\label{cha:classI} In this chapter a first estimation of the weight, wing planform and power characteristics of the aerobatic racing aircraft E-SPARC is given. These are based on statistical data found from reference aircraft and on basic performance equations.
\label{sec:wpws}
This section elaborates on the general method used, the input parameters for this method and the results of the analysis of the wing and power loading of the E-SPARC aircraft. Firstly, in Chapter \ref{sec:designpointmethod} the general method and goals of finding the \(\frac{W}{S}\) and \(\frac{W}{P}\) design point are explained. Secondly, Chapter \ref{sec:designpointresults} provides the inputs and results for the design point determination.
\label{sec:designpointmethod}
The general principle of the Class I determination of the E-SPARCs design point is based on constraining the possible combinations of \(\frac{W}{S}\) and \(\frac{W}{P}\), which is visualised in a \(\frac{W}{S}\)-\(\frac{W}{P}\) plot. The lines that constrain the design space are based on certain performance requirements, which are enumerated below, and some basic parameters like \(C_{L_{max}}\) in both clean and landing configuration. For each requirement simple equations \cite{Zerodragraymer} exist that relate \(\frac{W}{P}\) to \(\frac{W}{S}\). In order to meet the requirements the chosen design point should be below and also to the left side of the constraining curves resulting from the requirements.
2
Stall speed
takeoff distance
Landing distance
Climb rate
Climb gradient
Sustained turn
After all curves have been added to the \(\frac{W}{S}\)-\(\frac{W}{P}\) plot the design point can be chosen. In terms of propulsive and aerodynamic efficiency one wants to find an as high as possible wing loading and an as high as possible power loading. A higher wing loading means a smaller wing, which also has consequences on the cost and weight of the whole aircraft. Besides, a smaller wing also generates less drag because the wetted area is smaller. A higher power loading means that less power and therefore smaller batteries (or less fuel) is required to perform the mission.
Using the chosen design point and the weight estimation that will be performed in Section \ref{sec:class1weight} a first estimate of the wing surface and required power can be calculated.
\label{sec:designpointresults}
All input values needed to construct the \(\frac{W}{S}\)-\(\frac{W}{P}\) plot for the E-SPARC, which is given in Figure \ref{fig:wingpowerloading}, are tabulated in Table \ref{tab:wswpparameters}. Some of the values are purely based on reference aircraft while others are obtained from the mission requirements (Red Bull and CS-23 Experimental Aircraft Category). A more elabotare explanation of the used inputs is provided below.
\label{tab:wswpparameters}
| Value | Description | ||
|---|---|---|---|
| \(A\) | [-] | 6 | Aspect ratio |
| \(\rho_0\) | [kg/m\(^3\)] | 1.225 | Air density at sea level |
| \(V_{stall}\) | [m/s] | 31 | Stall speed |
| \(V_{takeoff}\) | [m/s] | 34.1 | takeoff speed |
| TOP | [Ns/m\(^3\)] | 38.6 | Takeoff parameter |
| \(\sigma\) | [-] | 1 | Air density ratio |
| \(s_{landing}\) | [m] | 500 | Landing distance |
| \(f\) | [-] | 1 | takeoff/landing weight ratio |
| \(\eta_{prop}\) | [-] | 0.8 | Propeller efficiency |
| \(c\) | [m/s] | 7 | Climb rate |
| \(\frac{c}{V}\) | [-] | 0.083 | Climb gradient |
| \(e\) | [-] | 0.8 | Oswald factor |
| \(C_{d0}\) | [-] | 0.025 | Zero lift drag coefficient |
| \(V_{max,turn}\) | [m/s] | 80 | Maximum speed during sustained turn |
| \(n_{max,turn}\) | [-] | 3.5 | Maximum load factor during sustained turn |
| \(C_{L,max,clean}\) | [-] | 1.5 | Maximum lift coefficient in clean configuration |
| \(\Delta C_{L,max,landing}\) | [-] | 0.3 | Maximum lift coefficient increase in landing configuration |
W/S versus W/P plot
\label{fig:wingpowerloading}
The aspect ratio is purely based on the reference aircraft that can be found in Appendix B. The requirements for stall speed indicate that this speed shall not exceed 61 knots or \(31\,m/s\) \cite{stallspeed}. \(f\), the takeoff weight over landing weight ratio, is generally given as top level requirement and is chosen to be \(1\) since the aircraft is electric. The density ratio is also taken equal to \(1\), because the landing distance requirements are applicable to sea level conditions. The lift-off speed for a normal takeoff is 1.1 times the stall speed. A \(C_{L,max,clean}\) of 1.5 is chosen based on the same reference aircraft. In order to allow for a more aerodynamically efficient aircraft it is assumed that E-SPARC will have some simple high lift devices (HLDs) in order to generate an increase in \(C_{L,max}\) of 0.3. In case of simple plain flaps this would require a flap wetted area ratio (\(\frac{S_{wf}}{S}\)) of approximately 0.37.
The zero drag coefficient of 0.025 is based on Raymer’s method which is more elaborately discussed in Section \ref{sec:airplanedrag}. It is based on a skin-friction drag coefficient for the category of “Light aircraft - single engine” multiplied with the ratio of wetted area over reference area \cite{Zerodragraymer}.
For a sustained turn the air race aircraft will experience a load factor close to 3.5 with a maximum speed of around \(80\,m/s\), which is equivalent to a turn at \(\,73\degree\) of bank according to Equation \ref{eq:turn}.
\[\phi= cos^{-1}\left(\frac{1}{n}\right) \label{eq:turn}\]
According to Equation \ref{eq:turn} the load factor goes to infinity as the bank angle approaches \(90\degree\). In reality this means that the aircraft will not perform a sustained turn, but will rapidly lose height as the lift only acts in the horizontal plane. During a so called knife-edge maneuver the only vertical lift contributions are generated by the fuselage and vertical tail (assuming the aircraft will have a vertical tail) or comparable vertical lifting surfaces.
The takeoff parameter is taken to be 38.6 for a takeoff distance of 500\(\,m\), which follows from the RBAR regulations \cite{committee2010}. A relationship has been found between the TOP for several reference aerobatic racing aircraft and their landing distance in Figure \ref{fig:TOP}. From this statistical data the TOP parameter was derived from the least square solution.
Statistical relation between landing distance and takeoff parameter
\label{fig:TOP}
Based on Figure \ref{fig:wingpowerloading} a design point can be chosen. In order to maximize the aerodynamic efficiency and minimize the required power, the design point should be as far to the upper right corner as possible. The resulting design point is \(\frac{W}{S} = 790\) and \(\frac{W}{P} = 0.043\). As follows from Figure \ref{fig:wingpowerloading} the design point is dictated by the maneuvering and landing distance requirements, therefore E-SPARC will perform better than required on the four remaining performance requirements. Combining the chosen design point and Equation \cite{Zerodragraymer} used to construct the curves the expected performance based on these requirements can be calculated. Table \ref{tab:wswpresults} shows all the expected performance values of E-SPARC.
| Required Value | Expected Value | ||
|---|---|---|---|
| \(n_{max,turn}\) | [-] | 3.5 | 3.5 |
| \(V_{max,turn}\) | [m/s] | 80 | 80 |
| \(s_{landing}\) | [m] | 500 | 500 |
| \(V_{stall}\) | [m/s] | 31.0 | 29.3 |
| TOP | [Ns/m\(^3\)] | 38.6 | 22.6 |
| \(s_{takeoff}\) | [m] | 500 | 303 |
| \(c\) | [m/s] | 7.0 | 15.3 |
| \(\frac{c}{V}\) | [-] | 0.083 | 0.518 |
\label{tab:wswpresults}
\label{sec:class1weight}
After the design point has been chosen as explained in Section \ref{sec:wpws}, the Class I weight estimation can be performed. The most popular method for a Class I weight estimation uses fuel fractions based on the Breguet range and endurance equations. However, these equations are only applicable to aircraft with conventional power systems based on the combustion of (fossil) fuels, which is not an option for the E-SPARC. Therefore, a new Class I method has been developed to estimate main weight characteristics of an electric aircraft, powered by either a battery or a hydrogen fuel cell. In Section \ref{subsec:weightestimationmethod} this method will be explained, while Section \ref{subsec:weightinputandresults} elaborates on the inputs and obtained results. The sensitivity of the results with respect to the main input values is discussed in Section \ref{subsec:weightsensitivity}.
\label{subsec:weightestimationmethod}
The main feasible power options are a battery or hydrogen fuel-cell powered aircraft. Because the trade-off between this power supplies has not been performed at the moment the Class I weight estimation is done, two separate weight estimation methods have been constructed. First the method for the battery powered aircraft will be presented.
Battery Class 1 Method
The basis for the Class I method is Equation \ref{eq:classIbat1}, where the maximum takeoff weight (\(W_{TO}\)) has been written as a function of its main fractions: the structural weight (\(W_{struct}\), which is assumed equivalent to the empty weight without the engine weight), the electric engine weight (\(W_{en}\)), the battery weight (\(W_{bat}\)) and the payload weight (\(W_{payload}\)). For conventional aircraft \(W_{struct}\) and \(W_{en}\) would be combined in the empty weight. Since electric power trains are significantly lighter than the conventional combustion engines the two weights are shown separately for the battery powered aircraft method.
\[W_{TO} = W_{struct}+W_{en}+W_{bat}+W_{payload} \label{eq:classIbat1}\]
In order to estimate \(W_{TO}\), all terms on the right side of Equation \ref{eq:classIbat1} will be replaced by either a number or a function of \(W_{TO}\) in order to solve the remaining equation for \(W_{TO}\). The structural weight is related to \(W_{TO}\) via Equation \ref{eq:Wstruct}, where \(A_{bat}\) and \(B_{bat}\) are coefficients based on a set of reference aircraft. \(W_{en}\) depends on the required power \(P\), the propeller efficiency \(\eta_{prop}\) and the specific power of the electric engine \(p_{en}\), as can be seen in Equation \ref{eq:Wen}. The required power can be replaced by the takeoff weight divided by the power loading \(\left(\frac{W}{P}\right)\), which was determined in Section \ref{sec:wpws}. Equation \ref{eq:Wbat} gives the relation between \(W_{bat}\) and \(W_{TO}\) which also includes the engine efficiency \(\eta_{en}\), the battery discharge efficiency \(\eta_{bat}\) and the specific energy of the battery \(e_{bat}\). The required energy \(E_r\) from Equation \ref{eq:Wbat} can also be rewritten in terms of the takeoff weight, power loading and mission time \(t\).
\[\begin{aligned} \label{eq:Wstruct} W_{struct} &= A_{bat} \cdot W_{TO} + B_{bat} & \\ \label{eq:Wen} W_{en} &= \frac{P}{\eta_{prop} \cdot p} = \frac{W_{TO}}{\left(\frac{W}{P}\right) \cdot \eta_{prop} \cdot p_{en}} \\ \label{eq:Wbat} W_{bat} &= \frac{E_{r}}{\eta_{prop} \cdot \eta_{en} \cdot \eta_{bat} \cdot e_{bat}} = \frac{W_{TO}\cdot t}{\left(\frac{W}{P}\right) \cdot \eta_{prop} \cdot \eta_{en} \cdot \eta_{bat} \cdot e_{bat}}\end{aligned}\]
Substituting the relations from Equation \ref{eq:Wstruct} \ref{eq:Wen} and \ref{eq:Wbat} in Equation \ref{eq:classIbat1} and rewriting to solve for \(W_{TO}\) results in Equation \ref{eq:classIbat2}. Therefore combining some statistics, efficiencies and required power loading and payload weight a first estimate of the battery powered E-SPARC can be determined.
\[W_{TO} = \frac{B_{bat}+W_{payload}}{1-A_{bat}-\frac{1}{\left(\frac{W}{P}\right)\cdot\eta_{prop}\cdot p_{en}}-\frac{t}{\left(\frac{W}{P}\right)\cdot\eta_{prop}\cdot\eta_{en}\cdot\eta_{bat}\cdot e_{bat}}} \label{eq:classIbat2}\]
Hydrogen Fuel Cell Class 1 Method
The Class I method for a hydrogen fuel-cell powered version of the E-SPARC is very similar to the method for a battery powered aircraft. The fundamental equations is Equation \ref{eq:classIH21}, where \(W_E\) is the empty weight and \(W_{H_2}\) the fuel weight. For the hydrogen fuel-cell aircraft it is assumed that the combination of electric engine and the fuel-cell are equivalent in weight to the conventional combustion engine, therefore \(W_e\) is used instead of \(W_{struct}\) and \(W_{en}\) separately.
\[W_{TO} = W_{e}+W_{H_2}+W_{payload} \label{eq:classIH21}\]
As was the case for \(W_{struct}\) for the battery powered aircraft \(W_{e}\) depends on \(W_{TO}\) via the statistical parameters \(A_{H_2}\) and \(B_{H_2}\), shown in Equation \ref{eq:WE}. \(W_{H_2}\) is calculated in a similar fashion to \(W_{bat}\) as shown in Equation \ref{eq:WH2}, where \(\eta_{FC}\) is the fuel-cell efficiency and \(e_{H_2}\) the specific energy of hydrogen.
\[\begin{aligned} \label{eq:WE} W_{E} &= A_{H_2} \cdot W_{TO} + B_{H_2} \\ \label{eq:WH2} W_{H_2} &= \frac{E_{r}}{\eta_{prop} \cdot \eta_{en} \cdot \eta_{FC} \cdot e_{H_2}} = \frac{W_{TO}\cdot t}{\left(\frac{W}{P}\right) \cdot \eta_{prop} \cdot \eta_{en} \cdot \eta_{FC} \cdot e_{H_2}}\end{aligned}\]
Combining Equations \ref{eq:classIH21}, \ref{eq:WE} and \ref{eq:WH2} results in Equation \ref{eq:classIH22}, which can be used to estimate \(W_{TO}\) for the hydrogen fuel-cell powered version of the E-SPARC.
\[\label{eq:classIH22} W_{TO} = \frac{B_{H_2}+W_{payload}}{1-A_{H_2}-\frac{t}{\left(\frac{W}{P}\right) \cdot \eta_{prop} \cdot \eta_{en} \cdot \eta_{FC} \cdot e_{H_2}}}\]
Finally some important additional comments that should be respected to make proper use of the class I methods explained above:
The reference aircraft used to derive the statistical coefficients should be a conventional aerobatic (racer) aircraft or general aviation aircraft of similar size. The weight difference between a conventional and electric drive-train is taken into account by using \(W_{struct}\) instead of \(W_E\) for the battery powered method. Electric reference aircraft are mainly used to verify the feasibility of the estimation, e.g. by comparing the \(\frac{W_{bat}}{W_{TO}}\).
The time \(t\) in Equation \ref{eq:Wbat}‘and \ref{eq:WH2} should be interpreted as the ‘equivalent flight time at full power’, e.g. when flying 20 minutes at 50% power and 10 minutes at full power, \(t\) should be 20 minutes. This includes the assumption that the efficiency of the electric engine is constant, not depending on the delivered power. For the ESPARC \(t\) should be substituted by \(t_{race} + \frac{1}{2}t_{loiter}\). Here \(t_{loiter}\) also accounts for takeoff, taxiing and landing etc.
\label{subsec:weightinputandresults}
The inputs for the developed Class I estimate of the weights of the E-SPARC are given in Table \ref{tab:classIinput}, followed by the results for both the battery powered and hydrogen fuel-cell powered version of E-SPARC in Table \ref{tab:classIoutput}. Some of the input values need some clarification, which can be found below.
| Input | Source | ||||
|---|---|---|---|---|---|
| \(W_{payload}\) | 929 | \(N\) | 94.7 | \(kg\) | \cite{RB-PartA} |
| \(A_{bat}\) | 0.430 | Figure \ref{fig:ABplot} | |||
| \(B_{bat}\) | 570.89 | \(N\) | 58.195 | \(kg\) | Figure \ref{fig:ABplot} |
| \(A_{H_2}\) | 0.476 | Figure \ref{fig:ABplot} | |||
| \(B_{H_2}\) | 1955.82 | \(N\) | 199.37 | \(kg\) | Figure \ref{fig:ABplot} |
| \(p_{en}\) | 530.1 | \(W/N\) | 5.2 | \(kW/kg\) | \cite{siemens1} |
| \(e_{bat}\) | 14.7\(\cdot\)10\(^4\) | \(J/N\) | 500 | \(Wh/kg\) | Section \ref{sec:trade_power} |
| \(e_{H_2}\) | 14.5\(\cdot\)10\(^6\) | \(J/N\) | 142 | \(MJ/kg\) | \cite{e_H2} |
| \(\eta_{prop}\) | 0.80 | \cite{n_prop} | |||
| \(\eta_{en}\) | 0.95 | \cite{siemens1} | |||
| \(\eta_{bat}\) | 0.90 | ||||
| \(\eta_{FC}\) | 0.50 | \cite{n_FC} | |||
| \(t_{race}\) | 180 | \(s\) | 3 | min | - |
| \(t_{loiter}\) | 1800 | \(s\) | 30 | min | - |
| \(\frac{W}{P}\) | 0.043 | \(N/W\) | 4.4 | \(kg/kW\) | Section \ref{sec:wpws} |
| \(\frac{W}{S}\) | 790 | \(N/m^2\) | 80.5 | \(kg/m^2\) | Section \ref{sec:wpws} |
| \(A\) | 6.0 | Section \ref{sec:wpws} |
\label{tab:classIinput}
| Output | Battery | Hydrogen | |
|---|---|---|---|
| \(W_{TO}\) | [\(kg\)] | 485.3 | 566.4 |
| \(W_{payload}\) | [\(kg\)] | 94.7 | 94.7 |
| \(P\) | [\(W\)] | 110.7 | 129.2 |
| \(S\) | [\(m^2\)] | 6.03 | 7.03 |
| \(b\) | [\(m\)] | 6.01 | 6.50 |
| \(W_{struct}\) | [\(kg\)] | 266.9 | |
| \(W_{en}\) | [\(kg\)] | 26.6 | |
| \(W_{bat}\) | [\(kg\)] | 97.1 | |
| \(V_{bat}\) | [\(L\)] | 88.3 | |
| \(W_E\) | [\(kg\)] | 469.1 | |
| \(W_{H_2}\) | [\(kg\)] | 2.6 |
\label{tab:classIoutput}
Scatter plot of empty weight, structural weight and takeoff weight for reference aircraft from Appendix B, including the plots of the corresponding linear regressions
\label{fig:ABplot}
First of all the input value for the payload weight was based on the Red Bull requirements \cite{RB-PartA}, which state a minimum pilot weight of 82.0\(\,kg\). An additional 12.7\(\,kg\) is added to account for the suit, helmet and parachute.
The statistical parameters \(A_{bat}\), \(B_{bat}\), \(A_{H_2}\) and \(B_{H_2}\) are the coefficients belonging to the linear correlation between \(W_{struct}\) (which is assumed equal to the empty weight without the engine) and \(W_{TO}\) for a battery powered aircraft, and \(W_E\) and \(W_{TO}\) for a hydrogen powered aircraft. The plot in Figure \ref{fig:ABplot} shows the relevant weights of the reference aircraft from Appendix B. To determine the coefficients two linear least-square solutions were found. The results from Table \ref{tab:classIoutput} are also plotted for comparison.
The battery performance parameters like \(e_{bat}\) were first based on average battery technologies. However, since the power subsystem trade-off is also included in this report, the current battery performance parameter values have been updated based on the results from the power subsystem trade-off (Section \ref{sec:trade_power}. To summarize, E-SPARC will have a lithium-sulfur battery with a specific energy of \(500\,Wh/kg\) \cite{e_bat}.
\label{subsec:weightsensitivity}
As some of the inputs from Subsection \ref{subsec:weightinputandresults} are estimated, minor changes to these parameter values is still possible. Therefore it should also be investigated how sensitive the results from the weight estimation are to minor changes in the input values. The resulting sensitivity analysis in summarized by the plot in Figure \ref{fig:classIsensitivity}. For all main input parameters it was investigated what \(W_{TO}\) would be if the value is 10% lower, 5% lower, 5% higher and 10% higher.
Results of the sensitivity analysis of the Class I
\label{fig:classIsensitivity}
Clearly \(W_{TO}\) is most sensitive to changes of the value of the statistical coefficient \(A_{bat}\). The only reason for \(A_{bat}\) to change would be using a new (bigger) set of reference aircraft. Since almost all aerobatic racers for which information is available are already in the list of reference aircraft in Appendix B there is little chance \(A_{bat}\) is going to be changed, therefore the high sensitivity of \(W_{TO}\) with respect to \(A_{bat}\) is not considered to be a big risk.
Two other parameters that clearly have a big influence on the Class I weight estimation are \(\eta_{prop}\) and \(\frac{W}{P}\). \(\eta_{prop}\) mainly depends on whether a contra-rotating propeller will be used, which could improve the propulsive efficiency and therefore decrease the weight. Luckily the current value of 0.8 is conservative and therefore there is little risk of a weight increase due to a change of \(\eta_{prop}\). The power loading \(\frac{W}{P}\) has been carefully selected based on Section \ref{sec:designpointmethod} so for now, no apparent reasons for a change of its value exist.
The estimated takeoff weight is clearly less sensitive to the remaining input parameters. For instance, a 10% decrease of the battery specific energy results in a 7.6% increase in \(W_{TO}\) and because the value for \(e_{bat}\) is based on the subsystem trade-off in Section \ref{sec:trade_power} it is quite definitive. To conclude the current weight estimation is definitely not set-in-stone because some input parameters could still change a little which certainly also changes the weight estimation. However, the expected changes in \(W_{TO}\) are expected to not exceed 10% of the current estimation.