# Hybrid Propulsion Efficiency Increment through Exhaust Energy Recovery—Part 1: Radial Turbine Modelling and Design

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## Abstract

**:**

## 1. Introduction

_{2}emissions cut by 2030 [1]. The data provided by the European Environment Agency (EEA) in 2018 showed that the transport sector is responsible for about 32% of the total CO

_{2}emissions in the European Union [2]. As a consequence, actions are being taken to achieve the reduction in pollutant emissions from road, rail, aviation, and waterborne transport [3]. In this framework, the authors performed a study aimed at investigating and designing a system that allows better exploitation of the chemical energy contained in the fuel used in internal combustion engines. In fact, as widely known, internal combustion engines based on Otto or Diesel cycles do not complete the in-cylinder gas expansion down to atmospheric pressure; as a result, hot and pressurized gas exits the cylinder during the exhaust process, thus reducing by about 19% the energy transfer to the piston. Several solutions have been implemented to recover the unexpanded gas energy; the Atkinson cycle, for instance, allows a theoretical efficiency increase of 19% compared to the Otto cycle with the same compression ratio. There is a major drawback though, as the Atkinson cycle would require an engine displacement four times greater than that of the Otto cycle engine, resulting in a 72% IMEP reduction. The Miller cycle, on the other hand, is meant to provide a compromise between engine efficiency increase and power density depletion; as example, adopting a compression ratio of 14 allows for attaining a theoretical 8% efficiency increase compared to the Otto cycle, with a 25% IMEP reduction. It is also possible to conceive more complex systems that involve turbomachinery, such as turbomechanical and turboelectrical compounding [4,5,6,7,8,9,10,11], and turbocharging. Concerning turboelectrical compounding, the approach usually followed for the automotive application is to install an electrical generator on the turbocharger shaft with the aim to convert into electric power the part of the mechanical power produced by the turbine that is not employed by the turbocompressor [4,5,6,7,8,9]. In all of these cases, maximum efficiency increments within 6% have been attained. In [9], an auxiliary turbogenerator was installed downstream from the first turbine, reaching a maximum fuel economy improvement of 4%. In [10,11], the implementation of an auxiliary turbogenerator in parallel with the main turbocharger allowed for attaining efficiency improvement up to 9%. In these two papers (see also Part 2 [12]), the authors investigated a different arrangement, which is particularly suitable for hybrid propulsion architectures.

_{T,tm}(i.e., the product of the total to static isentropic efficiency η

_{t,s}and the mechanical efficiency η

_{m}), i.e., independent of the turbine speed and pressure ratio; more in detail, in their preliminary analysis, the authors considered two different levels of thermomechanical efficiency, namely 0.70 and 0.75. These values are substantially higher than the efficiency of a common turbocharging turbine on account of the more favourable working condition already mentioned, which allows a design strategy aimed at maximizing the efficiency in steady operation, and also permits the control module of the electric generator to operate the turbine at its best efficiency speed ratio, regardless of the power produced. It is also worth noting that in the preliminary evaluations performed [13,14], the efficiency of the electric motor (0.90) employed to drive the supercharger was considered in the evaluation of the power drained by the compressor, since it constitutes an ancillary device, which burdens the overall energy balance of the engine; instead, concerning the turbine, the efficiencies of the electric generator and of the battery charging were not considered coherently with the approach followed for the main thermal engine, whose power output was not reduced by generator efficiency or by battery-charging efficiency; the reason for this approach is that the engine power split (i.e., part to the generator and part to the wheels) should depend on the particular mechanical transmission adopted and was not defined in the evaluation carried out. Moreover, both the generator and the battery-charging efficiency can also be considered the same for the comparative hybrid propulsion system equipped with the traditional turbocharged engine; on account of this, the authors decided to fairly base the comparison on the overall mechanical output power produced. The aim of the work reported in these two papers is to provide a more precise and reliable estimation of the performances of the separated electric compound system by evaluating the turbine efficiency in accordance with its operating condition [12]; this should improve the reliability of the results obtained, which would not be influenced any more by the simplifying assumption used in the preliminary studies, for which the turbine efficiency was assumed to be constant. To this end, the turbine was analysed and designed by means of a simple yet effective mean-line-based approach.

_{T}

_{.}For this reason the authors considered a variable nozzle turbine, capable to allow multiple mass-flow curves through the variation in the nozzle angle [21].

_{T}and rotational speeds n (and consequent mass flow G). Successively, the design algorithm was adopted to determine the main geometric dimensions of the turbine at the design-operating condition, as presented. As a final step, the authors present the realistic performance of the separated electric compound engine evaluated by means of a 1D turbine performance prediction model applied to the design geometry.

## 2. Radial Turbine Performance Calculation Model

#### 2.1. Analysis of the Volute

_{1}can be calculated from the application of the first law of thermodynamics to a perfect gas:

_{1}is the absolute gas velocity at the volute inlet Section 1, and ${c}_{p}\left(\overline{{T}_{T1-1}}\right)$ is the specific heat at constant pressure evaluated at the average temperature $\overline{{T}_{T1-1}}$ = (T

_{T}

_{1}+ T

_{1})/2 through the use of the Shomate equation, and the coefficients available in the NIST Chemistry WebBook [22]. It is worth noting that Equation (1) implies an iterative calculation, since ${c}_{p}\left(\overline{{T}_{T1-1}}\right)$ requires the value of T

_{1}, which is the output of the equation. As a starting value, ${c}_{p}\left(\overline{{T}_{T1-1}}\right)={c}_{p}\left(\overline{{T}_{T1}}\right)$ was assumed. Analogous iterations have been carried out for similar evaluations performed in this paper.

_{T}= P

_{T}

_{1}/P

_{5}and rotor speed n, the pressure levels in the Section 1, Section 2, Section 3 and Section 4 (i.e., P

_{1}, P

_{2}, P

_{3}, and P

_{4}) are not known a priori, all the absolute velocities in the same sections are initially unknown; for this reason, each calculation procedure was started assuming an initial first attempt value of 100 m/s for each of the unknown absolute velocities (c

_{1}, c

_{2}, c

_{3}, and c

_{4}); the values of these velocities were then updated by successive iterations performed with the aim to obtain the mass-flow convergence on all the turbine sections.

_{T1}and temperature T

_{T}

_{1}) to the static condition, it was possible to calculate the static pressure at the volute inlet as:

_{p}and c

_{v}. It is worth pointing out that, in the case of the isentropic transformation, the average temperature $\overline{{T}_{T1-1,is}}$ = (T

_{T}

_{1}+ T

_{1,is})/2 should be considered; however, since the difference between T

_{1}and T

_{1,is}not relevant, it results that the difference between $k\left(\overline{{T}_{T1-1,is}}\right)$ and $k\left(\overline{{T}_{T1-1}}\right)$ is negligible (less than 0.01%), as is the difference between ${c}_{p}\left(\overline{{T}_{T1-1,is}}\right)$ and ${c}_{p}\left(\overline{{T}_{T1-1}}\right)$; the same conclusion is obtained considering the turbine nozzle and the rotor, where the enthalpy drops are higher; according to this observation the authors adopted the approximation to use the average temperature of the actual evolution in place of the average temperature of the isentropic evolution when calculating the thermochemical properties of the gas, i.e., the specific heat at constant pressure and the isentropic coefficient.

_{1}was then calculated by means of the ideal gas law as function of P

_{1}and T

_{1}as:

_{1}is:

_{1}is the inlet volute area. The absolute gas velocity c

_{1}was then corrected with the aim to reduce the difference between the mass flow at the volute inlet G

_{1}and outlet G

_{2}(whose calculation is shown below), which is considered a mass-flow error:

_{C}is the factor adopted for the absolute velocity iterative corrections; the same approach was followed for the correction of all the other absolute velocities, each one correlated to a proper mass-flow error; once all the mass-flow errors reach a negligible value (i.e., less than 0.1% of the mass-flow), the solution is considered the final and the calculation procedure is stopped. The convergence on the mass flow was reached acting on the absolute velocities (c

_{1}, c

_{2}, c

_{3}, and c

_{4}) rather than on the static pressure levels (P

_{1}, P

_{2}, P

_{3}, and P

_{4}) since the iterative correction performed on the static pressure values revealed some calculation instabilities.

_{hyd,vol}and D

_{hyd,vol}are the hydraulic length and diameter of the volute respectively, while the coefficient of friction C

_{f}is defined as:

_{2u}. This correlation is based on an analysis performed by Stanitz [24] on the flow in the vaneless diffuser of a compressor, but being an analysis based on fundamental control volume, it could be adapted to the flows in the turbine volutes. Obviously, the volute loss produces a reduction in the actual fluid velocity at the volute outlet with respect to the ideal evolution, as shown by the simple application of the first law of thermodynamics between volute inlet and outlet sections:

_{is(vol)}and ΔH

_{re(vol)}are the isentropic and the actual enthalpy drop in the volute, respectively. As already mentioned, however, the pressure level at the volute outlet P

_{2}is initially unknown, which means that the isentropic enthalpy drop is also unknown and therefore the absolute velocity c

_{2}as well, which, in turn, is required for the calculation of the volute loss. The calculation procedure is hence based on successive iterative approximation; the initial value of 100 m/s was assumed for the absolute velocity at nozzle inlet c

_{2}, which, in turn, allowed for evaluating the gas temperature at the volute outlet T

_{2}by the application of the first law of thermodynamics:

_{re(vol)}:

_{1}+ T

_{2})/2. As done for the volute inlet (see Equation (4)), the fluid density at the volute outlet section ρ

_{2}was evaluated though the application of the ideal gas law as a function of the pressure P

_{2}and the temperature T

_{2}. The mass flow through the volute outlet section G

_{2}could be hence calculated as:

_{2m}was evaluated as a function of the tangential component of the absolute velocity c

_{2u}obtained by Equation (9):

_{2}was corrected with the aim to reduce the mass-flow error between the volute outlet and nozzle outlet, i.e., (G

_{2}− G

_{3}):

_{2}is updated, then the entire calculation procedure from Equation (7) to Equation (17) is repeated until the mass-flow error (G

_{2}− G

_{3}) becomes negligible (i.e., less than 0.1% of G

_{2}).

#### 2.2. Analysis of the Nozzle

_{hyd,nozzle}and D

_{hyd,nozzle}are the hydraulic length and diameter of the nozzle, respectively, $\overline{c}=\left({c}_{2}+{c}_{3}\right)/2$ is the mean passage velocity (average value between nozzle inlet and outlet), and f is the friction factor, evaluated as:

_{2,f}and the nozzle geometric inlet angle α

_{2,g}; a simple yet effective correlation for the calculation of the nozzle incidence loss is:

_{3}is related to the isentropic enthalpy drop in the nozzle ΔH

_{is(nozzle)}, to the actual enthalpy drop ΔH

_{re(nozzle)}and to the losses Δh

_{p(nozzle)}and Δh

_{i(nozzle)}by the first law of thermodynamics applied between the inlet and outlet sections of the nozzle (i.e., Section 2 and Section 3 of Figure 3):

_{3}and the isentropic enthalpy drop are initially unknown, the absolute velocity at the nozzle outlet c

_{3}could not be evaluated through Equation (21); the calculation procedure was also based on a successive iterative approximation in this case, adopting a first attempt value of 100 m/s also for the absolute velocity c

_{3}. The static temperature at nozzle outlet T

_{3}was hence calculated as:

_{3}at nozzle outlet was obtained through the isentropic transformation:

_{is(nozzle)}was obtained from the actual enthalpy drop and the losses in the nozzle:

_{3}was evaluated by applying the ideal gas law (see Equation (4)) as a function of the temperature T

_{3}and pressure P

_{3}, thus allowing evaluation of the mass flow at the nozzle exit G

_{3}:

_{3m}is the meridional component of the gas velocity in the nozzle outlet section:

_{3}the flow section normal to c

_{3m}. As already carried out for the volute, the absolute velocity c

_{3}was corrected on the basis of the mass-flow error:

#### 2.3. Analysis of the Nozzle–Rotor Interspace

_{3}and the rotor inlet radius r

_{4}is necessary (see Figure 4) to allow the nozzle blades to rotate without causing any impact on the rotor.

_{4}is related to the isentropic enthalpy drop in the radial gap ΔH

_{is(gap)}:

_{4}, and hence the isentropic enthalpy drop in the radial gap, the absolute velocity c

_{4}could not be evaluated through Equation (30); an iterative calculation procedure was employed in this case, also adopting a first attempt value of 100 m/s for the absolute velocity c

_{4}. The static temperature at the rotor inlet T

_{4}was then calculated as:

_{4}could be instead obtained through the isentropic evolution from the nozzle exit (condition 3) to rotor inlet (condition 4):

_{4}:

_{4}is the flow passage section at the rotor inlet, while the meridional component of the absolute velocity c

_{4m}was obtained as:

_{4}at rotor inlet was hence corrected on the basis of the mass-flow error (G

_{4}− G

_{5}):

_{5}is the mass flow through the outlet section from the rotor. The calculation procedure from Equation (29) to Equation (36) was repeated until the mass-flow convergence was obtained.

#### 2.4. Analysis of the Rotor

_{in}associated with the rotor incidence loss depends on the difference between the actual fluid-dynamic angle β

_{4,f}and the optimum incidence angle β

_{4,opt}:

_{4}represents the relative velocity of the fluid at the rotor inlet; it is worth mentioning that the optimum angle β

_{4,opt}is different from the geometric angle β

_{4,g}due to the motion that the rotor induces in the flow approaching the blades. The optimum angle β

_{4,opt}can be evaluated on the basis of the optimal tangential component of relative velocity w

_{4u,opt}:

_{4u,opt}in turn can be evaluated as a function of the peripheral linear velocity at the rotor inlet u

_{4}by means of an empirical equation proposed by Stanitz [28]:

_{4,opt}can be evaluated as:

_{p}since there is currently no way to isolate and measure their effects separately. In [21], the rotor passage losses are quantified through:

_{hyd,R}and D

_{hyd,R}are the hydraulic length and diameter of the rotor, respectively, β

_{5}and b

_{5}are the geometric angle and the blade height at rotor outlet, respectively, and ch

_{rot}is the rotor blade chord, which, according to [21] can be approximated as:

_{p}is a coefficient which, as suggested in [21], should be set to 0.11 on the basis of some experimental data.

_{r}than by the axial clearance ε

_{x}, and there appears to be a cross-coupling effect between the two parameters. The authors evaluated the enthalpy variation Δh

_{cl}due to the rotor clearance loss as reported in [21]:

_{R}is the number of blades in the rotor, C

_{x}and C

_{r}are geometrical parameters, and the three coefficients K

_{x}, K

_{r}, and K

_{xr}should be set to 0.4, 0.75, and −0.3, respectively, as indicated in [21], in agreement with the previously described influences of ε

_{r}and ε

_{x}on rotor clearance loss.

_{w}can be modelled as:

_{5}is the turbine mass flow, and the coefficient K

_{f}, as described in [30], depends on the Reynolds number evaluated at the rotor inlet (Section 4 in Figure 3) and on the turbine geometry:

_{b}is the clearance between the back face of the turbine disc and its housing, and b

_{4}represents the blade height at rotor inlet.

_{5,rel}(evaluated on the basis of the relative velocity at the rotor exit w

_{5}) approaches 1. The model here adopted by the authors calculates the enthalpy variation Δh

_{t}related to the trailing-edge loss as:

_{5}and T

_{5}are the static pressure and temperature at the rotor exit (Section 5), M

_{5,rel}is the previously mentioned relative Mach number in Section 5, k and C

_{p}are the isentropic coefficient and the specific heat, respectively (both evaluated at the temperature T

_{5}), and ΔP

_{rel}is the pressure drop caused by the sudden expansion, which, according to the model adopted [31], is assumed to be proportional to the relative kinetic energy at the rotor exit:

_{5s}and r

_{5h}are the shroud and hub radii at rotor exit, respectively (see Figure 3), N

_{R}the number of blades in the rotor, and t the blade thickness. In the calculation performed on the rotor, the outlet static pressure P

_{5}is known, being part of the boundary conditions adopted (as resumed in Part 2 [12]); the isentropic enthalpy drop in the rotor can be hence calculated as:

_{re(rot)}can be obtained:

_{rot}is the sum of all the losses in the rotor (=Δh

_{in}+ Δh

_{p}+ Δh

_{cl}+ Δh

_{w}+ Δh

_{t}) discussed from Equation (37) to Equation (49). The static gas temperature T

_{5}at the rotor outlet is hence:

_{5}obtained from the ideal gas law:

_{5}through the application of the first law of thermodynamics between Section 4 and Section 5, in the relative reference system of the rotor:

_{4}is the relative velocity at the rotor inlet, u

_{4}and u

_{5}are the peripheral linear velocities at the rotor inlet and outlet, respectively. Once calculated, the value of w

_{5}is updated in Equations (48) and (49), thus allowing an iterative solution. The meridional component of the velocity w

_{5}is:

_{5,g}is the geometric blade angle at rotor outlet (see Figure 5). The mass-flow rate can be hence calculated:

_{4}. Since all the losses were expressed in terms of enthalpy variations, it is possible to evaluate the total-to-static isentropic efficiency of the stage as follows:

_{id}is calculated considering an isentropic expansion from the inlet conditions (P

_{T}

_{1}, T

_{T}

_{1}) to the rotor exit static pressure P

_{5}:

#### 2.5. Mechanical Friction Losses

_{in,t}denotes the inner radius, determined by shaft diameter D; r

_{ext,t}is the outer radius calculated by making reference to the r

_{ext,t}/ r

_{in,t}ratios used in similar cases; and ε

_{th}is the axial clearance, for which a value of 0.095 mm was assigned according to [32]. Petroff’s equation reveals that the power dissipation is highly dependent on geometry, as for the journal bearing. Larger bearing contact surfaces result in higher power losses.

## 3. Design of the Radial Inflow Turbine

_{T1}, outlet static pressure P

_{5}, total inlet temperature T

_{T1}, mass-flow rate G, and rotational speed of the turbine rotor n.

#### 3.1. Rotor Design

_{T}

_{1}to the final exhaust pressure P

_{5}:

_{s}:

_{s}is initially unknown, in the design procedure the first attempt value of 0.5 was assumed, which allows for determining u

_{4}and, in turn, the radius of the rotor inlet section.

_{5u}= 0. As a result, the tangential component c

_{4u}of the absolute velocity at rotor inlet could be computed by the application of the Euler equation, once the stage total-to-static efficiency η

_{t,s}is known:

_{t,s}is unknown (Equation (57)); for this reason, in the design procedure, a first attempt value was obtained by the empirical formula proposed by Aungier [19]:

_{s}is:

_{5}is the ratio between the mass flow G and the rotor exit density ρ

_{5}:

_{5}and of the rotor outlet temperature T

_{5}, the latter being computed through the application of the first law of thermodynamics from Section 1 to Section 5 on the basis of the actual enthalpy drop ΔH

_{r}:

_{4g}was considered to be 90° in this application. Concerning instead the inlet absolute flow angle α

_{4}, in the best efficiency condition, it is substantially a function of the specific speed n

_{s}, as shown in [34]. The data reported in [34] have been employed to obtain a numerical correlation [19] used to obtain a first attempt value of the inlet absolute flow angle α

_{4}, which is successively updated on the basis of the results:

_{T}

_{4}can be estimated as:

_{4}is hence:

_{4}and the temperature T

_{4}at the rotor inlet may be evaluated as:

_{4}can be then calculated through the application of the ideal gas law, employing the pressure and temperature evaluated in Equations (76) and (78), respectively; the rotor inlet blade height can be hence obtained by the mass-flow rate equation:

_{5u}= 0) in the nominal design condition, thus minimizing the exit kinetic energy losses at the rotor outlet; as a consequence, the absolute velocity at rotor outlet is equal to the meridional component, i.e., c

_{5}

_{=}c

_{5m}. The section area at the rotor outlet is then evaluated:

_{5}(imposed as boundary condition) and the temperature calculated in Equation (70). The exit shroud radius r

_{s5}is therefore:

_{5g}at the rotor outlet (which in the design condition is equal to the flow angle β

_{5f}) is determined:

_{R}:

_{t,s}obtained from the turbine performance calculation model described in Equation (57) from the previous section. As already mentioned above, the value of ν

_{s}of the first iteration was imposed equal to 0.5, but successively optimized to obtain the maximum efficiency of the stage.

#### 3.2. Nozzle Design

_{3}represents the radius at the nozzle exit. As already mentioned and shown in Figure 4, this radius depends on the position of the distributor blades, i.e., on the angle α

_{3}: when the distributor blades are closed (i.e., for the minimum mass flow rates), the nozzle outlet flow angle α

_{3}is at its minimum value and hence the radial gap Δr reaches the maximum amplitude. When the mass-flow rates are high, instead, the distributor blades are in a “fully open” position, i.e., with the maximum angle α

_{3}, to which corresponds the minimum radial gap Δr. Since in this section the authors are giving the guidelines to define the turbine geometry for a particular design operating condition, the radius at the nozzle exit r

_{3}is given by:

_{D}denotes the radial gap value at the design operating condition, which can be roughly obtained as:

_{3}, which corresponds to a null radial gap; if it results that in this condition the turbine is unable to swallow the required maximum mass-flow rates, then the design must be corrected with a higher value for Δr

_{D}.

_{3}was evaluated assuming the absolute flow angle at nozzle outlet α

_{3}equal to the rotor inlet absolute flow angle α

_{4}(i.e., considering a null radial gap):

_{3}and the temperature of the gas T

_{3}at the nozzle outlet are computed as:

_{2f}= α

_{2g}); moreover, a constant density flow between Section 2 and Section 3 is also assumed, which allows for calculating the meridional component of the absolute velocity at the distributor inlet:

_{2}/r

_{3}lies between 1.1 and 1.7. In this work, the authors adopted a value of 1.3 to obtain a good compromise between performance and overall size of the nozzle. Assuming the symmetrical nozzle blade profile of Figure 6, simple geometric considerations based on the triangle generated by the nozzle geometry (shown in Figure 7), allowed for calculating the nozzle inlet blade angle α

_{2,g}as a function of the exit blade angle α

_{3,g}:

_{nozzle}:

_{2}, the tangential component of absolute velocity at the nozzle inlet, can be calculated as:

_{2m}obtained from Equation (99), an iterative correction process based on the conservation of mass-flow rate (G

_{2}= G

_{3}) is necessary to obtain a more accurate value for c

_{2m}. For this purpose, the static temperature and pressure at nozzle inlet are evaluated as:

_{2}of the working fluid at the nozzle inlet can hence be obtained from the ideal gas law as a function of the temperature T

_{2}and pressure P

_{2}. At this point the meridional component of the absolute velocity at the distributor inlet is given by:

#### 3.3. Volute Design

_{1}of the volute inlet section from the turbine axis is unknown in the initial phase of the design process, at first attempt it is assumed equal to r

_{2}. This assumption allows for evaluation of the static temperature at the volute inlet:

_{1}and T

_{1}. The area of the volute inlet section can hence be estimated:

_{x}is then:

_{1}is:

_{x}is calculated on the basis of the volute aspect ratio A

_{R}:

_{R}= 1, in accordance with the values of 0.75 ≤ A

_{R}≤ 1.5 recommended in [19]. Once the volute section aspect ratio is established, then the calculation loop from Equation (107) to Equation (113) is iteratively repeated until the value of the radius r

_{1}reaches convergence. Therefore, after the volute design is completed, it is possible to estimate the total axial length of the stage as:

#### 3.4. Design Calculation Procedure

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Symbols and Abbreviations

**Symbols**

A | Area | m^{2} |

AR | Aspect ratio | - |

A_{x} | Axial semi-axis | m |

b | Blade width | m |

B_{x} | Radial semi-axis | m |

c | Absolute velocity | m/s |

c_{0s} | Spouting velocity | m/s |

ch | Chord | m |

c_{p} | Constant pressure specific heat | $\left[\frac{J}{kg\xb7K}\right]$ |

c_{v} | Constant volume specific heats | $\left[\frac{J}{kg\xb7K}\right]$ |

D_{hyd} | Hydraulic diameter | m |

EVO | Exhaust valve open | |

G | Mass flow rate | kg/s |

H | Specific enthalpy | J/kg |

IMEP | Indicated mean effective pressure | bar |

IVC | Inlet valve closure | |

k | Specific heat ratio | - |

L_{hyd} | Hydraulic length | m |

l_{r} | Rotor axial length | m |

M | Mach number | - |

MFP | Mass flow parameter | - |

n | Rotational speed | rpm |

N_{R} | Number of rotor blades | - |

P | Pressure | Pa |

POW | Power | W |

r | Radius | M |

Re | Reynolds number | - |

RR | Wall relative roughness | M |

t | Trailing edge blade thickness | M |

T | Temperature | K |

u | Peripheral velocity | m/s |

w | Relative velocity | m/s |

**Greek letters**

α | Absolute flow angle | - |

β | Relative flow angle | - |

β_{T} | Pressure ratio | - |

η | Efficiency | - |

ρ | Density | kg/m^{3} |

ω | Rotational speed | rad/s |

Δh | Enthalpy losses | J/kg |

ΔH | Enthalpy drop | J/kg |

ε | Clearance gap | m |

ν | Kinematic viscosity | m^{2}/s |

ν_{s} | Velocity ratio | - |

**Subscripts**

1 | Total |

2 | Nozzle inlet |

3 | Nozzle exit |

4 | Rotor inlet |

5 | Rotor exit |

a | Axial |

b | Blade |

cl | Clearance |

f | Fluid-dynamic |

h | Hub |

id | Ideal |

in | Incidence |

is | Isentropic |

J,B | Journal bearing |

m | Meridional |

nozzle | Nozzle |

p | Passage |

r | Radial |

re | Real |

rel | Relative |

rot | Rotor |

s | Shroud |

t | Trailing edge |

T | Total |

tm | Thermomechanical |

T,B | Thrust bearings |

t,s | Total-to-static |

t,t | Total-to-total |

u | Peripheral |

vol | Volute |

w | Windage |

## References

- Eurostat, Energy, Transport and Environment Statistics-2020 Edition. Available online: https://ec.europa.eu/eurostat/web/products-statistical-books/-/ks-dk-20-001 (accessed on 14 November 2022). [CrossRef]
- Monitoring CO
_{2}Emissions from Passenger Cars and Vans in 2018; European Environment Agency: Copenhagen, Denmark, 2020. [CrossRef] - European Commission, Climate Action, EU Action, Transport Emissions, Road Transport: Reducing CO
_{2}Emissions from vehicles, CO_{2}Emission Performance Standards for Cars and Vans. Available online: https://ec.europa.eu/clima/eu-action/transport-emissions/road-transport-reducing-co2-emissions-vehicles/co2-emission-performance-standards-cars-and-vans_en (accessed on 14 November 2022). - Pasini, G.; Lutzemberger, G.; Frigo, S.; Marelli, S.; Ceraolo, M.; Gentili, R.; Capobianco, M. Evaluation of an electric turbo compound system for SI engines: A numerical approach. Appl. Energy
**2016**, 162, 527–540. [Google Scholar] [CrossRef] - Arsie, I.; Cricchio, A.; Pianese, C.; Ricciardi, V.; De Cesare, M. Evaluation of CO
_{2}reduction in SI engines with Electric Tur-bo-Compound by dynamic powertrain modelling. IFAC-PapersOnLine**2015**, 28, 93–100. [Google Scholar] [CrossRef] - Millo, F.; Mallamo, F.; Pautasso, E.; Ganio Mego, G. The Potential of Electric Exhaust Gas Turbocharging for HD Diesel Engines; SAE Technical Papers; SAE International: Warrendale, PA, USA, 2006. [Google Scholar] [CrossRef]
- Hopmann, U.; Algrain, M. Diesel Engine Electric Turbo Compound Technology; SAE Technical Paper; SAE International: Warrendale, PA, USA, 2003. [Google Scholar] [CrossRef]
- Mohd Noor, A.; Che Puteh, R.; Rajoo, S.; Basheer, U.M.; Md Sah, M.H.; Shaikh Salleh, S.H. Simulation Study on Electric Turbo-Compound (ETC) for Thermal Energy Recovery in Turbocharged Internal Combustion Engine. Appl. Me-Chanics Mater.
**2015**, 799–800, 895–901. [Google Scholar] [CrossRef] - Kant, M.; Romagnoli, A.; Mamat, A.M.; Martinez-Botas, R.F. Heavy-duty engine electric turbocompounding. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2015**, 229, 457–472. [Google Scholar] [CrossRef] - Cipollone, R.; Di Battista, D.; Gualtieri, A. Turbo compound systems to recover energy in ICE. Int. J. Engi-Neering Innov. Technol. (IJEIT)
**2013**, 3, 249–257. Available online: https://www.ijeit.com (accessed on 14 November 2022). - Zhuge, W.; Huang, L.; Wei, W.; Zhang, Y.; He, Y. Optimization of an Electric Turbo Compounding System for Gasoline Engine Exhaust Energy Recovery; SAE Technical Paper; SAE International: Warrendale, PA, USA, 2011. [Google Scholar] [CrossRef]
- Pipitone, E.; Caltabellotta, S.; Sferlazza, A.; Cirrincione, M. Hybrid propulsion efficiency increment through exhaust energy recovery—Part 2: Numerical simulation results. (under review).
- Pipitone, E.; Caltabellotta, S. Efficiency Advantages of the Separated Electric Compound Propulsion System for CNG Hybrid Vehicles. Energies
**2021**, 14, 8481. [Google Scholar] [CrossRef] - Pipitone, E.; Caltabellotta, S. The Potential of a Separated Electric Compound Spark-Ignition Engine for Hybrid Vehicle Application. J. Eng. Gas Turbines Power
**2022**, 144, 041016. [Google Scholar] [CrossRef] - Michon, M.; Calverley, S.; Clark, R.; Howe, D.; Chambers, J.; Sykes, P.; Dickinson, P.; Clelland, M.; Johnstone, G.; Quinn, R.; et al. Modelling and Testing of a Turbo-generator System for Exhaust Gas Energy Recovery. In Proceedings of the 2007 IEEE Vehicle Power and Propulsion Conference, Arlington, TX, USA, 9–12 September 2007; pp. 544–550. [Google Scholar] [CrossRef]
- Nonthakarn, P.; Ekpanyapong, M.; Nontakaew, U.; Bohez, E. Design and Optimization of an Integrated Turbo-Generator and Thermoelectric Generator for Vehicle Exhaust Electrical Energy Recovery. Energies
**2019**, 12, 3134. [Google Scholar] [CrossRef] [Green Version] - Haughton, A.; Dickinson, A. Development of an Exhaust Driven Turbine-Generator Integrated Gas Energy Recovery System (TIGERS®); SAE Technical Paper; SAE International: Warrendale, PA, USA, 2014. [Google Scholar] [CrossRef]
- Rahbar, K. Development and Optimization of Small-Scale Radial Inflow Turbine for Waste Heat Recovery with Organic Rankine Cycle. Ph.D. Thesis, School of Mechanical Engineering, University of Birmingham, Birmingham, UK, 2016. [Google Scholar]
- Aungier, R.H. Turbine Aerodynamics: Axial-Flow and Radial-Inflow Turbine Design and Analysis; ASME Press: New York, NY, USA, 2005; ISBN 0791802418. [Google Scholar]
- Wei, Z. Meanline Analysis of Radial Inflow Turbines at Design and Off-Design Conditions. Master’s Thesis, Carleton University, Ottawa, ON, Canada, 2014. [Google Scholar]
- Moustapha, H.; Zelesky, M.F.; Baines, N.C.; Japikse, D. Chapter 8. In Axial and Radial Turbines; Concepts ETI, Inc.: Plano, TX, USA, 2003; ISBN 0933283121. [Google Scholar]
- NIST Chemistry WebBook. Available online: https://webbook.nist.gov/chemistry/ (accessed on 16 July 2022).
- Kastner, L.J.; Bhinder, F.S. A Method for Predicting the Performance of a Centripetal Gas Turbine Fitted with a Nozzle-Less Volute Casing; ASME, United Engineering Center: New York, NY, USA, 1975. [Google Scholar]
- Stanitz, J.D. One-Dimensional Compressible Flow in Vaneless Diffusers of Radial- and Mixed-Flow Centrifugal Compressors. Including Effects of Friction, Heat Transfer and Area Change; National Advisory Committee for Aeronautics; Lewis Flight Propulsion Laboratory: Cleveland, OH, USA, 1952. [Google Scholar]
- Whitfield, A.; Baines, N.C. Design of Radial Turbomachines; Longman Scientific & Technical: Harlow, UK; Wiley: New York, NY, USA, 1990; ISBN 0582495016. [Google Scholar]
- Suhrmann; Peitsch, D.; Gugau, M.; Heuer, T.; Tomm, U. Validation and development of loss models for small size radial turbines. In Proceedings of the ASME Turbo Expo: Power for Land, Sea and Air GT2010, Glasgow, UK, 14–18 June 2010; 2010. [Google Scholar] [CrossRef]
- Wasserbauer, C.A.; Glassman, A.J. Fortran Program for Predicting Off-Design Performance of Radial-Inflow Turbines; Lewis Research Center, National Aeronautics and Space Administration: Washington, DC, USA, 1975. [Google Scholar]
- Stanitz, D. Some Theoretical Aerodynamic Investigations of Impellers in Radial and Mixed-Flow Centrifugal Compressors; ASME Paper; United Engineering Center: New York, NY, USA, 1952; Volume 74, pp. 473–497. [Google Scholar]
- Meroni, A.; Robertson, M.; Martinez-Botas, R.; Haglind, F. A methodology for the preliminary design and performance prediction of high-pressure ratio radial-inflow turbines. Energy
**2018**, 164, 1062–1078. [Google Scholar] [CrossRef] - Daily, J.W.; Nece, R.E. Chamber Dimension Effects on Induced Flow and Frictional Resistance of Enclosed Rotating Disks. ASME. J. Basic Eng.
**1960**, 82, 217–230. [Google Scholar] [CrossRef] - Ghosh, S.K.; Sahoo, R.K.; Sarangi, S.K. Mathematical Analysis for Off-Design Performance of Cryogenic Turboexpander. ASME J. Fluids Eng.
**2011**, 133, 031001. [Google Scholar] [CrossRef] - Sjöberg, E. Friction Characterization of Turbocharger Bearings. Master’s Thesis, KTH Industrial Engineering and Management, Stockholm, Sweden, 2013. MMK 2013:06 MFM 149. [Google Scholar]
- Shigley’s Mechanical Engineering Design, Tenth Edition in SI Units; McGraw-Hill: New York, NY, USA, 2015; ISBN 9780073398204.
- Rohlik, H.E. Analytical Determination of Radial Inflow Turbine Design Geometry for Maximum Efficiency, Technical Note TN D-4384; NASA: Washington, DC, USA, 1968. [Google Scholar]
- Ventura, C.; Peter, J.; Rowlands, A.; Petrie-Repar, P.; Sauret, E. Preliminary Design and Performance Estimation of Radial Inflow Turbines: An Automated Approach. J. Fluids Eng. Trans. ASME
**2012**, 134, 031102. [Google Scholar] [CrossRef] - Glassman, A.J. Computer Program for Design Analysis of Radial-Inflow Turbines; Lewis Research Center: Cleveland, OH, USA, 1976. [Google Scholar]

**Figure 2.**Operating points of the exhaust gas turbine for the separated electric compound engine studied in [4] (mass flow parameter as a function of the turbine pressure ratio β

_{T}).

**Figure 3.**Schematic representation of turbine geometry, the numbers refer to the following main flow sections: (1) volute inlet; (2) nozzle inlet; (3) nozzle outlet; (4) rotor inlet; (5) rotor exit.

**Figure 4.**Radial gap between nozzle outlet section (radius r

_{3}) and rotor inlet section (radius r

_{4}).

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**MDPI and ACS Style**

Pipitone, E.; Caltabellotta, S.; Sferlazza, A.; Cirrincione, M.
Hybrid Propulsion Efficiency Increment through Exhaust Energy Recovery—Part 1: Radial Turbine Modelling and Design. *Energies* **2023**, *16*, 1030.
https://doi.org/10.3390/en16031030

**AMA Style**

Pipitone E, Caltabellotta S, Sferlazza A, Cirrincione M.
Hybrid Propulsion Efficiency Increment through Exhaust Energy Recovery—Part 1: Radial Turbine Modelling and Design. *Energies*. 2023; 16(3):1030.
https://doi.org/10.3390/en16031030

**Chicago/Turabian Style**

Pipitone, Emiliano, Salvatore Caltabellotta, Antonino Sferlazza, and Maurizio Cirrincione.
2023. "Hybrid Propulsion Efficiency Increment through Exhaust Energy Recovery—Part 1: Radial Turbine Modelling and Design" *Energies* 16, no. 3: 1030.
https://doi.org/10.3390/en16031030