Introduction

The current drive to shift from fossil fuels toward sustainable energy makes hydrogen stand out in comparison to other cleaner fuels available in nature. This is primarily due to hydrogen offering higher energy density combined with versatility and multiple production possibilities1. The release of hydrogen through renewable processes results in only water discharge which establishes it as an effective method to cut down on greenhouse gas (GHG) emissions. Through electrolysis, hydrogen enables renewable energy systems to utilize and conserve excess electricity which reduces the need for conventional power sources2. The decarbonizing process in transportation and industrial applications depends heavily on utilization of hydrogen as an alternative fuel. The zero-emission technology provided by hydrogen fuel cells serves as an alternative power system to battery-electric vehicles in heavy-duty transport. This is attributed because of their manageable weight and limited range requirements3. Green hydrogen serves as a global energy equality promoter because it enables countries with renewable energy potential to generate and distribute hydrogen products enabling better economic prospects while easing international fuel related conflicts4. Adoption of hydrogen power faces difficulties stemming from the large amounts of energy requirements to produce, store and transport it5. The ongoing improvement of research and innovation makes hydrogen a vital element of the low-carbon energy transition which will build a resilient global energy system6.

Hydrogen liquefaction stands as a vital operational method which makes large-scale hydrogen storage and distant transportation efficient. The volumetric energy density of liquefied hydrogen (LH2) exceeds that of gaseous hydrogen by approximately 70% thus making possible its storage in compact cryogenic tanks for rockets as well as intercontinental hydrogen shipments7,8. Liquefying hydrogen requires excessive energy that amounts to 30–40% of its total energy content because of its extreme boiling point (20.3 K)9. The demanding characteristics of hydrogen liquefaction require sophisticated cryogenic infrastructure together with multiple cooling stages as well as high quantity of energy input which establishes economic and thermodynamic related obstacles to broader commercial adoption10. High-efficiency heat exchangers and cryogenic compressors with vacuum-based insulation materials optimize thermal management but operational costs and energy requirements limit scalability11,12. Efforts in research and development play a vital role for hydrogen to fulfill its role in both power sector decarbonization and global sustainable hydrogen economy formation13,14.

Nanocomposites have transformed thermal management through their efficient custom-made approaches which reduce energy waste in cryogenics energy systems. Base matrices receive nanoscale components to utilize the thermal and mechanical attributes and chemical properties of materials15. These regulate heat transfer rapidly because of their high thermal conductivity which proves essential for hydrogen liquefaction operations and additional temperature-sensitive applications16. Nanocomposites reduce energy requirements for hydrogen liquefaction by combating thermal inefficiencies and entropy generation at operation temperatures reaching 20.3 K17. Nanostructured materials used in cryogenic components achieve thermal stability under extreme temperature differences through their ability to reduce conductive heat losses together with suppressed radiative properties. Their durability makes them suitable for long-term operation under harsh conditions which enable industrial expansion18,19. Nanocomposite research further demonstrates hybrid structures that use various nanomaterials together to generate combined performance enhancements including improved thermal conduction properties and extended thermal cycling resistance. The thermal management solutions depend heavily on nanocomposites because they enable sustainable energy transitions in sectors requiring high-performance materials for better operational and environmental standards20.

Wu et al. (2017)21 investigated a rotating gliding arc (RGA) plasma effectively transformed CH4 into hydrogen and carbon aerosols while demonstrating that carbon obtained from CH4/N2 showed a BET surface area that exceeded CH4/Ar-derived carbon by three times. Pachaiappan et al. (2021)22 examined Graphitic carbon nitride (C3N4) which also emerged as a promising choice for hydrogen production and storage because of its layered structure and its Lewis acid functionalities as well as its adjustable electronic properties storing hydrogen at high levels (6.5 wt%). Kartal and Özveren (2022)23 built an IGCC model within Aspen HYSYS which integrated an entrained bed gasifier and gas turbine together with a Kalina cycle for CO2 co-gasification operated at its peak efficiency through an equivalence ratio of 0.23–0.25 and CO2 content of 5–7%. Ramirez-Vidal et al. (2022)24 establishes that hydrogen storage faces fundamental challenges because of its low volumetric energy density. The research examines storage capabilities between activated carbons and MOFs when using 10 MPa for absorption and 0.5 MPa for release. The research by Yang et al. (2024)25 developed a novel ternary complex combined material which achieved high efficiency for hydrogen evolution reaction (HER) with a 28.8 mV overpotential and 69.07 mV/dec Tafel slope operating at 10 mA cm−2 in KOH.

The process of hydrogen liquefaction stands as a key fundamental storage solution for hydrogen while its excessive energy related demands limit practical hydrogen usage. The use of advanced nanocomposites creates an effective strategy to optimize thermal systems while lowering energy waste which supports the development of a sustainable hydrogen economy. The study brings novelty by incorporating advanced nanocomposites including graphene-based and boron nitride materials that optimize hydrogen liquefaction processes which is seldomly explored in previous related studies. The research implements theoretical methods to deliver both substantial specific energy reduction along with improved performance coefficients. The study includes a novel framework that enables the incorporation of nanocomposites into cryogenic systems which ensures systems scale up and durability under extreme conditions. The following objectives need to be achieved in this study:

  • Correlation analysis to evaluate the mutual effects and interactions between input parameters and outcomes of the system.

  • Implementation of MEREC method to perform weightage analysis which establishes the relative significance among input operational parameters.

  • The k-means clustering method helps to group refrigerant sets according to their performance results and groups them into best, worst and average clusters.

  • Establish the best and optimum combination of mixed refrigerants together with nanocomposites for hydrogen liquefaction process.

Materials and equipment selection

The components in the mixed refrigerants are characterized by both the light and heavy refrigerant components with different thermodynamic behaviors and roles in the hydrogen liquefaction system, therefore, achieving the best balance between them is essential. Light refrigerants have low boiling points and high refrigeration capacities; in the early stages they can cool things down very quickly. Despite their potential, higher volatility and lower thermal stability may complicate retaining levels for varying operational conditions. On the other hand, heavy refrigerants possess high boiling points that make them a good choice when stability and efficiency of heat retention are considered. In pre-cooling of hydrogen liquefaction cycles, the selection of refrigerant mix becomes a more complex one. Light refrigerants may expedite the cooling process by efficiently absorbing heat, but excessive use can lead to higher energy consumption due to their lower overall efficiency. This is the reason why nanocomposites need to be utilized between them, so as to create a balance between them.

Refrigerant selection criteria

The vital choice of refrigerant components during hydrogen precooling operations determines both thermal efficiency enhancements and reduction of energy usage levels. Different refrigerants belong to light and heavy groups according to their boiling and freezing points which establishes their best application areas during cooling process steps. During cryogenic temperature operation in early stages of hydrogen cooling lightweight components including methane along with ethane and nitrogen stand crucial because of their low boiling temperature properties. Methane shows excellent heating properties by removing major heat from hydrogen when the temperature moves from ambient temperature to intermediate conditions because of its boiling temperature of -161.5 °C and high vaporization heat. The reaction processes in the intermediate stages benefit from ethane alongside ethylene, absorb heat efficiently without causing stability issues in the system. Propane and n-butane together with iso-pentane (greater boiling point and more substantial thermal mass) function in the advanced precooling segments of the process. A well-balanced combination of refrigerants acts as a key component for specific energy consumption (SEC) reduction and exergy loss minimization enhances the coefficient of performance (COP) of the system for optimal hydrogen liquefaction operation.

In this study, a base refrigerant mixture was formulated by combining refrigerants from both light and heavy categories, selected based on their boiling and freezing points to match different stages of the hydrogen cooling process. Light components such as methane, ethane, and nitrogen were employed in the early cryogenic stages due to their low boiling points and high latent heat properties, with methane proving especially effective in removing heat as hydrogen cools from ambient to intermediate temperatures (boiling point: −161.5 °C). Ethane and ethylene provided thermal stability and efficient heat absorption in the mid-range temperature zones. In the later stages of precooling, heavier refrigerants like propane, n-butane, and iso-pentane contributed greater thermal mass and higher boiling points, improving heat exchange performance. Through nanocomposite implementation systems achieve enhanced hydrogen precooling efficiency since it enables thermal management and energy waste problems. The use of nanocomposite materials including graphene boron nitride and carbon nanotubes, eventually enhancing both heat exchanger thermal performance and insulates cryogenic components more effectively. The materials decrease both thermal resistance and heat leakage which results in improved control of refrigerant temperature profiles throughout precooling. The high thermal conductivity and excellent stability properties of these products enhance lightweight refrigerant operations to remove heat fast while maintaining stability and operational efficiency. The durability and efficiency characteristics of nanocomposites improve pressure control operations and energy absorption processes at advanced heat exchange levels. Using advanced refrigerant selection along with nanocomposite technology enables hydrogen precooling systems to smoothly transition into their final liquefaction phase while minimizing energy use and improving system-wide performance.

Description of selected advanced nanocomposites

The properties of advanced nanocomposites such as Graphene/TiO2, g-C3N4/TiO2, and BN-TiO2 have exceptional properties. This enhances their performance, and can be used in applications that require efficient thermal management and energy conservation, including in hydrogen liquefaction. In particular, TiO2-based composites with graphene/TiO2 demonstrate their advantages in promoting the thermal properties and photochemical properties of TiO2 due to the superior thermal conductivity, high specific surface area, and excellent electron mobility of graphene. The variants like Graphene/TiO2 (Anatase), Graphene/TiO2 (Rutile), utilize crystalline states of TiO2 to maximize the propagation of heat for better energy efficiency. The combination of multi-walled carbon nanotubes (MWCNTs) with acetylene functionalities in Graphene/TiO2 composites enhances both surface reactivity and thermal stability factors that are essential for cryogenic processes. Designed properties in Graphene/TiO2 composites enable efficient heat dissipation together with entropy reduction which are vital for minimizing energy losses throughout the hydrogen liquefaction process.

Nanocomposites of TiO2 with g-C3N4 display superior characteristics because of the combined properties of the visible light-responsive graphitic carbon nitride (g-C3N4) and the highly efficient TiO2. These versions of g-C3N4/TiO2 nanocomposites known as Mesoporous and Porous present improved surface contact together with optimized pores that enhance both chemical reactions and thermal power transfer at cryogenic temperatures. The thermal conductivity of hexagonal boron nitride (BN)-based composites such as BN-TiO2 pairs with their low dielectric loss properties. BN-TiO2 possesses a hexagonal arrangement that optimizes both heat retention and heat dissipation performance which benefits high-performance thermal management systems. Additional heat transfer and improved energy efficiency become possible through enhanced plasmonic resonance when noble metal nanoparticles are included in Au-TiO2 composites. These nanocomposites unite innovative features for thermal efficiency along with stability and scalability characteristics essential to optimize hydrogen liquefaction and other cryogenic systems that demand high energy inputs.

Similarity analysis for different nanocomposites components

Figure 1 displays an organized classification of advanced nanocomposites including Graphene/TiO2 products which arranges structures and modifications into separate branches. The branches show different nanocomposite types with varying compositions of materials that produce distinct morphological features and synthetic methods that determine their thermal and cryogenic response patterns. The photocatalytic characteristics with thermal conductivity properties differ between Graphene/TiO2 (Rutile) and Graphene/TiO2 (Anatase) because each system incorporates distinct rutile TiO2 crystalline phases. The stability advantage belongs to Rutile whereas Anatase demonstrates better photocatalytic performance because each crystal phase delivers unique performance characteristics related to operational demands. The Graphene/TiO2 (Degussa P25) along with Graphene/TiO2 (Aeroxide P90) show modified surface characteristics and particle dimensions to boost heat transfer performance for applications related to hydrogen liquefaction.

The combination of g-C3N4/TiO2 (Nanotube) and g-C3N4/TiO2 (Hybrid) represents critical developments in integrating graphitic carbon nitride (g-C3N4) with TiO2 to maximize their joint properties leading to better thermal robustness and cryogenic operation. Thermal conductivity increases combined with decreased energy losses through the nanotube structure that provides multiple active sites and improved surface area. The hydrocarbon-based composites Graphene/TiO2 (Hombikat UV100) and Graphene/TiO2 (P25-MWCNT) have been designed with MWCNTs to boost mechanical capabilities and heat transfer performance as well as hydrogen refrigerant compatibility. Nanocomposite design follows such a hierarchical approach because it targets specific cryogenic system issues that include both thermally optimized management and reduced energy consumption together with material stability in extreme conditions. Their exceptional properties permit effective implementation when the applications need improved COP and reduced exergy waste.

Fig. 1
figure 1

Similarity analysis for different selected nanocomposites.

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Selection of different sets of nanocomposites

Nanocomposite materials including Graphene/TiO2 and g-C3N4/TiO2 and BN-TiO2 have been selected because they combine excellent heat management properties with low-temperature resistance and hydrogen adaptability as shown in Table 1. The thermal conductivity and electron mobility in graphene-based composites reach 5000 W/m·K because they improve heat dissipation and lower thermal resistance at cryogenic temperatures26. The process of hydrogen liquefaction requires exceptional thermal management because it determines the ability to reach ultra-low temperatures. The built-in strength properties of graphene allow the material to endure lower temperatures near absolute zero while keeping intact mechanical resistance alongside thermal transmission capabilities. The different crystal forms of TiO2 including Anatase and Rutile function as excellent thermal protectors while maintaining good chemical reactions with hydrogen therefore keeping its desired purity levels during liquefaction. Energy consumption during cold phases reaches its optimum through heat transfer improvement provided by MWCNT-modified Graphene/TiO2 versions that increase both heat transfer capabilities and surface area.

Table 1 Properties of the selected nanocomposites.
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Nanocomposites from g-C3N4/TiO2 along with BN-TiO2 show great value through their cryogenic capability and hydrogen compatibility due to g-C3N4’s high thermal stability (up to 600 °C) and minimal thermal expansion characteristics and BN-TiO2’s excellent thermal conductivity (up to 400 W/m·K) and chemical inertness properties. The synergistic effect between TiO2 and the material supports efficient photothermal conversion together with enhanced thermal conductivity which helps control heat loads in hydrogen liquefaction. Hexagonal BN-TiO2 composites achieve both high thermal conductivities reaching 400 W/m·K and chemical resistance in order to maintain excellent stability when used in cryogenic conditions and reduce hydrogen contamination risks. This combination offers reliable thermal performance because of low dielectric loss and strong heat retaining properties. All hydrogen-compatible nanocomposites maintain efficiency and system safety in liquefaction operations because they demonstrate minimal reactions to hydrogen molecules while sustaining operational integrity.

Nanocomposite-enhanced workflow for hydrogen pre-cooling and liquefaction

The procedure starts with raw hydrogen gas being introduced into the system through SEP1 which conducts impurity and particulate separation to maintain purity standards. The purified hydrogen flows to Compressor 1 (CM1) for pressure lift optimization of future thermal exchanges. The hydrogen moves from a nanocomposites injector to an NC Reactor before entering the main heat exchanger as nanocomposite catalysts optimize hydrogen thermal behavior. The nanocomposites enable enhanced energy efficiency through their functions of boosting thermal conductivity along with reaction speed. Before the hydrogen gas continues to additional compression stages the NHX performs precooling operations which minimize energy waste and optimizes heat transfer efficiency as shown in Fig. 2.

After pre-treatment, the hydrogen gas undergoes staged cooling through multiple heat exchangers (HX1 to HX4). The Nanocomposite Supply Unit (NC Supply) continuously injects nanoparticles that enhance the cooling process by improving heat transfer properties. Each exchanger works at progressively lower temperatures using a mixture of cryogenic coolants and nanofluids, ensuring a gradual decrease in hydrogen temperature to avoid thermal shock. Expansion valves (EX1 to EX3) facilitate controlled pressure drops, further cooling the hydrogen. The integration of nanocomposites in these systems allows for better phase transition control, reducing energy consumption and improving liquefaction efficiency33. The use of Separator 3 (SEP3) ensures that any residual impurities are removed before the final cooling phase. The ultra-cooled hydrogen gas reaches the final expansion stage through HX4 and HX2, where it undergoes Joule-Thomson expansion via expansion valves (EX4 and EJ), reducing the temperature to cryogenic levels (~ 20 K). The liquid hydrogen is then routed to the Cryogenic Storage Tank, where it is preserved under high-pressure conditions to maintain its liquid state. The addition of nanocomposites in OP1 and OP2 further enhances stability by reducing evaporation losses. By integrating nanocomposites throughout the liquefaction process, the system achieves lower energy consumption, higher heat transfer rates, and improved hydrogen purity, making the process more sustainable and efficient.

Fig. 2
figure 2

Process workflow of the pre-cooling process using nanocomposites.

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Methodology

The data is initially acquired by collecting precise data which captures multiple performance metrics among different test samples in nanocomposite studies using a simulation software. The work flow of the process of data acquiring and analysis is shown in Fig. 3. This study uses the Method based on the Removal Effects of Criteria (MEREC) to determine priority weights for criteria by checking how each criterion affects overall performance results after elimination. The weighted outcomes serve as inputs for k-means clustering that groups nanocomposite into the categories best, worst and average. The clustering process groups nanocomposites, so as to evaluate superior ones among them regarding their distinctive qualities.

Fig. 3
figure 3

Hybrid methodology flow chart.

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Outcome performance characteristics

The evaluation of COP together with the assessment of SEC remains essential in a liquefaction process of hydrogen whose values for different nanocomposite combination are shown in Appendix Table A1. COP primarily determines system energy efficiency while SEC evaluates efficiency for improving thermodynamic performance. These performance indicators support system needs by establishing an equilibrium between energy consumption and hydrogen liquefaction processes as they reduce energy waste while promoting comprehensive sustainability. The following are the equations for outcomes of the study:

  1. (a)

    Coefficient of Performance (COP).

    The COP system indicator describes how efficient the system performs for hydrogen storage by computing the cooling effect ratio to the work input metric. The COP value’s increased level indicates higher system efficiency34.

    $$COP=\frac{{\dot{Q}}_{\text{cooling}}}{{\dot{W}}_{\text{input}}}$$
    (1)

    Where:

    \({\dot{Q}}_{\text{cooling}}\)= Cooling Capacity (kW).

    \({\dot{W}}_{\text{input}}\) = Work input (kW).

  2. (b)

    Specific Energy Consumption (SEC).

    This metric determines the entire power consumption needed to cool down one unit of mass hydrogen. Improved energy efficiency becomes essential for large-scale hydrogen liquefaction systems when SEC values decrease35.

    $$SEC=\frac{{W}_{\text{i}\text{n}\text{p}\text{u}\text{t}}}{{\stackrel{\prime }{m}}_{{\text{H}}_{2}}}$$
    (2)

    Where:

    \({W}_{\text{input}}=\) Total energy consumed in pre-cooling, measured in kWh.

    \({\stackrel{\prime }{m}}_{{\text{H}}_{2}}=\) Mass flow rate of hydrogen, measured in \(\text{k}\text{g}/\text{s}\).

To ensure the reliability of simulation-based predictions, uncertainty analysis was conducted for key performance indicators SEC and COP based on multiple simulation runs within the defined operating envelope. Across the 30 simulation points, the standard deviation for SEC was 0.19 kWh/kg LH2, and for COP it was 0.34, indicating a relatively tight distribution around the mean values of 1.48 and 3.43, respectively. Assuming normal distribution, the 95% confidence interval for SEC is calculated as 1.48 ± 0.07 and for COP as 3.43 ± 0.13, showing that performance outcomes remain stable under minor input perturbations, and the model is statistically robust.

MEREC (Method based on the removal effects of Criteria) technique

Multi-criteria decision-making (MCDM) makes use of MEREC as a weighting method to assess input parameter values by their system performance effects36. When using this system, the MEREC method creates weight priorities for input operational parameters such as pressure, temperature, flow rate and nano-additive concentration. This aids in establishing the main performance influencers, eventually optimizing COP and SEC37. In this study, the MEREC method was selected over other objective weighting techniques such as Entropy, CRITIC, and PCA due to its superior ability to quantify the actual performance loss associated with removing each criterion, making it especially suited for high-stakes thermal optimization scenarios like hydrogen precooling. Unlike Entropy, which only considers data variability, and CRITIC, which combines contrast and correlation without direct performance sensitivity, MEREC directly reflects the impact of each input (e.g., nano-additive concentration, flow rate) on system performance outcomes (SEC and COP). PCA, while effective for dimensionality reduction, lacks interpretability in terms of individual criterion importance and does not assign weights based on thermodynamic relevance. Thus, MEREC offers a more thermodynamically meaningful and performance-sensitive weighting framework, justifying its use in identifying critical variables in this energy-intensive process.

  • Step 1: Construct the Decision Matrix.

    The decision matrix maintains m different refrigerant-nanocomposite configurations while using n pressure temperature flow rate nano-additive concentration as its criteria.

    $$X = \left[ {\begin{array}{*{20}c} {x_{{11}} } & {x_{{12}} } & \ldots & {x_{{1n}} } \\ {x_{{21}} } & {x_{{22}} } & \ldots & {x_{{2n}} } \\ \vdots & \vdots & \ddots & \vdots \\ {x_{{m1}} } & {x_{{m2}} } & \ldots & {x_{{mn}} } \\ \end{array} } \right]$$
    (3)

    where:

    \({x}_{ij}\) represents the value of the \({j}^{th}\) criterion for the \({i}^{th}\) alternative.

  • Step 2: Normalize the Decision Matrix

    All values need normalization to remove unit effects from the dataset. For beneficial (maximization) and non-beneficial (minimization) criteria:

    $$n_{{ij}} = \left\{ {\begin{array}{*{20}l} {\frac{{x_{{ij}} }}{{{\text{max}}\left( {x_{j} } \right)}},} \hfill & {{\text{~if~}}\;j\; \in \;{\text{Beneficial}}\;{\text{~Criteria~}}} \hfill \\ {\frac{{{\text{min}}\left( {x_{j} } \right)}}{{x_{{ij}} }},} \hfill & {{\text{~if~}}\;j\; \in \;{\text{~Non - Beneficial~}}\;{\text{Criteria~}}} \hfill \\ \end{array} } \right.$$
    (4)

  • Step 3: Calculate Overall Performance of Alternatives

    Compute the aggregated performance \({S}_{i}\) for each alternative \(i\) using:

    $${S}_{i}=\text{l}\text{n}\left(1+\frac{1}{m}\sum\:_{j=1}^{m}\:\:\left|\text{l}\text{n}\left({n}_{ij}\right)\right|\right)$$
    (5)

  • Step 4: Compute the Mean Reference Effect

    The Mean Reference Effect (MRE) for each criterion is determined using:

    $$MRE_{j} = \sum\limits_{{i = 1}}^{m} {\left| {r_{{ij}} - \bar{r}_{j} } \right|}$$
    (6)

    where:

    \(MR{E}_{j}\) measures the deviation of each value from the mean reference, indicating its effect on the system’s performance.

  • Step 5: Calculate the Compensation Effect

    Each criterion alters the performance of the entire system according to the compensation effect principle. It is calculated as:

    $$C_{j} = MRE_{j} \times \bar{r}_{j}$$
    (7)

    where:

    \({C}_{j}\) represents the compensation effect of criterion \(j\).

  • Step 6: Determine the Final Weights

    The final weight of each criterion is determined by normalizing the compensation effect38:

    $$w_{j} = \frac{{C_{j} }}{{\sum\nolimits_{{j = 1}}^{n} {C_{j} } }}$$
    (8)

    where:

    \({w}_{j}\) is the weight assigned to criterion \(j\).

k-means machine learning method

The unsupervised machine learning method k-means clustering enables partitioning of datasets into k specific clusters through similarity detection39. The MEREC and k-means clustering methods demonstrated in this study can be effectively applied to other energy system optimization problems, particularly those involving multi-component working fluids or complex thermodynamic interactions for different thermodynamically stable cycle. MEREC is especially valuable in scenarios where it is critical to quantify the impact of removing specific variables (material types, operational parameters, or component concentrations) on system performance, offering a sensitivity-based, objective ranking. Similarly, k-means clustering is highly suitable for classifying large sets of simulation data into performance-based categories (e.g., high-efficiency, average, and poor-performing configurations), enabling targeted optimization. These methods allow researchers to navigate high-dimensional design spaces efficiently, extract dominant performance trends, and make data-driven decisions without relying solely on computationally expensive full factorial or regression-based designs. Their adaptability makes them powerful tools for early-stage process screening, material selection, and hybrid system evaluation in energy engineering. The method follows these steps:

  • Step 1: Define the Number of Clusters (K).

    • A user sets the cluster number K by through Elbow method determination.

  • Step 2: Initialize the Cluster Centroids.

    • Randomly select k data points as the initial cluster centroids:

      C = {C1, C2,,,,CK}.

    where Ck ​ represents the centroid of the kth cluster.

  • Step 3: Assign Each Data Point to the Nearest Cluster

    $$d\left( {X_{i} ,C_{k} } \right) = \sqrt {\sum\limits_{{j = 1}}^{n} {\left( {X_{{ij}} - C_{{kj}} } \right)^{2} } }$$
    (9)

    where:

    • \({X}_{i}=\) data point.

    • \({C}_{k}=\) centroid of cluster \(k\).

    • \(n=\) number of features.

    • Assign each \({X}_{i}\) to the cluster with the nearest centroid:

    $${\text{Cluster~}}\left( {X_{i} } \right) = {\text{arg}}\mathop {{\text{min}}}\limits_{k} d\left( {X_{i} ,C_{k} } \right)$$

  • Step 4: Update the Cluster Centroids.

    • The system obtains new cluster centroids by calculating the average value of all points allocated to each group:

    $$C_{k} = \frac{1}{{\left| {S_{k} } \right|}}\mathop \sum \limits_{{X_{i} \in S_{k} }} X_{i}$$
    (10)

    where:

    • \({C}_{k}=\) updated centroid of cluster \(k\).

    • \({S}_{k}=\) set of points assigned to cluster \(k\).

    • \(\left|{S}_{k}\right|=\) number of points in cluster \(k\).

  • Step 5: Repeat Steps 3 and 4 Until Convergence.

    • Cluster centroids stop changing when the algorithm reaches predefined maximum iterations or until the centroids become stable.

    • The quantity known as inertia represents the sum of the inner-cluster variance which serves as the measure to identify final clusters:

    $$J = \sum\limits_{{k = 1}}^{K} {\sum\limits_{{X_{i} \in S_{k} }}^{{}} {d\left( {X_{i} ,C_{k} } \right)^{2} } }$$
    (11)

    where \(J\) is the objective function representing the sum of squared distances between each point and its centroid40.

The model presented in this study demonstrates strong potential for generality primarily due to the use of objective data-driven techniques using correlation, MEREC weighting, and k-means clustering which are not dependent on the identity of refrigerants and nanomaterials but rather on their quantified thermophysical impacts (e.g., thermal conductivity, boiling point, heat capacity). Additionally, the model’s clustering and optimization framework operates on normalized input-output relationships, making it adaptable to new datasets where refrigerants and nano-additives with similar performance characteristics are introduced. The use of the hybrid technique thus supports scalability across a broad range of refrigerant formulations and nanocomposite-enhanced systems, allowing the methodology to be tailored for future cryogenic and low-temperature thermal applications.

Results and discussion

Correlation analysis

The Pearson-r correlation analysis represents essential component discovering crucial operational analysis, which improves the performance of the determinants. Initial efficiencies combined with reliability through the analysis of system relations and operatory parameters enable investigators to achieve optimal system design.

Figure 4 shows the operational parameters correlation matrix which contains the inter-relationship between the operational parameters (pressure, temperature, flow rate, nano-additive concentration) and outcomes (SEC and COP). The scatter plots together with the histograms show parameter relationships and distributions through their values of Pearson’s correlation coefficient (r) and adjusted R2 indicating their statistical significance. System energy consumption rises with pressure increment, since an increased compressor work enables a higher-pressure atmosphere, thereby validating its strong positive correlation with SEC (Pearson’s r = 0.83889 and Adjusted R2 = 0.6667). An excessive pressure level shows a reverse relationship with the system energy efficiency because more pressure creates higher entropy generation (Pearson’s r = -0.14328, Adjusted R2 = -0.1019). Analysis further reveals that temperature affects energy consumption through SEC (Pearson’s r = 0.7117, Adjusted R2 = 0.45363) which is in accordance with the elevated temperatures leading to increased energy usage possibly because of increased heat exchanger thermal resistance. The analysis also indicates that temperature changes have a minimal effect on system efficiency because COP shows a weak negative correlation (Pearson’s r = -0.03859, Adjusted R2 = -0.12332).

Flow rate shows a robust positive association with SEC (Pearson’s r = 0.70353, Adjusted R2 = 0.43182) since this enables a faster fluid movement which requires higher energy consumption for pumping and compressive operations. The analysis also shows that enhanced flow rate has limited impact on performance because of pressure drop losses as reflected by the weak negative COP correlation (Pearson’s r = -0.27359, Adjusted R2 = -0.04716). The addition of nano-additives causes a minimal decrease in energy consumption according to the inverse relationship between concentration and SEC (Pearson’s r = -0.20073, Adjusted R2 = -0.07967). The overall system efficiency benefits from nano-additive concentration (Pearson’s r = 0.19001, Adjusted R2 = -0.08438) because nanocomposites improve thermal conductivity and reduce compressor workload. The calculations demonstrate a strong negative correlation between energy consumption and COP as SEC and COP (Pearson’s r = -0.994) show an Adjusted R2 of 0.98655 which proves that efficient nano-additive selection directly enhances system performance. The present research study relies heavily on nano-additives because their concentrations show direct effects on SEC and COP measurement values. The adjusted R2 values prove the reliability of linear relationships between the studied variables by confirming their strong correlation. The model fit between SEC and COP demonstrates remarkable accuracy because the adjusted R2 value reaches 0.98655 which shows precise matching between predicted and actual data.

Fig. 4
figure 4

Correlation analysis of nanocomposite inputs and their performance outcomes.

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MEREC priority analysis

The analysis of correlation data will further undergo the MEREC weighting process to establish systematic ranking of key variables as shown in Fig. 5. The results show that nano-additive concentration achieves the maximum priority value of 33%, followed by flow rate with 29%. Further pressure (23%) also achieves as the third most crucial parameter with temperature (14%) attaining the least priority. The weighting system guarantees appropriate representation of essential factors when they become input for machine learning clustering models. The priority assessment is crucial in hydrogen pre-cooling applications because hydrogen demonstrates unique characteristics like its − 252.87 °C boiling point and 0.839 cm2/s thermal diffusivity and 0.08988 g/L density at standard temperature and pressure. The special properties of hydrogen expose it to heavy thermal losses when pre-cooling occurs. The implementation of nano-composite materials creates additional optimization challenges because they possess excellent cryogenic stability along with thermal conductivity surpassing 1000 W/mK. Machine learning models achieve optimal outcomes when utilizing MEREC-weighted results in clustering algorithms to distinguish different refrigerant mixtures and operating parameters which maximize energy performance and minimize SEC and maintain system robustness.

Fig. 5
figure 5

Weightage analysis among input based on priority analysis.

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k-means cluster modelling analysis

k-means analysis is essential in this study as it enables the identification of optimal refrigerant-nanocomposite configurations by clustering high-dimensional simulation data into performance-based categories (e.g., best, average, worst) based on outcomes. Unlike traditional optimization techniques such as Response Surface Methodology or gradient-based solvers, which assume smooth and continuous objective functions, k-means can efficiently handle the nonlinear, discontinuous, and multidimensional nature of mixed refrigerant systems. In the context of this study, where input operational variables interact in complex thermodynamic patterns, k-means offers a robust data-driven classification that captures performance clusters without requiring an explicit functional relationship. This facilitates the isolation of high-efficiency refrigerant blends that may be overlooked by conventional methods, ultimately supporting more targeted optimization and system design.

ANOVA analysis presented statistical information regarding central tendency and variability as well as significant variable differences between clusters in Table 2. Table 3 showing initial cluster centers alongside Table 4 with final cluster centers. ANOVA examinations in Table 2 demonstrate that the outcomes of the hydrogen liquefaction process such as SEC and COP have a mean value of 2.91 and 4.65 respectively. The mean values operate as essential components for cluster development.

Table 2 ANOVA results by K-mean clustering.
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Table 3 Initial cluster center.
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Table 4 Final cluster center.
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The most promising circumstances emerge from Cluster 3 according to mean value assessments. The study groups Cluster 2 presents the average case situation and Cluster 1 outlines the worst-case situation according to Table 3. Dataset variability increases according to higher standard deviations while both SEC and COP follow closely the mean values. These parameters demonstrate F-values amounting to 22.01 as well as 32.39. The clustering technique demonstrates robustness because slight shifts in centroid coordinates do not alter the cluster classification results which identify the inherent data structures.

The analysis included Table 5 for the complete summary of final clusters where the average cluster data belongs to Cluster 2 while Cluster 1 stands out as the best cluster group. The algorithm shows its capability to identify fundamental patterns through gradual adjustments of centroid positions. During the final execution of centroid-based clustering the dataset displays noticeable distinct features. The classification stability proves the selected algorithm has an optimal performance even when centroid positions experience small variations.

Table 5 Cluster summary.
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A visual presentation of the cluster’s final centroids distances appears in Table 6 as it helps understand spatial patterns in the final machine-learning model. The Fig. 6 shows the heat map representation of cluster separation distances which helps understand the distance levels between clusters. The size of pairwise distance measurements reflects the separation status of cluster centers between different areas. Close cluster positions appear with small distance values.

Fig. 6
figure 6

Heatmap with dendrogram among different clusters.

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The spatial examination of cluster results allows researchers to evaluate their quality while guiding upcoming analytical operations or choosing suitable decisions.

A significant separation exists between Clusters 1 and 3 because their variables demonstrate a spacing of 0.81 in Table 6. The minimal distance of 0.40 between Clusters 1 and 2 along with Clusters 2 and 3 indicates substantial similarities between these two pairs. A relationship distance of 0.40 indicates Cluster 1 and 2 share stronger connections when compared to the relationship between Cluster 2 with Cluster 3. Although Cluster 1 and Cluster 3 demonstrate a larger separation in space their distinct characteristics demonstrate similarity as shown in Fig. 6.

Table 6 Distance between final cluster centers.
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To validate the reliability of the k-means clustering used for categorizing refrigerant combinations, the Elbow Method was applied. This technique evaluates the Within-Cluster Sum of Squares (WCSS) as the number of clusters (k) increases. The plot showed a noticeable “elbow” at k = 3, indicating that adding more clusters beyond this point results in marginal improvements to intra-cluster compactness. This justifies the selection of three optimal performance categories: best, average, and worst refrigerant blends. Additionally, Silhouette Score analysis confirmed the appropriateness of clustering by yielding values above 0.60, denoting strong separation and cohesion among clusters. Combined with the ANOVA results (significant p-values for COP), these validation techniques confirm that the clustering model is both statistically sound and meaningful for thermodynamic performance classification.

The hybrid K-means cluster analysis groups observations based on their shared features evident in both SEC and COP system. The membership distance values play a crucial role in Table 7 since they focus on the variables used for clustering in this particular analysis. The distances determine the member characteristics of each data point through their nearness to cluster centers. The cluster representativeness for data points increases proportionally to the length of distances between them. Lower values of membership correspond to data points that belong to varied clusters as visible in the scatter plot distributions. The data points belonging to group number two cluster tightly around their centroid while other groups show lower cluster concentration as shown in Fig. 7.

Table 7 Principal component scores for each cluster.
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Fig. 7
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Percentage contribution of principal components.

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PCA membership assignments demonstrate the method by which data points group according to principal component patterns in order to improve comprehensive data analysis through PCA results. The data has been analyzed through Table 7 showing PC1 and PC2 contributions and membership assignments with their corresponding impact on dataset variance measurements. Statistics indicate PC1 and PC2 explain 99.70% and 0.30% of data variability so they represent the key elements of dataset representation as shown in Fig. 7. Data spreads through dimensions to reveal hidden patterns and tendencies and this process is measured by total variance. Higher variability percentages in principle components indicate more substantial information content which makes these components more vital for the dataset.

From the above analysis, the optimal performance in the hydrogen precooling liquefaction process is achieved with combination 9, which integrated an input setting of 0.23 MPa pressure, 260 K temperature, 0.11 kg/s flow rate, and 0.24 wt% nanocomposite concentration. These input conditions created a balanced thermodynamic environment that maximized heat transfer while minimizing entropy generation during the precooling stage. The selected operating range ensured that the refrigerant mixture maintained high thermal conductivity without introducing excessive pressure drop or instability, thereby supporting efficient heat exchange between the mixed refrigerant and hydrogen across the temperature glide.

This optimal combination also featured a synergistic nanocomposite blend of Graphene/TiO2 (Hombikat UV100), g-C3N4/TiO2 (2D), and SiC/TiO2, which contributed significantly to the low SEC of 2.70 kWh/kg LH₂ and a high COP of 5.00. The Hombikat UV100 variant of Graphene/TiO2 enhanced surface area and photocatalytic properties, improving refrigerant flow behavior, while g-C3N4 (2D) offered excellent thermal dispersion and structural stability under cryogenic cycling. SiC/TiO2 added robustness with its high thermal conductivity and phase stability at low temperatures. Together, these materials created a composite with superior heat absorption and transfer capabilities, justifying the exceptional performance metrics observed in Combination 9.

Comparison of thermal properties among the best cluster outcomes

The Fig. 8 demonstrates an assessment of hydrogen pre-cooling behavior through evaluation of the selected four best data sets obtained from k-means clustering analysis. The study evaluates variation in COP and SEC when nano-additive concentration increases alongside pressure and temperature as well as flow rate variation. Figure 8a shows SEC decreases with rising nano-additive concentration until it reaches its minimum value at 0.22 wt% before stabilization occurs. The performance indicator value of COP achieves its highest point when nano-additive concentration reaches 0.18 wt%. SEC decreases in the beginning because nanocomposites enhance thermal conductivity which enables more efficient heat exchange and reduces the work required from the compressor. The heat transfer efficiency decreases through mixing effects when nano-additive concentrations exceed a specific threshold enabling a drop in COP. The optimal condition exists at a concentration between 0.18 wt% and 0.22 wt% because it creates an equal balance between decreased SEC and improved COP.

Figure 8b shows a double curve where pressure increases SEC until it reaches maximum at 0.20 MPa. Lower working pressures lead to higher compressed work requirements, resulting in increased SEC. The heat transfer rate enhances when pressure surpasses 0.20 MPa thus decreasing SEC values and enhancing the COP of the system. SEC reaches its lowest point, yet COP reaches its highest level when operation occurs between 0.22 and 0.23 MPa. A denser molecular arrangement of hydrogen molecules at elevated pressures creates more efficient heat exchange pathways that decrease the cooling energy requirements.

The relationship between SEC and COP with temperature in hydrogen liquefaction process using nanocomposites is shown in the Fig. 8c. Because the temperature 235 K to 260 K, SEC first reaches its peak value around 245 K (3.15 kWh/kg) then decreases down to 2.7 kWh/kg, and COP steadily increases between 4.3 and 5.0. This reverse trend means that the increase in temperatures will be more efficient in terms of the efficiency of the system due to the better thermal conductivity of such nanocomposites as graphene/TiO 2 and g -C 3 N 4 /TiO 2. At reduced temperatures, more energy is required as there is more load of the refrigerant and reduced efficiency of the heat exchange. But above 245 K, the nanocomposites enhance heat conduction and the compressor work, and thus the performance is enhanced. The given result is significant because it helps to locate the temperature range in which nanocomposites are able to deliver the best thermophysical advantages, utilizing less energy and increasing the COP. It confirms the benefit of nanocomposites in the optimization of hydrogen liquefaction and justifies its inclusion in the perspective high-tech cryogenic systems.

Figure 8d shows SEC stays consistent until 0.14 kg/s before increasing simultaneously with COP starting from its peak value at 0.14 kg/s and declining afterward. The nanocomposite-enhanced refrigerant reaches maximum heat absorption at an optimal flow rate because the heat exchanger operates most efficiently at that point without producing excessive turbulence. The cooling efficiency drops after 0.14 kg/s due to flow instabilities which result in increased SEC while COP decreases. A flow rate of 0.14 kg/s provides the highest COP alongside minimum SEC values thus making it the perfect operational point for hydrogen pre-cooling using nanocomposites. The findings thus prove that hydrogen pre-cooling reaches its maximum efficiency when nanocomposite concentration, pressure, temperature and flow rate are optimized.

Fig. 8
figure 8

Performance analysis based on the different input parameters.

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Validation and comparison with previous studies

The results of the current study, employing a nano-mixed refrigerant cycle with Peng–Robinson EOS, demonstrate a substantial improvement in thermodynamic performance when compared to other prior similar models as shown in Table 8. Specifically, the achieved SEC in the current study of 2.70 kWh/kgLH₂ and COP of 5.0 is better for the specific model. In previous researches such as Yin and Ju’s (2020) used Joule–Brayton cycle (SEC: 6.50, COP: 2.34), while Yuksel et al.‘s (2017) employed helium precooled cycle (SEC: 11.47), and Li et al.’s (2024) applied a binary precooling cycle (SEC: 4.44, COP: 2.70). This enhanced performance is attributed to the integration of nanocomposites, which significantly increase refrigerant thermal conductivity and improve heat exchange efficiency during precooling. Furthermore, the use of a data-driven optimization framework (MEREC + k-means) allowed precise tuning of refrigerant compositions, enabling effective thermal glide matching with hydrogen’s cooling profile. These improvements collectively validate the proposed model’s superiority over conventional systems, both in energy efficiency and thermodynamic stability.

Table 8 Comparison of previous literature for different models.
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The results of the study are based on simulation analysis using Aspen HYSYS. The thermodynamic modelling relies on the Peng–Robinson equation of state, which is well-established for accurately representing phase behaviour and property prediction of cryogenic gases, including hydrogen and mixed refrigerants. The output values SEC and COP generated by simulating refrigerant flow, energy exchange, and hydrogen cooling within the optimized operating envelope. Unlike experimental setups, the simulation environment ensures precise control over variables such as pressure, nano-additive concentration, and refrigerant composition, enabling detailed analysis and performance evaluation under idealized, reproducible conditions. While the results have not been validated through physical testing, they show good consistency with values reported in prior literature, confirming their technical reliability and alignment with existing cryogenic liquefaction principles.

Although simulations inherently lack real-world disturbances like material degradation, component inefficiencies, or system aging, Aspen HYSYS provides a robust platform for process design due to its rigorous thermodynamic solvers and ability to model complex multi-component refrigeration cycles. The use of advanced modelling tools, combined with high-quality input data for thermophysical properties, enables a highly realistic approximation of real-world performance, especially in early-stage process design. Furthermore, the model incorporates refrigerant behaviours across phase boundaries and temperature ranges relevant to hydrogen precooling, ensuring its applicability to actual liquefaction systems. Therefore, while experimental validation is a necessary future step for full implementation, the simulation-based approach presented here offers a scientifically grounded, cost-effective, and credible basis for optimizing mixed refrigerant cycles for hydrogen liquefaction.

Conclusion

The high energy requirements of hydrogen liquefaction stem mainly from its physical characteristics that dictate the process requirements. When used for cryogenic applications this substance requires large quantities of energy because of its low boiling point and its gaseous state with low density demands additional stages of compression to maintain its low pressure. The high specific heat capacity of hydrogen creates additional energy requirements because it demands considerable power to control small temperature fluctuations thus resulting in conventional liquefaction procedures becoming both expensive and highly energy consuming. The incorporation of nanocomposites brings together improved thermal conductivity and cryogenic stability with superior heat transfer properties which resolves various operational barriers. Machine learning algorithms work with nanocomposite-based systems to process extensive HYSYS Aspen simulation data along with operational data which helps identify optimal energy-efficient solutions and optimize cooling rates and manage refrigerant-nanocomposite mixtures. The initial stage of this study determined the vital connections between refrigerant components, nanocomposite concentrations and performance outputs to function as a foundation that enabled the development of a weight-based analysis and k-means machine learning model. The following outcomes are achieved from the study:

  • The Pearson correlation analysis reveals that SEC is strongly and positively correlated with pressure (r = 0.83889) and flow rate (r = 0.70353); in contrast, COP exhibits weak or negative correlations with these parameters.

  • MEREC method identifies nano-additive concentration (33%) and flow rate (29%) as top priorities, followed by pressure (23%) and temperature (14%).

  • The optimum operating conditions for hydrogen precooling were identified as 0.23 MPa pressure, 260 K temperature, 0.11 kg/s flow rate, and 0.24 wt% nanocomposite concentration, providing an ideal thermal balance for efficient liquefaction.

  • The best-performing nanocomposite blend—Graphene/TiO2 (Hombikat UV100), g-C3N4/TiO2 (2D), and SiC/TiO2—exhibited exceptional thermal conductivity and stability, contributing to enhanced refrigerant behavior and cryogenic performance.

  • This optimal configuration led to a significantly reduced SEC of 2.70 kWh/kg LH2 and a high COP of 5.00, validating the effectiveness of the hybrid nanocomposite-mixed refrigerant system in improving energy efficiency in hydrogen liquefaction.

While the present study offers promising insights into optimizing hydrogen precooling using nanocomposite-enhanced refrigerants, it has certain limitations that warrant future validation. The simulation-based results, though thermodynamically robust, lack long-term experimental testing to evaluate the durability, stability, and potential degradation of nanocomposites under cryogenic cycling. Moreover, the feasibility of scaling the process to an industrial level must be investigated, considering the real-world integration of nanoparticle dispersions in large-scale refrigeration loops. Additionally, the environmental sustainability and life-cycle impact of nanomaterials should also be assessed, including their sourcing, toxicity, and recyclability. Future work should therefore focus on experimental validation under operational conditions, techno-economic assessment at scale, and environmental impact analyses to ensure the practical viability of the proposed approach. Future studies could also explore the long-term durability of hydrogen liquefaction systems incorporating nano-additives, as sustained exposure to cryogenic conditions may affect material stability. Moreover, further analysis on agglomeration, degradation, or phase separation over repeated cycles is critical for ensuring operational reliability. Preliminary investigations into the aging behavior and thermal fatigue resistance of nanocomposite-enhanced refrigerants will support safe and scalable deployment.