The present work is focused only on the recuperation process and more specifically on the optimization of both the geometry of the heat exchangers and their installation within the exhaust nozzle, in order to maximize the recuperation benefits, specifically targeted for an IRA engine. More details about the IRA engine concept can also be found in [

2,

9,

14,

15]. The implementation of recuperation in an IRA engine is performed through the mounting of a number of heat exchangers inside the hot-gas exhaust nozzle, downstream of the low-pressure turbine. The basic heat exchanger (HEX) of the IRA engine, which was invented and developed by MTU Aero Engines AG and was used for the initial HEX performance studies, is presented in

Figure 1b. It consists of elliptically profiled tubes placed in a 4/3/4 staggered arrangement targeting high heat transfer rates and reduced pressure losses. Additional information of the HEX operation can be found in [

10]. The HEX tubes’ geometry and arrangement can significantly affect the turbine expansion and thus degrade the produced turbine work due to the imposed pressure losses. The overall heat exchanger design plays a critical role since the above mentioned pressure losses are linked directly to the available heat exchange surface and its geometry, which in turn strongly affects the HEX effectiveness and the exhaust gas waste heat exploitation. As a result, to achieve the maximum recuperation benefits, a compromise between the HEX design parameters is required.

Towards this direction, the development of accurate and validated numerical tools is of particular importance since they can provide time- and cost-efficient design solutions which can lead to the a priori estimation of the HEX major operational characteristics (i.e., pressure losses and effectiveness). These operational characteristics can then be integrated in a thermodynamic cycle analysis of the aero engine in order to assess the recuperation effects on the aero engine efficiency and fuel consumption. Thus, these tools can significantly contribute to the development, assessment and optimization of various innovative heat exchanger concepts which otherwise could not be affordable in laboratory (due to time and cost limitations).

#### 2.1. The Heat Exchanger–Recuperator Porosity Model Approach

The development and optimization of innovative heat exchanger concepts, focused on the Intercooled Recuperated Aero Engine, which are an evolution of the original MTU HEX design, are presented here. The investigation and the optimization reported in this work were performed with the use of 2D and 3D CFD computations, experimental measurements and thermodynamic cycle analysis, for a wide range of engine operating conditions. The optimization activities were mainly based on the use of an innovative customizable 3D numerical tool which could efficiently model the heat transfer and pressure loss performance of the heat exchangers of the IRA engine installation.

The numerical tool was based on an advanced porosity model approach in which the heat exchangers were modeled as porous media of predefined heat transfer and pressure loss behavior, which was determined by correlations specifically developed for the recuperator HEX. The use of a porous media methodology for modeling the heat exchangers provides significant advantages since it can facilitate the incorporation of the heat exchangers’ macroscopic heat transfer and pressure loss behavior in 3D CFD models of the overall aero engine installation (including the hot-gas exhaust nozzle and the precise mounting of the recuperator system inside the aero engine). In addition, the use of the numerical tool can be incorporated in CFD models, by being integrated to the fluid flow momentum and energy transport equations in the 3D CFD computations. This tool is able to provide consistent numerical solutions of the complicated flow inside the recuperator nozzle installation. Without the use of the presented numerical tool, 3D CFD computations of the precise recuperator detailed geometry could not be achieved due to the extremely high CPU and memory requirements since more than one billion computational points are estimated be necessary for the accurate representation of the overall nozzle and recuperator geometry.

This approach has been successfully applied in the past as presented in [

14,

15] and it was shown that the use of porous media methods in combination with CFD computations can lead to accurate and computationally affordable CFD models which can provide the basis for optimization of the overall geometrical configuration. In the present approach, the most important part is the accurate incorporation, through appropriate correlations, of the overall heat transfer and pressure losses in the macroscopic performance of the heat exchangers. Here, these correlations have been derived through detailed CFD computations and the use of experimental measurements, as presented in detail in [

16,

17], and were numerically incorporated in the CFD models of the overall aero engine installations by adding appropriate source terms in the momentum and energy transport equations. These correlations included the effect of both flow currents, i.e., inner-cold air and outer-hot-gas flows, on pressure losses, together with the achieved heat transfer between them.

#### 2.4. Heat Transfer Model

The inclusion of heat transfer in the modeling of the heat exchanger required the calculation of the inner and outer heat transfer coefficients. An approach based on a Nusselt–Prandtl–Reynolds number correlation was used, given by Equation (3):

where the coefficients

$C$,

$m$ and

$n$ were calibrated through detailed CFD computations and experiments separately for each of the inner and outer flow streams, while all properties were calculated at the mean flow temperatures. It must be mentioned that similar analysis was performed for various tube core geometries, corresponding to different number of tubes and staggered arrangement (e.g., heat exchanger cores of 3/2/3, 4/3/4, 5/4/5, 6/5/6 tubes staggered arrangement and for different tubes spacing have been tested, in which the tube spacing has been almost doubled in relation to the initial MTU design) and various pressure loss and heat transfer correlations were derived for each tube’s core configuration, which were exploited for the optimization of the recuperator geometrical characteristics. These analyses also provided data for the most efficient selection of the heat exchangers’ core for both CORN (COnical Recuperative Nozzle) and STARTREC (STraight AnnulaR Thermal RECuperator) concepts, in which different numbers of tubes and arrangement were used in the recuperators’ cores.

As the next step of the analysis, the pressure losses and heat transfer correlations were included in the momentum and energy Reynolds Averaged Navier–Stokes equations of the Fluent CFD software [

18] as additional source terms, as presented in

Figure 2, through specially programmed User Defined Functions (UDF).

The set of equations presented in

Figure 2 is solved for the outer flow. The thermal energy exchange of the two heat exchanger flow streams is achieved through the energy source term given by Equation (4),

where

${T}_{inner}$ refers to the temperature of the inner flow,

${S}_{exchange}$ is the heat exchange surface per unit volume of the heat exchanger (m

^{2}/m

^{3}), calculated as the ratio of the total outer surface of the heat exchanger tubes to the occupied volume of the heat exchanger, while

${U}_{overall}$ corresponds to the overall heat transfer coefficient and is calculated by Equation (5),

where

${h}_{inner}$ and

${h}_{outer}$ are the inner and outer heat transfer coefficients calculated separately for each flow current using Equation (3).

For the appropriate modeling of the inner flow (cold air flow), two additional 1D transport equations, Equations (6) and (7), were coupled with the equations presented in

Figure 2. These transport equations model the transport of the total specific enthalpy and the total pressure of the inner flow and were also implemented in the CFD computations through the use of UDF in the Fluent CFD software.

For the inner flow, the calculation of

$Re$ and

$Pr$ numbers at every computational cell and the temperature

${T}_{inner}$ must be known. The inner flow temperature,

${T}_{inner}$ is provided by the 1D transport Equation (6), via the calculation of the total specific enthalpy, given in Equation (8).

In Equation (6), the right hand side is calculated from the computations of the outer flow. This is a parameter reflecting the geometrical structure of the heat exchanger core and is directly linked to the selection of the tubes’ staggered arrangement and number.

The inner “cold” air velocity, ${u}_{inner}$, is calculated from the prescribed “inner” mass flow inside the tubes, taking into consideration the density variations due to temperature and pressure along the tubes. The inner flow total pressure losses along the tube were obtained by the numerical integration of Equation (7).

The main advantage of the presented porosity model in relation to previous porosity models used for similar setups, as presented in [

19,

20], is that both flow currents are included in the model equations and thus, the derived model can simultaneously provide both the inner and outer flow pressure losses.

In addition, the specific model can calculate the 3D distribution of the achieved heat exchange between the outer hot-gas and inner-cold air flow, something which cannot be computed through a literature-based effectiveness-NTU method. Furthermore, the direct effect of the heat exchanger core geometry, such as the shape and size of the tubes, on the achieved heat transfer can be straightforwardly computed; the actual heat exchanger effectiveness can be calculated and then be taken into consideration in the aero engine cycle thermodynamic analysis. Some additional details about the customizable numerical tool can be found in [

21].

Additionally, this innovative customizable 3D numerical tool can also incorporate major and critical heat exchanger design parameters in the CFD computations, by supporting the numerical integration of heat exchanger geometrical characteristics (e.g., tubes collector numbers, streams flow splitting and mixing, tubes core arrangement).

It should be mentioned again here that the use of the currently presented, porosity-model-based approach for the heat exchanger geometry, is almost obligatory since the inclusion of the precise heat exchanger geometry in a CFD model would result in an extremely high number of computational points which could not be computationally affordable in terms of CPU and memory (RAM) resources (more than 1 billion computational points would be required for the accurate modeling of the overall exhaust nozzle installation with the heat exchangers of the recuperators installation). On the other hand, the use of a porosity model approach where the recuperator heat exchangers are modeled as regions of predefined pressure loss and heat transfer characteristics, allows for the use of an affordable computational grid to obtain an accurate modeling of the overall exhaust nozzle configuration which can be used for further engineering analysis. Moreover, if someone attempted to model a small part of the heat exchanger core through the detailed CFD modeling of both inner and outer flows, together with the modeling of the tubes walls, and attempted to model the same part of the heat exchanger core with the use of the currently presented approach, the required time for the convergence of the detailed CFD model would be larger by a factor of 80 while both models would provide results of similar accuracy, as presented in detail in [

21].