ideal gas, the aumulor filling gas behaves as desribed hereafter. - T - diagram for onstant "k" values in ase of nitrogen Equion for an ideal gas At onstant temerure and isothermal ste hange: ,0 1,3 1,2 1,1 k = f (gas,, T) 0 T (K) By integring the Betie Bridgeman equion into our omuter rogram, we are able to take aount of the above mentioned behavioural differenes of the gas. k Boyle-Mariotte's Law At onstant temerure, the rodut of ressure and volume of the gas ontained in a vessel is always onstant. 1,3 1,2 T = 300 K Bridgeman As, however, no ideal gas is available, one must take the behaviour of the real gas into onsiderion. 1,1 1 ideal

2 Ste hanges of ideal gases The ste of a gas is defined by three fors: ressure, volume and temerure, also alled ste variables. A ste hange refers to the hange of two or all ste variables. Filling or emtying a hydrauli aumulor leads to an exhange of work aumulor gas level. A gas temerure differing from the ambient temerure leads to a thermal exhange. Proesses affeting the aumulor gas and linked to the work and thermal exhanges an be desribed by means of an isobari (onstant ressure), isothermi (onstant temerure), isohore (onstant volume), adiabi (without he transfer) or olytroi (between isothermi and adiabi) hange of ste. The following roesses inlude volume variions: Isothermi hanges of ste One refers to isothermi hanges of ste for a hydrauli aumulor when harging and disharging haen over a long eriod allowing for a full thermal exhange with the environment. During suh a ste hange, the gas exhanges work and he with the environment. 2 Relionshi between, and T The following rule governs this relionshi: χ = f (, T, gas) with χ = adiabi exonent = seifi he aaity onstant ressure v = seifi he aaity onstant volume Based on an ideal gas, the adiabi exonent deends on the number of gas oms of the gas. χ = 1,67 gas with 1 om } χ = 1,4 gas with 2 oms 0 C and 1 bar χ = 1,3 gas with 3 oms The higher the number of oms, the loser to 1 χ will be. The adiabi exonent of nitrogen is 1,4. As mentioned earlier, the adiabi exonents does not only deend on the gas, but also on ressure and temerure. This adiabi exonent an also exeed the value of 1,4. isothermi 1 T1 1 Relionshi between, and T Boyle-Mariotte governs following relionshi: T = onstant Adiabi hanges of ste One refers to adiabi hanges of ste for a hydrauli aumulor when harging and disharging haen in suh a short sae of time, th aart from an exhange of work no he exhange may take lae with the environment. Polytroi hanges of ste While harging and disharging the aumulor, the hange of ste rarely ours fully isothermially or fully adiabially. The gas ontained in the aumulor exhanges some of its he. This hange is alled olytroi and harerised by a mix of exhange of work and to a bigger or smaller extent of he. 2 adiabi 2 isothermi olytroi adiabi { 1 2

3 Relion between, and T The following rule governs this relion: OLAER owns this rogram! n = olytroi exonent Challenge us. n = f ( 1, 2, 0, 0, T oil, T gas, t, aumulor shae) By getting loser to the isothermi ste "n" tends to 1 and to the adiabi ste "n"tends to the adiabi exonent "χ". The following examles illustre the informion ontained in the hard oies rinted by the omuter. Hard oy 1 - Tehnial feures 20 C Relion between the adiabi exonent "χ" and the olytroi exonent "n" Examle of a tye EH hydrauli aumulor an oering ressure of 300 bar and an oering temerure of 10 C: x = 2,1 n = 1,7 adiabi olytroi Hard oy t diagram Pressure/time grah 1 0,001 se 1 se 120 min isothermi t (time) χ reresents the highest value on the urve. The olytroi exonent "n" is loed between the adiabi and isothermi exonent and deends on t, 1, 2, 0, 0, T oil, T gas and the aumulor shae. All these formulas and exlanions learly show th it is not ossible to aurely dimension hydrauli aumulors by means of diagrams and simle formulas. One really needs a rogram running on a owerful omuter taking all the above mentioned fors into aount. Hard oy t diagram Pressure/time grah Hard oy diagram Pressure/time grah 3

4 Aroxime aumulor alulion Parameters and abbreviions 0 = reharge ressure (bar) usually 20 C 1 = minimum oering ressure (bar) admissible minimum oering overressure 2 = maximum oering ressure (bar) admissible maximum oering overressure = returned volume (l) ( 1-2 ) volume of the aumuled or returned liquid T 1 = minimum oering temerure ( C) minimum ambient or liquid temerure T 2 = maximum oering temerure ( C) maximum ambient or liquid temerure t = harge time / disharge time (s) neessary liquid aumulion or return time 0 = ual aumulor volume (l) orresonds to the term "Caaity" in the da sheets 1 = aaity ressure 1 (l) aumuled gas volume ressure 1 2 = aaity ressure 2 (l) aumuled gas volume ressure 2 n = olytroi exonent oeffiient taking the thermal exhange into aount m = average oering (bar) used to dimension ulsion damers = 2 m For all aumulor dimensioning alulions absolute ressures (relive ressures + 1 bar) will be used. The temerures T1 and T2 are in Kelvin (T + 273). Power buffer dimensioning: Formula used to determine the aaity 0 : Gas filling ressure As a general rule, the ressures 1 and 2 are defined by the hydrauli system. The gas filling ressure must be hosen from ase to ase and aording to the aumulor shae. The gas filling ressure is always set for the maximum oering temerure (T 2 ). Gas filling is usually arried out a temerure of 20 C. All indiions onerning gas filling ressures issued by OLAER aly for 20 C. In general, the following formulas are used: Formula used to determine the returned volume : Energy buffering / seurity reserve aliions temerure T 2 Temerure influene The above formulas may be used only aroximely stable temerures. When the system is subjet to imortant temerure variion a orretion must be alied (also true in ase of aroximive alulions). Limits : 0 min. 0,2 x 2 0 max. = 1 (ont OLAER; aording to oering onditions) Weight balaning aliions temerure T 2 The Gay-Lussa law is used here: Pulsion daming aliions temerure T 2 4

5 Aumulor aliions with additional gas bottles ) Calulion of the aaity ' 0 Bladder aumulor temerure T 2 l re Piston aumulor d) Calulion of the gas filling ressure 0 20 C temerure T 2 Gas filling ressure formula 0 20 C T 2 relive Dimensioning examle: Known values: Oering ressure 2 max. 190 bar Oering ressure 1 min. 100 bar Returned volume 2 l Disharge time 1 s Oering temerure T 1 min. 25 C Oering temerure T 2 min. 45 C Polytroi exonent n 25 C = 1,638 (aording to our PC rogram) 45 C = 1,617 alues sought: The da sheets allow one to selet the desired aumulor in the requested ressure range with the aaity of 0 > 7,3 l. In our examle, the aumulor tye EH K or EH L will do the job (aording to the desired aumulor shae). Our omuter alulion gives a of 2,06 l 25 C and 2,26 l 45 C. Hydrauli aumulor aaity 0 Solution: a) Calulion of the gas filling ressure 0 the maximum oering temerure relive bar b) Calulion of the aaity 0 ( ) ( ) l re Warning! In the theoretial art of the resent doument we have mentioned the imortant influene of the temerure when dimensioning aumulors. At -10 C for instane, a 10 l aumulor will only return 1,71 l oil. OLAER (Shweiz) AG, Bonnstrasse 3, CH 3186 Düdingen, Tel , Fax , info@olaershweiz.h,