Advanced Modeling of Hydrogen Turbines Using Generalized Conformable Calculus

Abstract

This article addresses critical challenges in the transition to clean energy sources by highlighting the importance of advanced mathematical modeling and computational techniques in turbine design and operation. Specifically, we extend and generalize the work of Camporeale to advance the modeling of hydrogen turbine systems. By utilizing conformable calculus, we develop dynamic equations that analyze key aspects of turbine performance, including temperature variations in turbine blades, angular velocities of rotating shafts, and mass–energy balances within the plenum and combustion chamber. Furthermore, we incorporate Kirchhoff’s equation in its generalized conformable integral form, enhancing the precision of energy balance calculations and improving the representation of heat transfer processes in the combustion chamber. This methodology introduces novel perspectives in hydrogen turbine research, contributing to the advancement of sustainable and efficient technologies. Our comprehensive approach aims to provide more accurate and efficient predictions of turbine behavior, thereby impacting the design and optimization of hydrogen-based clean energy systems.

1. Introduction

With the rapid progression of climate change, the shift to fuels that produce lower greenhouse gas emissions is crucial for ensuring the long-term sustainability of humanity. In the realm of turbine development, significant strides have been made toward hydrogen-based technologies. For instance, Ref. [1] explores advancements in material design tailored for hydrogen turbines, while Ref. [2] demonstrates the implementation of hydrogen for sustainable development, particularly highlighting the economic advantages of hydrogen gas turbine (HGT) configurations in modernizing power plants, such as those in Bangladesh. Turbines are a cornerstone in the energy generation industry, predominantly powered by hydrocarbons—the most widely used fuels globally. However, hydrocarbons are a major contributor to ozone layer depletion and the greenhouse effect, posing a significant threat to environmental stability [3].

Hydrogen, being one of the cleanest energy sources, is now at the forefront of renewable energy research. When used in turbine systems, it provides a high-energy, low-emission alternative to traditional fossil fuels, offering a promising pathway for achieving sustainability targets. Hydrogen combustion, despite its benefits, introduces unique challenges, particularly in accurately modeling the combustion and flow dynamics in turbines. These dynamics are often highly non-linear, complex, and prone to deviations from ideal behavior, making them difficult to model using conventional techniques. Addressing these challenges requires advanced mathematical models that can capture the real-world, non-ideal behavior of turbine systems powered by hydrogen. Additionally, the optimization of energy extraction in hydrogen turbines can be achieved through novel approaches like the use of fractional calculus. By allowing models to incorporate varying degrees of memory and complex dynamics, fractional calculus provides the flexibility needed to represent non-linearities and oscillations that are typical in real-world turbine operations.

Implementing models based on conformable derivatives is justified by their ability to more accurately represent real-world dynamics compared to traditional idealized models. Despite their relative complexity, conformable derivatives offer a more precise modeling approach, capturing nuances that classical methods often overlook [4]. This advanced modeling technique has already shown promise in predicting the behavior of energy production systems, as demonstrated in previous works like [5], where conformable derivatives were employed for the prediction of electric power generation systems.

In addition to the use of conformable calculus, incorporating fuzzy logic could significantly enhance turbine models by addressing uncertainties and imprecise variations in critical parameters such as blade temperature, angular velocity, and mass flow within the combustion chamber. Unlike traditional models, which rely on rigidly defined parameters, fuzzy logic allows for a more adaptable and flexible representation of operating conditions, accommodating the variability and non-linearities that are typical in real-world turbine systems [6,7,8]. Although this study focuses on a deterministic approach, using conformable derivatives to capture anomalous behaviors in hydrogen-based turbines, future work could benefit from integrating fuzzy logic to further refine model precision and adaptability. However, the exploration of fuzzy methods falls outside the scope of this paper, as we concentrate on conformable calculus. Such an integration would provide a more robust framework for managing uncertainties, thus improving both predictive accuracy and operational flexibility in future studies.

This model can be effectively applied to predict anomalous combustion processes within a gas turbine. Dieguez et al. [9] present a methodology for identifying anomalies in combustion systems, which can lead to significant operational challenges. The model proposed in this work expands on these findings by incorporating generalized conformable derivatives, offering a more accurate prediction of such anomalous behaviors. Specifically, the fractional-order parameter $\alpha$ represents a delay coefficient that may describe thermal delay, spatial delay, or general temporal delay, depending on the parameters governing the conformable function. This allows the model to capture deviations from standard combustion dynamics with greater precision. Additionally, ${}^{\psi }D_t^{\alpha }f\left(t\right)=\psi (t,\alpha )f^{\prime }\left(t\right)$ represents an anomalous rate of change (an anomalous slope), both in the temporal and spatial dimensions, providing a comprehensive framework for understanding and predicting irregular combustion behavior. This approach significantly enhances the reliability and performance of gas turbines operating under non-ideal conditions.

Moreover, by incorporating the concept of generalized conformable derivatives, this study provides a novel method for detecting and predicting anomalous behaviors in turbine systems, particularly in the combustion process. This approach enhances the reliability and performance of gas turbines operating under non-ideal conditions, offering a valuable tool for both the design and optimization of future energy systems.

Our main findings indicate that incorporating conformable derivatives into turbine models results in simulations that closely mirror real-world dynamics. This enhanced modeling approach underscores the ability of conformable derivatives to refine system behavior predictions, making the models more representative of actual turbine operations. The key conclusion drawn from this study is that the implementation of conformable derivatives enables a sophisticated manipulation of the factors affecting turbine dynamics, allowing for adjustments through the careful choice of conformable functions rather than the extensive introduction of multiple parameters.

The structure of this paper is as follows: Section 2 introduces the preliminaries of generalized conformable derivatives, while Section 3 provides an overview of hydrogen turbines. In Section 4, we detail the modifications and development of the mathematical models. Section 5 presents the simulation results using Python. Section 6 discusses the results in detail. Finally, Section 7 concludes the paper.

2. Fundamentals and Preliminaries of the Conformable Derivative

This section aims to introduce the fundamental concepts and mathematical preliminaries of the conformable derivative. We will explore its definition, key properties, and how it differs from and extends traditional derivative concepts. Understanding these fundamentals is crucial for effectively applying the conformable derivative to model the dynamic behavior of hydrogen turbines. The conformable derivative is a recent advancement in the field of fractional calculus, designed to address some of the limitations of traditional integer-order derivatives. Inspired by the derivative introduced by Khalil et al. [10], the conformable derivative offers a flexible framework that can capture the memory and hereditary properties of various physical and engineering processes. Gateaux further extended this concept by referencing Khalil’s derivative to propose the linear extended Gâteaux derivative, as presented in [11], where the generalized conformable derivative was also introduced.

Definition 1 

(Conformable function). A conformable function is a function $\psi (t,\alpha )$ that is differentiable with respect to t and satisfies the following properties:

$$\begin{array}{cc}\hfill \psi (t,1)& =1,\hfill \\ \hfill \psi (t,\alpha )& \ne \psi (t,\beta ),\mathit{where}\alpha ,\beta \in (0,1],\hfill \\ \hfill \psi (t,\alpha )& \ne 0,\mathit{for}\mathit{all}\alpha \in (0,1].\hfill \end{array}$$

Definition 2 

(Generalized Conformable Derivative). Consider a conformable function $\psi (t,\alpha )$ and a function $f:\mathbb{R}\to \mathbb{R}$ that is differentiable at t. The generalized conformable derivative of order α at t, denoted ${\mathrm{a}\mathrm{s} }^{\psi }{D}_t^{\alpha }f\left(t\right)$, extends the concepts of classical integer-order derivatives and fractional derivatives to functions $\psi (t,\alpha )$ and $f\left(t\right)$. This derivative is defined by the formula

$$\begin{array}{cc}\hfill {}^{\psi }D_t^{\alpha }f\left(t\right)& =\underset{\epsilon \to 0}{lim}{\displaystyle \frac{f(t+\epsilon \psi (t,\alpha \left)\right)-f\left(t\right)}{\epsilon }}.\hfill \end{array}$$

(1)

Here, the parameter α in the interval $(0,1]$ determines the scale of the derivative. The generalized conformable derivative captures the immediate change rate of the function $f\left(t\right)$, taking into account its fractal characteristics and the variability introduced by the conformable function $\psi (t,\alpha )$ [12]. By applying L’Hôpital’s rule to (1), the expression can be reformulated as

$$\begin{array}{cc}\hfill {}^{\psi }D_t^{\alpha }f\left(t\right)& =\psi (t,\alpha ){\displaystyle \frac{d}{dt}}f\left(t\right).\hfill \end{array}$$

(2)

Definition 3 

(Generalized Conformable Partial Derivative). Consider a conformable function ψ and a multivariable function $f:{\mathbb{R}}^n\to \mathbb{R}$ that is differentiable at $(x_1,x_2,\dots ,x_n)$. The generalized conformable partial derivative of order α with respect to $x_i$, denoted as ${}^{\psi }P_{x_i}^{\alpha }f(x_1,x_2,\dots ,x_n)$, extends the concepts of classical integer-order partial derivatives and fractional derivatives by incorporating the conformable function $\psi (x_i,\alpha )$. This derivative is defined by the formula

$$\begin{array}{cc}\hfill {}^{\psi }P_{x_i}^{\alpha }f(x_1,x_2,\dots ,x_n)& =\underset{\epsilon \to 0}{lim}{\displaystyle \frac{f(x_1,\dots ,x_i+\epsilon \psi (x_i,\alpha ),\dots ,x_n)-f(x_1,x_2,\dots ,x_n)}{\epsilon }}.\hfill \end{array}$$

(3)

Here, the parameter α in the interval $(0,1]$ influences the scale of the derivative. The generalized conformable partial derivative captures the rate of change of the function $f(x_1,x_2,\dots ,x_n)$ with respect to the variable $x_i$, incorporating both fractal characteristics and variability introduced by the conformable function $\psi (x_i,\alpha )$.

Applying L’Hôpital’s rule to (3), the expression simplifies to

$$\begin{array}{cc}\hfill {}^{\psi }P_{x_i}^{\alpha }f(x_1,x_2,\dots ,x_n)& =\psi (x_i,\alpha ){\displaystyle \frac{\partial f(x_1,x_2,\dots ,x_n)}{\partial x_i}}.\hfill \end{array}$$

(4)

This formulation allows the conformable derivative to flexibly adapt the differentiation process based on the properties of $\psi (x_i,\alpha )$, thus generalizing the concept of partial derivatives to accommodate more complex system behaviors [12].

Theorem 1 

(Properties of the Generalized Conformable Derivative). Let $f:A\to B_1$ and $g:A\to B_2$ be functions defined on A, where $A,B_1,B_2\subset \mathbb{R}$ are α-differentiable functions, with $\alpha \in (0,1]$ as defined in Definition 2, for $t\in A\subset \mathbb{R}$. Consider $\phi (t,\alpha ):A\times (0,1]\to {\mathbb{R}}^{+}$, where ${\mathbb{R}}^{+}$ represents the set of non-negative real numbers including zero. Let a, b, and c be real constants. Then, the following properties hold:

1.

${}^{\psi }D_t^{\alpha }c=0$.

2.

${}^{\psi }D_t^{\alpha }(af\left(t\right)+bg\left(t\right))=a{}^{\psi }D_t^{\alpha }f\left(t\right)+b{}^{\psi }D_t^{\alpha }g\left(t\right)$.

3.

${}^{\psi }D_t^{\alpha }\left(f\left(t\right)g\left(t\right)\right)=f\left(t\right){}^{\psi }D_t^{\alpha }g\left(t\right)+g\left(t\right){}^{\psi }D_t^{\alpha }f\left(t\right)$.

4.

${}^{\psi }D_t^{\alpha }\left({\displaystyle \frac{f\left(t\right)}{g\left(t\right)}}\right)={\displaystyle \frac{g\left(t\right){}^{\psi }D_t^{\alpha }f\left(t\right)-f\left(t\right){}^{\psi }D_t^{\alpha }g\left(t\right)}{{\left[g\left(t\right)\right]}^2}}$.

5.

${}^{\psi }D_t^{\alpha }(f\circ g)\left(t\right)=f^{\prime }\left(g\left(t\right)\right){}^{\psi }D_t^{\alpha }g\left(t\right)$.

For the proof of this Theorem, refer to [12].

Definition 4 

(Generalized Conformable Integral). Let $a\ge 0$, $t\ge a$, f be a function defined on $(a,t]$, and $\psi (t,\alpha )$ be a conformable function defined on $(a,t]\times (0,1]$. Then, the generalized α-integral conformable of f is defined as

$$\begin{array}{c}\hfill {}_a{}^{\psi }I_t^{\alpha }f\left(t\right)={\int }_a^t{\displaystyle \frac{f\left(\tau \right)}{\psi (\tau ,\alpha )}}d\tau .\end{array}$$

(5)

The operators ${}^{\psi }D_t^{\alpha }$ and ${}_a{}^{\psi }I{t}^{\alpha }$ have been demonstrated to be inverses and to satisfy the Second Fundamental Theorem of Calculus, as proven in [12], Theorems 10 and 11.

Theorem 2 

(First Fundamental Theorem for Generalized Fractional Conformable Derivatives, Theorem 10 of [12]). Let $a>0$, $t>a$, and $\alpha \in (0,1]$. If f is a continuous function such that ${}^{\psi }aI{t}^{\alpha }f\left(t\right)$ exists, then

$${}^{\psi }D_t^{\alpha }\left({}_a{}^{\psi }I_t^{\alpha }f\left(t\right)\right)=f\left(t\right).$$

(6)

Theorem 3 

(Second Fundamental Theorem for Generalized Fractional Conformable Derivatives, Theorem 11 of [12]). Let $a>0$, $t>a$, and $\alpha \in (0,1]$. If f is differentiable on $[a,\infty )$, then

$${}_a{}^{\psi }I_t^{\alpha }\left({}^{\psi }D_t^{\alpha }f\left(t\right)\right)=f\left(t\right)-f\left(a\right).$$

(7)

Remark 1. 

The conformable functions used in these models, as defined in (41), are constructed such that each parameter is carefully selected to ensure the functions remain dimensionless. This approach prevents any interference with the dimensional analysis of the physical models, maintaining the integrity and consistency of the overall system.

Definition 5. 

A system of conformable differential equations is a system of equations involving conformable derivatives that describe the dynamics of a set of dependent variables $x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)$ with respect to an independent variable t. The system is given by

$$\begin{array}{cc}\hfill {}^{{\psi }_1}D_t^{{\alpha }_1}{x}_1\left(t\right)& =f_1(x_1,x_2,\dots ,x_n,t),\hfill \\ \hfill {}^{{\psi }_2}D_t^{{\alpha }_2}{x}_2\left(t\right)& =f_2(x_1,x_2,\dots ,x_n,t),\hfill \\ \hfill & ⋮\hfill \\ \hfill {}^{{\psi }_n}D_t^{{\alpha }_n}{x}_n\left(t\right)& =f_n(x_1,x_2,\dots ,x_n,t),\hfill \end{array}$$

subject to the initial conditions

$${(x_1\left(t_0\right),x_2\left(t_0\right),\dots ,x_n\left(t_0\right))}^T={(x_{1,0},x_{2,0},\dots ,x_{n,0})}^T,$$

Definition 6. 

A system of conformable partial differential equations is a set of equations involving conformable derivatives that describe the dynamics of a set of dependent variables $x_1\left(\mathbf{t}\right),x_2\left(\mathbf{t}\right),\dots ,x_n\left(\mathbf{t}\right)$, where $\mathbf{t}=(t_1,t_2,\dots ,t_m)$ represents a vector of independent variables. The system is defined by

$$\begin{array}{cc}\hfill {}^{{\psi }_1}P_{t_1}^{{\alpha }_1}{x}_1\left(\mathbf{t}\right)& =f_1(x_1,x_2,\dots ,x_n,\mathbf{t}),\hfill \\ \hfill {}^{{\psi }_2}P_{t_2}^{{\alpha }_2}{x}_2\left(\mathbf{t}\right)& =f_2(x_1,x_2,\dots ,x_n,\mathbf{t}),\hfill \\ \hfill & ⋮\hfill \\ \hfill {}^{{\psi }_n}P_{t_m}^{{\alpha }_n}{x}_n\left(\mathbf{t}\right)& =f_n(x_1,x_2,\dots ,x_n,\mathbf{t}),\hfill \end{array}$$

subject to the initial and/or boundary conditions, depending on the context and specific variables involved:

$${(x_1\left({\mathbf{t}}_0\right),x_2\left({\mathbf{t}}_0\right),\dots ,x_n\left({\mathbf{t}}_0\right))}^T={(x_{1,0},x_{2,0},\dots ,x_{n,0})}^T,$$

Definition 7. 

A system of conformable differential equations, whether partial or ordinary, is said to be homogeneous in its conformable function if all equations in the system use the same conformable function. Similarly, the system is homogeneous in its order if all equations use the same order of conformable derivatives.

3. Preliminaries of Hydrogen Turbines

Hydrogen is gaining attention as a versatile and sustainable energy carrier with the potential to significantly reduce emissions in various sectors, particularly in hard-to-abate industries such as heavy industry and transportation. The drive towards hydrogen-powered systems is motivated by the global need to reduce greenhouse gas emissions and transition towards cleaner energy sources, aligning with targets to achieve near-zero carbon dioxide (CO₂) emissions by 2050 [13].

In the context of future energy systems, hydrogen can play a key role by integrating electrolyzers, hydrogen storage, and gas turbines fueled by hydrogen, which together facilitate the decarbonization of the energy sector. A study by [13] employs a techno-economic optimization model to evaluate the competitiveness of hydrogen in electricity generation across 15 European countries by the years 2030, 2040, and 2050. The results indicate that hydrogen-based gas turbines, combined with emissions caps, can substantially contribute to reducing CO₂ levels, highlighting hydrogen’s critical role in future energy solutions.

Hydrogen turbines also offer a significant advantage in industrial processes by efficiently recovering energy. As noted by [14], hydrogen turbines can replace traditional pressure relief valves in refineries, which typically lead to energy loss. Instead, these turbines capture the expansion work of hydrogen-rich streams and convert it into useful energy, reducing the total power consumption and operational costs. For instance, implementing hydrogen turbines in a refinery network reduced total power consumption by 3.9% and annualized costs by 0.6%, showcasing both economic and environmental benefits.

The shift towards hydrogen in gas turbines addresses the pressing concerns associated with traditional fossil fuels such as oil and coal. Hydrogen offers a cleaner alternative with zero emissions and high energy content, making it a compelling option for power generation. As reviewed by [15], hydrogen-fueled gas turbines (GTs) significantly reduce greenhouse gas emissions and enhance turbine performance, influenced by factors such as operating conditions and ambient parameters. These findings underscore hydrogen’s potential to align power generation with global sustainability goals.

Figure 1 illustrates a hydrogen turbine model; the turbine was designed by Turbine Services Chromalloy, extracted from [16], providing a visual representation of the advanced technology being developed for hydrogen-based energy systems. Overall, hydrogen turbines represent a transformative technology with the capability to improve energy efficiency, reduce emissions, and support the transition to a carbon-neutral future.

4. Development and Modification of the Mathematical Model of the Turbine

In this section, we present the mathematical models for a gas turbine as developed by Camporeale et al. [17], which has been adapted for hydrogen fuel use. This model encompasses the main components of a gas turbine, including the plenum, combustion chamber, rotating shaft, blade temperatures, and turbine temperatures. Originally designed for hydrocarbon fuels, the modifications in our study reflect the unique characteristics of hydrogen, emphasizing its impact on the combustion process and turbine dynamics. The turbine temperature model, in particular, is adapted from the work of London et al. [18], demonstrating the thermal response within the turbine stages during gas expansion.

The turbine system is divided into several key stages: the compressor, which pressurizes the intake air; the plenum, which serves as a buffer to stabilize pressure variations; the combustion chamber, where the hydrogen fuel mixes with air and undergoes combustion; and the turbine itself, which converts the energy of the expanding gases into mechanical work. The combustion chamber and plenum models employ mass and energy balances to accurately represent the dynamics of hydrogen combustion and its influence on pressure and temperature fluctuations within the system.

Additionally, the model integrates the dynamics of the rotating shaft, which directly connects the compressor and turbine stages. This analysis ensures that the angular velocity and stability of the shaft are maintained, impacting overall turbine performance. The blade and turbine temperature models provide insights into the heat transfer processes occurring in these components, which are crucial for maintaining operational efficiency and preventing thermal stress on the turbine materials.

Figure 2 illustrates the various stages of the turbine, highlighting the flow of air and fuel through the system and the key components involved in the energy conversion process. This diagram provides a clear visualization of how the integrated components work together to facilitate the turbine’s operation using hydrogen as a fuel source.

Overall, this adapted model provides a comprehensive framework for analyzing the dynamic behavior of hydrogen-fueled turbines. By incorporating conformable derivatives, the model captures the complex interactions between different system components, enhancing the accuracy of simulations and supporting the development of more efficient and sustainable turbine technologies.

4.1. Balance of the Hydrogen Combustion Reaction

First, we perform the energy balance of the hydrogen combustion reaction. This is

$$\begin{array}{c}\hfill 2H_2+O_2\to 2H_{2}O.\end{array}$$

(8)

For calculating the energy released by the chemical reaction, we use the Kirchhoff equation [19,20,21], which is expressed as

$$\mathrm{\Delta }{H}_r\left(T\right)=\mathrm{\Delta }{H}_r^o+{\int }_{{25}^{o}C}^T\mathrm{\Delta }{C}_p\left(\tau \right)d\tau ,$$

(9)

$\mathrm{\Delta }{C}_p\left(T\right)$ represents the difference in heat capacities at constant pressure between the products and reactants and $\mathrm{\Delta }{H}_r^o$ denotes the enthalpy change of the reaction under standard conditions. They are given by

$$\begin{array}{cc}\hfill \mathrm{\Delta }{C}_p\left(T\right)& =\sum _{\mathrm{prod}}{\nu }_i{C}_{p_i}\left(T\right)-\sum _{\mathrm{react}}{\nu }_i{C}_{p_i}\left(T\right),\hfill \\ \hfill \mathrm{\Delta }{H}_r^o& =\sum _{\mathrm{prod}}{\nu }_i{H_r^o}_i-\sum _{\mathrm{react}}{\nu }_i{H_r^o}_i.\hfill \end{array}$$

where ${\nu }_i$ denotes the stoichiometric coefficients of the products and reactants and ${H_r^o}_i$ represents the standard enthalpy of formation of each component, measured in kJ/mol.

In the case of a hydrogen turbine, it is assumed that oxygen is sourced from the air, which contains traces of nitrogen that absorb energy. Thus, the enthalpy released can be calculated using the following specific terms in Equation (8):

$$\begin{array}{cc}\hfill \mathrm{\Delta }{C}_p\left(T\right)& =2{C_p}_{H_{2}O}\left(T\right)-2{C_p}_{H_2}\left(T\right)-{C_p}_{O_2}\left(T\right),\hfill \\ \hfill \mathrm{\Delta }{H}_r^o& =2{H_r^o}_{H_{2}O}-2{H_r^o}_{H_2}-{H_r^o}_{O_2},\hfill \end{array}$$

(10)

where the enthalpies are ${H_r^{\circ }}_{H_{2}O}=-241.83$ kJ/mol and ${H_r^{\circ }}_{O_2}={H_r^{\circ }}_{H_2}=0$ in the gas phase, resulting in $\mathrm{\Delta }{H}_r^{\circ }=-483.66$ kJ/mol. The specific heat capacities ${C_p}_{H_{2}O}\left(T\right)$, ${C_p}_{H_2}\left(T\right)$, and ${C_p}_{O_2}\left(T\right)$, provided by [20], are shown below:

$$\begin{array}{cc}\hfill {C_p}_{H_{2}O}\left(T\right)& =33.46\times {10}^{-3}+0.688\times {10}^{-5}T+0.7604\times {10}^{-8}{T}^2-3.593\times {10}^{-12}{T}^3,\hfill \\ \hfill {C_p}_{H_2}\left(T\right)& =28.84\times {10}^{-3}+0.00765\times {10}^{-5}T+0.3288\times {10}^{-8}{T}^2-0.8698\times {10}^{-12}{T}^3,\hfill \\ \hfill {C_p}_{O_2}\left(T\right)& =29.10\times {10}^{-3}+1.058\times {10}^{-5}T-0.6076\times {10}^{-8}{T}^2+1.311\times {10}^{-12}{T}^3.\hfill \end{array}$$

(11)

The expressions in (11) can be used in the temperature range from $0°\mathrm{C}$ to $1200°\mathrm{C}$ in the gas phase. On the other hand, we can write $\mathrm{\Delta }{C}_p\left(T\right)$. The units for $C_p$ are $\mathrm{kJ}/\mathrm{mol}\mathrm{K}$ or $\mathrm{kJ}/\mathrm{mol}°\mathrm{C}$, using (11) as follows:

$$\mathrm{\Delta }{C}_p\left(T\right)=-19.86\times {10}^{-3}+0.2027\times {10}^{-5}T+1.4708\times {10}^{-8}{T}^2-6.7674\times {10}^{-12}{T}^3.$$

(12)

Furthermore, we extend the model to consider transient scenarios induced by changes in fuel flow and compressor guide vane geometry. By analyzing these transient cases and obtaining a linearized model, we aim to identify the dynamic behavior of the hydrogen-based gas turbine and design a multivariable controller for enhanced performance.

If we introduce the conformable integral to model anomalous combustion, Equation (9) can be written as follows:

$$\begin{array}{c}\hfill \mathrm{\Delta }{H}_r\left(T\right)=\mathrm{\Delta }{H}_r^o+{\int }_{{25}^{o}C}^T{\displaystyle \frac{\mathrm{\Delta }{C}_p\left(\tau \right)}{\psi (\tau ,\alpha )}}d\tau ,\end{array}$$

(13)

where $\psi (T,\alpha )$ is a conformable function defined in $(0,1]\times {\mathbb{R}}^{+}$. In this context, the function $\psi (T,\alpha )$ is an anomalous combustion rate, which reflects how the system approaches thermal equilibrium as a function of temperature. The parameter $\alpha$ represents thermal delay coefficients, capturing deviations from standard combustion behavior. This adjustment enables the model to account for irregularities in the combustion process, providing a more accurate representation of how the system responds to changes in temperature and combustion dynamics.

4.2. Models for the Plenum and Combustion Chamber

The models for the plenum and combustion chamber are quite similar, both derived from performing mass and energy balances over control volumes. These models, originally obtained from [22], have been modified to incorporate generalized conformable derivatives. This modification allows for a more accurate representation of the dynamic behavior of the system.

The plenum in a turbine, a pressurized air chamber before the turbine’s inlet, serves to distribute airflow uniformly, increasing air pressure, and smoothing fluctuations, thereby enhancing turbine efficiency and stability. In some designs, it also integrates multiple air inlets for uniform distribution [23].

The combustion chamber in a gas turbine burns fuel with air from the compressor to release energy, causing air to expand and accelerate, providing a uniformly heated gas stream. This process minimizes pressure loss while maximizing fuel energy release. About 60–75% of the air flow is used to cool the gas and the flame tube walls, ensuring complete combustion before dilution air is introduced. An electric spark initiates combustion, which then becomes self-sustaining. Despite variations in design and fuel addition methods, the air flow distribution remains consistent [24].

For the plenum, the mass balance, accounting for anomalous behavior, can be expressed using generalized conformable derivatives as

$$\begin{array}{cc}\hfill V_p{}^{\psi }D_t^{\alpha }\left({\rho }_2\right)& =\dot{m_1}-\dot{m_2},\hfill \end{array}$$

(14)

where $V_p$ is the volume of the plenum, ${\rho }_2$ is the density at the exit of the plenum, $\dot{m_1}$ is the mass flow rate entering the plenum, and $\dot{m_2}$ is the mass flow rate leaving the plenum.

Similarly, the energy balance for the plenum is given by

$$\begin{array}{cc}\hfill V_p{}^{\psi }D_t^{\alpha }\left({\rho }_2{e}_2\right)& =\dot{m_1}{h}_1-\dot{m_2}{h}_2,\hfill \end{array}$$

(15)

where $e_2$ is the specific internal energy at the exit of the plenum, and $h_1$ and $h_2$ are the specific enthalpies at the inlet and outlet, respectively.

For the combustion chamber, the mass balance, accounting for anomalous behavior, can be formulated using generalized conformable derivatives as

$$\begin{array}{cc}\hfill V_{cc}{}^{\psi }D_t^{\alpha }\left({\rho }_{cc}\right)& =\dot{m_1}-\dot{m_2},\hfill \end{array}$$

(16)

which mirrors the mass balance equation for the plenum.

The energy balance for the combustion chamber, however, includes an additional term to account for the fuel input:

$$\begin{array}{cc}\hfill V_{cc}{}^{\psi }D_t^{\alpha }\left({\rho }_{cc}{e}_{cc}\right)& =\dot{m_1}{h}_1+\dot{m_f}(h_f+H_i)-\dot{m_2}{h}_2,\hfill \end{array}$$

(17)

where $\dot{m_f}$ is the mass flow rate of the fuel, $h_f$ is the specific enthalpy of the fuel, and $H_i$ is the heating value of the fuel.

The use of generalized conformable derivatives ${}^{\psi }D_t^{\alpha }$ in these equations allows us to model the fractional-order dynamics of the system, which can provide a more accurate depiction of the transient behavior in the plenum and combustion chamber. This approach enhances the predictive capability of the model, making it more suitable for analyzing complex, real-world systems.

These modifications, inspired by the work in [22], demonstrate the potential of fractional calculus in improving the fidelity of thermodynamic and fluid dynamic models in engineering applications.

We can consider two systems of generalized conformable differential equations. The first system consists of Equations (14) and (15). Additionally, a second system can be formulated using Equations (16) and (17). Both systems can be solved using classical methods. The solutions are as follows.

Plenum system:

$$\begin{array}{cc}\hfill V_p{\psi }_{p1}(t,{\alpha }_p){\displaystyle \frac{d{\rho }_2}{dt}}& =\dot{m_1}-\dot{m_2},{\rho }_2\left(t_0\right)={\rho }_{2,0},\hfill \\ \hfill V_p{\psi }_{p2}(t,{\alpha }_p){\displaystyle \frac{d\left({\rho }_2{e}_2\right)}{dt}}& =\dot{m_1}{h}_1-\dot{m_2}{h}_2,e_2\left(t_0\right)=e_{2,0}.\hfill \end{array}$$

(18)

The solution of (18) is

$$\begin{array}{cc}\hfill {\rho }_2\left(t\right)& ={\int }_{t_0}^t{\displaystyle \frac{\dot{m_1}-\dot{m_2}}{{\psi }_{p1}(\tau ,{\alpha }_{p1})V_p}}d\tau +{\rho }_{2,0},\hfill \\ \hfill e_2\left(t\right)& ={\displaystyle \frac{{\int }_{t_0}^t\frac{\dot{m_1}{h}_1-\dot{m_2}{h}_2}{V_p{\psi }_{p2}(\tau ,{\alpha }_{p2})}d\tau +{\rho }_{2,0}{e}_{2,0}}{{\int }_{t_0}^t\frac{\dot{m_1}-\dot{m_2}}{{\psi }_{p1}(\tau ,{\alpha }_{p1})V_p}d\tau +{\rho }_{2,0}}}.\hfill \end{array}$$

(19)

It is important to note that in the plenum system, if $m_1=m_2$, then ${\rho }_2=\mathrm{constant}$. On the other hand, if $m_1>m_2$, then ${\rho }_2$ is an increasing function. Finally, if $m_2<m_1$, then ${\rho }_2$ is a decreasing function. Additionally, $e_2$, the energy, is a variable that depends on time.

In an industrial plenum turbine, what enters the plenum is equal to what exits the plenum ($m_1=m_2$). This makes $\rho$ constant. Thus, the focus will be on the energy balance of the system, as $\rho$ being constant is of more interest.

Combustion chamber system:

$$\begin{array}{cc}\hfill V_{cc}{\psi }_{cc1}(t,{\alpha }_{cc1}){\displaystyle \frac{d{\rho }_{cc}}{dt}}& =\dot{m_1}+\dot{m_f}-\dot{m_2},{\rho }_{cc}\left(t_0\right)={\rho }_{cc,0},\hfill \\ \hfill V_{cc}{\psi }_{cc2}(t,{\alpha }_{cc2}){\displaystyle \frac{d\left({\rho }_{cc}{e}_{cc}\right)}{dt}}& =\dot{m_1}{h}_1+\dot{m_f}\left(h_f+H_i\right)-\dot{m_2}{h}_2,e_{cc}\left(t_0\right)=e_{cc,0}.\hfill \end{array}$$

(20)

The solution of (20) is

$$\begin{array}{cc}\hfill {\rho }_{cc}\left(t\right)& ={\int }_{t_0}^t{\displaystyle \frac{\dot{m_1}+\dot{m_f}-\dot{m_2}}{{\psi }_{cc1}(\tau ,{\alpha }_{cc1})V_{cc1}}}d\tau +{\rho }_{cc,0},\hfill \\ \hfill e_{cc}\left(t\right)& ={\displaystyle \frac{{\int }_{t_0}^t\frac{\dot{m_1}{h}_1+·m_f\left(h_f+H_i\right)-\dot{m_2}{h}_2}{V_{cc}{\psi }_{cc2}(\tau ,{\alpha }_{cc2})}d\tau +{\rho }_{cc,0}{e}_{cc,0}}{{\int }_{t_0}^t\frac{\dot{m_1}+\dot{m_f}-\dot{m_2}}{{\psi }_{cc1}(\tau ,{\alpha }_{cc1})V_{cc}}d\tau +{\rho }_{cc,0}}}.\hfill \end{array}$$

(21)

In this context, the functions ${\psi }_{p1}(t,{\alpha }_p)$ and ${\psi }_{p2}(t,{\alpha }_p)$ represent delay functions for achieving thermal equilibrium in the plenum, while ${\psi }_{cc1}(t,{\alpha }_{cc})$ and ${\psi }_{cc2}(t,{\alpha }_{cc})$ serve the same purpose for the combustion chamber. Specifically, the functions ${\psi }_{p1}(t,{\alpha }_p)$ and ${\psi }_{cc1}(t,{\alpha }_{cc})$ correspond to anomalous diffusion coefficients, whereas ${\psi }_{p2}(t,{\alpha }_p)$ and ${\psi }_{cc2}(t,{\alpha }_{cc})$ represent anomalous combustion coefficients. The parameters ${\alpha }_i$ denote temporal delay coefficients.

4.3. Simplified Modeling of Compressor Dynamics

In transient analysis, the compressor is assumed to exhibit quasi-steady behavior, allowing the application of steady-state compressor maps throughout the transient process. The compressor is equipped with variable inlet guide vanes (VIGVs).

However, detailed compressor maps including characteristic curves and surge lines for various VIGV positions are challenging to obtain from the open literature. Therefore, for the purposes of this study, a simplified approach is adopted. An axial compressor map is used without specific consideration of VIGVs, neglecting their effects on characteristic curves and surge line dynamics.

The variation in air mass flow due to VIGVs is expressed by

$${\dot{m}}_{a,\mathrm{VIGV}}=r_{\mathrm{VIGV}}{\dot{m}}_{a,1},$$

(22)

where $r_{\mathrm{VIGV}}$ represents the ratio of effective air mass flow to the air mass flow at the same rotational speed and pressure ratio when VIGVs are fully open ($r_{\mathrm{VIGV}}=1$).

4.4. Rotating Shaft

The rotational dynamics of the shaft, which connects the compressor, turbine, and electric generator, can be modeled using generalized conformable derivatives. This method provides a refined description of the shaft dynamic balance, as adapted from [22]. The equation governing the rotational acceleration, considering anomalous dynamics, can be expressed using generalized conformable derivatives as

$${}^{\psi }D_t^{\alpha }\left(\omega \right)={\displaystyle \frac{P^{*}}{J\omega }},\mathrm{and}{P}^{*}=P_T-P_C-P_E-P_F,$$

(23)

where J represents the moment of inertia of the shaft, $\omega$ is the angular velocity, $P_T$ is the power output from the turbine, $P_C$ is the power absorbed by the compressor stages connected to the shaft, $P_E$ is the power absorbed by the electric generator, and $P_F$ accounts for the power losses due to friction and other accessories.

For multi-shaft gas turbines, Equation (23) needs to be applied separately to each shaft to accurately describe the system’s dynamic behavior. This model is based on the dynamic balance principles presented in [22], modified to incorporate generalized conformable derivatives.

Equation (23) can be solved as follows:

$$\begin{array}{cc}\hfill {}^{\psi }D_t^{\alpha }\left(\omega \right)& ={\displaystyle \frac{P^{*}}{J\omega }},\omega \left(t_0\right)={\omega }_0,\hfill \\ \hfill \psi (t,\alpha ){\displaystyle \frac{d\omega }{dt}}& ={\displaystyle \frac{P^{*}}{J\omega }},\hfill \\ \hfill \omega d\omega & ={\displaystyle \frac{P^{*}}{J\psi (t,\alpha )}}dt,\hfill \\ \hfill {\displaystyle \frac{{\omega }^2}{2}}& ={\int }_{t_0}^t{\displaystyle \frac{P^{*}}{J\psi (\tau ,\alpha )}}d\tau +{\displaystyle \frac{{\omega }_0^2}{2}},\hfill \\ \hfill \omega \left(t\right)& =\sqrt{2{\int }_{t_0}^t{\displaystyle \frac{P^{*}}{J\psi (\alpha ,\tau )}}d\tau +{\omega }_0^2}.\hfill \end{array}$$

This can be expressed as follows:

$$\begin{array}{cc}\hfill \omega \left(t\right)& =\sqrt{2{}_{t_0}{}^{\psi }I_t^{\alpha }\left({\displaystyle \frac{P^{*}}{J}}\right)+{\omega }_0^2}.\hfill \end{array}$$

(24)

Here, $\psi (t,\alpha )$ represents the anomalous rotation coefficients, while $\alpha$ denotes the temporal delay coefficient.

4.5. Blade Temperature

The blade temperature estimation is carried out under the assumption that the blades maintain a uniform temperature, disregarding the temperature distribution of the hot gas. Due to the lack of detailed data regarding the internal geometry of the blades and the cooling system, this approach is considered sufficient to describe the thermal process [17,25,26].

For internal convection cooling, the unsteady heat flux balance across the blade wall with anomalous heat transfer can be described using the generalized conformable derivative of the blade temperature as

$$n_{bl}{C}_{bl}{}^{\psi }D_t^{\alpha }\left(T_{bl}\right)=h_{bl}{A}_{bl}(T_g-T_{bl})-{\dot{m}}_{cl}{c}_{p_{cl}}\mathrm{\Delta }{T}_{cl},$$

(25)

where $n_{bl}$ is the number of blades, $C_{bl}$ is the heat capacity of each blade, $A_{bl}$ is the external surface area of the blade, $h_{bl}$ is the convection heat transfer coefficient, $T_g$ is the gas temperature, $T_{bl}$ is the blade temperature, ${\dot{m}}_{cl}$ is the coolant mass flow rate, $c_{p_{cl}}$ is the specific heat of the coolant, and $\mathrm{\Delta }{T}_{cl}$ is the temperature rise of the coolant through the blade.

Camporeale et al. propose Equation (26) to calculate the mass flow rate [17]:

$${\dot{m}}_{cl}={\displaystyle \frac{p_2{K}_1}{\sqrt{T_1}}}\sqrt{1-{\displaystyle \frac{p_2^{\prime }}{p_1}}},$$

(26)

where $T_1$ and $p_1$ are evaluated at the point where the coolant is extracted, and the static pressure $p_2^{\prime }$ is evaluated at the row exit.

For nozzles, $T_g$ is represented by the absolute stagnation temperature at the stage inlet. For rotors, $T_g$ is represented by the rotor-relative stagnation temperature, determined from the velocity triangle.

In the case of blades cooled by both internal and film cooling, the gas temperature $T_g$ is replaced by the adiabatic wall temperature $T_{ad}$, which is derived from the blade film cooling effectiveness [17,25,26].

The coolant temperature rise, $\mathrm{\Delta }{T}_{cl}$, is related to the internal heat transfer efficiency by

$$\mathrm{\Delta }{T}_{cl}=\epsilon \left(T_{bl}-T_{cl,1}\right),$$

(27)

where $T_{cl,1}$ is the coolant temperature at the inlet of the blade’s internal cooling circuit, and $\epsilon$ is the heat transfer efficiency.

A non-dimensional approach, based on extensive experimental data, is utilized to evaluate the variations in heat transfer coefficients under off-design conditions. This methodology is particularly suitable for off-design analysis of gas turbines, providing a reliable tool for such evaluations [17]. If we combine Equations (25) and (27), we obtain

$$\begin{array}{cc}\hfill {}^{\psi }D_t^{\alpha }{T}_{bl}& ={\displaystyle \frac{A{T}_{bl}+B}{C_{bl}\left(T_{bl}\right)}},T_{bl}\left(t_0\right)=T_{bl,0},\hfill \end{array}$$

(28)

where A and B are defined as

$$\begin{array}{cc}\hfill A& ={\displaystyle \frac{h_{bl}{A}_{bl}-{\dot{m}}_{cl}{c}_{p_{cl}}\epsilon }{n_{bl}}},\hfill \\ \hfill B& ={\displaystyle \frac{h_{bl}{A}_{bl}{T}_g+{\dot{m}}_{cl}{c}_{p_{cl}}\epsilon T_{cl,1}}{n_{bl}}}.\hfill \end{array}$$

By considering the mean value of $C_{p_{bl}}$ as $C_{P_{bl}}^{*}$, the solution to Equation (28) is

$$T_{bl}\left(t\right)=\left(T_{bl,0}+{\displaystyle \frac{B}{A}}\right)exp\left({}_{t_0}{}^{\psi }I_t^{\alpha }\left({\displaystyle \frac{A}{C_{bl}^{*}}}\right)\right)-{\displaystyle \frac{B}{A}}.$$

(29)

Remark 2. 

It is important to note that B is positive, whereas A can take both positive and negative values, depending on the difference $h_{bl}{A}_{bl}-{\dot{m}}_{cl}{c}_{p_{cl}}\epsilon$. Furthermore, for metals, $C_p>0$, which implies that if $A<0$, $T_{bl}$ will decrease, whereas if $A>0$, $T_{bl}$ will increase.

Next, we consider $C_{bl}$ as a variable. Thus, the temperature of the blades can be expressed by the following generalized conformable differential equation:

$${}^{\psi }D_t^{\alpha }{T}_{bl}={\displaystyle \frac{A{T}_{bl}+B}{C_{bl}\left(T_{bl}\right)}}.$$

(30)

We can solve (30) as follows:

$$\begin{array}{c}\hfill {\int }_{T_{bl,0}}^{T_{bl}}{\displaystyle \frac{C_{bl}\left(\tau \right)}{A\tau +B}}d\tau ={}_{t_0}{}^{\psi }I_t^{\alpha }\left(1\right),\end{array}$$

(31)

where the conformable function $\psi$ represents an anomalous heat transfer coefficient, and $\alpha$ is the temporal delay coefficient.

The blades of wind turbines are commonly fabricated using high-performance composite materials, such as high-temperature resistant superalloys (HRSAs), carbon fiber-reinforced polymer (CFRP) composites, and titanium. HRSA materials, like Inconel and Waspaloy, offer excellent strength, fatigue resistance, and thermal stability, making them well suited for the demanding operating conditions of turbine blades. Additionally, CFRP composites provide a favorable strength-to-weight ratio and enable more aerodynamic blade designs, further improving the efficiency of wind turbines [27,28]. Titanium is also frequently used due to its exceptional strength, low density, and corrosion resistance, contributing to the overall durability and performance of wind turbine blades [29,30,31].

Remark 3. 

To evaluate heat capacity, it is common to use third-degree polynomials [20], such as

$$C_p\left(T\right)=A+BT+C{T}^2+D{T}^3,$$

(32)

These polynomials are obtained by fitting experimental data to provide accurate representations of the heat capacity over a range of temperatures.

Another model for the specific heat capacity $C_p^{\prime }$ can be expressed as follows [32]:

$$C_p\left(T\right)=3Rn\left(1+{\displaystyle \frac{k_1}{T}}+{\displaystyle \frac{k_2}{T^2}}+{\displaystyle \frac{k_3}{T^3}}\right)+(A+BT)+C_p^{\prime },$$

(33)

where R is the gas constant and n represents the number of atoms in the chemical formula. The parameters A and B are derived from the thermal expansion coefficient and isothermal bulk modulus data. The coefficients $k_i$ (for $i=1,2,3$) are obtained by fitting experimental low-temperature heat capacity data. The term $C_p^{\prime }$ accounts for deviations from the $3Rn$ limit in certain substances due to factors such as cation disorder, anharmonicity, and electronic contributions.

Another mathematical model that can be used is the one proposed by [33], which is given by

$$C_p\left(T\right)=a+bT+{\displaystyle \frac{c}{T}}$$

(34)

This model provides a simple and effective way to describe the heat capacity over a wide range of temperatures. The coefficients a, b, and c are obtained by fitting experimental data. According to the authors, this model has been successfully used to model the heat capacity of carbon fiber composites, providing a determination of the heat capacity of the decomposing composite with a relative standard deviation and of the heat of pyrolysis.

Another model frequently used is the one proposed in [34]:

$$C_p\left(T\right)=A+BT+{\displaystyle \frac{C}{T^2}}$$

(35)

This model describes the heat capacity with parameters A, B, and C obtained from fitting experimental data. It effectively captures the temperature dependence of the heat capacity for a variety of substances.

For this study, we use titanium and the heat capacity model proposed by [34]. The heat capacity is given by

$$C_p\left(T\right)=37.27944+4.76976\times {10}^{-3}T-{\displaystyle \frac{1811672}{T^2}}\left[{\displaystyle \frac{\mathrm{J}}{\mathrm{mol}·\mathrm{K}}}\right]$$

(36)

4.6. Turbine Temperature

The temperature distribution within the turbine is a critical factor influencing the performance and longevity of its components. The model presented here is derived from the work of Camporeale et al. [17], which in turn was based on the formulations originally introduced by London et al. [18]. This model describes the temperature dynamics of the hot gas, cooling air, and turbine walls through a set of coupled partial differential equations that consider the intricate heat transfer processes within the turbine system.

Equation (37) provides a mathematical framework for analyzing the temperature profiles within the turbine, employing a stage-by-stage approach to model the air-cooled turbine blades and the heat exchanger dynamics. The key variables in the model are as follows:

• $T_h^{*}$: Non-dimensional temperature of the hot gas, representing the temperature of the combustion gases flowing through the turbine.
• $T_c^{*}$: Non-dimensional temperature of the cooling air, which refers to the temperature of the air used to cool the turbine components.
• $T_w^{*}$: Non-dimensional wall temperature, representing the temperature of the turbine blades or walls in contact with the hot gas and cooling air.
• ${\theta }^{*}$: Non-dimensional time, defined as ${\theta }^{*}=t/t_d$, where t is the actual time and $t_d$ is the characteristic dwell time.
• $x^{*}$: Non-dimensional flow length, defined as $x^{*}=x/L$, where x is the axial position along the turbine with units of meters.

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial T_h^{*}}{\partial {\theta }^{*}}}& =a_1{\displaystyle \frac{\partial T_h^{*}}{\partial x^{*}}}+a_2{T}_h^{*}+a_3{T}_w^{*},\hfill \\ \hfill {\displaystyle \frac{\partial T_c^{*}}{\partial {\theta }^{*}}}& =b_1{\displaystyle \frac{\partial T_c^{*}}{\partial x^{*}}}+b_2{T}_c^{*}+b_3{T}_w^{*},\hfill \\ \hfill {\displaystyle \frac{\partial T_w^{*}}{\partial {\theta }^{*}}}& =c_1{T}_h^{*}+c_2{T}_w^{*}+c_3{T}_c^{*},\hfill \end{array}$$

(37)

and the parameters $a_1,a_2,a_3,b_1,b_2,b_3,c_1,c_2,c_3$ are as follows:

$$\begin{array}{ccc}{a}_1={\theta }_d^{*}& a_2=-{\theta }_d^{*}(R^{*}+1)N_{TU}{\displaystyle \frac{C_c}{C_h}}& a_3={\theta }_d^{*}(R^{*}+1)N_{TU}{\displaystyle \frac{C_c}{C_h}}\\ b_1=-1& b_2=-{\displaystyle \frac{N_{TU}(R^{*}+1)}{R^{*}}}& b_3={\displaystyle \frac{N_{TU}(R^{*}+1)}{R^{*}}}\\ c_1={\displaystyle \frac{(R^{*}+1)N_{TU}}{\overline{C_w^{*}}}}& c_2=-{\displaystyle \frac{{(R^{*}+1)}^2{N}_{TU}}{R^{*}\overline{C_w^{*}}}}& c_3={\displaystyle \frac{(R^{*}+1)N_{TU}}{R^{*}\overline{C_w^{*}}}}\end{array}$$

where:

• ${\theta }_d^{*}=t/t_d$: non-dimensional dwell time.
• $R^{*}={\alpha }_h/{\alpha }_c$: relative resistance ratio.
• $N_{TU}$: number of thermal units.
• $C_c=\mathrm{mass}\mathrm{flow}\mathrm{rate}\times \mathrm{specific}\mathrm{heat}$: heat capacity rate of the cold fluid or cold-fluid side.
• $C_h=\mathrm{mass}\mathrm{flow}\mathrm{rate}\times \mathrm{specific}\mathrm{heat}$: heat capacity rate of the hot fluid or hot-fluid side.
• $\overline{C_w^{*}}=\mathrm{wall}\mathrm{capacity}\mathrm{relative}\mathrm{to}\mathrm{the}\mathrm{flow}\mathrm{capacity}.$

Physically, the model captures the complex thermal interactions within the turbine, including heat transfer between hot gases, cooling air, and turbine walls. These temperature profiles are essential for predicting thermal stresses on turbine components, which impact performance and service life. Figure 2 illustrates the turbine stages, showing the flow of air and fuel and the key components involved in energy conversion.

The original model (37) accounts for turbine components using a stage-by-stage approach based on blade geometry and cooling system characteristics. It also includes a counter-flow surface heat exchanger, modeled with partial differential equations to describe the temperatures of air, hot gas, and metal.

Our goal is to adapt this model for hydrogen combustion, capturing the specific dynamics and unique properties of hydrogen as a fuel. To better represent these dynamics, we introduce the variable substitutions ${\displaystyle \frac{\partial T_c^{*}}{\partial x^{*}}}=z$ and ${\displaystyle \frac{\partial T_h^{*}}{\partial x^{*}}}=s$, simplifying the system into first-order partial differential equations. By incorporating the generalized conformable derivative, we account for anomalous behavior, refining the model to align more closely with the non-ideal and complex real-world combustion dynamics. The resulting modified system is expressed as

$$\begin{array}{cc}\hfill {}^{{\psi }_1}P_{{\theta }^{*}}^{{\alpha }_1}{T}_h^{*}& =a_{1}s+a_2{T}_h^{*}+a_3{T}_w^{*},\hfill \\ \hfill {}^{{\psi }_2}P_{{\theta }^{*}}^{{\alpha }_2}{T}_c^{*}& =b_{1}z+b_2{T}_c^{*}+b_3{T}_w^{*},\hfill \\ \hfill {}^{{\psi }_3}P_{{\theta }^{*}}^{{\alpha }_3}{T}_w^{*}& =c_1{T}_h^{*}+c_2{T}_w^{*}+c_3{T}_c^{*},\hfill \\ \hfill {}^{{\psi }_4}P_{{\theta }^{*}}^{{\alpha }_4}{T}_c^{*}& =z,\hfill \\ \hfill {}^{{\psi }_5}P_{{\theta }^{*}}^{{\alpha }_5}{T}_h^{*}& =s.\hfill \end{array}$$

(38)

where ${}^{{\psi }_i}P_y^{{\alpha }_i}$ represents the partial conformable generalized derivative, ${\psi }_i$ for $i=1,2,3,4,5$ is the conformable function, and ${\alpha }_i$ for $i=1,2,3,4,5$ is the order of the derivative.

These modifications allow the model to more accurately reflect the physical dynamics of hydrogen combustion, thereby enhancing its predictive capability. System (38) can be expressed in the following manner:

$$\begin{array}{cc}\hfill {\psi }_1({\theta }^{*},{\alpha }_1){\displaystyle \frac{\partial T_h^{*}}{\partial {\theta }^{*}}}& =a_{1}s+a_2{T}_h^{*}+a_3{T}_w^{*},\hfill \\ \hfill {\psi }_2({\theta }^{*},{\alpha }_2){\displaystyle \frac{\partial T_c^{*}}{\partial {\theta }^{*}}}& =b_{1}z+b_2{T}_c^{*}+b_3{T}_w^{*},\hfill \\ \hfill {\psi }_3({\theta }^{*},{\alpha }_3){\displaystyle \frac{\partial T_w^{*}}{\partial {\theta }^{*}}}& =c_1{T}_h^{*}+c_2{T}_w^{*}+c_3{T}_c^{*},\hfill \\ \hfill {\psi }_4(x^{*},{\alpha }_4){\displaystyle \frac{\partial T_c^{*}}{\partial x^{*}}}& =z,\hfill \\ \hfill {\psi }_5(x^{*},{\alpha }_5){\displaystyle \frac{\partial T_h^{*}}{\partial x^{*}}}& =s.\hfill \end{array}$$

(39)

System (39) is a linear partial system that can be effectively solved using numerical methods. The parameters ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$, and ${\alpha }_5$ can be adjusted to enhance the model’s accuracy. We specifically focus on the finite difference method, which is a widely used approach for numerically solving differential equations. This method transforms the differential equations into a system of algebraic equations that can be addressed using numerical techniques. The functions ${\psi }_1$, ${\psi }_2$, and ${\psi }_3$ serve as temporal relaxation coefficients for anomalous combustion, while ${\psi }_4$ and ${\psi }_5$ represent spatial relaxation coefficients for anomalous combustion. The coefficients ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ denote temporal delay coefficients, whereas ${\alpha }_4$ and ${\alpha }_5$ indicate spatial delays.

By implementing the finite difference method, we can effectively simulate the dynamic behavior of the hydrogen turbine’s heat exchanger, capturing the essential characteristics of the system under study. Using finite differences, the model (39) can be given by

$$\begin{array}{cc}\hfill {\psi }_1({\theta }_i^{*},{\alpha }_1){\displaystyle \frac{T_{h,i+1}^{*}-T_{h,i}^{*}}{\mathrm{\Delta }{\theta }_i^{*}}}& =a_1{s}_i+a_2{T}_{h,i}^{*}+a_3{T}_{w,i}^{*},\hfill \\ \hfill {\psi }_2({\theta }_i^{*},{\alpha }_2){\displaystyle \frac{T_{c,i+1}^{*}-T_{c,i}^{*}}{\mathrm{\Delta }{\theta }_i^{*}}}& =b_1{z}_i+b_2{T}_{c,i}^{*}+b_3{T}_{w,i}^{*},\hfill \\ \hfill {\psi }_3({\theta }_i^{*},{\alpha }_3){\displaystyle \frac{T_{w,i+1}^{*}-T_{w,i}^{*}}{\mathrm{\Delta }{\theta }_i^{*}}}& =c_1{T}_{h,i}^{*}+c_2{T}_{w,i}^{*}+c_3{T}_{c,i}^{*},\hfill \\ \hfill {\psi }_4(x_j^{*},{\alpha }_4){\displaystyle \frac{T_{c,j+1}^{*}-T_{c,j-1}^{*}}{2\mathrm{\Delta }{x}^{*}}}& =z_j,\hfill \\ \hfill {\psi }_5(x_j^{*},{\alpha }_5){\displaystyle \frac{T_{h,j+1}^{*}-T_{h,j-1}^{*}}{2\mathrm{\Delta }{x}^{*}}}& =s_j.\hfill \end{array}$$

(40)

5. Simulations

This section outlines the simulations conducted to analyze the turbine system’s behavior, focusing on temperature dynamics, angular velocity, and material balances. To model these phenomena accurately, we use two conformable functions, defined rigorously as follows:

$$\begin{array}{cc}\hfill {\psi }_1(u,\alpha )& =a^{b(1-\alpha )f\left(u\right)},\mathrm{defined}\mathrm{on}\left({\mathbb{R}}^{+}\cap \mathrm{Dom}\left(f\right)\right)\times (0,1],a\in {\mathbb{R}}^{+},b\in \mathbb{R},f\left(u\right)\in C^0,\hfill \\ \hfill {\psi }_2(u,\alpha )& ={(Asin\left(\theta u\right)+1)}^{1-\alpha },\mathrm{defined}\mathrm{on}{\mathbb{R}}^{+}\times (0,1],0<A<1,\theta \in \mathbb{R},\hfill \end{array}$$

(41)

where $f\left(u\right)=sin\left(\theta u\right)$. These conformable functions were selected heuristically, as they provided the best fit for capturing the variability and non-linearity of the turbine system’s physical processes. Their flexibility in modeling the non-linear and oscillatory dynamics made them ideal for this study. The parameters used in the simulations are detailed in

Table 1. Parameters used in the simulations, their values, and units.

Parameter	Value	Units
a	e	Dimensionless
A	$0.99$	Dimensionless
b	$2.8$	Dimensionless
$f\left(u\right)$	$sin\left(\theta u\right)$	Dimensionless
$\alpha$	$\{1/10,1/4,1/2,3/4,1\}$	Dimensionless
$\theta$	$5\pi$	Inverse units of u

, including their values and units. However, these functions could be adjusted or replaced in future studies, depending on the specific behaviors or dynamics researchers aim to model, making this conformable framework a versatile tool for various turbine applications.

The simulations are designed to enhance predictive accuracy and support the optimization of hydrogen-based clean energy systems. To achieve this, we use Python’s computational libraries for precise and efficient results. The methodology encompasses several key aspects.

First, we analyze temperature distributions within turbine blades by applying different heat capacity functions and conformable derivatives. This approach helps to capture the thermal dynamics of critical components. Second, we model the angular velocity of the turbine to evaluate system responsiveness under varying conditions, providing insights into dynamic behaviors. Third, we simulate mass–energy balances within the plenum and combustion chamber, incorporating combustion reactions along with conformable functions to accurately represent energy release and mass flow. Finally, finite difference methods are employed to model the entire turbine system, integrating conformable functions to offer a comprehensive understanding of system dynamics.

Remark 4. 

In all simulations, the conformable functions are consistently applied across the differential equations, ensuring homogeneity in both the order α and the conformable function used. Each equation employs the same conformable function and order, while adapting to the specific independent variables of each scenario.

To enhance the clarity of the theoretical model implementation in our simulations, we detail the specific parameters utilized in each scenario, as shown in the accompanying tables. These parameters—such as temperature, angular velocity, and mass flow rate—were chosen for their theoretical relevance to the phenomena being studied, ensuring their practical applicability. Furthermore, a systematic variation of the parameter $\alpha$ was employed to demonstrate the gradual transition in the dynamics of the models for each specific case. The conformable functions defined in (41) were selected in conjunction with the general parameters outlined in Table 1, as these reflect the real behaviors observed in such phenomena. This is evident in [35], which presents graphs of turbine boundary layer temperature, and in [36], where the behavior of angular velocity in the turbine shaft is depicted, exhibiting a semi-oscillatory dynamic that is effectively captured by the chosen conformable functions. Additionally, the article [37] was used in our simulations for the turbine temperatures.

This comprehensive approach allows us to capture the non-linear and complex dynamics inherent in hydrogen-based turbine systems. The insights gained from these simulations contribute significantly to the development of more efficient and sustainable energy technologies.

5.1. Simulation of Reaction Enthalpy

To simulate the reaction enthalpy, we utilize the conformable functions defined by Equation (13). This simulation investigates the variations in enthalpy throughout the reaction process under different conditions, aiming to understand how various factors influence energy release. The conformable functions employed in the simulation to represent the anomalous combustion rate are

$$\begin{array}{cc}\hfill {\psi }_2(T,\alpha )& ={(Asin\left(\theta T\right)+1)}^{(1-\alpha )}\hfill \\ \hfill {\psi }_1(T,\alpha )& =a^{b(1-\alpha )f\left(T\right)}\hfill \end{array}$$

(42)

where $f\left(T\right)=sin\left(\theta T\right)$. The parameters used in the simulation are summarized in

Table 2. Parameters for the reaction enthalpy simulation.

Parameter	Value	Units
$\mathrm{\Delta }{H}_r^0$	$-483.66$	kJ/mol
$\theta$	$5\pi$	1/°C
$T_0$	25	°C
$\mathrm{\Delta }{C}_p\left(T\right)$	(12)	kJ/mol°C

.

The introduction of the conformable derivative provides a broader range of behaviors that more closely resemble real-world phenomena. Figure 3 presents the simulation results of reaction enthalpy for different values of thermal rate delay $\alpha$: $1/10$, $1/4$, $1/2$, $3/4$, and 1. This figure illustrates how the reaction enthalpy can be decomposed into a spectrum of functions depending on the derivative order. Notably, the results suggest that increasing the temperature theoretically allows for more energy to be extracted from the reaction. This is because negative enthalpy represents energy leaving the system, and using derivative orders close to zero ($\alpha \to 0$) leads to a theoretical increase in the energy released. The observed behavior varies based on the conformable function used, which aids in modeling different dynamic behaviors.

The results from this simulation provide valuable insights into how the theoretical framework of conformable functions can predict an increase in energy extraction with rising temperatures. By taking derivative orders close to zero, we can model and potentially realize a greater release of energy from the reaction process. It is important to highlight that the behavior observed is similar to that of the enthalpy presented in [38], where a hydrogen-rich fuel stream was analyzed.

5.2. Blade Temperature Simulation

The blade temperature simulation employs Equation (31), varying the heat capacity and conformable derivative functions to analyze their effects on blade temperature dynamics. By incorporating the generalized conformable derivative, this approach enables a flexible modeling of the blades’ thermal behavior under various conditions. The conformable functions used to simulate the anomalous heat transfer coefficient are

$$\begin{array}{cc}\hfill {\psi }_2(t,\alpha )& ={(Asin\left(\theta t\right)+1)}^{(1-\alpha )}\hfill \\ \hfill {\psi }_1(t,\alpha )& =a^{b(1-\alpha )f\left(t\right)}\hfill \end{array}$$

(43)

where $f\left(t\right)=sin\left(\theta t\right)$. The parameters utilized in the simulations are detailed in

Table 3. Parameters used in the blade temperature simulation.

Parameter	Value	Units
$h_{\mathrm{bl}}$	10.0	W/(m2 · K)
$A_{\mathrm{bl}}$	5.0	m2
${\dot{m}}_{\mathrm{cl}}$	200	kg/s
$c_{p_{\mathrm{cl}}}$	1.0	kJ/(kg·K)
$ϵ$	0.8	Dimensionless
$T_g$	500	°C
$T_{{\mathrm{cl}}_1}$	300	°C
$n_{\mathrm{bl}}$	1.0	mol
$\theta$	5$\pi$	1/s

.

Figure 4 shows the simulation results of blade temperature dynamics for different values of $\alpha$ ($1/10$, $1/4$, $1/2$, $3/4$, and 1), illustrating the variety of thermal behaviors that can arise. The application of conformable fractional derivatives significantly enhances the modeling of the transient phase, capturing complex and atypical thermal behaviors that emerge before reaching a steady state.

The introduction of conformable fractional derivatives in the modeling of blade temperature dynamics provides a significant enhancement in capturing the transient phase, which is critical as this phase is where unusual and complex thermal behaviors can arise. The selection of the conformable function and the fractional order $\alpha$ allows for the modeling of a diverse array of atypical responses over short time scales, effectively capturing abnormal or transient thermal fluctuations that deviate from the steady state. In practice, it has been observed that the temperatures in the blades tend to transition from a transient phase to a stable state, a behavior that is well modeled by the inclusion of these derivatives. This approach not only aids in the accurate depiction of the progression from transient to steady state, but also offers valuable insights into predicting and managing future anomalous thermal behaviors that may emerge outside the steady-state regime, enhancing our understanding of unexpected dynamics in complex thermal systems.

It is important to note that the behavior shown in Figure 4 is similar to that observed in similar equipment, as demonstrated in [39], where the temperature increases before stabilizing. This can be seen in Figure 2, Figure 3 and Figure 4 of the cited article, where the real behavior of this phenomenon shows slight fluctuations, although the overall trend remains consistent. This same behavior is replicated by implementing the conformable derivative as the rate of change.

5.3. Angular Velocity Simulation

The angular velocity of the rotating shaft is simulated using Equation (24). This simulation explores how varying the initial conditions and conformable derivative functions influences the rotational dynamics, providing insights into the stability and performance of the turbine’s rotating components. Figure 5 illustrates the effects of changing the order $\alpha$, with values $1/10$, $1/4$, $1/2$, $3/4$, and 1, on the angular velocity.

The conformable functions employed to model the anomalous rotation coefficient in the simulations are

$$\begin{array}{cc}\hfill {\psi }_2(t,\alpha )& ={(Asin\left(\theta t\right)+1)}^{(1-\alpha )}\hfill \\ \hfill {\psi }_1(t,\alpha )& =a^{b(1-\alpha )f\left(t\right)}\hfill \end{array}$$

(44)

where $f\left(t\right)=sin\left(\theta t\right)$. The specific parameters are detailed in

Table 4. Parameters used in the angular velocity simulation.

Parameter	Value	Units
$P^{★}$	2	Dimensionless
J	1	kg·m2
${\omega }_0$	3	rad/s
$\theta$	5$\pi$	1/s

.

The implementation of conformable derivatives in the model reveals a spectrum of possible behaviors that deviate from ideal conditions, offering a more accurate approximation of real-world dynamics. As seen in Figure 5, the angular velocity can be increased theoretically by selecting conformable derivative orders close to zero ($\alpha \to 0$), highlighting the potential for enhanced rotational speeds. This approach provides valuable insights into how the system’s rotational dynamics can be optimized, and it emphasizes the importance of understanding the influence of fractional orders on performance. It is important to note that the behavior modeled with the conformable derivative is similar to that presented in [40] under non-steady-state conditions. This demonstrates that implementing conformable derivatives as the rate of change helps model anomalies in the rotational system.

5.4. Plenum Mass–Energy Balance Simulation

The mass–energy balance within the plenum is simulated using Equation (19). This simulation employs conformable derivative functions to assess their effects on the mass and energy dynamics of the airflow within the plenum. The conformable functions used to represent anomalous diffusion coefficients and anomalous combustion coefficients are

$$\begin{array}{cc}\hfill {\psi }_2(t,\alpha )& ={(A·sin(\theta ·t)+1)}^{(1-\alpha )}\hfill \\ \hfill {\psi }_1(t,\alpha )& =a^{b(1-\alpha )f\left(t\right)}\hfill \end{array}$$

(45)

where $f\left(t\right)=sin\left(\theta t\right)$. The values of $\alpha$ considered in the simulation are $1/10$, $1/4$, $1/2$, $3/4$, and 1, capturing a wide range of dynamic responses. The mass flow rate is assumed to be constant for simplification. The specific parameters used in the simulation are detailed in

Table 5. Parameters used in the plenum mass–energy balance simulation.

Parameter	Value	Units
$V_p$	1.0	m3
${\dot{m}}_1$	1.0	kg/s
${\dot{m}}_2$	1.0	kg/s
$h_1$	300	kJ/kg
$h_2$	500	kJ/kg
$\theta$	5$\pi$	1/s

.

Figure 6 shows the simulation results for the plenum’s mass–energy balance dynamics under different fractional orders $\alpha$. The simulations indicate that while the mass balance remains constant, the energy dynamics exhibit significant changes: lower values of $\alpha$ (closer to zero) correspond to increased energy extraction, as indicated by negative energy values. This suggests that orders close to zero can model scenarios with enhanced energy release from the system. Conversely, when $\alpha$ approaches 1, the behavior aligns more closely with ideal or steady-state conditions, illustrating how varying the conformable order provides a powerful tool for modeling a range of dynamic behaviors in energy systems.

5.5. Combustion Chamber Mass–Energy Balance Simulation

The mass–energy balance within the combustion chamber is simulated using Equation (21). This analysis utilizes conformable derivative functions to investigate the effects of varying fractional orders on the combustion process, mass flow rates, and overall chamber dynamics. The functions employed in this simulation to represent anomalous diffusion coefficients and anomalous combustion coefficients are

$$\begin{array}{cc}\hfill {\psi }_1(t,\alpha )& =a^{(1-\alpha )·f\left(t\right)}\hfill \\ \hfill {\psi }_2(t,\alpha )& ={(A·sin(\theta ·t)+1)}^{(1-\alpha )}\hfill \end{array}$$

(46)

where $f\left(t\right)=sin\left(\theta t\right)$. The simulation considers $\alpha$ values of 0.01, $1/3$, $2/3$, and 1 to observe the system’s response under various fractional orders. The parameters used for the simulation are outlined in

Table 6. Parameters used in the combustion chamber mass–energy balance simulation.

Parameter	Value	Units
$V_{cc}$	1.0	m3
${\dot{m}}_1$	0.8	kg/s
${\dot{m}}_2$	0.8	kg/s
${\dot{m}}_f$	0.02	kg/s
$h_1$	300.0	kJ/kg
$h_2$	500.0	kJ/kg
$h_f$	120.0	kJ/kg
$H_i$	142,000.0	kJ/kg
${\rho }_{c{c}_0}$	1.2	kg/m3
$e_{c{c}_0}$	300.0	kJ/kg
$\theta$	5$\pi$	1/s

.

Figure 7 shows the mass balance behavior within the combustion chamber, illustrating how the gas dynamics evolve over time. The figure demonstrates that the gas density within the chamber increases as time progresses, indicating a continuous accumulation of mass. The results suggest that by adjusting the conformable derivative order $\alpha$ closer to zero, the gas density theoretically increases further, leading to a more compressed state. Conversely, as the order $\alpha$ approaches 1, the behavior aligns more closely with ideal conditions, where the system operates near steady-state mass flow. This analysis highlights the potential of using fractional derivative orders to model scenarios where a denser, more compressed gas can be theoretically achieved, providing valuable insights into optimizing the combustion chamber’s performance beyond ideal assumptions.

Figure 8 illustrates the dynamic behavior of the combustion chamber’s energy balance for various $\alpha$ values, offering significant insights into the system’s performance. The positive difference in enthalpy between the incoming and outgoing air indicates that the turbine is effectively extracting energy from the fluid, performing work. The application of conformable derivatives reveals a spectrum of non-ideal behaviors. Lower $\alpha$ values, closer to zero, demonstrate enhanced energy extraction, suggesting that the system can theoretically achieve higher energy output. This approach provides a valuable framework for modeling atypical responses within the combustion chamber, offering a theoretical foundation for optimizing turbine performance under non-ideal conditions.

Overall, this simulation demonstrates how fractional calculus can be used to model and understand complex dynamics within the combustion chamber, revealing potential avenues for exploring non-ideal scenarios that may improve turbine efficiency.

5.6. Turbine Temperature

The integration of generalized conformable derivatives into turbine system simulations marks a significant advancement in modeling non-ideal scenarios. By employing fractional calculus, it is now possible to capture a broader spectrum of dynamic behaviors that extend beyond ideal conditions. The simulation incorporated conformable derivative functions ${\psi }_1$ and ${\psi }_2$, as defined in Equation (41), to model the temporal relaxation coefficients for anomalous combustion, allowing for a comprehensive exploration of their effects on turbine performance. This approach has demonstrated its effectiveness in modeling complex systems where traditional methods may prove inadequate.

The simulation results highlight how different conformable derivatives impact the system’s behavior. The use of these models, combined with the chemistry of hydrogen combustion, revealed a considerable potential for energy extraction from the chemical reaction. The energy output varied depending on the conformable derivative order, with lower orders demonstrating higher energy extraction capabilities.

The mathematical models developed provide valuable insights for approximating real-world turbine scenarios outside the idealized assumptions. By interpolating between values of $\alpha$, the behavior of various turbine systems can be generalized, offering a more flexible framework for performance optimization. This is particularly useful for adapting theoretical models to practical applications in the industry.

Table 7. Simulation parameters.

Parameter	Value	Units
Non-dimensional hot gas temperature ($T_h^{*}$)	298.15	K
Non-dimensional cooling air temperature ($T_c^{*}$)	400	K
Non-dimensional wall temperature ($T_w^{*}$)	298.15	K
Step size (${\delta }_{{\theta }^{*}}$)	0.01	-
Step size (${\delta }_{x^{*}}$)	0.01	-
Maximum domain (${\theta }_{\max }$)	15	-
Maximum domain ($x_{\max }^{*}$)	15	-
Mesh size ($N_{\theta }^{*}$)	$\mathrm{int}(15/0.01)$	-
Mesh size ($N_{x^{*}}$)	$\mathrm{int}(15/0.01)$	-
Input Parameter	Value	Units
${\theta }_d$	1.6	-
$R^{*}$	0.8	-
$N_{\mathrm{TU}}$	0.2	-
$C_c$	100.0	J/kg/K
$C_h$	250.0	J/kg/K
$C_{w^{*}}$	500.0	J/kg/K

presents the data used for the simulation, distinguishing between simulation parameters and input parameters for solving the model using numerical methods.

The following images illustrate the non-dimensional temperatures of the turbine system components, using conformable derivatives of order $1/2$. These thermal maps, rendered with color gradients, effectively demonstrate the temperature distributions for different turbine components.

The thermal maps displayed in Figure 9, Figure 10 and Figure 11 provide detailed insights into the temperature distributions within the turbine system. The wall temperature ($T_w^{*}$) ranges from 250 to 300, the hot gas temperature ($T_h^{*}$) ranges from 350 to 450, and the cooling air temperature ($T_c^{*}$) ranges from 250 to 350. The utilization of conformable derivatives of order $1/2$ in these simulations helps approximate the turbine’s thermal behavior effectively. The comparison of the two conformable functions, ${\psi }_1$ and ${\psi }_2$, reveals that both functions yield similar temperature distributions, indicating that their impact on the thermal behavior of the system is minimal. The simulations were executed using Python with the finite difference method, providing a robust analysis of the turbine’s temperature dynamics.

6. Discussion

This section aims to analyze the implications of our findings, emphasizing how the application of generalized conformable derivatives enhances the modeling of dynamics present in hydrogen turbines. Our primary objective was to improve the representation of these complex systems, capturing their non-linear behaviors and intricate energy extraction processes. By exploring specific subsections, we illustrate the theoretical and practical significance of our models, highlighting their relevance in advancing hydrogen turbine technology.

In Section 5.1, we demonstrate that it is theoretically feasible to extract greater energy by implementing conformable functions. This approach reveals a spectrum of various functions that can effectively model different energy extraction behaviors from the reaction phenomenon. As shown in [41], the dynamics typically associated with this reaction are non-linear and often exhibit subtle oscillations, which underscores the importance of our methodology in accurately representing these characteristics.

Further insights are provided in Section 4.2 and Section 4.3, where the utilization of a generalized conformable model in mass and energy balances results in distinct and varied system dynamics. This contrasts sharply with traditional approaches found in the literature that frequently rely on fixed constants and their variations, limiting the understanding of these balances by assuming static conditions. Our flexible modeling approach facilitates a deeper exploration of the system’s behavior, accommodating variations commonly present in real-world applications, thereby enhancing the predictive capabilities of turbine performance models.

The dynamics of angular velocity, detailed in Section 5.3, align closely with the actual behavior observed in turbine shafts. This correlation is supported by studies such as those in [36,42], which have shown that the behavior of angular velocity is characterized by oscillatory and non-linear traits. Notably, the application of a conformable function with periodic characteristics can effectively represent this behavior, simplifying simulation and enhancing understanding. By adjusting the values of $\alpha$, we can obtain more pronounced or subtle variations, transforming this approach into a powerful simulation tool for exotic scenarios, particularly in unsteady regimes or in the presence of anomalies.

Moreover, the implementation of the conformable derivative, in contrast to other fractional models or unconventional differential operators, significantly streamlines computational processes. This choice provides a manageable tool that can be interpreted physically as a variable scaling factor for the phenomena under study. Such adaptability aids in tailoring the required dynamics, and the demand for a dimensionless form mitigates the challenges associated with fractional unit interpretations. Consequently, we retain classical interpretations while merely scaling the dynamics, facilitating a more intuitive understanding of the underlying processes. This not only enhances model efficiency but also broadens its applicability in various engineering fields, paving the way for advancements in turbine technology and design optimization.

The conformable functions $\psi$ serve as anomalous coupling coefficients, providing insights into the interpretation of anomalies observed in mass transfer, energy dynamics, and rotational behaviors within the system. These coefficients enable a deeper understanding of the interactions among various physical phenomena, helping to characterize non-ideal behaviors that might arise in the turbine’s operational environment. This interpretation is crucial for advancing our comprehension of the complex dynamics at play in hydrogen turbines, allowing for more accurate predictions and optimizations. Moreover, the fractional orders $\alpha$ of the conformable derivative are interpreted as delay coefficients, whether thermal, spatial, or temporal, depending on the specific context, thereby enriching the model’s applicability to various scenarios.

Finally, in Section 5.6, we analyze the turbine temperature, which closely resembles behaviors documented in studies monitoring turbine temperatures, such as [37,43]. In our analysis, the parameters selected lead to relatively stable behavior, consistent with the findings reported by these authors. Importantly, minor adjustments to the chosen conformable functions could potentially expand the range of dynamics observed, paving the way for further refinements in future research and underscoring the transformative potential of our modeling approach in the hydrogen energy sector.

7. Conclusions

The main objective of this study was to improve the accuracy of hydrogen turbine modeling by using generalized conformable derivatives to account for anomalous behaviors. Hydrogen, as a clean fuel, plays a key role in the transition to sustainable energy systems. However, modeling its combustion in turbines is challenging due to the frequent occurrence of anomalies. Employing a model capable of representing these anomalies, and the intensity with which they appear through the order of the derivative $\alpha$, offers significant advantages for future research and technological development.

The functions used in this study were selected heuristically, which represents a clear advantage as it allows for adaptation to different system behaviors. The conformable functions provide the flexibility to model a range of anomalous phenomena in hydrogen turbine dynamics. For instance, the $\alpha$ parameters represent thermal delay coefficients in the calculation of enthalpy (13), temporal delays in mass and energy balances (19) and (21), and spatial delays in the heat transfer equations ${}^{{\psi }_4}P_{x_{*}}^{{\alpha }_4}{T}_c^{*}$ and ${}^{{\psi }_5}P_{x_{*}}^{{\alpha }_5}{T}_h^{*}$ (38), where the primary parameter depends on the system’s length.

The simulations of blade temperature (Section 5.2) revealed anomalous behavior in the transient regime, where varying the delay coefficient $\alpha$ allowed us to observe a spectrum of behaviors, including changes in the anomalous heat transfer coefficient. This demonstrates that the conformable derivative framework is capable of modeling complex thermal dynamics in turbine blades, which is essential for preventing material fatigue and optimizing performance.

In the simulations of the energy released during an anomalous combustion scenario (Section 5.1), the model captured how variations in the thermal delay coefficient ($\alpha$) and the anomalous combustion rate affected energy output. It was observed that, theoretically, it is possible to extract more energy by decreasing the thermal delay coefficient, indicating that the conformable derivative framework can be used to enhance energy efficiency in hydrogen combustion processes.

In the plenum and combustion chamber simulations (Section 5.4 and Section 5.5), the introduction of conformable derivatives perfectly modeled the anomalous behavior. The conformable functions helped represent the anomalous slope, enabling us to simulate a spectrum of behaviors that deviate from ideal conditions. Notably, for the plenum, the mass balance remained constant regardless of the derivative order, while in the combustion chamber, mass accumulation increased as the temporal delay coefficients decreased. This reflects the utility of conformable calculus in capturing mass–energy dynamics in real-world systems.

The angular velocity simulation (Section 5.3) demonstrated that as the order of the temporal delay coefficient $\alpha$ decreased, the anomalous behavior became more pronounced, leading to sharper increases in angular velocity. This finding underscores how adjusting the order of derivation can optimize turbine rotational performance, making conformable calculus a valuable tool for fine-tuning operational dynamics.

The simulation of turbine temperature (Section 5.6) showed that when both spatial and temporal derivatives were considered, the anomalous rate of variation exhibited both spatial and temporal components. The derivatives ${}^{{\psi }_5}P_{x_{*}}^{{\alpha }_5}$ (spatial) and ${}^{{\psi }_1}P_{{\theta }_{*}}^{{\alpha }_1}$ (temporal) confirmed that the derivative orders are effectively delay coefficients, helping to explain the anomalous heat transfer dynamics in the system. Despite the chosen conformable function, no significant variation was observed in the dimensionless wall temperature, while the dimensionless temperature of the hot gas fluctuated between 300 and over 400. In contrast, the dimensionless temperature of the cooling air, mainly between 200 and 250, rose above 400 only for small dimensionless times, revealing opposite behavior to the hot gas.

Ultimately, the generalized conformable derivative allows us to physically represent a spectrum of anomalous behaviors. The orders of derivation (delay coefficients) that are closer to zero indicate a higher degree of anomaly in the system, as they reflect the dynamics of conformable functions (anomalous slopes), progressively showing behaviors that deviate more from ideality. This capacity to model various degrees of anomalies through the delay coefficients offers great potential for the optimization of turbine systems, especially under non-ideal conditions.

Nevertheless, it is essential to acknowledge the limitations of the system. Being a local operator, the generalized conformable derivative does not account for memory effects, which could be a disadvantage when considering systems with significant hysteresis or memory-driven phenomena. Additionally, the model is deterministic, meaning it does not incorporate random or stochastic effects. Another limitation is the lack of fuzzy logic integration, which would allow the model to address uncertainties and imprecise variations in the turbine’s operating parameters. Without this, the model may struggle to capture the full range of variability present in real-world systems. Given that anomalies in a system can vary, understanding the typical behavior of the anomaly is critical to selecting the appropriate conformable function.

For future research, it would be valuable to extend this approach by incorporating memory effects into the conformable derivative framework, enabling the modeling of systems where past states influence current dynamics. Additionally, integrating stochastic variations into the model could capture random anomalies and enhance the model’s robustness in real-world applications. Another promising avenue is the incorporation of fuzzy logic to handle uncertainties and imprecise variations in operating parameters, which would improve the model’s adaptability to real-world conditions. This integration is planned for future investigations, as it could enhance the model’s precision and flexibility in predicting turbine performance. Furthermore, experimental validation of these models, combined with real-time anomaly detection and correction mechanisms, would represent an important step toward the practical implementation of these findings in the design of next-generation hydrogen turbines.

In conclusion, this study has successfully demonstrated that the use of generalized conformable derivatives can improve the modeling of hydrogen turbines by providing a flexible framework for representing anomalous behaviors. By varying the order of the derivative $\alpha$, researchers can simulate a wide range of real-world behaviors with high precision, which is crucial for the design and optimization of sustainable energy systems. The findings of this research lay the groundwork for further exploration and application of fractional calculus in hydrogen turbines, offering valuable insights for the advancement of sustainable and efficient energy technologies.