## 1. Introduction

In the realm of the global energy crisis and climate change, the aviation sector is a significant player. Since the 1980s, CO

${}_{2}$ emissions from aviation have increased by 3.6% per year, i.e., twice the world’s total growth rate [

1]. As a result, aviation currently accounts for 12% of transport-related CO

${}_{2}$ emissions and 2–3% of all anthropogenic emissions [

2] with a steadily increasing influence. In order to tackle this challenge, the European Commission formulated reduction targets in the “Flight Path 2050”: 75% CO

${}_{2}$ reduction per passenger kilometre relative to the capabilities of typical new aircraft in 2000, as well as 90% NO

${}_{x}$ and 65% perceived noise reduction [

3]. Similarly, two US government agencies, the International Air Transport Association (IATA), and the International Civil Aviation Organization (ICAO), pursue an average improvement in fuel efficiency of 1.5% per year until 2020, a cap on net aviation CO

${}_{2}$ emissions from 2020 (carbon-neutral growth) and a 50% CO

${}_{2}$ reduction until 2050 [

2].

Considering the increasing amount of air travel, these goals are unlikely to be reached by evolutionary improvement of existing aviation technology. Recently, Müller et al. analysed the influence of modern, fuel efficient aircraft and retrofit options on fleet planning and fleet emissions [

4]. The latest generation of aircraft has 15% less fuel burn by using geared turbofans and composite materials. Retrofit options of existing aircraft yield 9–12% less fuel burn by using blended winglets, cabin weight reduction, electric taxing, or re-engining. A low-cost carrier as well as a full service network carrier were analysed. While moderate CO

${}_{2}$ emission reductions of 12% and 7% compared to the “business as usual” scenario could be achieved with such measures, the overall CO

${}_{2}$ emissions are predicted to increase further until 2025.

A tenable solution would be the use of hydrogen as fuel, as already pointed out at the end of the 1970s by Bockris and Justi [

5] and references therein. Because of its three times higher gravimetric energy density, using hydrogen can reduce overall weight of the aircraft. This effect is pronounced in fuel-intensive aircraft like long-distance, large passenger number, and hypersonic aircraft [

5,

6,

7,

8]. Once hydrogen is used as a fuel, there is no better converter than a fuel cell. The increased efficiency of a fuel cell leads to a further reduction of the fuel load. Fuel cells enable further advantages of electric aircraft such as distributed propulsion, which increases the aerodynamic efficiency. Furthermore, multifunctional integration of the fuel cell into aircraft via harvesting by-products, such as water, heat or oxygen-depleted exhaust air, allows for using the fuel cell to provide vital processes like de-icing, cabin air conditioning, water supply or fire suppression of luggage compartment or fuel tanks [

9].

Besides fuel cell and hydrogen tank, fuel cell systems are composed of various auxiliary components for media and heat management. These components and their arrangement influence efficiency and weight as well. As in automotive applications, a good system design minimises weight, complexity and number of system components, while assuring operation over the whole load range and high efficiencies [

10]. Robustness to the strongly changing environmental conditions, especially temperature and humidity of air inlet, are furthermore crucial. Model-based analysis allows a systematic evaluation and quantification of effects of environmental conditions [

11] and of process integration for weight minimisation [

12]. Assessing the wide range of system design options is out of scope of this work; instead, the fuel cell is designed in the context of the whole system.

In order to design fuel cell systems for aviation, this article starts with general design considerations for fuel cells derived from the analysis of a physical model of a polymer electrolyte membrane fuel cell (PEMFC). It can be seen that the PEMFC can be designed to optimize performance, thus minimizing the size of the fuel cell, or can be designed to increase efficiency, thus minimizing fuel consumption and the size of the fuel tank and other system components like air supply compressors.

These design considerations are then applied to design fuel cell systems for the main energy supply of passenger transport aircraft. Starting with simplified models of fuel cell and tank, the design principles are explored. Since the optimum fuel cell design depends on the characteristics not only of the fuel cell but also of the other systems components, the model is refined by a more detailed liquid hydrogen tank model. In this tank model, the energy density is maximised for a specific operation profile, i.e., flight mission. In addition, estimates of future developments in lightweight fuel cells are discussed. With these advanced technologies, the potential of fuel cell systems in future aircraft is assessed.

## 2. Model

This analysis employs a PEMFC model that is as simple as possible while still covering all relevant processes in physically meaningful, macroscopic parameters. The charm of the model is that Andrei Kulikovsky was able to obtain an approximate but accurate analytical solution [

13]. This has three main advantages: (1) the functional relationship between parameters and interesting variables is directly revealed; (2) the model has no computational limitations and can be included into larger model hierarchies, e.g., for aircraft design, aircraft fleet models or future aviation scenario modelling; and (3) the effective, physical parameters of the model can be used to estimate the improvements due to future materials’ development.

In the model, the main losses of the fuel cell stem from the cathode catalyst layer (CCL). The ohmic losses are combined into one ohmic resistance, ${R}_{\mathsf{\Omega}}$, which covers membrane, electric, and contact resistances. The anode losses are considered negligible.

For the cathode performance, a well known model based on pioneering works of Perry, Newman and Cairns [

14] as well as Eikerling and Kornyshev [

15] is used. The model consists of governing equations for the rate of the oxygen reduction reaction (ORR) in the cathode catalyst layer, i.e., electrochemical current generation,

Ohm’s law for proton conduction,

with

$\sigma $ as ionic conductivity, as well as oxygen diffusion through gas diffusion layer (GDL) and CCL,

Here, j is the local proton current density, x is the distance from the membrane, ${i}_{*}$ is the volumetric exchange current density (per unit CCL volume, A cm${}^{-3}$), c is the local oxygen concentration ${c}_{\mathrm{ref}}$ is its reference (inlet) concentration, $\eta $ is the local overpotential, b is the Tafel slope, ${\sigma}_{\mathrm{t}}$ is the CCL proton conductivity and ${j}_{0}$ and ${\eta}_{0}$ are the cell current density and cell overpotential, respectively.

This CCL model is 1D in through-plane direction and solved in steady-state under isothermal conditions. It is valid for large overpotentials,

$\eta \gg b$. Since the diffusion coefficients in CCL and GDL, i.e.,

D and

${D}_{\mathrm{b}}$, are effective parameters, polarization curves for flooded conditions can be simulated. However, a changing water balance, i.e., a dependency of the diffusion coefficients on current, is not modelled in detail here. Under these assumptions, Kulikovsky [

13] was able to obtain the analytical solution for the polarization curve

${\eta}_{0}\left({j}_{0}\right)$,

with the three characteristic current densities

where

${l}_{\mathrm{t}}$ and

${l}_{\mathrm{b}}$ are the thickness of CCL and GDL, respectively. The parameter

$\beta $ is a solution to the equation

$\beta tan(\beta /2)={j}_{0}/{j}_{*}$, which can be accurately approximated by [

13]

The first term on the right-hand side of Equation (5) gives the overpotential due to the combined effect of ORR activation and proton transport. The second and third terms describe the potential losses due to oxygen transport in the CCL and GDL, respectively.

With Equation (5), the cell voltage

${V}_{\mathrm{cell}}$ can be calculated as

where

${V}_{\mathrm{oc}}$ is the open circuit voltage. The power density,

and electric efficiency,

complete the set of equations used in the analysis.

In addition to the fuel cell with mass

${m}_{\mathrm{fc}}$, the mass of the fuel,

${m}_{{\mathrm{H}}_{2}}$ and of the fuel tank,

${m}_{\mathrm{tk}}$ are considered. Since the scaling relation for the mass of the tank depends on its geometry, the limiting case of a spherical tank, i.e., the smallest surface-to-volume ratio, is considered. In a first step, a constant area-specific mass

${\varrho}_{\mathrm{tk}}$ (unit kg m

${}^{-2}$) of the walls of the tank is assumed. With the derivation of

${m}_{\mathrm{tk}}$ in

Appendix A, the total mass of the fuel cell system is

where

${P}_{\mathrm{rq}}$ and

${E}_{\mathrm{rq}}$ are the required output power and energy, respectively.

P is the power at which the cell is operated,

${P}_{\mathrm{max}}$ is the maximum power of the cell at the maximum power point (MPP), i.e.,

${P}_{\mathrm{max}}/P$ is the oversizing factor of the fuel cell.

${\rho}_{\mathrm{fc}}$ is the specific power of the fuel cell system including peripheral components at maximum net output power in kW/kg. This way,

${\rho}_{\mathrm{fc}}$ implicitly includes the mass of peripheral components (like compressors, pumps, valves, etc.) and their parasitic power demand.

${\omega}_{{\mathrm{H}}_{2}}$ is the specific energy of hydrogen, 33.3 kWh/kg, i.e.,

${E}_{\mathrm{rq}}/\nu $ is the energy that needs to be stored in the tank.

In a second step, a more detailed tank model from Winnefeld et al. [

16] was substituted. This model uses the equation of state of Leachman et al. [

17] to describe the properties of the stored hydrogen. Modules for geometrical, mechanical and thermal design are combined with a study mission. With this, the geometry, wall thickness and insulation thickness can be dimensioned. For the comparison in this paper, again a spherical geometry is chosen as a limiting case. The other parameters and the mid-range study mission are taken as described in [

16]. For a detailed description and discussion of the model and design process, the reader is referred to [

16]. All models were implemented in MATLAB

^{®} (MathWorks, Natick, MA, USA) using the parameters given in

Table 1.

## 4. Conclusions

In this paper, the design of fuel cells for the main energy supply of passenger transportation aircraft is discussed. For the description of the behaviour of the fuel cell, a physical model covering the main electrochemical and transport processes was employed. The model uses effective physical parameters, which allows for estimating future materials’ developments. From the general model-based analysis, it was seen that the optimal size of the fuel cell depends on the interplay of efficiency and power density. In any case, an oversizing of the fuel cell (or correspondingly operation below the maximum power point) is beneficial, since it increases efficiency.

A universal cost–benefit curve yielding the efficiency gain upon reduction of the operating power was derived. This curve proved to be insensitive to the parameters of the fuel cell and is thus of fundamental nature. To describe the trade-off between power and efficiency, a weight factor ${w}_{P}$ was introduced, which allows for incorporating technical as well as non-technical design objectives into the design process of the fuel cell. Possible design objectives are system mass and volume, fuel burn and emissions, direct operating costs, carbon tax, emission targets or restrictions, social acceptance or hype of new technologies, political factors, and many more.

Applying the presented design process for weight optimization of a fuel cell system for passenger aircraft revealed that the optimal fuel cell size cannot be determined from the fuel cell model alone, but the characteristics of the other system components affect the optimal fuel cell design. Hence, the fuel cell needs to be designed in the context of the whole energy system. It was found that in systems in which the mass of the fuel cell is determining, the optimal oversizing of the cell is very small, whereas if the system is heavy in fuel and/or tank, it is beneficial to apply larger oversizing. Hence, the ratio of energy-to-power determines the optimal oversize of the stack.

The fuel cell model was combined with a detailed liquid hydrogen tank model, which maximizes the energy density towards a specific desired flight mission. Such a tailored tank design significantly reduces the tank mass and allows reaching energy densities of the tank system surpassing that of kerosene. This mass advantage of the tank system can compensate mass disadvantages of the other components, like the power densities of current fuel cells and electrical engines that fall behind jet engines.

Similarly to the tank, the future potential for lightweight fuel cell developments was discussed. While in the automotive sector, power density goals are already reached and are thus out of the focus of current research and development efforts, the power density of future fuel cell stacks could be improved by a factor of at least 5. Key components are the bipolar plates and peripheral components like end plates, which could be made from better performing, lightweight materials. Further improvements can be achieved with advanced catalyst designs. With these estimates, the total mass of fuel cell systems for passenger aircraft employing both current and future fuel cell technologies was determined. While current fuel cell technology can power aircraft, future technologies have the potential for mass reductions compared to jet engine technology. This advantage is especially pronounced in fuel-heavy aircraft like long haul aircraft.

The current design process focuses on the main components, i.e., fuel cell and hydrogen tank. Other fuel cell system components, especially hydrogen recycling, air supply and conditioning have been neglected. Furthermore, the overall system design has to consider not only weight, but also feasibility for dynamic and robust operation. Additionally, system integration into the airplane, including heat management, reactant supply and product usage, is an important step to increase overall efficiency. This remains a future task.