Turbomachinery Lecture Notes

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1 Trbomachinery Lectre Notes KTH Corse MJ49/MJ41 Trbomachinery for Compressible Flids DRAFT version Compressors Damian M. Vogt KTH Heat and Power Technology

2 Trbomachinery Lectre Notes Axial Compressor Design Parameters Damian Vogt Corse MJ49 Nomenclatre Sbscripts Symbol Denotation Unit c Absolte velocity m/s h Enthalpy J/kg m& Mass flow rate kg/s r Radis m Tangential velocity m/s w Relative velocity m/s α Absolte flow angle deg β Relative flow angle deg ω Rotational speed rad/s C Normalized absolte - velocity C = c W Normalized relative - velocity W = w U Normalized tangential velocity U = = 1-0 Total 1 Inlet rotor Otlet rotor (inlet stator) 3 Otlet stator n Normal r Radial component x Axial component θ Tangential component

3 Trbomachinery Lectre Notes Compressor Stage Denotations and Conventions Stage denotations 1 rotor stator 3 1 rotor inlet stator inlet 3 stator otlet Reference radis ω Stage velocity triangles rotor stator w c w 3 w 1 c 1 c 3 Velocity triangles denotations and conventions Relative flow angle θ neg α, β neg Origin Axial velocity component (absolte and relative) Relative velocity x pos c x =w x Axial direction θ pos α, β pos Absolte flow angle α β w w θ Relative circmferential velocity component Absolte velocity c c θ Absolte circmferential velocity component Circmferential speed

4 Trbomachinery Lectre Notes Stage Velocity Triangles The stage velocity triangles are commonly employed as graphical measre to represent averaged kinetics of the flow at a reference radial position throghot the stage. Usally one of the following reference radial positions is sed: Mean radis rm rh + rs = Eq. 1 Eler radis ( radis that splits the annlar cross section in half) re rh + rs = Eq. The absolte frame of reference is bond to the stator and is therefore non-rotating. The relative frame of reference is bond to the rotor and rotates with the circmferential speed of the rotor at the reference radis obtained from = r ref ω Eq. 3 The relation between the velocities in the absolte frame of reference (denoted absolte velocities ) and the ones in the relative frame of reference (respectively denoted relative velocities ) is the following w x = c x w θ = c θ Eq. 4 Eq. 5 where c x and c θ are the axial and circmferential components of the respective velocity as follows c = cx + c θ w = wx + w θ Eq. 6 Eq. 7 The flow angles are defined as Note: c θ c x w θ w x tan α = Eq. 8 tan β = Eq. 9 The relative velocity is the velocity that an observer sees while sitting on the rotor The rotor blades ths see the relative flow velocities The direction of the absolte flow velocity at stator otlet corresponds approximately to the stator blade metal angle at the trailing edge The direction of the relative flow velocity at rotor otlet corresponds approximately to the rotor blade metal angle at the trailing edge

5 Trbomachinery Lectre Notes First Design Parameter: Degree of Reaction The degree of reaction relates the change in enthalpy effectated in the rotor to the change in enthalpy of the stage as follows rotor = Eq. 10 hstage R Δ Δh, which can be rewritten as R Δh Δh h h rotor 1 = = rotor = Eq. 11 Δhstage Δhrotor + Δhstator h h1 + h3 h The change in enthalpies in stator and rotor respectively are related to the velocities as follows In the rotor the rothalpy w I = h + is constant, ths w w1 1 h + = h1 + Eq. 1 leading to 1 h h1 = ( w1 w 1 + ) Eq. 13 Note 1 ( w1 w ) is the contribtion arising from the deceleration of the relative flow 1 ( 1 ) is the contribtion arising from the centrifgal effect In the stator the stagnation enthalpy c h 0 = h + is constant, ths c3 c h 3 + = h + Eq. 14 leading to h3 1 h = ( c c3 ) Eq. 15 Sbstitting these expressions into the eqation of stage reaction above leads to the following general expression R w1 w 1 + = Eq. 16 w1 w c c3

6 Trbomachinery Lectre Notes For a normal repetition stage with the following restrictions r r c 1 = c 3 Eq. 17 c x, 1 = c x, = c x, 3 = const Eq. 18 = Eq the expression of the degree of reaction can frther be simplified. Firstly it can be noted that the circmferential speed cancels ot. Secondly the velocities shall be written in terms of their components as c = cx + c θ and w = wx + w θ respectively. This yields the following expression wθ,1 wθ, R = Eq. 0 wθ,1 wθ, + cθ, cθ,3 The relative velocity components in the denominator shall be expressed by the absolte velocity components as w θ = c θ leading to wθ,1 wθ, R = cθ,1 cθ,1 + cθ, + cθ, + cθ, cθ,3 Eq. 1 After canceling ot elements the expression can be rewritten as wθ,1 wθ, R = ( cθ, cθ,3 ) Eq. At this stage the enmerator shall be expressed as w w = ( w w )( w + w ) θ, 3 θ, θ,3 θ, θ,3 θ,. The absolte velocity components in the denominator shall be expressed in terms of relative velocities as c = w θ + leading to θ ( ) ( ) wθ,1 wθ, wθ,1 + w R = ( w + w ) θ, θ,3 θ, 3 Eq. 3 Both the circmferential speeds in the denominator and the relative components wθ w cancel ot finally yielding, 1 θ, 1 R ( w θ,1 + w θ, = ) Eq. 4

7 Trbomachinery Lectre Notes At this position a more intimate analysis of the degree of reaction is appropriate. For this prpose the relative circmferential velocity component at position 1 shall be expressed in the absolte frame of reference as w = c θ yielding θ ( cθ,1 + wθ, ) = ( cθ,1 wθ, R = + ) Eq. 5 c By expressing the circmferential velocity components in terms of flow angles as tan α = θ the c x following expression is obtained for the degree of reaction 1 c x R = ( tanα 1 + tan β ) Eq. 6 In the above eqation the degree of reaction is expressed in terms of axial velocity component, circmferential speed and stator and rotor otflow angles respectively, which are approximately eqal to blade metal angles at trailing edge (note that the assmption of repetition stage shall still be valid). According to the convention of velocity components depicted above flow angle β is negative whilst flow angle α 1 is positive. Usally the stage inflow angle α 1 is close to zero, i.e. almost axial inflow. This leads to the following observations An increase in flow angle β leads to an increase in degree of reaction ( β R ), i.e. the contribtion of enthalpy change in the rotor to the total change in enthalpy in the stage gets larger An increase in flow angle α 1 leads to an decrease in degree of reaction ( α 1 R ), i.e. the contribtion of enthalpy change in the stator to the total change in enthalpy in the stage gets larger For compressor stages the degree reaction sally lies in the range [0.5 1]

8 Trbomachinery Lectre Notes Second Design Parameter: Loading Factor The loading factor relates the change in total enthalpy effectated in the stage to the rotational speed as follows Δh0 = Eq. 7 Under application of Eler s trbine eqation the change in enthalpy can be expressed as Δh = cθ c leading to 0, 1 θ,1 cθ, 1cθ,1 = Eq. 8 For a normal repetition stage with the following restrictions r r c 1 = c 3 Eq. 9 c x, 1 = c x, = c x, 3 = const Eq. 30 = Eq the expression of the loading factor can frther be simplified to cθ, cθ, 1 = Eq. 3 Expressing the absolte flow velocities in the relative frame of reference as loading factor can be expressed as c θ = w θ + the wθ, wθ, 1 = Eq. 33 An eqivalent expression can be obtained by sbstitting the relative velocity component at position 1 in the absolte frame of reference as c θ = w θ + yielding wθ, cθ, 1 = 1+ Eq. 34, which also can be expressed in terms of flow angles β and α 1 as follows c x ) Eq. 35 = 1+ ( tan β tanα1 According to the convention of velocity components depicted above flow angle β is negative whilst flow angle α 1 is positive. This leads to the following observation: Decrease in flow angles β and α 1 lead to increase in loading factor ( α ) β, 1

9 Trbomachinery Lectre Notes Third Design Parameter: Flow Coefficient The flow coefficient relate the axial velocity component to the circmferential speed as follows c φ = x Eq. 36 The only observation to make for this coefficient is that the higher the axial velocity in the stage the higher the flow coefficient. As can be recognized below the flow coefficient stretches the velocity triangles in the axial direction.

10 Trbomachinery Lectre Notes The Normalized Velocity Triangle At this position the normalized velocity triangle shall be introdced. The normalization consists therein that all velocity components are depicted with reference to the otlet circmferential velocity 3. The normalized velocity components are denoted by the respective capital letters and yield from c C = 3 Eq. 37 w W = 3 Eq. 38 U = Eq The special case of a normal repetition stage shall be regarded here for the sake of simplicity. The applied principle is however valid for all types of trbine stages. Conveniently the velocity triangle is drawn with a common origin for stator and rotor otlet. As a normal repetition stage with the condition c x, 1 = c x, = c x, 3 = const is considered the height of the cx triangle corresponds to C x = = φ, i.e. the flow coefficient. C W 1 Φ C 1 W U=1 R Note: The height of the velocity triangle corresponds to the flow coefficient Φ The loading coefficient corresponds to the circmferential distance between C and C 1. In the case of a repetition stage this eqals to the circmferential distance between W and W 1. The degree of reaction eqals to the distance between axial and half the midpoint between W and W 1.

11 Trbomachinery Lectre Notes Special Cases The special cases are here analyzed for the case of normal repetition stage. Similar analysis can be performed in a general manner for other types of stages. Degree of Reaction eqal to one half (R=0.5) The expression of the degree of reaction yields the following ( wθ,1 + wθ, ) wθ, + = wθ, R = = Eq. 40, which is eqivalent to cθ, = wθ,1 Sbstitting this expression into the eqation of loading coefficient yields wθ, wθ,1 wθ, cθ, = = + 1 = 1 Eq. 41 Velocity triangle C W 1 Φ C 1 W U=1 R Note: As cθ, = wθ,1 and normal stage it follows that c = w1 and with the assmption of repetition stage ( c 1 = c 3 ) conseqently Δ hrotor = Δhstator. The change in enthalpy in a reaction stage is ths eqally split on rotor and stator. Both stator and rotor effectate compression of the flid and ths p > p 1 and p 3 > p

12 Trbomachinery Lectre Notes Zero Exit Swirl (c Θ,3 =0) With cθ, 3 = 0 = wθ,3 + wθ, 3 = and w θ, 3 = wθ, 1 the degree of reaction writes to The loading coefficient yields from 1 1 wθ, R = ( wθ,1 + wθ, ) = Eq. 4 wθ, wθ,1 wθ, = = + 1 Eq. 43 Combining the two expressions leads after reformlation to a relationship between degree of reaction and loading coefficient as follows Velocity triangle = (1 R) Eq. 44 C W 1 Φ C 1 W U=1 R Note: The flow exists the stage prely axial, i.e. there is no swirl at stage exit For a zero exit swirl stage the degree of reaction and the loading factor are dependent

13 Trbomachinery Lectre Notes Simplified Off-Design Analysis At this position a simplified off-design analysis shall be carried ot. Ideally a compressor is operated at design point, i.e. the point at which the design has been carried ot. At this point the flow angles are eqal to the design flow angles (see velocity triangle) and the compressor operates at optimm performance. Off-design operation of a compressor might however occr nder the following circmstances: Change in compressor rotational speed Change in flow rate Note that the flow rate will establish as a fnction of the consmer after the compressor, i.e. the device that follows the compressor. In a gas trbine the consmer comprises combstion chamber and trbine. In case the compressor is spplying a network for pressrized air the consmer can be seen as the sm of a nmber of sb-consmers, e.g. pnematic tools. As has been shown above the loading coefficient can be represented by the following eqation nder the assmption of normal repetition stage c x ) Eq. 45 = 1+ ( tan β tanα1 In a rogh approximation the flow angles β and α 1 can be assmed eqal to the blade metal angles at trailing edge of rotor and stator respectively. As the compressor stage geometry does not change dring operation the parameters in the parenthesis above can be replaced by constant leading to c = 1+ x k 1 Eq. 46 c x Employing the flow coefficient φ = this expression can frther be simplified to = 1+ φ k 1 Eq. 47 When drawn in a flow rate loading coefficient diagram the loci of possible compressor operation ths lie on a straight line with inclination k 1 as depicted below. Note that the same analysis is also valid for trbines. k 1 >0 trbine k 1 =0 netral k 1 <0 compressor θ

14 Trbomachinery Lectre Notes The compressor operating line indicates that a higher flow rate leads to a lower loading factor and by this also to a lower increase in pressre. Note that this behavior is different for trbines as a higher flow rate also leads to a higher loading factor. To nderstand this easier trn the problem pside down: if yo connect a trbine to a pressre plenm yo will be able to increase the flow rate by increasing the pressre difference across the trbine. A simplified off-design analysis can be obtained by normalizing the above eqations by the φ vales at design point sch that the expression yields = 1 and = 1 at design point. Normalizing the above expression by 1 φ k + 1 D D D D leads to D = Eq. 48 φ D The inclination k 1 is expressed by the design parameters at design point as is D 1 k1 = Eq. 49 φ D finally leading to D 1 = D + D 1 D φ φ D Eq. 50 Graphically the simplified off-design analysis yields the following operating characteristic: Normalized loading parameter D 1 1 Note: 1 φ φ D Normalized flow coefficient This is jst a simplified off-design analysis and ths does not reflect the tre conditions. The overall characteristic is however captred, i.e. lower pressre ratio at higher flow rate Towards low flow rates the operation of a compressor is limited by instability phenomena (rotating stall, srge) Towards high flow rates the operation is limited by choke, i.e. sonic conditions (M=1) in narrowest cross section

15 Trbomachinery Lectre Notes Axial Compressor Effect of Degree of Reaction Damian Vogt Corse MJ49 Sorce: Trapel, W., 1977 Thermische Trbomaschinen Vol. 1, 3rd Ed. ISBN

Trbomachinery Lectre Notes 1 007-10-11 Compressor Design Aspects Damian Vogt

16 Trbomachinery Lectre Notes Compressor Design Aspects Damian Vogt Corse MJ49 Axial Compressor Rotor Types Rotors of drm and disk constrction Sorce [1]

17 Trbomachinery Lectre Notes Mltiple Rotors Sorce [1] Abbreviations: L.P. low pressre H.P. high pressre Advantages Single spool simple and robst constrction Twin spool optimm speed for LP and HP compressor; LP sally rotates slower than HP

Trbomachinery Lectre Notes 3 007-10-11 Sorce [1] Advantages of

compressor (large diameter gives high blade tip speeds spersonic de to low

18 Trbomachinery Lectre Notes Sorce [1] Advantages of triple-spool engine: Optimm fan speed; fan speed is lower than LP and HO compressor (large diameter gives high blade tip speeds spersonic de to low speed of sond) Avoid having a geared fan (a gear box for these power ratings is heavy)

19 Trbomachinery Lectre Notes Different problem for power generation trbine: Power shaft mst rotate at constant speed. At part load behavior the compressor wold however work in a better operating point at part speed. Soltion: free power trbine Solar Trbines mechanical drive Sorce []

Trbomachinery Lectre Notes 5 007-10-11 Rotor Blade Mont Sorce [1]

tree Compressor: dove tail Reason: higher load at higher temperatres

20 Trbomachinery Lectre Notes Rotor Blade Mont Sorce [1] Comparison of compressor and trbine blade fixations: Trbine: fir tree Compressor: dove tail Reason: higher load at higher temperatres for trbines, fir tree distribtes loads more efficiently bt is more costly

Trbomachinery Lectre Notes 6 007-10-11 Stator Vane Mont Variable Stage Geometry Variable gide vanes: control of stager angle

21 Trbomachinery Lectre Notes Stator Vane Mont Variable Stage Geometry Variable gide vanes: control of stager angle of all blades in a blade row Sorce [1] Sorce [1] Prpose: Minimize incidence at off-design operation Pre-swirl for load reglation

22 Trbomachinery Lectre Notes Inlet gide vanes Rotor Stator Zero swirl Negative swirl Positive swirl Sorce [3]

23 Trbomachinery Lectre Notes Srge Control (Bleed Valves) Sorce [1] Bleed valves are sed to avoid compressor srge dring fast acceleration. Their se dring steady operation however wold be very wastefl and is therefore not taken into consideration.

24 Trbomachinery Lectre Notes Bondary Layer Control Sorce [3]

25 Trbomachinery Lectre Notes Aspiration Sorce [4]

26 Trbomachinery Lectre Notes Blade Twist Sorce [1]

27 Trbomachinery Lectre Notes References [1] Rolls-Royce, 199, The Jet Engine, ISBN , Derby, UK, 199 [] Boyce, M.P., 00, Gas Trbine Engineering Handbook, Glf Professional Pblishing, ISBN , Hoston, 00 [3] Eckert, B., 1953, Axialkompressoren nd Radialkomprssoren, Springer Verlag, Berlin/Göttingen/Heidelberg, 1953 [4] Merchant, A., 1999, Design and Analysis of Axial Aspirated Compressor Stages, PhD thesis, Massachsetts Institte of Technology, USA, Gas Trbine Laboratory, Boston, 1999

Axial Compressor Design Parameters

Trbomachinery Lectre Notes 007-09-9 Axial Compressor Design Parameters Damian Vogt Corse MJ49 Nomenclatre Sbscripts Symbol Denotation Unit c Absolte velocity m/s h Enthalpy J/kg m& Mass flow rate kg/s