Soave-Redlich-Kwong Adiabatic Equation for Gas-loaded Accumulator *

Transcription

1 65 研究論文 SoaveedlichKwong Adiabatic Equation for Gasloaded Accumulator * Shuce ZHANG **, Hiromu IWASHIA ***, Kazushi SANADA **** In this paper, a SoaveedlichKwong adiabatic equation of real gas charged in accumulators is proposed. he Soave edlichkwong equation is the most accurate real gas model for representing accumulator behavior, but it is relatively complex mathematical equation. For simulating accumulator dynamic behavior, it has been used by combining with a thermal time constant model. he mathematical procedure for the SoaveedlichKwong equation for the adiabatic equation is described in this paper. he SoaveedlichKwong adiabatic equation is discussed by comparing the van der Waals adiabatic equation. By comparing representative gas models, the SoaveedlichKwong adiabatic equation is found to be valid. Key words: Gasloaded accumulator, Adiabatic process, SoaveedlichKwong equation, van der Waals equation, Benedict Webbubin equation, hermal time constant.introduction Selection of accumulator size and prefill pressure have been discussed by many researchers. A typical practice is to assume that the process involved is isothermal, adiabatic, or polytropic. Isothermal operation maintains a constant temperature, which requires the rate of expansion and compression of the gas at such a rate that the gas temperature inside the accumulator remains relatively constant throughout the entire cycle. For the adiabatic operation, the gas temperature inside the accumulator increases or decreases at a high rate due to the rapid compression or expansion of the gas. However, in practical processes, the temperature is increasing during charge process and decreasing during discharge process. A concept of polytropic change is used to describe the phenomena, where the polytropic exponent n is used to compensate the inaccuracy for calculating accumulator. For the polytropic exponent of the currently employed gas, nitrogen, the value has been predicted by experiment according to charging and discharging time and average working pressure. Simultaneously, for taking intermolecular forces and * Manuscript received September,, 07 ** Graduate School of Engineering Yokohama National University (okiwadai 795 Hodogayaku, Yokohama, Kanagama, , Japan ( zhangshucehc@ynu.jp *** College of Engineering Science Yokohama National University **** Faculty of Engineering Yokohama National University molecular volume into account, real gas models have been presented and developed by researchers. Modeling of thermodynamic behavior of fluid over the whole range of temperatures, densities and pressures is not an easy task. At low pressure, the isotherms are fairly easily modeled with something like the ideal gas model. However, at higher density the behavior becomes more difficult to model with a simple equation 3. As the first correction to the ideal gas equation, the van der Waals equation was developed for engineering purposes 4. he two most commonly used cubic equations of state are the edlichkwong 5 and Soave s modification of the edlichkwong equation of state (Soave edlichkwong equation 6. hese models were developed by modifying the van der Waals equation, and it has been applied in a commercial software. It is necessary to consider the energy balance in accumulator. he nitrogen gas in accumulator has internal energy which receives heat energy from the environment and performs work to the working fluid. Predicting thermodynamic loss is necessary. Otis has presented a thermal time constant model which has been combined with the BeattieBridgman equation of state to describe the heat transfer from the accumulator wall 78. It is relatively easy to use the van der Waals equation of state. Its isothermal model and the adiabatic model are also easy to obtain mathematically. However, the SoaveedlichKwong equation has relatively complex mathematical expression. he adiabatic equation has not been found. herefore, for calculating accumulator, only one possibility was to combine the SoaveedlichKwong equation with the thermal time constant model to carry out numerical integration. he purpose of this paper is to propose an adiabatic expression of 第 49 巻第 3 号

2 66 日本フルードパワーシステム学会論文集 the SoaveedlichKwong equation. Section is the nomenclature and section 3 gives introduction of representative real gas models. In section 4, the SoaveedlichKwong adiabatic equation is derived. In section 5, the proposed adiabatic equation is validated by comparing with representative real gas models. he conclusion is made in section 6..Nomenclature a vdw a( b b vdw C C v H m n P Q S U v κ ω : Parameter of van der Waals equation : Parameter of SoaveedlichKwong equation : Parameter of SoaveedlichKwong equation : Parameter of van der Waals equation : Constant : Specific heat at constant volume : Adiabatic exponent : Amount of gas : Polytropic exponent : Absolute pressure : Heat energy : Gas constant : Entropy : Absolute temperature : Internal energy : Gas volume : Mole volume : Parameter of SoaveedlichKwong equation : Acentric factor Subscript 0 : Initial value c : Critical property 3.eal gas models 3. van der Waals equation of state Gas equation of state properly expresses thermal motion with considering mass of molecule. he accumulator gas is often modeled as ideal gas which does not exist. Ideal gas model fails to account for intermolecular force and molecular volume. he gas in the accumulator assumed to be ideal in many cases has shown to result in large errors 90. o account for the volume that a real gas molecule takes up and the intermolecular attraction, the van der Waals equation of state is formulated as avdwm ( P+ 3 avdw= vc, bvdw= (mbvdw=m 9 cvc 8 It is known that the entropy is constant in adiabatic condition. ⑴ According to this condition, the van der Waals adiabatic equation has been founded: avdwm P+ (mbvdw H =const ⑵ ( where the adiabatic exponent is expressed by: H= +. ⑶ Cv 3. BenedictWebbubin equation of state Gas pressure is related to gas temperature and gas volume by the BenedictWebbubin equation of state 3 which was shown to be in remarkable agreement with the data published by the NBS for nitrogen 4 for the entire range of interest for accumulator applications 9. he expression of this equation of state is as follows: where A 0, B 0, C 0, a 0, b 0, c 0, α and γ are BenedictWebbubin parameters. 3.3 SoaveedlichKwong equation of state In physics and thermodynamics, the edlichkwong equation of state is an empirical, algebraic equation that relates temperature, pressure, and volume of gases. It is generally more accurate than the van der Waals equation and the ideal gas equation at temperatures above the critical temperature. It was presented by Otto edlich and Joseph Neng Shun Kwong in 949. It showed that a twoparameter, cubic equation of state could well reflect reality in many situations, standing alongside the much more complicated BeattieBridgeman model 5 and BenedictWebbubin equation that were used at the time. he edlichkwong equation has undergone many revisions and modifications, for either improving its accuracy in terms of predicting gasphase properties of more compounds, as well as in better simulating conditions at lower temperatures, including vaporliquid equilibria. he SoaveedlichKwong equation fitted experimental vaporliquid data well and could predict phase behavior of mixtures in the critical region. he Soaveedlich Kwong equation of state is given by the expression: where m( B0A0 C0 m m(b0a0 3 P= m6 a0α + 6 m 3 c0( + m a(m P=, ⑸ mb (+mb c a(= k, Pc m γ e 3 mγ ( c ⑷ ⑹ 第 49 巻第 3 号

3 67 c b= , and ⑺ Pc k= ω0.76ω. he Pitzer s acentric factor ω of nitrogen gas is assumed as he supercritical cohesion parameters for the Soave edlichkwong equation of state were adjusted by fitting cohesion parameters to the reliable Joulehomson inversion curve data by Mehdi Ghanbari and Gholam eza Check 6 : k= ω0.76ω ; c ( ω0.76ω ; > 5 c he modified parameter k was fit with a coefficient of /5 for supercritical region. For the nitrogen gas using in gasloaded accumulator, the operating temperature is commonly larger than its critical temperature (6.K, thus, the function of k with /5 is employed in this paper. 3.4 hermal time constant model he Otis s thermal time constant model introduced a thermal time constant τ to express the functional relation of temperature with time: d dt w P dv = τ, cv v dt ⑻ ⑼ ( ⑽ where the definition of thermal time constant is given by the expression: mcv τ=. ⑾ ha It is necessary to employ a gas model to evaluate the term of ( P/ v in the second part of the right side of Eq. (0. For the nitrogen gas sealed in accumulator, the mole number of nitrogen gas m is constant. Because the mole volume is defined as: therefore v= m, ⑿ dv dt = d m dt. ⒀ he condition of constant mole volume is identical to the constant volume condition. Hence, P P ( ( ⒁ =. v hen the Otis s model can be written as: d dt ( ⒂ w P d =. τ cv m dt hus, for calculating Otis s model, the term ( P/ v can be expressed by the SoaveedlichKwong equation of state: P m m ( = mb a(, (+mb ⒃ where Pc ( c ( c a( k c = k. ⒄ Combining Otis s model of Eq. (5with Eq. (6and (7, it is possible to mathematically analyze the accumulator gas charge and discharge processes by numerical integration. 4.Derivation of adiabatic model of SK he SoaveedlichKwong adiabatic equation is derived in this section as follows. First law of thermodynamics is expressed as: du=dqpd. he entropy is defined as: dq ds=. ⒆ Combining Eq. (8and Eq. (9yields: du=dspd. he volume partial differential of Eq. (0under constant temperature is: U = S ( ⒅ ⒇ ( P. From Maxwell relations, we know that S P ( ( =. hen the Eq. (can be deformed as: U = P ( ( P. Substituting Eq. (6and the Eq. (5into Eq. (3yields: U ( = m a( m a( (+mb (+mb he total differential of internal energy is expressed by: ( ( U U du= d+ d. he definition of specific heat at constant volume is: m( U Cν=. Substituting Eq. (4and Eq. (6to Eq. (5yields: ( ( U U du= d+ d m =mcνd+ a( (+mb a( Combining Eq. (0with Eq. (7to delete du and solving for the total differential of entropy ds yield: dspd=mcνd+ (+mb a( m a( d d 第 49 巻第 3 号 3

4 68 日本フルードパワーシステム学会論文集 ds= mcν ( d+ a( a( m P+ (+mb Substituting the pressure P of Eq. (5into Eq. (9yields: mcν m ds= d d mb + ( m +mb d m a( d b Integrating both sides of the above equation yields: S=mCν C ν ln ( (mb ( +mb a( +S0 he temperature expression by the SoaveedlichKwong equation is: mb a(m = P+. m ( (+mb Substituting Eq. (3into Eq. (3yields: a(m S=mCν ln P+ ( + m (+mb (mb ( a( +S0 +mb It is known that the entropy is constant for adiabatic condition. In Eq. (33, the constant entropy means that the term inside the natural logarithm brackets is constant. According to this condition, the following formula is founded for adiabatic process: a(m P+ ( + (+mb (mb =const ( +mb C ν C ν a( According to this expression form and substituting the definition of adiabatic exponent H of Eq. (3into above equation, the SoaveedlichKwong adiabatic equation can be written as: a(m ( H (mb (+mb ( +mb a( P+ =const he volume term of the SoaveedlichKong adiabatic equation (mb H is identical to that of the van der Waals adiabatic equation. he pressure term of the Soaveedlich Kwong adiabatic equation is slightly different from the pressure term of the van der Waals adiabatic equation. However, both adiabatic equations include the product of a pressure term and a volume term. In the Soaveedlich Kwong adiabatic equation, an additional term Z is included: ( +mb a( Z=. For calculating charge and discharge processes by the SoaveedlichKwong adiabatic equation, it is necessary to calculate temperature during the processes. o calculate the temperature, a function can be built by the Soaveedlich Kwong equation: mb a(m P+ ( g( =m (+mb he temperature value can be obtained by computing the root of the nonlinear equation: g(=0 he constant C in Eq. (35can be calculated by the adiabatic initial condition: C= P ( 0+ a(0m 0(0+mb 0 ( 0+mb (0mb ( a( Eq. (35can be rewritten as: =0 P+ a(m C = (+mb H (mb +mb he pressure P is expressed by: a(m P= + (+mb H ( C a( H ( (mb +mb a( Combining the solution of the Eq. (38for temperature, it is possible to calculate the pressure of adiabatic process if the corresponding volume is known. 5.Considerations Calculation of adiabatic process by the Soaveedlich Kwong adiabatic equation proposed in this paper is carried out and the results are shown in Fig.. As an assumption of the calculation, it is assumed that the initial condition at the point A is 34MPa, abs. in gas pressure, m 3 (.67L in gas volume, and 93.5K in gas temperature. he top plot presents gas absolute pressure as a function of gas volume. By gas expanding, the gas pressure decreases to the point B. In the middle plot, gas absolute temperature is shown. It is a discharge process and gas absolute temperature also decreases. In the bottom, the additional term Z of Eq. (36is plotted. he term Z shows around 0.9 during this process. Parameters a(, b, and a(/ of the Soaveedlich Kwong equation are plotted in Fig.. In the figures, the parameter values of the van der Waals equation are noted. During the discharge process, while the van der Waals 第 49 巻第 3 号 4

5 69 Fig. Discharge process of gasloaded accumulator calculated by the Soave edlichkwong adiabatic equation parameters a vdw and b vdw are constant, the Soaveedlich Kwong parameters a(increases. he parameter b of the SoaveedlichKwong equation is constant. he value of a (/ t is negative from the initial point A to the final point B. In the Otis s thermal time constant model, the heat losses are considered 7. his means the process is no longer assumed as adiabatic. In this study, the thermal timeconstant model is couple with the SoaveedlichKwong equation, Eq. (5, and the BenedictWebbubin equation, Eq. (4. he BenedictWebbubin coefficients predicted by ASAMI are applied in this paper 8. A Pdiagram of charge process is shown in Fig. 3. Parameter values of experiment of charge and discharge processes carried out by Miyashita, et. al. 9 are referred for simulation in this paper. Operating time is 0 seconds and the surrounding temperature is 93K. he initial condition A is 4.4MPa, abs. and m 3 (7.4L. he thermal time constant τ is assumed to be 0 seconds for the Otis s models coupled with the SoaveedlichKwong equation and the BenedictWebbubin equation. According to Otis s research, the value of τ were determined referring to experimental results 8. Circle symbols and triangle symbols are calculated results by the SoaveedlichKwong adiabatic equation (SKand the van der Waals adiabatic equation (vdw, respectively. he dot line and dash line represent the Otis s thermal time constant models combined with the SoaveedlichKwong equation and the BenedictWebbubin equation, respectively. By charging, the gas volume decreases and the gas pressure increases from the initial point A to the point B. he calculated results of the SoaveedlichKwong adiabatic equation show a good agreement with the other representative gas models. he comparison results of discharge process are plotted in Fig. 4. he initial condition A is 34.5MPa, abs. and m 3 (.65L. he operating time is 0 seconds and the surrounding temperature is 30K. he thermal time constant τ is assumed Fig. Comparison of parameters during discharge process Fig. 3 Comparison of charge process of accumulator estimated by different gas models 第 49 巻第 3 号 5

6 70 日本フルードパワーシステム学会論文集 Fig. 4 Comparison of discharge process estimated by different gas models operation are significant topics. In this paper, the adiabatic expression of SoaveedlichKwong was presented. A lack of mathematical expression for the SK equation of state, that is, a lack of SK adiabatic equation, has been solved by this research. his contribution is important for mathematical modeling of accumulator. he combination of Soaveedlich Kwong equation with Otis s thermal time constant model was presented as well. Adiabatic models (van der Waals equation and SoaveedlichKwong equationand thermal timeconstant model that considered heat losses were compared with each other. It has been found that the Soave edlichkwong adiabatic equation shows consistency. to be 5 seconds for the SoaveedlichKwong equation and 7 seconds for the BenedictWebbubin equation. he value of τ was also determined referring to experimental results. In this discharge case, thermal time constant values of the Soave edlichkwong equation and the BenedictWebbubin are different, because of curve fitting. By discharge, the gas volume increases and the gas pressure decreases. he calculated results of the SoaveedlichKwong adiabatic equation show a good agreement with the other representative gas models. he SoaveedlichKwong adiabatic equation shows accurate calculated results of charging and discharging processes. From comparison of these results, the proposed SoaveedlichKwong adiabatic equation is successfully validated. In the thermal time constant model, heat transfer is considered. herefore, this model cannot simulate pure adiabatic process. Adiabatic assumption of gas equation of state, which describes gas thermodynamic properties without transfer of heat, provides a conceptual basis in thermodynamics. In some cases, the thermodynamic properties vary very similarly as adiabatic process, the physical process occurs so rapidly that there is no enough time for the transfer of energy, thus the Soaveedlich Kwong adiabatic equation can be employed to express the assumption models. In this paper, as the comparison of the SoaveedlichKwong equation with thermal time constant model shows, this equation can be used as a reference to study thermal time constant model combined with various gas models. 6.Conclusion For commercial applications, according to equation of state and operating conditions, accurate prediction of accumulator performance and design of accumulator size for efficient eferences Wasbari, F., Bakar,.A., Gan, L.M. et al.:precharge pressure effects on isothermal and adiabatic energy storage capacity for dual hybrid hydropneumatic passenger car driveline, Proceedings of Mechanical Engineering esearch Day 07, (07:34, (07 Japan Hydraulics & Pneumatics Society:New hydraulics & Pneumatics handbook (in Japanese, Ohmsha, p (989 3 Mohamed, K., Paraschivoiu, M.:eal gas simulation of hydrogen release from a highpressure chamber, International Journal of Hydrogen Energy, ol. 30, No. 8, p (005 4 an der Waals J D.:he equation of state for gases and liquids, Nobel lectures in Physics, ol., p (90 5 edlich, O., Kwong, J.N.S.:On the thermodynamics of solutions; an equation of state; fugacities of gaseous solutions, Chem. ev., ol. 44, No., p (949 6 Soave, G.:Equilibrium constants from a modified edlich Kwong equation of state, Chem. Eng. Sci., ol. 7, No. 6, p (97 7 Otis, D..:New developments in predicting and modifying performance of hydraulic accumulators, National conference on fluid power, p (974 8 anaka, Y., Nakano, K.:Energy balance of bladder type hydraulic accumulator ( st eport:experimental investigation of thermal time constant, Journal of the Japan hydraulic & pneumatics society, ol., No. 6, p (99 9 Pourmovahed, A., Otis, D..:An algoritm for computing nonflow gas processes in gas springs and hydropneumatic accumulators, ransactions of the ASME, Journal of Dynamic Systems, Measurement and Control, ol. 07, p (985 0 Mohamed, K. Paraschivoiu, M.:eal gas simulation of hydrogen release from a highpressure chamber, 第 49 巻第 3 号 6

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