PARAMETRIC CYCLE ANALYSIS OF IDEAL ENGINES
`
`UTC-2912.601
`
`GE v. UTC
`
`Trial IPR2016-00952
`
`0.0
`
`0.0
`
`0.5
`
`1.0
`
`-
`
`2.0
`
`2.5
`
`3.0
`
`1.5
`M0
`
`FIGURE 5-28c
`
`turbofan performance
`Ideal
`Versus M0:
`for
`75¢ =24 and
`yrf = 3: thrust ratio.
`
`for Figs. 52751 and 5-271) and at a value of 3 for Figs. 5-286! and 5-28b. Figures
`5—27a and 5—28a show that specific thrust decreases with increasing flight Mach
`number and with increasing bypass ratio. These four figures also show that the
`high—bypass-ratio engines are limited to lower flight Mach numbers and that a
`low-bypass—ratio engine is required for the higher flight Mach numbers. ’
`Propulsive, thermal, and overall efficiencies are plotted versus the flight
`Mach number for different values of the bypass ratio in Figs. 5—.27c, 5—27d, 5—28c,
`and 5—28d. We can see that propulsive efficiency increases with flight Mach
`number and that there is an optimum bypass ratio for each flight Mach number
`that gives maximum propulsive and overall efficiencies.
`‘
`The thrust ratio is plotted versus flight Mach number for difierent Values of
`‘ the bypass ratio in Figs. 5—27e and 5-28:2‘. These plots show that thrust ratio
`decreases with increasing bypass ratio and generally decreases with increasing
`flight Mach number.
`
`IDEAL TURBOFAN WITH
`5-10
`OPTIMUM BYPASS RATIO
`
`For a given set of flight conditions To and M0 and design limit 1',\, there are
`three design Variables:
`7rC,_{_zf_,_ and oz. In Figs. 5-23a through 5-233 and 5-2-4a
`through 5-246, we can see that by increasing the compressor ratio no we can
`increase the thrust per unit mass flow and decrease the thrust specific fuel
`consumption. Increases in the compressor pressure ratio above a Value of 20 do
`not increase the thrust per unit mass flow but do decrease the thrust specific
`fuel consumption. In Figs. 5-25a through 5-25d, We can see that an optimum
`
`‘‘
`
`UTC-2012.001
`
`GE V. UTC
`
`Trial IPR2016-00952
`
`GE v. UTC
`Trial IPR2016-00952
`
`UTC-2012.001
`
`

`

`300
`
`GAS TURBINE
`
`fan pressure ratio exists for all other parameters fixed. This optimum fan
`pressure ratio gives both the maximum thrust per unit mass flow and minimum
`thrust specific fuel consumption. In Figs. 5~26a throug‘h 5-26a‘, we
`that an
`optimum bypass ratio exists for all other parameters
`This optinmm
`bypass ratio gives the minimum thrust specific fuel consumption. ‘W’e will look.
`at this optimum bypass ratio first. The optimum fan pressure ratio will be
`analyzed in the next section of this chapter.
`
`Optimum Bypass Ratio 04*
`
`When Eq. (5-58]‘) for the fuel/ air ratio and Eq. (5—58z') for specific thrust are
`‘inserted into the equation for specific fuel consumption [Eq.
`(5-58k)], an
`expression in terms of the bypass ratio oz and other prescribed variables results.
`For a given set of such prescribed variables (7,, yrc, 73,:-, 1;, V};,), we may locate
`the minimum 5 by taking the partial derivative of S with respect to tlie bypass
`ratio—oz. Since the fuel/ air ratio f is not a function of the bypass ratio, We have
`
`S:
`
`._,..._2f_._,.__l_.
`(1 4” a)(F/Vi’l0)
`
`3:: ”" aaa [(1 + a)];F/n'10)] = O
`" [(1 + a)?£'/n'10)]2 aaa [(1 +
`
`Z 0
`
`Thus 65/80: = O is satisfied by
`
`h
`were
`
`F
`V9
`8c
`<—~—>=-1
`»-1
`VO( +04) mo
`V0
`
`V19
`<——>
`+a V0
`1
`
`Then the optimum bypass ratio is given by
`
`— —-—1+ —-—-1
`
`[V9
`3
`aavo.
`
`<V19
`“V0
`
`However,
`
`Thus Eq. (i) becomes
`
`1.
`
`-6-
`
`2V9/Vo Ba
`
`=-—~— — ———1=O
`
`V19
`(V9)
`5
`aat/0+0
`
`2
`
`] == —<—3——
`
`662 V0
`
`Va
`
`6 W1
`1
`2V9/VO 801
`V0
`
`VQ
`
`.._. _ + __ __ 1 :
`
`UTC-2012.002
`
`.
`(1)
`
`(11)
`
`UTC-2012.002
`
`

`

`PARAMETREC CYCLE ANALYSES 01: IDEAL ENGINES
`
`301
`
`$mfi
`
`V0
`
`Mg
`
`610
`
`1
`(Kfi3mig(Vfi2:
`W T/\ _ 7’-r[Tc _' 1 + C“(Tf —
`1, —— 1
`
`'—
`
`“‘
`
`Cfiy
`‘ TA/(cilia)
`
`-4330}
`
`then
`
`1
`
`601
`
`V0
`
`_
`
`_ a {Q — rr[7:C — 1 + am) — 1)] —— rs)/(~1—_,__;g£)}
`
`'5, W 1
`
`60:
`
`.
`-
`5’
`V912” W
`”‘7r('5.f ” 1)
`gm 5; [(7/3)
`~ ~ r,-~1
`.
`V19>2
`1 <V19>2
`1”“
`1/;
`Mg
`:20
`
`S
`
`<
`
`1
`
`__
`
`1
`V
`V
`]
`my ~— 1)/2](e::,. —~ 1.)
`
`‘
`1
`Tr’? ""1
`1
`1-, — 1
`
`<V19>2
`1....
`are
`
`1
`
`<m>
`.
`(IV)
`
`then Eqs. (iii) and (iv), substituted into Eq. (ii), give
`
`1
`
`2V9/V0
`
`[_2Qg:Q]1V@g;1_1=O
`
`*5, — 1
`
`"17,. — 1
`
`Substitution of the equation for V9/V0 gives
`1
`""Z.‘,(7.',c —- 1)
`M
`]_\/r
`2
`11,] -- 1:,[*cC — 1 + Q{*(Tf —— 1)] — Q/('L',1:C)}/(*5, — 1)
`
`-W 1”
`/*L',.*;,--
`+ -
`/
`\~ .7,“ ~ 1
`
`__
`—— 1 — O
`
`1.',T_,~ ,— 1
`r, —— 1
`
`r,(1jf —- 1)
`1
`§\/{:1} - »:,[»cC — 1 + a>«(cf — 1)] — 1,]/(m;)"}/(7,-, — 1) K \/
`or
`Squaring both sides, we have A
`:(¢%n~l_02
`1
`1
`U%@~DF
`13 — r,[rC — 1 —I— a*(17 ; 1)] —- r,\/(r,1:C)
`1
`[r,(1:- — 1)]2
`n-3
`“ Mvmw~nXa—fi—u’"
`5 W)
`An expression for the bypass ratio giving the minimum fuel consumption
`is obtained by solving Eq. (V) for cu*. Thus
`
`4{rA — ’L',.[’KC - 1 -5- oz*(7:f -— 1)] -~ TA/(’L',TC)}/(T, - 1)
`
`I, — 1
`
`’
`
`:5: __
`
`Q
`
`1, -11
`'7:,.(1-J». ~— 1)
`
`1
`
`[TA — 17,('rC -— 1) -— TA/(ETC)
`1', —- 1
`
`1, (
`4
`
`’
`
`,,“_,_m__,,___,W____
`
`/14, ~— 1
`1*, — 1 +
`
`\/
`
`1
`
`or
`
`
`
`04* =
`
`1 ~—-[1,] — 1-(1-.— 1) —— 1“ »1~3(v‘«: 1 ~51 + x«’»ET”:-””i)2]
`r,,(*cf -- 1)
`’
`°
`I‘,/17¢.
`4
`r f
`'
`
`(5-59)
`
`Now note that we may write
`
`1,(r_,: - 1)
`1', — 1
`
`:5,1.“aj,—. —-~ ~15; +11.
`‘L’, — 1
`
`-1» 1 _ *z',1:,« — 1 _ 1
`7, — 1
`
`UTC-2012.003
`
`UTC-2012.003
`
`

`

`302
`
`GAS TURBINE
`
`So then Eq. (V) becomes
`
`’
`(
`
`1
`4
`
`/'5,/z;'~~1‘2
`
`2
`
`1:
`_ ’Z',\—T,[’C'c"-1+0d*(‘L'f"‘1]’Tr;C
`1', —— 1
`
`«
`‘-1,
`/«Q4-f. M 1 ~ 1
`*r,g-fl
`g
`Taking the square root of both sides of this equation gives
`_1(
`/1,*z:,« - 1 + 1) = \/1',‘ — 'c,[t,_. —- 1 + a*(Tf — 1)] — 1,‘/(iiyrc)
`2( V ’L“,.-1
`1 W \/
`1
`'z:,—1
`TA _’ Tr[1."c _ 1 + a*(7'-f _—
`)_____
`T«'.»7='f W 1 __
`1
`
`2
`
`'L',-1
`
`':,,-1
`
`or
`
`” TA/(Trrc) _
`
`1
`
`(V1)
`-
`
`Noting from Eqs. (5-50) and (5-53) that the term within the square root on the
`left side of the equals sign is the Velocity ratio V19/V0, and that the term within
`the square root on the right side is the Velocity ratio V9/V0, We see that Eq. (vi)
`becomes
`
`(5—60)
`
`Thus
`
`We thus note from Eq. (5-60) that when the bypass ratio is chosen to give the
`minimum specific fuel consumption, the thrust per unit mass flow of the engine
`core is one—half that of the fan. Thus the thrust ratio of an optimum-bypass
`ratio ideal turbofan is 0.5. Using this fact, we may write the equation for the
`specific thrust simply as
`
`(5-61)
`
`where a* is obtained from Eq. (5~59).
`One can easily show that the propulsive efficiency at the optimum bypass
`ratio is given by
`
`[ (7'IP)mins—(3 +4a*)VO+ (1 +4a*)Vl9 J
`
`4(1 + 2a*)V0
`
`__
`
`(5-62)
`
`UTC-2012.004
`
`UTC-2012.004
`
`

`

`PARAMETRIC CYCLE ANALYSIS OF IDEAL ENGINES
`
`303
`
`Summary of Equations---Optimum-Bypass-Ratio
`Ideal Turbofan
`
`INPUTS:
`
`OUTPUTS:
`
`EQUATIONS:
`
`Y;4(K9 °R), 7%, 75f
`
`F
`
`N
`
`lbf
`
`""3: (flkg/sec‘°lbm/sec)’f’ S< NM.’
`
`mg/sec lbm/hr
`
`lbf
`
`)9 TIT) TIP: 770: 05*
`
`1 .
`
`,_.——».——..
`
`Equations (5-58a) through (5-58h) plus
`__
`1
`_
`_
`_TA
`a*“1:,(4:f—1)T’\
`“(TC D 'c,7:C
`— Z (vrxcf — 1 + V1, — 1?]
`__1:__@1+2a* [\/2
`._'_ J
`mo
`gc2(1+cu*)
`13/—1(T-*7"
`1) M0
`41-+2 *M
`(
`a )
`°
`"P = (3 + 4a*)M¢ + (1 + 4oz*)(V1g/a_0)
`and Eqs. (5—58j), (5-58k), (5-581), (5—58n)
`
`(5~63a)
`
`(56%)
`(5—63c)
`
`Example 5-5. The engine we looked at in Sec. 5-9 is used again in this section for
`an example. The following Values are held Constant for all plots:
`c,, = 1.004 kJ/(kg - K)
`YI;=216.7K
`7 = 1.4
`2:4 = 1670 K
`
`hp,;: = 42,800 kJ/kg
`
`12
`
`11
`
`10
`
`10
`
`15
`”C
`
`20
`
`25
`
`V
`
`30
`
`FIGURE 5-29a
`for 7c,v=2 and
`05* versus
`7:6,
`M0 ‘ 09'
`UTC-2012.005
`
`UTC-2012.005
`
`

`

`304
`
`GAS TURBINE
`
`20
`
`15
`
`10
`
`5
`
`0
`
`1
`
`am
`
`'
`
`2
`
`3
`
`itf
`
`4
`
`5
`
`5
`
`FIGURE 5-2915
`a* versus 7rf,
`M0 = 0.9.
`
`for
`
`75¢ = 24 and
`
`72'}: = 2
`
`75f:-_ 3
`
`00
`
`0.5
`
`1.0
`
`1.5
`M0
`
`2.0
`
`2.55
`
`3.0
`
`FIGURE 5-290
`a’*‘ versus M0, for 7:6 = 24 arid
`75f = 2 and 3.
`UTC-2012.006
`
`14
`
`12
`
`10
`
`8
`
`6
`
`4
`
`2 0
`
`(I3:
`
`UTC-2012.006
`
`

`

`PARAMETRIC CYCLE ANALYSIS OF IDEAL ENGINES
`
`305
`
`1
`
`The optimum bypass ratio is plotted in Fig. 5-2951 versus the compressor
`pressure ratio for a flight Mach number of 0.9 and fan pressure ratio of 2. From
`this plot, we can see that the optimum bypass ratio increases with the compressor
`pressure ratio. The optimum bypass ratio is plotted in Fig. 5-291) versus the fan
`pressure ratio, and this figure shows that optimum bypass ratio decreases with fan
`pressure ratio.
`The optimum bypass ratio versus the flight Mach number is plotted in Fig.
`5—29c for fan pressure ratios of 2 and 3. From these plots, we can see that the
`optimum bypass ratio decreases with the flight Mach number. Note that the
`optimum engine is a turbojet at a Mach number of about 3.0 for a tan pressure
`ratio of 3.
`'
`The plots of specific thrust and thrust specific fuel consumption. for an ideal
`turbofan engine with optimum bypass ratio versus compressor pressure ratio are
`superimposed on Figs. 5—23a through 5—23e by a dashed line marked a*. The
`optimum-bypass-ratio ideal
`turbofan has
`the minimum thrust
`specific fuel
`consumption.
`The plots of specific thrust and thrust specific fuel consumption for the
`optimum-bypass-ratio ide'al turbofan versus fan pressure ratio are superimposed
`on Figs. 5-255: through 5-2503 by a dashed line marked ct*. As shown in these
`figures, the plot for a* is the locus of the minimum value of S for each rzrf.
`The plots of specific thrust and thrust specific fuel consumption for the
`optimum—bypass-ratio ideal turbofan versus flight Mach number are superimposed
`on Figs. 52752 through 5—27e for a fan pressure ratio of 2 and on Figs. 5—28a .
`through 5-288‘ for a fan pressure ratio of 3. As shown in these figures, the plots for
`a* are the locus of the minimum value of S for fixed values of me and 75,.
`The thermal efficiency of an optimum-bypass—ratio ideal turbofan is the
`same as that of an ideal turbojet. The bypass ratio of an ideal turbofan afifects
`only the propulsive efficiency. The propulsive efficiency of an optimum—bypass—
`ratio ideal turbofan is superimposed on those of Figs. 5-23d, 5-25¢, 5-2”/c, and
`5—28c. As can be seen, the optimum bypass ratio gives the maximum propulsive
`eficiency. Likewise, as shown in Figs. 5—23d, 5~25c, 5-27d, and 5-28d,
`the
`optimum bypass ratio gives the maximum overall efficiency. The thrust ratio of an
`optimum—bypass—ratio~ ideal turbofan is superimposed on those of Figs. 5-236,
`5—25d, 5—27e, and 5-288 and is equal to 0.5.
`
`IDEAL TURBOFAN WITH
`5-11
`OPTIMUM FAN PRESSURE RATIO
`Figures 5-25a and 5-2529 show that for given flight conditions 72) and MO, design
`limit 'z:,\, compressor pressure ratio 7:6, and bypass ratio oz, there is an optimum
`fan pressure ratio rtf which gives the minimum specific fuel consumption and
`maximum specific thrust. As will be shown, the optimum fan pressure ratio
`corresponds to the exit velocity V19 of the fan stream, being equal to the exit
`velocity of the core stream V9. It is left, as a reader exercise, to show that equal
`exit velocities (V9 == V19) correspond to maximum propulsive efficiency.
`
`Optimum Fan Pressure Ratio 7rf*
`‘EA, V0, or), we may locate the
`For a given set of prescribed variables (Tr, 715C,
`optimum fan pressure ratio by taking the partial derivative of the specific
`UTC-2012.007
`
`"‘?
`
`UTC-2012.007
`
`