SOME FUNDAMENTAL ASPECTS OF COMPRESSIBLE FLOW

Transcription

1 SOE FUNDAENAL ASECS OF CORESSIBLE FLOW ah number gas veloity mah number, speed of sound a a R < : subsoni : transoni > : supersoni >> : hypersoni art three : ah Number 7

2 Isentropi flow in a streamtube In order to illustrate the importane of the ah number in determining the onditions under whih ompressibility must be taken in aount, isentropi flow, i.e., fritionless adiabati flow, through a streamtube will be first onsidered. From previous hapter, we know that ; d d and a the above equation an be written as : d d a d () his equation shows that the magnitude of the frational pressure hange, indued by a given frational veloity hange, depends on the square of ah number. art three : ah Number 8

3 Next, onsider the energy equation. Sine adiabati flow is being onsidered ; d p d R p d Sine; R v p and R p Above equation an be written as ; d ( ) d () Lastly, onsider the equation of state; d d d ombining above equation with eq.() and eq.() art three : ah Number 9

4 d ( ) d d his equation indiates that: d d (negative sign means, density derease when veloity inreased) at., at.33, d d d d % % At low mah number, density hanges will be insignifiant. art three : ah Number 3

5 Normally at <.3, the fluid is assumed inompressible. It should also be noted that above equation an we written as ; d d ( ) Similarly, the temperature differene is negleted at lower value of ah number. art three : ah Number 3

6 ah waves Disturbanes tend to propagated ahead of the body in motion to warn the gas of the approah of the body. his is due to pressure at the surfae is higher than surrounding gas and pressure waves spread out from the body. he pressure waves spread out at the of sound Effet of the veloity of the body relative to the speed of sound (pressure wave veloity) on the flow field. art three : ah Number 3

7 Consider for subsoni flow <, figure (). Speed of the body u and speed of sound a, where u<a. Body position at a, b, and d at time interval t. Waves generated at time, t, t and 3t. Sine u<a, a body moves slower than the waves and therefore a body will never overtake it. art three : ah Number 33

8 If u>a, then >, the flow is supersoni, a body moves faster than the waves and will overtake it, (figure ()). he waves lie within a one whih has its vertex at the body at the time onsidered. On gas within this one aware of the presene of the body. ertex angle α is alled ah angle, where ; a sin α u he one is therefore termed a onial ah wave. art three : ah Number 34

9 art three : ah Number 35

10 art three : ah Number 36

11 ONE-DIENSIONAL ISENROIC FLOW INRODUCION An adiabati flow (a flow in whih there is no heat exhange) in whih visous losses are negligible, i.e., it is an adiabati fritionless flow. Although no real flow is entirely isentropi, there are many flows of great pratial importane in whih the major portion of the flow an be assumed to be isentropi. art four : One-Dimensional Isentropi Flow 37

12 For example, in internal dut flows there are many important ases where the effets of visosity and heat transfer are restrited to thin layers adjaent to the walls, i.e., are only important in the wall boundary layers, and the rest of the flow an be assumed to be isentropi. Even when non-isentropi effets beome important, it is often possible to alulate the flow by assuming it to be isentropi and to then apply an empirial orretion fator to the solution so obtained to aount for the non-isentropi effet, for example, in the design nozzle. art four : One-Dimensional Isentropi Flow 38

13 GOERNING EQUAION By definition, the entropy remains onstant in an isentropi flow. (:onstant) (4.) From equation (4.) art four : One-Dimensional Isentropi Flow 39

14 Hene, sine the general equation of state gives ; or It follows that in isentropi flow ; Realling that R a, that ; a a eq.(4.5) he steady flow adiabati energy equation is next applied between the point and point. his gives ; p p art four : One-Dimensional Isentropi Flow 4

15 It an be written as ; ) ( ) ( p p From ; R R p p So, it follows that ; ) ( ) ( eq.(4.6) his equation applies in adiabati flow. If frition effets are also negligible, i.e., if the flow is isentropi, eq.(4.6) am be used in onjuntion with the isentropi state relations given in eq.(4.5) to obtain ; art four : One-Dimensional Isentropi Flow 4

16 ) ( ) ( and ) ( ) ( Lastly, it is alled that the ontinuity equation gives ; A A whih an be rearranged to give ; A A art four : One-Dimensional Isentropi Flow 4

17 SAGNAION CONDIIONS Stagnation onditions are those that would exist if the flow at any point in fluid stream was isentropially brought to rest. If the entire flow is essentially isentropi and if the veloity is essentially zero at some point in the flow, then the stagnation onditions will be those existing at the zero veloity point. art four : One-Dimensional Isentropi Flow 43

18 However, even when the flow is non-isentropi, the onept of the stagnation onditions is still useful, the stagnation onditions at a point the being the onditions that would exist if the loal flow were brought to rest isentropially. If the equations derived in the previous setion are applied between a point in the flow where the pressure, density, temperature and ah number are,,, respetively, then if the stagnation onditions are denoted by the subsript, the stagnation pressure, density and temperature will, sine the ah number is zero at the point where the stagnation onditions exist, be given by ; art four : One-Dimensional Isentropi Flow 44

19 ( for the partiular ase of 4. ) art four : One-Dimensional Isentropi Flow 45

20 CRIICAL CONDIIONS he ritial onditions are those that would exist if the flow was isentropially aelerated or deelerated until the ah number was unity, ( ) hese ritial onditions are usually denoted by an asterisk. By setting, we found ; * * a a * art four : One-Dimensional Isentropi Flow 46

21 * By setting, we found ; * * a a * * For the ase of air flow ;.833 *,.58 *,.634 * art four : One-Dimensional Isentropi Flow 47

22 AXIU DISCHARGE ELOCIY Also known as maximum esape veloity, is the veloity that would be generated if a gas was adiabatially expanded until its temperature has dropped to absolute zero. Using the adiabati energy equation gives the maximum disharge veloity as : ˆ p his an be rearranged to give ; ) ( ˆ p ) ( ˆ a a art four : One-Dimensional Isentropi Flow 48