APPENDIX A
APPENDIX A
m NOMENCIATURE a Local Speed of Sound ft/sec a = _goRT s *2 2¥ a _ Critical Speed of Sound a ft/sec - _+i goRTT A Area in b Axial Blade Width inch Ab Axial Distance Between Blade Rows inch C Blade Chord inch C ft/sec Isentropic Velocity C2 o = 2goJ _H' o e Specific Heat at constant Pressure BTU/lb-°R P C Specific Heat at Constant Volume BTU/lb-°R v d Throat Width inch D Diameter inch D Diffusion Parameter Diffusion Parameter of Suction Side D S Diffusion Parameter of Pressure Side Dp ibm-ft Proportionality Factor in Newton Second Law 32.17 2 go ibf-sec h Blade Height inch h Specific Static Enthalpy BTU/lb i Incidence degree J Mechanical Equivalent of Heat (778.2) ft lb/BTU K R Rotor Velocity Coefficient W4/W _ k Axial Clearance inch a k Radial Clearance inch r Relative Roughness K_/c k Leakage Factor BTU k Thermal Conductivity (ft-HR-°R) Page A-I Length of Point on Blade Surface from Point A inch on Suction Side L Length of Point on Blade Surface from Point A inch on Pressure Side I, Specific Work in Blading U BTU/LB m T _o Specific internal Work BTU/LB m M Absolute Mach Number M Molecular Weight Mach Number Relative to Rotor Blade Mass Flow ]om/s e c n Number of Blades in a Row n >_o!ytropic Exponent Nm_ N Rotational Speed rpm Mass Ratio 0xidizer/Fuel 0/ F }.
Separation Parameter Static Pressure psia S Total Pressure ?F: psia Total PrEssure Relative to Rotor Blade psia Prandtl-Number i P V 2 Dynamic Head.
psi
2 gol-
ibf-ft R Gas Constant ibm-_ r Radius ft _e Reynolds Number Reyr_olds Number based on Blade Chord c Peynoids Number based, on Hydraulic Diameter k)n T _ _T , s2 s4 R Degree of Reaction_ R = rrj I X x TTI - s S Blade _itch inch Page A-2 / / t Maximum Blade Thickness inch t inch Trailing Edge Thickness e oR Effective Blade Temperature T B °R Total Temperature T T T °R Static Temperature S oR Total Temperature Relative to Rotor Blade TTR U Wheel, Velocity ft/sec V Absolute Velocity ft/sec W Velocity Relative to Rotor ft/sec (I (alpha ) Angle Between Axial Direction and Absolute degrees Gas Velocity (beta) Angle Between Axial Direction and Relative degrees Gas Velocity c Y (gamma) v Deviation'
(delta)
degrees A
(delta)
Prefix to Indicate Change < Loss Coefficient (zeta) Blading Efficiency, total to total _u Blading Efficiency, total to static Internal Efficiency_ total to total Internal Efficiency, total to static Hi (eta) Turbine Efficiency Based on Total to Static Pressure Ratio Mean Politropic Efficiency rap Flow Efficiency in Nozzle Blading n_ n Flow Efficiency in Rotor Blading Z (zeta) Stagger Angle degrees Page A-3 fr degrees Camber Angle Absolute Viscosity
(m_)
LBm/(HR- ft )
._ ° ibm/f t3
P (rho) _e._s mty
Blade Solidity = c/s (sigma) SUBSCRIRT. S Inlet Stator :L Out.let Stator !:elet Rotor
4 O_Ltle t Rotor
At Blade Throat d h At Blade Root At Mean Blade Height m t At Blade Tip u Tangential Component x Axial Component Subscript preceding a symbol indicates the number of the stage° SUPERo CRIF_, Attached, to temperature or enthalpy means value for isentropic expansion° Page A-4
NOI.L03_IC -_I'XV
..... j CD
\
kH o !
N?
CD Z A
k-
APPENDIX B
APPENDIX B ESTIMATING LOSSES IN A _TURBINE STAGE Traupel _lj# _ presents a complete and Consistent system for estimating losses in a turbine stage. Because this loss system was used for the M-I oxidizer turbine design, the pertinent portion of Chapter 8.4 (1) is abstracted and presented herein. Nomenclature deviated from that used in the report proper with respect to station numbers, which are defined in Figure BI, and angles, which are defined in Figure B2. Those parameters .used solely in this appendix are defined as appropriate.
i. Blading Efficiency Deft niti on*:
(1)
2go_ - qn ah_ + 2go_/ _ 2gJ = qr A h' + Wl2
r 2go_ where: 1 -_ =
+ _ )
(2)
_n or _r i- (_p + _w + <r zus with: : Total loss coefficient (p = Profile loss coefficient = Wall loss coefficient w : Secondary loss coefficient r = Damping wire loss coefficient (not used for M-I turbine) zus The profile loss coefficient _ is obtained from: P
(3)
X5 + _m + <F <p = <po " × p Xm with* : from Figure B3 <po = f C_o ; _l) _F'or the rotor, the absolute angles should be replaced by the relative angles and the flow conditions at the stator exit by the flow conditions at the rotor exit.
(1)_raupel, Walter, Thermische Turbomaschinen, Erster Band, Springer Verlag, iBerlin/Goettingen, Heidelberg, 1958, Pages 269 through 298.
Page B-I !
IVI c Pl
from Figure B5
=
; ks/c )
from Figure B4
Xm : f (Vl/a l)
from Figure B6
x_ = f (1 - f)
= f (i - f) from Figure B6 m
<F = f(hl%) from Figure B7
The wall loss coefficient <w' which is due to the friction loss on hub and tip annuli, is estimated as follows: Sl sinai _b
_w = _po ×p h + Cf h. sin_1 7 (4)
The second term gives the loss due to gas friction in the gap between stator and rotor. It usually is negligible.
For the secondary loss coefficient _ we set: r
r : _'l<ro + _s (5)
,_ V I sina I from Figure B8**
w_+h_ _ro = f (_l - U I )
from Figure B9
Xl/Xp = f (c/h)
from Figure BIO
<s : f (hs/h)
(not applicable for present design) *See footnote on Page B1.
_In Figure B8 the band for _ro : f (_) was extrapolated. The band was assumed to become horizontal, similar to the loss coefficients given in VAVBA.
Page B-2 is obtained from: The dampening wire loss coefficient zus
cw sin<(
(6)
D d . d 2 2 D t - Dh C = 1.2 - 2.4 w c - ._ - o8 w 4d 2. Velocity Triangle Efficiency and K ( q = K 2) known, the velocity triangle With q n and qr or K n r can be calculated and with it the velocity triangle efficiencies:
(7)
= Ah/Ah' Static to static su 2 V° - V2 2 Ah+ Lu 2go'D :: = Total to total
(8)
qu 2 V° - V2 2 Vo - V2 2 Ah' + A h' + 2god 2goO 2 V2 2 V ° - Ah+ 2goU * Lu Total to static
(9)
u : ah' + Vo2/2g# Ah' + Vo2/2go U 3o Stage Efficiency The internal or stage efficiencies are defined as, follows: *See footnote on Page _i.
Page B-3 Ah - ZAL _ Ah EAL : qsu - g< _]si : Ah' Ah v Ah' _si : _su - ((Ln + (Lr + _'<R + (v + (B) Static to static (i0) Witht Ln = Leakage loss coefficient for nozzle (stator) < = Leakage loss coefficient for rotor Lr Sum of disc - and shroud friction loss coefficients =: Blade windage loss coefficient v Moisture loss coefficient (not applicable for present design) As in section 2 the following additional efficiencies are defined Ah' B ) 2 "qi : _]U - ((Ln + <Lr + X_R + _v + _ V ° - V2 2 Ah' + 2go_ total to total (ii) * _ Ah' : _u - -(<Ln + <Lr + _]<R + < + <B )- i
v Ah'+ Vo2/2go
total to static (12) -The leakage loss coefficients (Ln' <Lr for blades without shrouds are calculated from: _Ln KI ALn ALr = _ _Lr : K I (13) _i sin_l _2 sin_2 from Figure :BI.I With : K I = f ( a _ ) '_See footnote page BI.
Page B_L
AL = Leakage area (ALn, nozzle; ALr, rotor)
= Annulus area=?rDh
and for blades with shrouds from: ALr
KII ALn KII
- _-- ; _Lr - _Ln _J Zn r _2 sin _2 With*: Z = Number of laybrinths g = arc of admission from Figure BI2 KII = f (ah'/Vl2/2gofi]) The loss coefficients _ for disc - and shroud friction are expressed by equations R (15) and (16).
<__ 2._4 cM (%/I)m) 4 (Dh/h) (Disc) (l_)
Cf (Dt/Dm)4 (b/h)
<R : _ (Shroud) (16)
With :
Dh%P
) from Figure B5 C M : f (Recto V m _ X _ U U2 g_2 P2 = ah'/(U2/2go _ )
ut_ p
) from Figure B5 Cf =
: f (_eof
*See footnote page BI.
Page B-5
The blade windage loss coefficient _v can be calculated from equation (17)
I -e .30 Zb b
(17)
<v = C + With_
c : .04 + .52 h/Dm
blades free, upstream I
c = .o19 + 1.1 (.iS5 - h/Dm) e
blades covered, upstream C = .88 - 13 (h/Dm) 2 blades free, downstream C = .02 + 3.0 (.125 - h/Din) 2 blades covered downstream number of admission segments which Z_ : result in the total admission g .
Make if possible Zb = i.
Frequently it is convenient to calculate the power loss due to friction (NR) or ventilation (Nv) , rather than the loss coefficient _R or _v"
_R = 4 cM p__m 3 (Dh/_)3 Dh 2 (18)
go
Nv - 2 c (1-_) p D m h _m (19)
go The moisture losses being negligible in the present application are not discussed further.
According to Ref. 3 the loss system described above is applicable to subsonic turbines having blades of the general shape of Figure BI with a solidity according to Figure B2.
® @
lIJ
<<
<< <%
®
NOZZLE ROTOR FIGURE BI Page B-6 F IG[_E BI _4ous_N¢).
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