# Experimental water boiling in macroscopic horizontal and slightly tilted tubes: Flow pattern map and associated heat transfer analysis

Aguilera-Cortes Israël, Duval Hervé, Hernandez Emmanuel, Huc Jean-Jaques, Toutant Adrien, Mer Samuel

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## ► To cite this version:

Aguilera-Cortes Israël, Duval Hervé, Hernandez Emmanuel, Huc Jean-Jaques, Toutant Adrien, et al.. Experimental water boiling in macroscopic horizontal and slightly tilted tubes: Flow pattern map and associated heat transfer analysis. Applied Thermal Engineering, 2026, 292, pp.130400. 10.1016/j.applthermaleng.2026.130400. hal-05526493

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Submitted on 25 Feb 2026

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# Experimental water boiling in macroscopic horizontal and slightly tilted tubes: Flow pattern map and associated heat transfer analysis.

AGUILERA-CORTES Israël<sup>a,b</sup>, DUVAL Hervé<sup>a,b</sup>, HERNANDEZ Emmanuel<sup>a,b</sup>, HUC Jean-Jaques<sup>a,b</sup>,
TOUTANT Adrien<sup>a,b</sup>, MER Samuel<sup>a,b</sup>

<sup>a</sup>PROcédés Matériaux et Energie Solaire - PROMES - CNRS UPR 8521, 66100 Perpignan, France

<sup>b</sup>Université Perpignan Via Domitia, 66100 Perpignan, France

## Abstract

Direct Steam Generation (DSG) is gaining relevance in Concentrating Solar Thermal (CST) technologies due to its potential to reduce system complexity and costs by eliminating conventional working fluids (molten salts or synthetic oils). However, DSG introduces complex two-phase flow patterns in horizontal or slightly inclined receivers, which can pose operational risks if not properly understood. This study aimed to investigate flow patterns and heat transfer in a small-scale DSG experimental setup, focusing on the influence of tube inclination on flow regime transitions. An experimental facility, CONBO (CONvective BOiling flow loop), was developed, featuring a heating test section (TS) of 1 m length with an internal diameter of 49.3 mm. Water is vaporized using induction heating, and the section can be inclined to evaluate tilt effects. Various flow patterns are observed and recorded with a high-speed image acquisition system, while wall-mounted temperature sensors and internal thermocouples measure thermal distributions. A total of 231 tests were analyzed to classify flow regimes and assess heat transfer coefficients. Results indicate that tube inclination significantly affects flow regime transitions, from stratified to intermittent flows, and strongly influences wall temperature distribution and heat transfer performance. Flow pattern classification enabled the identification of regime transitions under different operating conditions. For tilted cases, a counterintuitive temperature distribution is observed in the TS, with the upper part of the tube being cooler than its bottom, which significantly improves heat transfer for the same section by delaying local dryout at the top. This inverse stratification is explained by a mechanistic model in accordance with the literature. These findings provide a detailed understanding of two-phase flow behavior in DSG receivers. The insights gained can inform the design and optimization of CSP receivers for industrial process heat applications, contributing to more efficient and reliable solar thermal systems.

**Keywords:** Horizontal boiling flow, Direct steam generation, Water-steam flow patterns, Two-phase flow

## 1. Introduction

The current climate emergency has made it imperative to explore alternative approaches to energy generation. Among these alternatives, concentrating solar power (CSP) has emerged as a promising technology that provides a viable substitute for conventional energy sources [42].

The fundamental principle of CSP relies on concentrating solar radiation onto a focal point (e.g., solar tower or Dish Stirling systems) or along a focal line (e.g., parabolic trough or Fresnel technologies) using large mirrors [27]. The objective is to increase the local temperature, enabling the use of the generated thermal energy either for electricity production (through steam generation and turbine operation) or for direct utilization in industrial processes that require heat (SHIP).
Nomenclature

Abbreviations

ConBo Convective Boiling flow loop

CSP Concentrating Solar Power

DSG Direct Steam Generation

HTC Heat Transfer Coefficient

HTF Heat Transfer Fluid

I Intermittent

PDF Probability Density Function

S Stratified

S1, S2, S3 Different measurement sections

SHIP Solar Heat for Industrial Processes

SW-S Stratified-Wavy Smooth

SW-U Stratified-Wavy Unstable

Greek symbols

Δ Increment [-]

δ Film thickness [-]

γ Adjustment factor [-]

μ Dynamic viscosity [kg.m¹.s⁻¹]

ν Kinematic viscosity [m².s⁻¹]

ρ Density [kg.m⁻³]

σ Surface tension [N.m⁻¹]

θ Tilt angle [°]

ε Void fraction [-]

Roman symbols

ṁ Mass flow rate [kg.s⁻¹]

Q̇ Electrical power [W]

q̇ Heat flux [W.m⁻²]

ẋ Steam quality [-]

Bo Boiling number [-]

cₚ Thermal capacity [J.K⁻¹.kg⁻¹]

c Wave propagation velocity [m.s⁻¹.K⁻¹]

D Diameter [m]

Fr Froude number [-]

f Friction factor [-]

G Mass flux [kg.m⁻².s⁻¹]

g Gravity [m.s⁻²]

h Heat transfer coefficient [W.m⁻².K⁻¹]

k Thermal conductivity [W.m⁻¹.K⁻¹]

Nu Nusselt number [-]

Pr Prandtl number [-]

P Pressure [Pa]

Re Reynolds number [-]

T Temperature [K]

u Velocity [m.s⁻¹]

Subscripts

δ Film

bot Bottom

cb Convective boiling

in Inlet

lv Latent heat of vaporization

l Liquid

out Outlet

sat Saturation

surf Surface

top Top

ts Test section

v Vapor

wall Wall

wet Wetted

Superscripts

* Dimensionless

TOP, MID, BOT Top, mid and bottom

In conventional CSP plants, the incident solar energy absorbed by the receiver is typically transferred to a heat transfer fluid (HTF), such as synthetic oils or molten salts. However, these HTFs present significant drawbacks: they are environmentally hazardous, can cause corrosion, and require periodic replacement, all of which increase operating costs. It has been reported that these costs can reach up to 5% of the initial
investment [44].

Direct Steam Generation (DSG) has emerged as a promising solution to enhance the sustainability of CSP plants [21]. In this configuration, steam is generated directly in the solar field receiver, eliminating the need for intermediate HTFs. Subcooled water is introduced into the receiver, where it is first heated to the saturation temperature and then evaporated. The generated steam can subsequently be directed either to a turbine for electricity production (as implemented in the eLLO plant [38]) or to industrial applications (as demonstrated at the Heineken facility in Seville [34]). In recent years, the number of industrial processes integrating solar heat for industrial processes (SHIP) has increased steadily, and demand is expected to continue growing. The main advantages of DSG technology can be summarized as follows:

* Improved efficiency: By using water as the working fluid, components such as heat exchangers are eliminated, thereby enhancing the overall efficiency of the plant [17].

* Cost reduction: The elimination of purchase, operation, and maintenance costs associated with conventional HTFs increases the long-term profitability of the plant.

* Lower environmental impact and greater social acceptance: The carbon footprint is reduced by simplifying the plant design and avoiding the use of highly polluting substances, which also contributes to higher public acceptance.

Nevertheless, the integration of DSG into CSP plants introduces additional challenges, particularly concerning system operation and component design [3]. The prediction of two-phase flow patterns is of paramount importance in the design and operation of thermal-fluid systems, such as boilers, condensers, and heat exchangers. Flow regimes govern key parameters including pressure drop, void fraction, and heat transfer coefficient, which directly affect system efficiency and reliability. Despite decades of research, accurate modeling of regime transitions remains a challenge due to the complex interplay between hydrodynamics, phase change, and geometry.

Under adiabatic conditions, early investigations focused on the mechanisms driving regime formation in gas-liquid flows. Dukler et al. [12] were among the first to describe slug flow instabilities, predicting the front and tail velocities of slugs as well as their frequency. Mandhane et al. [37] later proposed a comprehensive air-water flow pattern map, which has since become a widely adopted reference for horizontal two-phase flows. Complementing these efforts, Taitel and Dukler [48] developed a theoretical model for flow regime transitions, incorporating pipe inclination effects based on stability and momentum balance criteria, although their predictions lacked direct experimental validation.

Diabatic studies extended this framework by accounting for phase change phenomena. Kattan et al. [32] introduced a diabatic flow map for boiling in small-diameter tubes, validated with 702 experimental data points and five refrigerants. Their method relied on iterative void fraction calculations while preserving the influence of inclination. Building on this work, El Hajal et al. [14] adopted the Rouhani-Axelsson correlation [47] to simplify void fraction estimation, though at the expense of losing tilt dependence. A further refinement was proposed by Wojtan et al. [51], who developed the latest diabatic flow pattern map based on 1250 data points for R-22 and R-410A, combining advanced image acquisition techniques with manual classification.

More recent efforts have sought to address limitations related to geometry and heating conditions. Araujo et al. [2] investigated flow boiling in a novel experimental facility, analyzing the impact of non-uniform wall heating using R-141b in a 4.57 mm tube. Dirker et al. [10] performed a similar experiment with R245fa in an 8.5 mm tube. Yin et al. [52] further expanded diabatic studies to small-diameter water flows in U-shaped horizontal tubes, emphasizing how geometry-induced acceleration and deceleration influence regime transitions.

Overall, the literature shows significant progress in characterizing flow regime transitions under adiabatic and diabatic conditions. However, important gaps remain regarding the combined influence of tube inclination, non-uniform heating, and the study of flow patterns in boiling water flows. The aim of this study is to present the new experimental facility (ConBo) that has been built to investigate evaporation phenomena in water flows in pipes with diameters similar to those of solar receivers. The focus is on the
techniques developed for image acquisition, flow regime classification, and transition formulation. Finally, the results of the heat transfer coefficients are presented.

## 2. Materials and methods

### 2.1. *Experimental setup*

Figure 1 presents a schematic of the ConBo experimental facility, which was designed and constructed to investigate water boiling flows in horizontal or slightly tilted tubes (a more detailed picture of the CAD model could be found in Appendix B). The facility is primarily composed of three loops, each serving a specific function.

<figure><img src="image_3.png" /><figcaption>Figure 1: ConBo setup layout.</figcaption></figure>

The main loop (highlighted in salmon) feeds the test section where boiling occurs. Distilled water is stored in a custom-built cylindrical tank with a capacity of 35 L, which is sealed and can be pressurized. The tank is equipped with a safety valve (model 532T12MP-0.2 bar) to regulate the internal pressure. Water from the tank, at an approximate temperature of $T_{\text{tank}} = 85^{\circ}\text{C}$, is driven by a centrifugal pump (model MDT20 HCA, Iwaki) toward the preheater. Before entering the preheating section, the water passes through a Coriolis-effect mass flow sensor (R025S Micromotion, Emerson), which measures the mass flow rate and enables its subsequent control via the partial opening or closing of manual valves $v_1$ and $v_2$. The mass flow rate can be adjusted up to 0.2 kg/s.

In the preheater (model 10630, Vulcanic), the water is heated to the desired temperature, which can reach the saturation point. The flow then continues through a 3 meter long establishment section before reaching the test section. Both have an inner diameter of $D_{\text{in}} = 49.3$ mm and an outer diameter of $D_{\text{out}} = 60.3$ mm. The test section is 1 meter in length, made of Duplex 1.4462 steel and equipped with an induction heating system. The inductor consists of a copper cable with Litz-wire technology and insulating coating, helically wound around the tube. An alternating current of high intensity and high frequency is passed through the coil, enabling tube heating by induction. The induction coil has a total thickness of 15 mm, consisting of 4 mm of insulation, a 1 mm functional clearance, and a 2 mm composite casing around
<figure><img src="image_4.png" /><figcaption>Figure 2: Visualization box scheme.</figcaption></figure>

which an 7,4 mm diameter Litz wire is wound. The coil extends over a length of 1 m. For an external casing diameter of 74.3 mm, this geometry corresponds to 135 turns, resulting in a total Litz wire length of approximately 31.5 m. Because its aspect ratio $L_{coil}/r_{coil} \sim 13$, it can be considered long enough that there are no significant effects of magnetic field decay at the edges. The frequency of the inductor ranges from 35 to 108 kHz, and the current intensity passing through it ranges from 10 to 93 A. The inductor-transformer assembly was custom manufactured by ID Partner.

The test section, the establishment section, and the visualization box are mounted on a support arm with an adjustable tilt mechanism.

Due to the heat flux generated by the induction system, the water is heated until it reaches the saturation temperature and starts boiling. The resulting two-phase flow then passes through the visualization box, shown in detail in Figure 2, which give an optical access to the flow. The purpose of the visualization box is to capture images of the two-phase water-steam flow. For this reason, after the test section the tube material is replaced by borosilicate glass, which is transparent. This glass tube is enclosed within a stainless steel box through which a secondary flow circulates at a temperature close to saturation, preventing steam condensation. The box is equipped with two viewing windows, one at the front and one at the rear. A backlight is placed at the rear window, allowing high-quality photographs to be taken from the front window (see Fig. 4). Once the flow passes through the visualization section, the steam condenses, and the water is returned to the tank to restart the process.

<figure><img src="image_5.png" /><figcaption>Figure 3: Instrumentation on the test section.</figcaption></figure>
The secondary, or thermalization loop (highlighted in green), is designed to circulate water from the tank to the visualization box in order to prevent steam condensation. For this purpose, the flow is driven by a circulator (model *UPM3 15-75, Grundfoss*) and the resulting flow rate is measured by an electromagnetic flowmeter (model *AF-E 400, Krohne*).

The cooling loop (highlighted in blue on Fig. 1) is used to condense the vapor. Cold laboratory water is passed through a counterflow plate heat exchanger (model *B12MTx20, SWEP*). The cold leg of the condenser is regulated by an adjustable thermostatic valve (type *AVTA, Danfoss*).

Figure 3 shows the instrumentation installed in the test section. A total of six PT100 class 1/10 sensors were installed at different positions inside the tube, in direct contact with the water-vapor flow. For this purpose, special fittings were manufactured to allow insertion of the sensor tip 10 mm into the tube. In addition, nine PT100 class-A sensors were installed on the outer wall of the tube. These sensors were distributed across three different axial locations, positioned 450, 750, and 950 mm from the inlet of the test section. At each axial position, sensors were arranged at the top, middle, and bottom of the tube.

An absolute piezoelectric pressure sensor (model *PAA33X, Keller*) was also installed. Furthermore, a differential pressure sensor (model *Rosemount 2051, Emerson*) was included to measure the pressure drop across the section where vapor is generated. Additional thermocouples were installed to monitor process-related temperatures, such as at the outlet of the preheater, the inlet and outlet of the visualization box, the inlet and outlet of each branch of the condenser, and within the tank (see Fig. 1).

## 2.2. Image calibration and acquisition technique

In order to extract variables of interest, a camera (model *PROMON U1000*, AOS Systems) was installed, capable of acquiring images at a maximum frame rate of 151 fps. A mirror, oriented at 45° in front of one of the visualization box windows (see Figure 4), was used to position the camera further away from the visualization box. This setup minimizes image distortion caused by the fisheye effect when using short focal length lenses.

<figure><img src="image_6.png" /><figcaption>Figure 4: Top view scheme of the camera and mirror.</figcaption></figure>

For image processing, a python code using the OpenCV library was developed. Each stage of the process is briefly described below; for a more detailed explanation of the method the reader is referred to Appendix A:

* **Preprocessing:** vibration induced by certain flow patterns in the facility is corrected, and the images are rotated and cropped.

* **Correction:** a pattern of points with known dimensions is introduced inside the tube. This allows the determination of the distorted pattern dimensions, calculation of a transformation function, and subsequent correction of all images.

* **Processing:** the image is eroded and smoothed, normalized to minimize the effects of non-uniform illumination, binarized, and closed contours are filled.

* **Postprocessing:** quantitative variables are calculated, which are then used in the analysis of the results.

At the end of the process, quantitative variables of interest are obtained, such as the vapor-phase velocity $U_G$ (for intermittent flow patterns) or the volumetric fraction along a line, $\varepsilon_{line}$.
## 2.3. Data reduction

### 2.3.1. Heat transfer calculation

The induction system supplies an oscillating current of high intensity (up to 40 A) and high frequency (up to 140 kHz) in order to generate an induction heat flux in the tube. The induction generator provides the absorbed electrical power, $Q_{abs}$, in watts. The heat flux generated by induction is often treated as a uniformly distributed volumetric heat source; however, a large fraction of the volumetric heating occurs close to the surface near the inductor. This phenomenon is known as the skin depth effect, $\delta$, which can be calculated as :

$$ \delta = \sqrt{\frac{2}{\mu \cdot \sigma \cdot \omega}}, \qquad (1) $$

where $\sigma$ is the electrical conductivity, $\sigma = 1.25 \times 10^6$ S/m, and $\omega = 2\pi f$, is the angular frequency of the electromagnetic field. The operating frequency of the inductor is $f = 100$ kHz. $\mu = \mu_r \cdot \mu_0$ represents the effective magnetic permeability of the TS, where $\mu_r = 50$ is the relative permeability (dimensionless) of Duplex 1.4462 [23] and $\mu_0 = 4\pi \times 10^{-7}$ H/m is the vacuum permeability. An exponential model can be used to estimate the normalized volumetric heat generation as a function of the distance from the tube surface, $x_{surf}$ [9]:

$$ q_v(x) \propto e^{-2x_{surf}/\delta}. \qquad (2) $$

Figure 5 shows the normalized heat generation versus penetration depth. It can be observed that most of the heat is generated within the first 0.2 mm from the outer surface of the tube. For this reason (keeping in mind the tube thickness is 5.5 mm) the heat source generated by the inductor is considered as a surface heat flux boundary applied on the outer of the test section.

<figure><img src="image_7.png" /><figcaption>Figure 5: Normalized volumetric heat flux versus distance from the outer wall skin.</figcaption></figure>

The net heat flux entering the fluid region $\dot{q}$ is calculated on the inner surface of the tube as follows:

$$ \dot{q} = \frac{Q_{abs}}{\pi D_{in} L_{ts}}, \qquad (3) $$

where $Q_{abs}$ is the absorbed electrical power, $D_{in}$ the inner diameter, and $L_{ts}$ the length of the test section. For two-phase flow calculations, the local bulk temperature corresponds to the saturation temperature at a given axial position $z$:

$$ T_{fluid}(z) = T_{sat}[p_{sat}(z)], \qquad (4) $$
where the local saturation pressure is obtained through linear interpolation:

$$p_{\text{sat}}(z) = p_{\text{out}} - (z - L_{\text{ts}}) \frac{\Delta p}{L_{\text{ts}}}, \qquad (5)$$

The local pressure along the test section is calculated as $p(z) = p_{\text{out}} + \Delta p(L_{\text{ts}} - z)/L_{\text{ts}}$, with $p_{\text{out}}$ being the pressure measured at the outlet of the heated section, $\Delta p$ the pressure drop measured by the differential sensor, $L_{\text{ts}}$ the length of the test section, and $z$ the axial coordinate measured from the inlet. It is assumed that the pressure drop varies linearly along the test section. This assumption, also adopted in previous studies [2, 51], is justified by the relatively small pressure drop involved.

The extended thermodynamic vapor quality, $x^{\text{th}}$, at any axial location $z$ is a fundamental dimensionless parameter in two-phase flow. It originally represents the mass fraction of the vapor phase in a saturated liquid-vapor mixture and can be reformulated to also indicate subcooled or superheated states. For this purpose, it is calculated based on a steady-state energy balance from the inlet of the test section as given by Eq. 6.

$$x^{\text{th}}(z) = \frac{h_{\text{in}} - h_{\text{sat}}(p(z))}{\Delta h_{\text{lv}}(p(z))} + \frac{\dot{q}\pi D_{\text{in}}z}{\dot{m}\Delta h_{\text{lv}}(p(z))} \qquad (6)$$

where $h_{\text{in}}$ is the specific enthalpy of the fluid at the inlet of the test section, $h_{\text{sat}}(p(z))$ is the specific enthalpy of saturated liquid at the local pressure $p(z)$, $\Delta h_{\text{lv}}(p(z))$ is the latent heat of vaporization at the local pressure $p(z)$ and $\dot{m}$ is the mass flow rate.

In Eq. 6, the first term, accounts for the inlet subcooling. It is negative, $x^{\text{th}} < 0$, if the inlet fluid is subcooled (liquid below its saturation temperature), meaning a certain amount of sensible heat must be supplied before saturation is reached ($x^{\text{th}} = 0$) and boiling can begin. The second term, represents the heat power supplied by the electrical heater over the length $z$ normalized by the latent heat power required for phase change. Thus, $x^{\text{th}}(z)$ provides a linear profile along the tube (for constant fluid properties and heat flux), mapping the thermodynamic state of the fluid from inlet conditions to the outlet. For most of the experiments, $0 < x^{\text{th}} < 1$ indicating the presence of a two-phase mixture (the core region of boiling/condensation). This parameter serves as a crucial reference state for analyzing heat transfer coefficients, flow patterns, and pressure drop in convective boiling studies.

<figure><img src="image_8.png" /><figcaption>Figure 6: Thermal resistance diagram.</figcaption></figure>

The heat transfer coefficient between the wall and the flow is calculated as the ratio between the heat flux and the temperature difference between the wall and the fluid. A simple conduction scheme was followed, as illustrated in Figure 6. An average heat transfer coefficient was determined for each section. First, the mean section outer surface temperature of the tube was obtained as:

$$T_{S_i}^{\text{out}} = \frac{T_{S_i}^{\text{top}} + 2T_{S_i}^{\text{mid}} + T_{S_i}^{\text{bot}}}{4}, \qquad (7)$$

where the superscripts indicate the circumferential position of the temperature sensors (see Fig. 3). Once the average outer surface temperature is known, the mean inner surface temperature is calculated using a
simple radial conduction model:

$$T_{S_i}^{\text{int}} = T_{S_i}^{\text{out}} - \frac{\dot{q} \ln \left( \frac{D_{\text{out}}}{D_{\text{in}}} \right)}{2\pi k_{\text{wall}} L_{\text{ts}}}, \qquad (8)$$

where $k_{\text{wall}} = 17$ W/(mK) is the thermal conductivity of the tube. Finally, the heat transfer coefficient for each section and the corresponding local Nusselt number are calculated as:

$$h_{\text{exp}} = \frac{\dot{q}}{T_{S_i}^{\text{int}} - T_{\text{fluid}}(z)}, \qquad (9)$$

$$\text{Nu}_{\text{exp}} = \frac{h_{\text{exp}} D_{\text{in}}}{k_{\text{fluid}}}, \qquad (10)$$

where $k_{\text{fluid}}$ is the thermal conductivity of the working fluid.

### 2.3.2. Uncertainty assessment

To calculate the uncertainty of the heat transfer coefficient, the method of uncertainty combination described by the Joint Committee for Guides in Metrology [29] was used. This method expresses the propagation of error as:

$$\sigma_f = \sqrt{\sum_{i=1}^{n} \left( \frac{\partial f}{\partial x_i} \sigma_{x_i} \right)^2}, \qquad (11)$$

where $f = f(x_1, x_2, \dots, x_n)$ denotes the function that depends on the measured variables $x_i$, while $\sigma_f$ represents the uncertainty (standard deviation) of the result $f$. The symbol $n$ corresponds to the total number of independent input variables. The term $\frac{\partial f}{\partial x_i}$ is the partial derivative of $f$ with respect to the variable $x_i$, which quantifies the sensitivity of $f$ to changes in $x_i$. Finally, $\sigma_{x_i}$ is the uncertainty (standard deviation) associated with the variable $x_i$. In words, the total uncertainty of $f$ is obtained by taking the square root of the sum of the squared contributions, where each contribution is given by the product of the sensitivity of $f$ to the variable $x_i$ and the corresponding uncertainty $\sigma_{x_i}$.

As an example, the uncertainty associated with the heat transfer coefficient can be calculated by assuming that the uncertainty in $\dot{q}$ is negligible, and that both $T_{S_i}^{\text{int}}$ and $T_{\text{fluid}}(z)$ have equal and independent uncertainties:

$$\delta T_{S_i}^{\text{int}} = \delta T_{\text{fluid}}(z) = \delta T = 0.08^{\circ}\text{C} \qquad (12)$$

as given for a class 1/10 PT100 sensor. The uncertainty in $h$, denoted $\delta h$, is then obtained from:

$$\frac{\partial h}{\partial T_{S_i}^{\text{int}}} = - \frac{\dot{q}}{\left( T_{S_i}^{\text{int}} - T_{\text{fluid}}(z) \right)^2}, \quad \frac{\partial h}{\partial T_{\text{fluid}}(z)} = \frac{\dot{q}}{\left( T_{S_i}^{\text{int}} - T_{\text{fluid}}(z) \right)^2}. \qquad (13)$$

Substituting and simplifying yields:

$$\delta h = \frac{\dot{q}}{\left( T_{S_i}^{\text{int}} - T_{\text{fluid}}(z) \right)^2} \cdot \sqrt{\left( \delta T_{S_i}^{\text{int}} \right)^2 + \left( \delta T_{\text{fluid}}(z) \right)^2}. \qquad (14)$$

Since both temperature uncertainties are equal, this reduces to:

$$\delta h = \frac{\dot{q}}{\left( T_{S_i}^{\text{int}} - T_{\text{fluid}}(z) \right)^2} \cdot \sqrt{2 \delta T}, \qquad (15)$$

This provides the final expression for the uncertainty in the heat transfer coefficient due to the measurement uncertainties of both temperature readings.
## 2.4. Experimental procedure

The procedure for data acquisition, both from the sensors and the camera, is as follows. First, the combinations of operating parameters to be investigated is selected: mass flow rate $\dot{m}$, inclination angle $\theta$, inlet subcooling $\Delta T_{sub}$, and electrical power supplied to the inductor $\dot{Q}$. After selecting the desired operating conditions, a transient period occurs during which vapor is generated and the temperatures measured by the tube sensors increase. This temperature rise tends to stabilize once a thermodynamic equilibrium is reached between the supplied heat flux and the heat absorbed by the flow. Once statistical steady state is reached, temperature fluctuations remain within $\pm 0.5^{\circ}C$ and data acquisition begins.

Acquisition is performed using two data acquisition cards (NI-9217 and NI-9213, National Instruments). The sampling frequency is 20 Hz for temperature sensors, mass flow, and inclination, and 67 Hz for absolute and differential pressure sensors. The total acquisition time for each experiment is 60 s, after which the results are processed and averaged for further analysis.

For image acquisition, a physical time span of 7 s is recorded, corresponding to 1057 frames at an acquisition rate of 151 fps. The stored images have a resolution of 1984 × 1264 pixels. The final image sequence for a single experiment requires approximately 10 GB of storage, resulting in more than 2 TB for all experiments conducted.

A total of 12 single-phase experiments were carried out to validate the heat transfer coefficient calculation method, along with 231 two-phase water-vapor experiments. Table 1 summarizes the experiments performed.

<table><caption>Table 1: Experimental conditions of Aguilera et al. dataset</caption><thead><tr><td>Name</td><td>Fluid</td><td>n° points</td><td>G [kg/s·m²]</td><td>p [bar]</td><td>q̇ [kW/m²]</td><td>xth [%]</td><td>Configuration</td></tr></thead><tbody><tr><td>Aguilera et al.</td><td>Water-steam</td><td>231</td><td>30-120</td><td>1.2</td><td>up to 100</td><td>-1-5</td><td>0°and up to +5°</td></tr></tbody></table>

## 2.5. Non-dimensional numbers

Dimensionless numbers are fundamental parameters in the study of fluid dynamics and heat transfer, as they allow the characterization of flow regimes and transport mechanisms independently of the system's scale. Dimensionless numbers used in this study are the following. The Reynolds number is defined as:

$$Re = \frac{\rho uD}{\mu} \qquad (16)$$

where $\rho$ is the fluid density, $u$ the characteristic velocity, $D$ the characteristic length (e.g., the tube diameter), and $\mu$ the dynamic viscosity. This number expresses the ratio between inertial and viscous forces, determining the transition between laminar and turbulent flow.

The Froude number is expressed as:

$$Fr = \frac{u}{\sqrt{gD}} \qquad (17)$$

where $g$ is the gravitational acceleration. This number relates inertial to gravitational forces and is particularly relevant in two-phase or free-surface flows.

The Grashof number is defined as:

$$Gr = \frac{g \beta (T_s - T_\infty) D^3}{\nu^2} \qquad (18)$$

where $\beta$ is the thermal expansion coefficient, $T_s$ is the surface temperature, $T_\infty$ is the fluid temperature far from the surface, and $\nu$ is the kinematic viscosity. This number expresses the ratio between buoyancy and viscous forces, and it characterizes natural convection phenomena.

The Prandtl number is defined as:

$$Pr = \frac{c_p \mu}{k} \qquad (19)$$
where $c_p$ is the specific heat at constant pressure and $k$ the thermal conductivity of the fluid. This number compares momentum diffusivity to thermal diffusivity and indicates the relative thickness of the hydrodynamic and thermal boundary layers.

The Boiling number is expressed as:

$$Bo = \frac{\dot{q}}{Gh_{lv}} \qquad (20)$$

where $\dot{q}$ is the applied heat flux, $G$ the mass flux, and $h_{lv}$ the latent heat of vaporization. This number represents the ratio between the imposed heat flux and the mass flux required for phase change.

Finally, the Nusselt number is defined as:

$$Nu = \frac{hD}{k} \qquad (21)$$

where $h$ is the convective heat transfer coefficient. This number expresses the ratio between convective and conductive heat transfer, providing a measure of the enhancement in heat transfer due to fluid motion.

<table><caption>Table 2: Ranges of dimensionless numbers considered in the present study.</caption><thead><tr><th>Dimensionless Number</th><th>Range in Present Study</th></tr></thead><tbody><tr><td>Re</td><td>1.5 × 10<sup>3</sup> – 2.6 × 10<sup>4</sup></td></tr><tr><td>Fr</td><td>2 × 10<sup>-3</sup> – 20 × 10<sup>-2</sup></td></tr><tr><td>Gr</td><td>4.0 × 10<sup>5</sup> – 1.0 × 10<sup>7</sup></td></tr><tr><td>Pr</td><td>1.6 – 1.7</td></tr><tr><td>Bo</td><td>2.8 × 10<sup>-5</sup> – 6.0 × 10<sup>-4</sup></td></tr></tbody></table>

## 3. Results and discussion

### 3.1. Single phase validation

Single-phase experiments were carried out with the objective of validating the methodology used to calculate the heat transfer coefficients and the Nusselt number. A total of 32 tests were performed, varying the mass flow rate between 50 and 200 g/s, the inclination fixed at 2.5°, with an inlet temperature fixed at 25 °C. The applied heat flux ranged from 10.1 to 38.8 kW/m². That makes a total of 96 experimental points to be evaluated.

The evaluation of the Nusselt number is challenging, as single-phase experiments revealed a strong interaction between natural and forced convection, with ratios $Gr/Re^2$ in the range of 10–800. Furthermore, the flow was within the transition regime between laminar and turbulent conditions, with Reynolds numbers ranging from 1800 to 8000. Due to this complexity – and the relative failure of existing correlation to predict our experimental data – a new correlation is proposed in the following. It consists of a weighted combination of the Nusselt numbers for forced ($Nu_{Gnie}$) and natural convection ($Nu_{Morcos}$):

$$Nu_{New} = \sqrt{\gamma Nu_{Gnie}^2 + (1 - \gamma) Nu_{Morcos}^2} \qquad (22)$$

where $\gamma$ is a ponderation factor fitted from the experimental data which represents the ratio of the forced over de natural convection.

For the forced convection contribution, the correlation of Gnielinski [22] is employed:

$$Nu_{Gnie} = \frac{(f/8) (Re - 1000) Pr}{1 + 12.7 (f/8)^{1/2} (Pr^{2/3} - 1)}, \qquad (23)$$
<figure><img src="image_9.png" /><figcaption>Figure 7: Comparison between experimental and theoretical Nusselt numbers for different correlations.</figcaption></figure>

where Re is the Reynolds number, Pr the Prandtl number, and f the Darcy–Weisbach friction factor for smooth pipes ($\epsilon = 0$). The latter depends on the flow regime. For laminar flows (Re < 2300), f = 64/Re as originally described by Poiseuille [46]. In transitional flow (2300 < Re < 4000), an explicit approximation of the Colebrook equation by Haaland [24] applies, which for smooth pipes reduces to $f = 1/[-1.8 \log_{10}(\frac{6.9}{Re})]^2$. Finally, in turbulent conditions, different correlations are applied depending on the Reynolds number. For moderate turbulence (Re ≤ 10⁵), the empirical Blasius correlation [7] giving f = 0.3164/Re⁰.²⁵ is widely used.

For the Nusselt number in natural convection, the correlation of Morcos [39] is employed:

$$Nu_{\text{Morcos}} = 0.145 \left( \frac{Gr_f \cdot Pr_f^{1.35}}{Pw_f^{0.25}} \right)^{0.265} \quad (24)$$

Where Pw_f is the modified wall tube parameter defined as:

$$Pw_f = \frac{k_l D_{in}}{k_w t} \quad (25)$$

Where k_l and k_w are the thermal conductivities of the liquid and the wall and t is the thickness of the wall. The weighting parameter $\gamma$ was adjusted based on the available experimental data, $\gamma = (Re^2/Gr)^{0.4}$. Figure 7 presents a comparison between the experimental results and the theoretical Nusselt numbers calculated as described in Section 2.3. The results are presented for the different correlations that were evaluated.

It can be observed that the Gnielinski correlation exhibits the largest deviations. The Morcos correlation generally follows the trend, but some flattening of the data points is noticeable, as this correlation does not account for the effect of forced convection. The newly proposed correlation accurately follows the experimental trend and achieves better agreement, capturing 96.8% of the data within a ±10% error band.

## 3.2. Steam velocity calculation

Velocities were post-processed for 32 experiments – for which small or elongated bubble appears in the VB (such regime are classified as intermittent in Fig. 10) – with inlet mass flow rates ranging from $\dot{m} = 0.05$ kg/s to $\dot{m} = 0.2$ kg/s and tilt angles between $\theta = 1^\circ$ and $5^\circ$.

Figure 8 presents the vapor-phase velocity obtained through image processing, normalized by the inlet liquid velocity ($U_G/U_{in}$), as a function of the Froude number. Error bars account for the uncertainty of the image acquisition method, which depends on both spatial resolution and acquisition frequency. The data were fitted using a drift-flux type law, expressed as:
<figure><img src="image_10.png" /><figcaption>Figure 8: Velocity ratio as a function of the inlet Froude number.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 9: Flow pattern images from the experimental work and their classification.</figcaption></figure>

$$u_v = C_0 U_m + C_1 \sqrt{gD}, \qquad (26)$$

where $U_m$ is the mixture velocity, here approximated by the inlet liquid velocity $U_{in}$. The coefficients $C_0$ and $C_1$ were determined through a least-squares fit, yielding $C_0 = 0.81$ and $C_1 = 0.58$. These values are in close agreement with those reported in the literature [19, 5], where typical values of $C_0 \approx 1$ and $C_1 \approx 0.5$ are observed for horizontal or slightly inclined intermittent/plug flow regimes.

### 3.3. Visual flow pattern classification

A total of 231 experiments were conducted, which were first classified visually. Figure 9 shows the flow patterns observed and their corresponding labels. The following regimes were identified:

* **Stratified (S):** characterized by the complete separation of the two phases, with a smooth and stable interface free of oscillations.

* **Intermittent (I):** groups all cases in which the vapor phase appears intermittently in bubbles of varying size. This regime includes both plug and slug flow, which differ mainly in bubble length and frequency.

* **Smooth Stratified-Wavy (SW-S):** similar to the stratified regime, but with interfacial oscillations generated by the relative velocity increase between phases, leading to the formation of surface waves.
* **Unstable Stratified-Wavy (SW-U):** comparable to the SW-S regime, but strongly influenced by channel inclination. In this case, the vapor phase accelerates under gravity, while the liquid tends to flow backwards. This increases the relative velocity between phases, and, combined with boiling in the liquid phase, produces strong instabilities with poorly defined interfaces and high upward liquid motion.

These regimes are consistent with those found in classical flow pattern maps [48, 37, 14, 51]. However, the present study introduces a distinction between the SW-S and SW-U regimes. This differentiation is justified by the experimental configuration, which allows the inclination angle to vary. Combined with the large tube diameter and boiling conditions, these factors promote the development of distinct unstable interfacial behaviors not typically captured in conventional maps. Initially, the flow patterns were manually classified through direct visual inspection of the recorded sequences corresponding to each experiment.

<figure><img src="image_12.png" /><figcaption>Figure 10: Flow patterns classification.</figcaption></figure>

Figure 10 illustrates the identified flow patterns for both the horizontal configuration and the tilted cases. For horizontal configurations (Figure 10a), only S and SW-S patterns are observed. The transition between these regimes occurs as the mass flux in the pipe increases, and consequently the vapor velocity rises. Once the relative velocity between the two phases reaches a critical value, interfacial instabilities caused by shear forces lead to the appearance of the SW-S pattern.

In the inclined configuration (Figure 10b), the stratified (S) flow pattern disappears and is replaced by I, SW-S, and SW-U regimes. The absence of stratified flow is attributed to buoyancy, which accelerates the generated vapor and destabilizes the free surface. This increased vapor velocity reduces its residence time in the tube, promoting the formation of bubbles of different sizes. Furthermore, in the inclined cases, the transition is not governed by variations in mass flux, but rather by changes in the outlet vapor quality.

Figure 10 shows that two-phase flow can be observed in both plots, even for negative values of $x_{out}^{th}$, indicating that boiling may occur in subcooled liquid conditions. It should be recalled that $x^{th}$ is computed from a one-dimensional energy balance over the test section (Eq. 6). In practice, when $x_{out}^{th} < 0$, local vapor generation may occur due to wall superheating, despite the one-dimensional energy balance predicting subcooled liquid throughout the section. Given that the operating pressure of the facility is approximately 1 bar, even a small locally generated vapor mass fraction can occupy a significant volumetric fraction.

### 3.4. Automatic flow pattern classification

The classification of flow regimes is of great importance because accurate characterization and prediction of regime transitions are essential to understand heat transfer mechanisms and pressure drop. Previous studies in horizontal channels have employed a variety of sensors and signal-analysis techniques, including pressure and temperature fluctuation analysis and direct visual observation with high-speed imaging.
Time-frequency and wavelet-based methods applied to differential pressure or impedance/void-fraction signals have proven effective to identify flow patterns and transitions [25, 15, 41]. More recent work has explored machine learning approaches (SVM, CNNs, deep networks) trained on pressure, ultrasonic/Doppler, ECT or image-derived features for objective flow-regime classification in horizontal pipes [49]. While ML methods often yield high classification accuracy, they typically require facility-specific training data and large datasets, which can limit generalizability. For these reasons, high-speed imaging followed by image-based processing remains a widely used and complementary approach; however, purely visual and human based classification is inherently subjective.

In the following we develop an image-based metric, $\varepsilon_{line}$ (the void fraction over a line see Appendix A.4 and Eq. A.6) computed at the center of the images from the 7 s/151 fps sequences acquired for each experiment, to assist and partially automate the visual classification of flow patterns. Despite known limitations (interface detection affected by light refraction and relatively short acquisition time), the approach yields promising results when used as a decision-support tool for visual labelling.

The analysis of the probability density functions (PDFs) associated with the variable $\varepsilon_{line} \in [0, 1]$ was carried out in three main steps: normalization, interpolation and feature matrix construction, and hierarchical clustering. Each dataset is initially given as a discrete set of pairs,

$$ \text{Dataset } i: \left\{ (\varepsilon_{line,k}, \text{PDF}_{i,k}) \right\}_{k=1}^{n_i} \qquad (27) $$

Where each vector of $\varepsilon_{line}$ has an initial length of 1057 data points, and each vector of the PDF of $\varepsilon_{line}$ has a length of 200. Thus, each PDF$_i$ constitutes a valid probability density function. For a systematic comparison, a common axis was defined as

$$ \varepsilon^* = \{\varepsilon_1^*, \varepsilon_2^*, \dots, \varepsilon_m^*\}, \quad \varepsilon_1^* = 0, \quad \varepsilon_m^* = 1, \qquad (28) $$

where $\widetilde{\text{PDF}}_i$ represents the normalized PDF with $m$ equally spaced points. Each normalized distribution was interpolated onto this axis, yielding a feature vector (see Eq. 29) and stacking all such vectors results in the feature matrix given by Eq. 29, where $N$ is the number of PDFs.

$$ \mathbf{p}_i = (\widetilde{\text{PDF}}_i(\varepsilon_1^*), \widetilde{\text{PDF}}_i(\varepsilon_2^*), \dots, \widetilde{\text{PDF}}_i(\varepsilon_m^*)) \in \mathbb{R}^m, \qquad (29) $$

$$ X = \begin{bmatrix} \mathbf{p}_1 \\ \mathbf{p}_2 \\ \vdots \\ \mathbf{p}_N \end{bmatrix} \in \mathbb{R}^{N \times m}, \qquad (30) $$

The similarity between rows of $X$ was quantified using the Euclidean distance $d_{ij} = \|\mathbf{p}_i - \mathbf{p}_j\|_2$ and based on this distance metric, hierarchical clustering was performed using Ward's method [50], which merges clusters by minimizing the increase in within-cluster variance. For two clusters $C_i$ and $C_j$ with centroids $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$, and sizes $|C_i|$ and $|C_j|$, the Ward linkage criterion is defined as

$$ \Delta(C_i, C_j) = \frac{|C_i| \cdot |C_j|}{|C_i| + |C_j|} \|\boldsymbol{\mu}_i - \boldsymbol{\mu}_j\|_2^2. \qquad (31) $$

This procedure generates a dendrogram that reveals the hierarchical grouping structure of the PDFs, thereby enabling the identification of natural clusters within the data, except for the flow patterns in the horizontal configuration, a threshold value of PDF($\varepsilon_{line}$) > 15 was empirically defined to identify the stratified (S) regime. This value was determined by analyzing the variation of the $\varepsilon_{line}$ variable, above which the flow exhibits sufficient interfacial oscillations to be classified as SW-S rather than S. The interface thickness along the $\varepsilon_{line}$ was measured, as it appears as a band rather than a line in the image due to the meniscus effect near the wall.

Assuming that the probability density function (PDF) of $\varepsilon_{line}$ follows a perfect Gaussian distribution, the
maximum height was calculated by integrating the Gaussian function. This yield Eq. 32, where $\varepsilon_{interface} \approx 0.075$ was measured directly from the images.

$$PDF(\varepsilon_{line}) = \frac{2\sqrt{\ln 2}}{\varepsilon_{interface}\sqrt{\pi}} \approx 15 \qquad (32)$$

For the flow patterns in tilted cases, the dendrogram results are shown in Figure 11. The flow patterns were regrouped into three distinct clusters in order to identify the I, SW-S, and SW-U flow regimes.

<figure><img src="image_13.png" /><figcaption>Figure 11: Dendogram results for 3 clusters.</figcaption></figure>

Figure 12a shows the confusion matrix for the automated flow pattern detection method. It can be observed that the S and SW-U flow patterns are perfectly identified. However, the I and SW-S patterns are the ones most frequently confused due to visual classification. This occurs for two reasons: first, these flow patterns are very similar and can be subjectively misidentified; second, the image acquisition technique is based on a 7-second video (for storage reasons), and there are additional limitations associated with interface detection, as the interface has a finite thickness in the plane where the images are taken. Figure 12b shows the matches for all cases, highlighting that the most problematic region occurs in the transitions between I $\rightarrow$ SW-S and SW-S $\rightarrow$ SW-U, as shown in Figure 10b. Overall, the method is considered satisfactory and promising, as despite the limitations and the limited amount of data used, it is able to correctly predict the visually identified flow pattern in 79% of the cases.

<figure><img src="image_14.png" /><figcaption>Figure 12: Automatic flow pattern classification results.</figcaption></figure>

### 3.5. Flow pattern transitions

To determine the transitions between flow patterns, the results of the visual classification and the variable $\varepsilon_{line}$, obtained from the image acquisition technique, were used. As shown in Figure 10, the flow patterns
vary depending on whether the configuration is horizontal or inclined. It is also observed that the stratified flow pattern is exclusive to the horizontal configuration. Therefore, the transitions have been formulated as a function of the configuration, either horizontal or inclined.

In the case of the horizontal configuration, the transition between the S and SW-S patterns is governed by the appearance of waves caused by the increase in vapor velocity, which becomes sufficient to generate waves but not yet an annular flow. The phenomenon of wave generation has been previously studied, and in this work we adopt the concept introduced by Jeffreys [30]:

$$ (u_v - c)^2 c > \frac{4\nu_L g(\rho_l - \rho_v)}{s \rho_v} \qquad (33) $$

where c is the wave propagation velocity, $\nu_l$ the kinematic viscosity and s is a sheltering coefficient, for which Jeffreys proposed a value of 0.3. However, other studies suggest values of a lower order of magnitude [6]. In this study, a value of c = 0.05 has been used. When the transition between two regimes occurs, it can be expected that $u_v \gg c$. Previous studies [8, 20] show that the ratio between the wave velocity and the average liquid velocity, $c/u_L$, increases with the Reynolds number, approaching $c/u_L \approx 1$. In this study, the relation $u_L = c$ has been adopted, which leads to the following expression:

$$ u_v \geq \left[ \frac{4\nu_L g(\rho_l - \rho_v)}{s \rho_v u_l} \right]^{1/2} \qquad (34) $$

Using the continuity relations $u_v = Gx/(\rho_v\epsilon)$ and $u_L = G(1-x)/(\rho_l(1-\epsilon))$, the transition criterion between stratified (S) and smooth stratified–wavy (SW-S) flow can be written as:

$$ G_{S \to SW-S} = \left( \frac{4g\nu_l(\rho_l - \rho_l)\rho_l\rho_v}{s} \cdot \frac{\epsilon^2(1-\epsilon)}{x^2(1-x)} \right)^{1/3} \qquad (35) $$

where the void fraction $\epsilon$ is calculated using the Rouhani correlation [47], which has been widely used and validated:

$$ \epsilon = \frac{x}{\rho_v} \left[ (1 + C_0(1-x)) \left( \frac{x}{\rho_v} + \frac{1-x}{\rho_l} \right) + \frac{C_1(1-x) (g\sigma(\rho_l - \rho_v))^{0.25}}{G \rho_L^{0.5}} \right]^{-1} \qquad (36) $$

Here, $C_0 = 0.12$ is the distribution parameter accounting for non-uniform phase distribution, and $C_1 = 1.18$ is an empirical coefficient related to drift flux and interfacial effects. It can be noted that Rouhani's equation contains a term related to the mass flux G; however, other studies [51] have shown that its influence is minor.

To finalize the formulation of the transition, the inlet vapor quality $x_{IA}$ from Kattan [32] has been used:

$$ x_{IA} = \left\{ \left[ 0.34^{1/0.875} \left( \frac{\rho_v}{\rho_L} \right)^{-1/1.75} \left( \frac{\mu_L}{\mu_v} \right)^{-1/7} \right] + 1 \right\}^{-1} \qquad (37) $$

Hence, the transition is given by $G_{S \to SW-S}(x_{IA})$ when $x < x_{IA}$, and by $G_{S \to SW-S}(x)$ when $x > x_{IA}$. Figure 13 shows the experimental points together with the calculated transition for the horizontal configuration. It can be observed that most of the points fall within the regions defined by the transition.

In the case of inclined flow patterns, it has been observed that the void fraction is strongly correlated with the flow pattern transition. To visualize this influence, an extrapolation was first performed between the height detected in the images and the corresponding void fraction in the cross section, as shown in Figure 14.

First, the contact angle was calculated as:

$$ \theta_{\text{wet}} = 2 \cos^{-1} \left( \frac{D_{in} - 2H_L}{D_{in}} \right) \qquad (38) $$

where $H_L$ is the height obtained from image processing. Subsequently, the void fraction was computed
<figure><img src="image_15.png" /><figcaption>Figure 13: Transition for horizontal flow patterns.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 14: Camera shot and corresponding cross-section projection.</figcaption></figure>

using:

$$ \varepsilon = 1 - \frac{1}{2\pi} (\theta_{\text{wet}} - \sin \theta_{\text{wet}}) \qquad (39) $$

Figure 15 shows the results and the transition bands, which take the values $0.15 < \varepsilon_{I \to SW-S} < 0.25$ and $0.35 < \varepsilon_{SW-S \to SW-U} < 0.55$.

Due to the influence of the void fraction on the transitions, Rouhani's equation has been used to formulate both transitions, which has been rearranged as follows:

$$ G = \frac{C_1(1-x)[g\sigma(\rho_L - \rho_V)]^{0.25}}{\rho_L^{0.5} \left[ \frac{x}{\rho_V} - \varepsilon_{\text{band}}(1+C_0(1-x)) \left( \frac{x}{\rho_V} + \frac{1-x}{\rho_L} \right) \right]} \qquad (40) $$

where $\varepsilon_{\text{band}}$ represents the transition bands described previously. The parameter $C_0$ has a value of 0.12 according to the original equation. The parameter $C_1$ has been empirically adjusted to $C_1 = 2$ for $\varepsilon_{I \to SW-S}$ and $C_1 = 3$ for $\varepsilon_{SW-S \to SW-U}$. Figure 16 shows the transitions and the corresponding flow pattern map for the conducted experiments. Overall, a good agreement is observed.

### 3.6. Inception of the local dryout

During the experimental campaign with the ConBo facility, a critical thermal-hydraulic phenomenon was identified: the occurrence of local dryout at the upper wall of the test section. This region is characterized by intermittent and partial wetting due to the gravitational separation of phases in horizontal and inclined flows. The dryout manifests as a sudden and sharp increase in the wall temperature measured by the top-mounted PT100 sensors, often exceeding the safety threshold of 250 °C. When this occurs, the automated safety system immediately terminates power to the induction heater to prevent tube damage, resulting in an abrupt cessation of the heat flux.
<figure><img src="image_17.png" /><figcaption>Figure 15: Influence of the void fraction on the transition between flow patterns.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure 16: Flow pattern transitions.</figcaption></figure>
This dryout condition is triggered when the local heat flux and the corresponding vapor quality exceed a critical threshold, beyond which the liquid film on the upper wall can no longer be sustained, leading to direct contact between the wall and the low-conductivity vapor phase. The transition to dryout represents a fundamental limit for the safe and efficient operation of boiling heat transfer equipment.

The propensity for dryout was found to be strongly dependent on both the mass flux (G) and the tube inclination angle (θ). Figure 17 illustrates how the operational envelope and flow pattern distribution shift with varying tilt angles. A key observation is that a positive inclination (upward tilt) significantly delays the onset of dryout, allowing the system to reach higher vapor qualities before the upper wall dries out.

<figure><img src="image_19.png" /><figcaption>Figure 17: Experimentally observed flow patterns mapped in the vapor quality (x) vs. mass flux (G) plane for different tube inclination angles θ.</figcaption></figure>

The physical mechanism behind this delay is related to improved liquid distribution. Under positive inclination, the axial component of gravity assists in redistributing liquid towards the upper wall, either through increased wave action, more frequent bubble impingement, or the formation of a more stable liquid film, thereby extending the wetted period and delaying the dryout transition.

In the analysis of two-phase flows, dryout is typically treated as a transition boundary within flow pattern maps, marking the shift from intermittent or annular flows (where the wall remains wetted) to a mist or dry-wall regime. Two prominent correlations for predicting the dryout vapor quality (xdi), Mori et al. [40] (Eq. 41) and more recently by Wojtan et al. [51] (Eq. 42):

$$x_{di} = 0.58 \exp \left[ 0.52 - 2.1 \times 10^{-5} We_v^{0.96} Fr_v^{-0.02} \left( \frac{\rho_v}{\rho_l} \right)^{-0.08} \right] \quad (41)$$

$$x_{di} = 0.58 \exp \left[ 0.52 - 0.235 We_v^{0.17} Fr_v^{0.37} \left( \frac{\rho_v}{\rho_l} \right)^{0.25} \left( \frac{q}{q_{crit}} \right)^{0.7} \right] \quad (42)$$

where We_v and Fr_v are the vapor-phase Weber and Froude numbers, respectively. The primary advancement in Wojtan's formulation is the inclusion of a dimensionless heat flux term (q/q crit), recognizing that the driving thermal force directly influences the dryout condition.

However, a significant limitation of both models is their development for horizontal configurations; they do not account for the effect of tube inclination, which the present experiments have shown to be a first-order parameter.

To address this gap, a new empirical correlation is proposed that explicitly includes the tube inclination angle θ. The correlation was derived by multivariate regression analysis of the experimental dryout data, using dimensionless groups that capture the relevant forces:
$$x_{di} = 0.71 \ln \left[ Re_v^{-0.02} \left( \frac{\rho_l}{\rho_v} \right)^{0.58} \left( \frac{\mu_l}{\mu_v} \right)^{-1.22} + \sin^{1.35}(\theta) \right] \quad (43)$$

where $Re_v = GD/\mu_v$ is the vapor-phase Reynolds number. The term $\sin^{1.35}(\theta)$ introduces the inclination effect, with the exponent obtained from fitting.

For practical application in designing operating limits, Eq. (43) can be rearranged to express the critical mass flux $G_{dryout}$ as a function of vapor quality and inclination. This form is more convenient for plotting the dryout boundary on a conventional G vs. x flow pattern map:

$$G_{\text{dryout}} = \left\{ \left[ \exp\left(\frac{x}{0.71}\right) - \sin^{1.35}(\theta) \right] \left(\frac{\rho_l}{\rho_v}\right)^{-0.58} \left(\frac{\mu_l}{\mu_v}\right)^{1.22} \left(\frac{\mu_v}{D}\right)^{-0.02} \right\}^{0.02} \quad (44)$$

The dryout boundaries predicted by the new correlation for various inclination angles are presented in Figure 18. The map is divided into two distinct zones:

* Safe Operating Region: Located below and to the left of the dryout curve. In this zone, the upper wall remains sufficiently wetted, and heat transfer is effective without risk of temperature excursion.

* Dryout Region: Located above and to the right of the curve. Entering this region leads to partial or complete drying of the upper wall, causing a drastic reduction in the local heat transfer coefficient and triggering the safety shutdown due to excessive wall temperature.

<figure><img src="image_20.png" /><figcaption>Figure 18: Predicted dryout boundaries for different tube inclination angles using the proposed correlation (Eq. (44)). The shaded area represents the safe operating region for $\theta = +5^\circ$.</figcaption></figure>

The correlations of Wojtan and Mori have not been plotted, as the dryout transition lines are around $x_{out}^{th} = 80\%$. The plot clearly demonstrates the expansion of the safe operating envelope with increasing positive inclination. For instance, at a fixed mass flux of 100 kg s$^{-1}$ m$^{-2}$, the allowable vapor quality before dryout increases by more than 100% when the inclination changes from 0° to +5°.

### 3.7. Heat transfer coefficients

The experimental heat transfer coefficients (HTCs) are calculated as described in Section 2.1, but under two-phase (water-vapor) conditions. Figure 19a shows the results of the heat transfer coefficients for the different measurement sections as a function of vapor quality. A general trend can be observed in which the
HTC increases as the measurement section is located further away from the inlet of the TS (see Figure 3 for the location of each measurement section). For sections $S_1$ and $S_2$, a nearly linear increase in HTC with vapor quality is observed, whereas section S3 shows a different behavior. It can also be seen that for negative vapor qualities (where nevertheless local evaporation occurs), the HTC is lower.

<figure><img src="image_21.png" /><figcaption>Figure 19: HTC results.</figcaption></figure>

Figure 19b presents the HTCs for section S3 (the one closest to the outlet for which we can reasonably consider that the flow pattern is the same as in the VB) as a function of the identified flow patterns. It is observed that for the SW-U flow pattern, the HTC is generally higher. This is attributed to the increased evaporation rate and the higher vapor-phase velocity. The higher vapor velocity, combined with the fact that the SW-U pattern occurs when the tube is tilted, leads to the formation of a fast-moving liquid wave, which significantly enhances the heat transfer coefficient. Conversely, when the flow pattern is stratified (S), the HTC decreases substantially. The HTCs shown in Figure 19 correspond to cases with an uncertainty below 15%, as calculated using Eq. 15.

Figure 20 shows a comparison between the experimental heat transfer coefficients and several correlations available in the literature [16, 31, 28]. It can be observed that none of these correlations accurately reproduce the experimental results. The tested correlations are widely used empirical models for boiling flow applications.

<figure><img src="image_22.png" /><figcaption>Figure 20: HTC comparison with literature correlations.</figcaption></figure>
The correlation proposed by Fang et al. [16] was developed using more than 20,000 data points from various sources; however, all of them correspond to refrigerants (R134a, R22, R410A...) in small geometries, typically microchannels. The correlation of Kandlikar [31] was developed using 5246 experimental points, including both refrigerants and water. Nevertheless, most of the water-based data corresponded to vertical orientations. The only subset of horizontally oriented experiments is that of Mumm [43], which involve mass fluxes and pressures far from the present experimental conditions and consisted of only 158 data points—approximately 3% of the total database used in Kandlikar’s correlation. Similarly, the correlation by Jabardo [28] was developed from experimental data obtained using three refrigerants (CFC-12, HFC-134a, and HCFC-22).

These correlations fail to correctly predict the experimental results for two main reasons. First, they were developed using fluids—mainly refrigerants—whose thermophysical properties differ significantly from those of water. They also correspond to smaller geometries and different mass flux ranges compared with those in the present study. The second reason is that these correlations include the Martinelli parameter [36], given by:

$$X_{tt} = \left(\frac{1-x}{x}\right)^{0.9} \left(\frac{\rho_g}{\rho_l}\right)^{0.5} \left(\frac{\mu_l}{\mu_g}\right)^{0.1} \qquad (45)$$

In the experimental data of this study, the vapor quality $x$ ranges from -1 to 5. Negative vapor quality values correspond to local evaporation (calculated using Eq. 6). Because of these small and sometimes negative vapor quality values, the correlations available in the literature (Kandlikar) are unable to accurately estimate the heat transfer coefficient under the present experimental conditions.

It can be observed that Fang’s correlation is the one that approximates the experimental heat transfer coefficients most closely. Therefore, it was proposed to readjust the coefficients of this equation to better fit the available experimental data. Fang’s correlation for water is given by:

$$h_{\text{theo}} = \frac{k_l}{D} F_f M^{-0.18} Bo^{0.98} Fr^{0.48} Bd^{0.72} \left(\frac{\rho_l}{\rho_g}\right)^{0.29} \left[\ln\left(\frac{\mu_{lf}}{\mu_{lw}}\right)\right]^{-1} Y \qquad (46)$$

$$Y = \begin{cases} 1 & \text{for } p_r \le 0.43 \\ 1.38 - p_r^{1.15} & \text{for } p_r > 0.43 \end{cases} \qquad (47)$$

where $M$ is the molecular mass, $Bo$ is the Boiling number, $Fr_{lo}$ is the Froude number based on liquid properties, $Bd$ is the Bond number, $\mu_{lf}$ and $\mu_{lw}$ are the dynamic viscosities at the liquid and wall temperatures, $F_f$ is a parameter equal to 2035 for water, and $p_r$ is the reduced pressure.

As can be observed, Fang’s original correlation lacks dimensional consistency, as it introduces the molecular mass $M$ raised to a power, resulting in units that are not consistent with a heat transfer coefficient in W/(m²·K). Therefore, a correlation with a new set of parameters was proposed, removing the molecular mass term, density ratio, Bond number, Froude number, $F_f$ and $Y$ parameters. The Reynolds number, the steam quality and the $H = (h_{in} - h_{sat})/\Delta h_{lv}$ were introduced in the correlation. The proposed correlation, valid for the experimental data, is expressed as:

$$h_{\text{theo}} = \frac{k_l}{D} Bo Re^{1.6} (x^{th} - H)^{0.8} \left[ \ln \left( \frac{\mu_{lf}}{\mu_{lw}} \right) \right]^{-1.55} \qquad (48)$$

Figure 21 shows the results of the new correlation, which fits the experimental data well, with a mean percentage error (MPE) of -2.05%, a mean absolute percentage error (MAPE) of 10.43% and 96.9% of the points in the ± 30% range.

### 3.8. Cross sectional temperature differences

The temperature difference along the same cross-section has been analyzed, since it is well known that in horizontal two-phase flows, the separation of the two phases due to gravity causes the gas phase to accumulate in the upper part of the tube. As the heat transfer coefficient of the vapor phase is lower
<figure><img src="image_23.png" /><figcaption>Figure 21: HTC comparison with proposed correlation.</figcaption></figure>

$(h_v < h_l)$, the upper wall temperature becomes higher than the lower wall temperature ($T_{S_i}^{top} > T_{S_i}^{bot}$). This behavior has been confirmed both numerically [1, 45] and experimentally [2, 13]. Figure 22a shows the temperature difference for the three measurement sections, while Figure 22b presents the same difference for the third section as a function of the observed flow patterns.

It can be observed that, in most cases, the opposite behavior occurs, as $T_{S_i}^{top} < T_{S_i}^{bot}$. This phenomenon generally appears across all sections, with section $S_3$ showing a more pronounced negative temperature difference, reaching values as low as $T_{S_i}^{top} - T_{S_i}^{bot} = -15$ K. It can also be seen that for horizontal and slightly inclined cases ($\theta = 0°$ and $1°$), the temperature difference becomes positive.

This counter intuitive temperature difference can be explained by the formation of liquid film on the upper part of the TS, as well as the corresponding liquid acceleration inducted by a bubble. This phenomena has been studied in intermittent air-water flows [26, 18], where it has been discovered that the coefficient of the upper part is slightly increased due to the acceleration of the liquid when a bubble passes through, in addition to the effects of agitation. For most of flow patterns, the upper wall intermittently comes into contact with a thin liquid film, of thickness $\delta_{top}$, which modifies the local heat transfer distribution. Figure 23 schematizes the thickness of this liquid film at both the upper and lower parts of the tube.

The convective boiling heat transfer coefficient for the upper liquid film can be empirically estimated using the correlation proposed by Kattan [33]:

$$h_{cb}^{top} = 0.0133 Re_{\delta_{top}}^{0.69} Pr_l^{0.4} \frac{k_l}{\delta_{top}} \quad (49)$$

Similarly, the convective heat transfer coefficient for the lower part of the tube in contact with the liquid can be estimated using the Dittus-Boelter correlation [11]:

$$h_{cb}^{bot} = 0.023 Re_{D_{in}}^{0.8} Pr_l^{0.4} \frac{k_l}{D_{in}} \quad (50)$$

The ratio of the convective coefficients $h^*$ between the upper and lower liquid films within the same section is computed as follows, considering two characteristic velocities $u_{bot}$ and $u_{top}$:

$$h^* = \frac{h_{cb}^{top}}{h_{cb}^{bot}} = 0.59 \left( \frac{\mu_l}{\rho_l} \right)^{0.11} \frac{u_{top}^{0.69} D_{in}^{0.2}}{u_{bot}^{0.8} \delta_{top}^{0.31}} \quad (51)$$

In the following, we will consider $u_{bot} \sim u_{in}$ and that the upper film is advected with the vapor phase velocity. The velocity ratio $u^*$, between the steam and liquid velocity is defined as $u^* = u_s/u_l$. Figure 8 shows that $u^*$ can reach values up to 10 for intermittent flow, and possibly even higher for other regimes. Figure 24 shows the calculated convective coefficients ratio $h^*$ as a function of the normalized film thickness
<figure><img src="image_24.png" /><figcaption>(a) Cross sectional temperature difference for the measurement sections.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(b) Cross sectional temperature difference for $S_3$.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Figure 23: Schematic model showing the liquid film thicknesses at the upper and lower walls.</figcaption></figure>

for different values of $u^*$.

It can be observed that this ratio increases as the film thickness ratio decreases, meaning that a thinner upper film results in a higher heat transfer coefficient. The influence of the velocity ratio is also evident, as
<figure><img src="image_27.png" /><figcaption>Figure 24: Ratio of the top and bottom theoretical convective boiling heat transfer coefficients.</figcaption></figure>

higher $u^*$ values lead to a greater enhancement of the heat transfer ratio.

However, the liquid film is not always in contact with the top wall of the tube. For example, for intermittent flow patterns, the upper wall is wetted by a film only in the region occupied by the bubble. For SW-S or SW-U regimes, it can reasonably be assumed that a liquid film forms on the upper wall after the passage of a wave, and that its characteristic drainage time is shorter than the wave period. This ratio over time can be simplified as a step function, with the cycle time $t_{cycle}$ as the independent variable and the duration during which the film contacts the top wall $t_{wet}$. Figure 25 illustrates this simplification, along with the time-averaged value that can be calculated according to the following equation:

$$h_{avg}^* = 1 + \frac{t_{wet}}{t_{cycle}} (h^* - 1) \qquad (52)$$

<figure><img src="image_28.png" /><figcaption>Figure 25: Step-like signal of the convective boiling HTC ratio for the top wall when waves pass and a liquid film forms.</figcaption></figure>

Once the time-averaged ratio $h_{avg}^*$ is obtained, a normalized film thickness of $\delta_{top}/D_{in} = 0.004$ is proposed, corresponding to $\delta_{top} = 0.2$ mm. This value of $\delta_{top}$ is within the appropriate order of magnitude, as indicated by Eq. 53, proposed by Aussillous et al. [4]:

$$\frac{\delta_{top}}{D_{in}} \approx \frac{Ca^{2/3}}{2(1 + Ca^{2/3})} \qquad (53)$$
where Ca is the capillary number, defined as $Ca = \mu v/\sigma$. Figure 26 shows the evolution of the time-averaged heat transfer coefficient ratio and the fraction of the cycle during which the upper wall remains covered by the liquid film.

<figure><img src="image_29.png" /><figcaption>Figure 26: Average HTC ratio over time for different wetting durations when $\delta_{top}/D_{in} = 0.004$.</figcaption></figure>

It can be observed that the longer the upper wall remains wetted by the film, the higher the heat transfer coefficient ratio becomes. An increase in the velocity ratio $u^*$ also enhances this effect.

Based on a thermal resistance model, the average temperature difference between the top and bottom of the tube can be approximated. This expression, given by Eq. 54, involves the heat transfer coefficient at the bottom part as well as the average ratio of the top-to-bottom coefficients.

$$T_{S_i}^{top} - T_{S_i}^{bot} \approx \dot{q} \left( \frac{1}{h_{cb}^{top}} - \frac{1}{h_{cb}^{bot}} \right) \approx \frac{\dot{q}}{h_{cb}^{bot}} \left( \frac{1 - h_{avg}^*}{h_{avg}^*} \right) \quad (54)$$

$h_{cb}^{bot}$ is calculated with Eq 50, with a Reynolds number based on $u_{in}$ and $D_{in}$. Figure 27 shows the estimated temperature difference for various $q/h_{cb}^{bot}$ ratios, which experimentally range between 1 and 20. The observed trend is consistent with Figure 22. When the ratio is negative, it indicates the presence of a liquid film at the top wall, and the temperature difference exhibits an asymptotic behavior between 0 and -15 K, as observed in Figure 22a for most experimental points. Furthermore, when the heat transfer coefficient ratio is negative, it implies that the film gradually disappears, leading to a rapid increase in temperature. This effect is particularly evident in Figure 22 for stratified flow patterns and near-horizontal inclinations ($\theta = 0^\circ$ and $1^\circ$).

This simple mechanistic model is compared with the experimental results shown in Figure 28 for the 32 experimental cases identified within the intermittent flow regime. For this regime, image analysis makes it possible to determine (i) the bubble velocity — and thus $u^*$ — as well as (ii) the time during which the upper wall is in contact with a bubble — and hence the ratio $t_{wet}/t_{cycle}$.

Figure 28a shows that all experimental points fall within the range of theoretically expected temperature differences. The points are colored according to the ratio $q/h_{cb}^{bot}$ calculated experimentally. Figure 28b presents the confrontation between the experimental values and the predictions obtained from the proposed model. Good agreement is observed, highlighting the robustness of the approach despite the inherent assumptions and simplifications.

## 4. Summary and concluding remarks

In this study, the boiling water flows in a tube with a diameter similar to that of a solar power plant have been experimentally investigated. An experimental facility was constructed capable of evaporating water under different operating conditions, including mass flow rate, heat flux, inlet temperature, and inclination.
<figure><img src="image_30.png" /><figcaption>Figure 27: Temperature difference versus convective boiling HTC ratio for different operating conditions at $\delta_{top}/\delta_{bot} = 0.004$.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Figure 28: Comparison of temperature differences for the intermittent flow pattern.</figcaption></figure>

A high-speed image acquisition technique was developed to capture the relevant variables of the water-vapor flows. Subsequently, the flow patterns were analyzed, a method for their visual classification was developed, and existing flow pattern transitions were updated. The heat transfer coefficients were evaluated, compared with literature correlations, and an empirical correlation was proposed. Additionally, the phenomenon of negative temperature differences was qualitatively and theoretically explained. The main findings are summarized below:

* Using the image acquisition technique, it was found that in intermittent flow patterns, bubble velocities can reach up to 10 times the inlet velocity, which enhances heat transfer.

* Four flow patterns were visually identified: stratified (S), intermittent (I), smooth stratified-wavy (SW-S), and unstable stratified-wavy (SW-U). An automated classification technique based on the variable $\epsilon_{line}$ was developed. Using hierarchical clustering, this method correctly classifies 78.78% of the cases in agreement with visual observations. Limitations of the technique include the duration of the sequences analyzed and difficulties in detecting the interface in certain cases.

* The flow pattern transition was traced for horizontal cases following the criteria proposed by Jeffreys
and Kattan. For inclined cases, two transitions were proposed based on changes in the void fraction, using the Rouhani equation to account for experimental conditions.

* Heat transfer coefficients were evaluated and first compared with correlations found in the literature. Due to the inability of existing correlations to accurately predict heat transfer coefficients under the experimental operating conditions, a new correlation was proposed, which predict 96.9% of the data within a ±30% error range.

* Temperature differences between the top and bottom of the tube were analyzed. In most cases, the temperature difference was found to be negative. A thin-film model was proposed to qualitatively explain this phenomenon, demonstrating the possible existence of a liquid film causing the negative temperature difference within the same cross-section. This model was validated with the data corresponding to an intermittent flow pattern.

**Declaration of Competing Interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

**Credit authorship contribution statement**

**Israel Aguilera Cortes**: Conceptualization, Methodology, Software, Validation, Formal analysis, Data curation, Investigation, Writing - original draft. **Hervé Duval**: Experimental setup conceptualization and construction. **Emmanuel Hernandez**: Experimental setup conceptualization and construction. **Jean-Jaques Huc**: Experimental setup conceptualization and construction. **Adrien Toutant**: Conceptualization, Supervision, Writing - review. **Samuel Mer**: Funding acquisition, Project administration, Conceptualization, Supervision, Writing - review.

**Acknowledgments**

This work was funded by the European Union as part of the TOPCSP project (HORIZON MSCA Doctoral Network, Project number 101072537) and the French National Research Agency (ANR) for the purpose of the STEAMSUN project under contract ANR-25-CE51-1459. We gratefully acknowledge the technical support of Sébastien Cazin and Catherine Colin for their help with the image processing technique and its calibration.

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## Appendix A. Image calibration and acquisition technique

The steps followed for image processing are described below. Figure A.29 illustrates the different stages of the procedure.

<figure><img src="image_32.png" /><figcaption>Figure A.29: Image processing code workflow.</figcaption></figure>
## Appendix A.1. Preprocessing

When the images are acquired, a preprocessing stage is required prior to correction and analysis. This step is essential to mitigate one of the main issues of the experimental facility: the vibration induced by certain flow regimes.

As mentioned above in Section 2.1, the mechanical arm supporting the heating and visualization section can be tilted, since it is anchored to a pivot and adjusted using a crank mechanism. The anchorage is solid and reinforced with vertical bars to stiffen the structure. However, when specific flow regimes occur, the test and visualization section starts vibrating, which poses a problem for subsequent image analysis. To address this issue, a control point was fixed on the visualization box in order to track its position over time. Figure A.30 shows the displacement of this control point as a function of time.

<figure><img src="image_33.png" /><figcaption>Figure A.30: Displacement of the images over time.</figcaption></figure>

To correct this effect, the position of the control point is tracked over time, and a translation is applied to all images for $t > t_1$, where $t_1$ is the first frame of the sequence. This translation is expressed as a spatial transformation:

$$ 
\begin{aligned}
X' &= x + \Delta x \\
Y' &= y + \Delta y
\end{aligned} 
\qquad (A.1) 
$$

where $(x, y)$ are the coordinates of the point in the first frame, $(X', Y')$ are the corrected coordinates in frame $t$, and $(\Delta x, \Delta y)$ are the displacements induced by the vibration.

## Appendix A.2. Correction

Due to the arrangement of the image acquisition system (arm with a mirror at 45°) and the fact that the transparent tube has a thickness of 5 mm, it is necessary to correct the image for the following reasons:

* A misalignment of the arm holding the mirror or the arm holding the camera could lead to a deformation in the longitudinal direction if the mirror is not perfectly aligned at 45°.

* Due to the thickness of the tube the dimensions near both the bottom and the top of the tube may be altered due to the curvature of the tube.

The error due to the curvature and thickness of the transparent tube is more difficult to quantify by analytical methods. This is why we have proceeded to make a calibration with a target to correct the deformation errors. The aim of this type of calibration is to introduce a known pattern at the place where the images are to be taken, so that when an image is taken of the known pattern it is possible to calculate the deformation induced in the image due to both the position of the image acquisition system and that induced by the curvature of the glass tube [35].

For image correction, a calibration target measuring 300 × 50 × 1 mm was manufactured, consisting of an anodized plate engraved with a dot pattern (2 mm diameter, 2 mm spacing). A support structure was also built to position the plate inside the tube and act as a stopper for the water circuit.
<figure><img src="image_34.png" /><figcaption>Figure A.31: Checkboard target calibration.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>Figure A.32: Checkboard target distortion.</figcaption></figure>

The procedure consists of three main steps. First, the original coordinates of the calibration pattern were obtained by scanning the plate with high resolution and detecting the point centers (Figure A.31, left). Second, the same pattern was photographed inside the visualization box under operating conditions (Figure A.31, right), yielding lower resolution due to the acquisition system (1920×1200 pixels). Coordinates were again extracted, and both sets (scanned vs. photographed) were compared to quantify distortions introduced by tube curvature and camera position (Figure A.32). The maximum deviation reached 0.5 mm, expressed by the deformation modulus:

$$ \delta = \sqrt{\delta_x^2 + \delta_y^2}. \qquad (A.2) $$

Finally, a transformation matrix $T$ was determined to map the distorted coordinates $(X, Y)$ to the corrected ones $(X', Y')$:

$$ \begin{bmatrix} X' \\ Y' \end{bmatrix} = T \begin{bmatrix} X \\ Y \end{bmatrix}. \qquad (A.3) $$

This mapping was subsequently applied to all acquired images to correct geometric distortion.

## Appendix A.3. Processing

After the images have been corrected, they have been processed to differentiate the vapour phase from the liquid phase and to extract the velocities in the case of a bubble flow regime. First, the image has to be corrected, as the backlight illumination is not perfectly uniform over the window, which can cause problems when applying a binarisation threshold to distinguish the vapour phase. This non-uniformity of the backlighting can be seen in Figure A.33.

To correct this problem, the image has been normalised. Basically, to detect the outlines of the vapour phase, instead of performing a subtraction operation from an image without vapour to an image with vapour, a division operation has been carried out.

After the normalisation of the image, the image is inverted and binarised by applying a threshold, in order to subsequently detect the vapour phase. This binarisation is done because when the vapour phase
<figure><img src="image_36.png" /><figcaption>Figure A.33: Non uniform light issue and normalization.</figcaption></figure>

is present, the liquid-vapour interface deviates the light by refraction, so it can be detected and thus the vapour phase can be detected. Figure A.34 shows the vapour detection process for a stratified flow regime. It can be seen that when the image is segmented, it is possible to differentiate the liquid-vapour interface, but it is not possible to distinguish the liquid and vapour phase, since as can be seen in the graph they have the same intensity. This problem also occurs when the vapour bubbles are large enough so that the light coming from the backlight cannot be refracted inside them.

<figure><img src="image_37.png" /><figcaption>Figure A.34: Image for a stratified flow regime after hole filling.</figcaption></figure>

To solve this problem, a hole-filling algorithm has been implemented, so that when the interface is detected, it is possible to detect the vapour. For this purpose, when the algorithm detects a closed contour, usually generated by the liquid-vapour interface of a bubble, it fills it with white pixels as if it were vapour. This process can be seen in Figure A.34.

## Appendix A.4. Postprocessing

After binarising the image and detecting the vapour phase, it is possible, for bubbly flow regime, to detect the position of the nose and tail of the bubble, and subsequently calculate their respective velocities. A schematic of the process is shown in Figure A.35.

After detecting the position of the nose and tail of the bubble, their respective positions are stored over time, after which it is possible to calculate the respective velocities using a second-order finite-difference discretisation as follows:
<figure><img src="image_38.png" /><figcaption>Figure A.35: Steam velocity detection.</figcaption></figure>

$$v_{\text{Nose}}(t) = \frac{-3x_{\text{Nose}}^{t} + 4x_{\text{Nose}}^{t+1} - x_{\text{Nose}}^{t+2}}{2\Delta t} \qquad (A.4)$$

$$v_{\text{Tail}}(t) = \frac{-3x_{\text{Tail}}^{t} + 4x_{\text{Tail}}^{t+1} - x_{\text{Tail}}^{t+2}}{2\Delta t} \qquad (A.5)$$

Where $x_{\text{Nose}}^t$ and $x_{\text{Tail}}^t$ are the respective positions of the nose and tail at time instant $t$.

It is also interesting to plot, as a function of time, the evolution of the vapour volume fraction over a vertical line. To do this, what has been done is to obtain in vertical lines the amount of white pixels (after binarisation they represent the vapour phase) over the total number of pixels in the line. This fraction is calculated as follows:

$$\varepsilon_{\text{line}} = \frac{N_{\text{White Pixels}}}{N_{\text{White Pixels}} + N_{\text{Black Pixels}}} \qquad (A.6)$$

Where $N_{\text{White Pixels}}$ and $N_{\text{Black Pixels}}$ are the number of white and black pixels respectively on that line.

Additionally, it is useful to analyse the probability distribution of $\varepsilon_{\text{line}}$ to characterise its statistical behaviour over time. The probability density function (PDF) of $\varepsilon_{\text{line}}$ is calculated using:

$$f(\varepsilon_{\text{line}}) = \frac{1}{N \Delta \varepsilon} n(\varepsilon_{\text{line}}) \qquad (A.7)$$

where $n(\varepsilon_{\text{line}})$ is the number of occurrences of $\varepsilon_{\text{line}}$ in a given bin of width $\Delta\varepsilon$, and $N$ is the total number of data points collected along the vertical line over time. This normalisation ensures that the area under the PDF equals one.

## Appendix B. CAD model of the experimental setup

The following image shows the CAD model of the ConBo experimental installation, detailing the layout of each of the auxiliary systems.
<figure><img src="image_39.png" /><figcaption>Figure B.36: CAD model of ConBo setup highlighting the auxiliary components.</figcaption></figure>