Three dimensional flow analysis in Axial Flow Compressors

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May Jenkins
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2 Thee dimensional flow analysis in Axial Flow Compessos 2

3 The ealie assumption on blade flow theoies that the flow inside the axial flow compesso annulus is two dimensional means that adial movement of the fluid while passing though the blade passages is ignoed. 3

4 Radial flow can appea due to following easons 1. Centifugal action on being impated otational motion is expeienced by the fluid 2. Convegence of the annulus flow tack intoduces adiality in highly loaded compesso stages. 3. Twist and tape (chod and thickness-wise) of the blade intoduces adial component to the fluid; 4. Tip cleaance effects - effect of tip flow aound the open tip of the blade oto 4

5 5 Passage votex fomation inside the blade passages; 6. Tempeatue/ Enthalpy / Entopy gadient in the adial diection (due to 1 to 4 above); 7. Blade solid body thickness blockage (including the effect of cambe and stagge) 8. End wall (casing and hub) bounday laye blockage effects, that deflect the flow inwad, in addition to educing the main flow ate 5

6 The adial equilibium theoy is based on the pemise that the adial gadients of foces expeienced by all the fluid contibutes to the adial movement of the flow and hence those foces must be balanced by the static foces exeted by the pessue gadient existing in the flow, so that at any instant of time the fluid system is in adial balance of foces i.e. in adial equilibium. 6

7 Motion of a paticle w..t. two co odinate systems O Assume that a fluid paticle, p is moving in an abitay path within the two coodinate systems o In axial flow compessos otos need a otating co-odinate system, wheeas the stato may use a static co-odinate system 7

8 o O Velocity of the paticle P with espect to fixed axes system fom the figue is : The two efeence systems have elative motion epesented by Ṙ (motion of vecto R w..t. fixed oigin, o). ω is otation of the paticle with espect to moving axes system xyz, and v xyz is the tanslational motion of the paticle with espect to moving axes xyz. V d = dt 8

9 Velocity of P w..to small xyz, v xyz Vectoially, motion of P is summation of the motion of the moving system w..t the fixed system and paticle P w..t the moving system O, Velocity of P w..t the fixed system d ρ' dr d = + dt dt dt XyZ. V R V Lect - 9 = + + ω.ρ' xyz = dρ' dt xyz =R+ρ ' 9 9

10 Definitions of the acceleations Acceleation of P w..t. Fixed coodinates, a = dv dt And, acceleation of P w..t. Rotating coodinates xyz, d a = V xyz xyz dt xyz 10 10

11 Thus, total acceleation of P w..t the fixed coodinate system, a V = d dt = d ( ) V ** xyz dω ρ' + R + dt dt xyz 11

12 The acceleation of P w..t the body fitted otating coodinates xyz, is a summation of its tanslational and otating motions dv dt xyz = dv dt xyz xyz + ω.v xyz Rotation in xyz system may be captued in the eqns dω.ρ' ( ) dρ' dω = ω + ρ' dt dt dt dρ' dρ' dt dt = + ' xyz ω ρ 12 12

13 Acceleation of the paticle p w..t the fixed oigin O may be witten as a =.. ** xyz xyz + ρ' + aω R 2ω vω ω ρ' Fo motion with constant angula velocity ω a ω = a xyz ω+ 2ω v + ρ' xyz 13

14 Fo a compesso blade passage, the flow velocities ae Diffeentiating, a V V V xyz = V = elative velocity ( ) C ( absolute velocity ) = + = = V ( ) dω ( ρ' ) d V = - dt xyz Tanslational motion ω ρ' V + ω V + u dt Rotation Final accn. a ω xyz xyz ( ) = + + ρ' a ω2 ωv 14 14

15 Consideing the equilibium of foces along abitay flow diection, s diection, we get, between any two axial stations, sepaated by a small distances s whee aea, A i is constant. P. Foce, p.a i = A i.ρ. s. a (Mass x acceleation) Hence, 1Δp = ρ Δs a Thus, the acceleation equation fom the last slide may be witten as : 1 Dv 2. p= ω + 2ωV ρ Dt 15

16 As the flow in compesso blade is diffusing DV Dt is negative 1 Dv 2. p = ω +2 - ω V ρ Dt x x 16

17 New of axis notations whee, w and a ae the adial, whil(peipheal) and axial diections espectively. W 17 17

18 Assumptions made ae:- The fluid is fictionless The oto is igid and otates with constant angula velocity ω The flow is steady elative to the oto The adial vaiation of density is neglected This still leaves enough scope fo fomation of i) voticity, ii) entopy gadients, and iii) stagnation enthalpy gadients in the flow field. 18

19 Then fom the definition of unit vectos D i t = i t = i ωdt V t and D Lect - 9 i w V t wt =- i =- i ωdt w using, By definition V =V i +V i +V i D wt wt a a ( ) D( ) =V + Ds dt Dt ds s is length in any diection as ds =0 dt Fo Steady State flow

20 Now, the equation fo flow inside the compesso blade passage may be ecast using, θ and z coodinate system and modified to : DV DV DVt = - + w t + + w dθdθ DV i a w a Dt Dt V i dt Dt V i dt Dt Using the coodinate systems the flow velocity in elative fame is V and its components may be shown as V a, V, V w 20

21 Now, the equation may be esolved in its thee components, using, θ and z coodinate system, ( ) + 1 D V V t ω- = w V p -ρ (a) 1 -. = + + p 2.ω.V Vρ. D V a V t. V w (b) 1 - p V =ρ D V a (c) 21 21

22 v w Using the coodinate systems the flow velocity in elative fame is V and its components may be shown as V a, V, V w 22

23 The velocity tiangle fo the flow gives us C V t = t + w w Then equation (a) and (b) fom slide 21 can be ewitten as 1 p DV - = V - ρ Ds ( ) w C t (d) 1 p V -. D s θ=ρ ( ) D.C t w (e) 23 23

24 Now, we can wite the kinematic elation as, V D ( ) D ( ) Ds = V a Da Whee V a and a ae axial the components of V and s espectively. Define a meidional diection by D i =D i +D i m m a a φ Now, equation (d) fom the last slide can be ewitten as : 1 - = V - p ρm ( ) DV C w t

25 ρlect - 9 Meidional diection may be defined as tan.φ= V V Z and V = V m sinϕ Hence the foce balance equation only in the adial diection is e-witten as : 1 = -- p Dsin.φsin.φC 2 wt V 2 m V m DV m Now, by ou ealie definition V D ( ) D ( ) Ds = V m Dm 25 25

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27 Now, D sin. φ D φ = cos. φ Dm D m Lect - 9 Dφ 1 =- Dm Whee m is the adius of cuvatue of the meidional flow The negative sign is abitay. But, fo axial flow compesso the flow tack inside geneally moves towads lesse φ o highe m, i.e. the flow late on flattens out. Hence, 1 p = + cos.φ-ρρ and 2 2 D C t V m V w V Dm m This is the full adial equilibium Equation fo cicumfeentially aveaged (blade to blade) flow popeties inside of a tubo machine blade ow m m 21 27

28 Fo old fashioned compesso designs V m (instead of constant axial velocity V a / C a ) is consideed constant and the last tem is eliminated. In the vey ealy design of compesso the flow path was consideed linea and hence even the 2nd tem vanishes, giving us back the simple adial equilibium equation p = = ρ. ρ C w t C Fo moden compesso, this simple adial equilibium equation elationship is inadequate and it becomes necessay to utilize the full adial equlibium equation. w 23 28

29 Wheeve the flow is not expeiencing the centifugal foce, the adial equilibium can not be applied. Expeiments have shown that in between the blade ows, in the axial gaps between the oto and the stato, thee could be adial shift of the meidional path. Hence fo accuate design flow analysis the full adial equilibium equation is be used. Fo using the full REE, fo computational puposes, futhe steps need to be taken. i) The R.E.E is to be tansfomed into a fom that contains patial deivatives of all paametes with espect to and θ ii) Next, the cicumfeential aveage of those paametes is taken by integating ove θ fom pessue side of one blade to the suction side of the othe blade. iii) The flow is analysed at vaious axial stations with a) Enegy equation, b) Continuity condition and c) R.E.E. 29

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31 It is necessay that flow popeties obtained in this manne at vaious axial stations be consistent with one anothe as the flow popeties ae evaluated fom hub to tip at each station. That means adial acceleation of the fluid paticle is to be accounted fo in the R.E.E. This can be achieved by assuming shapes fo the meidional steamlines consistent with the continuity condition expessing the adial acceleation in tems of the steamline slope and cuvatue. This implies an iteative method of solution. This method, in which sufaces ae used to build up flow inside a tubomachiney blade, has been widely used. The equations on the blade-to-blade suface and those on the meidional plane need to be solved sepaately. 31

32 The 3-D flow computations has povided immense assistance to engine designes. It has cut down on design time and has educed dependence on costly expeimental analysis. The 3-D methods have helped undestand vaious flow phenomena e.g. seconday flow development, chocking in the stages, effects of end-wall flows etc. Howeve, the designe uses these solutions in conjunction with many empiical elations and expeimental data to make the design. Thee is still scope fo impovement in these methods and fo educing dependence on empiical elations 32

33 Next Class Poblem Solving and Tutoial poblems using simple 3-D flow theoies on Axial Flow Compesso 33

Chapter 7-8 Rotational Motion

Chapter 7-8 Rotational Motion Chapte 7-8 Rotational Motion What is a Rigid Body? Rotational Kinematics Angula Velocity ω and Acceleation α Unifom Rotational Motion: Kinematics Unifom Cicula Motion: Kinematics and Dynamics The Toque,