1、REPORT 959Oh-DIMENSK)NAL FLOWS OF AN L%IPEIU?ECT DIATOMIC GASBy A. J. EGGKRS,Jr.SUM3MRY%h the assumptions that B.wthelofs equation of statenccountsfor molecularrize and intermolecularforce e-fleck, andthat changes in. the le is considerablyW-w and hence the density rise is well abore that predicteda
2、s.wuwingideal gas behauior. It is-further shown that only thecaloric imperfection in air has am appreciable eect on. thepre.mure.sderelopedin the shock process considered. fie eectsqf gaseous imperfectionns on obue shock$ows we stuclied2fromthe standpoint of their inj%ience on the iijlt and pressure
3、 dragofa plate operaingat :?fach numbers of 10 mnd90. Thei-n.uence is found to be small.1%PRODUCTIONA wide -m.rietyof problems in compressible flow has beensolved on the assumption that air behaves as an iclecilclia-iomic gas. This assumption is justifiecl, provided thepressure and ternperaturerange
4、 of interest is small ancl nearatmospheric. It is an experimental fact, home-rer, thatw-henair is subjected to Iarge changes in state at pressurestmcl temperatures far remo-eclfrom atmospheric, it ceasesto obey the simple gas law ancl eshibits other properties not,cha.racterist.icof an ideal gas. Co
5、nsequently, flow processesin which air is subjecied to these extreme conditions can beexpectecl t-odepart from perfect gas behavior. It. is knownthat such flow-s wiH be encountered in hypersonic wincltunnek ancl by a.ircraft fly at high supersonic airspeeds;hence, the nature anc extent of this depar
6、ture have becomeimportant considerations in aeroclamics.Chissical theories and experiments have shown that threeproperties of a.reaI gas first cause it to exhibit characteristicsunlike those of an iclea.1gas. These properties may beckssified as thermal and caloric imperfections. Thermalimperfections
7、 in the form of intermolecular forces cmclmolecular size effects are signilic.antlly manifest at. lowtemperatures and high pressures. Changes in the vibrationalheat capacities become an important. caloric imperfection atrehitively high temperat.ures. Circumstances under whicheffects of molecular cks
8、ociation anclorelectronic excitationbecome important (e. g., temperatures apprec.iabFyabove.5000R) maybe neglectecl for the present. Insofar m gasesin equilibrium are concernecl, it is usutdIy suilicient to ac-count for intermolecular force anclmolecular size effects withadcLitioncdterms in the equu
9、tion of state. Simikrly, changesin the vibrut.ions.1heat capacities of the molecules may beaccounted for with a function of temperature in the e.spres-sions for the spectic heats.Tsien (reference I) investigateed the effects of gaseous _iruperfect.ionson air flows w=ingTan ckr TTaals state equa-tion
10、. Appro.simat.e solutions to the one-climensiomd isen-tropic and normal shock equations were obtained. (Tsienpoints out two ery early papers of limited from equation (1) ykddsT dp=o (5 ,c,dT-p(l-bp),Now the dtierentia.1expression for c, is(%9r=T(aProvided by IHS Not for ResaleNo reproduction or netw
11、orking permitted without license from IHS-,-,-OXE-DILIEN%1ONAL I?LOTVS cM Al%-LERl?EC?T DLkT0311C! GAS 241which, upon substituting from equation (1), may be inte-grated to give(6)where c is a function which describes the quanhnn-mechan- iical variations of cwith temperature. The second term onthe ri
12、ght of equation (6) represents the effects of gaseousimperfections on cThe function chosen for G is determined by the molecularstructure of the gas under consideration and the temperaturerange over which accurate predictions of c. are desired. Foraerodynamic purposes, diatomic gases are of primary i
13、ntcrest.The import-ant temperature range extends from liquefa.ct,iontemperatures to several thousand ddgrees Rmkine. A rela-tively simple function may be written for c, in this case, asthe number of translational and rotational degrees of freedomis constant, and only the variation with temperature o
14、f thetibrational heat capacity need be cotiidered. This functionis ()02 (emc, =Coi 1-!- (yru 1 _g(m)2(7)The second term in the brackets, essentially a Pla.nck term,accounts for the tibra.tional contribution to the specificheat at constant vo1ume2 The assumption is that the mole-cules of the gas beha
15、ve Likelinear harmonic oscillators insofaras the vibrat,ionaI degrees of freedom are concerned. (Seereference 8.)b expression governing isentropic expansion of an imper-fect c!.iatomic gas may now be obtained by substitutingequat-iom (6) and (7) into equation (5) and integrating fromstagnation to st
16、atic states. Since()d +, = TdPZPdTPThis fs a common method of accounting for the vm”atfon wfh temperature of the vibro-tfonal heat capacities. M MS been adopted hy Donoldson and others for imperfect gsstiudfes.thwe results the relation(8)b order to determine the Mach number of a stream, it isnecessa
17、ry to fid the velocity of flow and speed of sound inthe stream. These quantities may be found by employingthe one-dimensional energy equation,().(P) k). (P)du+d(Pu) + VdV= * d u dT+d - +VdV=O(9)Substituting equations (l), (4), (6), and (7) into equation (9)and integrating from stagnation to static-
18、tempe.rtureanddensity Ms for the velocityR6r+c(w+(%wzs=2 Cni(TO T) + 1 _e(e/ To(lo)The corresponding speed of sound is determined by substitut-ing equations (1), (5), (6), and (7) into the general equation“=%=(3.+(3.%The resulting expression isRP2T +2+P(1 /)p)1“= (l:;p)+Zcp%f 1-1-(%- Q ($) 15;T)2+1(
19、11)Combining equations (10) and (11) yields the foIIowing equation for the Mach number .A(;-)+($)lJ(,.Io+)l312=2 )(Yi-l) (+*2 2 _ 2cp + (13P) 2“1+ (-fru ($) g(:= (7),2+2, %f?- (7, 1)ISENTROPIC FLOW EQUATIONS(i)i=(:Y=(+-M2)-%(:)i”(i)=(+=)+7-1(%)i=(;)T=(+7+-17A-1)2(7;-1),9%+ 1 2+w-l M2.A* (-)At =MP,()
20、=27,M,2 (7* 1)-Zi Yt+ 1()hence, it has been employed in this paper to determine band c, as well as e, for air. It is sufficient for the ilhmka.tiveapplications presented here to determine the former con-stants only for the approximate flow equations. Separatevalues of 8 are found for the exact and a
21、pproximate Planckterms.As pointed out previously (note the development of approx-imate expressions for Cfl,cU,and ), it is a simple matter toobtain the characteristic gas constants appearing in the app-roximate flow equations. In this example, c and 6 werechosen fist to give agreement between experi
22、mental andtheoretical values of c? and , the latter values being, calcu-lated from equations (33) and (34), respectively. Data onthe variation of these quantities with pressure and tempera-ture at high temperatures were obtained from references 12through 19? A comparison of these data with theory, s
23、etting*ThesedatahtwebeenemalatedbytheResearchDepartmentofthePittsburgh-DesMofnesSteelCompeng,and the results of thle correlation are presented here.24sc=2.25 X 108 R, ft6/slug, sec2 and 8=5800 R, is shown infigures 3 and 4 for pressures of O tmcl 144X 10s pounds persquare foot absohde. The agreement
24、 is observed to be goodup to temperatures of 30000 R. Using equations (15) and(16) to calculate c= and , respectively, and a vrduc of8=5500 R, it is seen tha,t at zero pressure excellent rqpe-ment with the correlated data is obtained up to 50000 R.The approximate theory was then chcckecl al low temp
25、era-tures with experimental wdues of given in reference 20.At temperatures above liquefaction and pressures up to 25atmospheres, the difference between theoretical ancl cxpmi-mental values of was found not to exceecl 3 pcrccnt. P 8000 I;$ ,c $4! Q 70001.jjkO. I /c %Q I 1. Corretafed da+o - - .% 7000
26、 ; ,ck. .%(b) -%_ 6000c, 2- 3 57/0 ”-2030 5oxlo.Absolufe femperufure, T, “R(b) P=144X10; pounds per square foot abeolutc.Figmw 3.Variation ofc,wibh temperature at absoluto pressures of Oand 144XW pourrdsper square foot absolute,II-;111,1Provided by IHSNot for ResaleNo reproduction or networking perm
27、itted without license from IHS-,-,-ONE-DIM2Z_XSIOXAIIFLOWS OF .422 IMPERFECT DIATO.MIC GAS 249(a) P=O, rounds per square foot 18. Ellenwood, Frank O., IMik, Nicholas, and Gay, Norman R.:The Speci6c Heats of Certain Gases over Wide Ranges ofPressures and Temperatures; Air, CO, C02, Cm, C&H4, H?,N, an
28、d 02. Cornell University Engineering Experiment StationBuUet.inno. 30, Oct. 1942.19. Hubbard, J. C., and Hedge, A. H.: Ratio of Specific Heats of Air.Jour. Chem. Phys., I-O1.5, Dec. 1937, p. 978.20. ?i-ational Research Council of the United States of America,InternationsJ Critical Tables, VO1. 5, Is
29、t cd., McGraw-HiUBook Co., Inc., X. Y., 1929, pp. 81-82.21. National Research Council of the United States of America,International Critical Tables. VO1. 3, let cd., McGraw=HillBook Co., Inc., N. Y., 1928, pp. 9-10.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
