Machine Learning for Fluid-Agnostic Laminar Heat Transfer Predictions Under Supercritical Conditions

Abstract

Machine learning was employed to make fluid agnostic laminar heat transfer prediction in supercritical conditions, encompassing three fluids (sCO₂, sH₂O, sC₁₀H₂₂) representing a wide range of operating conditions. High-fidelity training data, consisting of both non-dimensional and dimensional (operating parameter) as inputs and Nu and T_wall as outputs, were generated from grid-converged, steady-state, computational fluid dynamic (CFD) simulations. The Random Forest (RF) algorithm outperformed the artificial neural networks (ANNs) across all scenarios on the small multi-fluid dataset (~1600 data points) employed during the training process. When using non-dimensional parameters as inputs, Nu prediction fidelities were better than T_wall predictions for both ML algorithms across both horizontal and vertical configurations. The RF model trained on data from a specific flow configuration (horizontal/vertical) could predict T_wall within an accuracy of +/−1% with dimensional, operational parameters as inputs while being agnostic to the working fluid. Furthermore, by including the gravity vector as an additional variable during the training process, the RF model could predict T_wall accurately in a mixed, multi-fluid dataset containing data from both horizontal and vertical configurations.

1. Introduction

1.1. Motivation for Fluid Agnostic Heat Transfer Correlations

Heat transfer under supercritical conditions plays a central role in a wide range of energy, aerospace, and process systems, including regenerative cooling systems in rocket engines and hypersonic vehicles (sHydrocarbon) [1], advanced nuclear reactors (sH₂O) [2] and supercritical Brayton cycles (sCO₂) [3]. In these applications, fluids operate in regimes where thermo-physical properties exhibit strong nonlinearity and sharp gradients, leading to heat transfer behavior that departs fundamentally from classical correlations developed for subcritical flows that were formulated for constant or mild property variations. Accurately predicting laminar heat transfer in such regimes remains a long-standing challenge. A bewildering number of supercritical fluid-specific heat transfer correlations (for turbulent regimes, in particular) have been proposed in recent years [4,5]. However, many emerging energy and thermal management applications and system architectures are being increasingly designed to accommodate multiple candidate working fluids, including supercritical CO₂, water, hydrocarbons, refrigerants, and molten salts. In such settings, fluid selection is treated as an optimization variable rather than a fixed design choice, and the goal may be to assess material limits, minimize pressure drop and pumping power, corrosion reduction, turbomachinery complexity reduction and environmental regulations. This creates a pressing need for a fluid agnostic heat transfer prediction tool as an enabling technology that can be generalized across fluids and thermodynamic regimes. To fill this void, this study explores a comprehensive evaluation of fluid agnostic laminar heat transfer prediction in supercritical conditions encompassing three fluids (sCO₂, sH₂O, sC₁₀H₂₂) transitioning through their respective pseudocritical temperatures (T_PC). By restricting ourselves to the laminar regime, high-fidelity data were generated from grid-converged, steady-state, computational fluid dynamic (CFD) simulations encompassing wide variations in operating pressures, inlet temperatures, wall heat fluxes and mass fluxes in both horizontal and upward flow tube configurations. After attempting to fit the data using traditional heat transfer correlation methods employing non-dimensional parameters, machine learning (ML) approaches are considered using both non-dimensional and dimensional (operating parameter) inputs.

1.2. Wall Temperature and Heat Exchanger Design Under Constant Heat Flux

We restrict ourselves to constant heat flux (heating) scenarios in this study. Heat exchangers or cooling channels experience constant wall heat fluxes in several scenarios, including regenerating cooling in rocket engines, where combustion gases provide nearly uniform heat fluxes to the combustion chamber wall [1], nuclear reactors, where volumetric heat generation is nearly constant within a fuel rod resulting in constant wall heat flux (q) along a cooling channel section [2], and supercritical Brayton cycles, where the working fluid is heated with an imposed heat flux from an external source, such as a solar receiver or a reactor [3]. In these scenarios, the goal is to design cooling passages consisting of channels, pipes or tube banks that can remove a specified amount of heat (q) while keeping the wall temperatures (T_wall) within operational limits to prevent material failure due to thermal fatigue, corrosion reduction or fuel pyrolysis and deposit formation in regenerative cooling systems. In order to assess T_wall variations under different operating conditions, the following procedure is followed:

The local convective heat transfer coefficient (h) in these scenarios can be expressed in terms of the bulk fluid temperature (T_bulk) as:

$$\mathrm{h}\left({\mathrm{T}}_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}-{\mathrm{T}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}\right)=\mathrm{q}$$

(1)

Equation (1) can be rewritten in terms of the Nusselt number (Nu), hydraulic diameter (D_h) and bulk thermal conductivity (K_bulk) as:

$${\mathrm{T}}_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}-{\mathrm{T}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}={\displaystyle \frac{\mathrm{q}{\mathrm{D}}_{\mathrm{h}}}{\mathrm{N}\mathrm{u}{\mathrm{K}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}}}$$

(2)

Most heat transfer correlations are expressed in terms of Nu. However, in Equation (2), and for subcritical flows, Nu is a function of T_wall, T_bulk and K_bulk (with q and D_h held constant); this equation needs to be solved iteratively as follows:

(a)

Calculate the variation in T_bulk along the length of the channel (z) using the relation:

$$\dot{\mathrm{m}}{\mathrm{C}}_{\mathrm{p},\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}{\displaystyle \frac{{\mathrm{d}\mathrm{T}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}}{\mathrm{d}\mathrm{z}}}=\mathrm{q}\mathrm{P}$$

(3)

where $\dot{m}$ is the mass flow rate, C_p,bulk is the specific heat based on the local temperature and P the perimeter of the channel.

(b)

Calculate the properties or non-dimensional variables that are relevant to Nu calculation (in Equation (2)) based on T_bulk: K, dynamic viscosity (µ), C_p, K, Reynolds number (Re), Prandtl number (Pr), flux-based Grashof number (Gr*), etc.

(c)

Use established correlations based on the variables determined in step (b) to compute Nu.

(d)

Compute T_wall from Equation (2) and check to ensure that it falls within operation limits. If it does not meet the limits, then changes to any of the operating conditions can be made to ensure compliance: q, D_h, P and $\dot{m}$

In supercritical conditions, thermophysical properties can change by several orders of magnitude across the thermal boundary layer near T_PC. Therefore, many of the correlations for Nu (step c) under supercritical conditions are also in terms of T_wall [3,4]. In such scenarios, one needs to iteratively solve for T_wall by looping from steps (b–d), i.e., guessing T_wall in step (b) to compute Nu and ensuring that it converges with the T_wall obtained in step (d). In constant flux scenarios where buoyancy effects are important (large-diameter tubes, upward/downward flows), it is customary to employ a flux-based Grashof number (Gr*) on the Nu correlation, thereby avoiding any dependencies on the wall temperature:

$${\mathrm{G}\mathrm{r}}^{\mathrm{*}}={\displaystyle \frac{\mathrm{g}{\mathrm{q}{\mathsf{\beta }}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}{\mathrm{D}}_{\mathrm{h}}}^4}{{{\mathrm{k}}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}\mathsf{\nu }}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}^2}}$$

(4)

where ν is the kinematic viscosity and $\mathsf{\beta }$ is the coefficient for thermal expansion (often tabulated and can be computed based on density variation with temperature). On the other hand, if the wall temperature is known, a density-based Grashof number defined as:

$${\mathrm{G}\mathrm{r}}_{\mathrm{B}\mathrm{U}\mathrm{L}\mathrm{K}}={\displaystyle \frac{\mathrm{g}{{\mathrm{D}}_{\mathrm{h}}}^3(1-\frac{{\mathsf{\rho }}_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}}{{\mathsf{\rho }}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}})}{{\mathsf{\nu }}_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}^2}}$$

(5)

may take the place of Gr* in Nu correlations. What should be clear from the aforementioned methodology is that accurate estimation of T_wall under different operational scenarios hinges upon the accuracies of the Nu correlation (step c) employed (generally +/−20% is deemed acceptable).

1.3. Empirical Approaches to Predict Nu in Laminar Supercritical Heat Transfer Conditions

Under laminar conditions, when fluid properties are constant or varying weakly, the Nusselt number (Nu) can be obtained from the bulk Reynolds (Re), Prandtl numbers (Pr) and/or Grashof numbers (Gr) from well-established correlations. However, formulating similar fluid-agnostic Nu correlations that are applicable across the entire supercritical fluids spectrum has been challenging due to heat transfer enhancement (HTE) or heat transfer deterioration (HTD) that has been observed when the bulk temperature is in the vicinity of T_PC due to the strong property variations across the thermal boundary layer. This issue is further compounded by the effects of gravity. HTE/HTD in the presence of buoyancy has been extensively studied in vertical tubes, and various criteria for HTE/HTD onset have been proposed [6,7,8,9,10]:

$${\displaystyle \frac{{\mathrm{G}\mathrm{r}}^{\mathrm{*}}}{{\mathrm{R}\mathrm{e}}^{2.7}}}\ge {10}^{-5}$$

(6)

Since Gr* and Re vary along the flow path, a criterion for HTD has also been proposed for the minimum heat flux q_DHT (kW/m²) at which deteriorated heat transfer (HTD) occurs as a function of mass flux G (in Kg/m²-s) [11]:

$${\mathrm{q}}_{\mathrm{D}\mathrm{H}\mathrm{T}}=-58.97+0.745\mathrm{G}$$

(7)

1.4. Machine Learning Approaches for Supercritical Heat Transfer Prediction

Recent advances in machine learning (ML) offer an alternative paradigm for modeling these complex transport phenomena. By learning directly from high-fidelity data, ML models can capture nonlinear relationships that are difficult to encode in closed-form Nu correlations. A summary of several studies that have demonstrated promising results using ML for heat transfer prediction under supercritical conditions is provided in

Table 1. A summary of previous ML studies for supercritical heat transfer prediction.

Reference	Training Dataset [Fluid]	Input/Output Parameters	ML Algorithms
Wu et al. [12]	8192 data points from direct numerical simulations (DNSs) [sCO2]	A total of 44 feature parameters. A new dimensionless parameter (based on Re) showed the strongest importance, followed by Pr for Nu prediction.	CatBoost algorithm showed best performance.
Wen et al. [13]	11,032 experimental data points [sCO2]	Mass flow rate, wall heat flux, pressure, fluid enthalpy, and tube diameter to predict Twall.	Random Forest (RF), Extreme Gradient Boosting (XGBoost), Support Vector Machines (SVR), and artificial neural networks (ANNs). ANNs resulted in the best prediction performance
Shi et al. [14]	Multifidelity data: Low-fidelity data (empirical correlations), medium-fidelity data (RANS with the shear stress transport (SST) k–ω model), and high-fidelity data from (experiments) [sCO2]	Pressure, mass flux, wall heat flux, inner diameter and local specific enthalpy to predict Twall.	A transfer learning model based on multi-fidelity data performed the best.
Xiao et al. [15]	4737 measurement data points [sCO2, sH2O]	Binary classification to predict HTD using inlet temperature, inlet pressure, diameter, mass flow rate, heat flux and bulk to pseudocritical enthalpy ratios as inputs.	K-fold cross-validation used to train and test the machine learning model. Several machine learning models (SVM, KNN, EC and ANN) were used to solve the binary classification of HTD and no-HTD.
Zhang et al. [16]	11,589 data points, including circular and non-circular tubes [sCO2, sH2O]	A total of 17 feature variables (experimental conditions, thermo-physical property ratios, and dimensionless parameters) were normalized and used to predict both Twall and Nu.	ANN results were independent of the channel shape (circle/non-circle) and fluid type.
Zhu et al. [17]	11,589 data points, including circular and non-circular tubes [sCO2].	Effects of mass flux, heat flux, inlet temperature, and tube diameter on HTD and Nu prediction.	An ANN model with all features in the input parameters performed best. (number of input features ranged from six to 14)
Rajendra Prasad et al. [18]	134,698 CFD-generated data points [sCO2]	Input parameters, including Pr, Re, Gr*, q, and P/Pc, used to predict Nu	A deep neural network with four hidden layers and 15 neurons in each layer gave the best performance.
Sun, et al. [19]	5780 measurement data points [sCO2]	Inputs, including pressure, mass flux, heat flux, an inner diameter of tube, and bulk-specific enthalpy, used to predict Twall and the heat transfer coefficient.	ANN predictions were better than empirical correlations
Sun, et al. [20]	5895 measurement data points [sCO2]	Inputs, including pressure, mass flux, heat flux, inner diameter of tube, and bulk-specific enthalpy, used to predict Twall.	GA-BP (Genetic Algorithm–Back Propagation) predictions were better than empirical correlations.
Ye, et al. [21]	4354 filtered experimental data points out of 7313 data points collected from the published literature [sCO2]	Heat flux, mass flux, tube diameter, pressure and bulk-specific enthalpy used to predict Twall.	An ANN model can be used to predict heat transfer without significant buoyancy force and flow acceleration.
Cao, et al. [22]	Experimental measurements [sCO2]	Non-dimensional parameters including buoyancy and flow accelerations numbers used to predict Nu.	Ratio of near wall and bulk fluid properties are important; Re at wall and bulk are also important to predict Nu.
Yibo et al. [23]	2071 experimental data points in internally rifled tubes [sH2O]	A combination of dimensional parameters, ratios of bulk to wall properties, non-dimensional parameters (including rib geometry) used to predict Nu and Twall.	An ANN model after optimization was able to perform well.
Li et al. [24]	1598 data points [sH2O]	Dimensionless numbers to predict Nu and Twall.	Rebulk and Prbulk play an important role in heat transfer for HTE, while buoyancy and acceleration factors contribute to heat transfer deterioration.
Lopes et al. [25]	25,010 individual data points from k-omega SST simulation [sCO2] (horizontal orientation)	TBulk, q, G both Twall and Nu predicted independently.	Emphasized the effort required to develop such a model—collecting and processing data, training, validating, and testing.
Li et al. [26]	SST K-omega CFD 8500 data points [sHydrocarbon]	Artificial neural network (ANN), Random Forest (RF), support vector regression (SVR), and k-nearest neighbor (kNN), used for local heat transfer coefficient prediction.	A one-dimensional simulation model incorporating machine learning algorithms was developed, enabling rapid prediction of heat transfer coefficient distributions along the channel.
Gong et al. [27]	6275 data points, 2D slices of CFD simulation [sHydrocarbon]	Deep learning (DL) employed to map relationship between outer wall temperature Twall to Tbulk, velocity V and h.	The flow field and heat transfer predicted by the DL model match well with the CFD simulation while being orders of magnitude faster.
Tao et al. [28]	4185 Nu data points, 327 friction factor data points, horizontal tube [sHydrocarbon]	T/Tc, P/Pc, mass flux, axial position used to predict Nu and friction factor.	Different combinations of transfer functions explored. ANN topology for high prediction accuracy was presented.

.

The following points are noteworthy in Table 1:

• Almost all of the studies involved turbulent regimes. But laminar flows may be encountered during thermal management in compact, high-power-density systems like data centers and advanced electronic devices [sCO₂], mini/microchannels in specific regenerative cooling [sDecane], or nuclear reactors [sH₂O].
• All the studies considered a single fluid except [15,16], which considered both sCO₂ and sH₂O.
• Most studies considered only a single tube orientation (horizontal/vertical). The challenges with directly extending the trained ML algorithm to a different orientation or configuration were highlighted in [21].
• Large datasets (>1000 data points) were employed to train ANNs or other ML algorithms.
• Both non-dimensional and dimensional (operating parameters) have been employed successfully to predict T_wall or Nu using ML approaches. Therefore, the need to use non-dimensional/scaled variables to predict other challenging heat transfer phenomena, such as boiling/condensation, does not appear to be necessary [29].

The novel contributions of this study, which builds on and distinguishes itself from these previous studies, are highlighted next.

Contribution #1: Fluid agnostic predictions of Nu, T_wall.

In many emerging applications, multiple working fluids (e.g., CO₂, water, hydrocarbons, refrigerants, and molten salts) may be considered interchangeably during design optimization. The lack of fluid-agnostic predictive capability, either in the form of correlations or ML algorithms, poses a critical bottleneck for system-level modeling and rapid design exploration, necessitating the need for fluid-specific correlations or ML models. However, this seriously undermines the wide-scale adaptability of these models/correlations, as well as their integration into system-level codes such as ASPEN Plus or RELAP [30]. We attempt to overcome this shortcoming in this study by training ML algorithms on mixed data from three supercritical fluids [sCO₂, sH₂O, sDecane], encompassing a wide range of operating temperatures, pressures and thermophysical property variations, and assessing their ability to predict T_wall and Nu.

Contribution #2: Prediction using the Random Forest (RF) algorithm with small datasets in the laminar regime.

While most previous studies in Table 1 use large datasets to train ML models, we assess the performance of the RF model in predicting Twall and Nu in as few as 1600 mixed fluid datasets using dimensional or non-dimensional inputs, as well as horizontal and vertical orientations. RF algorithms are particularly well suited for small datasets, as they combine ensemble averaging with decision-tree learners that are inherently low-bias and nonparametric [31]. By aggregating predictions across many decorrelated trees, random forests reduce variance without requiring large training samples, enabling robust performance in data-scarce regimes without the risk of overfitting.

The high-fidelity data needed to train the ML algorithms were generated from steady-state, laminar flow computational fluid dynamic simulations of three supercritical fluids (sCO₂, sH₂O, sC₁₀H₂₂) transitioning through their respective T_PC. The simulations were carried out at sufficient spatial resolutions to generate grid-independent, “ground truth” training data for the ML algorithms. The focus on laminar flows is deliberate so as not to invoke the uncertainties associated with turbulence modeling. At the same time, laminar regimes are directly relevant to microchannels, startup and transient operations, and low-Reynolds-number thermal systems increasingly encountered in advanced energy technologies.

Contribution #3: Extendibility to varying tube orientations/gravity environments.

Supercritical fluids in realistic heat exchanger geometries may encounter varying gravity environments such as serpentine/S-shaped channels or different tube orientations (when deployed in hypersonic vehicles, for instance). The challenges with training a single ML model that encompasses data across different gravity regimes in turbulent flow conditions have already been highlighted previously due to the effect of buoyancy [21]. However, in order to integrate the ML model developed in this study as a predictive layer in realistic engineering workflows, this shortcoming needs to be overcome. Towards filling this void, we also explore and assess the extendibility of our machine learning model trained on a “mixed” multi-fluid dataset encompassing both horizontal and vertical tube orientations.

2. Materials and Methods

2.1. Fluids and Conditions Investigated

The thermophysical properties (Φ) of the fluids investigated in this study, normalized by their maximum values across the temperature ranges investigated, are shown in Figure 1. sCO₂ and sH₂O properties were obtained from REFROP [32], and sC₁₀H₂₂ properties were obtained from Liu et al. [33].

The T_PC values of each fluid are readily identified by the location where their specific heats reach their highest values. The different operating conditions investigated in this study are listed in

Table 2. The operating conditions investigated in this study (upward flow, vertical orientation).

Case Number	Fluid	Tube Diameter (m)	Mass Flow Rate, G (kg/s)	Inlet Temperature Tinlet (K)	Wall-Flux (W/m2)	Pressure (MPa)	q > qDHT? (Equation (7))
sC10H22—Case II	sC10H22	0.000375	1.0 × 10−5	350	15,000	3	Yes
sC10H22—Case II	sC10H22	0.000375	1.0 × 10−5	498	35,000	3	Yes
sC10H22—Case III	sC10H22	0.0003	1.0 × 10−5	600	3685	5	No
sC10H22—Case IV	sC10H22	0.000375	8.0 × 10−6	350	30,000	5	Yes
sCO2—Case I	sCO2	0.001	3.63 × 10−5	280	3000	8.2	Yes
sCO2—Case II	sCO2	0.002	1.19 × 10−4	300	3000	8.2	Yes
sH2O—Case I	sH2O	0.001	5.39 × 10−5	600	20,000	24	Yes
sH2O—Case II	sH2O	0.002	1.11 × 10−4	625	20,000	24	Yes

(upward flow, vertical orientation) and

Table 3. The operating conditions investigated in this study (horizontal orientation).

Case Number	Fluid	Tube Diameter (m)	Mass Flow Rate, G (kg/s)	Inlet Temperature Tinlet (K)	Wall-Flux (W/m2)	Pressure (MPa)	q > qDHT? (Equation (7))
sC10H22—Case I	sC10H22	0.000375	1.0 × 10−5	498	8500	3	Yes
sC10H22—Case II	sC10H22	0.001	1.0 × 10−5	350	15,000	3	Yes
sC10H22—Case III	sC10H22	0.0015	1.1 × 10−4	350	17,550	5	Yes
sC10H22—Case IV	sC10H22	0.0015	4.84 × 10−5	600	3500	5	Yes
sCO2—Case I	sCO2	0.001	3.63 × 10−5	280	3000	8.2	Yes
sCO2—Case II	sCO2	0.002	1.19 × 10−4	300	3000	8.2	Yes
sH2O—Case I	sH2O	0.001	5.39 × 10−5	600	20,000	24	Yes
sH2O—Case II	sH2O	0.002	1.11 × 10−4	625	20,000	24	Yes

(horizontal orientation), respectively. In the horizontal-orientation simulations, gravity was turned off to avoid any effects of buoyancy-induced vortices that might result in unequal temperature profiles between the top and bottom portions of the tube. Although such effects are generally seen only at tube diameters > 2 mm (greater than the maximum tube diameter considered in this study), turning off gravity in the horizontal orientation was deemed to be important since T_wall is the key variable obtained from the CFD simulations (Equation (2)), which were being employed to compute Nu. Even small differences in heating profiles between the top and bottom portions of the tube will impact the T_wall profiles along the tube length from CFD depending on the azimuthal co-ordinate at which the data is extracted.

In all of the scenarios, we ensured that the fluids transitioned through their respective T_PC values.

2.2. Generation of Ground Truth Data

Ground truth data for training the ML algorithms were obtained from laminar, steady-state simulations carried out employing the commercial computational fluid dynamic (CFD) code ANSYS FLUENT 19.1 [34]. 3D geometries of circular pipes were resolved using 485,000 hexahedral cells and simulated after ensuring that grid-independent results were obtained at this resolution across all scenarios (i.e., inlet Re, tube diameters, and heat fluxes). A representative mesh layout at the tube cross-section is shown in [8]. Fully developed velocity profiles and constant temperature conditions were imposed at the inlet boundaries. The computed wall temperature profiles (the primary simulation output of interest) were used to assess convergence by ensuring that they did not change with any further increase in mesh resolution across all scenarios.

In order to compute the Nusselt number (Nu) and heat transfer coefficient (h), the local wall temperature (T_wall) must be obtained first. For a specified heat flux q, T_wall was computed by ANSYS FLUENT as:

$${\mathrm{T}}_{\mathrm{W}\mathrm{A}\mathrm{L}\mathrm{L}}={\displaystyle \frac{\mathrm{q}\mathsf{\Delta }\mathrm{n}}{{\mathrm{K}}_{\mathrm{f}}}}+{\mathrm{T}}_{\mathrm{f}}$$

(8)

where q is in W/m², Δn is the normal distance from the wall to the centroid of the first cell adjacent to the wall, and K_f and T_f are the fluid thermal conductivity and the fluid temperature adjacent to the wall, respectively. The local heat transfer coefficient (h) was then estimated from T_wall using Equation (1).

T_Bulk, the local fluid bulk temperature, was estimated from the local bulk enthalpy (H_z) as:

$${\mathrm{H}}_{\mathrm{z}}={\mathrm{H}}_{\mathrm{i}\mathrm{n}}+{\displaystyle \frac{\mathrm{q}\mathrm{*}\mathsf{\pi }\mathrm{*}\mathrm{d}\mathrm{*}\mathrm{z}}{\dot{\mathrm{m}}}}$$

(9)

where $\dot{m}$ is the mass flow in Kg/s, H_in is the enthalpy corresponding to the inlet temperature, z is the distance from the inlet, and d is the tube diameter (in m). Note, that Equation (9) is simply the energy balance represented by Equation (3) written in terms of enthalpy. The local bulk temperature (T_Bulk) was evaluated from the estimated local bulk enthalpy (H_z) (cf. Figure 1). The thermal conductivity (K_Bulk) corresponding to the bulk temperature was then estimated (cf. Figure 1) and used to calculate the Nusselt number (Nu) from the expression:

$$\mathrm{N}\mathrm{u}={\displaystyle \frac{\mathrm{h}{\mathrm{D}}_{\mathrm{h}}}{{\mathrm{k}}_{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}}}}$$

(10)

The different bulk non-dimensional numbers of interest in heat transfer scenarios, including the Reynolds number (Re), Prandtl number (Pr), and modified Grashof number, (Gr*) were also computed at each axial spatial location along the tube (200 locations per simulation). Consequently, simulations of the entire set of operating scenarios listed in Table 2 and Table 3 generated 1600 data points at each operating condition.

To demonstrate the challenge with obtaining a correlation for Nu using statistical regression techniques, Nu variations for the simulations carried out at vertical orientation (cf. Table 2) are shown as a function of various combinations of non-dimensional parameters in Figure 2, with each fluid represented by a distinct color. Nu approaches a value of 4.36 at fully developed flow conditions for constant property fluids [35], and this value is represented by horizontal lines in Figure 2a–d. Therefore, the data indicates that HTE was present in vertical flows involving sCO₂ and sH₂O, whereas HTD was encountered with sDecane. HTD in sDecane for laminar flows in the vertical tube orientation has also been observed previously by Liu et al. [36].

For laminar, mixed convection, vertical flows of sCO₂, Viswanathan and Krishnamoorthy proposed a power-law relationship for Nu in terms of Gr/Re [8] that was functionally similar to that of Jackson et al. [10], where Nu was expressed in terms of Gr*/Re. However, the scatter in Nu data in Figure 2a,b clearly shows that a unified fluid-agnostic correlation for Nu in such simple functional forms may not be feasible. Since a constant temperature boundary condition was imposed at the inlet and the thermal boundary layer was not fully developed, Nu is expected to be greater than 4.36 at developing flow conditions. However, HTE may also occur due to buoyancy effects. Figure 2c,d attempt to delineate these two effects by employing commonly employed buoyancy-enhancement criteria [6,36,37,38], marked as vertical lines in Figure 2c,d. Nu data points that are above 4.36 but to the left of the vertical lines in these figures may be attributed to the developing thermal boundary layer, whereas those to the right of the vertical lines and above 4.36 fall in the buoyancy-augmented heat transfer regimes. A few studies have also employed the Richarson (Ri) number (Gr/Re²) to characterize the importance of buoyancy. However, there is no consensus on the lower limit associated with Ri associated with buoyant effect onset (this has been reported to vary from 0.001 to 1) [6]. Nu versus Ri variation is shown in Figure 2e. Regardless of the criterion employed, Figure 2c–e indicate definitively that buoyancy-induced HTE was observed with sCO₂ and sH₂O.

While the scatter in the data encompassing all three supercritical fluids clearly highlights the challenge of formulating a unified correlation using statistical regression alone, an attempt to uncover the relationship among the different non-dimensional variables, including Nu, Re, Pr, Gr and Gr*, was undertaken.

2.3. Attempts to Fit Ground Truth Data Using Correlations

First, Spearman rank correlation coefficients were determined to quantify the strength and direction of a monotonic relationship between Nu and other non-dimensional parameters. In addition to the non-dimensional variables shown in Figure 2, the Graetz (Gz) number, defined as:

$$\mathrm{G}\mathrm{z}=\left({\displaystyle \frac{\mathrm{D}}{\mathrm{z}}}\right)\mathrm{R}\mathrm{e}\mathrm{P}\mathrm{r}$$

(11)

was also employed in the Spearman analysis. The Spearman coefficients (r_s) corresponding to two sets of non-dimensional variables were computed separately for both horizontal and vertical orientations as:

$${\mathrm{r}}_{\mathrm{s}}=1-{\displaystyle \frac{6\sum _{\mathrm{i}=0}^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}{{\mathrm{d}}_{\mathrm{i}}}^2}{\mathrm{n}({\mathrm{n}}^2-1)}}$$

(12)

where d_i is the difference between two ranks of each observation and n is the number of observations.

Heat maps of r_s are shown in Figure 3a and Figure 3b, corresponding to the horizontal and vertical orientations, respectively. Values of r_s range from −1 to +1, where +1 indicates a perfect monotonic increasing relationship, −1 indicates a perfect monotonic decreasing relationship, and 0 indicates no monotonic association. In the horizontal orientation (cf. Figure 3a), a weak correlation is observed between Nu and all of the non-dimensional variables with |r_s| < 0.75. In the vertical orientation (cf. Figure 3b), Nu exhibits a strong monotonic increasing correlation with Re, Gr* and Gz.

The PySR symbolic regression module [38] was initially employed to obtain a best fit to the Nu data shown in Figure 2. For horizontal, constant property, laminar flows, Nu is a function of Gz only. The following relationship was ascertained from PySR for the horizontal orientation that offered the best combination of simplicity and accuracy:

$$\mathrm{N}\mathrm{u}={\displaystyle \frac{4.06+0.068{\mathrm{G}\mathrm{z}}^{0.73}}{1+1.13{\mathrm{G}\mathrm{z}}^{-1.05}}}$$

(13)

The corresponding equation for vertical orientation was determined to be:

$$\mathrm{N}\mathrm{u}={\left({\displaystyle \frac{\mathrm{G}\mathrm{r}\mathrm{*}}{\mathrm{R}\mathrm{e}}}\right)}^{0.27}$$

(14)

Nu predictions from Equations (13) and (14) are compared against actual Nu values ascertained from the CFD simulations in Figure 3c and Figure 3d, respectively. The poor agreement between the actual and predicted Nu in Figure 3c,d highlights the need to employ machine learning (ML)-based approaches in order to improve predictability.

2.4. Machine Learning (ML) Algorithms Employed

Artificial neural networks (ANNs) and Random Forest (RF) algorithms were employed in this study to predict output variables of interest (Nu or T_wall) as a function of input parameters (either in dimensional or non-dimensional form), as shown in Figure 4. These correspond to 8 unique algorithmic scenarios (2 input variable types × 2 ML algorithms × 2 output variables of interest) for each tube orientation (horizontal/vertical). The notable difference being Gr*, which captures the effects of buoyancy and is an input parameter in vertical flows, whereas it is replaced by z/D in horizontal flows. In addition, the ability of ML algorithms to predict T_wall directly was assessed as opposed to predicting Nu first and then computing T_wall based on Equation (2) via the iterative procedure outlined in Section 1.

A brief description of the ML algorithms is provided below.

2.4.1. ANN

An artificial neural network (ANN) was implemented as a unidirectional, fully connected feed-forward network, in which information propagates strictly from the input layer to the output layer without any recurrent or feedback connections. The network consists of an input layer followed by one to three hidden layers, each containing 32–128 neurons, determined through hyperparameter optimization. All hidden layers employed the rectified linear unit (ReLU) activation function to capture nonlinear dependencies between inputs and heat transfer response. The output layer consists of a single neuron with linear activation, yielding a continuous prediction of the output variables. Prior to training, all input features were standardized to zero mean and unit variance to ensure numerical stability and balanced gradient updates. The nonlinear input–output mapping in the ANN is of the form:

$$\mathrm{y}={\mathrm{f}}_{\mathrm{L}}({\mathbf{W}}_{\mathrm{L}}{\mathrm{f}}_{\mathrm{L}-1}\left(..{\mathrm{f}}_1\left({\mathbf{W}}_1\mathbf{x}+{\mathbf{b}}_1\right)\dots \right)+{\mathbf{b}}_{\mathrm{L}})$$

(15)

where y is the output of interest, x is the vector of inputs, W_L and b_L are the weight matrices and bias vectors of layer L, and f_L(·) denotes the ReLU activation function for hidden layers.

Hyperparameters including the number of hidden layers, number of neurons per layer, and learning rate were selected using random search optimization based on validation loss. To prevent overfitting, early stopping was employed, with training terminated when validation performance failed to improve for a predefined number of training cycles (epochs).

During training, the network parameters (weights and biases) were updated by computing gradients of the loss function with respect to each parameter using the chain rule of calculus (backpropagation). These gradients were then used to iteratively adjust the weights using the Adam optimization algorithm, which adaptively controls the learning rate for each parameter based on first- and second-order moment estimates of the gradients. The training objective was to minimize the mean squared error (MSE) between predicted and reference values.

2.4.2. RF

A Random Forest (RF) regression model was employed to predict the target thermal response from the input feature set. Random Forest is an ensemble learning method that constructs a collection of decision trees during training and outputs predictions as the average of the individual tree predictions. Each decision tree in the ensemble was trained on a bootstrap sample of the training data, while a randomly selected subset of input features was considered at each split. During bootstrap sampling, a new dataset is created by randomly selecting data points from the original dataset with replacement, so some points may appear multiple times while others may not appear at all. Since each decision tree in a Random Forest is trained on a different bootstrap sample, the trees are diverse, which reduces overfitting and introduces controlled randomness into the training process, ensuring that individual trees learn slightly different patterns from the same data, thereby improving ensemble robustness and generalization.

Key model hyperparameters, including the number of trees, maximum tree depth, minimum samples required for node splitting, minimum samples per leaf node, and the number of features considered at each split, were optimized using a randomized hyperparameter search. The search space spanned wide ranges to balance model expressiveness and computational efficiency.

Hyperparameter tuning was performed using randomized search with 5-fold cross-validation, where model performance was evaluated using the negative mean squared error as the objective metric. This approach efficiently explores the hyperparameter space while mitigating overfitting to a single train–validation split.

The optimized Random Forest model was trained using the full training dataset without feature scaling, as tree-based methods are invariant to monotonic transformations of the input variables. Model predictions on the test dataset were obtained by averaging the outputs of all trees in the ensemble.

The RF prediction y for an input vector x is given by:

$$\mathrm{y}={\displaystyle \frac{1}{\mathrm{N}}}\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{T}}_{\mathrm{i}}\left(\mathbf{x}\right)$$

(16)

where T_i denotes the prediction of the ith decision tree and N is the total number of trees in the ensemble.

The horizontal and vertical configuration datasets were handled separately, and an 80/20 data split was applied for training and testing for both the ANN and RF models.

3. Results and Discussion

3.1. Nu Predictions (Non-Dimensional Inputs)

Nu predictions are compared against their true values in Figure 5 when using non-dimensional inputs. The RF-tuned model using hyperparameter optimization achieved strong predictive performance for the dataset with inputs Re, Pr, and Gr*. The best model configuration used 1441 trees and a maximum depth of 18. This model produced an excellent fit, with R² values of 0.99 and 1.0 for the horizontal and vertical orientations, respectively, indicating that it explains over 99% of the variance in the target variable. Overall, these metrics confirm that the tuned RF models provide a reliable and precise mapping from the CFD-based non-dimensional inputs (cf. Figure 4) to the predicted output (Nu).

The optimized artificial neural network (ANN) also achieved strong predictive capability using a multilayer architecture consisting of two hidden layers, with 64 ReLU-activated units in the first layer followed by two successive hidden layers of 96 ReLU units each. This configuration provided a balance between nonlinearity and model capacity, enabling the network to capture complex patterns in the input variables. The final model demonstrated solid performance, achieving R² values of 0.97 and 0.98 for the horizontal and vertical configurations, respectively, indicating that it explains over 97% of the variance in the target data. Overall, the trained ANN offers a flexible nonlinear predictor, performing well but with slightly more variability than the tree-based RF approach.

Absolute Percent Error (APE) values were computed as

$$\mathrm{A}\mathrm{P}\mathrm{E}=100\times {\displaystyle \frac{\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e} \mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} \mathrm{C}\mathrm{F}\mathrm{D}-\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}}{\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e} \mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} \mathrm{C}\mathrm{F}\mathrm{D}}}$$

(17)

Histograms of APE corresponding to the predictions shown in Figure 5 are shown in Figure 6. Again, the improved predictability of RF over the ANN is noticeable across both the horizontal and vertical configurations. It is interesting to note that while the R² of 0.98 associated with the ANN in the vertical orientation (cf. Figure 5d) is higher than the R² of 0.97 associated with the ANN in the horizontal orientation (cf. Figure 5c), a comparison of Figure 6c,d shows higher APE magnitudes associated with the vertical orientation. This is because R² primarily reflects variance capture and is dominated by high-magnitude Nu responses, whereas APE emphasizes relative deviations and disproportionately penalizes errors in low-magnitude Nu regimes. As a result, improved trend prediction R² at higher response levels can coexist with increased percentage errors for smaller values.

3.2. T_wall Predictions (Non-Dimensional Inputs)

Analogous to Figure 5 and Figure 6, where Nu prediction fidelities are assessed, T_wall predictions are compared against their true values in Figure 7 when using non-dimensional inputs, with their corresponding APE values represented as histograms in Figure 8. A comparison of Figure 5, Figure 6, Figure 7 and Figure 8 clearly shows that Nu prediction fidelities are better than T_wall predictions when using non-dimensional input parameters for both ML algorithms, as well as both the horizontal and vertical configurations. While having T_wall as a predictive output may be more desirable than having Nu when the objective relates to thermal operating limits of materials, the larger errors in T_wall, especially at the extremes (maximum, minimum values, cf. Figure 7), may be attributed to the scale of variation (temperature range spans 500 K, while the maximum value of Nu was 25). This can be appreciated by examining the variations in T_wall and Nu as a function of two non-dimensional parameters (Re, Pr) in the horizontal configurations considered in this study, as shown in Figure 9. Notably, multiple values of Nu and T_wall are encountered at a given Re and Pr value due to the attempt to uncover a fluid-agnostic relationship. However, ML algorithms are able to circumvent this challenge as long as the scale of variation between the multiple values is not too large. However, this scale of variation issue may potentially be overcome via careful normalization of temperature (or using log (T_wall)) as the output variable of interest, for instance) to create smoother gradients for ML algorithms to learn from.

3.3. Dimensional Inputs for Nu Predictions

In an attempt to minimize the scale of variation in T_wall as a function of input parameters (cf. Figure 9b,d), dimensional variables in the form of operating conditions were employed to predict Nu and T_wall. The variations in T_wall and Nu as a function of two of the dimensional parameters (G, q) in the horizontal configurations considered in this study are shown in Figure 10 for illustrative purposes. While multiple values of Nu and T_wall are observed at each G and q, representing their corresponding variation along the flow direction, the variations are smooth and monotonic without the abrupt gaps/jumps in their variation seen in Figure 9b,d. Therefore, an improved predictive performance of the ML algorithms is to be expected with the use of dimensional inputs.

Analogous to Figure 5 and Figure 6, where Nu prediction fidelities are assessed with non-dimensional inputs, Nu predictions are compared against their true values in Figure 11 when using dimensional inputs, with their corresponding APE represented as histograms in Figure 12. A comparison of the corresponding histograms in Figure 6 and Figure 12 shows no significant differences in terms of Nu prediction fidelities when using dimensional or non-dimensional input parameters for both the horizontal and vertical configurations, with the exception of Figure 6d and Figure 12d, where a noticeable improvement in the predictive performance of the ANN is observed in the vertical flow configuration. A superior performance from RF in comparison to the ANN is again observed across all flow configurations.

3.4. T_wall Predictions (Dimensional Inputs)

Analogous to Figure 7 and Figure 8, where T_wall prediction fidelities are assessed with non-dimensional inputs, T_wall predictions are compared against their true values in Figure 13 when using dimensional inputs, with their corresponding APE represented as histograms in Figure 14. A comparison of Figure 7 and Figure 13 clearly shows a significant improvement in T_wall predictions when using dimensional input parameters for both ML algorithms across both horizontal and vertical configurations. In addition, the histograms of APE shown in Figure 14 indicate that T_wall can be predicted within an accuracy of +/−1% with the dimensional, operational parameters shown in Figure 4 as inputs while being agnostic to the working fluid. This greatly facilitates the RF ML model integration into existing engineering workflows and control loops since operational parameters of interest can be directly obtained using measurement/sensor data to output T_wall. In the event T_wall exceeds the threshold for material compliance, the operational parameters (G or q) can be varied accordingly.

The summary of results showcased so far, therefore, indicates that RF algorithms trained on dimensional/operational parameter inputs can predict T_wall in a specific flow configuration while being agnostic to the working fluid. However, as indicated in Section 1, realistic heat exchanger configurations may involve varying tube orientations and varying gravity environments. The prediction of HTE/HTD in these scenarios is compounded by the effects of buoyancy. The challenges with training and deploying a single ML algorithm across all tube orientations in the turbulent regime, even for a single fluid, have been highlighted previously [21]. However, whether this is true even in the laminar regime remains to be ascertained. Therefore, a combined dataset encompassing both horizontal and vertical flow data (in their dimensional form) was employed to train the RF model, and its ability to predict T_wall was assessed. It is worth noting from Figure 4 that the input parameters when using the dimensional form are the same operating parameters for both the horizontal and vertical orientations, with no specific additional input that takes into effect the direction of gravity. This independence of the gravity direction may be deemed to be important in realistic scenarios where flow direction orientation changes are sharp.

3.5. T_wall Predictions (Dimensional Inputs) Across All Flow Configurations

T_wall predictions encompassing the entire combined dataset (both horizontal and vertical orientations) are compared against their true values in Figure 15a–c when using dimensional inputs. These predictions were made employing RF algorithms trained exclusively on horizontal orientation data (in Figure 15a), vertical orientation data (in Figure 15b) and the combined dataset in Figure 15c. The corresponding histograms of the APE are shown in Figure 15d–f, respectively. The results in Figure 15 indicate that even in the laminar regime, the flow direction/tube orientation plays a significant role in ML prediction fidelities. In other words, T_wall prediction fidelities made using dimensional inputs are accurate only in the specific flow configuration corresponding to the data used to train the ML algorithm (cf. Figure 13a,b).

In order to remedy this shortcoming, gravity direction was used as an input parameter to train the RF ML algorithm. Specifically, we used the cosine of the gravity direction (with respect to the vertical axis) as an input parameter to enable its extension to other orientations as well. This resulted in the gravity vector being a binary input in the configurations that were investigated in this study: taking the value of unity (cosine of zero) at vertical orientations and zero (cosine of 90 degrees) at horizontal orientations. Figure 16 shows T_wall predictions using the mixed multi-fluid dataset (horizontal and vertical) that included gravity as an additional input parameter. While this significantly enhanced the predictive capability of the RF ML algorithm in comparison to Figure 15c,f, the applicability of this methodology to other tube orientations will be investigated in the future.

4. Conclusions

Fluid-agnostic laminar heat transfer prediction in supercritical conditions encompassing three fluids (sCO₂, sH₂O, sC₁₀H₂₂), each transitioning through their respective pseudocritical temperatures (T_PC), is carried out in this study as an enabling technology for emerging energy and thermal management applications. By restricting ourselves to the laminar regime, high-fidelity data were first generated from grid-converged, steady-state, computational fluid dynamic (CFD) simulations encompassing wide variations in operating pressures, inlet temperatures, wall heat fluxes and mass fluxes in both horizontal and upward flow tube configurations. The results from the CFD simulations showed very complex inter-relationships among the input and output variables of interest (T_wall and Nu), deeming the development of a single unified, fluid-agnostic correlation via statistical/symbolic regression techniques to be challenging. Therefore, artificial neural network (ANN) and Random Forest (RF) machine learning (ML) approaches were employed to uncover these relationships. The training consisted of both non-dimensional and dimensional (operating parameter) inputs, and the predictability of both Nu and T_wall was assessed. Based on the results of this study, the following conclusions can be drawn:

• The RF algorithm outperformed the ANN across all scenarios on the small mixed fluid datasets (~1600 data points) employed during the training process.
• When using non-dimensional parameters as inputs, Nu prediction fidelities were better than T_wall predictions for both ML algorithms across both the horizontal and vertical configurations. The larger errors in T_wall, especially at the extremes (maximum, minimum values), may be attributed to the scale of variation in the output parameters (temperature range spanned 500 K, while the maximum value of Nu was 25).
• To alleviate this shortcoming, dimensional variables in the form of operating conditions were employed to predict Nu and T_wall. The RF model trained on data from a specific flow configuration (horizontal/vertical) could predict T_wall within an accuracy of +/−1% with dimensional, operational parameters as inputs while being agnostic to the working fluid. This greatly facilitates the RF ML model integration into existing engineering workflows and control loops since operational parameters of interest can be directly obtained using measurement/sensor data to output T_wall. In the event T_wall exceeds the threshold for material compliance, the operational parameters (G or q, for instance) can be varied accordingly.
• While a fluid-agnostic ML model can be trained and developed in the laminar regime, the flow direction/tube orientation impacts its prediction fidelities. T_wall prediction fidelities made using dimensional operational parameters as inputs are accurate only in the specific flow configuration corresponding to the data used to train the ML algorithm. This was ascertained by attempting to train and develop an ML model on a combined dataset encompassing both horizontal and vertical flow directions. This resulted in large errors. However, by including the gravity vector as an additional variable during the training process, the RF model could predict Twall accurately in a mixed, multi-fluid dataset containing data from both horizontal and vertical configurations.