A Fast-Time MATLAB Model of an Aeronautical Low-Temperature PEM Fuel Cell for Sustainable Propulsion and Compressor Behavior at Varying Altitudes

Abstract

The aviation sector significantly contributes to environmental challenges, including global warming and greenhouse gas emissions, due to its reliance on fossil fuels. Fuel cells present a viable alternative to conventional propulsion systems. In the context of light aircraft applications, proton exchange membrane fuel cells (PEMFCs) have recently attracted growing interest as a substitute for internal combustion engines (ICEs). However, their performance is highly sensitive to altitude variations, primarily due to limitations in compressor efficiency and instability in cathode pressure. To address these challenges, this research presents a comprehensive numerical model that couples a PEMFC system with a dynamic air compressor model under altitude-dependent conditions ranging from 0 to 3000 m. Iso-efficiency lines were integrated into the compressor map to evaluate its behavior across varying environmental parameters. The study examines key fuel cell stack characteristics, including voltage, current, and net power output. The results indicate that, as altitude increases, ambient pressure and air density decrease, causing the compressor to work harder to maintain the required compression ratio at the cathode of the fuel cell module. This research provides a detailed prediction of compressor efficiency trends by implementing iso-efficiency lines into the compressor map, contributing to sustainable aviation and aligning with global goals for low-emission energy systems by supporting cleaner propulsion technologies for lightweight aircraft.

1. Introduction

1.1. Background and Motivation

The aviation sector accounts for nearly 5% of global greenhouse gas emissions, primarily due to fossil fuel combustion in aircraft. In 2013, the aviation sector released roughly 700 million tons of CO₂, and if no mitigation measures are taken, this amount is expected to triple by 2050. This process releases pollutant emissions, including CO₂, NO_x, and water vapor, exacerbating global warming and radiative forcing [1,2,3]. To tackle these environmental challenges, electric propulsion architectures have become feasible replacements for conventional ICEs by utilizing emission-free energy sources throughout the entire flight mission. Electrified lightweight aircraft can draw power from sources such as batteries and fuel cells. However, batteries alone lack the energy capacity required for long-endurance missions. The prospect of increasing the number of batteries proves impractical due to weight and spatial limitations [4]. In contrast, fuel cells (FCs) offer extended flight durations compared with lithium–ion batteries, owing to their higher energy density and continuous fuel supply.

Sustainable aviation fuels (SAFs), derived from renewable sources, are currently the most scalable solution and are compatible with existing aircraft infrastructure. However, SAF production is costly and faces feedstock and policy challenges [5]. Given the limitations of both battery–electric systems for long-range flights and the high production costs of SAFs, PEMFCs represent a promising middle-ground solution, combining the benefits of high energy density, zero emissions at the point of use, and operational scalability.

Despite these advantages, a major challenge for PEMFC systems in aviation is maintaining a sufficient oxygen supply under varying altitude conditions. Air properties, including pressure and density, differ significantly in high-altitude environments compared with sea level [6,7], leading to greater demands on the air supply system. Compressors serve as essential components of the air supply system, ensuring sufficient oxygen delivery to the cathode of the fuel cell stack. In addition to this, altitude variations introduce dynamic operational stresses that can accelerate the degradation of PEMFC components. This degradation, driven by the combined effects of operational stress and material aging, compromises system durability and reduces the effective lifetime of PEMFCs [8]. Therefore, this study focuses on understanding how compressor efficiency varies across different altitudes and how these variations impact the overall fuel cell system performance.

1.2. Literature Review

A wide range of studies have been conducted on PEMFC system performance, specifically examining the impact of altitude on air supply systems and compressor efficiency, as briefly outlined below.

Haraldsson and Alvfors [9] studied the influence of altitude on PEMFC performance, specifically focusing on air compressor power demand and fuel consumption. They found that, as altitude increases, the compressor workload rises due to decreasing ambient pressure, consuming up to 40% of the net FC output at 3000 m. This increased parasitic load reduces the overall efficiency and net power available for propulsion. Liu et al. [10] conducted an experimental analysis on the effects of water flooding and pressure drop in the anode and cathode channels of a PEMFC. Their study indicates that liquid water accumulation on the cathode side is of paramount importance compared with the anode side, leading to decreased cell performance. Atkinson et al. [11], in an experimental study, investigated the effect of altitude on a PEM fuel cell system in unmanned aerial vehicles (UAVs). They concluded that the oxygen partial pressure decreases at higher altitudes, reducing fuel cell efficiency. Zhao et al. [7] analyzed the performance of a centrifugal compressor at varying altitudes, focusing on ambient conditions such as air pressure and density. Their results show that increasing altitude reduces absolute pressure and compressor mass flow rate (e.g., from 0.024 kg/s at sea level to 0.008 kg/s at 12,000 m). González-Espasandín et al. [12] numerically examined PEMFC behavior with a particular emphasis on the polarization curves for UAVs. Their findings indicate that the reduction of the polarization curve with altitude is due to atmospheric temperature, decreased atmospheric pressure and changes in atmospheric relative humidity. Chen et al. [13] experimentally analyzed the pressure drop in baffled and straight channels of a PEMFC under varying cell temperature conditions. They found that higher temperatures, ranging from 333 K to 343 K, improve fuel cell efficiency, whereas lower temperatures lead to decreased efficiency. Their findings demonstrate that higher pressure drops occur at higher current densities. Schröter et al. [14] investigated the impact of low ambient pressure on PEMFC efficiency and compressor performance in an aircraft. They found that, as ambient pressure decreases, the compressor outlet pressure drops from 2.4 bar to 1.6 bar, leading to a decline in PEMFC stack efficiency. Gong et al. [15] developed a model based on a coupled electrochemical–thermal approach to investigate the performance of an air-cooled PEMFC in UAVs. Their results show that increasing the altitude from 0 m to 4000 m leads to an accompanying decrease in electric power ranging from 4.7% to 6.5%. Moreover, the degradation of stack voltage is approximately 0.25 V/km as altitude increases. Meng et al. [16] developed and validated a zero-dimensional model of an air-cooled PEMFC to evaluate its performance under varying altitude conditions. Their simulation results show that both output voltage and power density significantly decline with altitude, primarily due to the reduced oxygen partial pressure, with a notable performance degradation of approximately 20% at 3000 m. Donateo et al. [17] analyzed the feasibility of hybrid PEMFC-battery systems for UAVs and identified 3000 m as a critical operational altitude, where the reduction in air density causes a substantial voltage drop and efficiency loss in the fuel cell system. Their results emphasize that, above this altitude, the fuel cell stack output power significantly diminishes due to oxygen scarcity, making compressor support and adaptive control strategies essential. Willich et al. [18] conducted experimental tests on a 100 kW PEMFC system under simulated high-altitude conditions to assess compressor and stack performance. They found that oxygen shortages below 867 mbar significantly reduced power output, requiring adaptive current limitation. The study emphasized the need for pressure-resilient compressor control strategies in aviation PEMFC systems. Their findings provide valuable insights into altitude-induced limitations on air supply and system efficiency. Gao and Wang [19] proposed an A3C-based intelligent control strategy for regulating the air compressor and back pressure valve in PEMFC systems under high-altitude conditions. Their model significantly improved dynamic response (up to 43.7%) and electrical efficiency (up to 5.8%) compared with conventional logic-based control. The study demonstrated robust adaptability by training the controller under real road profiles with altitude variation. This work highlights the potential of deep reinforcement learning in enhancing PEMFC performance in thin-air environments. Wei et al. [20] developed a cathode-side model and altitude-adaptive air control strategy for PEMFC systems using a transfer-learning algorithm. The proposed method ensures optimal oxygen stoichiometry and efficient air supply under varying ambient pressures. Simulation results showed improved stack voltage stability and reduced performance loss in high-altitude conditions. This study underscores the importance of intelligent control in maintaining PEMFC efficiency during altitude fluctuations. Piqueras et al. [21] analyzed the performance of a Roots-type positive-displacement compressor integrated into a PEMFC system for high-altitude applications. Simulation results indicated that the Roots compressor maintained higher efficiency across a range of altitudes compared with conventional centrifugal compressors. The proposed design limited system efficiency losses to about 3% up to 5 km altitude. This work highlights the viability of novel compressor architectures for altitude-resilient PEMFC systems.

1.3. Research Contributions and Objectives

This research focuses on analyzing compressor performance under altitude-dependent conditions for a PEMFC system. The study specifically addresses the integration of a compressor efficiency map, with particular emphasis on iso-efficiency line visualization, which represents the key contribution of this research. To the best of our knowledge, no previous studies have reported a compressor efficiency map for a PEMFC system in the literature. The numerical analysis comprehensively explains compressor behavior under various oxygen conditions resulting from altitude-dependent changes. Furthermore, it provides a detailed visualization of key PEMFC system parameters, including stack voltage, oxygen partial pressure, and net power, under different oxygen conditions. This offers valuable insights into the efficiency trends observed in the compressor performance map.

The structure of this paper is organized as follows: Section 2 presents the integration of iso-efficiency lines into the compressor map and the implementation of the fuel cell system. Section 3 describes the results of the developed numerical model, including cathode pressure, compressor efficiency and current stack. Section 4 concludes this paper.

2. Materials and Methods

2.1. Fuel Cell Stack Model

In the current study, the fuel cell module comprises four main components: a compressor, hydrogen storage tank, a humidifier, and a fuel cell stack, as illustrated in Figure 1. The compressor plays a critical role in supplying pressurized air (oxygen) to the system. Hydrogen is regulated by a control valve to achieve the desired flow rate. The humidifier adjusts the humidity levels of both hydrogen and oxygen streams before they enter the fuel cell stack, where the electrochemical reaction occurs. This reaction generates electricity as the final output.

2.2. Methodological Framework Overview

Figure 2 presents the methodological framework adopted in this study to evaluate the performance of a PEMFC air compressor system under altitude-dependent conditions. The framework begins with identifying the motivation and background, highlighting the environmental importance of fuel cells in aviation and the operational challenges imposed by high-altitude environments. The research objective is then defined as assessing system performance across varying altitudes. This is followed by the development of a numerical model that simulates how altitude variations affect ambient conditions, compressor behavior, and PEMFC performance. Key variables and relationships are then explored, with an emphasis on compressor efficiency mapping, iso-efficiency lines, and the identification of critical performance trends under altitude variation. These variables form the basis for an in-depth analysis and interpretation of system behavior. Finally, the framework results in generating insights and outcomes aimed at guiding optimization strategies and informing the design of more robust and altitude-resilient PEMFC systems for lightweight aircraft applications.

2.3. The Effect of Altitude on Air Consumption in PEM Fuel Cell

The homosphere consists of three layers: the first layer is the troposphere, the stratosphere is the second, and the last layer is the mesosphere. The main gases in the homosphere are nitrogen (78%), oxygen (21%) and other gases such as ozone, carbon dioxide, water vapor, etc. (1%) [22].

The increase in altitude leads to changes in air pressure, temperature, and density. With increasing altitude, air density and pressure reduce exponentially. The maximum values of pressure and density occur at sea level. Moreover, temperature has a reduction trend with increasing altitude. The range of the troposphere layer is between 0 to 11 km. The international standard atmosphere (ISA) model is utilized to calculate air density, pressure and temperature within the troposphere layer. According to the ISA model, pressure and temperature at a specific altitude are given as follows:

$$p\left(h\right)=p_0.{\left(1-\left(0.0065\times {\displaystyle \frac{h}{288.15}}\right)\right)}^{5.2561}$$

(1)

where $h$ is the altitude in meters and $p_0$ is the pressure at sea level.

$$T\left(h\right)=T_0-\left(279.65\times {\displaystyle \frac{h}{1000}}\right)$$

(2)

where $T_0$ is the temperature of the air at sea level.

$$\rho \left(h\right)={\displaystyle \frac{ P\left(h\right)}{R.T\left(h\right)}}$$

(3)

The density of air, $\rho \left(h\right)$ is considered a function of pressure and temperature at any altitude. $R$ is the specific gas constant [22,23,24,25].

2.4. Compressor Model and Iso-Efficiency Map Integration

2.4.1. Compressor Design

As shown in Figure 3, the schematic of the compressor model consists of two main components: the compressor map and the compressor motor. The compressor map defines the thermodynamic properties, including density, pressure and temperature in the inlet, as a function of altitude. A motor provides the necessary power to drive the compressor, ensuring it meets the required performance demands.

In the design of the compressor, the ISA model is used to estimate the air density, temperature and pressure at the compressor inlet. As discussed in the previous section, pressure, temperature and density vary with altitude, therefore, at the compressor inlet, pressure $P_{cp,in}$, temperature $T_{cp,in}$ and density ${\rho }_{cp,in}$ are calculated by ISA. The outlet pressure $P_{cp,out}$ is determined by the supply manifold pressure $P_{sm}$. The voltage of the compressor motor $v_{cm}$ is another input in the compressor model. A PI controller adjusts the compressor motor voltage to maintain the oxygen excess ratio within the desired range. Purkruspan [26] concluded that the highest net power is achieved when this ratio is around 2, which is considered optimal and is used as the set point for the PI controller. The oxygen excess ratio is calculated using the following equation:

$${\lambda }_{O_2}={\displaystyle \frac{W_{O_2,ca,in} }{W_{O_2,reacted}}}$$

(4)

where $W_{O_2,ca,in}$ represents the oxygen mass flow rate at the cathode inlet, and $W_{O_2,reacted}$ represents the rate of reacted oxygen. Based on this set point, the PI controller then adjusts the compressor voltage $v_{cm }$ to maintain the desired oxygen excess ratio.

Using the Jensen and Kristensen model, presented in [27], compressor characteristics such as dimensionless pressure head $\Psi$, the normalized compressor flow rate $\Phi$ and inlet Mach number $M$ are defined as follows:

$$\mathit{\Psi }={\displaystyle \frac{{\mathit{C}}_{\mathit{p}}{\mathit{T}}_{\mathit{c}\mathit{p},\mathit{i}\mathit{n}}\left[{\left(\mathit{P}\mathit{R}\right)}^{\frac{\mathit{{\rm Y}}-1}{\mathit{{\rm Y}}}}-\mathbf{1}\right]}{\frac{\mathbf{1}}{\mathbf{2}}{\mathit{U}}_{\mathit{c}}^{\mathbf{2}}}}$$

where

$$\begin{gathered} U_c={\displaystyle \frac{\pi }{60}}{d}_c{N}_{cr} \\ PR={\displaystyle \frac{P_{cp,out}}{P_{cp,in}}} \end{gathered}$$

(5)

$$\Phi ={\displaystyle \frac{W_{cr}}{{\rho }_a\frac{\pi }{4}{d}_c^2{U}_c}}$$

(6)

$$M={\displaystyle \frac{U_c}{\sqrt{\mathsf{{\rm Y}}R{T}_{cp,in}}}}$$

(7)

In the aforementioned equations, $U_c$, $PR$ and $d_c$ represent the compressor blade tip speed $\left(m/s\right)$ pressure ratio and compressor diameter $\left(m\right)$, respectively. In this study, the compressor diameter is set to $0.2286 \mathrm{m}$.

Corrected characteristics in the compressor, including the corrected compressor speed $N_{cr} \left(rpm\right)$, the corrected mass flow $W_{cr} \left(kg/s\right)$, corrected temperature $\theta$ and corrected pressure $\delta$ are given by the following:

$$N_{cr}=N_{cp}/\sqrt{\theta }$$

(8)

$$W_{cr}=W_{cp}\sqrt{\theta }/\delta$$

(9)

$$\theta =T_{cp,in}/288K$$

(10)

$$\delta =P_{cp,in}/1atm$$

(11)

where $N_{cp}$ and $W_{cp}$ are compressor speed and compressor air mass flow rate, respectively.

The dimensionless compressor flow rate can be correlated with the exponential function as follows:

$$\Phi ={\Phi }_{max}\left(1-ex{p}^{\beta \left({\displaystyle \frac{\Psi }{{\Psi }_{max}}}-1\right) }\right)$$

(12)

where coefficients ${\Phi }_{max}$, $\beta$ and ${\Psi }_{max}$ can be defined by the following polynomial function Mach number:

$$\begin{gathered} {\Phi }_{max}=a_4{M}^4+a_3{M}^3+a_2{M}^2+a_{1}M+a_0 \\ \beta =b_2{M}^2+b_{1}M+b_0 \\ {\Psi }_{max}=c_5{M}^5+c_4{M}^4+c_3{M}^3+c_2{M}^2+c_{1}M+c_0 \end{gathered}$$

(13)

In the above equation, the a, b and c coefficients are determined by the curve fitting scheme, as reported by Pukrushpan [26]. The specific values of these coefficients are listed in

Table 1. Compressor map curve-fit coefficients [26].

Parameter	Value
$a_4$	$-3.69906\times {10}^{-5}$
$a_3$	$2.70399\times {10}^{-4}$
$a_2$	$-5.36235\times {10}^{-4}$
$a_1$	$-4.63685\times {10}^{-5}$
$a_0$	$2.21195\times {10}^{-3}$
$b_2$	$1.76567$
$b_1$	$-1.34837$
$b_0$	$2.44419$
$c_5$	$-9.78755\times {10}^{-3}$
$c_4$	$0.10581$
$c_3$	$-0.42937$
$c_2$	$0.80121$
$c_1$	$-0.68344$
$c_0$	$0.43331$

.

The compressor torque ${\tau }_{cp}$ and the compressor speed ${\omega }_{cp}$ are represented as follows:

$${\tau }_{cp}= {\displaystyle \frac{C_p}{{\omega }_{cp}}}{\displaystyle \frac{T_{cp,in}}{{\eta }_{cp}}}\left[{\left(PR\right)}^{{\displaystyle \frac{\gamma -1}{\gamma }}}-1\right]W_{cp}$$

(14)

$$J_{cp}{\displaystyle \frac{d{\omega }_{cp}}{dt}}=\left({\tau }_{cm}-{\tau }_{cp}\right)$$

(15)

where $J_{cp}$ represents the combined inertia of the compressor and the motor $\left(kg/m^2\right)$, and ${\tau }_{cm}$ is the compressor motor torque input $\left(Nm\right)$. ${\eta }_{cp}$ is the compressor efficiency, which is described in the subsequent section on its implementation in MATLAB R2024b.

The torque of the compressor motor is defined by the below equation:

$${\tau }_{cm}={\eta }_{cm}{\displaystyle \frac{k_t}{R_{cm}}}\left(v_{cm}-k_v{\omega }_{cp}\right)$$

(16)

where the parameters of the compressor motor torque input are given in the following table (

Table 2. Compressor motor specifications.

Symbol	Description	Value	Unit
${\eta }_{cm}$	Motor mechanical efficiency	0.98	$J/kg.K$
$k_t$	Torque constant of the motor	0.0153	$Nm/Amp$
$k_v$	Speed constant of the motor	0.0153	$V/\left(rad/sec\right)$
$R_{cm}$	Resistance of the compressor motor	0.82	$\mathsf{\Omega }$

):

The output temperature of the compressor is calculated using the following thermodynamic equation:

$$T_{cp,out}=T_{cp,in}+{\displaystyle \frac{T_{cp,in}}{{\eta }_{cp}}}\left[{\left(PR\right)}^{{\displaystyle \frac{\gamma -1}{\gamma }}}-1\right]$$

(17)

2.4.2. Polynomial Fit for Iso-Efficiency Lines

In the present research, to implement iso-efficiency lines in the compressor map, the compressor efficiency ${\eta }_{cp}$ is expressed as a function of two independent variables: the corrected mass flow rate $W_{cr}$ and the pressure ratio $PR$.

$${\eta }_{cp}=f\left(W_{cr},PR\right)$$

(18)

A data-driven approach was utilized to digitize data points, namely $W_{cr}$ and $PR$ from the compressor map reported by Purkrushpan [26]. The extracted data cover a wide range of corrected mass flow rates and pressure ratios across different speed lines. In this study, compressor efficiency values ranging from 75% to 83% were assumed and assigned to the corresponding flow–pressure data points. Based on this approach, a fourth-order polynomial was employed to fit the assigned data and create iso-efficiency lines in the compressor performance model as follows:

$${\eta }_{cp}=f\left(x,y\right)=P_{00}+ P_{10}x+P_{01}y+P_{20}{x}^2+P_{11}xy+P_{02}{y}^2+P_{30}{y}^3 +P_{21}{x}^{2}y+P_{12}x{y}^2+P_{03}{y}^3+P_{40}{x}^4+P_{31}{x}^{3}y+P_{22}{x}^2{y}^2+P_{13}x{y}^3+P_{04}{y}^4$$

(19)

where $x=W_{cr}$, $y=PR$ and $P_{ij}$ represent the polynomial coefficients, which are determined through a curve-fitting technique in MATLAB R2024b.

2.5. PEMFC Model

The numerical modeling of a PEMFC necessitates a comprehensive set of equations to capture the system’s dynamic behavior. It is well documented in the literature that the characteristic time constant of electrochemical reactions is on the order of 10−19 s, reflecting an ultrafast process. In contrast, the dominant system dynamics are governed by flow control components, with characteristic time scales of around 100 s. As such, the high-frequency transients associated with electrochemical kinetics and electrode charge dynamics exert minimal influence under aeronautical operating conditions and may be neglected in dynamic simulations. On the other hand, transient phenomena arising from manifold filling behavior, membrane water transport, operation of supercharging devices, and thermal dynamics can substantially affect system performance and must be rigorously incorporated into the model.

2.5.1. Steady-State Modeling

Ideal Voltage

In an ideal case, each cell in a fuel cell stack is assumed to have the same voltage. Therefore, the total stack voltage can be computed as the product of the single-cell voltage E and the number of cells $N_{cell}$:

$$E_{stack}=E·N_{cell}$$

(20)

For this reason, the following equations refer to a single cell.

The ideal voltage of a single fuel cell can be calculated using the Nernst equation, which represents the open-circuit voltage under thermodynamic equilibrium, i.e., when no current flows through the external circuit:

$$E=E_0+{\displaystyle \frac{RT}{2F}}·\ln \left({\displaystyle \frac{p_{H_2}·p_{O_2}^{0.5}}{p_{H_{2}O}}}\right)$$

(21)

where $E_0$ is the electromotive force (EMF) at standard temperature and pressure with pure reactants. This is given by the following:

$$E_0=-{\displaystyle \frac{\Delta G}{2F}}$$

(22)

The Gibbs free energy change ∆G is the difference between the Gibbs free energy of the products and the Gibbs free energy of the reactants. Under isothermal conditions (constant temperature $T$) in the fuel cell, it is expressed as follows:

$$\Delta G=\Delta H-T·\Delta S$$

(23)

where $\Delta H$ and $\Delta S$ are the variation in molar enthalpy and entropy between reactants and products, respectively.

The partial pressures of the gases can be represented as follows:

$$p_{H_2}=0.5· p_{H_{2}O}^{sat}\left[\exp \left({\displaystyle \frac{1.635j}{T^{1.334}}}\right)\left({\displaystyle \frac{p_a}{p_{H_{2}O}^{sat}}}\right)\right]$$

(24)

$$p_{O_2}=p_{H_{2}O}^{sat}\left[\exp \left({\displaystyle \frac{4.192j}{T^{1.334}}}\right)\left({\displaystyle \frac{p_c}{p_{H_{2}O}^{sat}}}\right)\right]$$

(25)

where

• $p_a$ and $p_c$ are the partial pressures at the anode and cathode, respectively.
• $j$ is the current density in [A/cm²].
• $p_{H_{2}O}^{sat}$ is the saturation pressure of water, evaluated with the following equation:

  $${\log }_{10}\left(p_{H_{2}O}^{sat}\right)=-2.18+2.95e^{-2}T-9.18e^{-5}{T}^2+1.44e^{-7}{T}^3$$

  (26)

Voltage Losses

The cell voltage is then computed by subtracting various losses from the Nernst voltage evaluated from Equation (21). These include activation, ohmic and concentration overpotentials. However, as the model used [28] includes oxygen concentration within the activation losses, only activation and ohmic losses are subtracted here.

The activation overpotential is modeled as follows:

$$V_{act}=B_1+B_{2}T+B_{3}T\ln \left(C_{O_2}\right)+B_{4}T\ln \left(j\right)$$

(27)

where $B_1$, $B_2$, $B_3$, $B_4$ are parameters opportunely tuned to match experimental data.

The ohmic overpotential is given by the following:

$$V_{ohm}=j\left(R_{ohm}+R_C\right)$$

(28)

where $R_C$ is the electron resistance (assumed 0.008 Ω cm²) and $R_{ohm}$ is the proton resistance, which can be evaluated as follows in accordance with Springer et al. [29]:

$$R_{ohm}={\displaystyle \frac{t_m}{{\sigma }_m}}$$

(29)

Here, $t_m$ is the membrane thickness (cm) and ${\sigma }_m$ represents the membrane conductivity, which is a function of the membrane water content ${\lambda }_m$ and the stack temperature.

In this way, the dependency of the fuel cell voltage on the molar concentration of oxygen is properly accounted for.

3. Results and Discussion

The altitude boundary condition, ranging from 0 to 3000 m, was imposed on the fuel cell system. Variations in altitude affect environmental parameters such as ambient air pressure and density, resulting in reduced oxygen availability at higher elevations. Evaluating compressor performance under these conditions is essential to ensure an adequate oxygen supply to the fuel cell cathode, which directly influences system efficiency.

3.1. Fuel Cell Stack Voltage, Current and Power

Figure 4 illustrates the time-dependent behavior of key fuel cell stack parameters, including voltage, current, and power. In Figure 4a, the voltage remains stable at approximately 215 V. In contrast, Figure 4b shows a slight increase in current demand, reaching approximately 187 A over the time interval. Consequently, based on the relationship $P=VI$, Figure 4c demonstrates that the fuel cell’s power output is around 40 kW.

3.2. Compressor Performance

High-altitude conditions significantly impact both the fuel cell and compressor performance, as variations in altitude affect air pressure, temperature, and density. Figure 5 presents the compressor performance parameters, including power, voltage, efficiency, mass flow rate, compression ratio, and compressor map. Figure 5a illustrates the variations in compressor power over time. As altitude rises, compressor power demand rises, showing an increasing trend from 2.6 kW at sea level to 4.2 kW at 3000 m. This is primarily due to oxygen scarcity at higher altitudes, which forces the compressor to work harder to supply sufficient oxygen to the fuel cell cathode. Consequently, compressor power consumption increases, and, as shown in Figure 5b, the motor voltage rises from 123 V to 162V. Figure 5c depicts the compressor efficiency, which declines slightly with increasing altitude due to the reduced oxygen availability, leading to an increase in compressor work to provide the desired pressure at the cathode. Figure 5d,e illustrate the mass flow rate and pressure ratio of the compressor, respectively. According to Figure 5d, the mass flow rate remains constant due to a balance between air density and compressor speed. In other words, as altitude increases, air density decreases, and the compressor compensates by increasing its speed to maintain a constant mass flow rate over time. Figure 5e shows that the compression ratio follows an increasing trend. This is due to the reduction in ambient pressure at the compressor intake, which forces the compressor to operate at a higher pressure ratio to maintain performance.

Figure 5f illustrates the compressor performance map as part of the fuel cell system designed for altitudes ranging from 0 to 3000 m. In this study, the compressor model has been analyzed across a range of rotational speeds, spanning from 30,000 rev/min to 105,000 rev/min. The characteristic behavior of the compressor is represented by the characteristic curve, which consists of multiple curves obtained by varying the rotational speed. The surging line and the choking line bound these curves. The surge line defines the minimum stable flow rate through the compressor. Operating near or beyond this line can cause backflow and pressure oscillations, leading to potential system instability or damage. On the other hand, the choking line represents the maximum flow capacity of the compressor at a given pressure ratio. Beyond this line, the mass flow rate cannot increase, even with a reduction in the pressure ratio. Contours between the surging and choking lines correspond to interpolated iso-efficiency lines, indicating regions of constant efficiency. In the present study, interpolated efficiency surfaces were implemented in the compressor model to achieve accurate predictions of compressor performance. Furthermore, with the inclusion of operating points, as shown in Figure 5f, this study enables the determination of compressor efficiency values.

3.3. Net Power and Stack Power

As demonstrated in Figure 6, net power and stack power are compared over time. It is evident that net power is lower than stack power due to the power consumption of auxiliary components, particularly the compressor, which ensures an adequate oxygen supply for the fuel cell system. As previously mentioned in Section 3.1, given that the current demand and voltage output remain almost constant, the stack power follows a steady trend over the simulated period. In contrast, net power gradually decreases due to the increasing power consumption of the compressor motor in the air supply system.

3.4. Cathode Pressure and Oxygen Pressure

Figure 7 illustrates the variation in cathode pressure over time. As altitude rises, cathode pressure—comprising both nitrogen and oxygen partial pressure—decreases due to the decline in ambient pressure at higher altitudes.

At the molecular level, the decline in ambient pressure with altitude reduces the number of air molecules—including oxygen-per unit volume. This directly decreases the oxygen partial pressure at the cathode. According to the Nernst equation, a drop in $O_2$ availability reduces the reversible voltage of the cell. As a result, the overall fuel cell performance degrades due to the reduced driving force for the electrochemical reactions.

Figure 8 illustrates the oxygen partial pressure in the cathode of the fuel cell system over time. It is clear that, with increasing altitude, oxygen pressure gradually decreases throughout the time interval due to oxygen starvation at higher altitudes.

4. Conclusions

In the present study, iso-efficiency lines were integrated into the compressor model, and a PEMFC was employed as an electrochemical power source. Additionally, altitude was imposed as a boundary condition, varying from 0 to 3000 m. Under altitude-dependent conditions, key performance characteristics, including compressor efficiency, cathode pressure, and the compressor map, were analyzed and reported.

The findings indicate that, as altitude increases, compressor power consumption rises due to the reduction in oxygen pressure at higher altitudes. Ambient parameters, such as air pressure and density, decrease at higher altitudes. Consequently, the compressor in the air supply system plays a critical role in compensating for oxygen pressure deficiencies by increasing the compression ratio. Despite the reduction in air density, the mass flow rate remains constant due to a compensatory increase in compressor speed. The pressure ratio increases due to the declining oxygen pressure at the compressor inlet, leading to higher compressor work to maintain an adequate oxygen supply.

The integration of iso-efficiency lines into the compressor map provides valuable insights into efficiency variations under different environmental conditions. Furthermore, tracking operating points within the compressor map enables a detailed evaluation of efficiency trends, particularly in relation to surge and choke lines.

Future work should explore a broader sensitivity analysis, incorporating humidity effects, extended temperature ranges, and transient flight profiles, to better evaluate system behavior under more realistic and extreme aviation scenarios.