1 Introduction

Aviation is responsible for approximately 2% of the global CO2 emissions. Considering also non-CO2 effects, the contribution of the aviation sector to the current anthropogenic climate change is rather 5% [1]. This number is expected to increase since aviation passenger transport is projected to grow by 4% per year whilst other sectors continue to decarbonize. To reduce the environmental footprint, the Advisory Council for Aviation Research and Innovation in Europe (ACARE) has set targets for the reduction of emissions in its Flightpath 2050 document [2]. Among these goals is a 75% reduction in CO2 per passenger kilometer till 2050 compared to an aircraft in 2000. A mid-term reduction of 30% is aimed for the year 2035. These goals are difficult to achieve with the evolutionary developments of current technology. Zheng and Rutherford [3] report an average reduction in fuel consumption per passenger kilometer at the global fleet level of 1.3% per year over the years 1960–2019. Without any further specific measures this reduction is expected to continue at a similar rate until 2037 [4]. Therefore, radical approaches are required for the propulsion system for commercial aviation.

One way to potentially improve the efficiency of the gas turbine is by replacing the constant pressure combustion process with a constant volume combustion process. Combustion at constant volume yields a total pressure increase. Pressure gain combustion (PGC) enables higher specific work thereby improving thermal efficiency [5]. There are commonly two thermodynamic representations used. The Humphrey cycle and the Zeldovich, von Neumann, Doering cycle (ZND). The former assumes an isochoric change of state along the combustion process whereas the latter models the PGC process by an initial detonation followed by an isochoric heat release. Both are depicted in an enthalpy-entropy diagram in Fig.  1.

Fig. 1
figure 1

Specific entropy enthalpy diagram of Joule, Humphrey (’) and ZND (”) cycles

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Many studies have confirmed this statement for the simple cycle gas turbine [6,7,8,9,10] and for the turbofan engine [11,12,13,14]. However, PGC detrimentally affects both adjacent turbomachines - namely the downstream high-pressure turbine (HPT) and the upstream high-pressure compressor (HPC) - and the secondary air system (SAS) of the engine. This is because PGC is inherently unsteady. The combustion process is periodic and causes fluctuating boundary conditions at the inlet and outlet of the combustor. This reduces the efficiency of the turbo components [15, 16]. Furthermore, the pressure gain across the combustor requires the implementation of a SAS compressor to enable film cooling of the turbine blades and sealing of the HPT. To date, studies referred to above either lack the consideration of such detrimental effects or replace the constant pressure combustor with a PGC device without any adaptation of the thermodynamic cycle.

This gap is closed by evaluating the technical potential of PGC for a single-aisle aircraft engine for a short-range mission. In this thrust class, PGC is expected to give higher efficiency benefits than in large turbofan engines. This is because small engines typically have a lower overall pressure ratio (OPR), which is generally beneficial for PGC. The resulting efficiency of an engine with PGC is compared to an engine based on the same technology level for an entry into service (EIS) 2035 with conventional constant pressure combustion. Furthermore, both engines have the same fan diameter to allow for a comparison at a constant propulsive efficiency [17].

The required methods and models are presented in Sect. 2. The results of the simulations, which includes the comparison of an optimized Joule engine with an optimized PGC engine, are presented in Sect. 3.

2 Methods and engine models

This section presents the underlying methods and models of the simulations. First, the design methodology is outlined. This comprises the workflow of the optimization and the selected constraints. This is followed by the introduction of the engine model. Furthermore, details of the modeling of PGC, the secondary air system and the estimation of engine weight are provided.

2.1 Design methodology

The paper aims for a preliminary design of the engine. The methodology accounts for the design and off-design operating points of the engine. The simulations are performed with GTlab-Performance [18]. The aero engine design point is defined at the max climb rating (MCL) at the top of the climb, where the highest reduced mass flows are present. The specific fuel consumption at cruise (CR) serves as the objective function in the optimization. However, cruise does not present the engine with the most challenging conditions. Thermo-mechanical limits are approached at the take-off rating (TO) at the end of the runway with high temperatures and high pressures. The operating points are defined in terms of altitude, temperature deviation from ISA condition, flight Mach number and thrust requirement in Table 1. The thrust requirements are projected for a future single-aisle aircraft from [19].

Table 1 Definition of the operating points and fan diameter according to [19]
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The design methodology for both, the conventional aero engine which serves as a reference and the PGC aero engine, is conducted in IPSM. IPSM is a software package developed at the Chair for Aero Engines to manage generic workflows. More information can be found in the work of Woelki [20]. The design methodology can be split into three levels (see Fig. 2) which are introduced starting from the core, the lowest level.

Fig. 2
figure 2

Flow chart of the design methodology

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Level 1

At this level, the performance software GTlab-Performance is executed. Within GTlab-Performance, the Newton–Raphson solver secures an engine design that complies with the requirements regarding thrust and fan diameter. That is achieved by matching pairs for the engine design point and for off-design operation. The solver satisfies ten matching pairs for the design operating point MCL listed in Table 2.

Table 2 GTlab-Performance design point matching pairs
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Matching pair 1 computes the intake mass flow from a given fan diameter and assumed hub-to-tip-ratio and axial Mach number. Matching pair 2 finds the appropriate bypass ratio, to achieve the required thrust and matching pair 3 alters the pressure ratio of the HPC to match the OPR coming as an input from the optimizer.

Matching pair 4 varies the bypass fan pressure ratio to yield optimum SFC. That is achieved with Eq. 1. Equation 1 links the velocity ratio of the core and bypass nozzles with the energy transfer efficiency from core to bypass (compare Guha [21]).

$$\begin{aligned} \frac{v_{\textrm{core}}}{v_{\textrm{byp}}} = \frac{\eta _{\textrm{Fan,byp}}}{\eta _{\textrm{LPT}} \cdot \eta _{\mathrm {LP-shaft,mech}}} \end{aligned}$$
(1)

The fan core pressure ratio (matching pair 5) is calculated from Eq. 2 and the pressure loss in the bypass duct (matching pair 6) is a function of bypass ratio as suggested by Kurzke and Halliwell [17].

$$\begin{aligned} \pi _\text {Fan,core} = 1 + 0.8 \left( \pi _\text {Fan,byp} -1 \right) \end{aligned}$$
(2)
$$\begin{aligned} \frac{p_{16}}{p_{13}} = 1 - \frac{0.05}{(\textrm{BPR}+1)^{0.4}} \end{aligned}$$
(3)

Matching pair 7 accounts for high-pressure turbine efficiency losses due to the mixing of main and secondary air. The exact procedure is outlined in the model section. Matching pair 8 fixes the position of the HPC bleed port for the low-pressure turbine (LPT) secondary air. The total pressure at the HPC bleed port shall always be 2% higher than the LPT inlet total pressure. Matching pair 9 accounts for losses in the gearbox. They are assumed to be 1% of the total fan power [22]. The last matching pair accounts for size effects on the polytropic efficiency of the HPC.

The engine model with PGC requires two additional matching pairs. For the PGC engine, secondary air for the first HPT stage is further compressed. This increases the secondary air temperature and requires power for the SAS compressor. An increase of 4% above combustor exit pressure is assumed for the PGC cycle, resulting in identical pressure ratios across the SAS for Joule and PGC engines. The resulting temperature after additional compression as well as the power for the SAS compressor, which is provided by the high-pressure shaft, are considered via additional matching pairs presented in Table 3.

Table 3 Additional GTlab-Performance design and off-design matching pairs for PGC
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Level 2

The second level of the methodology consists of a fixed-point iteration to compute the required secondary flows for the HPT and LPT. GTlab-Performance simulates the three operating points starting with default cooling air flows for the HPT and LPT. The actually required cooling air flows are calculated using data from the TO operating point. The calculation method is briefly presented in Sect. 2.2.2. The newly calculated value is set as input for the subsequent iteration in GTlab-Performance. This procedure is repeated till an acceptable level of convergence is achieved.

Level 3

Level 3 consists of an outer optimizer. The objective function of the optimization is the SFC of the cruise operating point. The optimizer varies the remaining engine design parameters: TET, OPR and booster pressure ratio. To cope with the highly nonlinear solution space, both stochastic and gradient-based algorithms are used to identify the global minimum and avoid local minima.

The available design space is constrained to obtain realistic results. The following constraints are considered in the outer optimization:

  • HPC exit blade height \(h_\text {bl} \ge {0.012}\) m: Both high BPR and high OPR potentially reduce SFC but result in small core sizes. That goes hand in hand with reduced turbo component efficiencies as the clearance losses increase. To restrict the optimization to reasonable cycles, a minimum required compressor exit blade height of 0.012 m is set as a constraint. The compressor exit blade height is calculated assuming a hup-to-tip ratio of 0.92 and a flow Mach number of 0.3 at the HPC exit. [17]

  • TET \(\le {2000}\) K: Higher combustion temperatures cause the dissociation of nitrogen resulting in NOx. The amount of NOx emissions during a landing and take-off cycle are already limited by legislation. In view of future emission specifications, the requirements for NOx are expected to become more stringent. Hence, the TET is limited to 2000 K during take-off. This assumption is in line with other researcher’s estimate. For example, Samuelsson et al. [23] estimated a maximum TET of 2030K.

  • Turbine pressure ratio \(\le {4.5}{}\): Some single-aisle aircraft engine such as the CFM56 engine family have a single-stage HPT design. That limits the tolerable pressure ratio of the turbine. A single-stage design is adopted for this study. Thus, a maximum allowable turbine pressure ratio of 4.5 is set as a constraint according to [17].

2.2 Engine model and modeling of relevant components

The engine architecture and model assumptions are presented first. The PGC model is presented below. Details of the SAS are given, including the SAS flow distribution and the effect on turbine efficiency. Finally, the method for estimating the engine weight is outlined.

The modeled engine is an ultra-high bypass geared turbofan with separate nozzles. A single-stage fan is mounted on a shaft connected with a gearbox. A gearbox is required to reduce the rotational speed of the large fan compared to the low-pressure shaft speed. The LP-shaft also drives the booster compressor. The high-pressure system consists of a multi-stage compressor followed by the combustor and a cooled single-stage turbine. The HPT cooling air is extracted at the HPC exit. The LPT features cooled blades in the front blade rows. Required sealing air is provided through an HPC bleed. Core and bypass flows are accelerated in separate nozzles. Pressure losses are accounted for in the inlet diffusor, bypass duct, swan neck duct, combustor, inter-turbine duct and nozzles. Furthermore, the model accounts for mechanical losses and windage losses through a constant mechanical shaft efficiency. Gearbox losses are also considered. The engine has no customer bleed and does not have a shaft power off-take. This is in line with new more-electric aircraft.

The component efficiencies and pressure losses are assumed for an EIS 2035. The assumptions are summarized in Table 4. Component efficiencies are taken from [24]. Pressure losses are taken from [17]. These specifications are used for both engine configurations. The polytropic efficiency of the HPC is derived from an approach by Grieb [25]. This method accounts for the component size and EIS. Polytropic efficiencies are selected as metrics for compressors to represent a certain technology level. That results in a lower isentropic efficiency for high-pressure ratios and vice versa. Isentropic efficiency is selected for turbines. For the HPT, this efficiency corresponds to an uncooled turbine. The effect of cooling and sealing air on the HPT efficiency is captured by an exchange rate. LPT efficiency is not affected by changes in its secondary air flows. Furthermore, the effect of pressure gain combustion on HPC and HPT efficiencies is not yet considered and will be superimposed.

Table 4 Specification of engine models
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2.2.1 Pressure gain combustion model

This study simplifies the highly unsteady process of PGC into a steady-state process using a 0D component representation. Accordingly, a transfer function is required to compute the combustor outlet temperature, outlet pressure and outlet mass flow as a function of inlet conditions and fuel flow. Outlet mass flow and temperature are found from mass and energy conservation, respectively. The outlet pressure is calculated from a model proposed by this author [6]. This model assumes that the pressure gain combustion process can be split into a deflagration and a detonation. A portion of the chemical energy of the fuel denoted r is burned at constant pressure and the remainder is burned isochorically. Furthermore, non-ideal detonation combustion is modeled by the introduction of a pressure loss \(dp_\textrm{Loss}\), which is associated with the deflagration. The model adapts the real operating behavior within a pulsed detonation combustion tube. After ignition, the fuel-air mixture burns subsonically in a deflagration until a transition to detonation is induced at a shock-focusing nozzle. The combustion up to this transition is isobaric and the remainder of fuel is burned isochorically.

A good agreement with outlet total pressure computed by a 1D-CFD code from Paxson [26] was achieved setting the fuel ratio parameter to \(r= 0.3\) and the pressure loss to \(dp_\mathrm {{Loss}} = {17.5}\%\). The calculation of the combustor outlet pressure is repeated here. As with conventional combustion, \(T_\textrm{out}\) is the result of an energy balance:

$$\begin{aligned} q_\text {supp} = \textrm{FAR} \cdot \textrm{LHV} = c_p \left( T_\text {out} - T_\text {in} \right) \end{aligned}$$
(4)

A similar energy balance may be set up for the deflagration combustion alone:

$$\begin{aligned} q_\text {cp}=r \cdot \textrm{FAR} \cdot \textrm{LHV} = c_p \left( T_\text {cp} - T_\text {in} \right) \end{aligned}$$
(5)

Here, r accounts for the fuel which is burned at constant pressure condition in order to accelerate the flame. By rearranging Eq. 5 the temperature after the deflagration \(T_\text {cp}\) can be calculated. The pressure after deflagration is reduced because of pressure losses as defined by Eq. 6.

$$\begin{aligned} p_\text {cp} = p_\text {in} \cdot ( 1 - dp_\textrm{Loss}) \end{aligned}$$
(6)

Finally, the outlet pressure is calculated using Eq. 7 assuming an isochoric change of state.

$$\begin{aligned} p_\text {out}= \frac{T_\text {out}}{T_\text {cp}} \cdot p_\text {cp} \end{aligned}$$
(7)

This PGC combustor model is compared to other models from the literature in Fig. 3. Below a temperature ratio of 1.3, no pressure increase is observed for the developed model as pressure losses prevail. At moderate temperature ratios, the model yields a linear dependency comparable to the other models. Towards high-temperature ratios, a maximum pressure ratio is obtained. A further increase in temperature ratio does not result in any further pressure ratio gain, as the simulations by Paxson [26] suggest.

Fig. 3
figure 3

Combustor pressure ratio over combustor temperature ratio for the developed model [6] and selected models of the literature

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2.2.2 SAS and turbine cooling model

Assuming a single-stage HPT, cooling air is required for the stator, the rotor and for sealing. It is assumed that 60% of the secondary air for the HPT is used to cool the nozzle guide vane (NGV), 30% are for rotor cooling and 10% for sealing. Furthermore, the secondary flows can be divided into chargeable and non-chargeable flows (see Table 5). It is assumed that stator cooling air and 50% of the sealing air provides work in the rotor. The remainder provides no work in the HPT. These assumptions are cast into the single-stage turbine model used in GTlab-Performance.

Table 5 Specification of HPT cooling flows
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The absolute value of the required secondary air is calculated by an approach presented by Wilcock et al. [27]. Ultimately, the required cooling air ratio \(\xi\) of coolant (\(\dot{m}_\text {c}\)) to hot air (\(\dot{m}_\text {h}\)) can be calculated for a blade row from

$$\begin{aligned} \xi = \frac{\dot{m}_\text {c}}{\dot{m}_\text {h}} = C \cdot W^+ , \end{aligned}$$
(8)

where C is a constant indicative of the level of technology. \(W^+\) is called a temperature difference ratio. \(W^+\) is determined by different equations depending on the specific cooling technology. Here, the equation for advanced convective and film cooling with TBC is used:

$$\begin{aligned} W^+ = \frac{1}{1+ \text {B} } \frac{ \epsilon _0 - (1 - \eta _\text {c} ) \epsilon _\text {F} - \epsilon _0 \cdot \epsilon _\text {F} \cdot \eta _\text {c} }{ \eta _\text {c} (1 - \epsilon _0)} \end{aligned}$$
(9)

The overall cooling effectiveness \(\epsilon _0\) is defined as

$$\begin{aligned} \epsilon _0 = ( T_\text {h} - T_\text {bl} ) /( T_\text {h} - T_\text {c} ), \end{aligned}$$
(10)

where \(T_\textrm{bl}\) is the blade material temperature, \(T_\text {h}\) is the hot gas temperature and \(T_\text {c}\) is the coolant temperature. In Eq. 9 B is defined as:

$$\begin{aligned} \text {B} = \text {Bi}_{\textrm{TBC}} - \frac{\epsilon _0 - \epsilon _\text {F} }{1- \epsilon _0} \cdot \text {Bi}_{\textrm{met}}, \end{aligned}$$
(11)

where \(\text {Bi}_{\textrm{TBC}}\) and \(\text {Bi}_{\textrm{met}}\) are the Biot numbers of the TBC and the metal, respectively. The remaining two parameters are internal cooling efficiency \(\eta _\textrm{c}\) and film cooling effectiveness \(\epsilon _\text {F}\).

The presented calculation of cooling air requires various input parameters. Their values are adopted from Wilcock et al. [27] presented as super advanced in the year 2005. Table 6 lists the specific values of the cooling model parameters. Currently, allowable blade material temperatures are approximately 1150 K. Kaiser [24] assumed an allowable temperature improvement of 50 K from the state-of-the-art till 2035. Accordingly, an allowable blade metal temperature of 1200 K is assumed for this study.

Table 6 Specification of cooling model [27]
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The presented approach uses turbine entry quantities for hot gas and cooling air temperatures for the cold gas as input. This permits the calculation of the required cooling mass flow of the downstream blade row, i.e. the NGV. This value is scaled to obtain the total cooling air required for the whole turbine. Since 60% of the secondary air is assumed for NGV cooling, the cooling ratio for the complete HPT is found by

$$\begin{aligned} \xi _\text {SAS} = \frac{\xi _{\textrm{NGV}} }{ 0.6}, \end{aligned}$$
(12)

where \(\xi _{\textrm{NGV}}\) is computed from Eq. 8.

Lastly, the modeling of the effect of secondary air on turbine efficiency is outlined. Thus far, an uncooled isentropic turbine efficiency has been assumed for the HPT in Table 4. The mixing of the secondary flow with the main flow generates additional entropy causing pressure losses resulting in a lower efficiency. According to [28, 29], each percentage point of secondary air (\(\dot{m}_\textrm{SAS}/\dot{m}_{25}\)) lowers the turbine efficiency by the exchange rates listed in Table 7. The values assume different cooling technologies for each blade row, categorized by Kurzke and Halliwell [17].

Table 7 Exchange rates on turbine isentropic efficiency in % points according to [28, 29]
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For the PGC engine, secondary air for the HPT requires further compression to overcome the pressure increase across the PGC combustor. The necessary power of the compressor is provided by the HP-shaft. It is assumed that the additional SAS compressor has a single-stage radial design with an isentropic efficiency of 80%. The SAS compressor pressure ratio is adjusted to the pressure gain of the main flow during combustion.

2.3 Engine weight estimation

The engine weight is inferred from an semi-empirical approach proposed in [30]. The mass of the turbo components is assumed proportional to their volume. The volume is derived from the change in reduced corrected mass flow \(\dot{m}_\text {red,cor} = \frac{\dot{m} \sqrt{T_\text {t}/T_\text {t,ref}}}{p_\text {t}/p_\text {t,ref}}\) according to Eq. 13.

$$\begin{aligned} m = c \cdot \bigl | \dot{m}_\text {red,cor,out}^{3/2} - \dot{m}_\text {red,cor,in}^{3/2} \bigl | \end{aligned}$$
(13)

Appropriate calibration factors c have been determined as presented in Table 8 based on mass prediction by more sophisticated methods [31]. The factors reflect the higher specific weight of high-pressure turbo components due to lower axial Mach number, higher aspect ratio, higher solidity, heavier disks and use of materials with higher density.

The combined specific weight of the remaining bare engine components (i.e. gearbox, combustion chamber, ducts, bearings, seals, shafts, casings, accessories and mounts) was estimated with 16kg per kN sizing thrust. The nacelle weight was estimated with 680kg per meter of fan diameter [30].

Table 8 Mass calibration constants c for turbo components according to [31]
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3 Results

This section presents the results of the simulations. First, the results of a turbofan with a conventional Joule cycle are presented. That will serve as a reference and is followed by the results of the PGC engine. Furthermore, the weights of both the Joule and PGC engine are estimated and fuel burn deltas for a representative short-range mission are computed.

3.1 Reference engine

This section presents the results of the reference engine. First, a pre-study explores the solution space and gives first indication regarding an optimum cycle of the conventional Joule engine for an EIS 2035. Then, the results of the optimization are presented.

3.1.1 Pre-study

Prior to the application of a comprehensive optimization, some light is shed on the resulting solution space. For that, the design methodology as depicted in Fig. 2 is executed for an OPR within the range [20; 70] and TET within a range of [1400; 2000 K] for a constant booster pressure ratio of 2. Figure 4 depicts relevant parameters over TET and OPR taken at the design point MCL.

Increasing OPR and TET decreases cruise SFC throughout the selected parameter space. Lowest SFCs are located outside the allowable parameter space. The blade height of the last compressor stage is estimated to be below 12mm. The TET constraint of 2000 K at take-off is not exceeded. The HPT pressure ratio is not considered here, as it depends on the pressure ratio split between booster and HPC. Taking into account the limit introduced for compressor exit blade height, maximum cruise efficiency is approximately 13g/(kNs). Moving to higher TET or higher OPR results in a constraint violation as the HPC exit blade height falls below 12mm. The resulting bypass ratio for optimum SFC is between 18 and 22 at the design point MCL. Higher TETs increase the specific work of the core and hence enable higher bypass ratios. Moving to a higher OPR reduces the required bypass ratio to meet the specified thrust.

Fig. 4
figure 4

Isolines for SFC, TET, compressor exit blade height and BPR over TET and OPR of the Joule engine

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3.1.2 Optimization

After this initial parameter study, the design methodology with optimizer is used as introduced earlier to identify an optimum engine configuration. Of course, the booster pressure ratio is now a degree of freedom.

Changing the booster pressure ratio affects many engine parameters. A higher booster pressure ratio leads to a reduced HPC pressure ratio at constant OPR. That has an effect on the LPT entry temperature. A high HPC pressure ratio requires more work delivered by the HPT. Thus, the temperature drop across the HPT is larger and the entry temperature into the LPT is reduced. Moreover, the temperature after compression is affected by the selected booster pressure ratio as well. That is because polytropic efficiencies are defined. The resulting isentropic efficiency is not only a function of polytropic efficiency but also pressure ratio. The isentropic efficiency of the HPC with an increased pressure ratio is lower compared to an HPC with a lower pressure ratio. This results in higher HPC outlet temperatures for low booster pressure ratios and affects, furthermore, the cooling air requirement of the HPT. Overall, lower booster pressure ratios lead to higher HPC exit temperatures with increased HPT cooling air requirements but lower LPT inlet temperatures and a lower LPT cooling air requirements.

The output of the optimization is presented below. Optimum SFC of the cruise operating point under compliance with all constraints is achieved with the engine configuration summarized in Table 9.

Table 9 Optimum turbofan specification for the Joule cycle
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The optimization results in a cruise SFC of 12.97 g/(kNs) for a conventional engine with EIS in 2035. That is an improvement of 22.5% compared to a state-of-the-art engine of the year 2000 such as the IAE V2500 engine with a cruise SFC of 16.7 g/(kNs). The SFC improvement is due to a high thermal efficiency of 48.25% and a very high propulsive efficiency of 87.78%. The SFC benefit is put into context of flight path 2050 goals. The Strategic Research and Innovation Agenda (SRIA) targets further significant improvements on engine level: the fuel consumption should decrease by 30% by 2035 compared to the year 2000 technology standard [32]. Since the presented baseline engine yields a SFC reduction of 22.5% it is just not able to comply with the requirements for the year 2035.

The engine design is positioned at the limit for HPC last stage blade height and at the limit for HPT pressure ratio. The booster pressure ratio is 2.5. A reduction in booster pressure ratio would yield improved SFC but results in higher HPT pressure ratios in order to drive the HPC. The bypass ratio is at 19.52, which is substantial increase compared to current engines. The required high specific work of the core is achieved by both high OPR and TET. However, the TET limit at take-off is not reached. That is because of the coupling of OPR and TET. The former is limited due to the specified minimum blade height. This would be different in a design study for a large turbofan engine, where rather TET than blade height is the limiting factor. The cooling flows are comparatively low. That is because the cooling model parameters reflecting a very advanced technology level.

3.2 Turbofan with PGC

This section covers results of a turbofan with PGC. The conventional combustor is replaced by a PGC module in the performance model. Furthermore, the turbofan model is extended by an additional compressor in the SAS.

The impact of PGC on HPC and HPT is mentioned in the introduction of this paper. The effect on the component efficiencies is adopted from the literature. A penalty of 1% point in compressor polytropic efficiency is assumed according to [33]. For the turbine efficiency penalty, values are as well taken from the literature. Bakhtari and Schiffer [34] conclude that turbine isentropic efficiency losses remain below 1.1% for a pressure amplitude of up to 10%. Grönstedt et al. and Xisto et al. [13, 15] reduced high pressure turbine efficiency from 91.4 to 87.6% because of PGC. Based on these findings, the HPT isentropic efficiency is reduced by 3%-points for this study. The LPT efficiency is assumed not to be affected by PGC.

3.2.1 Pre-study

The solution space is again presented for varying OPR and TET with a constant booster pressure ratio of 2.0. Figure 5 presents cruise SFC as solid lines over TET and OPR at MCL conditions. Optimum SFC is approx. 12.4 g/(kNs) and is located at an OPR of 50 and TET of 1900 K. That is outside the allowable parameter space. Taking into account the constraints introduced earlier, optimum cruise SFC is around 12.55 g/(kNs), exploiting a minimum blade height at the last HPC stage of 12 mm. Hence, core size is still a limiting factor even in an engine with PGC. The integration of PGC reduces the OPR for optimum efficiency. Unlike the Joule engine, OPRs beyond 50 do not further increase the efficiency of the engine. The bypass ratio is expected to be around 23 for optimum cruise SFC. That increase originates from the potentially higher specific work of the PGC cycle.

Fig. 5
figure 5

Isolines for SFC, TET, compressor exit blade height and BPR over TET and OPR of the PGC engine

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3.2.2 Optimization

As a next step, the optimizer is used to identify optimum engine specifications for PGC. The booster pressure ratio is now again a degree of freedom. The results of the optimization are summarized in Table 10 for the cruise operating point.

PGC improves cruise SFC from 12.97 g/(kN s) of the Joule engine to 12.54 g/(kN s) of the PGC engine, which equals to an improvement of +3.29%. This improvement is in the same order of magnitude for MLC but one percentage point higher for take-off (not depicted in this paper). The BPR is increased by 24%, which results in a smaller core mass flow. The HPC exit blade height is again the limiting constraint, despite a reduction in OPR by \(-\)17.6%. The increased temperature ratio across the combustor is typical for PGC as the pressure increase depends on the available temperature ratio. The HPT pressure ratio is 4.3 at MCL. That is not at the limit in contrast to the Joule engine. Interestingly, the reduction in OPR is realized via a reduction only in booster pressure ratio. The HPC pressure ratio is indeed increased. That is because the optimizer favors low LPT entry temperatures to maintain a lower LPT cooling air requirement.

In view of flight path goals, the turbofan engine with PGC yields a SFC improvement of 25% compared to a year 2000 turbofan. Again, that would just not be sufficient for the SRIA goal of 30% [32].

Table 10 Optimum turbofan specification for the PGC cycle of the cruise operating point including relative delta to Joule cycle
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The performance of the PGC combustor is further analyzed in Table 11. The combustion pressure ratio is approximately 1.24 at a temperature ratio of around 2.0 similar for all three operating points. Comparing these results with the combustor performance characteristic in Fig. 3, it can be inferred that temperature ratios and pressure ratios are moderate. A pressure ratio of 1.24 is just in the linear region of the graph. Higher pressure ratios are possible by higher temperature ratios considering just the combustor performance. However, that is not possible in this engine configuration as a higher TET increases the BPR, which leads to smaller blade heights of the last compressor stage. The power consumption of the SAS compressor is equal to 1% of the HPC power requirement. The SAS compressor increases the secondary air temperature by almost 70 K, which explains the increased cooling air requirement of the HPT of almost +30%.

Table 11 Combustor and SAS compressor performance at the cruise operating point
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3.3 System mass

The optimized engine with PGC achieves relevant SFC benefits. However, the increased bypass ratio and the resulting smaller core engine with reduced OPR could lead to weight savings that result in an additional reduction in fuel consumption. This will be investigated next.

From the results of both the Joule engine optimization in Table 9 and the PGC engine optimization in Table 10 a few observations can be made which affect components’ weights:

  1. (1)

    Similar fan pressure ratio

  2. (2)

    Smaller PGC core engine

Both engines have the same fan diameter. Hence, assuming no or just little changes in engine length, nacelle weight is not affected by the implementation of PGC. Furthermore, fan weight is also not affected as fan mass flow and fan pressure ratio are very similar (observation 1). Hence, the implementation of PGC will mainly affect the weight of the core components booster, HPC, combustor, HPT and LPT. On the one hand, all turbo components are expected to be lighter, because of observation 2. Additionally, weight savings are expected for the booster. On the other hand, the weight of the HPC could increase due to the higher pressure ratio and the LPT could also gain weight due to the higher bypass ratio. The reduced booster pressure ratio could likewise result in a lower power requirement, i.e. mass of the LPT.

The PGC combustor is expected to increase in weight compared to a conventional annular combustor. That is due to both larger wetted area with multiple tubes and more complex fuel metering units for designs employing a PDC. Jones and Paxson [12] considered a doubling of combustor weight as conservative. They accounted roughly 70 kg increase in weight for a 122 kN engine. Furthermore, the engine with PGC requires a SAS compressor. The additional weight of the SAS compressor is however neglected.

The resulting engine mass breakdowns for both Joule and PGC engines are depicted in Fig. 6. Only absolute values for the engine core components are presented. The rubric MISC contains the combustor, gearbox, ducts, bearings, seals, shafts, casings, accessories and mounts. Since their weight scales linearly with sizing thrust according to [30], the only difference between Joule and PGC engine is the combustor weight difference of 70 kg.

Fig. 6
figure 6

Engine core components mass breakdown

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The method estimates the total weight of the Joule engine of 3735 kg. This number seems reasonable, as the ENOVAL project estimates an engine mass of 4000 kg for a turbofan with a similar TO thrust rating and an identical fan diameter [35].

Looking at the masses of the core components. The booster, HPC and HPT masses are comparatively low. For instance, according to Donus et al. [36] the relative weight of the booster is between 5 and 11% of the total engine weight (here: 4%), the HPC between 7 and 13% (here: 4%) and the HPT between 5 and 11% (here: 2%). Especially, the calculated values for the HPC and the HPT are rather low. That can be explained by the high BPR leading to a small core size.

Regarding the LPT, Donus et al. provide data that suggests that its mass amounts to 9–17% of the engine mass. That would be in line with the results presented here (10%). However, the weight estimation methods does not take into account if the LPT rotational speed is constrainted by the fan or if a gearbox enables higher rotational speeds leading to less stages. Hence, it is strongly advised not to focus on the absolute values of the Joule or PGC engine weights alone but concentrates on deltas between the two configurations. For that, the results of the PGC engine are discussed next.

The PGC turbofan is predicted to weight 3700 kg, i.e. a marginal reduction of 35 kg or by \({-1.0}\%\). The weight of the booster (\({-33}\%\)), HPT (\({-27}\%\)) and LPT (\({-16}\%\)) is reduced but HPC (\({+7}\%\)) and combustor (\({+18}\%\)) gain in weight. The relative changes observed for booster, HPT and LPT can partially by attributed to the reduced core size. HPT and LPT weights are lower due to PGC. PGC leads to a high total pressure at the HPT inlet, which allows for a smaller design. Furthermore, the LPT of the PGC engine benefits from the fact that the incoming fluid carries more energy (higher \(T_{45}\)). That allows for a reduced size despite the increased power requirement due to a higher BPR.

3.4 Mission fuel burn

The difference in SFC and engine mass is now used to infer changes in mission fuel burn. Linear exchange factors from [37] applicable for a short-range mission of 1500 km are used. They are presented in Table 12.

Table 12 Linear trade factors influencing fuel burn [37]
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Using the values from Tables 9 and 10 yields a \(\Delta \textrm{SFC}\) of \(-\)3.29%. Together with the weight saving of 35kg the fuel burn difference is obtained with factors summarized in Table 12. The resulting fuel burn benefit is \(-\)3.93%. Roughly \(-\)0.15% comes from weight savings and the majority is gained from improved SFC of the PGC engine.

4 Conclusion

This paper evaluated the potential efficiency and fuel burn benefits of a turbofan with pressure gain combustion. Typical thrust ratings are applicable for a single-aisle aircraft were specified. The engines were optimized and simulated for take-off, climb and cruise using widespread design limitations as constraints. The effects of PGC on the adjacent turbo components were accounted for by efficiency penalties. An additional compressor was integrated into the secondary air system to supply the HPT with secondary air. The effect on secondary air temperature was considered as well as the impact on the cooling air requirement. A comparison with a reference turbofan based on the Joule cycle with an identical technology level provided insights into the technical potential of PGC. The following conclusions can be drawn:

  1. (1)

    PGC improves cruise SFC by 3.29%.

  2. (2)

    PGC enables an increase in BPR by 17.64%, as the specific work of the core engine increases.

  3. (3)

    Maintaining a minimum blade height for the last compressor stage is a design limitation for a PGC engine; as it is for the conventional Joule engine. Both optimum engines are designed at the constraint limit. The reduction in OPR for the PGC engine, potentially resulting in larger blade heights, is offset by a higher BPR, which leads to a smaller core mass flow.

  4. (4)

    PGC does not yield remarkable engine weight savings. Overall, the PGC engine is estimated to be just 1% lighter.

  5. (5)

    PGC reduces the mission fuel burn by 3.93%. This is mainly due to improved SFC rather than weight savings.

Overall, SFC and fuel burn benefits are modest compared to the high expectations for PGC. There are two reasons for this. First, current and even more future aero engines are expected to have a high OPR because that yields high thermal efficiency in the Joule cycle. However, the theoretical improvement from integrating PGC diminishes at higher OPR. This is because PGC thrives at high combustion temperature ratios that are only available at lower OPR. Second, PGC has detrimental effects on the turbo components and the SAS. These effects have been modeled here and reduce the efficiency of a PGC engine. In general, PGC would flourish in applications where specific thrust is more important than efficiency and where a reduced component life can be accepted.