Trefftz Method for Time-Dependent Boiling Heat Transfer Calculations in a Mini-Channel Utilising Various Spatial Orientations of the Flow

Abstract

The main objective of this study was to investigate boiling heat transfer during refrigerant flow in a mini-channel heat sink. The test section consisted of multiple parallel mini-channels, each with a depth of 1 mm. The working fluid was heated by a thin layer of Haynes-230 alloy with a thickness of 0.1 mm. The outer surface temperature of the heater was measured using infrared thermography, while other thermal and flow-based parameters were recorded via a dedicated data acquisition system. The mini-channel heat sink was tested in seven different spatial orientations, with inclination angles relative to the horizontal plane of 45°, 60°, 75°, 90°, 105°, 120°, and 135°. The analysis focused on the early stage of the experiment, corresponding to the forced convection, boiling incipience, and subcooled boiling region. A time-dependent, two-dimensional model of heat transfer during flow boiling of a refrigerant in asymmetrically heated mini-channels was developed. The temperatures of both the heating foil and the working fluid (Fluorinert FC-770) were described using appropriate forms of the Fourier–Kirchhoff equation, subject to relevant boundary conditions. Two sets of time-dependent Trefftz functions were employed to solve the governing equations: one set corresponding to the two-dimensional Fourier equation and another, newly derived, for the energy equation in the fluid. The results include thermographic images of the heated surface, temperature distributions, fluid temperatures, local heat-transfer coefficients, and boiling curves. A comparison of the heat-transfer coefficients obtained using the Trefftz-based approach and those calculated using Fourier’s law demonstrated satisfactory agreement.

1. Introduction

Miniaturised industrial and domestic devices require efficient cooling systems due to the high heat fluxes generated during operation. This is especially important for power electronics and microprocessors, which emit significant amounts of heat. One solution involves mini-channels with flow boiling, which can be directly integrated into heat-generating structures. This approach enables effective heat dissipation, with a relatively small temperature difference between the heating surface and the saturated liquid. The high efficiency of heat exchange in such systems results from a large heat-transfer-area-to-volume ratio and a high heat-transfer coefficient (HTC). In addition, some cooling systems focus on minimising pumping power requirements, leading to the use of flows with Reynolds numbers below 2000.

Flow boiling in mini-channels has emerged as a highly effective mechanism for thermal management in compact and high-power-density systems, including microelectronics, avionics, and renewable energy modules. As device miniaturisation continues and heat fluxes surpass conventional cooling limits, researchers have turned to microscale channel configurations, where boiling regimes can offer superior heat removal capabilities. However, challenges such as flow instability, early dryout, and limited critical heat flux have highlighted the need for innovative enhancement strategies to ensure reliable and efficient operation. The physical behaviour of two-phase mini-channel flow boiling in mini-channels is fundamentally different from that in conventionally sized channels. The increased influence of surface tension, capillary forces, and microscale confinement alters bubble dynamics, pressure drop characteristics, and the distribution of boiling regimes.

Recent research has explored a variety of techniques aimed at intensifying boiling heat transfer while maintaining flow stability and structural simplicity. These include surface engineering methods (e.g., porous and coated surfaces), geometric channel modifications (e.g., grooves, fins), and integrated diagnostic techniques (e.g., infrared thermography, high-speed flow visualisation). Studies have also addressed the roles of working fluid properties and temporal–spatial flow behaviour in local and overall HTC and boiling heat flux limits. The presented review of the literature, based on selected works, synthesises and analyses the findings of several research studies [1,2,3,4,5], offering a comprehensive perspective on the strategies currently used to analyse boiling heat transfer in mini-channel systems. These works provide insight into the mechanisms of heat transfer in mini-channels and microscale heat exchangers.

Bian et al. [6] designed and experimentally investigated a mini-channel heat sink with inclined bottom grooves (MC-BG) used to enhance the two-phase flow boiling performance. The heat sink, with channel dimensions of 200 mm length and 2 mm × 2 mm cross-section, incorporated grooves 0.5 mm deep, inclined 45° to the flow direction. Experiments were carried out with deionised water at two inlet temperatures (70 °C and 80 °C) and mass fluxes of 138.8, 230.3, and 331.9 kg/(m²·s). The results were benchmarked against a smooth mini-channel configuration (SMC). The study demonstrated that the inclined grooves enhanced nucleate boiling by increasing the density of the active nucleation sites and suppressing excessive bubble growth and coalescence. This effect delayed the transition from bubbly to annular flow, thus improving boiling stability. Compared to the smooth channel, the MC-BG showed earlier onset of nucleate boiling and required lower levels of wall superheating and effective heat flux to initiate boiling. At an inlet temperature of 80 ° C and mass flux of 138.8 kg/(m²·s), the maximum average heat-transfer coefficient in the two-phase region for MC-BG reached 30.70 kW/(m²·K), representing an improvement of 18.31% over the SMC. Furthermore, MC-BG exhibited a lower pressure drop across most test conditions, confirming superior net heat-transfer performance and reduced boiling-induced instability. Flow instability analysis showed that groove structures mitigated fluctuation amplitudes. Instability increased with increasing heat flux and decreased with higher mass flux but remained largely unaffected by inlet temperature. The authors stated that, overall, the use of inclined grooves in mini-channel heat sinks presents a viable thermal management enhancement strategy, one with potential for further improvement through geometric optimisation of groove angle, height, and channel layout. In their experimental study, He et al. [7] developed a new interconnected biporous mini-channel (IBPM) to enhance both heat-transfer performance and flow stability in flow boiling applications. The design featured a biporous coating on the channel base and periodically distributed interconnections between channels. Flow boiling tests were performed at mass fluxes of 161.2, 255.2, and 349.2 kg/(m²·s). High-speed visualisation revealed that confined elongated bubbles in the IBPM expanded and partially transferred vapour into adjacent channels at interconnection sites, promoting heat and mass transfer. The onset of nucleate boiling occurred at lower wall superheats in IBPM (from 1.1 to 2.7 °C), compared to biporous mini-channels without interconnections (from 2.3 to 3.4 °C), due to the modified flow field. The two-phase heat-transfer coefficient of IBPM was significantly higher across all mass fluxes, up to 1.84 times higher than BPM in the subcooled regime; this was attributed to an increase in the number of nucleation sites. Unlike BPM, IBPM exhibited strong single-phase heat-transfer sensitivity to mass flux. Importantly, no inlet backflow was observed in IBPM, and the pressure deviation remained between 0.10 and 0.34 kPa, confirming the improved flow stability. The study demonstrates that combining interconnected structures with biporous surfaces was effective in mitigating flow instabilities while enhancing boiling heat transfer, making IBPM a promising design for advanced thermal management systems.

Kumar and Kothadia [8] conducted a comparative thermal analysis of subcooled flow boiling in mini and conventional channels, with the use of water as a working fluid. The experiments were carried out in tubes of 2 mm (a mini-channel), 4 mm, and 11.7 mm diameter (conventional channels), maintaining a constant length-to-diameter ratio of 150. The study evaluated the wall temperature distribution, the local heat-transfer coefficient, and the temporal fluctuations, utilising mass fluxes of 150 to 2400 kg/(m²·s) and heat fluxes from 109 to 1080 kW/m². In the subcooled boiling region, HTC was found to be strongly dependent on heat flux but not on mass flux in conventional channels. On the contrary, saturated flow boiling in these channels showed an increase in HTC with increasing mass flux. In mini-channels, significant wall temperature fluctuations emerged at boiling numbers exceeding 1.76 × 10⁻⁴; this was associated with abrupt disturbances caused by the growth affecting the narrow flow passage. At higher boiling numbers, HTC in mini-channels increased but also became more unstable. Maximum HTC was observed in the 2 mm mini-channel up to Bo = 2.24 × 10⁻⁴, beyond which the risk of dryout increased. On the contrary, 4 mm channels maintained circumferential temperature uniformity and stability across all boiling regimes. The study indicated the critical role of boiling number in determining the thermal stability and operational limits of mini-channel flow boiling. The need for geometry-specific models and better predictive methods tailored to small diameter channels was highlighted. Shang et al., in [9], investigated flow boiling heat transfer in smooth (SPMC) and porous (PPMC) parallel mini-channel heat sinks fabricated from 6061 aluminium alloy, targeting server chip cooling applications. Experiments with HFE-7100 were conducted under low-to-moderate mass fluxes in the range 121.0–241.9 kg/(m²·s), and effective heat fluxes up to 9.1 W/cm². The PPMC structure, produced by surface etching, provided improved nucleation due to additional surface cavities, reducing wall temperatures by up to 2.39% (1.7 K) at 181.4 kg/(m²·s) and 7.3 W/cm². Average wall temperatures remained below 75 °C under all tested conditions. Compared to SPMC, the PPMC also showed slightly higher local heat-transfer coefficients (HTCs), with the difference decreasing along the flow direction. Visualisation of flow revealed that slug and churn flow dominate the middle and downstream regions; no dryout occurred, up to 9.1 W/cm². These results offer practical insights for the optimization of mini-channel heat sink performance in electronics cooling under varying operating conditions.

In their study [10], Charnay et al. examined the flow boiling of R-245fa in a 3 mm inner-diameter tube at saturation temperatures in the range of 100–120 °C. The heat-transfer coefficient was analysed with respect to vapour quality, heat flux, and mass flux. At 60 °C, during intermittent flow, the coefficient was independent of the vapour quality and the mass flux but increased with the heat flux, indicating dominant nucleate boiling. In annular flow, it rose with both vapour quality and mass flux, suggesting combined boiling mechanisms. At 120 °C, the coefficient decreased in both regimes, reflecting reduced nucleate boiling activity. The study also compared its results with existing predictive models. Wang et al., in [11], proposed a counter-flow interconnected mini-channel (CFIM) heat sink and compared its performance with a conventional co-current configuration (CCM). Experiments using R1233zd(E) under varying thermal conditions demonstrated that the CFIM effectively suppressed backflow and dryout, improving flow stability and wall temperature uniformity. The design resulted in a 51.2% increase in the heat-transfer coefficient, a 56% improvement in the coefficient of performance (COP), and a 48.5% reduction in pressure drop, achieving heat fluxes up to 230.4 W/cm². These findings underscore the potential for high performance in two-phase thermal management applications.

The studies highlight that modifications to channel geometry (e.g., grooves, interconnections), surface enhancements, and the modification of flow configurations significantly influence both the heat-transfer performance and the hydrodynamic stability of mini-channel heat sinks. The selection of working fluid, inlet parameters, and thermal boundary conditions further impacts boiling dynamics, bubble behaviour, and flow regimes. At the same time, it must be emphasised that the choice of a computational method for analysing experimental or simulation results plays a crucial role in ensuring the accuracy and reliability of the findings. A properly selected and validated modelling approach enables a confident interpretation of the observed thermal and flow-related phenomena. These insights offer valuable guidance for the future design and optimisation of compact thermal management systems, particularly in high-heat-flux applications such as electronic device cooling or renewable energy systems. This study focusses on developing and solving a two-dimensional, and this is worth emphasising, time-dependent, model of heat transfer during boiling in asymmetrically heated mini-channels in different spatial orientations. The governing energy equations with their boundary conditions result in two inverse Cauchy-type problems, one concerning the solid heating foil and the other addressing the mini-channel with flowing fluid. Inverse problems are inherently ill-posed and highly sensitive to input data, often leading to unstable numerical solutions [12]. To counteract these challenges, it is crucial to use computational techniques that prioritise accuracy and stability, such as the Trefftz function-based approach.

In 1926, Erik Trefftz published [13]. The Ritz method belongs to the class of internal methods for solving partial differential equations, i.e., those in which the solution satisfies the initial and boundary conditions exactly and the equation approximately. The Trefftz method belongs to the class of external methods, in which the equation is strictly satisfied and the initial-boundary conditions approximated. From a numerical point of view, the Trefftz method has an advantage over the Ritz approach, because the latter requires the minimization of an n-dimensional integral for an n-dimensional differential equation, whereas the Trefftz method requires only an n-1-dimensional integral. The dynamic development of the method began in the 1970s. The works of Herrera, Jirousek, Sabina, and Zieliński [14,15,16,17,18] are worth mentioning. The shared aim of these works was to develop methods for obtaining the Trefftz function and to investigate the completeness of the resulting systems. In 1956, ref [19] was published, in which one-dimensional thermal polynomials were derived that strictly satisfy the non-stationary heat-conduction equation. It should be emphasized that the variable describing time is treated as continuous spatial variables.

The Trefftz method offers several significant advantages for solving both direct and inverse engineering problems. It is a meshless semi-analytical technique that approximates solutions to the governing differential equations by representing them as linear combinations of Trefftz functions (T-functions). The coefficients for linear combinations are determined based on known boundary conditions, ensuring stability and precision. Its primary strength lies in its ability to provide solutions that strictly satisfy the governing differential equations, thus ensuring high precision. Additionally, the method simplifies computations, as it typically employs polynomial T-functions. It also exhibits considerable flexibility, accommodating various types of boundary conditions, whether related to temperature distributions, fluid flow, or a combination of discrete and continuous conditions.

In [1,2,3,4], the Trefftz method was used to determine time-dependent solutions of partial differential equations, which include equations of the parabolic, hyperbolic, and parabolic–hyperbolic types. The classical Trefftz method is also used in [1] to solve time-dependent problems of the parabolic type. In [2], the authors, using Laplace and Laguerre transforms, reduce time-dependent partial differential equations to the Helmholtz equation, which they solve by combining the Trefftz method with the method of fundamental solutions. In [3], the heat equation is first discretised with respect to time and then the resulting set of elliptic equations is solved using the appropriate time-independent hybrid Trefftz-element approach. The spacetime collocation Trefftz method was adopted in [4] to solve the inverse heat-conduction problem by approximating numerical solutions using Trefftz base functions. In [1,2,3,4], the validity of the methods proposed was established by using a number of test problems to demonstrate the accuracy and effectiveness of the computational techniques based on T-functions. In these papers, the methods employing T-functions allowed for the obtaining of highly accurate semi-numerical and numerical solutions, with the maximum error in solving the differential equation not exceeding a value of one at any point.

A mathematical model of time-dependent heat transfer during the flow boiling of ecological refrigerants in mini-channels is presented in [5]. A hybrid combination of T-functions with FEM and the iterative Picard method allowed the determinations of the temperature distribution of the heating surface, the heat flux transferred to the fluid, and the heat-transfer coefficients at the heating wall surface–fluid interface. The model and calculation methods with T-functions were verified based on the experimental data. The results of the numerical calculation were compared with the results obtained from known correlations in the literature, giving satisfactory results. Hybrid calculation techniques using T-functions are also discussed in [1,20,21,22].

It is well established that numerous studies in scientific research focus on CFD modelling. For example, the study reported in [23] proposed a CFD–DEM-based strategy to investigate multi-field coupled particle flows in microreactors. Ultrasonic excitation was applied to control particle clustering, while fractal analysis was used to characterize distribution irregularities within the channels. An experimental observation platform was developed to validate the simulations. The results indicated that ultrasound significantly enhanced flow uniformity and particle dispersion, with inlet velocity and excitation frequency exerting strong influences on distribution patterns.

This paper proposes a two-dimensional approach to analysing time-dependent heat transfer during the flow boiling of a working fluid, Fluorinert FC-770, utilising asymmetrically heated mini-channels with different spatial orientations. The novelty of the study is in its introduction of new T-functions, which allow one to obtain a semi-analytical solution of the fluid energy equation under the assumption of a fixed form of the velocity vector. These functions, used alongside the classical T-functions for the two-dimensional Fourier equation, provide a basis for solving inverse heat-transfer problems and reconstructing the thermal-boundary conditions from transient temperature measurements. The calculations use experimental data, and the results are verified based on solutions obtained using a simplified approach based on Fourier’s law.

It should be emphasised that this study has two main objectives. The first is to experimentally investigate the effect of channel inclination on boiling heat transfer during refrigerant flow in a mini-channel heat sink. The time-dependent measurements include the temperatures of the heated foil and the working fluid, electrical input parameters, and mass flow rate. The second objective is to demonstrate the applicability of the Trefftz method as a reliable semi-analytical tool for solving time-dependent direct and inverse heat-conduction problems in both the heating foil and the refrigerant. The resulting temperature fields and local heat-transfer coefficients are compared with those obtained from simplified models. By combining experiments with Trefftz-based analysis, the study provides a validation of the computational approach that cannot be achieved by CFD simulations alone. This methodological integration is rarely applied in mini-channel flow boiling research, which highlights the novelty and significance of the contribution.

2. Materials and Methods: Experimental Stand, Methodology, and Raw Data

2.1. Experimental Stand and Methods

A schematic diagram of the circulating loop for the working fluid (Fluorinert FC-770) implemented in the experimental setup is shown in Figure 1.

The main components of the flow loop include the following: the test section with five parallel mini-channels, a gear pump, a compensation tank, a heat exchanger, and a Coriolis mass flow meter (Figure 1). The core component of the experimental setup is the test section, which contains a group of five parallel mini-channels with rectangular cross-sections. Each channel has a fixed depth of 1 mm, a width of 6 mm, and a length of 43 mm. The working fluid flows through the mini-channels in a laminar regime and is heated by a thin heating foil made of Haynes-230 alloy, which is tensioned between two metallic frames. The temperature distribution on the outer surface of the foil is monitored using infrared thermography. The opposite side of the foil is observed through a glass plate to visually identify the flow structures. A measurement is recorded every second, for a duration of one minute. A detailed schematic of the test section is presented in Figure 2.

2.2. Direct Experimental Data

The experimental dataset comprises measurements of the fluid temperature at the inlet and outlet of the mini-channel, fluid pressure, temperature distribution on the external surface of the heating foil, and mass flow rate. The principal experimental parameters, including their respective measurement ranges and absolute uncertainties, are summarised in

Table 1. Measurement ranges of key experimental parameters and the corresponding absolute measurement errors.

Measured Parameter	Measuring Device	Measuring Range	Absolute Error	Additional Information
Fluid temperature at the inlet, [K]	K-type Thermocouple, type K 221 b, Czaki Thermo-Product (Raszyn-Rybie, Poland)	286.05–295.45	0.34 K	The details associated with the calculation of the absolute error of the fluid temperature are described in [25]
Fluid temperature at the outlet, [K]		290.05–309.25
Absolute pressure at the inlet, [kPa]	Gauge pressure: Cerabar S PMP71 m type, manufactured by Endress + Hauser; Atmospheric pressure: A-10 m type, manufactured by WIKA Polska (Wloclawek, Poland)	93–115	1 285 Pa	(1) A maximum measurement error of ±0.05% of reading (2) A maximum measurement error of 0.5% of full scale Details associated with the calculation of the absolute error for the fluid temperature are presented in [25]
Absolute pressure at the outlet, [kPa]		85–106
Temperature of the heating foil, [K]	Infrared camera A655sc FLIR (resolution 640 × 480)	307.59–390.04	2 K	±2 K or ±2% in the range of 273.15 ÷ 393.15 K
Mass flow rate [kg/s]	Coriolis mass flowmeter Proline Promass A 100, Endress + Hauser (Wroclaw, Poland)	4.05 × 10−3–4.53 × 10−3 kg/s	4 × 10−6 kg/s	±0.1% of the reading

. Additionally, the electric current and voltage drop across the heating foil were used in the calculation of the heat flux. The resulting mean relative errors, which were on the order of a few percent under subcooled boiling conditions, are discussed in detail in [24].

It should be emphasised that the early stage of the experiment was analysed in detail (the first 60 s of a 300 s test in this experimental series); this section corresponds to the forced convection, boiling incipience, and subcooled boiling region. Figure 3 presents the raw data obtained from the experiment, including the temperature of the heating foil, the fluid temperature at the inlet and outlet of the mini-channel, and the heat flux, for a selected inclination angle of the mini-channel relative to the horizontal plane (135°). Heat flux was calculated on the basis of the measured current and voltage drop across the heating foil and the known dimensions of the foil.

3. Mathematical Model

The mathematical model presented in the following is a modification of the model presented in [5]. All further considerations and calculations concern only one selected central mini-channel, for which two dimensions were taken into account: one in the flow direction (x), and one perpendicular to the flow (y). These relate to the thickness of the heating foil thickness ${\delta }_h$ and the mini-channel depth ${\delta }_M$. In addition, it was assumed that the physical parameters of the module elements do not depend on the temperature, heat loss to the surroundings through the outer walls of the module, which are negligible—the entire measurement module is insulated, convective heat exchange takes place in the mini-channel, and the fluid flow is laminar with a known mass flow rate and known temperatures at the inlet and outlet that extend to and from the mini-channel. The temperatures of the heating foil T_h and the working fluid T_f satisfy the appropriate Fourier–Kirchhoff equation:

$${\lambda }_h{\nabla }^2{T}_h={\rho }_h{c}_{p,h}{\displaystyle \frac{{\partial T}_h}{\partial t}}-{\displaystyle \frac{q\left(t\right)}{{\delta }_h}}$$

(1)

$${\lambda }_f{\nabla }^2{T}_f={\rho }_f{c}_{p,f}{\displaystyle \frac{{\partial T}_f}{\partial t}}+c_{p,f}{\rho }_{f}v\left(y,t\right){\displaystyle \frac{\partial T_f}{\partial x}}-g{\rho }_{f}v\left(y,t\right)sin\omega$$

(2)

where differential operator ${\nabla }^2={\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}$, q(t) denotes heat flux supplied to the heater and $v\left(y,t\right)$ means the velocity of the fluid in a single mini-channel. The velocity is determined based on the mass flow rate known from the measurements.

For the heating foil, it was assumed that the surface temperature of the heater not in contact with the fluid (i.e., for y = 0), determined from infrared camera measurements as $T_{ij}^m$, is prescribed, while the heater walls perpendicular to the fluid-contacting surface are considered insulated.

By analogy to [5], for the fluid energy equation, the following boundary conditions were adopted:

• The fluid temperatures at the inlet and outlet of the mini-channel are known as

$$T_f\left(0,y,t_j\right)=T_{f,in}\left(y,t_j\right)$$

(3)

$$T_f\left(L,y,t_j\right)=T_{f,out}\left(y,t_j\right)$$

(4)

• The temperatures of the fluid and the heating foil are equal at the contact surface, i.e.,

$$T_h\left(x,{\delta }_h,t\right)=T_f\left(x,{\delta }_h,t\right)$$

(5)

In addition, the boundary conditions for Equations (1) and (2) are shown in Figure 4.

The effect of the assumptions made is the existence of two inverse problems relating to the heat transfer in the heating foil and the mini-channel. In the first step, the temperatures of the heating foil and the fluid were determined using the Trefftz method, similar to [26]. In the next step, using knowledge of the temperature distributions of the heating foil and the working fluid, the HTC at the contact surface was determined from the Robin condition:

$${\alpha }_{2D}\left(x,t\right)={\displaystyle \frac{-{\lambda }_h\frac{\partial T_h}{\partial y}\left(x,{\delta }_h,t\right)}{T_h\left(x,{\delta }_h,t\right)-T_{f,ave}\left(x,{\delta }_h,t\right)}}$$

(6)

where the values for the reference fluid temperature T_f,ave are calculated by analogy to [5].

4. Calculations Method

4.1. Trefftz Functions for the Fourier–Kirchhoff Equations

Let us consider a one-dimensional Fourier equation of the form

$$\lambda {\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}=\rho c_p{\displaystyle \frac{\partial T}{\partial t}}$$

(7)

for which we make the substitution $\tau ={\displaystyle \frac{\lambda t}{\rho c_p}}$ and obtain

$${\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}={\displaystyle \frac{\partial T}{\partial \tau }}$$

(8)

In [19], for Equation (8), the Trefftz functions are given in the form

$$u_m\left(x,\tau \right)={\sum }_{k=0}^{\left[{\displaystyle \frac{m}{2}}\right]}{\displaystyle \frac{x^{m-2k}{\tau }^k}{\left(m-2k\right)!k!}}\mathrm{for}m=\mathrm{0,1},\dots$$

(9)

where [.] means the floor function. As shown in [27], the products of the form

$$V_{k,0}(x,y,\tau )=u_m\left(x,\tau \right)u_n\left(y,\tau \right)$$

(10)

where $k=0.5\left(1+\left(m+n\right)\right)\left(m+n\right)+m$ for $m,n=\mathrm{0,1},\dots$, are the Trefftz functions for the equation

$${\nabla }^{2}T={\displaystyle \frac{\partial T}{\partial \tau }}$$

(11)

Table 2. Properties of the Trefftz functions.

${\displaystyle \frac{\partial }{\partial x}}\left(u_m\left(x,\tau \right)u_n\left(y,\tau \right)\right)=\left\{\begin{array}{cc}m{u}_{m-1}\left(x,\tau \right)u_n\left(y,\tau \right)& \mathrm{f}\mathrm{o}\mathrm{r}m=\mathrm{1,2},\dots ,n=\mathrm{0,1},\dots \\ 0& \mathrm{f}\mathrm{o}\mathrm{r}m=0,n=\mathrm{0,1},\dots \end{array}\right.$	(12)
${\displaystyle \frac{\partial }{\partial y}}\left(u_m\left(x,\tau \right)u_n\left(y,\tau \right)\right)=\left\{\begin{array}{cc}n{u}_m\left(x,\tau \right)u_{n-1}\left(y,\tau \right)& \mathrm{f}\mathrm{o}\mathrm{r}m=\mathrm{0,1},\dots ,n=\mathrm{1,2},\dots \\ 0& \mathrm{f}\mathrm{o}\mathrm{r}m=\mathrm{0,1},\dots ,,n=0\end{array}\right.$	(13)
${\displaystyle \frac{\partial }{\partial t}}\left(u_m\left(x,\tau \right)u_n\left(y,\tau \right)\right)=$ $\left\{\begin{array}{cc}m\left(m-1\right)u_{m-2}\left(x,\tau \right)u_n\left(y,\tau \right)+n\left(n-1\right)u_m\left(x,\tau \right)u_{n-2}\left(y,\tau \right)& \mathrm{f}\mathrm{o}\mathrm{r}m,n=\mathrm{2,3}\dots \\ 0& \mathrm{f}\mathrm{o}\mathrm{r}m,n=\mathrm{0,1}\end{array}\right.$	(14)

presents the properties of the function described in (10).

These functions will be used to define the Trefftz function for the Fourier–Kirchhoff equation in the following form:

$$\mathcal{L}T=v(y,\tau ){\displaystyle \frac{\partial T}{\partial x}}$$

(15)

where $\mathcal{L}T={\nabla }^{2}T-{\displaystyle \frac{\partial T}{\partial t}}$.

In the further transformations we assume that the velocity $v\left(y,\tau \right)$is a polynomial. First, we define the inverse operator of the operator $\mathcal{L}$ for a function of the form $x^n{y}^m{\tau }^k$ where n, m, and k are natural numbers, i.e.,

$$\begin{gathered} {\mathcal{L}}^{-1}\left(x^n{y}^m{\tau }^k\right)={\displaystyle \frac{1}{\left(m+2\right)\left(m+1\right)}}{x}^n{y}^{m+2}{\tau }^k-{\displaystyle \frac{n\left(n-1\right)}{\left(m+2\right)\left(m+1\right)}}{\mathcal{L}}^{-1}\left(x^{n-2}{y}^{m+2}{\tau }^k\right) \\ +{\displaystyle \frac{k}{(m+2)(m+1)}}{\mathcal{L}}^{-1}\left(x^n{y}^{m+2}{\tau }^{k-1}\right) \end{gathered}$$

(16)

Let us define the new functions $V_n(x,y,\tau )$ as follows:

$$V_n\left(x,y,\tau \right)={\sum }_{k=0}^m{V}_{n,k}(x,y,\tau )$$

(17)

where

$$V_{n,k}=\left\{\begin{array}{c}{V}_{n,0}\mathrm{for}k=0\\ {\mathcal{L}}^{-1}\left(v\left(y,\tau \right){\displaystyle \frac{\partial V_{n,k-1}}{\partial x}}\right)\mathrm{for}k=\mathrm{1,2},\dots ,m\end{array}\right.$$

(18)

In (18), $V_{n,0}$ is a T-function of the form (10), and m is the degree of the polynomial $V_{n,0}$ with respect to the variable x. Note that the function $V_{n,m}$ depends only on the variables y and $\tau$. Then, it is easy to show that the functions $V_n\left(x,y,\tau \right)$ satisfy Equation (15), which means that they are Trefftz functions for this equation. The presented method for defining the Trefftz function for Equation (15) can be generalised to any form of velocity $v\left(y,\tau \right)$. One necessary condition is then to define the inverse operator ${\mathcal{L}}^{-1}$ for all summands in $v\left(y,\tau \right)x^n{y}^m{\tau }^k$.

4.2. Trefftz Method

The idea of the Trefftz method is to present the solution of the governing differential equation as a linear combination of functions that exactly satisfy this equation. To determine the solutions of Equations (1) and (2), two sets of the Trefftz functions defined by formula (10) and formula (17) will be used. Similarly to [5], the temperatures of the heating foil $T_h$ and refrigerant $T_f$ are approximated by the sum of the linear combination of the Trefftz functions defined by formulas (10) and (17), respectively, and the corresponding particular solutions of Equations (1) and (2), i.e.,

$$T_h(x,y,t)=V_{par,h}\left(x,y,t\right)+{\sum }_{n=1}^{N_h}{f}_n{V}_{n,0}\left(x,y,t\right)$$

(19)

$$T_f(x,y,t)=V_{par,f}\left(x,y,t\right)+{\sum }_{n=1}^{N_f}{g}_n{V}_n\left(x,y,t\right)$$

(20)

where the particular solutions are defined as follows:

$$V_{par,h}\left(x,y,t\right)=-{\displaystyle \frac{q\left(t\right)}{{{2\lambda }_h\delta }_h}}{y}^2$$

(21)

$$V_{par,f}\left(x,y,t\right)={\mathcal{L}}^{-1}\left(g{\rho }_{f}v\left(y,t\right)sin\omega \right)$$

(22)

The linear combination coefficients $f_n$ and $g_n$ are determined based on the minimisation of the functional describing the mean square error for which the functions $T_h$ and $T_f$ satisfy known initial and boundary conditions. The Trefftz method can be used to solve both direct and inverse problems, whether time-dependent or time-independent, yielding a differentiable solution that exactly satisfies the governing differential equation and approximately satisfies the boundary conditions. It imposes no limitations on the number or type of the boundary conditions (temperature-related, flow-related, continuous, or discrete), and any number of T-functions can be used in the construction of the solution.

4.3. Simplified Approach

The simplifications adopted below allow us to determine the HTC based on Fourier’s law, thereby verifying the calculation results obtained using the Trefftz method. It was assumed that the temperature of the heating foil, $T_h$, at each measurement point $\left(x_i,t_j\right)$ satisfies the Fourier equation in the form

$${\lambda }_h{\displaystyle \frac{{dT}_h}{dy}}=-q\left(t_j\right)$$

(23)

For y = 0, the temperature of the heater surface is known from the measurements of the infrared camera $T_{ij}^m$, i.e.,

$$T_h\left(x_i,0,t_j\right)=T_{ij}^m$$

(24)

The fluid temperature is calculated using the following relationship:

$$T_f\left(x_i,{\delta }_h,t_j\right)=\left\{\begin{array}{c}{T}_{lin}\left(x_i,t_j\right)if{T}_h\left(x_i,{\delta }_h,t_j\right)\le T_{sat}\left(x_i,t_j\right)\\ T_{sat}\left(x_i,t_j\right)if{T}_h\left(x_i,{\delta }_h,t_j\right)>T_{sat}\left(x_i,t_j\right)\end{array}\right.$$

(25)

where $T_{lin}$ is a function that changes linearly from the temperature of the fluid at the inlet of the mini-channel to the temperature of the fluid at the outlet. The local HTC can be determined using the equation

$${\alpha }_{1D}\left(x_i,t_j\right)={\displaystyle \frac{q\left(t_j\right)}{T_{ij}^m-\frac{q\left(t_j\right)}{{\lambda }_h}{\mathsf{\delta }}_h-T_f\left(x_i,{\delta }_h,t_j\right)}}$$

(26)

5. Results

Numerical calculations were performed for the heat flux q, ranging from 26.8 kW/m² to 42.6 kW/m², and the mass flow rate of the working medium, ranging from 4.05 × 10⁻³ kg/s to 4. × 10⁻³ kg/s. The remaining experimental parameters, including their measurement uncertainties, are listed in Table 1. The HTC was determined through a three-step process. First, the two-dimensional temperature of the heating foil was calculated using the T-functions defined by Equation (10) as shown in Figure 5. Next, based on the known constant fluid velocity and the temperature of the heating foil, the two-dimensional temperature of the fluid was determined using the Trefftz method, with new T-functions defined by formula (17). Finally, the HTC was calculated using Equation (6).

The results are illustrated through the HTC as shown in Figure 6, Figure 7, Figure 8 and Figure 9:

-

As a function of time and the distance from the inlet to the mini-channel for selected inclination angles, Figure 6 and Figure 9;

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Versus distance to the mini-channel inlet for selected times and selected inclination angles, Figure 7;

-

For all inclination angles, as a function of time, for two distances from the inlet to the mini-channel, Figure 8.

5.1. The Trefftz Method with New T-Functions

Figure 5 show the two-dimensional temperature distributions of the heating foil and the working fluid, respectively, at the contact surface as a function of time and the distance from the inlet to the mini-channel for a selected inclination angle.

The new T-functions, given by formula (17), made it possible to obtain a solution to the Fourier–Kirchhoff equation for the fluid that satisfies the adopted boundary conditions with high accuracy. The mean differences between the fluid temperature T_f and the assumed boundary conditions (3)–(5) are presented in

Table 3. The mean differences between the calculated fluid temperature T_f and the assumed boundary conditions (3)–(5).

	Mean Differences, [K]
$\mathbf{Inclination}\mathbf{Angle}\mathit{\omega }$	Conditions (3)	Conditions (4)	Conditions (5)
45°	6.74	8.32	1.23
60°	6.78	8.36	1.05
75°	7.10	8.37	1.19
90°	6.22	7.86	0.99
105°	6.41	7.26	1.49
120°	6.73	7.57	1.31
135°	6.31	7.39	0.98

. The largest mean differences 7.10 K (2.5%) and 8.37 K (2.86%) were obtained for angle of 75° and for boundary conditions (3) and (4), respectively. Condition (5) was fulfilled with the greatest accuracy for each inclination angle.

5.2. The HTC Analysis

Figure 6 and Figure 7 show the heat-transfer coefficient ${\alpha }_{2D}\left(x,t\right)$, which is calculated using formula (6), as a function of time and the distance from the mini-channel inlet, for two selected inclination angles, 60° (Figure 6a) and 135° (Figure 6b), and as a function of distance from the inlet for two selected times (Figure 7), taking into account all seven inclination angles.

In Figure 6, a decreasing trend in HTCs is observed with increasing distance from the mini-channel inlet. This behaviour is consistent across all inclination angles considered. The HTC values range from approximately 0.5 kW/(m²·K) to 3 kW/(m²·K), with the highest values consistently recorded near the channel inlet, regardless of time and inclination angle. The profiles of the heat-transfer coefficient associated with the early times (e.g., t = 20 s) show a negligible dependence on the inclination angle (Figure 7a). However, more pronounced differences are observed toward the end of the heating period, as represented by t = 60 s (Figure 7b). Analysis of the results presented in Figure 6 and Figure 7 indicates that, in general, the highest heat-transfer coefficients are achieved in both time intervals for the inclination angle of 120°.

The heat-transfer coefficient values ${\alpha }_{2D}\left(x,t\right)$ for all selected inclination angles are illustrated in Figure 8, which shows the continuous variation of ${\alpha }_{2D}\left(x,t\right)$over time at two selected distances from the mini-channel inlet x = 0.02 m (Figure 8a) and x = 0.04 m (Figure 8b).

When analysing the results presented in Figure 8a, it can be observed that at the location x = 0.02 m, the heat-transfer coefficients for the mini-channel inclination angles of 60°, 75°, and 90° exhibit only minor differences throughout the entire time interval. In each case, the temporal evolution of the HTC is characterised by an initial increase followed by a gradual decline. This behaviour becomes more pronounced at x = 0.4 m (Figure 8b), where a distinct effect is observed in temperature increase over time (and hence heat flux) and the influence of channel inclination becomes more apparent. At the downstream location x = 0.4 m, which corresponds to the outlet region, the highest HTC values are once again recorded with an inclination angle of 105°, particularly during the later stages of the heating process. Generally, higher HTCs are obtained at x = 0.4 m, compared to x = 0.2 m, with the exception of the inclined position of 135°. Furthermore, for the two orientations of 45° and of 135°, the heat-transfer behaviour exhibits a clear reversal: at x = 0.2 m, higher HTC values are observed for 135° than for 45°, while at x = 0.4 m, the opposite trend is seen, and the position of 45° yields higher HTC values than the position of 135°. These variations are most evident in further increase in heat flux, corresponding to the advancing time.

The overall trends indicate that the inclination angle of the mini-channel significantly influences surface-heat-transfer performance and the flow pattern, even in an asymmetrically heated mini-channel, in which only one wall is heated and the working fluid flows above it. Higher HTC values are obtained at steeper inclination angles approaching the vertical, namely 105° and 135°, at which more effective vapour detachment from the heating surface occurs at the incipience of boiling, primarily due to gravitational forces.

This trend becomes more pronounced as the heat flux supplied by the heating foil increases, leading to the formation of vapour agglomerates within the liquid–vapour mixture; this is characteristic of the vertical position (the position of 90°, with fluid upward flow) and higher inclination angles (e.g., 105°, 120°) of the mini-channel. The gravitational component, acting in the direction of vapour buoyancy, facilitates a more rapid removal of vapour from the heated surface. This, in turn, reduces the residence time, lowers thermal resistance at the liquid–solid interface, and maintains the wettability of the heated wall, which are conditions beneficial to efficient nucleate boiling heat transfer. Furthermore, such spatial orientations of the mini-channel improve surface rewetting and promote liquid backflow, enabling continuous replenishment of liquid at the heated interface.

The angle of the mini-channel’s inclination influences the overall heat-transfer behaviour. The observed intensification in heat transfer at specific inclination angles, particularly in downstream regions, is the result of a synergistic combination of improved vapour removal, reliable liquid supply, and favourable flow-regime evolution. These phenomena are fundamentally governed by gravity-driven effects acting within the geometrically confined and asymmetrically heated mini-channel specification.

In order to verify the calculations performed using the Trefftz method with newly defined T-functions, the HTC was determined from a simplified model based on Fourier’s law. Figure 9a shows the HTC, calculated by formula (26), as a function of the time and distance from the mini-channel inlet (Figure 9a), and the differences between the HTCs calculated from formulas (6) and (26), i.e., ${\alpha }_{2D}\left(x,t\right)$ − ${\alpha }_{1D}\left(x,t\right)$ (Figure 9b), which are also functions of time and distance. The average differences between the HTCs ${\alpha }_{2D}\left(x,t\right)$ and ${\alpha }_{1D}\left(x,t\right)$ do not exceed 0.3 kW/(m²K) for all inclination angles. Figure 9b shows that the HTC for the two-dimensional approach achieves higher values than the HTC for the one-dimensional approach.

5.3. Boiling Curves

The boiling curves presented in this article illustrate the relationship between the heat flux and the temperature difference between the heated wall and the bulk fluid, as evaluated at selected positions along the mini-channel wall. These curves are based on the complete set of experimental data obtained during a gradual increase in heat flux. Boiling curves were generated at four axial locations relative to the mini-channel inlet (0.01 m, 0.02 m, 0.03 m, and 0.04 m), with results corresponding to the following inclination angles of the mini-channel: ω = 60° (Figure 10a), ω = 90° (Figure 10b), and ω = 135° (Figure 10c).

The course of the boiling curves reveals several important observations. As a subcooled liquid enters the asymmetrically heated mini-channel, heat transfer initially proceeds through single-phase forced convection. With increasing heat flux, the liquid near the heated wall becomes superheated, whereas the bulk of the fluid remains subcooled. The inception of spontaneous vapour bubble formation results in a sudden drop in wall temperature during ONB (onset of boiling). These vapour bubbles act as internal heat sinks, absorbing significant amounts of thermal energy from the heated surface. As the heat flux continues to increase, fully developed nucleate boiling occurs.

Analysis of the boiling curves obtained for ω = 60° (Figure 10a) reveals a distinct temperature drop corresponding to the onset of nucleate boiling (ONB), which shifts toward higher values for the temperature difference between the heating foil and the bulk fluid as the distance from the mini-channel inlet increases. The boiling curves constructed for ω = 90° (Figure 10b) exhibit a slightly smaller temperature drop at ONB compared to those observed for ω = 60°. Although the temperature difference between the heated wall and the bulk fluid remains of comparable magnitude, the data points show significantly greater scatter. On the contrary, for ω = 135° (Figure 10c), the temperature drop typically associated with ONB is not observed in the course of the boiling curves.

The differences in boiling curve profiles and the characteristics of the ONB region observed at various inclination angles of the mini-channel can also be attributed to gravity-driven effects influencing vapour bubble behaviour and liquid replenishment near the asymmetrically heated wall.

At a lower angle of the mini-channel (ω = 60°), due to relatively stable bubble dynamics and efficient vapour evacuation, the temperature drop typically associated with ONB is distinct. As the distance from the inlet increases, development of the thermal boundary layer leads to a gradual shift of the boiling curve toward higher wall-to-fluid temperature differences. For ω = 90°, where the mini-channel is in vertical orientation and the heated surface is at the bottom, buoyancy acts symmetrically with respect to gravity and the geometry of the channel. Continuous upward removal of the vapour is promoted, but increases in bubble interactions and vertical coalescence may result in a more chaotic and fluctuating detachment. As a consequence, the ONB remains observable, but is less sharply defined, due to temporal instability in the local bubble dynamics. However, at ω = 135°, at which the heated surface is positioned above the liquid and orientated more horizontally, gravitational forces act more strongly on the liquid phase, significantly influencing the dynamics of the two-phase flow. In this configuration, the vapour bubbles generated on the heated wall tend to accumulate and coalesce beneath the surface because of limited buoyancy-driven detachment. As a result, larger and more oval-shaped vapour agglomerates are observed, which adhere for longer durations to the heated surface. This vapour accumulation inhibits the prompt departure of bubbles and delays the onset of nucleate boiling (ONB). Consequently, the boiling curves exhibit a smoother and more gradual transition into the nucleate boiling regime, without a clearly defined ONB point. Moreover, such spatial orientation of the heated surface reduces gravitational support for liquid backflow and rewetting of the wall, thereby limiting the replenishment of cooler liquid to the heated interface. This not only suppresses bubble nucleation due to insufficient wall wetting but also enhances the formation of a quasi-stable vapour layer that acts as a thermal barrier. The combined effects of impaired vapour detachment and weakened liquid renewal reduce the intensity of local heat transfer and mask the thermal signature of the ONB. As a result, boiling heat transfer under this orientation is less efficient, with poorer surface wetting and delayed establishment of fully developed nucleate boiling.

6. Conclusions

This study presents a time-dependent two-dimensional model for analysing boiling heat transfer during refrigerant flow in asymmetrically heated mini-channels. Particular attention is given to the results obtained in the early stage of the experiment that correspond to the forced convection, boiling incipience, and subcooled boiling region.

The mathematical formulation included two inverse Cauchy-type problems. The solution uses two sets of Trefftz functions, classical for the heating foil domain and newly developed for the working fluid domain. The application of new T-functions for the fluid domain marks a key innovation, enabling a semi-analytical solution under defined flow conditions.

It should be emphasised that the study pursues two intertwined objectives:

• Experimental analysis—this aim was to cover the investigation of the influence of channel inclination on boiling heat transfer during Fluorinert FC-770 flow in an asymmetrically heated mini-channel heat sink. The measurements focused on time-dependent data, including local foil temperature, inlet and outlet fluid temperatures, electrical parameters (current and voltage) of the supplied power, and mass flux. The strength of this approach lies in its ability to directly verify and validate the proposed computational model—an aspect that conventional CFD simulations alone cannot provide.
• Computational analysis—the Trefftz method was applied and validated for analysing coupled, time-dependent heat conduction and fluid flow. This analytical–numerical approach was used to solve both direct and inverse heat-conduction problems in the heated foil–mini-channel system. It enabled the determination of two-dimensional, transient temperature distributions in the heater and refrigerant, as well as local heat-transfer coefficients. The obtained results were subsequently compared with those derived from a simplified approach.

On the basis of these results and the associated analysis, the following insights have been derived:

• The mathematical model of time-dependent heat transfer allowed the effective determination of (i) the temperature distributions of the heating foil and the fluid flowing through the mini-channel, (ii) the temperature gradient of the heating foil, and, subsequently, (iii) the heat-transfer coefficient (HTC).
• Using the Trefftz method, two time-dependent inverse Cauchy-type problems were solved in two contact areas with different geometries and physical properties. The newly defined Trefftz functions were applied to solve the Fourier–Kirchhoff equation relative to the fluid.
• The inclination angle of the mini-channel has a substantial impact on local heat-transfer coefficients and two-phase flow development. Although early-stage HTC values remain comparable across tested angles of the mini-channel, differences become more pronounced with increasing distance from the inlet and higher heat flux. Steeper inclinations (such as 105° and 120°) promote enhanced vapour detachment, improved surface rewetting, and liquid backflow, leading to higher HTCs. The highest HTC values occur closer to the outlet of the mini-channel, and at an angle of 105°.
• The HTCs obtained from the two-dimensional and simplified approaches were similar. The two-dimensional model gave higher HTC values than the one-dimensional model, and the differences do not exceed 0.3 kW/(m² K).
• The inclination angle of the mini-channel notably affects the boiling curve profile and the onset of nucleate boiling (ONB), mainly through gravity-driven influences on vapour bubble dynamics and liquid replenishment. At lower mini-channel inclination angles (e.g., ω = 60°), stable bubble detachment and efficient vapour removal produce a distinct ONB point. At ω = 90°, buoyancy aids vapour escape but introduces greater detachment fluctuations. Furthermore, at ω = 135°, the heated surface facing upward delays the release of the vapour and limits wall rewetting, leading to the agglomeration of the vapour and a smoother and more gradual transition to nucleate boiling, without a clearly defined ONB point.