Nomenclature

A _e

sum of engine core and bypass jet exit cross sectional areas = (A ₉ + A ₁₉ )

A ₉

engine core jet exit cross sectional area

A ₁₉

engine bypass jet exit cross sectional area

a _∞

ambient sound speed = (γℜT _∞ )^1/2

BPR

engine nominal bypass ratio = ( ${\dot m_B}$ / ${\dot m_C})$

Cd

airframe drag coefficient = D/(q _∞ S _ref )

C _L

overall lift coefficient = L/(q _∞ S _ref )

C _P

specific heat at constant pressure for air (1,005 J/(kg K))

C _T

aircraft total net thrust coefficient = n.F _n /(q _∞ S _ref )

C _t

engine net thrust coefficient = F _n /(q _∞ A _e )

D

aircraft total drag force

FL

flight level

F _G

engine gross thrust

F _n

engine net thrust

F ₀₀

maximum thrust at zero speed and at sea level

h _0-4

engine dependent functions – Equations (19), (28), (29), (30) and (33)

L

aircraft lift force

LCV

lower calorific value of fuel ( $\approx$ 43 × 10⁶ J/kg for kerosene)

L/D

lift-to-drag ratio

M _∞

flight Mach number = V _∞ /a _∞

${\dot m_{air}}$

intake air mass flow rate = ( ${\dot m_B}$ + ${\dot m_C})$

${\dot m_B}$

air mass flow rate through bypass duct

${\dot m_C}$

air mass flow rate through engine core

${\dot m_f}$

fuel mass flow rate

n

number of engines

p

static pressure

$q_{\infty}$

freestream dynamic pressure $= 0.5 \rho_{\infty}(V_{\infty})^{2} = 0.5\gamma\rho_{\infty}(M_{\infty})^{2}$

R ^ac

characteristic aircraft Reynolds number

ℜ

gas constant for air (287.05 J/(kg K))

S _ref

aerodynamic reference wing area (Airbus definition)

SFC

specific fuel consumption

T

static temperature

TR

ratio of total temperature at turbine entry to freestream total temperature

T _o

total temperature – Equation (11)

TET

total temperature at the entry to the engine turbine section

V _∞

true air speed

γ

ratio of specific heats for air (=1.4)

$\eta$ _o

propulsion system overall efficiency – Equation (1)

$\eta$ ₁ , $\eta$ ₂

constants in Equation (16)

μ

dynamic viscosity

$\phi$

angle between the thrust line and the flight direction

$\rho$

air density = p/(ℜT)

$\Sigma$

engine parameter – Equation (19)

Superscripts

ac

whole aircraft value

M

at constant Mach number

P

at constant pressure or flight level

Subscripts

B

best, or local maximum, value

DO

at the design optimum conditions

EC

engine characteristic value

ICAO

given in the ICAO engine data base

ISA

in the International Standard Atmosphere

max

maximum value

min

minimum value

o

when ( $\eta$ _o L/D) has its optimum value

ref

reference

SLS

at sea level static conditions

$\eta$ B

when ( $\eta$ _o L/D) has its best value at a given Mach number

∞

flight, or freestream, value

2.0 Fundamental aero-thermodynamic relations

An aircraft’s speed and rate of climb are controlled by adjusting the engines’ net thrust. This is achieved by changing the throttle setting. Even though, superficially, a modern turbofan engine is an extremely complex machine, in essence, it is a simple, freely rotating system with the principal controlling parameter being the rate at which fuel is supplied to the combustion chamber, ${\dot m_f}$ . This determines the total temperature of the mixture of air and combustion products entering the engine’s turbine stages, i.e. the so-called turbine entry (total) temperature or TET. Therefore, the throttle parameter may be taken to be either the fuel flow rate itself, or, more conveniently, the TET.

The purpose of the engine is to drive the aircraft along a specified flight trajectory and the useful power being delivered at any point is equal to the product of the net thrust, F _n , and the aircraft speed, V _∞ . In aero-thermodynamic terms, the engine’s overall efficiency is the product of the thermal efficiency, $\eta$ _th , and the propulsive efficiency, $\eta$ _p . Thermal efficiency quantifies the rate of conversion of energy in the fuel to useable mechanical energy and is dependent upon the turbine entry temperature, the polytropic efficiencies of the compressor and turbine and, to a lesser extent, the overall pressure ratio. Propulsive efficiency quantifies the conversion of this useable mechanical energy into useful work and is primarily dependent upon the ratio of the mass of air passing through the fan, ${\dot m_B}$ , to that passing though the engine core, ${\dot m_c}$ , i.e. the bypass ratio, BPR. Hence, if LCV is the lowerFootnote ¹ calorific value of the fuel, the engine’s overall efficiency, $\eta$ _o , is given by

(1) \begin{align}{\eta _o} = {\eta _{th}}.{\eta _p} = \frac{{{F_n}{V_\infty }}}{{{{\dot m}_f}.LCV}} = \frac{{{V_\infty }}}{{SFC.LCV}},\end{align}

where SFC is the specific fuel consumption.

Consider an engine in which the core and bypass flows are separate. This arrangement may be idealised as shown in Fig. 1. Here the engine core provides shaft power to drive the fan in the bypass duct and the core and bypass exit jets do not mix.

Figure 1. The idealised turbofan engine with separate core and bypass effluxes.

If the rate at which air and fuel enters the engine is steady, control volume analysis indicates that the net forceFootnote ² developed by the bypass flow is

(2) \begin{align}{\left( {{F_n}} \right)_{bypass}} \approx {\dot m_B}\left( {{V_{19}} - {V_\infty }} \right) + \left( {{p_{19}} - {p_\infty }} \right){A_{19}},\end{align}

whilst that developed by the core flow is

(3) \begin{align}{\left( {{F_n}} \right)_{core}} \approx {\dot m_c}\left( {\left( {1 + \frac{{{{\dot m}_f}}}{{{{\dot m}_c}}}} \right){V_9} - {V_\infty }} \right) + \left( {{p_9} - {p_\infty }} \right){A_9},\end{align}

where p _∞ is the ambient atmospheric pressure, V ₁₉ and V ₉ are the bypass and core exit velocities, p ₁₉ and p ₉ are the values of static pressure in the bypass and propelling nozzle exit planes and A ₁₉ and A ₉ are the exit cross-sectional areas. Hence, the total net thrust is given by

(4) \begin{align}{F_n} \approx {\dot m_B}{V_{19}} + {p_{19}}{A_{19}} + {\dot m_c}\left( {1 + \frac{{{{\dot m}_f}}}{{{{\dot m}_c}}}} \right){V_9} + {p_9}{A_9} - {\dot m_{air}}{V_\infty } - {p_\infty }{A_e},\end{align}

where ${\dot m_{air}}$ is the total amount of air entering the engine and A _e is the sum of the core jet and bypass jet exit cross sectional areas.

By definition, the engine’s total gross thrust, F _G , is given by

(5) \begin{align}{F_G} = {F_n} + {\dot m_{air}}{V_\infty }.\end{align}

Hence, neglecting the fuel-to-air ratio, which is typically about 0.02 and so very small in comparison to unity,

(6) \begin{align}{F_G} + {p_\infty }{A_e} \approx {\dot m_B}{V_{19}} + {p_{19}}{A_{19}} + {\dot m_c}{V_9} + {p_9}{A_9}.\end{align}

In the case where the bypass and core flows are mixed and a single jet emerges from the exit nozzle, A _e is equal to A₉ and V₁₉ is equal to V₉ . Hence, and, again, ignoring the contribution from the fuel flow rate, the result is

(7) \begin{align}{F_G} + {p_\infty }{A_e} = {F_G} + {p_\infty }{A_9} \approx \left( {{{\dot m}_B} + {{\dot m}_c}} \right){V_9} + {p_9}{A_9}.\end{align}

As described in detail in chapter 8 of Cumpsty and Heyes [Reference Cumpsty and Heyes11], the key point to note is that the terms on the right-hand side of both Equations (6) and (7) only depend upon conditions inside the engine. Therefore, in general, all the quantities on the right side of these equations, plus the TET, are functions of the fuel flow rate, conditions in the intake, i.e. the freestream total pressure, (p ₀ ) _∞ , and the freestream total temperature, (T ₀ ) _∞ and the freestream static pressure, p _∞ .

This being the case, dimensional analysis reveals that, if the only control variable is the fuel flow rate, engines with both separated and mixed exhaust flows are described, to a very good approximation by the following relations

(8) \begin{align}\frac{{{F_G} + {p_\infty }{A_e}}}{{{A_e}{{\left( {{p_o}} \right)}_\infty }}} \approx {f_1}\left( {\frac{{TET}}{{{{\left( {{T_o}} \right)}_\infty }}},{{{M}}_\infty }} \right),\end{align}

(9) \begin{align}\frac{{{{\dot m}_{air}}\sqrt {{C_p}{{\left( {{T_o}} \right)}_\infty }} }}{{{A_e}{{\left( {{p_o}} \right)}_\infty }}} \approx {f_2}\left( {\frac{{TET}}{{{{\left( {{T_o}} \right)}_\infty }}},{{{M}}_\infty }} \right),\end{align}

and

(10) \begin{align}\frac{{{{\dot m}_f}LCV}}{{{A_e}{{\left( {{p_o}} \right)}_\infty }\sqrt {{C_p}{{\left( {{T_o}} \right)}_\infty }} }} \approx {f_3}\left( {\frac{{TET}}{{{{\left( {{T_o}} \right)}_\infty }}},{{{M}}_\infty }} \right),\end{align}

where the functions f ₁ , f ₂ and f ₃ are characteristic of the particular engine. Here, C _p is the constant pressure specific heat for air and the freestream, total pressure and total temperature are given by

(11) \begin{align}{\left( {\frac{{{T_0}}}{T}} \right)_\infty } = 1 + \left( {\frac{{\gamma - 1}}{2}} \right)M_\infty ^2 = \left( {\frac{{{p_0}}}{p}} \right)_\infty ^{\frac{{\left( {\gamma - 1} \right)}}{\gamma }},\end{align}

where $\gamma$ is the ratio of specific heats for air.

Consequently, from Equations (5), (8) and (9), net thrust is given by

(12) \begin{align}\frac{{{F_n}}}{{{A_e}{p_\infty }}} \approx {\left( {\frac{{{p_0}}}{p}} \right)_\infty }\left( {{f_1} - {f_2}{{{M}}_\infty }{{\left( {\left( {\gamma - 1} \right){{\left( {\frac{T}{{{T_0}}}} \right)}_\infty }} \right)}^{1/2}}} \right) - 1 = function\left( {\frac{{TET}}{{{{\left( {{T_o}} \right)}_\infty }}},{{{M}}_\infty }} \right),\end{align}

whilst, from Equations (1), (10) and (12), engine overall efficiency is

(13) \begin{align}{\eta _0} & \approx {M_\infty }{\left( {\left( {\gamma - 1} \right){{\left( {\frac{T}{{{T_0}}}} \right)}_\infty }} \right)^{1/2}}\left( {\frac{{{f_1}}}{{{f_3}}} - \frac{1}{{{f_3}}}{{\left( {\frac{p}{{{p_0}}}} \right)}_\infty } - \frac{{{f_2}}}{{{f_3}}}{M_\infty }{{\left( {\left( {\gamma - 1} \right){{\left( {\frac{T}{{{T_0}}}} \right)}_\infty }} \right)}^{1/2}}} \right)\nonumber\\& = function\left( {\frac{{TET}}{{{{\left( {{T_o}} \right)}_\infty }}},{{{M}}_\infty }} \right).\end{align}

Examples of these relations for a typical, high bypass ratio, turbofan engine are given in Figs 2 and 3, where the functions f ₁ , f ₂ and f ₃ have been constructed using data given in Cumpsty and Heyes [Reference Cumpsty and Heyes11].

Figure 2. Variation of normalised net thrust with turbine inlet to free stream total temperature ratio and Mach number for a typical, high bypass ratio, turbofan engine. Data from Cumpsty and Heyes [Reference Cumpsty and Heyes11].

Figure 3. Variation of overall efficiency with turbine inlet to free stream total temperature ratio and Mach number for a typical, high bypass ratio, turbofan engine. Data from Cumpsty and Heyes [Reference Cumpsty and Heyes11].

At a given flight level (p _∞ = constant) and fixed Mach number, M _∞ , the net thrust varies almost linearly with TET/(T ₀ ) _∞ , with the slope steepening slightly as the Mach number increases. On the other hand, the engine overall efficiency increases rapidly with increasing flight Mach number. At fixed Mach number, $\eta$ _o exhibits a weak local maximum at a particular value of TET/(T ₀ ) _∞ and this variation is shown in Fig. 4. The temperature ratio for best $\eta$ _o also goes through a maximum at a particular value of M _∞ . In this example, the largest value of TET/(T ₀ ) _∞ for maximum $\eta$ _o is 5.61 and this occurs at a Mach number of about 0.73. The variation is very close to being parabolic as indicted by the dashed line in Fig. 4.

Figure 4. Variation of the ratio of total turbine-entry-temperature to total freestream temperature for best engine overall efficiency as a function of Mach. Data from Fig. 3. Note the expanded scale.

For aircraft applications, it is convenient to introduce a thrust coefficient, C _t , defined as

(14) \begin{align}{C_t} = \frac{{{F_n}}}{{\left( {\gamma /2} \right){p_\infty }M_\infty ^2{A_e}}} \approx function\left( {\frac{{TET}}{{{{\left( {{T_0}} \right)}_\infty }}},\;{{{M}}_\infty }} \right).\end{align}

Hence, from Equations (1), (10) and (14),

(15) \begin{align} \eta_o = \frac{\gamma }{2}\sqrt {\gamma - 1} \left( {\frac{{{A_e}{p_\infty }\sqrt {{C_p}{T_\infty }} }}{{{{\dot m}_f}LCV}}} \right){C_t}{{M}}_\infty ^3 \approx function\left( {\frac{{TET}}{{{{\left( {{T_0}} \right)}_\infty }}},\;{M_\infty }} \right) \approx function\left( {{C_t},\;{{{M}}_\infty }} \right).\end{align}

The data from Figs 2 and 3 are cross plotted in this form in Fig. 5. When the Mach number is fixed, $\eta$ _o goes through a local maximum as C _t increases and this ‘best’ value is a strong function of Mach number. This variation is shown in Fig. 6. It is clear that ( $\eta$ _o) _B is a strong function of the Mach number and, as suggested in Poll and Schumann [Reference Poll and Schumann5], it is generally well represented by a simple power law, i.e. for M _∞ greater than 0.2Footnote ³ ,

(16) \begin{align}{\left( {{\eta _o}} \right)_B} \approx {\eta _1}{\left( {{{{M}}_\infty }} \right)^{{\eta _2}}}.\end{align}

This result is to be expected, since $\eta$ ₁ is essentially an approximation for the thermal efficiency, whilst the Mach number raised to a power element is an approximate form of the propulsive efficiency.

Figure 5. Variation of overall efficiency with thrust coefficient and Mach number. Data from Figs 2 and 3.

Figure 6. Variation of the best overall efficiency with Mach number. Data from Fig. 5.

The variation of the thrust coefficient for best $\eta$ _o , (C _t ) ${}_{\eta B}$ , with Mach number is given in Fig. 7. This dependency is strong, particularly at the lower Mach numbers, and it may be well represented by an empirical relation of the form

(17) \begin{align}{\left( {{C_t}} \right)_{\eta B}} \approx constant\left( {\frac{{1 + 0.55{{{M}}_\infty }}}{{{{M}}_\infty ^2}}} \right).\end{align}

3.0 Approximate representation of the engine overall efficiency as a function of net thrust and Mach number

As described in Poll [Reference Poll and Schumann5], Cumpsty and Heyes [Reference Cumpsty and Heyes11] provide information for an engine with a nominal bypass ratio of 4.5, whilst Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes8] present comprehensive, graphical data for a family of engines with nominal bypass ratios of 3, 6.5, 8 and 13Footnote ⁴ . In addition, the ICAO Aircraft Engine Emissions Data Bank [13] contains some detailed, performance information for stationary engines operating at sea level.

Figure 7. Variation of thrust coefficient for best engine overall efficiency with Mach number. Data from Fig. 5.

A source not used in our previous work is PIANO-X [Reference Simos14]. This is the freely available, public domain version of the PIANO commercial aircraft performance software. It is a ‘black box’ method whose accuracy has been discussed, e.g. by Owen et al. [Reference Owen, Lee and Lim15], Vera-Morales and Hall [Reference Velásquez-SanMartín, Insausti, Zárraga-Rodríguez and Gutiérrez-Gutiérrez16] and Velásquez-SanMartín et al. [Reference Vera-Morales and Hall17]. However, whilst the PIANO development team states that the method has been calibrated using ‘private and public sources’, true accuracy, as determined by comparison with manufacturers’ data, does not appear to have been fully demonstrated in the open literature. Nevertheless, it is widely used, with results appearing frequently in open-source studies. PIANO-X uses unspecified aerodynamic models with data from real, but again, unspecified, engines and it covers a very wide range of aircraft types.

To support this work, PIANO-X [Reference Simos14] was used to generate over 2,000 values of $\eta$ _o for M _∞ in the range 0.6 to 0.9 and thrust coefficients (C _t ) between about 1 and 7 for 50 of the aircraft considered in Poll and Schumann [Reference Poll and Schumann5] operating in the International Standard Atmosphere [18]. This data set is comprehensive, covering nominal bypass ratios ranging from 1 to 9, with in-service dates varying from 1969 to 2013. PIANO-X does not give the identify the particular engine associated with each aircraft type. However, since the range of engines offered for each aircraft is available from internet sources, it is easy to estimate average values of the nominal overall pressure ratio and the nominal bypass ratio associated with each aircraft type using the ICAO engine emissions data bank data [13]. This information is listed in Table 1.

Table 1. Approximate characteristics of the typical turbofan engines powering a range of civil transport aircraft. The characteristics are averaged over all engines appropriate to the aircraft type and the sea-level, static thrusts are total aircraft values, i.e. summed over all engines.

From Figs 6 and 7, it is clear that, for a given Mach number, there is a value of C _t at which $\eta$ _o has its maximum, or best, value, i.e. ( $\eta$ _o) _B and (C _t ) $_{\eta B}$ . However, when the values of $\eta$ _o are normalised with ( $\eta$ _o) _B and C _t is normalised with (C _t ) $_{\eta B}$ , the resulting curves exhibit a very weak dependence upon bypass ratio and, provided that M _∞ is greater than 0.4, almost no dependence upon Mach number. Data for the engines given in Refs [Reference Jenkinson, Simpkin and Rhodes8] and [Reference Cumpsty and Heyes11] are shown in Fig. 8 and data from PIANO-X are shown in Fig. 9.

Figure 8. Variation of normalised engine overall efficiency with normalised thrust coefficient at Mach numbers greater than 0.4 and a range of bypass ratios. Data from Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes8] and Cumpsty and Heyes [Reference Cumpsty and Heyes11].

Figure 9. Variation of normalised engine overall efficiency with normalised thrust coefficient at Mach numbers greater than 0.4 and a range of bypass ratios. Data from PIANO-X and the solid line is given by Equation (19).

When all the data are included, the resulting variation may be approximated by a single curve, i.e. if

(18) \begin{align}0.3 \le \frac{{{C_t}}}{{{{\left( {{C_t}} \right)}_{\eta B}}}} \lt 1.8,\end{align}

then

(19) \begin{align}\frac{{{\eta _o}}}{{{{\left( {{\eta _o}} \right)}_B}}} = {h_0} \approx \left( {1 - 0.43{{\left( {\frac{{{C_t}}}{{{{\left( {{C_t}} \right)}_{\eta B}}}} - 1} \right)}^2}} \right)\left( {1 + {\rm{\Sigma }}{{\left( {\frac{{{C_t}}}{{{{\left( {{C_t}} \right)}_{\eta B}}}} - 1} \right)}^2}} \right).\end{align}

For M _∞ greater than 0.4, $\Sigma$ is zero and, for

(20) \begin{align}0.2 \le {{{M}}_\infty } \le 0.4,\end{align}

(21) \begin{align}{\rm{\Sigma }} \approx 1.30\left( {0.4 - {{{M}}_\infty }} \right).\end{align}

If values of normalised overall efficiency are required for normalised thrust coefficients below 0.3, the variation may be represented by a fourth order polynomial, passing through zero when the thrust coefficient ratio is zero and matching Equation (19) for value, first derivative and second derivative when the normalised thrust coefficient is equal to 0.3. This extension is described in Appendix A.

Notwithstanding the difficulties of extracting values from small graphs, which may have been poorly drafted, or distorted during the reproduction process, the RMS difference between the data and the estimates from Equation (19) is less that 0.7%, with the maximum difference between any individual data point and the mean lines being less than 5%.

For any given engine, if the values of ( $\eta$ _o) _B and (C _t ) $_{\eta B}$ are known at a single value of M _∞ , i.e. there is a known reference condition, M _ref , (C _t ) _ref and ( $\eta$ _o) _ref , then from Equations (16) and (17),

(22) \begin{align}\frac{{{{\left( {{\eta _o}} \right)}_B}}}{{{{\left( {{\eta _o}} \right)}_{ref}}}} = {h_1} \approx {\left( {\frac{{{{{M}}_\infty }}}{{{{{M}}_{{\rm{ref}}}}}}} \right)^{{\eta _2}}},\end{align}

and

(23) \begin{align}\frac{{{{\left( {{C_t}} \right)}_{\eta B}}}}{{{{\left( {{C_t}} \right)}_{ref}}}} = \;{h_2} \approx \left( {\frac{{1 + 0.55{M_\infty }}}{{1 + 0.55{M_{ref}}}}} \right){\left( {\frac{{{{{M}}_{{\rm{ref}}}}}}{{{{{M}}_\infty }}}} \right)^2}.\end{align}

The data suggest that the power law approximation in Equation (22) is valid for all the engines. Since the thermal efficiency parameter, $\eta$ ₁ , is removed by the normalisation of Equation (16), Equation (22) is, in effect, a normalised form of the propulsion efficiency, which is closely related to the engine bypass ratio, BPR. The values of $\eta$ ₂ are plotted against the nominal BPR in Fig. 10. Whilst there is a large amount of scatter, the variation is reasonably well represented by a least-squares fit to all the data, i.e.

(24) \begin{align}{\eta _2} \approx 0.65\left( {1 - 0.035\left( {BPR} \right)} \right)\end{align}

Figure 10. The variation of engine parameter $\eta$ ₂ with nominal bypass ratio. Circles are the data from PIANO-X and diamonds from Refs [Reference Jenkinson, Simpkin and Rhodes8] and [Reference Cumpsty and Heyes11] and the solid line is Equation (24).

This equation supersedes the relation given in Poll and Schumann [Reference Poll and Schumann5] that was based on far fewer data points.

The accuracy of Equation (23) is demonstrated in Fig. 11, where, for the purposes of illustration, the reference Mach number has been taken to be 0.78. Clearly, the normalised thrust coefficient is strongly dependent upon M _∞ , but any dependence upon bypass ratio is weak. Therefore, the data suggest that the functions, h ₀ , h ₁ and h ₂ are near universal. Consequently, for a given engine, once the reference values are specified, Equations (19), (22) and (23) can be used to estimate $\eta$ _o for any value of Mach number and thrust coefficient that fall within the stated ranges of validity of Equation (19).

Figure 11. Variation of the normalised thrust coefficient for best $\eta$ _o with Mach number. Diamond symbols are data are from Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes8] and Cumpsty and Heyes [Reference Cumpsty and Heyes11]. Circles are PIANO-X data. The design optimum Mach number is taken to be 0.78 and the solid line is Equation (23).

Figure 12 shows the comparison between estimates for $\eta$ _o obtained from this simple model and the 2,000+ data points from PIANO-X. The agreement is found to be excellent, with the RMS deviation being only 1.2% and the maximum deviation on any single point is 6.5%.

Figure 12. Comparison between the current estimate for $\eta$ _o and the data from PIANO-X. Reference conditions are those for an M _∞ of 0.78. Dashed lines give the ±5% deviations.

4.0 Specification of the engine reference conditions

As discussed in detail in Poll and Schumann [Reference Poll and Schumann6], for an aircraft with a given mass, the relations that determine the Mach number and flight level (or p _∞ ) at which ( $\eta$ _o L/D) has its largest value are

(25) \begin{align}{\left( {\frac{{{C_L}}}{{Cd}}\frac{{\partial Cd}}{{\partial {C_L}}}} \right)^{{M}}}\left( {1 - {{\left( {\frac{{{C_T}}}{{{\eta _o}}}\frac{{\partial {\eta _o}}}{{\partial {C_T}}}} \right)}^{{M}}}} \right) - 1 = \;0,\end{align}

and

(26) \begin{align}{\left( {\frac{{{M_\infty }}}{{{\eta _o}}}\frac{{\partial {\eta _o}}}{{\partial {M_\infty }}}} \right)^p} - {\left( {\frac{{{M_\infty }}}{{Cd}}\frac{{\partial Cd}}{{\partial {M_\infty }}}} \right)^p} - 2 = 0.\end{align}

Here, C _L is the lift coefficient, Cd is the drag coefficient and C _T is the aircraft’s total thrust coefficient, defined as

(27) \begin{align}{C_T} = {C_t}\left( {\frac{{n{A_e}}}{{{S_{ref}}}}} \right),\end{align}

where n is the number of engines and S _ref is the aircraft’s aerodynamic reference wing area.

If the airframe and the engine are to be perfectly matched, the partial derivatives of $\eta$ _o with respect to both thrust coefficient and Mach number will be zero when the airframe’s lift-to-drag ratio is also at a local maximum. If this condition occurs in straight and level flight, the thrust coefficient, (C _T ) _o , must be equal to the drag coefficient, (Cd) _o , which is, in turn, a function of the airframe Reynolds number, R ^ac . As discussed in Section 2, the internal components of the engine are aware of the total pressure and the total temperature at the inlet and (sometimes) the atmospheric static pressure, but not the airframe Reynolds number. Therefore, irrespective of the value of R ^ac , at a given value of M _∞ , the engine can only deliver the local maximum overall propulsive efficiency at one value of the thrust coefficient. Consequently, perfect matching is only possible at one value of R ^ac , i.e. the optimum flight condition at one aircraft total mass in an atmosphere with a specified variation of temperature with pressure. Poll and Schumann [Reference Poll and Schumann6] refer to this as the “design optimum” and it is a fundamental characteristic of the engine and airframe combination. Assuming it to be close to a typical mid-cruise condition, Poll and Schumann [Reference Poll and Schumann6] define the design optimum to be that for an aircraft at 80% of the maximum take-off mass, cruising in the International Standard Atmosphere [18]. Estimates of the design optimum parameters have been produced for 53 civil transport aircraft and these are given in tabular form in Ref. (Reference Poll and Schumann6).

As stated above, at the design-optimum condition, the partial derivatives of $\eta$ _o with respect to both thrust coefficient and Mach number will be zero. Therefore, at the design optimum Mach number, M _DO , both ( $\eta$ _o) _DO and (C _t ) _DO lie on the curves h ₁ and h ₂ . This means that the design optimum conditions can be used as reference conditions for the engine, i.e. Equations (22) and (23) become

(28) \begin{align}\frac{{{{\left( {{\eta _o}} \right)}_B}}}{{{{\left( {{\eta _o}} \right)}_{DO}}}} = {h_1} \approx {\left( {\frac{{{{{M}}_\infty }}}{{{{{M}}_{{{DO}}}}}}} \right)^{{\eta _2}}},\end{align}

and

(29) \begin{align}\frac{{{{\left( {{C_t}} \right)}_{\eta B}}}}{{{{\left( {{C_t}} \right)}_{DO}}}} = \frac{{{{\left( {{C_T}} \right)}_{\eta B}}}}{{{{\left( {{C_T}} \right)}_{DO}}}} = \;{h_2} \approx \left( {\frac{{1 + 0.55{{{M}}_\infty }}}{{1 + 0.55{{{M}}_{DO}}}}} \right){\left( {\frac{{{{{M}}_{DO}}}}{{{{{M}}_\infty }}}} \right)^2}.\end{align}

In summary, when the design optimum values for ( $\eta$ _o) _DO , M _DO , and (C _T ) _DO , are known, the normalising values ( $\eta$ _o) _B and (C _T ) $_{\eta B}$ for any other operating condition are obtained from Equations (28) and (29). These are then used in Equation (19) to give the corresponding value of $\eta$ _o.

Finally, since engines are rarely produced for a single aircraft type, perfect matching may not always be possible. Consequently, (C _T ) _DO , may not coincide precisely with the value for maximum $\eta$ _o at M _DO at any aircraft weight and the point (M _DO , (C _t ) _DO ) may never lie exactly on the locus of maximum efficiency. Nevertheless, provided the airframe and the engine are well matched, M _DO will be close enough to the maximum of $\eta$ _o for the effects of any small discrepancies to be ignored.

5.0 Updating the PS data base

The PIANO-X data also provide information on the design optimum Mach number and the engine parameter $\eta$ ₁ (Equation (16)) that can be compared with the latest PS data set given in Poll and Schumann [Reference Poll and Schumann6].

Data for M _DO are presented in Fig. 13. As indicated by the ±2% lines, the overall agreement between these two, independent, sets is quite good. However, there are nine cases for which the difference exceeds 2%. As described in Poll and Schumann [Reference Poll and Schumann5], the PS estimates were obtained, primarily, from internet sources and, consequently, some errors are likely. Therefore, where the agreement between PS and PIANO-X is better than ±2% the original values PS are retained and in the other cases the PIANO-X result is used. The revised values are given in Table 1.

Figure 13. Comparison between the original PS estimates for M _DO from Ref. (Reference Poll and Schumann6) and PIANO-X. Dashed lines give the ±2% deviations.

The comparisons between the original PS values and the PIANO-X results for $\eta$ ₁ are shown in Fig. 14. In this case, there is found to be a small systematic difference, with PS being, on average, 1% lower than PIANO-X. On top of this, there is a scatter of about ±10%. As described in Poll and Schumann [Reference Poll and Schumann5], since engine information hardly ever appears in the public domain, a calculation scheme was devised to obtain estimates of the engine characteristics from payload range diagrams. This required several assumptions to be made and the accuracy of some of the diagrams is unknown. Nevertheless, comparisons with the PIANO-X data suggests that, whilst the original method is reliable, the PS values can be further improved by applying a blanket 1% increase to the original values.

Figure 14. Comparison between original PS estimates for $\eta$ ₁ from Ref. (Reference Poll and Schumann6) and PIANO-X. Dashed lines give the ±10% deviations.

A second problem arises because most aircraft listed in Table 1 have been powered by more than one engine type and this believed to be the cause of most of the scatter in Fig. 14. Therefore, to compensate for these potential mismatches, if the difference between PS and PIANO-X values exceeds 5%, it is assumed that the engine types are different and the PS value is replaced with the PIANO-X value. The revised values of $\eta$ ₁ and $\eta$ ₂ are given in Table 1.

As is clear from the analysis in Poll and Schumann [Reference Poll and Schumann6], changes to M _DO and $\eta$ ₁ mean that all the other parameters at the design optimum condition are also changed. Therefore, an updated set of design optimum is given in Table 2. These values supersede those given in Ref. [Reference Poll and Schumann6].

In addition, Poll and Schumann [Reference Poll and Schumann6] describe an approximate method for estimating the operating optimum condition for an aircraft of any given weight operating in a completely general atmosphere. This requires values to be assigned of a range of constant parameters for each aircraft. These are designated as $\psi$ ₀ to $\psi$ ₇ and $\tau$ with the definitions being given in Appendix B. Some of these are interdependent. Consequently, when M _DO (= $\psi$ ₄) is changed, $\psi$ ₂, $\psi$ ₃, $\psi$ ₇ and $\tau$ are also changed, changes to ( $\eta$ _o) _DO result in changes to $\psi$ ₁, whilst $\psi$ ₀, $\psi$ ₅ and $\psi$ ₆ remain the same. Therefore, for completeness, a full set of revised parameters is given in Table 3. Once again, these supersede the values given in Poll and Schumann [Reference Poll and Schumann6].

Table 2. Revised estimates of the performance characteristics at the design optimum condition. The mass at the design optimum, m _DO , is taken to be 80% of the nominal $\textrm{MTOM}$ and the atmosphere is the $\textrm{ISA}$ . These values supersede those given in Table 2 of Poll and Schumann [Reference Poll and Schumann6].

6.0 Approximate representation of the engine overall efficiency as a function of throttle setting and Mach number

As shown in Figs 2 and 3, engine net thrust is adjusted by varying the turbine entry total temperature and when this in changed the engine overall efficiency also changes. The maximum thrust that an engine can produce in each phase of flight is usually determined by the upper limit, or rating, placed upon TET, with different values being specified for take-off, climb and cruise. In addition, and in order to extend engine operating life, derated settings may be applied, i.e. the TET is restricted to fixed values that are below the maximum permitted ones. To model this situation, relations linking engine overall efficiency and thrust to the turbine entry temperature, Mach number and altitude are required.

Table 3. Revised estimates of the PS characteristic parameters. These values supersede those given in Table 3 of Poll and Schumann [Reference Poll and Schumann6].

Firstly, the variation of normalised engine overall efficiency with normalised thrust coefficient, i.e. Equation (19), still applies. Secondly, as shown in Fig. 3, at a given Mach number, $\eta$ _o has a local maximum at a particular value of the turbine entry to freestream total temperature ratio. Thirdly, as shown in Fig. 4, this temperature ratio has a near parabolic variation with Mach number, passing through a maximum value at a particular value of M _∞ . These two quantities are engine characteristics and will be referred to as TR _EC and M _EC respectively. When these parameters are used to normalise the turbine entry to freestream total temperature ratio versus Mach number relationship for the engine data given in Refs [Reference Jenkinson, Simpkin and Rhodes8] and [Reference Cumpsty and Heyes11], the result is close to a single curve, as shown in Fig. 15.

Figure 15. Variation of the normalised turbine entry to freestream total temperature ratio for maximum $\eta$ _o with flight Mach number. Engine data are from Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes8] and Cumpsty and Heyes [Reference Cumpsty and Heyes11]. The solid line is the variation given by Equation (30).

This relation, designated h ₃ , may be approximated by the parabola,

(30) \begin{align}{\left( {TET/{{\left( {{T_0}} \right)}_\infty }} \right)_{\eta B}} = {h_3}.T{R_{EC}} \approx \left( {1 - 0.53{{\left( {{{{M}}_\infty } - {{{M}}_{EC}}} \right)}^2}} \right)T{R_{EC}}.\end{align}

Clearly, TR _EC is closely related to the maximum permitted turbine entry temperature, which, along with the overall pressure ratio, determines the engine’s thermal efficiency, i.e. TR _EC is primarily a function of technology level and, hence, of the date when the engine was designed. By contrast, the parameter M _EC plays an important role in determining the variation of thrust with flight Mach number and, consequently, is closely related to the engine’s propulsive efficiency, i.e. M _EC is primarily a function of bypass ratio. This relationship is shown in Fig. 16, where

(31) \begin{align}\;{{{M}}_{EC}} \approx 0.515 + 0.042\left( {BPR} \right) - 0.0016{\left( {BPR} \right)^2}.\end{align}

Figure 16. Variation of the engine characteristic Mach number, M _EC , with bypass ratio. Engine data are given by the circles and are taken from Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes8] and Cumpsty and Heyes [Reference Cumpsty and Heyes11]. The solid line is the variation given by Equation (31).

Using the data from Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes8] and Cumpsty and Heyes [Reference Cumpsty and Heyes11] once again, when C _t and the corresponding total temperature ratio are normalised with the values at the best $\eta$ _o, i.e. C _t /(C _t ) $_{\eta B}$ and T _R , where T _R is the engine throttle setting parameter given by

(32) \begin{align}{T_R} = \frac{{TET/{{\left( {{T_0}} \right)}_\infty }}}{{{{\left( {TET/{{\left( {{T_0}} \right)}_\infty }} \right)}_{\eta B}}}} \approx \left( {\frac{1}{{T{R_{EC}}}}} \right)\frac{{\left( {TET/{T_\infty }} \right)}}{{\left( {1 - 0.53{{\left( {{{{M}}_\infty } - {{{M}}_{EC}}} \right)}^2}} \right)\left( {1 + 0.2{{M}}_\infty ^2} \right)}},\end{align}

values covering a wide range of conditions almost collapse onto a single curve – see Fig. 17. Whilst there is clearly some variation in the data, there do not appear to be any significant residual trends with either Mach number, or bypass ratio. The resulting function is almost linear and may be approximated by

(33) \begin{align}\frac{{{C_t}}}{{{{\left( {{C_t}} \right)}_{\eta B}}}} = \frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}} = {h_4} \approx f\!\left( {{T_R}} \right) \approx 1 + 2.50\!\left( {{T_R} - 1} \right),\end{align}

where the gradient of the line is subject to a maximum uncertainty of ±20%.

Figure 17. The variation of normalised thrust coefficient with the throttle parameter T _R for a range of values of Mach number and bypass ratio. Solid line is Equation (33) and the dashed lines show gradient changed by ±20%.

Assuming that the functions h ₃ and h ₄ are also true for all turbofan engines, if TR _EC and M _EC are known, Equations (32) and (33) can be combined with Equations (19), (28) and (29) to obtain estimates for both the thrust coefficient and the engine overall efficiency as a function of TET for any combination of speed and altitude in any atmosphere.

7.0 Ratings determined by maximum values of TET and the flat thrust rating

As already noted, maximum permitted values of the turbine entry temperature are assigned to the different phases of the flight. The highest values apply at take-off and they can only be maintained for a short time, usually about 5 mins in routine service and up to 10 mins in an emergency. This extreme case is determined by the technology level available at the time of initial design. Data given in Figure 5.5 of Cumpsty and Heyes [Reference Cumpsty and Heyes11] show that the maximum TET at take-off has been increasing steadily over time as turbine blade material, coatings and cooling technologies have improved. This trend may be represented, very approximately, by the asymptotic relation

(34) \begin{align}\;\;{\left( {TET} \right)_{\max T/O}}\; \approx 2000\left( {1 - EXP\left( {62.8 - 0.0325\left( {Year} \right)} \right)} \right) \pm 75\;\left( K \right),\end{align}

and this is shown in Fig. 18. In this simplified relation, the limit for future improvements has been set at 2,000 K. This is considerably lower than the constant pressure, adiabatic flame temperature of 2,600 K for the stochiometric combustion of kerosene. However, since turbine entry temperature is a major factor in the generation of NO _x , which is an important, indirect contributor to global warming, it appears likely that, in the long term, the permitted upper limit for TET will be significantly lower that the theoretical maximum value.

Figure 18. Approximate variation of the maximum turbine entry temperature, (TET) _max , with the year of entry into service. Data are taken from Cumpsty and Heyes [Reference Cumpsty and Heyes11] and internet sources.

In addition to the take-off condition, ratings are specified for maximum continuous climb and maximum continuous cruise. To a first approximation, these can be related to the maximum take-off value and, once again, using the engine data given in Refs [Reference Jenkinson, Simpkin and Rhodes8] and [Reference Cumpsty and Heyes11],

(35) \begin{align}\;\;\;\frac{{{{\left( {TET} \right)}_{\max climb}}}}{{{{\left( {TET} \right)}_{\max T/O}}}} \approx 0.92 \pm 0.015,\end{align}

and

(36) \begin{align}\;\;\;\frac{{{{\left( {TET} \right)}_{\max cruise}}}}{{{{\left( {TET} \right)}_{\max T/O}}}} \approx 0.88 \pm 0.025.\end{align}

There is also a maximum go-around thrust and a maximum continuous rating. However, if the maximum continuous rating was to be used in routine operations, the service life of the engine would be reduced significantly. Therefore, this is normally reserved for emergencies only, e.g. an en-route engine failure.

Finally, to extend engine life and, hence, reduce operating costs, the engine control system usually has a specified flat-rating value for the maximum thrust in each of the regimes listed earlier. For example, in the case of the maximum climb, to meet hot day requirements, the rated thrust at a given Mach number and flight level may be chosen to be the value generated with the maximum climb TET under ISA+10°C conditions. Therefore, for the same flight conditions, if the actual atmospheric temperature is below the ISA+10°C level, the engine control system will deliver the fixed, rated thrust by reducing the turbine entry temperature below the maximum given in Equation (35). Consequently, the maximum available rate of climb will be reduced. It is this reduction in TET that gives the extra engine life. The resulting operating boundary is such that, for atmospheric temperatures less than, or equal to, ISA+10°C, the maximum thrust available is equal to the rated thrust, i.e. the rating is flat for the lower temperatures. However, when the atmospheric temperature is higher than ISA+10°C, the maximum available thrust is once again determined by the maximum TET limit and so it drops below the rated value.

8.0 Estimation of TR _EC

The engine model is complete when values have been assigned to the parameters TR _EC and M _EC . These are available, at least in principle, from a complete thermodynamic model of the engine, but such models are generally only available in industry. Nevertheless, it can be shown that, to the level of accuracy expected from this method, estimates of C _T and, hence, $\eta$ _o are not particularly sensitive to errors in the value of M _EC . Therefore, at this stage in the development of the method, it is proposed that Equation (31) is assumed to be applicable to all engines and the values are given in Table 1. This leaves TR _EC to be determined and, since it has a significant influence, a reasonably accurate estimate is essential.

During the take-off run and the initial climb phase, the turbine entry total temperature is assumed to have its maximum value. As the aircraft accelerates, the net thrust decreases as the Mach number increases. Since this is an important element in the determination of take-off performance, this initial thrust lapse is often given in standard books on air vehicle design and it is known to be, primarily, a function of the engine bypass ratio. Using the information given by Shevell [Reference Shevell7] (Figure 17.16) and Jenkinson et al. [Reference Jenkinson, Simpkin and Rhodes8] (Figure 9.2), when the flight Mach number has reached 0.4, i.e. just after take-off, the thrust lapse is found to be

(37) \begin{align}{\left( {\frac{{{{\left( {{F_n}} \right)}_{M = 0.4}}}}{{{{\left( {{F_{{\rm{00}}}}} \right)}_{ICAO}}}}} \right)_{ISA}} \approx 0.97\left( {0.825 - 0.018\left( {BPR} \right)} \right),\end{align}

where F ₀₀ is maximum, uninstalled, static thrust at sea level from the ICAO Emissions Data Bank [13] and the factor 0.97 allows for the small installation loss when an engine is integrated into an aircraft. In addition, from the definition of the thrust coefficient given in Equation (27), when the Mach number is equal to 0.4,

(38) \begin{align}\left( {\frac{{{{\left( {{C_T}} \right)}_{M = 0.4}}}}{{{{\left( {{C_T}} \right)}_{DO}}}}} \right) = \frac{{n{{\left( {{F_n}} \right)}_{M = 0.4}}}}{{\left( {\gamma /2} \right){{\left( {{p_{SL}}} \right)}_{ISA}}{{\left( {0.4} \right)}^2}{S_{ref}}{{\left( {{C_T}} \right)}_{DO}}}} \approx 8.66\left( {0.825 - 0.018\left( {BPR} \right)} \right)\left( {\frac{{n{{\left( {{F_{00}}} \right)}_{ICAO}}}}{{{{\left( {{p_{SL}}} \right)}_{ISA}}{S_{ref}}{{\left( {{C_T}} \right)}_{DO}}}}} \right).\end{align}

However, from Equations (29) and (33),

(39) \begin{align}\frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{DO}}}} = \left( {\frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}}} \right)\left( {\frac{{{{\left( {{C_T}} \right)}_{\eta B}}}}{{{{\left( {{C_T}} \right)}_{DO}}}}} \right) = {h_4}.{h_2},\end{align}

and so, when M _∞ is equal to 0.4,

(40) \begin{align}{\left( {\frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}}} \right)_{{{M}} = 0.4}} = {\left( {{h_4}} \right)_{{{M}} = 0.4}} = 0.1311\left( {\frac{{1 + 0.55{M_{DO}}}}{{{{M}}_{DO}^2}}} \right)\left( {\frac{{{{\left( {{C_T}} \right)}_{m = 0.4}}}}{{{{\left( {{C_T}} \right)}_{DO}}}}} \right).\end{align}

From Equations (32) and (33),

(41) \begin{align}{\left( {{T_R}} \right)_{M = 0.4}} \approx \left( {\frac{{0.969}}{{T{R_{EC}}}}} \right)\frac{{\left( {{{\left( {TET} \right)}_{\max T/O}}/{{\left( {{T_{SL}}} \right)}_{ISA}}} \right)}}{{\left( {0.915 + 0.424{{{M}}_{EC}} - 0.530{{M}}_{EC}^2} \right)}} = \frac{3}{5}\left( {1 + \frac{2}{3}{{\left( {{h_4}} \right)}_{{{M}} = 0.4}}} \right),\end{align}

and, finally,

(42) \begin{align}T{R_{EC}} \approx \left( {\frac{{1.615}}{{1 + 0.667{{\left( {{h_4}} \right)}_{M = 0.4}}}}} \right)\frac{{\left( {{{\left( {TET} \right)}_{\max T/O}}/{{\left( {{T_{SL}}} \right)}_{ISA}}} \right)}}{{\left( {0.915 + 0.424{M_{EC}} - 0.530M_{EC}^2} \right)}}.\end{align}

In addition, when the aircraft is operating at the design optimum condition, $\eta$ _o has a local maximum value. Consequently, the parameter (T _R ) _DO is equal to one, and so

(43) \begin{align}{TR_{EC}} \approx \frac{{{{\left( {TET/{T_0}} \right)}_{DO}}}}{{\left( {1 - 0.53{{\left( {{{{M}}_{DO}} - {{{M}}_{EC}}} \right)}^2}} \right)}}.\end{align}

As described in Section 7, there is a maximum allowable value for the turbine entry temperature for continuous operation in the cruise and, clearly, the TET at the design optimum should not exceed this value. Moreover, for safety reasons, an aircraft should have sufficient thrust available to allow continuous operation at Mach numbers approaching the maximum operational value, M _MO , over a wide range of altitudes. Therefore, the design optimum condition must be achieved with a value of TET that is less than the maximum continuous cruise value, i.e.

(44) \begin{align}{\left( {TET} \right)_{DO}} \approx \beta {\left( {TET} \right)_{max\;cruise}} \approx \beta \left( {0.88{{\left( {TET} \right)}_{\max T/O}}} \right),\end{align}

where $\beta$ must be less than unity and so

(45) \begin{align}T{R_{EC}} \approx \frac{{\beta \left( {0.88{{\left( {TET} \right)}_{\max T/O}}} \right)/{{\left( {{T_0}} \right)}_{DO}}}}{{\left( {1 - 0.53{{\left( {{{{M}}_{DO}} - {{{M}}_{EC}}} \right)}^2}} \right)}}.\end{align}

The value of $\beta$ is obtained by combining Equations (42) and (45) and, as shown in Fig. 19, the data suggest that the best, average value is about 0.91. Given the large uncertainty in the values of the sea-level, static thrust, the best estimates for TR _EC have been obtained from Equation (45) with $\beta$ equal to 0.91 and the values of the other parameters being taken from Table 1. The resulting values are given in Table 1.

Figure 19. Comparison between estimates of TR _EC from Equations (42) and (44) with β equal to 0.91.

9.0 Summary

For a given aircraft and engine combination, the values of the engine characteristics $\eta$ ₁ , $\eta$ ₂ , M _EC , TR _EC are found in Table 1, together with the design-optimum values of Mach number, M _DO and thrust coefficient, (CT) _DO . The method can then be used in one of two ways, depending upon the input information.

Firstly, if the total thrust, Mach number, flight level and the variation of atmospheric temperature with pressure are specified and using S _ref from Table 2, from Equations (14) and (27),

(46) \begin{align}{C_T} = \frac{{n{F_n}}}{{\left( {\gamma /2} \right){p_\infty }{{M}}_\infty ^2{S_{ref}}}}.\end{align}

Equations (28) and (29) are then used to obtain ( $\eta$ _o) _B and (C _T ) $_{\eta B}$ and since, from Equation (27),

(47) \begin{align}\frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}} = \frac{{{C_t}}}{{{{\left( {{C_t}} \right)}_{\eta B}}}},\end{align}

the engine overall efficiency comes from Equation (19), i.e.

(48) \begin{align}{\eta _o} = {\left( {{\eta _o}} \right)_B}\left( {1 - 0.43{{\left( {\frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}} - 1} \right)}^2}} \right)\left( {1 + {\rm{\Sigma }}{{\left( {\frac{{{C_T}}}{{{{\left( {{C_T}} \right)}_{\eta B}}}} - 1} \right)}^2}} \right).\end{align}

The specific fuel consumption comes from Equation (1), i.e.

(49) \begin{align}SFC = \frac{{{{{M}}_\infty }\sqrt {\gamma \mathfrak{R}{T_\infty }} }}{{{\eta _o}LCV}},\end{align}

and the fuel flow rate per engine is

(50) \begin{align}{\dot m_f} = SFC.{F_n}\;.\end{align}

Secondly, if the turbine entry temperature, Mach number, flight level and the variation of atmospheric temperature with pressure are specified, T _R follows from Equation (32) and C _T is obtained from Equation (33). The thrust is then given by Equation (46) and the overall efficiency, SFC and fuel flow rate come from Equations (48), (49) and (50).