Flow Characteristics of High-Pressure Hydrogen Gas in the Critical Nozzle

Transcription

1 Flow Charateristis of High-Pressure Hydrogen Gas in the Critial Nozzle Shigeru MATSUO *1, Soihiro KOYAMA *2, Junji NAGAO *3 and Toshiaki SETOGUCHI *4 *1 Saga University, Det. of Mehanial Engineering, 1 Honjo-mahi, Saga-shi, Saga 84-82, Jaan Hydrogen gas has been extensively reeiving muh interest as one of alternative energy soures in future. Of the ratial usages of hydrogen gas, a fuel ell ar makes use of high-ressure hydrogen gas nearly at atmosheri temerature. In suh an aliation, metering tehnique of hydrogen gas is of ratial imortane for mileage and ower outut of the ar. However, the reise measurement of flow rate of hydrogen gas is extremely diffiult due to omressibility and real gas effets. For the urose of ratial use of high-ressure hydrogen gas, systemati researh is required to larify the ritial nozzle flow of high-ressure hydrogen gas. In the resent study, a omutational method has been alied to redit the ritial nozzle flow of high-ressure hydrogen gas. Key Words : Critial Nozzle, Comressible Flow, Hydrogen Gas, Real Gas Effets, Internal Flow 1. Introdution Critial nozzle is one of the flow metering devies whih are being extensively used in industrial area dealing with gases. It makes use of the onet of flow hoke that ours at the nozzle throat (1). Under the hoked flow onditions, ressure variations in the flow field downstream of the nozzle have negligible influene on the mass flow rate, and the oeffiient of disharge is easily obtained only by the flow roerties measured ustream of the nozzle. Reently, Kim et al. (2),(3) have reorted the disharge oeffiients of a variety of gases for a quite wide range of Reynolds number, using a omutational fluid dynamis method. The flow harateristis through the ritial nozzle have been well doumented at both onsiderably low and high Reynolds number regimes. Of many kinds of working gases emloyed in industrial field, hydrogen gas is one of the most romising gases as alternative energy. *1 佐賀大学理工学部 ( 佐賀県佐賀市本庄町 1) *2 佐賀大学大学院工学系研究科 *3 佐賀大学大学院工学系研究科. *3 佐賀大学海洋エネルギー研究センター matsuo@me.saga-u.a.j For instane, a hydrogen fuel ell, whih is being reeived as the driving ower system of vehiles, yields high-ressure hydrogen gas nearly at atmosheri temerature. In suh an aliation, reise measurement of flow rate is of ratial imortane for mileage and ower outut of the vehile. Only a few researhes have been to date made on the mass flow rate of the high-ressure hydrogen gas through ritial nozzle. Reently, Nakao (4) has onduted the flow rate measurement of hydrogen gas using a ritial nozzle, and has found that the disharge oeffiient of hydrogen gas exeeds unity in a seifi Reynolds number regime. No detailed exlanation has been made for this abnormal disharge oeffiient of high-ressure hydrogen gas. The resent study aims at investigating the detailed flow of high-ressure hydrogen gas through a ritial nozzle, with the hel of a omutational fluid dynamis method. The omutational results were validated with some exerimental data available. 2. Comutational analysis The high-ressure hydrogen gas flow through the ritial nozzle is simulated using a Comutational Fluid Dynamis method. The

2 governing equations are given by the onservation forms of mass, momentum and energy. The axisymmetri, mass averaged, time-deendent Navier-Stokes equations, whih use a k-ε turbulent model, are emloyed in the resent omutations (1),(2). Hydrogen gas has little attration between moleules so that it is inert in harater until the moleules are disruted. From the oint of view of kineti theory or statistial thermodynamis of hydrogen gas, the omressibility fator (Z) is frequently given as a olynomial funtion of seifi volume v, (1) where B(T), C(T) and D(T) are the virial oeffiients that are deendent only on temerature. Similar equation of state of real gas was also reorted by Benedit, Webb and Rubin (BWR equation of state) (), and later on, Lee & Kesler (6) modified BWR equation of state. Redlih and Kwong have imroved the van Der Waals equation to obtain the omressibility fator more aurately, as follows, where, (2) In ase of hydrogen gas, the n value in the funtion of a(t) is given by n=.31., T and v are the ressure, temerature and seifi volume, resetively, based on the ritial oint. Z = a v = ~ v b v ( T ) a, = B( T ) C( T ) D( T ) = v v v a( T ) ( v + b ) n T T 2 R T a = = + v ~ b = b,, a b =. 8664, + b v ( v + b ) The seed of sound of real gas is determined from the following thermodynami relation, = v { C /( R C )} /( v / ) The dynami visosity is given as (7), 7 w r ( T ) = T Tr +. 8 (3) (4) where T r is the redued temerature (=T/T ) and M w is the moleular weight of real gas. Using this visosity, the thermal ondutivity is obtained from Euken formula (8). In the resent study, the thermodynami roerties obtained by the above equations are inororated into the governing equation system and numerially solved to assess the real gas effets of high-ressure hydrogen gas. In the resent study, Redlih-Kwong s equation of state is seleted to simulate the high-ressure hydrogen gas through a ritial nozzle, sine it bears a better agreement with the virial equation of state In order to srutinize the deendene of temerature and ressure on the omressibility fator of hydrogen gas, Figure 1 shows the alulation results using the Comressibility fator [Z] Pressure inlet µ M T T 1. Redlih-Kwong's equation of state (Eq. 1 3) ) T = K a, atm Fig. 1 Deendene of ressure on the omressibility fator of hydrogen gas R= 2.D D/2 Flow diretion -1. D. D D =.6 mm 3 o minimum oint Adiabati no-sli wall 4 o 1 Pressure outlet Fig. 2 Shemati diagram of ritial nozzle flow field 16 2

3 Redlih-Kwong s equation of state. As the ressure is quite low, the omressibility fator aroahes 1., being the same to that of ideal gas, regardless of the temerature. However, as the ressure inreases, the omressibility fator beomes a strong funtion of temerature: At low temerature, the omressibility fator dereases and then inreases with ressure, after reahing a minimum value. It seems that the minimum value is deendent on the gas temerature. However, at high temerature, suh a trend is no longer found, and the omressibility fator is a simly inreasing funtion of the ressure. This is beause the moleular weight of hydrogen gas is very low and the intermoleular attration fore is less, omared with the other gases. The ritial nozzle emloyed in the resent study is of a tyial onial tye (1),(2) and its diameter D at the throat is D=.6 mm. A onvergent art with a radius of urvature of 2.D is given ustream of the nozzle throat. The straight divergent art has a half angle of 4 degrees and its axial length is.d. The omutational domain and boundary onditions used in the resent study are illustrated in Fig.2. Inlet total ressures and bak ressure of nozzle are o and a, resetively. The symmetri onditions are assumed at the axis of ritial nozzle. The adiabati, no-sli onditions are alied to the solid walls. In the resent study, the nozzle ressure ratio is defined as a / o and its value maintains onstant at., but the inlet total ressure is varied in the range from 2. bar to 3. bar at a fixed total temerature, T =288 K. A strutured grid system with about 3, grid oints was emloyed in the resent omutations. 3. Results and disussion The resent omutations were validated with the exerimental results (9) using hydrogen gas, where the Reynolds number is based on the diameter of nozzle throat and the total s / o Fig. 3 Predited stati ressure distributions along the nozzle axis roerties at the inlet of nozzle. It is found that the oeffiient of disharge is a strong funtion of Reynolds number, and the redited oeffiients of disharge were in good agreement with the exerimental results. It is believed that the resent omutation method redits well the gas flow through the ritial nozzle. Figure 3 desribes the stati ressure ( s ) distributions of hydrogen gas flow along the nozzle axis. The ressure ratio ( a / o ) is fixed at.. The stati ressure dereases with the distane and then suddenly inreases when the flow meets the shok wave. The shok wave moves downstream as the inlet total ressure inreases. It is noted that for the same o, the real gas effet auses the shok wave to be loated further downstream, omared with the ideal gas. This is qualitatively the same as what the inlet total ressure influenes the shok loations. In this figure, it should be noted that the real gas effet redues the stati ressure at the nozzle throat, omared with the ideal gas. This imlies that the mass flow through a ritial nozzle an be different due to the real gas effet. Meanwhile, the oeffiient of disharge C d is usually defined as, C = m& & d m theo o = 1.1 MPa 3.46 MPa () where m& is the theoretial mass flow rate through a ritial nozzle and m& means the theo 1.1 MPa 3.46 MPa

4 mass flow rate alulated by one-dimensional gasdynamis theory. Figure 4 shows the omarison of the redited and exerimental (4) disharge oeffiients. At resent, the only exerimental data for high-ressure hydrogen gas are available in Ref.(4). In 196, Johnson (1) alulated the ratio of seifi heats using the Virial equation of state and tabulated the thermodynami roerties of the hydrogen gas flow through a ritial nozzle. It is interesting to note that the exerimented data show a Disharge oeffiient [C d ] Re ( 1 - ) Fig. 4 Variation of the redited and exerimental disharge oeffiients with Reynolds number Density level Comressibility fator [Z] ( kg / m 3 ) Normalized mass flux [ρu / ρu theo ] throat ( Z=1. ) Present CFD () Present CFD () Ref. ( (4) 14 ) Ref. ( (1) ) mass flux omressibility fator Fig. Distributions of redited omressibility fators and mass fluxes along the nozzle axis ( o = 2.27 MPa) higher oeffiient of disharge than unity, in the range of Reynolds number below At resent, this unreasonable trend is not well understood, whether it omes from the real gas effet or from the exerimental error. The resent omutation taking aount for the real gas effet is qualitatively similar to Ref. (1) and (4), but still fails in rediting the oeffiient of disharge higher than unity. Unfortunately, a lear and ersuasive reasoning for this is, at resent, not known. More study is needed to eluidate this ambiguous roblem. In order to, in more detail, investigate the real gas effet of hydrogen gas flow, the redited omressibility fator and mass flux along the nozzle axis are resented in Fig. ( =2.27MPa), together with the omuted iso-density ontours. Ustream of the shok wave, the omressibility fator greatly dereases with the distane, and at the shok wave loation, it sharly inreases due to the omression effet of shok wave. There is also some differene in the mass flux distributions ustream of the shok wave. Thus, it is onluded that at high ressure onditions, the omressibility fator of the real gas an be one of the reasons for the oeffiients of disharge that were disussed in Fig.4. Figure 6 shows the axial distributions of the omuted values of the ratio of seifi heats of hydrogen gas. The ratio of seifi heats for ideal gas is onstant at The seifi heat Seifi heat ratio [γ] o =.2 MPa (γ = 1.41) Fig. 6 Influene of inlet total ressure on the seifi heat ratio of hydrogen gas

5 ratio for real gas inreases as the flow is aelerated through the nozzle throat, and it sharly dereases at the loation of the shok wave. Suh a tendeny aears similar, regardless of P, but the sudden dro in the ratio of seifi heats seems to more signifiant when P inreases. It is interesting to note that at the nozzle throat, the ratio of seifi heats remains nearly onstant at 1.41, when is less then 1.1MPa, but it signifiantly dereases as inreases. At the nozzle throat, this variation in the ratio of seifi heats with has an areiable imortane on the mass flow rate of hydrogen gas through a ritial nozzle. Therefore, the real gas effet should be inluded in evaluating the erformane and auray of ritial nozzle as flow metering devie. 4. Conluding remarks Several imortant and meaningful onlusions obtained from the resent study are summarized; Redlih-Kwong s equation of state redits the real gas effets of high-ressure hydrogen gas omaratively well. However, unlike the oeffiient of disharge of ideal gases whih have been obtained to date, the oeffiient of disharge of real gas through ritial nozzle dereases with an inrease in Reynolds number, as Reynolds number exeeds a ertain value. It is believed that this mainly results from the thermodynami roerties of real gas, suh as the omressibility fator and the ratio of seifi heats, whih aears more remarkable as the ressure of hydrogen gas inreases. Aknowledgement This study was artially suorted by Harada Memorial Foundation. Referenes (1) Kim, H. D., Kim, J. H., Park, K. A., Setoguhi, T., and Matsuo, S., Comutational Study of the Gas Flow through a Critial Nozzle, IMehE J Mehanial Eng Siene, 23, 217(1), (2) Kim, H. D., Kim, J. H., Park, K. A., Setoguhi, T., and Matsuo, S., Study of the Effets of Unsteady Downstream Conditions on the Gas Flow through a Critial Nozzle, IMehE J Mehanial Eng Siene, 24, 218, (3) Kim, J. H., Kim, H. D., Park, K. A., Setoguhi, T., and Matsuo, S., A Fundamental Study of a Variable Critial Nozzle Flow, Exeriments in Fluids, 26, 4(1), (4) Nakao, S., Develoment of ritial nozzle for High ressure Hydrogen Gas Flow Measurements, Proeedings of JSME, Fluid Dynamis Setion, 2. () Giorgio, S. S., An Effetive Modifiation of the Benedit-Webb-Rubin Equation of State, Fluid Phase Equilibria, 1999, 164, (6) Jyisy, Y., Peter R, G., Anthony, R. H. G., Comarison of Methods for Calulating Thermodynami Proerties of Binary Mixtures in the Sub and Suer Critial State: Lee-Kesler and Cubi Equations of State for Binary Mixtures Containing either CO2 or H2S, J. Chemial Thermodynamis, 23, 3, (7) Cheremisinoff, N. P., Fluid Flow Poket Handbook, Gulf Publishing Co., Houston, TX., (8) Ekert, ERG., Drake, RM., Analysis of Heat and Mass Transfer, MGraw-Hill Co., (9) Nakao, S., Irayama, T. and Takamoto, M., Relations between the disharge oeffiients of the soni venture nozzle and a kind of gas, JSME J Mehanial Engineering, Ser. B, 2, 66(642), (1) Johnson R. C., Real-Gas Effets in Critial-Flow-through Nozzles and Tabulated Thermodynami Proerties, NASA TN D-26, 196.