2.1 Power saving estimation method

The power saving coefficient (PSC), first introduced by Smith^{(Reference Smith11)}, is defined as the reduction of the mechanical power $P_K$ required by the propulsion system due to the ingestion of the boundary-layer. The mechanical power $P_K$ is defined as the increase in kinetic energy through the propulsor, as shown by Equation (1) for a conventional propulsor.

(1)\begin{equation}P_{K} = \frac{1}{2} \dot{m} \left(V_{jet}^2 - V_{0}^{2} \right).\end{equation}

The analytical method is developed by applying the power balance method^{(Reference Drela7)}, on the conventional aircraft configuration illustrated in Fig. 1, and on the generic propulsive fuselage configuration represented in Fig. 2.

Figure 1. Illustration of the baseline aircraft power terms for the fuselage and propulsor.

Figure 2. Propulsive fuselage aircraft power terms.

It is noted that primed variables refer to the propulsive fuselage configuration. The implementation of the power balance method presented below has been inspired by Sato’s work on a hybrid-wing-body aircraft^{(Reference Sato8)} and Hall et al’s work on the D8 aircraft^{(Reference Hall, Huang, Uranga, Greitzer, Drela and Sato9)}.

The power balance equation^{(Reference Drela7)} establishes the equilibrium between the power produced by the engine $P_{K}$ and the different dissipation terms, corresponding to profile, vortex, jet, and wave drag losses, as shown in Equation (2).

(2)\begin{equation}P_{K}-\Phi_{jet} = \Phi = \Phi_{p} + \Phi_{vortex} + \Phi_{wave}.\end{equation}

The energy dissipation $\Phi^{\prime}_p$ caused by shear stress losses on the body surface, i.e. in the boundary-layer, and in the wake of the propulsive fuselage configuration, can be split into the internal (ingested) and external part as shown by Fig. 2 and Equation (3). Although, there is no ingestion in the baseline configuration of Fig. 1 the same break-down can be applied on $\Phi_p$; in that case the ingested part $\Phi_{p,int.}$, is the one that would be ingested if there was a propulsive fuselage.

(3)\begin{equation}\Phi^{\prime}_{p} = \Phi^{\prime}_{p,int.} + \Phi^{\prime}_{p,ext.}.\end{equation}

At this point, one can introduce the following ratios:

(4)\begin{equation}\beta = \frac{\Phi_{p,int.}}{\Phi_{p}},\end{equation}

(5)\begin{equation}\psi = \frac{\Phi^{\prime}_{p,int.}}{\Phi_{p,int.}}.\end{equation}

The $\beta$ ratio represents the amount of the ingested viscous dissipation relative to the total viscous dissipation of the baseline. The $\psi$ ratio corresponds to the dissipation occurring upstream of the propulsive fuselage propulsor, relative to the total dissipation that would occur inside the ingested stream-tube if there was no BLI. For a propulsor located at the trailing edge of the fuselage, the numerator is equal to the ingested part of the dissipation occurring on the fuselage surface $\Phi_{p,surf}$^{(Reference Sato8)}, which is assumed unchanged between the conventional and the propulsive fuselage configurations^{Footnote 1}. The denominator would be the dissipation that would occur on the surface and in the wake of the stream-tube if this was not ingested. By introducing the above assumptions and definitions into Equation (3), and by considering that the dissipation that is not ingested remains unchanged ($\Phi^{\prime}_{p,ext.}=\Phi_{p,ext.}$), one derives the following equation:

(6)\begin{equation}\Phi^{\prime}_p = \Phi_p \left[ 1-\beta \left( 1- \psi \right)\right].\end{equation}

The above equation can be translated into total dissipation terms by introducing the parameters $\lambda$ and f, which represent respectively the ratio of wave and vortex dissipation to the total profile dissipation (Equation (7)), and the ratio of ingested dissipation to the total aircraft dissipation (Equation (9)). More specifically, Equation (7) is introduced in the first part of Equation (8), in order to express the total dissipation as a function of the profile dissipation and $\lambda$. Introducing Equation (6) into the left part of Equation (8) and assuming that the wave and vortex dissipation remain unchanged between the BLI and conventional configurations, Equation (10) is obtained.

(7)\begin{equation}\lambda = \frac{\Phi_{vortex}+\Phi_{wave}}{\Phi_{p}},\end{equation}

(8)\begin{equation}\Phi = \Phi_{p} + \Phi_{vortex} + \Phi_{wave}= \Phi_{p} \left( 1+ \lambda \right),\end{equation}

(9)\begin{equation}f = \frac{\Phi_{p,int.}}{\Phi} = \frac{\beta}{1+\lambda},\end{equation}

(10)\begin{equation}\Phi^{\prime} = \Phi \left[ 1-f \left( 1- \psi \right) \right].\end{equation}

The power-saving coefficient is defined as follows:

(11)\begin{equation}\mathrm{PSC} = 1- \frac{P^{\prime}_{K}}{P_{K}},\end{equation}

where $P^{\prime}_{K}$ is the propulsive power of the BLI configuration. As implied by the above equation, a positive PSC corresponds to a reduction in the required propulsive power.

Using the power balance method the propulsive efficiency is defined as the net propulsive power (taking the total power and subtracting the jet dissipation which is related to the propulsor), divided by the total propulsive power^{(Reference Sato8,Reference Hall, Huang, Uranga, Greitzer, Drela and Sato9)} , Equation (12).

(12)\begin{equation}\eta_{pr} = \frac{P_{K}-\Phi_{jet}}{P_{K}}.\end{equation}

By introducing the propulsive efficiency definition, PSC is derived in Equation (13).

(13)\begin{equation}\mathrm{PSC} = 1- \frac{P^{\prime}_{K}-\Phi^{\prime}_{jet}}{P_{K}-\Phi_{jet}} \frac{\eta_{pr}}{\eta^{\prime}_{pr}}.\end{equation}

Combining Equation (13) with Equations (2) and (10) gives:

(14)\begin{equation}\mathrm{PSC} = 1- \left[ 1-f \left( 1- \psi \right) \right] \frac{\eta_{pr}}{\eta^{\prime}_{pr}},\end{equation}

where $\eta^{\prime}_{pr}$ is the propulsive efficiency of the BLI configuration.

The above equation clearly shows that part of the power saving comes from the propulsive efficiency improvement and part from the aircraft dissipation reduction; i.e. even without an increase in propulsive efficiency, the BLI can produce a power saving due to the reduction of the aircraft mechanical energy dissipation. The power saving increases for increasing BLI, as this increases $\beta$ and f. For a given $\beta$, the power saving increases when the wave and induced drag represent a smaller share of total drag, leading to a lower $\lambda$ and higher f. Steiner et al^{(Reference Steiner, Seitz, Wieczorek, PlÖtner, Isikveren and Hornung12)} also reached to a similar conclusion in their study. In a case without wave and induced drag ($f=\beta$), where all the boundary-layer is ingested by a propulsor located on the trailing edge, and assuming no change in propulsive efficiency, the power saving coefficient is equal to $\left( 1 - \psi \right)$; i.e. a higher wake dissipation relative to the surface dissipation, leads to a lower $\psi$, and a higher power saving. As shown by Hall^{(Reference Hall13)} in a similar analysis, the $\psi$ parameter is half the kinetic energy shape factor, $\psi = \frac{H^{*}}{2}$, which for a typical attached turbulent flow gives a value of around $\psi = 0.875$^{(Reference Drela7)}. For such a case, the above assumptions would result in a PSC of 12.5%.

Until now, no assumption was made with respect to the system architecture. In order to calculate the propulsive efficiency change, the rest of the paper will focus on the propulsive fuselage configuration shown in Fig. 2 as compared to the baseline of Fig. 1.

Writing the power balance equation (Equation (2)) for the tube-and-wing baseline gives:

(15)\begin{equation}P_{K}-\Phi_{jet} = \dot{m} V_{0} \left(V_{jet} - V_{0}\right) = \Phi.\end{equation}

The propulsive efficiency is defined as follows:

(16)\begin{equation}\eta_{pr}=\frac{P_{K}-\Phi_{jet}}{P_{K}}= \frac{2V_{0}}{V_{jet}+V_{0}}= \frac{2}{V_{jet}/V_{0}+1}.\end{equation}

By introducing the power split ratio^{Footnote 2}$q=\Phi^{\prime}_{\mathrm{BLIF}}/\Phi^{\prime}$, the power balance of the turbofans represented in Fig. 2 is written as:

(17)\begin{align}P^{\prime}_{K,\mathrm{TF}}-\Phi^{\prime}_{jet,\mathrm{TF}} & = \overbrace{\frac{1}{2}\dot{m}^{\prime}_{\mathrm{TF}} \left({V^{\prime2}_{jet,\mathrm{TF}}} - V_{0}^2\right)}^{P^{\prime}_{K,\mathrm{TF}}} -\frac{1}{2}\dot{m}^{\prime}_{\mathrm{TF}} \left(V^{\prime}_{jet,\mathrm{TF}} - V_{0}\right)^2 = \dot{m}^{\prime}_{\mathrm{TF}} V_{0} \left(V^{\prime}_{jet,\mathrm{TF}} - V_{0}\right) \nonumber \\ & = \left(1-q\right) \Phi^{\prime} = \left(1-q\right) \Phi \left[ 1-f \left( 1- \psi \right) \right].\end{align}

Combining Equations (17) and (15) gives the following expression for the turbofan jet velocity of the propulsive fuselage BLI configuration:

(18)\begin{equation}\frac{V^{\prime}_{jet,\mathrm{TF}}}{V_{0}} = \frac{\left(1-q\right)\left[ 1-f \left( 1- \psi \right) \right]}{\alpha \left(1-\alpha_{\mathrm{BLIF}}\right)}\left(\frac{V_{jet}}{V_{0}}-1\right)+1,\end{equation}

where $\alpha$ is the ratio of total propulsor mass flow for the BLI configuration relative to the total propulsor mass flow of the conventional non-BLI baseline:

(19)\begin{equation}\alpha = \frac{\dot{m}^{\prime}}{\dot{m}}= \frac{2\dot{m}^{\prime}_{\mathrm{TF}}+\dot{m}^{\prime}_{\mathrm{BLIF}}}{2\dot{m}_{\mathrm{TF}}},\end{equation}

and $\alpha_{\mathrm{BLIF}}$ is the ratio of BLI propulsor fan mass flow to total propulsor mass flow for the BLI configuration:

(20)\begin{equation}\alpha_{\mathrm{BLIF}} = \frac{\dot{m}^{\prime}_{\mathrm{BLIF}}}{\dot{m}^{\prime}}.\end{equation}

Similarly, the power balance of the BLI fan of the configuration presented in Fig. 2 is written as:

(21)\begin{align}P^{\prime}_{K,\mathrm{BLIF}}-\Phi^{\prime}_{jet,\mathrm{BLIF}} & = \overbrace{\frac{1}{2}\dot{m}^{\prime}_{\mathrm{BLIF}} \left({V^{\prime2}_{jet,\mathrm{BLIF}}} - V_{0}^2\right) + \beta\psi\Phi_{p}}^{P^{\prime}_{K,\mathrm{BLIF}}} -\frac{1}{2}\dot{m}^{\prime}_{\mathrm{BLIF}} \left(V^{\prime}_{jet,\mathrm{BLIF}} - V_{0}\right)^2 \nonumber \\[5pt] & = q \Phi \left[ 1-f \left( 1- \psi \right) \right].\end{align}

The term $\beta\psi\Phi_{p}$ corresponds to the mechanical power loss between the free-stream $V_{0}$ and the inlet of the BLI fan. The above equation is subsequently combined with Equation (15) to obtain the following expression for the propulsive fuselage BLI fan jet velocity:

(22)\begin{equation}\frac{V^{\prime}_{jet,\mathrm{BLIF}}}{V_{0}} = \frac{q\left[ 1-f \left( 1- \psi \frac{q-1}{q}\right) \right]}{\alpha \alpha_{\mathrm{BLIF}}}\left(\frac{V_{jet}}{V_{0}}-1\right)+1.\end{equation}

The propulsive efficiency of the propulsive fuselage configuration is given by the following equation:

(23)\begin{equation}\eta^{\prime}_{pr}=\frac{P^{\prime}_{K,\mathrm{TF}}-\Phi^{\prime}_{jet,\mathrm{TF}}+P^{\prime}_{K,\mathrm{BLIF}}-\Phi^{\prime}_{jet,\mathrm{BLIF}}}{P^{\prime}_{K,\mathrm{TF}}+P^{\prime}_{K,\mathrm{BLIF}}}.\end{equation}

Using the definitions of $P_K$ and $\Phi_{jet}$, in the same way as for Equations (17) and (21), the following equation is obtained for the propulsive efficiency of the propulsive fuselage configuration:

(24)\begin{equation}\eta^{\prime}_{pr}=\frac{A}{\frac{V^{\prime}_{jet,\mathrm{BLIF}}+V_{0}}{2V_{0}}qB+\frac{V^{\prime}_{jet,\mathrm{TF}}+V_{0}}{2V_{0}}\left(1-q\right)A+f\psi},\end{equation}

(25)\begin{equation}A = 1-f \left( 1- \psi \right),\end{equation}

(26)\begin{equation}B = 1-f \left( 1- \psi \frac{q-1}{q}\right).\end{equation}

Equations (24) and (16) can now be combined with Equation (14) to calculate the power saving coefficient:

(27)\begin{equation}\mathrm{PSC} = 1 - \frac{\left(\frac{V^{\prime}_{jet,\mathrm{BLIF}}}{V_{0}}+1\right)qB+\left(\frac{V^{\prime}_{jet,\mathrm{TF}}}{V_{0}}+1\right)\left(1-q\right)A+2f\psi}{\frac{V_{jet}}{V_{0}}+1}.\end{equation}

The above equation is applicable to a BLI configuration, where the BLI fan provides a part ($q \lt 1$), or all of the propulsive power ($q=1$). In this case, Equation (27) reduces to the following simpler expression:

(28)\begin{equation}\mathrm{PSC} = 1 - \frac{\left(\frac{V^{\prime}_{jet,\mathrm{BLIF}}}{V_{0}}+1\right)\left(1-f\right)+2f\psi}{\frac{V_{jet}}{V_{0}}+1}.\end{equation}

The sensitivity of PSC against the parameters q, $\psi$, $\alpha$, and $\frac{V_{jet}}{V_{0}}$ is evaluated for a variation of f in Fig. 3. The relation between PSC and f indicates that the larger the f value, the higher the potential BLI benefits. With increasing q, more thrust is produced by the more efficient BLI propulsor and as a result PSC increases. By decreasing $\psi$, less dissipation occurs ahead of the propulsor leading to an increase in PSC. According to Equations (18) and (22), a larger value of $\alpha$ results in a decrease in the jet velocities of the propulsion system, which results in a higher propulsive efficiency and a BLI gain. Finally, for a given q, a variation of $\frac{V_{jet}}{V_{0}}$ reflects a change in the propulsive efficiency of the baseline propulsors. Deteriorating the propulsive efficiency of the baseline configuration by increasing $\frac{V_{jet}}{V_{0}}$ results in a higher PSC and BLI benefit.

Figure 3. Sensitivity of PSC to different input parameters.

In Equations (27) and (28), the jet velocities $V^{\prime}_{jet,\mathrm{BLIF}}$ and $V^{\prime}_{jet,\mathrm{TF}}$ can either be chosen directly, or indirectly by defining $\alpha$ and $\alpha_{\mathrm{BLIF}}$ in Equations (18) and (22), as the propulsor mass flows and jet velocities are interrelated. The BLI fan mass flow is also related to the amount of boundary-layer ingested. The relation between the aforementioned mass flows, jet velocities and the BLI parameter $\beta$ (and hence f) is the last missing element. The determination of this relation requires the physical sizing of the BLI propulsor^{Footnote 3}, which is the subject of the following section.

2.2 Propulsor sizing for a propulsive fuselage architecture

Assuming an axi-symmetric aircraft fuselage geometry, for given BLI fan diameter $D^{\prime}_{\mathrm{BLIF}}$, hub diameter $D^{\prime}_{hub,\mathrm{BLIF}}$, and known inlet boundary-layer profile, the sizing starts with the calculation of the boundary-layer related parameters, i.e. the average BLI fan inlet velocity and the amount of ingested mechanical energy defect corresponding to the fuselage, Equations (29)–(31). The amount of viscous aircraft mechanical energy defect ingested is then calculated by Equation (32), for a given ratio of fuselage profile drag to total profile drag. For a given ratio $\lambda$, between vortex/wave and viscous dissipation, Equation (9) gives the ratio of ingested mechanical power defect over the total aircraft power dissipation.

(29)\begin{equation}h^{\prime}_{\mathrm{BLIF}} = \frac{1}{2}\left(D^{\prime}_{\mathrm{BLIF}}-D^{\prime}_{hub,\mathrm{BLIF}}\right),\end{equation}

(30)\begin{equation}V^{\prime}_{inlet,\mathrm{BLIF}} = \frac{\int_{0}^{h^{\prime}_{\mathrm{BLIF}}} \rho u^2 \left( 0.5 D^{\prime}_{hub,\mathrm{BLIF}} + y \right) \, \mathrm{d}y}{\int_{0}^{h^{\prime}_{\mathrm{BLIF}}} \rho u \left( 0.5 D^{\prime}_{hub,\mathrm{BLIF}} + y \right) \, \mathrm{d}y},\end{equation}

(31)\begin{equation}\beta_{fuselage} = \frac{\Delta \mathrm{KE}\left(h^{\prime}_{\mathrm{BLIF}}\right)}{\Delta \mathrm{KE}\left(\delta\right)} = \frac{\int_{0}^{h^{\prime}_{\mathrm{BLIF}}} \left( V_{e}^2 - u^2 \right)\rho u \left( 0.5 D^{\prime}_{hub,\mathrm{BLIF}} + y \right) \, \mathrm{d}y}{\int_{0}^{\delta} \left( V_{e}^2 - u^2 \right)\rho u \left( 0.5 D^{\prime}_{hub,\mathrm{BLIF}} + y \right) \, \mathrm{d}y},\end{equation}

(32)\begin{equation}\beta = \beta_{fuselage} \frac{C_{Dp,fuselage}}{C_{Dp}}.\end{equation}

For a given BLI nacelle mass flow ratio (MFR), Equations (33) and (34) give the BLI fan mass flow. Subsequently, for given velocity ratio $\frac{V_{jet}}{V_{0}}$ and inlet mass flow of the conventional baseline, known propulsive power split q, and considering the same velocity ratio ($\frac{V^{\prime}_{jet,\mathrm{TF}}}{V_{0}}=\frac{V_{jet}}{V_{0}}$) for the under-wing turbofans of the propulsive fuselage configuration, Equations (18)–(20), define the parameters $\alpha$, $\alpha_{\mathrm{BLIF}}$, the total mass flow $\dot{m}^{\prime}$ and the under-wing turbofan mass flow $\dot{m}^{\prime}_{\mathrm{TF}}$. The BLI fan jet velocity is then calculated using Equation (22) and the power saving coefficient using Equation (27). The sizing ends by calculating the under-wing turbofan diameter, using Equations (35) and (36).

(33)\begin{equation}A^{\prime}_{\mathrm{BLIF}} = \frac{\pi}{4} {D^{\prime}_{\mathrm{BLIF}}}^2 \left[ 1- \left(\frac{D^{\prime}_{hub,\mathrm{BLIF}}}{D^{\prime}_{\mathrm{BLIF}}}\right)^2\right],\end{equation}

(34)\begin{equation}\dot{m}^{\prime}_{\mathrm{BLIF}} = \mathrm{MFR} \cdot \rho \cdot V^{\prime}_{inlet,\mathrm{BLIF}} \cdot A^{\prime}_{\mathrm{BLIF}},\end{equation}

(35)\begin{equation}A^{\prime}_{\mathrm{TF}} = \frac{\dot{m}^{\prime}_{\mathrm{TF}}}{2 \cdot \mathrm{MFR} \cdot \rho \cdot V^{\prime}_{0}},\end{equation}

(36)\begin{equation}D^{\prime}_{\mathrm{TF}} = \sqrt{\frac{4 \cdot A^{\prime}_{\mathrm{TF}}}{\pi \cdot \left[ 1- \left(\frac{D^{\prime}_{hub,\mathrm{TF}}}{D^{\prime}_{\mathrm{TF}}}\right)^2\right]}}.\end{equation}

2.3 Fuel burn evaluation for a turbo-electric propulsive fuselage architecture

In order to estimate the BLI impact at aircraft level, the change in fuel burn needs to be calculated. Such a computation includes the change in installed propulsion system weight and drag. The whole process is presented for the partial turbo-electric propulsive fuselage architecture shown in Fig. 4, where two under-wing turbofans generate the electricity that powers the electric BLI fan.

Figure 4. Simplified turbo-electric architecture including two generators, an inverter and a motor.

The first step consists in translating the propulsion system kinetic energy into the energy generated by the two turbofan cores. This transformation includes: (1) the transmission losses from the BLI propulsor kinetic energy output $P^{\prime}_{K,\mathrm{BLIF}}$, back to the under-wing turbofan core exit $P^{\prime}_{core}$; (2) the transmission losses between the under-wing turbofan nozzles kinetic energy output $P^{\prime}_{K,\mathrm{TF}}$, and their core exit as represented in Equation (37) using the conventional definition of the transmission efficiency. It is reminded that the conventional turbofan transmission efficiency gives the ratio between the kinetic energy delivered to the exit of the core and bypass nozzles ($P^{\prime}_{K,\mathrm{TF}}$ in Fig. 4), relative to the maximum power that can be extracted from the core outlet by an ideal turbine that expands the gas to atmospheric pressure ($P^{\prime}_{core}$ in Fig. 4).

(37)\begin{equation}\eta_{tr} = \frac{P_{K,\mathrm{TF}}}{P_{core}},\end{equation}

(38)\begin{equation}P^{\prime}_{core} = \frac{P^{\prime}_{K,\mathrm{TF}}}{\eta_{tr}} + \frac{P^{\prime}_{K,\mathrm{BLIF}}}{2 \cdot \eta_{e} \cdot \eta_{\mathrm{LPT,TF}} \cdot \eta^{\prime}_{fan, \mathrm{BLIF}}}.\end{equation}

For deriving Equation (38), the turbofan fan isentropic efficiency, $\eta_{fan, \mathrm{TF}}$, and low-pressure turbine isentropic efficiency, $\eta_{\mathrm{LPT,TF}}$, were assumed constant and equal to the baseline values for the sake of simplification. The BLI fan isentropic efficiency, $\eta^{\prime}_{fan, \mathrm{BLIF}}$, can easily be related to its pressure ratio using a relation similar to the one given by Felder et al^{(Reference Felder, Kim, Brown and Kummer14)}. The BLI fan pressure ratio is calculated iteratively by matching the desired nozzle jet velocity $V^{\prime}_{jet,\mathrm{BLIF}}$ calculated by Equation (18). The baseline transmission efficiency can be approximated using Equation (39)^{(Reference Giannakakis, Laskaridis and Pilidis15)}.

(39)\begin{equation}\eta_{tr} = \frac{1+\mathrm{BPR}}{1+\mathrm{BPR}/(\eta_{fan,\mathrm{TF}} \cdot \eta_{\mathrm{LPT,TF}})}.\end{equation}

The BPR of the underwing turbofans is defined as:

(40)\begin{equation}\mathrm{BPR} = \frac{\dot{m}^{\prime}_{\mathrm{TF}}-\dot{m}^{\prime}_{core}}{\dot{m}^{\prime}_{core}}.\end{equation}

Assuming that the turbofan core characteristics do not change, we can consider that the core specific power $(\mathrm{P}_{core}/ \dot{m}_{core})$ is constant and equal to the one of the baseline configuration turbofans. Hence, the core mass flow can be calculated from the core power $P^{\prime}_{core}$ and the constant specific power as follows:

(41)\begin{equation}\dot{m}^{\prime}_{core} = \frac{P^{\prime}_{core}}{(\mathrm{P}_{core}/ \dot{m}_{core})}.\end{equation}

Introducing Equation (41) into Equation (40), gives:

(42)\begin{equation}\mathrm{BPR} = \dot{m}^{\prime}_{\mathrm{TF}} \cdot (\mathrm{P}_{core}/ \dot{m}_{core})/ P^{\prime}_{core} -1.\end{equation}

A simple iterative scheme is required for the resolution of Equations 38, 39 and 42. A first guess of the BPR allows the calculation of the underwing turbofan transmission efficiency with Equation (39), which is subsequently used to calculate the core power with Equation (38). A new BPR is then calculated with Equation (42) and the process continues until the BPR values converge.

Equation (38) highlights the main loss mechanism of this turbo-electric configuration, which counterbalances the BLI benefits; the higher the power sent to the BLI propulsor, the higher the losses due to the inefficiencies of the electric transmission chain. These additional losses tend to counteract the beneficial BLI effect of reducing the $P^{\prime}_{K,\mathrm{BLIF}}$ term. It is highlighted to the reader that for constant core characteristics a change to the core exit power is directly translated to an equal change of fuel power. For a higher accuracy, the podded fan transmission efficiency can also be multiplied by a coefficient in order to add a degradation due to scaling effects, as a function of the core mass flow $\dot{m}_{core}$.

Having determined the impact on fuel power ($\Delta \mathrm{TSFC} \approx \Delta P_{core}$), the fuel burn evaluation can be completed by estimating the change in propulsion system drag and weight. Starting with the weight component, one needs to calculate the change in the under-wing turbofans weight and the added mass of the BLI propulsor. As the purpose of this paper is not to propose a new method of weight estimation, the reader is free to use whichever method they prefer. A few interesting methods, which would be compatible with the preliminary design approach developed in this work, are proposed by Guha et al^{(Reference Guha, Boylan and Gallagher16)} and Greitzer et al^{(Reference Greitzer, Bonnefoy, DelaRosaBlanco, Dorbian, Drela, Hall, Hansman, Hileman, Liebeck, Lovegren, Mody, Pertuze, Sato, Spakovszky, Tan, Hollman, Duda, Fitzgerald, Houghton, Kerrebrock, Kiwada, Kordonowy, Parrish, Tylko and Lord17)}, and can be calibrated to better represent the designer’s target. For this work, the mass of the underwing turbofans and of the BLI fan have been calculated as linear functions ($\mathrm{M} = a \cdot D_{fan} + b$) of their respective fan diameters, based on the detailed mechanical integration studies by Giannakakis et al^{(Reference Giannakakis, Maldonado, Tantot, Frantz and Belleville6)}. With respect to the underwing turbofans, the correlation approximately gives the variation in the weight of the low-pressure components and nacelle, as the core components weight stays fairly constant.

Subsequently, the total weight of the electric transmission chain is calculated by dividing the BLI fan power (estimated as $P^{\prime}_{K,\mathrm{BLIF}} / \eta^{\prime}_{fan, \mathrm{BLIF}}$) with the specific power of the transmission ${\mathrm{SP}}_e$ given in kW/kg.

Finally, the impact of the propulsion system drag change can be estimated by calculating the change in the propulsion system wetted area, based on the previously calculated propulsor diameters and assuming fixed length-to-diameter ratios.

Table 1 Validation case inputs

The change in fuel burn at aircraft level is derived based on exchange rates given for the baseline aircraft configuration. The final fuel burn calculation is a linear combination of the fuel flow, weight and drag changes multiplied by appropriate exchange rates. It must be underlined that a different weight exchange rate has been used for the underwing and the rear installations.

(43)\begin{equation}\Delta \mathrm{FB} = \Delta \mathrm{TSFC} \cdot \mathrm{XR_{TSFC}} + \Delta \mathrm{M} \cdot \mathrm{XR_{M}} + \Delta \mathrm{S_W} \cdot \mathrm{XR_{S}}.\end{equation}