Subject: Introduction to Component Matching and Off-Design Operation % % ( (1) R T % (

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1 16.50 Leture 0 Subjet: Introdution to Component Mathing and Off-Design Operation At this point it is well to reflet on whih of the many parameters we have introdued (like M, τ, τ t, ϑ t, f, et.) are free for the pilot to ontrol, and what the inter-relationships are that determine the others. This onnetivity is in part mehanial, like the shaft power balane (Eq. 9 of Leture 18), but it also omes via flow ontinuity among omponents. This topi is usually relegated to the very end of the study of engine omponents, where it is introdued under the rubri of Component Mathing (Leture 31 in our NOTES). We find it advantageous to move most of it forward to this point. The prie to pay for the insight to be gained is the need to introdue one assumption at this point (to be justified later). This is the assumption that the stators leading to the turbine (the turbine nozzles are hoked. This means the mass flow rate an be written as # " +1 & Pt 4 (" ) A # & 4 (" m = m 4 = % = " )1) ( % ( (1) R T % ( t 4 $ " +1' $ ' Where A 4 is the effetive flow area of these nozzles. But in addition, we have already shown in Let. 18 that the exhaust nozzle is also hoked. Passing the same flow through two hoked apertures in series imposes very strong onstraints on the flow onditions. The flow an be expressed at the throat as and equating (1) and (), P A m = m = t (" ) () R T t 7 P t 7 T t 4 A = 4 (3) P t 4 T t 7 A 7 For a non-afterburning turbojet, P t 7 P t 5 and T t 7 = T t 5 A t = 4 (4) " t A 7 # and if the turbine is ideal, t = " t # $1, and we obtain (( )1) " A % ( +1 4 t = $ ' (5) # A 7 & 1

2 ( " A %( +1 and then t = $ 4 ' (6) # A 7 & This is a strong result: as long as both, the turbine nozzles and the exhaust throat remain hoked, the turbine maintains the same pressure and temperature ratios (same operating point), regardless of fuel flow, Mah number, altitude, et. We an now trae the variability of other quantities: (1) Compressor ratios. In terms of ϑ = T t4 /T to, Eq. (9) of Leture 18 gives / =1 + " (1# ) ; = " # # $1 t (7) Thus τ and π do vary, but only as a funtion of the single quantity ϑ: τ = τ (ϑ) for a given engine. () Mah number at ompressor inlet (M ). The flow at ompressor inlet is generally subsoni, so we express the flow rate there as m (M ) 1 m = m " +1 ' (" #1) $ " +1 P A & ) (M & ) (8) R T 1 " # 1 t & + M ) % ( = t m (M ) ; m ) = M The dimensionless flow funtion m (M ) inreases to a maximum of 1 when M = 1, then dereases again. Equating (8) to (1), we see that m (M ) P T A = t 4 t 4 P t T t 4 A 0 1 M # For an ideal ombustor, P t 4 = P, and so, using /# $1 t 3 = ", T t = T to, " /" #1 m (M ) = A 4 (9) $ A

3 Sine = (" ), we see now that M = M ( ) as well (the supersoni solution for M given m an be disregarded). (3) Dimensionless air flow. Returning to (8), we see that the dimensionless mass flow m m (10) # P A " & % to $ R T ( to ' (flow rate as a fration of what the ompressor would pass if its inlet were hoked), is one more a unique funtion of ϑ. This is very useful for saling from one operating ondition to another. (4) Fuel/air ratio. The ombustor heat balane is and using T t = T to and f m f /m, m f h = m p (T t 4 T t 3 ) = m p T t (" # ) (11) fh = " # ( ) (1) p T to so the quantity f /T to is another funtion of ϑ alone. But notie that f itself does depend on M o at a fixed T o. (5)Throat pressure (normalized) P 7 P = P P P P P t t 3 t 4 t 5 t1 7 P t P t 3 P t 4 P t 5 P t 7 P7 1 $ ' / "1 P P and = = t / "1 & ). Also 3 = = " # /# $1 t, 5 = = " # /# $1 t t ; Pt 7 $ "1 ' % +1( Pt Pt 4 &1 + #1) % ( whih is yet another funtion of ϑ alone. P 7 $ ' = & " t " (# ) *1 ) (13) Pto % +1 ( (6) Thrust (mathed nozzle). We already have Eq. (10) of Leture 18, but it is sometimes better to normalize thrust by the total free-stream pressure on the ompressor inlet, A, 3

4 whih is known from flight onditions. If P e = P o (variable nozzle, or just design point for a fixed nozzle), F m (u e u o ) " = A A (14) P A u # u = m " to e o RT to A For u e, we go bak to Leture 18, Eq. (7): # u e = a o (# o$ $ t "1) " 1 $ ao R / T and o = =. All together then, R / T to " o RT to # ' $ ($ o & & t %1) * = "m ) % M o, (15) $ o () # %1 &,+ Here the quantities m and depend on ϑ only, but we an see that the Mah number " #1 M o appears expliitly (as M o and as o =1 + M o ), so the normalized thrust depends on both ϑ and M o. (7) Thrust (trunated soni nozzle). We now have m e = m 7 =1, but P e = P 7 > P o, so F m (u e ( " P )A = = " u o ) + P e o e A A (16) u " u $ e o P e P ' o A = #m e + & " ) R T to % ( A and this time M e = 1, so u e = RT e = R T = RTto"# +1 t 5 t +1 ue Pe so that = "# t depends on ϑ alone. Sine we also know that m and are R T to +1 funtions of ϑ alone, it makes sense to separate out Eq. (16) in the form 4

5 # u P A & e e e uo Po Ae = %"m + ( ) " m ) $ R T t o A ' R T t o A (17) & # ) ( # & )#,1 A - 0 e % + # 1 A = m e $ ( t + ( % t % +,/"m M # +1 '# +1 * A + # /#,. $ o $ ' * o A 1 ######"######$ (18) * ($ ) One again, the normalized thrust depends on both, ϑ and M o, but the struture is fairly simple, and in partiular, the portion * of (negleting the inoming momentum and the external pressure) is a funtion of ϑ alone. This portion an be very easily saled between onditions, and the rest an be subtrated separately. A note on ϑ: the near-onstany of the engine operating point Two important points in the flight envelope of an airraft engine are (a) Take-off onditions (M o 0.5, T o 90K ), and (b) End-of-limb onditions (M , T 0 0K. The total temperatures are T to = 90( ) = 94K (take-off) and T to = 0( ) = 5K (end of limb). Suppose the engine is dimensioned for end-of-limb, whih is ommon, and that the peak temperature T t4, whih will have to be maintained for many hours of ruise, is seleted at a onservative T t4 = 1600K. We then 1600 have = = 6.35 at this ondition. If we now deided to maintain ϑ = 6.35 also for 5 take-off, we would need then T t 4 = =1868K. While this is too high for longterm operation (reep, orrosion), it may be aeptable for the few minutes per yle that the engine will be at take-off maximum power. As a seond example, onsider a ommerial jet in a long ruise. As the fuel is onsumed and the weight dereases, so must the lift L =1/ 0 u 0 A w L. Now, the lift oeffiient will be kept lose to that for optimum L/D, and the Mah number M 0 is unlikely to hange muh, as it will stay just below the transoni drag peak, and so u0 will be proportional to T 0 due to the speed of sound variation. Together with the density part of lift, we an see that the ambient pressure p 0 must be dereasing in proportion to the airplane s weight, i.e., the plane must be limbing gradually. Turning now to the forward fore balane, given a onstant L/D, the drag, and hene the engine thrust, must also be dereasing in time in the same proportion as the ambient pressure. Therefore, from Eq. (14), the nondimensional thrust (",M 0 ) will remain onstant, and sine M 0 does too, the peak temperature ratio θ will also remain onstant, and with it all the important ratios like τ, M, et. In other words, ϑ may not vary muh among (important) flight onditions, and the engine will be operating at a fixed nondimensional ondition (onstant ompression ratio, nondimensional flow, ompressor inlet Mah number, et.). But of ourse, the 5

6 dimensional quantities (flow rate, peak pressure, et.) will be different, depending on p o, et. (8) The Operating Line in the ompressor map. Compressor performane is typially presented as a map of vs. m, with lines of onstant normalized rotational speed and " superimposed. The details are the subjet of later Letures, but the general shape is as shown below. (The flow and speed variables are renormalized by the Design values): = m (m ) des Kerrebrok, Jak L. (199). Airraft Engines and Gas Turbines (nd Edition). MIT Press, Massahusetts Institute of Tehnology. Used with permission. Atually the nominal operating line shown in the figure is not a property of the ompressor, but rather of the rest of the engine. We an alulate this line with the information we have now, before deiding what partiular ompressor to use. From Eqs. (11) and (9), and from the shaft power balane (Eq. 7), " /" #1 A m = 4 (19) A $ " #1 = 1# " t (0) where we reall that t is fixed for a fixed geometry. Eliminating ϑ, 6

7 A 1# m = 4 " /" #1 t (1) A #1 or, in terms of, A 1" # m = 4 t $ "1 () A $ "1 whih is the equation for the operating line (written in reverse). If the ompressor is already available, we see from () that we an adjust the nozzle area A 4 to plae this line in a good plae on the map, i.e., below the stall line and through the best effiieny points. T Sine m depends on = t 4, varying Tt4 moves the operating point along the operating Tto line, and this is what the pilot does with the throttle stik to power the engine up or down. At eah seleted ϑ, the engine settles to a, a M, a (normalized) rotation rate, et. Effets of Mah number If we look at operation of a given engine at different flight Mah numbers, we may try to maintain the same non-dimensional onditions throughout, whih, as we have seen, an be done by maintaining for example a onstant ompressor inlet Mah, M. This, in turn T guarantees a onstant = t 4, but sine now we have a varying Mah number, so that T to T t0 inreases with M 0, we may find that the turbine inlet temperature T t4 needs to beome too high at the higher Mah numbers. For example, T t4 would have to be 1.8 times higher at M 0 =.0 than at stati onditions, and.5 times at M 0 =.5. A more reasonable assumption is that the ratio θ t =T t4 /T 0 an be maintained the same at all Mah numbers, sine at least in the stratosphere, T 0 is almost invariant. The ompressor 1 " t (1# t ) temperature ratio now follows from = +, where the numerator is a " 0 onstant; thus, τ will be lowered as the Mah number inreases, but less strongly than would be required to maintain maximum thrust per unit flow ( = " t /" 0 ). The flow parameter m is now determined by Eq. (9), i.e. ompressor-turbine flow mathing, and then the ompressor-inlet Mah number from Eq. (8). One these parameters are known, we an use Eq. (15) to alulate the normalized thrust; sine we are interested in the effet of Mah number, it makes sense to re-normalize thrust by p 0 A, or F # = # " 0 $1. 0 p A A numerial example 7

8 We take now θ t =7, or T t4 =1540K in the stratosphere. The geometry of the engine must have been speified in advane. This means that the turbine temperature ratio (Eq. 5) is a known fixed number. For the example, we selet τ t suh as to obtain maximum thrust at M 0 =1. From the shaft balane equation, # 0 t =1" # t ( "1) and we put now θ 0 =1. and τ = 7/1.=.048 (at M 0 =1). This fixes τ t = Similarly, the area ratio A 4 /A must have been fixed, and we selet it here so as to obtain at M 0 =1 a ompressor-fae Mah number M =0.5, whih, from Eq. (8) implies m = From Eq. (9) then, " m = ( ) and the rest of the steps are as desribed above. The table below summarizes the results: M θ τ m M F/(p 0 A ) We find that at a fixed altitude the thrust is nearly onstant up to Mah 1, then it inreases rapidly. Atually the inrease is less rapid than this simple model predits, beause of losses in the supersoni flow in the engine inlet. Finally to this point, we should note that an airraft normally flies at inreasing altitude as the Mah Number inreases, so that dynami pressure p 0 M is roughly onstant. In this 0 ase the hange in F between Mah 1 and Mah is atually a thrust redution. 8

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