APPENDIX A
APPENDIX A
SYMBOLS
sum of orifice areas A3 and A4, in.2; cm 2 AC1 AC2 sum of orifice areas A5 and As, in.2; cm 2 general orifice area, in.2; cm 2 Ai spray b a r nozzle area, in. 2 ; cm 2 AN governor piston area, in. 2; cm 2 AP bypass orifice area, in. 2 ; cm 2
pt
reference orifice area, in. 2 ; cm 2 A1 area, in. 2; cm governor A2 maximum governor area, in. 2; cm 2 x2 orifice area, f(P3, P2), in. 2 ; cm 2 A3 orifice a r e a , f(P2), in. 2 ; cm 2 A4 orifice a r e a , f(p3, p2), in. 2; cm 2 A5 orifice area, f(P2), in. 2 ; cm 2 lb6 area coefficient "31 area coefficient "32 area coefficient a4 1 area coefficient a42 area coefficient "5 1 a r e a coefficient "52 area coefficient "6 1 a r e a coefficient "6 2
constants (i = 1, . . . , n)
' i D damping coefficient, (lbf)(sec)/in. ; (N)(sec)/m c d pump flow coefficient, (lb) (hr)/% speed; (kg)(hr)/R speed bias force, lbf: N Fb
I$
i
k orifice flow conversion constant, lbm/(hr)(in. 2 ) (lbf/in. 2, ; kg/(hr)(cm2) 2 1/2 "cm ) spring constant, lbf/in. ; N/cm kl spring constant, lbf/in. ; N/cm k2 M m a s s of governor spool, lbm; kg N engine speed, percent of rated speed
*
pressure, lbf/in. 2; N/cm ' a pressure, lbf/in. 2; N/cm i '; pressure, lbf/in. 2; N/cm2 ' b rate of change of pressure, (lbf)(in.2)/sec; N/cm /sec ' b pressure, lbf/in. 2; N/cm pC r a t e of change of pressure, (lbf/in.2)/sec; (N/cm )/see ' C A Pi general differential p r e s s u r e , lbf/in. 2; N/cm reference p r e s s u r e , lbf/in. 2; N/cm ' r compressor inlet pressure, lbf/in. 2; N/cm ' 2 compressor discharge pressure, lbf/in. 2; N/cm I p3 burner pressure, lbf/in. 2; N/cm '4 Laplace operator S OR; K inlet temperature, T2 t time, s e c 3 3 volume of chamber b, in. ; cm vb volume of chamber c, in. 3; cm vC AWB net fuel flow, lbm/hr; kg/hr AW, net fuel flow, lbm/hr; kg/hr a general fuel flow, lbm/hr; kg/hr wi i spray bar fuel flow, lbm/hr; kg/hr I wN i ' , total fuel flow, lbm/hr; kg/hr wT acceleration fuel flow, lbm/hr; kg/hr wT (max) deceleration fuel flow, lbm/hr; kg/hr wT (min) bypass fuel flow, lbm/hr; kg/hr wV 2 7 I , .. . .. . . --... - ... .. . --. _. .. . . . ..--.-. .
fuel flow through orifice A1, lbm/hr; kg/hr fuel flow through orifice A2, lbm/hr; kg/hr fuel flow through orifice A3, lbm/hr; kg/hr fuel flow through orifice A4, lbm/hr; kg/hr fuel flow through orifice A,-, lbm/hr; kg/hr fuel flow through orifice A6, lbm/hr; kg/hr equivalent linear displacement of throttle, in. ; cm linear displacement of governor spool, in.; cm maximum displacement of governor spool, in. ; cm C Y angular throttle motion, rad ha! change of throttle motion, r a d bulk modulus, lbf/in. 2; N/cm2 P P2/14.7; P2/10.1 fuel density, lbm/in. 3; kg/cm P
APPENDIX B
APPENDIX B
SYSTEM EQUATIONS
The weight flow a c r o s s the various orifices of the fuel control can be described by equations of the form I
? wi =kAi +F 1 (B1)
where the density t e r m is assumed constant and included in the coefficient k. For the six orifices of the system shown in the schematic of figure 1, this equation becomes
w1 = k A 1 d m (B2 )
w2 =kA2 (B3 )
w, = k A 3 034)
035)
w5 = k A 5 i F b (Be)
w 6 = k A 6 d R 037)
and the final fuel flow through the spray bar nozzle into the engine is
w , =kAN d m (B8)
Since the flow through the governor orifice A2 passes through A3 and A,, w2 = w3 -+ w, , i Equations (B3) to (B5) can be combined with equation (B9) to give an expression for the i intermediate p r e s s u r e Pc : Y- = Defining A3 + A4 as AC1 AiPa + A t l P b P = C A i + AE1 and If the flows W5 and w 6 are summed where AC2 = A5 + A6 The flow through the control pump is proportional to the engine speed, hence W1 = dN Combining equations (B2) and (B13) yields the differential pressure from equation (B14) into equations (B11) and (B12) Substituting results in
W5 + W6 = AC2 - dN
A1 The total fuel flow to the engine can now be computed from equations (B13), and (B16) as Equation (B17) relates the engine fuel flow, shaft speed, and the various area terms.
The flow areas are determined as illustrated in figure 4, by the force balance between I the pressure and spring forces applied across the spools of the fuel control.
The a r e a s A4 and A6, which a r e determined by the position of the second spool, L can thus be described by Similarly, A3 and A5 a r e determined by the position of the third spool: AC1 and AC2 of equation (B17) can be formed from equa- The combined a r e a t e r m s tions (B18) to (B21), where AC1 = A3 + A4 = C1 + C2P2 + CS(P3 - P2) The coefficients Ci can be determined from the a r e a s , spring constants, and flow a r e a against stroke characteristic of the system. The area characteristics for the orifices are linear functions of spool position, but the controlled by the second and third spools governor orifice area is a nonlinear function of the stroke of its spool.
It can be seen from figure 5 that, as the throttle is advanced through an angle ha!, the throttle cam converts the rotation to a linear advance against the throttle spring.
The spring, in turn, raises the forces on the spool tending to increase the governor ori- fice area A2. If the throttle advance Aa! is sufficiently large, the spool motion is limited by a mechanical stop. The spool response to throttle position can be described by " (Pb - Pa)Ap - Fb + kl(XL, - Xs) - k2Xs = MXs + DX, (B2 4) Neglecting the dynamics of the m a s s spring system reduces equation (B24) t o
(Pb - %)Ap - Fb + kl(X, - xs) - k2Xs = 0
(B2 5 1
Since
(Pa - Pb) a N2
The equation reduces to A nonlinear relation could exist between power lever and throttle spring position. How- Equation (B26) becomes ever, f o r this analysis, this function w a s assumed to be linear.
The governor orifice a r e a A2 is a logarithmic function of the throttle position Xs: The nonlinear characteristic w a s selected to compensate for the higher ratio of engine speed to fuel gain encountered for the lower speed region. By this method the overall system gain can be modified to prevent instability.
APPENDIX C
APPENDIX C
FUEL CONTROL ORIFICE SIZING
Once the basic design of the fuel control is established, the control components must be sized to satisfy the fuel requirements of the engine. An expression for the total flow from the control to the engine was developed in appendix B as For minimum flow the governor area A2 is fully closed and expression (Cl) re- duces to Similarly, for maximum flow the governor area A2 is fully open and equation (Cl) becomes
= Nd- kl + AC2
wT (max) A1 - where A2 =Agmax
Assuming that x i >> AC1, equation (C3) reduces to
Nd
WT(max) F Z - (A1 + AC2 + AC1)
A1 The fuel-control concept is based on the assumption that the minimum and maximum fuel-flow limits to corrected speed ratio can be expressed as a linear function of com- 1 . pressor pressure ratio. Specifically, it is assumed that t
iij7j - p3 (minimum flow)
N clo p, -
M
-- p3 (maximum flow) - c11 - + 5 2 ~ N p2
l b
Rewriting equations (C5) and (C6) yields N wT (min) = - ( ' 1 0'3) ' r N
wT(max) -- - h o P 3 + c12p2 + - cio)p3]
' r A comparison between the fuel-control equations (C2) and (C4) and the engine fuel requirements (C7) and (C8) yields the following expressions from which the orifice areas and AC2 may be computed AC1
% 1 = - 6 L 12 P 2 + (Cll - Cl0)P3]
'rd AC2 =-
I '10'3 - A1
' r The coefficients Cl0, Cll, and C12 must be evaluated from the minimum and maxi- mum engine fuel requirements. For the 585-13 engine considered in the analysis of this report, these requirements were approximated by selecting Cl0 = 1 . 5 4 (lbm/hr)/% speed U. S. customary units SI units = 0 . 7 0 (kg/hr)/% speed U. S. customary units 4 . 1 5 (lbm/hr)/% speed c 1 1 = SI units = 1 . 8 8 (kg/hr)/% speed C12 = 8.195 (lbm/hr)/% speed U. S. customary units SI units = 3 . 7 2 (kg/hr)/% speed Substituting these values into the a r e a relations of equations (C9) and (C10) and using values for d, P,, V, and AI from table I1 results in TABLE II. - CONSTANTS Reference orifice area, A1, in.2; c m 3 . 2 4 ~ 1 0 - ~ ; 2 0 . 9 ~ 1 0 - ~ - 2 2 Maximum governor area, A2, in. ; cm 0.15; 0.968 Governor piston area, A in.2; cm 1; 6.45 P’ Conversion constants : cl, in.2; c m2 2 . 2 ~ 1 0 - ~ ; 1 4 . 2 ~ 1 0 - ~ C2, in.2/(lbf/in. 2 ); c m2/(N/cm 2 ) 1 . 4 7 ~ 1 0 - ~ ; 13. 8X10-3 C3, in. 2/(lbf/in. 2 ); c m2/(N/cm 2 ) 3 . 5 5 ~ 1 0 - ~ ; 3 3 . ~ x I O - ~ c+, in.2; c m 2 i . 2 2 ~ 1 0 - ~ ; 7.87~10-3 C5, in. 2/(lbf/in. 2 ); c m2/(N/cm 2 ) 2 . 1 0 ~ 1 0 - ~ ; 1s. ~ X I O - ~ C6, in. 2/(lbf/in. 2 ); c m2/(N/cm 2 ) 2. i o ~ i o - ~ ; 1s. ~ X I O - ~ Cl0, (lbm/hr)/% speed; (kg/hr)/% speed 1.54; 0.70 Cll, (lbm/hr)/% speed; (kg/hr)/o/o speed 4.15; 1.88 C12, (lbm/hr)/% speed; (kg/hr)/% speed 8.195; 3.72 Damping coefficient, D, (lbf)(sec)/in. ; N-sec/cm 0.18; 0. 315 Pump flow coefficient, d, (lbm/hr)/o/o speed; (kg/hr)/o/o speed 1.62; 0.737 Bias force, Fb, lbf; N 5; 22.24 3rifice flow conversion constant, k , (Ibm/hr)in.’(lbf/in. 2 ) 1/2 ; 1 . 0 2 ~ 1 0 ~ : 863 (kg/hr)cm2(N/cm 2 ) 1/2 Spring constant, kl, lbf/in. ; N/cm 75; 131.3 Spring constant, k2, lbf/in. ; N/cm 25; 43.8 Mass of governor spool, M, Ibm; kg 0. 32’i?dO-3; 0.0574 Reference pressure, Pr, lbf/in.2; N/cm2 14.7; 10.1
Maximum linear displacement of governor spool, xs, in. ; cm
0.20; 0.508 Volume of chamber b, Vb, in. 3; cm3 6; 98. 32 Volume of chamber c, VC, in. 3; cm3 20; 327.7 3ulk modulus, p, ibf/in. 2; N/cm 2 1 . 5 ~ 1 0 ~ ; 1 . 0 3 4 ~ 1 0 ~ p , lbm/in.3; kg/cm3 Fuel density, 0.0291; 0 . 8 0 6 ~ 1 0 - ~
AC1 = 3.551X10- 4 (P3 - P2) + 14.7X10- 4 P2
U. S. customary units (C11)
= 33.2X10-4 (P3 - P2) + 1 3 8 ~ 1 0 - ~ P2
SI units
AC2 = 2 . 0 9 5 ~ 1 0 ~ ~ (P3 - P2) + 2 . 0 9 5 ~ 1 0 - ~ P2 - 3 . 2 4 ~ 1 0 ~ ~
U. S. customary (C12)
= 19. ? ’ X ~ O - ~ (p3 - P2) + 19.7x10- 4 p2 - 2 0 . 9 ~ 1 0 - ~
SI units Equations (C11) and (C12) show that the equivalent areas AC1 and AC2 can be formed from the combination of an area proportional to compressor pressure rise and an area I.
proportional to compressor inlet pressure. These expressions can be used with equa- tions (B23) and (B24) of appendix B t o compute the following fuel control orifice areas: (C13)
AC1 = A3 + A4
where
A3 = 3 . 5 5 ~ 1 0 - ~ (P3 - P2) + a31 U. S. customary units
= 3 3 . 2 ~ 1 0 - ~ (p3 - p2) + "31 SI units
A4 = 14. ~ x I O - ~ P2 + "41 U. S. customary units
SI units = 1 3 8 ~ 1 O - ~ P2 + "41
AC2 = A5 + A6
where
A6 = 2 . 0 9 5 ~ 1 0 - ~ P2 - 3.2&10-3 U. S. customary units
SI units = 19. 7X10-4 p2 - 2 0 . 9 ~ 1 0 - ~
A5 = 2 . 0 9 5 ~ 1 0 - ~ (P3 - P2) + a51 U. S. customary units
SI units = 19. ~ x I O - ~ (p3 - P2) + a51
In the simulation of the actual fuel control, the a r e a expressions (C11) and (C12) were not exactly duplicated. The corresponding expressions for the design are:
U. S. customary units 3 . 5 5 ~ 1 0 ~ ~ (p3 - P2) + 1 4 . 7 ~ 1 0 - ~ P2 - 2 2 ~ l O - ~
1 AC1 =
"
= 33. 2 x 1 r 4 (P3 - P2) + 1 3 8 ~ 1 0 - ~ P2 - 1 4 1 . 9 ~ 1 0 - ~
SI units
U. S. customary AC2 = 2. 10X10-4 (P3 - P2) + 2. 10X10-4 P2 - 12. 2X10m4
(C16)
= 19. ~ x I O - ~ (P3 - P2) + 1 9 . 7 ~ 1 0 - ~ P2 - 78. 7x10-
SI units Equations (C15) and (C16) are equivalent t o the following engine minimum and maximum fuel -flow requirements : For minimum flow L
-
' 3 14.85
-- 6fi - 1.544 - + -
U. S. customary units N
-
p2 p2 ' 3 + 4.63 =0.7- - SI units
I
For maximum flow wT
-
p3 1 . 32
! . @ = 4.153 - + 8.195 - -
U. S. customary units
- N p2 p2
.
fi
p3 0.43
= 1.88 - + 3.72 - -
SI units p2 p2 It can be seen that equations (C15) and (C16) are close to equations (C11) and (C12).
Equations (C17) and (C18) are close to equations (C5) and (C6) except that a t e r m in- versely proportional t o inlet p r e s s u r e P2 is included. The effect of this t e r m is illus- trated by the scatter shown in figure 7. Since the purpose of the simulation was t o dem- onstrate the feasibility of the design, the analytical work of this report is based on equa- tions (C15) and (C16).
REFERENCES 1. Zalmanzon, L. A. ; and Cherkasov, B. A. : Control of Gas-Turbine and Ramjet En- gines. NASA TT F-41, 1961.
2. Sobey, Albert J. ; and Suggs, Alfred M. : Control of Aircraft and Missile Power- plants. John Wiley & Sons, Inc., 1963.
3. Willoh, R. G. ; and Seldner, K. : Multistage Compressor Simulation Applied to the Prediction of Axial Flow Instabilities. NASA TM X-1880, 1969. 1 4. Batterton, Peter G. ; and Zeller, John R. : Dynamic Performance Analysis of a Fuel- Control Valve for Use in Airbreathing Engine Research. NASA TN D-5331, 1969.
I .
NASA-Langley, 1970 - 28 E-5263