A technique was developed for determining the minimum specific fuel consumption of a two-circuit turbojet engine using statistical data on the polytropic efficiency of individual compressor and turbine stages. The proposed method allows the presence of a technological advantage in specific fuel consumption compared to similar engines to be identified at the initial stage of designing a two-circuit turbojet engine.

The maximum achievable level of technical excellence determined for an engine with the selected design arrangement at the initial stage of design allows to assess in advance its competitive position compared to the analogues. To assess the level of technical excellence of the power unit of an aircraft (AC), the following two parameters are used: specific fuel consumption CR and specific engine weight удв [1]. Specific parameters of the designed engine are determined starting with thermodynamic calculation of “reference” operation mode. At that, the efficiency factor (EFF) of the main assemblies and the level of losses in the gas-air path of an engine are either based on the previous design experience (data of analogues or previous versions) or determined during individual calculations of a compressor, turbine and combustion chamber. The direct analytical relation between the parameters of thermodynamic cycle and EFF of the main assemblies is impossible for the given engine. Thus, selection of thermodynamic parameters, such as gas temperature inside the combustion chamber Тг*, total compression ratio πΣ*, by-pass ratio y and subsequent analysis of dependencies CR = f(Тг*, πΣ*, y,...), is carried out at constant loss and EFF values of assemblies.

When determining the maximum achievable technical level of an engine, a relation between parameters Тг*, πΣ*,, y and maximum possible EFF of assemblies can be established. The method is based on the usage of dependencies of the maximum possible polytropic EFF of the compressor or turbine stage on the stage load values, preliminary obtained based on the statistics. Then, adiabatic EFF of the entire compressor ηк* or turbine ηт* is calculated using the thermodynamic cycle parameters. The method of calculation of the maximum possible EFF of the main engine assemblies is described in detail in [2].

For the considered calculation methodology, the following assumptions and limitations are established:

Initial data for determining maximum possible ηк* of the axial compressor include the following parameters: reduced air flow rate GВПР 0, total temperature at the compressor inlet Твх*, as well as rate of total pressure rise in the compressor πк* and selected number of compressor stages z. At the beginning of calculation, the load rate per one stage Δi*ст0 and the rate of stage pressure rise π*ст0 are determined as a first approximation using the equalities:

where Δiк*ад, kcal/kg, is adiabatic change in enthalpy downstream of the compressor, determined using thermodynamic functions bonded to values πк* and Твх*; Δiад* ст, kJ/kg, is adiabatic change in enthalpy at the stage. Dependence of the maximum possible EFF of the compressor axial stage on the change in enthalpy Δiст* is shown in Figure 1.

For each compressor stage with sequence number s, pressure Δiст*(s) and maximum polytropic EFF can be determined as follows:

Where kα - correction for the stage pressure loss, and kH(s) is the coefficient determining pressure change across stages. For small-size TCTE, the number of axial stages in the compressor usually doesn’t exceed 2. In this case, in contrast with multistage compressors with predetermined pressure distribution, it can be assumed that kн(1) = kн(2) = 1.

Polytropic EFF, considering correction for the stage size, is determined by the following equations:

where GВПР(s), kg/s, is reduced air flow rate at the inlet of stage s, Δηпол* is correction for polytropic EFF determined based on dependence diagram shown in Figure 2.

Dependence diagrams for and Δηпол*, shown in Figures 1 and 2, are obtained through processing of experimental statistics on the axial and centrifugal compressor stages based on the data taken from [1][3][4].

Adiabatic EFF of a stage

Air parameters at the stage outlet:

where i*вх(s), S*вх(s) are air enthalpy and entropy at the stage inlet; Δiст*ад(s) is adiabatic head of the stage; iст*ад(s), T*ст*ад(s), S*ст*ад(s) are air enthalpy, temperature and entropy at the stage outlet, calculated using thermodynamic functions.

Basic parameters of axial compressor are determined based on the following relations:

Joint solution of equations (1)-(11) allows to determine adiabatic EFF, pressure of each compressor stage and total EFF of the compressor.

Similarly, the calculation methodology for a compressor consisting of several centrifugal or mixed flow compression stages can be worked out based on the above dependencies. A single centrifugal stage is used in most modern small-size TCTE. For centrifugal stage, dependency shown in Figure 1 should be used. Additionally, initial data used for calculation include reduced air flow rate Gв прц and braking temperature Твх* at the stage inlet. For single-stage centrifugal compressor, Gв прц = Gв прц0, Твх * is predetermined. For the end stage of axial-centrifugal compressor, Gв прц = Gв пр(z), Твх*= Тст*(z). With that, the calculation of adiabatic EFF of a stage is essentially simplified:

Change in enthalpy and air parameters at the outlet of centrifugal stage:

where i*вх, S*вх are air enthalpy and entropy at the stage inlet, determined by Твх*; Δiц*ад ад is adiabatic head of the stage; iц ад, Tц*ад, Sц*ад ад are air enthalpy, temperature and entropy at the outlet of centrifugal stage.

For a single-stage centrifugal compressor, the stage parameters are at the same time the compressor parameters. Basic parameters of axial-centrifugal compressor are determined taking into account the axial section parameters:

The methodology of determining the maximum possible adiabatic EFF for the compressor turbine was compiled taking into account the air bleed for cooling of nozzle assembly (NA) and impeller (IMP) for one or several stages. The following parameters from the reference mode calculation are used as initial data: change in enthalpy inside the compressor Δiк*, reduced air flow rate Gв пр0, braking temperature Т*г and total pressure Рг* of gas at the turbine inlet, air enthalpy downstream of the compressor iK*, relative fuel consumption in combustion chamber qт кс = Gт / (3600 · Gв кс). Dependencies for determining mechanical EFF ηmK = f(Gв пр0) on the compressor turbine shaft taking into account accessory drive and dependency for determining relative air bleed Δ охл ст(s) = f (Твх*) for cooling of one turbine stage are given in [2]. Relative air bleed for cooling of body disc and turbine discs Δ охл к = 0,005...0,01.

Air and gas flow rates at the inlet of the compressor turbine:

At the beginning of calculation, values μв, μг, Δ охл Σ are determined as a first approximation.

To determine the change in gas enthalpy in compressor turbine Δi*тк and in separate stage Δi*ст at predefined number of stages z (in accordance with arrangement option shown in Fig. 5), the following relations are used:

In cross-sections downstream of NA and IMP of the turbine, flow rates for each stage s are recalculated using the relations:

Where j – 1 indicates cross-section at the inlet of NA or IMP; j indicates cross-section at the outlet of NA or IMP; ψса, ψρκ are proportional factors of relative air flow rate for cooling of NA and IMP, respectively.

Thermodynamic parameters at the outlet of NA are determined using thermodynamic functions:

where iвх*'(s), Твх*'(s), Sвх*'(s) are, respectively, enthalpy, total temperature and entropy of gas downstream of NA, i. e., at the inlet of IMP; and cp, Rr, кг are, respectively, heat capacity, gas constant and adiabatic exponent of the gas.

Polytropic EFF of a stage η*пол(s) is determined using the dependencies:

Δη*пол = f(Аст), если Аст ≤ 40, Δη*пол = 0, если Аст > 40.

Where η*maxпол - is maximum possible polytropic EFF determined based on the dependency shown in Figure 3, Δη*пол - is correction for polytropic EFF of a stage depending on the capacity Аст, determined based on the dependency shown in Figure 4, P*вх(S) is total gas pressure at the inlet of the stage impeller. Dependencies η*maxпол are obtained through processing of statistics taken from [3]. Dependency for Δη*пол is taken from paper [4].

Fig. 3. Maximum possible polytropic EFF of compressor turbine stage

Fig. 4. Correction for polytropic EFF of turbine stage

Adiabatic parameters downstream of IMP and adiabatic EFF of a stage η*ад(s) are determined using the equations:

where i*ст ад(s), T*ст ад(s), S*ст ад(s) are, respectively, adiabatic enthalpy, total temperature and entropy of gas at the inlet of IMP determined using thermodynamic functions; Δi*ст ад(s) is adiabatic drop at the stage IMP; π*ст (s) is IMP total pressure ratio.

Gas enthalpy at the stage outlet is determined by enthalpy drop in IMP and cooling air flow rate

where i*ст (s) is gas enthalpy at the outlet of IMP.

Fig. 5. TCTE arrangement: a) first design arrangement, b) second design arrangement 1 – fan (opt. a), two-stage fan (opt. b), 2 – HP compressor axial stage (opt. a), two-stage axial booster stage (opt. b), 3 – HP compressor centrifugal stage, 4 – combustion chamber, 5 – HP turbine, 6 – LP turbine (opt. a), two-stage LP turbine (opt. b), 7 – second circuit nozzle, 8 – first circuit nozzle, NA – cross-section at the outlet of nozzle assembly, IMP – cross-section at the outlet of impeller

Total temperature and gas pressure at the turbine stage outlet:

Tст*(s) = f (qт(j), iст*(s), Pст*(s) = Pвх*(s) / πст*(s) . (32)

Since equalities i*вх (s + 1) = iст*(s) и Pвх*(s + 1) = Pст*(s) are true for a multistage turbine, the above equations allow to calculate the main parameters for each turbine stage z subject to their joint solution.

Then, basic parameters of the compressor turbine - turbine total pressure ratio π*тк and adiabatic EFF η*тк - are determined:

Calculations of low pressure turbine connected with a fan are carried out in a similar way, at that, for determining values η*maxпол и Δη*пол the dependencies shown in Figures 3, 4 are used. If temperature at the inlet of turbine or stage is T*вх(s) < 1200 K, it is assumed that Δохлс(s) = 0.

The proposed procedures for calculating adiabatic EFF of the compressor and turbine are used in this case as parts of thermodynamic calculation of the engine reference mode, implemented in the form of individual subprograms.

Other parameters that characterize losses in the gas-air path and combustion efficiency, as a rule, have a narrow range of possible values. Their quantity and numeric values are determined by the engine type (TE, TCTE, etc.) and can be adopted from [5][7]. When determining the maximum achievable level of technical excellence of an engine with minimum possible CR, parameters that characterize losses in the gas-air path may be set as constant values. The methodology of thermodynamic calculation of the reference mode is well known, consequently, it is not considered. To calculate thermodynamic functions of air and gas in the temperature range from minus 50 to 1500 °C, data of [6] are used, and for the temperatures exceeding 1500 °C - approximating dependencies as per NASA sp-273.

To validate the developed methodology, minimum possible CR were calculated with respect to small-size TCTE. The calculations were carried out for standard atmospheric conditions at the engine inlet of Н = 0, М = 0, TH = 288.15 K. The range of varying main parameters of thermodynamic cycle was selected based on the statistics for TCTE manufactured by Teledyne CAE, Williams International [8]: πΣ* = 10-13,8, Тг* = 1150-1400 K, y = 1. In all cases, the reduced air flow rate through the first circuit was equal to GВПР 0 = 2.5 kg/s. Based on the preset task, instead of the values of engine thrust the value of average specific engine thrust was calculated for all options as follows I = (Rуд1 + Rуд2 · y)/ (1 + У), где Rуд1 Rуд2 are specific nozzle thrust of the first and second circuits, respectively.

The results of variable calculations of TCTE reference mode with the maximum possible EFF of the assemblies are shown in Figures 6, 7. Figure 6 shows calculated dependencies CR = (Тг*, πΣ*, I) for the first TCTE design arrangement with a single-stage fan, high pressure (HP) compressor consisting of axial and centrifugal stages, ring straight-flow combustion chamber, single-stage turbine of high and low pressure (LP). The first design arrangement is shown in Figure 5а. The plotted lines represent the calculation results for multiple options of TCTE reference mode with selected constants of thermodynamic cycle Tr* = const πΣ* = const. Each dot on the diagram represents minimum possible CR value that is reached at preset Tг*, πΣ*, y and external conditions.

Similar dependencies by CR are shown in Figure 7 for the second arrangement of TCTE with a two-stage fan, two booster stages of LP cascade, HP compressor consisting of centrifugal stage, ring straight-flow combustion chamber, HP single-stage turbine and LP two-stage turbine. The second design arrangement is shown in Figure 5b. Additionally, Figure 7 shows engine data of small-size TCTE WR-19 family manufactured by Williams International and design data of these engines obtained at the same parameters of thermodynamic cycle with maximum possible EFF of compressor and turbine stages (dotted with the same marker). The analysis of presented data demonstrates the possibility of reducing CR for these engines by 7-10 % with increasing the polytropic EFF of component stages to the currently maximum possible level (data on Fig. 1, 3). It should be noted that the line of joint operation in the field of compressor characteristics, with a view to ensuring sufficient stall margin, may be shifted to a region where EFF is by 1-2 % lower than the maximum value line. Consequently, the maximum potential of CR reduction for a finally designed and manufactured engine in this case should be reduced to 5-8 %.

The methodology can also be used for comparing TCTE of different arrangements and having different parameters of thermodynamic cycle. Dependencies shown in Figures 6 and 7 may be represented in the form of an area with boundary lines for a fixed range of values Tг*, π*Σ. In this case, an overlapping of two such areas obtained for TCTE of the first and second arrangement with the same range of values Tг*, π*Σ, y allows to compare them visually by minimum achievable values CR, as shown in Figure 8.

Quantitative analysis can also be carried out. For example, a transition from the first TCTE arrangement to the second one (see Fig. 8) at constant values Tr* = 1300 K, πΣ* = 12.25, y = 1 allows to reduce specific fuel consumption by value δCR = -1.2 % with simultaneous increase of total specific impulse δΐ = 1.0 %. A reduction of CR is generally caused by an LP turbine EFF increase at the transition from the single-stage arrangement to two-stage one.

An advantage of the developed methodology, compared to the standard thermodynamic calculation of the reference mode, consists in the possibility to calculate minimum achievable values CR of the engine taking into account the relations between the change of the main parameters of thermodynamic cycle πΣ* and Tr*, the change of assemblies EFF and amount of air bleed for cooling. The methodology allows to assess the potential for improving cost-effectiveness of existing TCTE, limited by the achieved technical characteristics of the main assemblies. For the engine of a new design arrangement, at the initial design stage it is possible to reveal whether there is an advantage over analogous engines in the expected operating conditions in terms of minimum possible specific fuel consumption, or not.