北京工业大学-高等工程热力学－第5讲 可压缩流体在管道内的流动

1、by Professor Liu Zhongliang2by Professor Liu Zhongliang3n引言n管流，或管流，或流动的通道内的流动的通道内的稳定流动稳定流动n一维：一维：V/A很大，很大，L/de1，by Professor Liu Zhongliang4by Professor Liu Zhongliang5by Professor Liu Zhongliang6n根据稳定流动能量方程：Adiabatic stagnation(1) 2102hchby Professor Liu Zhongliang7by Professor Liu Zhongliang8pby P

2、rofessor Liu Zhongliang9n该量与微弱压力扰动的传播速度（声速，音速）a有关：(2) 2ssvpvpaFor perfect gases,(3) kRTkpvaby Professor Liu Zhongliang10n其中，by Professor Liu Zhongliang11nIt can be shown that for , the sound speed can be expressed as the following(2a) lnln1TvzkzRTaby Professor Liu Zhongliang12nDefinition(4) acaspeed

3、soundlocalvelocity cfluidMnClassification of Compressible flows: by Professor Liu Zhongliang13n理想气体，一维绝热流动Tchchhp 2120kRTMMaccTcTcpp 2120(5) 21120MkTTkkcRMckRTTpp1 2120by Professor Liu Zhongliang14100kkTTpp(6) 211120kkMkppkppvv100(7) 2111120kMkvv(5) 21120MkTTBasic Equationsby Professor Liu Zhonglian

4、g16 n一维可压缩流，无化学反应xAdx(cA)x(cA)x+dxby Professor Liu Zhongliang17n研究质量守恒情况nd时间内在x处流入CV的质量为： mx=cA d (8)nd时间内在x+dx处流出CV的质量为：mx+dx.! 21222dxxmdxxmmmxxxdxxdxxmmmxxdxxby Professor Liu Zhongliang18nd时间内CV质量增加了，(9) )(dxcAdxmmxdxx(10) )( AdxdmCVn按质量守恒原理，(11) CVdxxxmmmby Professor Liu Zhongliang19n将式（9）和式（10）

5、代入式（11），有，)( )(AdxddxcAdxmmxx(12) 0)(cAxA(11) CVdxxxmmmby Professor Liu Zhongliang20nFor steady or incompressible flows,0(13) 0)(cAxOr,(14) ConstvcAcADifferentiating Eq. (14) to give,(15) cdcvdvAdAby Professor Liu Zhongliang21nFor incompressible flows, dv=0, therefore(15) cdcvdvAdA(15a) cdcAdA That

6、is, the velocity c always the flow area AnFor ?by Professor Liu Zhongliang22nSame assumptions as continuity equationnForce balance analysis: xAdxSSby Professor Liu Zhongliang23n基本原理：冲量定理(16) )(mcddFxWhere Fx Forces acting along x-direction d(mc)/d Increment of the x-direction momentum per unit time

7、by Professor Liu Zhongliang24n单位时间内x 处流入系统的动量：(17) )(2cAccAMxn单位时间内x+dx 处流出系统的动量：.! 21222dxxMdxxMMMxxxdxx(18) dxxMMMxxdxxby Professor Liu Zhongliang25将式（17）代入式（18）得到，(19) )(2dxcAxMMxdxxn单位时间内CV内流体的动量了，(20) )( )(AdxccAdxMCVby Professor Liu Zhongliang26n单位时间内，流体流过CV后，x方向的动量实际上增加了：+()x dxCVxdmcMMMd将式（2

8、0）和式（19）代入上式得到：xxMAdxcdxcAxMmcdd)()()(2(21) )()()(2dxAccAxmcddby Professor Liu Zhongliang27按冲量定理，它应该。作用在x方向上的力包括：(21) )()()(2dxAccAxmcddby Professor Liu Zhongliang28Pressure forcesnPressure forces:nPressure at xnPressure at x+dxxAdxSSAdxxpppAFp)(22) dxxpAby Professor Liu Zhongliang29摩阻力S shear stres

9、sA flow areaD Hydraulic diameterxAdxSSdxDASFS4perimeter Wetted 4DA(23) 212fcS(24) 4212dxDAfcFSf 摩阻系数（skin friction factor)by Professor Liu Zhongliang30体积力（body force)nF 单位质量物质的体积力n F与x正方向之间的夹角(25) cos dxFAFbby Professor Liu Zhongliang31作用在x方向上的合力将（25）和（21）代入（16），整理后得到，bSpxFFFF(25) cos4212dxFAdxDAfcd

10、xxpA(26) cos421)()(22FADAfcxpAcAxAcby Professor Liu Zhongliang32或者写成(27) cos2)(1)(22FfcDxpcAxAc如果忽略体积力，(28) 2)(1)(22fcDxpcAxAcby Professor Liu Zhongliang33能量方程（Energy equation）nSteady flow systems onlynThe equation is:(30) )(2(29) 2122gzdcddhwqzgchwqss If shaft work and potential energy change are n

11、eglected, thenby Professor Liu Zhongliang34Energy equation(30a) 2(29a) 2122cddhqchq If shaft work and potential energy change are neglected, thenby Professor Liu Zhongliang35熵方程(Entropy equation)n开口系统稳定流动系统(31) TQmsmsddSddSiieeCVg对于稳定流动系统，(31a) 0mmmddSieCVby Professor Liu Zhongliang36Entropy equatio

12、n有，(32) 1TqssddSmieg其中，J/kg 热量，单位质量工质所吸收的mQq令， )Kkg/(J 1单位质量工质的熵产，ddSmsggby Professor Liu Zhongliang37Entropy equationn通道内的流动，在微元管段dx上，有(33) Tqsssieg(33) Tqdsdsgby Professor Liu Zhongliang38Entropy equationn按热力学基本方程，dpdhTds代入方程（33），TqdpTdhTdsg11dpTqdhT1)(1(30) )(22gzdcddhwqsdpTgzdcdT1)(2112(33) Tqds

13、dsgby Professor Liu Zhongliang39Entropy equationn讨论：根据熵增原理(34) 01)(212dpgzdcdTdsg 0gds于是，(35) 01)(212dpgzdcdby Professor Liu Zhongliang40Entropy equationn如果忽略位能的变化，那么 01212dpcd221cddp by Professor Liu Zhongliang42(34) gases idealfor velocity sound kRTkpva(35) number Mach local acM (36) equation cont

14、inuity ConstcA (36a) equation continuity cdcvdvAdA(37) equation momentum 0)(12xpcAxAIsentropic, frictionless, zero body forceby Professor Liu Zhongliang43n特点：n待求变量多n方程数目多n求解方法也多(38) equation energy 0 22cddh(40) equation State (39) equation process RTpvconstpvkby Professor Liu Zhongliang44One-dimensi

15、onal flow in nozzlesn喷管：变截面流道n研究重点：对流动参数的影响n对于理想气体，h=cpT=cppv/R所以，(41a) )(vdppdvRcdhpby Professor Liu Zhongliang45One-dimensional flow in nozzles由过程方程，pvk=const(41b) 0vdpkpdv代入方程（41a)，并注意到，1kkRcp(41c) kpdvdh代入能量方程（38），(41a) )(vdppdvRcdhp(38) 0 22cddhby Professor Liu Zhongliang46One-dimensional flow

16、in nozzlesn由声速的定义(42) 0cdcvdvkpv(34) kRTkpva 02cdcvdva 02cdcacvdv(35) acM (43) 2cdcMvdvn代入连续性方程(36a) cdcvdvAdAby Professor Liu Zhongliang47One-dimensional flow in nozzles(44a) ) 1(2cdcMAdAn再将（43）代入（44a）(44b) 122vdvMMAdA(39) constpvkn由过程方程 (45) 1pdpkvdv(44c) 122pdpkMMAdA (43) 2cdcMvdvby Professor Liu

17、 Zhongliang48One-dimensional flow in nozzles(44a) ) 1(2cdcMAdAn于是我们得到：(44b) 122vdvMMAdA(44c) 122pdpkMMAdAby Professor Liu Zhongliang49讨论n可以看出：nA c , v , p （扩压管,pressure increaser）nA c , v , p (喷管，nozzles）nA c , v , p (喷管，nozzles）nA c , v , p （扩压管,pressure increaser)by Professor Liu Zhongliang50结论by

18、Professor Liu Zhongliang51M1by Professor Liu Zhongliang52Aby Professor Liu Zhongliang53 212220chh )(2202hhc Tchp 120202TTTccp(46) 112102002kkppvpkkcby Professor Liu Zhongliang54222vcAm kppvv/12002(47) 11210200/10202kkkppvpkkppvAm by Professor Liu Zhongliang55 1102/102kkkrppppmm 000212vpkkvAmrby Prof

19、essor Liu Zhongliang5610.80.60.40.20p2/p000.10.20.3m/ mrby Professor Liu Zhongliang57n流速当地声速 临界截面*vkpac(48) *1121000kkppvpkkkppvv/100*by Professor Liu Zhongliang58(49) 12*10kkkppby Professor Liu Zhongliang59(50) 12*00vpkkac 12 ak入口(48) *1121000*kkppvpkkcby Professor Liu Zhongliang60(51) 121100minmax

20、kkkvpkAm by Professor Liu Zhongliang61 范诺流by Professor Liu Zhongliang63基本假设by Professor Liu Zhongliang64基本特性by Professor Liu Zhongliang65基本特性n基本方程(52) equation continuity ConstAmc(53) equation energy 202const hch(54) equation State RTpby Professor Liu Zhongliang66基本特性由状态方程，(55) TdTdpdp由连续性方程，(56) 0c

21、dcd由能量方程，(57) 0 212dcdh由热力学基本方程，(58) dpdhTds(54) equation State RTp(52) equation continuity ConstAmc(53) equation energy 202const hchby Professor Liu Zhongliang67基本特性方程（55）和（56）联立，消去密度项，(59) TdTcdcpdp将方程（59）代入（58）， cdcpdTTpdhTds(60) cdcRTRdTdhTds(58) dpdhTdsby Professor Liu Zhongliang68基本特性下面设法消去速度项

22、。按能量方程（57）(57) 0 212dcdh(61) 0 2cdccdh代入方程（60）dhcRTRdTdhTds2dhcakRdTdh221(61) 112RdTdhkM(60) cdcRTRdTdhTdsby Professor Liu Zhongliang69基本特性n理想气体的焓：hcpTdhcRdhkMTdsp211kcRp11dhMkTds2111(63) 122MkTMshby Professor Liu Zhongliang70基本特性n于是(63) 122MkTMsh then 1, if 0 then 1, if0 then 1, ifshMshMshMby Profe

23、ssor Liu Zhongliang71基本特性shby Professor Liu Zhongliang72基本特性by Professor Liu Zhongliang73基本特性by Professor Liu Zhongliang74基本特性by Professor Liu Zhongliang75范诺流熵增的计算n由方程（60），(60) cdcRTRdTdhTdscdcRdTTRdhTds1cdcTdTRRcRdsp11kRRcp 11cdcTdTkRdsn对上式从x0积分至x，得by Professor Liu Zhongliang76范诺流熵增的计算n另一方面，按能量方程 (

24、64) ln01100 xkxxccTTRssRs02002 2 2hchchxx22200020 xxchhchhby Professor Liu Zhongliang77范诺流熵增的计算n于是0000 xxhhhhcc(65) 0000 xxTTTTcc代入（64） (66) ln210001100 xkxxTTTTTTRssRs (64) ln01100 xkxxccTTRssRsby Professor Liu Zhongliang78范诺流熵增的计算n由于绝热摩擦流动过程，于是， 0 ln01100 xkxxccTTRssRs 1 0110 xkxccTTby Professor L

25、iu Zhongliang79范诺流极限管长的计算(28) 2)(1)(22fcDxpcAxAc 2)(122fcDdxdpcAdxdA(68) 2)(22fcDdxdpcdxdby Professor Liu Zhongliang80范诺流极限管长的计算n按连续性方程， c=常数(69) 22dxfcDdpcdc(70) 22122dxRTcDfpdpRTdc由连续性方程， c=常数cRTpc aMRTpMkRTRTpConstpMRTk(68) 2)(22fcDdxdpcdxdby Professor Liu Zhongliang81范诺流极限管长的计算n对上式取微分，得到(71) 21T

26、dTMdMpdpn在式（70）中，)(1222aMdRTRTdc)(12kRTMdRT(72) 22TdTkMkdM (70) 22122dxRTcDfpdpRTdcby Professor Liu Zhongliang82范诺流极限管长的计算将（71）、（72）和（73）代入（70）(73) 2222kMRTaMRTc(74) 2) 1(21) 1(2122222dxkMDfTdTkMMdMkM同除kM2，得到，(74) 21122242dxDfTdTkMkMdMkMkM(70) 22122dxRTcDfpdpRTdcby Professor Liu Zhongliang83范诺流极限管长的

27、计算n按能量方程(53) 202const hch(53a) 202const TccTcpp(53b) 212const kRTMTcp(53c) 0212122 kRTdMdTkRMdTcp(75) 2112122 MkdMkTdTby Professor Liu Zhongliang84范诺流极限管长的计算或者，(76) )211 (142242dMMkkMMdxDf(76a) 1)211 (42242MMkkMDfdxdMn代入方程（74），整理化简后得到，(74) 21122242dxDfTdTkMkMdMkMkMby Professor Liu Zhongliang85范诺流极限管

28、长的计算对方程（76）积分，(77) )211 ()211 (4002222420MMMMxxMkkMdMMkkMdMfdxD定义，(78) 10dxfxfxxby Professor Liu Zhongliang86范诺流极限管长的计算得到(78) 21121ln211142020202020MkMkMMkkMMkMxDf其中，nM0 入口处(x=0)的Mach数nM 任意位置x处的Mach数by Professor Liu Zhongliang87范诺流极限管长的计算00.050.10.150.26869704fLmax/D(Mo/M)max0.20.40.60.81Mo/M1530456

29、0754fx/Dby Professor Liu Zhongliang88范诺流极限管长的计算4fLmax/D(Mo/M)max48121620Mo/M-0.4-0.20.20.40.60.81.04fx/Dby Professor Liu Zhongliang89范诺流极限管长的计算by Professor Liu Zhongliang90范诺流极限管长的计算n在（78）中令M1，即得到对应入口条件下的极限管长Lmax的计算公式(79) 21121ln211420202020maxMkMkkkkMMLDfby Professor Liu Zhongliang91范诺流极限管长的计算02468

30、10M00123454fLmax/Dby Professor Liu Zhongliang92范诺流极限管长的计算n根据方程（79）：(79) 21121ln211420202020maxMkMkkkkMMLDfby Professor Liu Zhongliang93 1)211 (42242MMkkMDfdxdM 2dxdMby Professor Liu Zhongliang94by Professor Liu Zhongliang95by Professor Liu Zhongliang9600.20.40.60.811.24fx/D00.511.522.53MM0=2.0M0=4M0

31、=6M0=0.5M0=0.75M0=0.625by Professor Liu Zhongliang97by Professor Liu Zhongliang98n温度温度by Professor Liu Zhongliang99范诺流中其它参数的变化n将方程（将方程（71）改写）改写(71) 21TdTMdMpdp(80) 212122TdTMdMpdp(75) 2112122 MkdMkTdTn将方程（将方程（75）代入，整理后可以得到）代入，整理后可以得到(81) 211) 1(1212222MdMMkMkpdpby Professor Liu Zhongliang100范诺流中其它参数

32、的变化n这说明，即： by Professor Liu Zhongliang102(82) 0 )2(02dTccddhqp(83) cdcq by Professor Liu Zhongliang103by Professor Liu Zhongliang104根据声速和Mach数的定义：ConstkRTaacM (84) 2222cdcMdMcdcMdM连续性方程：连续性方程： c常数常数(85) 021 22cdcdcdcdby Professor Liu Zhongliang105压力沿管长的变化由动量方程(27) cos2)(1)(22FfcDxpcAxAc02)(22fdxcDdp

33、cd(86) 022fdxcDdpcdcdcdc由方程（85）(87) 0222fdxcDdpdc(85) 021 22cdcdcdcdby Professor Liu Zhongliang106压力沿管长的变化按过程方程，Tp/(R)=常数(88) dpdp代入方程（87），02)(22fdxcDpdpcp(89) 0212fdxDpdpcpby Professor Liu Zhongliang107压力沿管长的变化注意到p/RT，1122cRTcp12kckRT122kca112kM221kMkM 02122fdxDpdpkMkM (90) 1222dxkMkMDfpdpby Profes

34、sor Liu Zhongliang108速度沿管长的变化n由方程（85）、（88）， 021 22cdcdcdcd dpdp(91) 2122cdcpdpn代入方程（90）， (92) 142222dxkMkMDfcdcby Professor Liu Zhongliang109n按理想气体绝热滞止压力的计算公式，(6) 211120kkMkpp(93) 21121222200MdMMkkMpdppdp(84) 2222cdcMdMn按方程（84），(94) 21211222200cdcMkkMpdppdpby Professor Liu Zhongliang110n将方程（91）代入，有(

35、91) 2122cdcpdp(95) 21112200pdpMkkMpdp (90) 1222dxkMkMDfpdp(96) 211) 1(2112222200dxMkkMMkkMDfpdp将方程（90）代入，有by Professor Liu Zhongliang111 (90) 1222dxkMkMDfpdp (92) 142222dxkMkMDfcdc(96) 211) 1(2112222200dxMkkMMkkMDfpdpby Professor Liu Zhongliang112n如果如果kM2-10，即，即， (90) 1222dxkMkMDfpdp (97) 1kM 0dxdp

36、02dxdc即，x，p，cby Professor Liu Zhongliang113n压力、速度的变化规律n如果kM2-10，即， (90) 1222dxkMkMDfpdp (98) 1kM 0dxdp 02dxdc即，x，p，cby Professor Liu Zhongliang114M 1kby Professor Liu Zhongliang115(99) 121122kMMkJby Professor Liu Zhongliang116 121122kMMkJ 01 021122kMMk 1 12kMkM (100) 12 1kMkby Professor Liu Zhonglia

37、ng117n在该区域内，显然有：M 1k 12kby Professor Liu Zhongliang118 121122kMMkJ 01 021122kMMk 1 12kMkMby Professor Liu Zhongliang119 121122kMMkJ 01 021122kMMk 1 12kMkM 12 1kk(101) 1kM by Professor Liu Zhongliang120n在该区域内，显然有：M 1k 12kby Professor Liu Zhongliang121 121122kMMkJ 01 021122kMMk 1 12kMkM 12 1kk(102) 12

38、kMby Professor Liu Zhongliang122n在该区域内，显然有：M 1k 12kby Professor Liu Zhongliang123M 1k 12kby Professor Liu Zhongliang124by Professor Liu Zhongliang125由方程（84）和（92），知， (92) 142222dxkMkMDfcdc(84) 2222cdcMdMcdcMdM (103) 142222dxkMkMDfMdM(104) 114224dMMkMdxDfby Professor Liu Zhongliang126积分，MMxxdMMkMdxDf0

39、2240114，得到，(105) 11ln4202020MMkMMMxDfby Professor Liu Zhongliang1270102030405060704fx/D00.20.40.60.81M0/ MM0=0.1k=1.4 1 ,1 . 00kMkMM 4maxLDfby Professor Liu Zhongliang12800.511.522.534fx/D1471013M0/ MM0=5k=1.4 1 ,50kMkMM 4maxLDfby Professor Liu Zhongliang129(106) 1)ln(4202020maxkMkMkMLDf(105) 11ln42

40、02020MMkMMMxDfby Professor Liu Zhongliang1300123456789100.00010.0010.010.111010020kMmax4LDfby Professor Liu Zhongliang131by Professor Liu Zhongliang132by Professor Liu Zhongliang133(107) ln1ppRs)(21202ccq(108) 2202TccTqsfby Professor Liu Zhongliang134fgsss(109) 2ln2021TccppR因为，)(202202MMkRTccMMccRTR

41、Tpp0011by Professor Liu Zhongliang135(110) 0 12ln20200MMkMMMRsgby Professor Liu Zhongliang137(111) equation energy 212dcdhq(112) 1热力学基本方程dpdhTdsqby Professor Liu Zhongliang138n由方程（112）得到， 11dhdphsT(113) 11dTdpchsTpn由动量方程得到，(28) 2)(1)(22fcDxpcAxAc0)(12dxdpcAdxdA0)(2 cpdxd(112) 1热力学基本方程dpdhTdsqby Prof

42、essor Liu Zhongliang139所以，(114) 2ConstcpRTcppcp22Ideal gases:p/RT)1 (2RTcp)1 (2kMp(115) )1 (2constkMp微分，(116) 122kMkdMpdpby Professor Liu Zhongliang140cRTpc constc ckRTkpkRTckRTkpconstMkRTkp(117) )(2constTpM(118) 222TdTMdMpdpby Professor Liu Zhongliang141(119) 222pdpTdTMdM 2122pdpTdTkMkMpdp(120) 122

43、TpkMkMdTdp(116) 122kMkdMpdpby Professor Liu Zhongliang142(113) 11dTdpchsTp 1122kMkMTcphsTp 1122kMkMcRhsTp 1kkcRp 11122kMkMkkhsTby Professor Liu Zhongliang143整理后得到，(121) 1122kMMhsT(121a) 1122MkMTsh由方程（121）可以看出，by Professor Liu Zhongliang144(121a) 1122MkMTshby Professor Liu Zhongliang145shabcdM1, 加热加热

44、M1, 加热加热M1kM1M1, 冷却冷却164by Professor Liu Zhongliang146n从上面的图可以看出：by Professor Liu Zhongliang147n假定：假定：n单位管长对单位流体的加热量单位管长对单位流体的加热量为为qLn按能量方程按能量方程(122) 212dcdTcdxqpL(123) 2动量方程Constcp）（连续性方程124 constc ）（状态方程125 RTpby Professor Liu Zhongliang148将（125）代入（123）有，）（状态方程125 RTp 2ConstcRT两边同除 cconst，(126) Co

45、nstccRT取微分，02dcdccRTdTcR 12122dccRTRdT 1222kMkckRTcRT(123) 2动量方程Constcpby Professor Liu Zhongliang149将（127）代入（122） (127) 112122dckMRdT 1112122dckMRcdxqpL 1kkcRp (128) ) 1(1222LqkMMdxdc(122) 212dcdTcdxqpLby Professor Liu Zhongliang150n该式告诉我们： (128) ) 1(1222LqkMMdxdc0 , 0 0 , 0dxdcqdxdcqLL流体被冷却，流体被加热，0 , 0 0 , 0dxdcqdxdcqLL流体被冷却，流体被加热，by Professor Liu Zhongliang151(122) 212dcdTcdxqpL）管道入口处流体的速度）管道入口处流体的温度( ( 1010ccTTxxby Professor Li