where $F_{D,1}$ is the drag force in climbing. However, the drag force is different in climbing and in cruising, and the aircraft does not necessarily fly at its optimum lift-over-drag ratio. As a result, the drag force must be evaluated. The overall drag of the aircraft is as follows:

$$F_{D,1} = \frac{1}{2} \cdot \rho_{\infty} \cdot S_{\text{ref}} \cdot C_{\text{D}} \cdot v_{1}^{2}, \qquad (\text{A10})$$

where $S_{\text{ref}}$ is the reference area of the aircraft, $C_{\text{D}}$ is the drag coefficient of the aircraft, and the density, $\rho_{\infty}$, can be obtained with the International Standard Atmosphere (IISA) model. In the ISA model, in the troposphere (below 11 km of altitude), the temperature decreases by $6.5 \times 10^{-3} \text{ K} \cdot \text{m}^{-1}$ of altitude from 288.16 K on the ground. Consequently, at an altitude, $h$, in meters, the temperature, $\Psi_{\infty}$, in K and density, $\rho_{\infty}$, in kg·m$^{-3}$ are as follows [27]:

$$\Psi_{\infty} = 288.16 - 6.5 \times 10^{-3} h, \qquad (\text{A11})$$

and

$$\rho_{\infty} = 1.225 \left( \frac{\Psi_{\infty}}{288.16} \right)^{4.2579}, \qquad (\text{A12})$$

For instance, an average density, $\bar{\rho}_{\infty}$, can be chosen between the ground level, $h = 0$, and the cruise altitude, $h = h_2$:

$$\bar{\rho}_{\infty} = \frac{1}{h_2} \left( \int_{0}^{h_2} \rho_{\infty} \, dh \right). \qquad (\text{A13})$$

If the velocity in climbing is known, the drag coefficient still needs to be evaluated. When using the drag polar, the following applies:

$$C_{\text{D}} = C_{\text{D},0} + \frac{C_{\text{L}}^2}{\pi \cdot e \cdot AR}, \qquad (\text{A14})$$

where $C_{\text{L}}$ is the lift coefficient, $AR$ the aspect ratio, $e$ the span efficiency coefficient (that is, the Oswald coefficient), and $C_{\text{D},0}$ the zero-lift drag coefficient. It can be noted that using Equation (A14) for cruising with a known lift-over-drag ratio and Oswald efficiency coefficient leads to the determination of the zero-lift drag coefficient of the aircraft. The lift force in climbing, $F_{L,1}$, is determined as follows [27]:

$$F_{L,1} = F_{W} \cdot \cos(\beta). \qquad (\text{A15})$$

Moreover, the standard formula for the lift is the following:

$$F_{L,1} = \frac{1}{2} \cdot \rho_{\infty} \cdot S_{\text{ref}} \cdot C_{\text{L}} \cdot v_{1}^{2}. \qquad (\text{A16})$$

As a result, a known climb angle, $\beta$, allows the lift force to be derived with Equation (A15). Then, the lift coefficient is obtained with Equation (A16), which results in the determination of the drag coefficient using Equation (A14). Finally, Equation (A10) and then Equation (A9) lead to the total thrust required for the climb. For such calculations, it is possible to solve the equations for $\beta$ below a certain angle (e.g., 45 degrees) for a flight velocity between the take-off and cruising velocities. Conditions and optimization can be determined to select the best climb angle.

Finally, the distance covered by aircraft over the ground, $s_1$, and the time taken to perform a climb, $\tau_1$, are related as follows:

$$s_1 = \tau_1 \cdot v_1 \cdot \cos(\beta) = h_1 \cdot \cotan(\beta). \qquad (\text{A17})$$

## Appendix B.4. Take-Off Performance

As explained previously, only small UAVs were considered in the current estimations. Thus, they could be catapulted instead of taking off autonomously. However, it is still
where $F_{D,1}$ is the drag force in climbing. However, the drag force is different in climbing and in cruising, and the aircraft does not necessarily fly at its optimum lift-over-drag ratio. As a result, the drag force must be evaluated. The overall drag of the aircraft is as follows:

$$F_{D,1} = \frac{1}{2} \cdot \rho_{\infty} \cdot S_{\text{ref}} \cdot C_{D} \cdot v_{1}^{2}, \qquad (A10)$$

where $S_{\text{ref}}$ is the reference area of the aircraft, $C_D$ is the drag coefficient of the aircraft, and the density, $\rho_{\infty}$, can be obtained with the International Standard Atmosphere (IISA) model. In the ISA model, in the troposphere (below 11 km of altitude), the temperature decreases by $6.5 \times 10^{-3} \text{ K} \cdot \text{m}^{-1}$ of altitude from 288.16 K on the ground. Consequently, at an altitude, $h$, in meters, the temperature, $\Psi_{\infty}$, in K and density, $\rho_{\infty}$, in kg·m$^{-3}$ are as follows [27]:

$$\Psi_{\infty} = 288.16 - 6.5 \times 10^{-3} h, \qquad (A11)$$

and

$$\rho_{\infty} = 1.225 \left( \frac{\Psi_{\infty}}{288.16} \right)^{4.2579}, \qquad (A12)$$

For instance, an average density, $\bar{\rho}_{\infty}$, can be chosen between the ground level, $h = 0$, and the cruise altitude, $h = h_2$:

$$\bar{\rho}_{\infty} = \frac{1}{h_2} \left( \int_{0}^{h_2} \rho_{\infty} \, dh \right). \qquad (A13)$$

If the velocity in climbing is known, the drag coefficient still needs to be evaluated. When using the drag polar, the following applies:

$$C_D = C_{D,0} + \frac{C_L^2}{\pi \cdot e \cdot AR}, \qquad (A14)$$

where $C_L$ is the lift coefficient, $AR$ the aspect ratio, $e$ the span efficiency coefficient (that is, the Oswald coefficient), and $C_{D,0}$ the zero-lift drag coefficient. It can be noted that using Equation (A14) for cruising with a known lift-over-drag ratio and Oswald efficiency coefficient leads to the determination of the zero-lift drag coefficient of the aircraft. The lift force in climbing, $F_{L,1}$, is determined as follows [27]:

$$F_{L,1} = F_W \cdot \cos(\beta). \qquad (A15)$$

Moreover, the standard formula for the lift is the following:

$$F_{L,1} = \frac{1}{2} \cdot \rho_{\infty} \cdot S_{\text{ref}} \cdot C_L \cdot v_1^2. \qquad (A16)$$

As a result, a known climb angle, $\beta$, allows the lift force to be derived with Equation (A15). Then, the lift coefficient is obtained with Equation (A16), which results in the determination of the drag coefficient using Equation (A14). Finally, Equation (A10) and then Equation (A9) lead to the total thrust required for the climb. For such calculations, it is possible to solve the equations for $\beta$ below a certain angle (e.g., 45 degrees) for a flight velocity between the take-off and cruising velocities. Conditions and optimization can be determined to select the best climb angle.

Finally, the distance covered by aircraft over the ground, $s_1$, and the time taken to perform a climb, $\tau_1$, are related as follows:

$$s_1 = \tau_1 \cdot v_1 \cdot \cos(\beta) = h_1 \cdot \cotan(\beta). \qquad (A17)$$

## Appendix B.4. Take-Off Performance

As explained previously, only small UAVs were considered in the current estimations. Thus, they could be catapulted instead of taking off autonomously. However, it is still
possible to estimate the distance required for lifting off, s₀. The take-off distance is here approximated with the lift-off distance, i.e., the distance covered on the runway by the aircraft for the lift force to equal the weight.

For an accurate assessment of the lift-off distance, s₀, many parameters should be considered, such as the friction between the aircraft and the runway or the altitude of the runway. Nevertheless, assuming that the thrust greatly exceeds the drag and the friction, and that the lift-off velocity, v₀, is 120% of the stall velocity, vst, s₀ is as follows [27]:

$$s_0 = \frac{1.44 F_W^2}{g \cdot \rho_\infty \cdot S_{\text{ref}} \cdot C_{L,\text{max}} \cdot F_{T,0}}, \quad (A18)$$

where F₀ is the lift-off thrust, and C₀ is the maximum lift coefficient of the aircraft. Assuming F₀ = F₀ and that the lift-off is performed at altitude h = 0, Equation (A18) can be simplified as follows:

$$s_0 \approx \frac{11.5 m_0^2}{S_{\text{ref}} \cdot C_{L,\text{max}} \cdot F_{T,0}}. \quad (A19)$$

The maximum lift coefficient can be found by solving a steady flight at stall velocity:

$$F_{L,\text{st.}} = F_W = \frac{1}{2} \cdot \rho_\infty \cdot S_{\text{ref}} \cdot C_{L,\text{max}} \cdot v_{\text{st.}}^2. \quad (A20)$$

Finally, the lift-off time, τ₀, can be determined by dividing the lift-off distance, s₀, by the average take-off velocity. However, the speed is not constant during take-off. It can be approximated that the average speed on the runway is around 70% of the lift-off speed, v₀ [27]. Then the following applies:

$$\tau_0 = \frac{s_0}{0.7 v_0}. \quad (A21)$$

The previous equations were used in the algorithm schematized in Figure 10.

## Appendix C. Process of the Algorithm

This section details the process of the algorithm represented schematically in Figure 10, which was used for the derivation of the results presented in Section 5.

### Appendix C.1. Input Data

Three categories of input data can be determined. They can be seen in the blue frame on top of Figure 10. First, the design and performance of the aircraft must be selected (see Appendix B.1). This accounts for general design parameters such as the mass, the wingspan, and the aspect ratio, but it also accounts for general aircraft performance characteristics such as the cruise lift-over-drag ratio, the Oswald coefficient, and the energy density of the battery. Additionally, some mission parameters must be given. These include the cruise endurance, speed and height, the maximal allowed take-off distance, and the stall velocity.

Secondly, the geometric and aerodynamic characteristics of the single-stage thruster must be provided. For a given configuration (single or dual collector), they concern the geometry of the electrodes and thrusters (see Figure 6) with the thickness and chord of the electrodes and the distances between the electrodes (the inter-electrode gap and inter-collector separation). Moreover, the aerodynamic coefficients of the electrodes need to be set. For example, using the cruise velocity brought with the aircraft inputs, and when the chord of an electrode is known, the Reynolds number can be calculated. Then, simple tools such as airfoil databases or programs such as XFoil can be used to derive the coefficients. Because stacking stages in series strengthens the interference between successive stages, it can be preferred to favor an increase in the number of stages in parallel. However, increasing the number of stages in parallel increases the total height of the thruster (see Equation (25)). Thus, a maximal suitable height of the thruster must be defined.
possible to estimate the distance required for lifting off, $s_0$. The take-off distance is here approximated with the lift-off distance, i.e., the distance covered on the runway by the aircraft for the lift force to equal the weight.

For an accurate assessment of the lift-off distance, $s_0$, many parameters should be considered, such as the friction between the aircraft and the runway or the altitude of the runway. Nevertheless, assuming that the thrust greatly exceeds the drag and the friction, and that the lift-off velocity, $v_0$, is 120% of the stall velocity, $v_{st.}$, $s_0$ is as follows [27]:

$$s_0 = \frac{1.44 F_W^2}{g \cdot \rho_{\infty} \cdot S_{\text{ref}} \cdot C_{L,\text{max}} \cdot F_{T,0}}, \quad (A18)$$

where $F_{T,0}$ is the lift-off thrust, and $C_{L,max}$ is the maximum lift coefficient of the aircraft. Assuming $F_{T,0} = F_{T,1}$ and that the lift-off is performed at altitude $h = 0$, Equation (A18) can be simplified as follows:

$$s_0 \approx \frac{11.5 m_0^2}{S_{\text{ref}} \cdot C_{L,\text{max}} \cdot F_{T,1}}. \quad (A19)$$

The maximum lift coefficient can be found by solving a steady flight at stall velocity:

$$F_{L,\text{st.}} = F_W = \frac{1}{2} \cdot \rho_{\infty} \cdot S_{\text{ref}} \cdot C_{L,\text{max}} \cdot v_{\text{st.}}^2. \quad (A20)$$

Finally, the lift-off time, $\tau_0$, can be determined by dividing the lift-off distance, $s_0$, by the average take-off velocity. However, the speed is not constant during take-off. It can be approximated that the average speed on the runway is around 70% of the lift-off speed, $v_0$ [27]. Then the following applies:

$$\tau_0 = \frac{s_0}{0.7 v_0}. \quad (A21)$$

The previous equations were used in the algorithm schematized in Figure 10.

## Appendix C. Process of the Algorithm

This section details the process of the algorithm represented schematically in Figure 10, which was used for the derivation of the results presented in Section 5.

### Appendix C.1. Input Data

Three categories of input data can be determined. They can be seen in the blue frame on top of Figure 10. First, the design and performance of the aircraft must be selected (see Appendix B.1). This accounts for general design parameters such as the mass, the wingspan, and the aspect ratio, but it also accounts for general aircraft performance characteristics such as the cruise lift-over-drag ratio, the Oswald coefficient, and the energy density of the battery. Additionally, some mission parameters must be given. These include the cruise endurance, speed and height, the maximal allowed take-off distance, and the stall velocity.

Secondly, the geometric and aerodynamic characteristics of the single-stage thruster must be provided. For a given configuration (single or dual collector), they concern the geometry of the electrodes and thrusters (see Figure 6) with the thickness and chord of the electrodes and the distances between the electrodes (the inter-electrode gap and inter-collector separation). Moreover, the aerodynamic coefficients of the electrodes need to be set. For example, using the cruise velocity brought with the aircraft inputs, and when the chord of an electrode is known, the Reynolds number can be calculated. Then, simple tools such as airfoil databases or programs such as XFoil can be used to derive the coefficients. Because stacking stages in series strengthens the interference between successive stages, it can be preferred to favor an increase in the number of stages in parallel. However, increasing the number of stages in parallel increases the total height of the thruster (see Equation (25)). Thus, a maximal suitable height of the thruster must be defined.
The third and final category of input data relates to the EAD performance of the thrusters (see Section 4.2). An initial array of available thrusts per unit span can be set. With the original data used in the derivation of Equations (32) and (33) [5], using a range of 20 to 125 mN·m⁻¹ under quiescent conditions is more suitable. Then, the equations provide a corresponding array of power consumption per unit span. Moreover, when assuming an average induced velocity of 0.5 m·s⁻¹, Equation (9) allows the calculation of the self-induced drag per unit span $\hat{F}_{D,a} = F_{D,a}/L$ of the selected design. Here, the choice is $\alpha = 0$ for a dual-collector design and $\alpha = 1$ for a single-collector design in Equation (10). As a consequence, it is possible to estimate the thrust at any specific velocity with Equations (5) and (9).

## Appendix C.2. Thruster Design and Cruise Performance

The next step of the process is to agree on a given design of the thruster (e.g., finding the height, width, and span). In order to reduce the size, mass, and drag of the thruster, it was decided to select its final design (stages in series, span, and width) based on cruise conditions. In the previous section, the thrust per unit span under quiescent conditions was defined as an array (the default range is 20 to 125 mN·m⁻¹). Moreover, it was translated into a range of thrust per unit span under cruise conditions. Using the balance of thrust and drag in the cruising Equation (A6), the necessary thrust to hold steady cruising can be calculated for the selected aircraft. Consequently, for any thrust per unit span in the array, it is possible to determine the necessary span and width of the thruster. The algorithm starts by fixing the height of the thruster to the value given as an input. For each thrust per unit span in cruise and the number of stages in parallel, the algorithm then calculates the necessary span. If the calculated span exceeds the full span of the wing, then the algorithm starts adding series of stages. A loop runs until a span and the number of stages in series are met. The limits for the loop are the chord of the wing for the width of the thruster, the span of the wing for the span of the thruster, and the input height for the height of the thruster. If these limits cannot be met, the algorithm returns an error.

Since the aforementioned process is carried out over a range of possible thrusts per unit span, the final design must be selected based on additional conditions. First, for each possible thrust per unit span, the algorithm verifies that the sizes found previously achieve the required thrust in cruising (Equations (28) and (29)). Another condition is added here. The additional drag of the thruster (Equation (26)) must not exceed a given ratio of the total drag generated by the aircraft in cruising. In cruising, it should be recalled that the drag of the aircraft has the same amplitude as the thrust. As a result, a ratio $\kappa$ can be written as follows:

$$ \kappa = \frac{F_{D,t}}{F_{T,2}}. \qquad (A22) $$

$\kappa$ mostly depends on the size of the thruster. If the thrust is so large that it replaces a large portion of the wing (depending on the achieved lift force), then its share of the total drag can increase. In the study, $\kappa$ was kept at a value of 25%.

The final condition in cruising is the minimization of the consumed power. Since reduced power consumption is linked to a lower thrust generation, this condition translates to selecting the minimal thrust per unit span that meets the first two conditions.

## Appendix C.3. Take-Off and Climbing Performance

From the previous step of the solving process, a new range of possible thrust per unit span can be defined. Two values are now retained. First, the nominal thrust per unit span selected in cruising reveals the capabilities in cruise conditions. Second, the maximum thrust per unit span defined in the inputs is used during climbing and take-off. The power consumption per unit span is also known for each of these thrusts.

For take-off, the maximum lift coefficient can be determined with the balance of the lift force and weight under the stall condition (Equation (A20)). Then, the lift-off distance is derived from Equation (A19). If the calculation reveals a distance greater than the
The third and final category of input data relates to the EAD performance of the thrusters (see Section 4.2). An initial array of available thrusts per unit span can be set. With the original data used in the derivation of Equations (32) and (33) [5], using a range of 20 to 125 mN·m⁻¹ under quiescent conditions is more suitable. Then, the equations provide a corresponding array of power consumption per unit span. Moreover, when assuming an average induced velocity of 0.5 m·s⁻¹, Equation (9) allows the calculation of the self-induced drag per unit span $\hat{F}_{D,a} = F_{D,a}/L$ of the selected design. Here, the choice is $\alpha = 0$ for a dual-collector design and $\alpha = 1$ for a single-collector design in Equation (10). As a consequence, it is possible to estimate the thrust at any specific velocity with Equations (5) and (9).

## Appendix C.2. Thruster Design and Cruise Performance

The next step of the process is to agree on a given design of the thruster (e.g., finding the height, width, and span). In order to reduce the size, mass, and drag of the thruster, it was decided to select its final design (stages in series, span, and width) based on cruise conditions. In the previous section, the thrust per unit span under quiescent conditions was defined as an array (the default range is 20 to 125 mN·m⁻¹). Moreover, it was translated into a range of thrust per unit span under cruise conditions. Using the balance of thrust and drag in the cruising Equation (A6), the necessary thrust to hold steady cruising can be calculated for the selected aircraft. Consequently, for any thrust per unit span in the array, it is possible to determine the necessary span and width of the thruster. The algorithm starts by fixing the height of the thruster to the value given as an input. For each thrust per unit span in cruise and the number of stages in parallel, the algorithm then calculates the necessary span. If the calculated span exceeds the full span of the wing, then the algorithm starts adding series of stages. A loop runs until a span and the number of stages in series are met. The limits for the loop are the chord of the wing for the width of the thruster, the span of the wing for the span of the thruster, and the input height for the height of the thruster. If these limits cannot be met, the algorithm returns an error.

Since the aforementioned process is carried out over a range of possible thrusts per unit span, the final design must be selected based on additional conditions. First, for each possible thrust per unit span, the algorithm verifies that the sizes found previously achieve the required thrust in cruising (Equations (28) and (29)). Another condition is added here. The additional drag of the thruster (Equation (26)) must not exceed a given ratio of the total drag generated by the aircraft in cruising. In cruising, it should be recalled that the drag of the aircraft has the same amplitude as the thrust. As a result, a ratio $\kappa$ can be written as follows:

$$ \kappa = \frac{F_{D,t}}{F_{T,2}}. \qquad (A22) $$

$\kappa$ mostly depends on the size of the thruster. If the thrust is so large that it replaces a large portion of the wing (depending on the achieved lift force), then its share of the total drag can increase. In the study, $\kappa$ was kept at a value of 25%.

The final condition in cruising is the minimization of the consumed power. Since reduced power consumption is linked to a lower thrust generation, this condition translates to selecting the minimal thrust per unit span that meets the first two conditions.

## Appendix C.3. Take-Off and Climbing Performance

From the previous step of the solving process, a new range of possible thrust per unit span can be defined. Two values are now retained. First, the nominal thrust per unit span selected in cruising reveals the capabilities in cruise conditions. Second, the maximum thrust per unit span defined in the inputs is used during climbing and take-off. The power consumption per unit span is also known for each of these thrusts.

For take-off, the maximum lift coefficient can be determined with the balance of the lift force and weight under the stall condition (Equation (A20)). Then, the lift-off distance is derived from Equation (A19). If the calculation reveals a distance greater than the
maximum distance given in the inputs, the take-off is ignored by the algorithm. In such a case, it is assumed that the UAV is catapulted and that the take-off does not contribute to the energy consumption.

Finally, the climb performance of the aircraft (the rate-of-climb, climb angle, and velocity...) can be derived with Equations (A8)–(A16). First, a matrix of thrust values that solves the balance of the forces under climb conditions (Equation (A9)) is built using given ranges of velocities and angles. It was chosen to select a velocity between the take-off velocity (120% of the stall velocity) and the cruise velocity. The climb angle can be allowed to range from a value strictly greater than 0° up to a large value of 45°. From all of these possible solutions, the algorithm filters the ones that exceed the maximal thrust that can be achieved by the thruster at the maximal thrust per unit span (125 mN·m⁻¹ in the example). Then, calculating the time to climb, the energy consumption can be derived from the known power at the maximum thrust. The algorithm focuses on the solution that exhibits the lower energy usage. If several solutions are found by the algorithm, only the solution with the lower time to climb is retained. Consequently, the algorithm determines the climb velocity and angle. This choice is primarily driven by the need to reduce the energy stored in the battery in order to decrease the required mass of the battery.

### Appendix C.4. Output Parameters

With the determined performance of the aircraft in the different flight phases, it is possible to compute the times and distances covered by the different phases. The total energy consumed by the propulsion system is estimated based on the EAD performance of the thruster. Thus, the masses of the thruster and the battery are calculated using the thruster geometry and the energy density of the battery. In the final evaluation, the user needs to assess the share of the total mass that is not occupied by the battery and the thruster and determine whether it suffices for the remainder of the airframe with its systems. The analysis of these results allows for the determination of whether the available mass is sufficient for a feasible aircraft design.

## References

1. Xu, H.; He, Y.; Strobel, K.L.; Gilmore, C.K.; Kelley, S.P.; Hennick, C.C.; Sebastian, T.; Woolston, M.R.; Perreault, D.J.; Barrett, S.R.H. Flight of an aeroplane with solid-state propulsion. *Nature* **2018**, *563*, 532–535. [CrossRef]

2. Poon, H.S.; Lam, M.K.K.; Chow, M.; Li, W.J. Noiseless and vibration-free Ionic Propulsion technology for indoor surveillance blimps. In Proceedings of the 2009 IEEE International Conference on Robotics and Automation, Kobe, Japan, 12–17 May 2009; IEEE: Piscataway, NJ, USA, 2009; pp. 2891–2896. [CrossRef]

3. Drew, D.; Contreras, D.S.; Pister, K.S.J. First thrust from a microfabricated atmospheric ion engine. In Proceedings of the 2017 IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS), Las Vegas, NE, USA, 22–26 January 2017; pp. 346–349. [CrossRef]

4. Drew, D.S.; Lambert, N.O.; Schindler, C.B.; Pister, K.S.J. Toward Controlled Flight of the Ionocraft: A Flying Microrobot Using Electrohydrodynamic Thrust With Onboard Sensing and No Moving Parts. *IEEE Robot. Autom. Lett.* **2018**, *3*, 2807–2813. [CrossRef]

5. Moreau, E.; Benard, N.; Alicalapa, F.; Douyère, A. Electrohydrodynamic force produced by a corona discharge between a wire active electrode and several cylinder electrodes–Application to electric propulsion. *J. Electrost.* **2015**, *76*, 194–200. [CrossRef]

6. Monrolin, N.; Plouraboué, F.; Praud, O. Electrohydrodynamic Thrust for In-Atmosphere Propulsion. *AIAA J.* **2017**, *55*, 4296–4305. [CrossRef]

7. Monrolin, N.; Praud, O.; Plouraboué, F. Electrohydrodynamic ionic wind, force field, and ionic mobility in a positive dc wire-to-cylinders corona discharge in air. *Phys. Rev. Fluids* **2018**, *3*, 063701. [CrossRef]

8. Dremin, D.V.; Khomich, V.Y.; Rebrov, I.E. Thrust and thrust-to-power ratio in electrohydrodynamic propulsion electrode systems. *J. Physics Conf. Ser.* **2017**, *927*, 012015. [CrossRef]

9. Corke, T.C.; Post, M.L.; Orlov, D.M. SDBD plasma enhanced aerodynamics: Concepts, optimization and applications. *Prog. Aerosp. Sci.* **2007**, *43*, 193–217. [CrossRef]

10. Corke, T.C.; Enloe, C.L.; Wilkinson, S.P. Dielectric Barrier Discharge Plasma Actuators for Flow Control. *Annu. Rev. Fluid Mech.* **2010**, *42*, 505–529. [CrossRef]

11. Wang, J.J.; Choi, K.S.; Feng, L.H.; Jukes, T.N.; Whalley, R.D. Recent developments in DBD plasma flow control. *Prog. Aerosp. Sci.* **2013**, *62*, 52–78. [CrossRef]
maximum distance given in the inputs, the take-off is ignored by the algorithm. In such a case, it is assumed that the UAV is catapulted and that the take-off does not contribute to the energy consumption.

Finally, the climb performance of the aircraft (the rate-of-climb, climb angle, and velocity...) can be derived with Equations (A8)–(A16). First, a matrix of thrust values that solves the balance of the forces under climb conditions (Equation (A9)) is built using given ranges of velocities and angles. It was chosen to select a velocity between the take-off velocity (120% of the stall velocity) and the cruise velocity. The climb angle can be allowed to range from a value strictly greater than 0° up to a large value of 45°. From all of these possible solutions, the algorithm filters the ones that exceed the maximal thrust that can be achieved by the thruster at the maximal thrust per unit span (125 mN·m⁻¹ in the example). Then, calculating the time to climb, the energy consumption can be derived from the known power at the maximum thrust. The algorithm focuses on the solution that exhibits the lower energy usage. If several solutions are found by the algorithm, only the solution with the lower time to climb is retained. Consequently, the algorithm determines the climb velocity and angle. This choice is primarily driven by the need to reduce the energy stored in the battery in order to decrease the required mass of the battery.

### Appendix C.4. Output Parameters

With the determined performance of the aircraft in the different flight phases, it is possible to compute the times and distances covered by the different phases. The total energy consumed by the propulsion system is estimated based on the EAD performance of the thruster. Thus, the masses of the thruster and the battery are calculated using the thruster geometry and the energy density of the battery. In the final evaluation, the user needs to assess the share of the total mass that is not occupied by the battery and the thruster and determine whether it suffices for the remainder of the airframe with its systems. The analysis of these results allows for the determination of whether the available mass is sufficient for a feasible aircraft design.

## References

1. Xu, H.; He, Y.; Strobel, K.L.; Gilmore, C.K.; Kelley, S.P.; Hennick, C.C.; Sebastian, T.; Woolston, M.R.; Perreault, D.J.; Barrett, S.R.H. Flight of an aeroplane with solid-state propulsion. *Nature* **2018**, *563*, 532–535. [CrossRef]

2. Poon, H.S.; Lam, M.K.K.; Chow, M.; Li, W.J. Noiseless and vibration-free Ionic Propulsion technology for indoor surveillance blimps. In Proceedings of the 2009 IEEE International Conference on Robotics and Automation, Kobe, Japan, 12–17 May 2009; IEEE: Piscataway, NJ, USA, 2009; pp. 2891–2896. [CrossRef]

3. Drew, D.; Contreras, D.S.; Pister, K.S.J. First thrust from a microfabricated atmospheric ion engine. In Proceedings of the 2017 IEEE 30th International Conference on Micro Electro Mechanical Systems (MEMS), Las Vegas, NE, USA, 22–26 January 2017; pp. 346–349. [CrossRef]

4. Drew, D.S.; Lambert, N.O.; Schindler, C.B.; Pister, K.S.J. Toward Controlled Flight of the Ionocraft: A Flying Microrobot Using Electrohydrodynamic Thrust With Onboard Sensing and No Moving Parts. *IEEE Robot. Autom. Lett.* **2018**, *3*, 2807–2813. [CrossRef]

5. Moreau, E.; Benard, N.; Alicalapa, F.; Douyère, A. Electrohydrodynamic force produced by a corona discharge between a wire active electrode and several cylinder electrodes–Application to electric propulsion. *J. Electrost.* **2015**, *76*, 194–200. [CrossRef]

6. Monrolin, N.; Plouraboué, F.; Praud, O. Electrohydrodynamic Thrust for In-Atmosphere Propulsion. *AIAA J.* **2017**, *55*, 4296–4305. [CrossRef]

7. Monrolin, N.; Praud, O.; Plouraboué, F. Electrohydrodynamic ionic wind, force field, and ionic mobility in a positive dc wire-to-cylinders corona discharge in air. *Phys. Rev. Fluids* **2018**, *3*, 063701. [CrossRef]

8. Dremin, D.V.; Khomich, V.Y.; Rebrov, I.E. Thrust and thrust-to-power ratio in electrohydrodynamic propulsion electrode systems. *J. Physics Conf. Ser.* **2017**, *927*, 012015. [CrossRef]

9. Corke, T.C.; Post, M.L.; Orlov, D.M. SDBD plasma enhanced aerodynamics: Concepts, optimization and applications. *Prog. Aerosp. Sci.* **2007**, *43*, 193–217. [CrossRef]

10. Corke, T.C.; Enloe, C.L.; Wilkinson, S.P. Dielectric Barrier Discharge Plasma Actuators for Flow Control. *Annu. Rev. Fluid Mech.* **2010**, *42*, 505–529. [CrossRef]

11. Wang, J.J.; Choi, K.S.; Feng, L.H.; Jukes, T.N.; Whalley, R.D. Recent developments in DBD plasma flow control. *Prog. Aerosp. Sci.* **2013**, *62*, 52–78. [CrossRef]
12. Xu, H.; He, Y.; Barrett, S.R.H. A dielectric barrier discharge ion source increases thrust and efficiency of electroaerodynamic propulsion. *Appl. Phys. Lett.* **2019**, *114*, 254105. [CrossRef]

13. Bayoda, K.; Benard, N.; Moreau, E. Elongating the area of plasma/fluid interaction of surface nanosecond pulsed discharges. *J. Electrost.* **2015**, *74*, 79–84. [CrossRef]

14. Moreau, E. Airflow control by non-thermal plasma actuators. *J. Phys. Appl. Phys.* **2007**, *40*, 605–636. [CrossRef]

15. Zhang, Y.; Liu, L.; Chen, Y.; Ouyang, J. Characteristics of ionic wind in needle-to-ring corona discharge. *J. Electrost.* **2015**, *74*, 15–20. [CrossRef]

16. Moreau, E.; Benard, N.; Lan-Sun-Luk, J.D.; Chabriat, J.P. Electrohydrodynamic force produced by a wire-to-cylinder dc corona discharge in air at atmospheric pressure. *J. Phys. D Appl. Phys.* **2013**, *46*, 475204. [CrossRef]

17. Christenson, E.A.; Moller, P.S. Ion-neutral propulsion in atmospheric media. *AIAA J.* **1967**, *5*, 1768–1773. [CrossRef]

18. Masuyama, K.; Barrett, S.R.H. On the performance of electrohydrodynamic propulsion. *Proc. R. Soc. Math. Phys. Eng. Sci.* **2013**, *469*, 20120623. [CrossRef]

19. Kiousis, K.N.; Moronis, A.X.; Fruh, W.G. Electro-Hydrodynamic (EHD) Thrust Analysis in Wire-Cylinder Electrode Arrangement. *Plasma Sci. Technol.* **2014**, *16*, 363–369. [CrossRef]

20. Stuetzer, O.M. Ion Drag Pressure Generation. *J. Appl. Phys.* **1959**, *30*, 984–994. [CrossRef]

21. Gilmore, C.K.; Barrett, S.R.H. Electrohydrodynamic thrust density using positive corona-induced ionic winds for in-atmosphere propulsion. *Proc. R. Soc. Math. Phys. Eng. Sci.* **2015**, *471*, 20140912. [CrossRef]

22. Xu, H.; Gomez-Vega, N.; Agrawal, D.R.; Barrett, S.R.H. Higher thrust-to-power with large electrode gap spacing electroaerodynamic devices for aircraft propulsion. *J. Phys. D Appl. Phys.* **2020**, *53*, 025202. [CrossRef]

23. Robinson, M. Movement of air in the electric wind of the corona discharge. *Trans. Am. Inst. Electr. Eng. Part I Commun. Electron.* **1961**, *80*, 143–150. [CrossRef]

24. Chapman, S. Corona point current in wind. *J. Geophys. Res.* **1970**, *75*, 2165–2169. [CrossRef]

25. Grosse, S.; Benard, N.; Moreau, E. Electroaerodynamic thrusters: Influence of a freestream on the current, ionic wind and force produced by a DC corona discharge. *J. Electrost.* **2024**, submitted

26. Gilmore, C.K.; Barrett, S.R. Electroaerodynamic Thruster Performance as a Function of Altitude and Flight Speed. *AIAA J.* **2018**, *56*, 1105–1117. [CrossRef]

27. Anderson, J.D. Elements of Airplane Performance. In *Introduction to Flight*, 8th ed.; McGraw-Hill Higher Education: New York, NY, USA, 2016; pp. 441–588.

28. Khomich, V.Y.; Rebrov, I.E. In-atmosphere electrohydrodynamic propulsion aircraft with wireless supply onboard. *J. Electrost.* **2018**, *95*, 1–12. [CrossRef]

29. Martins, A.A.; Pinheiro, M.J. On the influence that the ground electrode diameter has in the propulsion efficiency of an asymmetric capacitor in nitrogen gas. *Phys. Plasmas* **2011**, *18*, 033512. [CrossRef]

30. Khomich, V.; Malanichev, V.; Rebrov, I. Electrohydrodynamic thruster for near-space applications. *Acta Astronaut.* **2021**, *180*, 141–148. [CrossRef]

31. Nelson, C.L.; Drew, D.S. High Aspect Ratio Multi-Stage Ducted Electroaerodynamic Thrusters for Micro Air Vehicle Propulsion. *IEEE Robot. Autom. Lett.* **2024**, *9*, 2702–2709. [CrossRef]

32. Gomez-Vega, N.; Brown, A.; Xu, H.; Barrett, S.R.H. Model of Multistaged Ducted Thrusters for High-Thrust-Density Electroaerodynamic Propulsion. *AIAA J.* **2023**, *61*, 767–779. [CrossRef]

33. Barrett, S.; Xu, H.; Brown, A.; Vega, N.G.; Perovich, N. Surface-Integrated Electroaerodynamic Thrusters. U.S. Patent US20230322368A1, 22 June 2023

34. Gomez-Vega, N.; Kambhampaty, J.D.; Barrett, S.R.H. Mitigating reverse emission in electroaerodynamic thrusters. *J. Phys. D Appl. Phys.* **2022**, *55*, 505202. [CrossRef]

35. Hepperle, M. Electric Flight–Potential and Limitations. In *Proceedings of the Energy Efficient Technologies and Concepts of Operation*, Lisbon, Portugal, 22–24 October 2012; p. 30. Available online: https://elib.dlr.de/78726 (accessed on 1 March 2024).

36. Belan, M.; Arosti, L.; Polatti, R.; Maggi, F.; Fiorini, S.; Sottovia, F. A parametric study of electrodes geometries for atmospheric electrohydrodynamic propulsion. *J. Electrost.* **2021**, *113*, 103616. [CrossRef]

37. He, Y.; Woolston, M.; Perreault, D. Design and implementation of a lightweight high-voltage power converter for electroaerodynamic propulsion. In *Proceedings of the 2017 IEEE 18th Workshop on Control and Modeling for Power Electronics (COMPEL)*, Stanford, CA, USA, 9–12 July 2017; pp. 1–9. [CrossRef]

38. Ilit', T.; Vary, M.; Valko, P. On the Thrust of a Single Electrode Electrohydrodynamic Thruster. *J. Electr. Eng.* **2015**, *66*, 117–120. [CrossRef]

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
12. Xu, H.; He, Y.; Barrett, S.R.H. A dielectric barrier discharge ion source increases thrust and efficiency of electroaerodynamic propulsion. *Appl. Phys. Lett.* **2019**, *114*, 254105. [CrossRef]

13. Bayoda, K.; Benard, N.; Moreau, E. Elongating the area of plasma/fluid interaction of surface nanosecond pulsed discharges. *J. Electrost.* **2015**, *74*, 79–84. [CrossRef]

14. Moreau, E. Airflow control by non-thermal plasma actuators. *J. Phys. Appl. Phys.* **2007**, *40*, 605–636. [CrossRef]

15. Zhang, Y.; Liu, L.; Chen, Y.; Ouyang, J. Characteristics of ionic wind in needle-to-ring corona discharge. *J. Electrost.* **2015**, *74*, 15–20. [CrossRef]

16. Moreau, E.; Benard, N.; Lan-Sun-Luk, J.D.; Chabriat, J.P. Electrohydrodynamic force produced by a wire-to-cylinder dc corona discharge in air at atmospheric pressure. *J. Phys. D Appl. Phys.* **2013**, *46*, 475204. [CrossRef]

17. Christenson, E.A.; Moller, P.S. Ion-neutral propulsion in atmospheric media. *AIAA J.* **1967**, *5*, 1768–1773. [CrossRef]

18. Masuyama, K.; Barrett, S.R.H. On the performance of electrohydrodynamic propulsion. *Proc. R. Soc. Math. Phys. Eng. Sci.* **2013**, *469*, 20120623. [CrossRef]

19. Kiousis, K.N.; Moronis, A.X.; Fruh, W.G. Electro-Hydrodynamic (EHD) Thrust Analysis in Wire-Cylinder Electrode Arrangement. *Plasma Sci. Technol.* **2014**, *16*, 363–369. [CrossRef]

20. Stuetzer, O.M. Ion Drag Pressure Generation. *J. Appl. Phys.* **1959**, *30*, 984–994. [CrossRef]

21. Gilmore, C.K.; Barrett, S.R.H. Electrohydrodynamic thrust density using positive corona-induced ionic winds for in-atmosphere propulsion. *Proc. R. Soc. Math. Phys. Eng. Sci.* **2015**, *471*, 20140912. [CrossRef]

22. Xu, H.; Gomez-Vega, N.; Agrawal, D.R.; Barrett, S.R.H. Higher thrust-to-power with large electrode gap spacing electroaerodynamic devices for aircraft propulsion. *J. Phys. D Appl. Phys.* **2020**, *53*, 025202. [CrossRef]

23. Robinson, M. Movement of air in the electric wind of the corona discharge. *Trans. Am. Inst. Electr. Eng. Part I Commun. Electron.* **1961**, *80*, 143–150. [CrossRef]

24. Chapman, S. Corona point current in wind. *J. Geophys. Res.* **1970**, *75*, 2165–2169. [CrossRef]

25. Grosse, S.; Benard, N.; Moreau, E. Electroaerodynamic thrusters: Influence of a freestream on the current, ionic wind and force produced by a DC corona discharge. *J. Electrost.* **2024**, submitted

26. Gilmore, C.K.; Barrett, S.R. Electroaerodynamic Thruster Performance as a Function of Altitude and Flight Speed. *AIAA J.* **2018**, *56*, 1105–1117. [CrossRef]

27. Anderson, J.D. Elements of Airplane Performance. In *Introduction to Flight*, 8th ed.; McGraw-Hill Higher Education: New York, NY, USA, 2016; pp. 441–588.

28. Khomich, V.Y.; Rebrov, I.E. In-atmosphere electrohydrodynamic propulsion aircraft with wireless supply onboard. *J. Electrost.* **2018**, *95*, 1–12. [CrossRef]

29. Martins, A.A.; Pinheiro, M.J. On the influence that the ground electrode diameter has in the propulsion efficiency of an asymmetric capacitor in nitrogen gas. *Phys. Plasmas* **2011**, *18*, 033512. [CrossRef]

30. Khomich, V.; Malanichev, V.; Rebrov, I. Electrohydrodynamic thruster for near-space applications. *Acta Astronaut.* **2021**, *180*, 141–148. [CrossRef]

31. Nelson, C.L.; Drew, D.S. High Aspect Ratio Multi-Stage Ducted Electroaerodynamic Thrusters for Micro Air Vehicle Propulsion. *IEEE Robot. Autom. Lett.* **2024**, *9*, 2702–2709. [CrossRef]

32. Gomez-Vega, N.; Brown, A.; Xu, H.; Barrett, S.R.H. Model of Multistaged Ducted Thrusters for High-Thrust-Density Electroaerodynamic Propulsion. *AIAA J.* **2023**, *61*, 767–779. [CrossRef]

33. Barrett, S.; Xu, H.; Brown, A.; Vega, N.G.; Perovich, N. Surface-Integrated Electroaerodynamic Thrusters. U.S. Patent US20230322368A1, 22 June 2023

34. Gomez-Vega, N.; Kambhampaty, J.D.; Barrett, S.R.H. Mitigating reverse emission in electroaerodynamic thrusters. *J. Phys. D Appl. Phys.* **2022**, *55*, 505202. [CrossRef]

35. Hepperle, M. Electric Flight–Potential and Limitations. In *Proceedings of the Energy Efficient Technologies and Concepts of Operation*, Lisbon, Portugal, 22–24 October 2012; p. 30. Available online: https://elib.dlr.de/78726 (accessed on 1 March 2024).

36. Belan, M.; Arosti, L.; Polatti, R.; Maggi, F.; Fiorini, S.; Sottovia, F. A parametric study of electrodes geometries for atmospheric electrohydrodynamic propulsion. *J. Electrost.* **2021**, *113*, 103616. [CrossRef]

37. He, Y.; Woolston, M.; Perreault, D. Design and implementation of a lightweight high-voltage power converter for electroaerodynamic propulsion. In *Proceedings of the 2017 IEEE 18th Workshop on Control and Modeling for Power Electronics (COMPEL)*, Stanford, CA, USA, 9–12 July 2017; pp. 1–9. [CrossRef]

38. Ilit', T.; Vary, M.; Valko, P. On the Thrust of a Single Electrode Electrohydrodynamic Thruster. *J. Electr. Eng.* **2015**, *66*, 117–120. [CrossRef]

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.