Time-Dependent Heat Transfer Calculations with Trefftz and Picard Methods for Flow Boiling in a Mini-Channel Heat Sink

The intensification of heat transfer using two-phase boiling flow in mini-channels is widely used to dissipate the high heat fluxes in miniaturized electronic devices. However, the process itself is not fully recognized and still requires experimental studies and developing computation methods appropriate for them. The main aim of this work was the mathematical modeling of time-dependent heat transfer process in FC-72 flow boiling in a mini-channel heat sink with five parallel mini-channels of 1 mm depth. Channels have an asymmetrically heated wall while its outer temperature was measured by infrared thermography. The opposite wall of the mini-channels was transparent, helping to record flow patterns due to a high-speed digital camera. The objective of the numerical calculations was to determine the heat transfer coefficient on the wall-fluid contact surface from the Robin boundary condition. The problem was solved using methods based on the Trefftz-type functions. Three mathematical methods were applied in calculations: the FEM with Trefftz type basis functions, the Classical Trefftz Method, and the Hybrid Picard-Trefftz Method. The results were compared with the values of the heat transfer coefficient obtained from theoretical correlations from the literature.


Introduction
There has been an increased interest in mini-channel heat sinks for over 20 years. Heat sinks can be categorized into micro-channels, mini-channels, and conventional ones. According to Kandlikar classification, the channels of hydraulic diameter in the range from 200 µm to 3 mm are named as mini-channels. Change of phase during flow has drawn considerable interest in the engineering community due to the use of latent vaporization heat. The use of boiling phenomena is one of the ways of intensification of heat transfer. Nowadays, there are increasing interests on heat exchangers with mini-or micro-channels due to these advantages mentioned above, resulting in high process efficiency and compactness of technological solutions. Several recent applications of mini-channel heat sinks cover the computer and IT industry, miniature refrigeration systems, domestic micro ORC, and cooling of PEM fuel cells. It is worth mentioning that miniaturization has become a requirement in the design of energy and process systems. The results of the scale reduction of the channels for heat transfer have not been recognized yet; therefore, further experimental and theoretical studies are required.
State-of-the-art focused on heat transfer in small confined spaces was presented in [1]. Physical phenomena accompanying flow boiling, such as flow maldistributions and other instabilities of pressure and temperature, are often also the subject of researchers' concern [2][3][4]. It is worth mentioning that lots of work deal also with condensation heat transfer [5,6] or pool boiling [7][8][9]. Heat transfer during fluid flow in mini gaps is the main interest of various technological aspects such as seals [10,11]. A lot of previous this method for nth-order linear differential equations, but the most comprehensive form of the method was developed by Emile Picard [44]. The convergence analysis of the method was presented in [45]. The first researchers to use the iteration method were Clenshaw and Norton [46], who proposed combining Chebyshev series with Picard's iteration.
The present paper proposes a mathematical model of time-dependent heat transfer in flow boiling of the FC-72 refrigerant in a rectangular mini-channel with an asymmetrically heated wall. The formulated model concerns a laminar flow with void fraction not exceeding 10% and constitutes the starting point for a mathematical description of two-phase flow with higher values of void fraction. In the presented approach, it was assumed that the heater temperature distribution was described by the Fourier equation, whereas in the flowing fluid by the energy equation with a known profile of one component of the fluid velocity (the other velocity components were assumed equal to zero).
A novelty of this publication is the formulation of a time-dependent mathematical model of heat transfer in a mini-channel and solving it using three methods based on the Trefftz functions: (i) the FEM with the Trefftz-type basis functions (FEMT), (ii) the Classical Trefftz Method (CTM), and (iii) the Hybrid Picard-Trefftz Method (HPTM). Importantly, the calculations were performed using the experimental time-dependent data. Two types of T-functions were used in the numerical calculations: time-independent functions for the Laplace and energy equation [47] and time-dependent T-functions for the Fourier [27] and for the Fourier-Kirchhoff equations (Maciejewska and Piasecka, 2019). Depending on the experimental results, the fluid velocity, void fraction, and the negative heat sources were taken into account in both energy equations. The system of partial differential equations was solved subject to a set of initial and boundary conditions only partially known. In addition, for the validation of our own results, theoretical correlations from the literature were used, to which own experimental data were introduced.

Experimental Apparatus
The experimental apparatus is shown in Figure 1. During the experiment, Fluorinert FC-72 is recirculated in the flow loop. This loop consists of the following elements: a test section with mini-channels (1), a recirculating pump (2), a compensating tank (3), a heat exchanger (4), a filter (5), a flow meter (6) and a deaerator (7). The test section is depicted in Figure 2. It contains five mini-channels, each 1 mm deep, 6 mm wide, and 38 mm long. The main parts of the test section: a front cover and a channel body are made of aluminum alloy, the internal side walls of the channels are formed by Teflon plate. A plate 0.45 mm thick, made of Haynes-230 alloy, constitutes a common heated wall of the mini-channels. The outer plate surface temperature is recorded by an infrared camera. The temperature is measured from the outer surface of the The data and image acquisition system, designed to collect the measurement data, includes: an infrared camera (9) and a high-speed digital camera (10). Infrared thermography is applied for measuring of the temperature distribution on the mini-channel heated wall. Simultaneously, a high-speed camera helps to identify flow structures in mini-channels during fluid flow. The PC computer (8) and two data acquisition stations complete this system. The supply and control system contains a power supply unit.
The test section is depicted in Figure 2. It contains five mini-channels, each 1 mm deep, 6 mm wide, and 38 mm long. The main parts of the test section: a front cover and a channel body are made of aluminum alloy, the internal side walls of the channels are formed by Teflon plate. A plate 0.45 mm thick, made of Haynes-230 alloy, constitutes a common heated wall of the mini-channels. The outer plate surface temperature is recorded by an infrared camera. The temperature is measured from the outer surface of the test section, contacting ambient air. The fluid temperature and pressure are controlled at the inlet and outlet manifolds. During the experiment, the test section was placed at an angle of 75 degrees to the horizontal plane, as shown in Figure 2a.  The experimental apparatus: 1-test section, 2-gear pump, 3-compensating tank,  4-heat exchanger, 5-filter, 6-Coriolis mass flow meter, 7-deaerator, 8-PC computer,  9-infrared camera, 10-high-speed digital camera.
The test section is depicted in Figure 2. It contains five mini-channels, each 1 mm deep, 6 mm wide, and 38 mm long. The main parts of the test section: a front cover and a channel body are made of aluminum alloy, the internal side walls of the channels are formed by Teflon plate. A plate 0.45 mm thick, made of Haynes-230 alloy, constitutes a common heated wall of the mini-channels. The outer plate surface temperature is recorded by an infrared camera. The temperature is measured from the outer surface of the test section, contacting ambient air. The fluid temperature and pressure are controlled at the inlet and outlet manifolds. During the experiment, the test section was placed at an angle of 75 degrees to the horizontal plane, as shown in Figure 2a. The test section: (a) cross-section, (b) front view, 1-front cover, 2-mini-channel manifold, 3-glass plate, 4-mini-channel, 5-heated plate, 6-channel body.

Experimental Methodology
During the experiment, the subcooled fluid (Fluorinert FC-72) flows into the mini-channels inlet manifold. Then, the heating power supplied to the heated plate, being the common mini-channel wall, is increased. When it is observed the first bubble

Experimental Methodology
During the experiment, the subcooled fluid (Fluorinert FC-72) flows into the minichannels inlet manifold. Then, the heating power supplied to the heated plate, being the common mini-channel wall, is increased. When it is observed the first bubble departure, it is the evidence of the boiling initiation. Basic experimental thermal and flow parameters are recorded in the selected time interval: from 300 s to 500 s. The temperature of the outer side of the heated wall of the mini-channels is measured by an infrared camera (FLIR, model A655SC). Other experimental parameters are also collected. Simultaneously, two-phase flow images are captured by a high-speed digital camera (JAI, model SP-5000M-CXP2, Denmark/Japan). The visualizations are thoroughly analyzed to better understand the heat transfer process. Based on the saved images during the experiment, after data processing using own software, void fraction in 10 areas along the flow are determined.

Experimental Parameters and Errors
Basic experimental thermal and flow parameters for the selected time moments of the set as well as the maximum errors are listed in Table 1.

Raw Experimental Data
The raw data recorded during the experiment are as follows: the temperature distribution on the heated plate, inlet and outlet temperatures and pressures of the working fluid, supplied heat flux (calculated on the basis of current intensity and voltage drop across the heated plate), atmospheric pressure, and mass flow rate. The plate temperature data have to be reduced with the application of the solution to the inverse heat conduction problem. The sought are local values of the heat transfer coefficient in the heated wall-fluid contacting surface. Figure 3 presents the raw temperature data concerning the outer surface of the heated plate, obtained from infrared measurement versus time and the distance from the mini-channel inlet.   Table 1.

Void Fraction Determination
Images of two-phase patterns were recorded at a speed of 1 fps with resolution of 360 × 360 pixels. For further processing, only a part of the images corresponding to the central mini-channel were taken. It was necessary to load images to our specialist software environment based on LabVIEW functions. Each digital photo was loaded and analyzed as a binarized image. A set of pixels contained in ten areas evenly divided along each image was taken sequentially. To determine void fraction during the next stage of Figure 3. Temperature of the heated plate from infrared measurement versus time and distance from the mini-channel inlet; experimental parameters are listed in Table 1.

Void Fraction Determination
Images of two-phase patterns were recorded at a speed of 1 fps with resolution of 360 × 360 pixels. For further processing, only a part of the images corresponding to the central mini-channel were taken. It was necessary to load images to our specialist software environment based on LabVIEW functions. Each digital photo was loaded and analyzed as a binarized image. A set of pixels contained in ten areas evenly divided along each image was taken sequentially. To determine void fraction during the next stage of the program, the number of pixels corresponding to the gas phase were counted and divided by the total number of pixels. For the areas partially filled with vapor, the procedure of closing bubbles was used. It is worth mentioning that two-phase pattern images correspond to the images recorded by an infrared camera, which are taken for further processing and computation. A scheme showing the distribution of areas corresponding to two-phase flow patterns, assumed for void fraction calculation is illustrated in Figure 4. In Figure 5, void fraction obtained from the calculations according to the authors' own program, versus time and the distance from the mini-channel inlet is presented.        Table 1.

Basic Mathematical Model
The mathematical model of heat transfer in a mini-channel suggested in this paper combines and modifies two models: a time-independent model described in [49] and a time-dependent model from [14]. Similarly to [49], two dimensions were taken into account in the approach: one in the direction of the flow (x) and one in the perpendicular direction (y) referring to the thickness of the heated plate and the depth of the mini-channel. The heat transfer process in the heating wall was described using two-dimensional Fourier equation complemented with an appropriate system of boundary conditions [50]: where The boundary conditions took into account the time-dependent measurements of the heated plate temperature conducted using infrared thermography, i.e.,: Additionally, it was assumed that the remaining boundaries of the heated plate, having no contact with the fluid, are isolated.
Assumptions for the flow of the working fluid were similar to [49]: With these assumptions, the energy equation written for the liquid phase takes the form: where D M = (x, y) ∈ R 2 : 0 < x < L, δ H < y < δ H + δ M and the negative heat source was calculated from the formula [51]: and the liquid superheat ∆T(t) was calculated in analogy to [52]. The bubble diameter d b in subcooled flow boiling was calculated using Tolubinski and Kostanchuk correlation [53,54]: For Equation (3), it was assumed that the fluid temperature at the inlet T f,in and outlet T f,out of the mini-channel is known and additionally, the fluid temperature at the contact with the heater surface mainly depends on temperature of the plate, i.e.,: In further calculations, the boundary conditions have been transformed so that they can be used in the numerical method. This remark applies mainly to the CTM.

Solution Methods
The system of energy equations with a set of adopted initial and boundary conditions defines two inverse heat conduction problems (IHCPs) in the heated surface and the flowing fluid. Three methods using the Trefftz functions were applied to solve these inverse problems: the CTM, the HPTM, and the FEMT.

The Classical Trefftz Method (CTM)
In the CTM, the time-dependent heat conduction problems described by Equations (1) and (3) were reduced to quasi-time independent problems. Additionally, the heater and fluid temperature, T H,n (x,y) and T f,n (x,y), respectively, were calculated at discrete times without dividing the domains D H and D M into subdomains. This approach allows us to use T-functions (harmonic functions) for the Laplace equation [24,55] and the T-functions for the time-independent energy equation in the fluid in analogy to [47]. As in [47], the heater and fluid temperature were approximated as a sum of a particular solution and a linear combination of the T-functions. Determining the coefficients of a linear combination is described in detail in [24].

The Hybrid Picard-Trefftz Method (HPTM)
The HPTM was used only to determine the temperature distribution of the fluid and the heater temperature was calculated by CTM. The main idea of Picard's iterations is a decomposition of a nonlinear operator into a linear and nonlinear part. Equation (3) can be rewritten as: where N = − ρ f c p, f λ f ∂ ∂t + w x (y) ∂ ∂x denotes a differential operator. Next, Picard's iterations method can be used to solve Equation (9) and for k = 2,3,... it gives the following equations to be solved: with the initial equation (for k = 1) in the form: In each step, the solution is approximated by linear combination of the time-dependent Trefftz functions V j (x, y, t) and the particular solution T sol (x, y, t) of the Equation (10), i.e.,: The particular solution T sol (x, y, t) and (∇ 2 ) −1 is described in detail in [56].

The FEM with Trefftz Type Basis Functions (FEMT)
The numerical calculations by the FEMT were carried out in subintervals [(n − 1)∆t, n∆t], for n = 1, 2, . . . , N, where ∆t denotes the length of the time interval, like in [14]. A novelty in the presented work is the solution of the non-homogeneous Equation (3) with the source term given by Equation (4) and adopted to the FEMT calculations in the form: where void fraction ϕ EL is constant in an element.
In each element, the approximate fluid temperature was defined as follows: where u(x, y, t)-the particular solution of the non-homogeneous Equation (3)

Heat Transfer Coefficient Determination
Knowledge of the temperature distribution of both surfaces (the heated plate and the fluid) allows to determine the heat transfer coefficient from the formulas included in Table 2.   (18) The determination of the temperature distributions of the film and fluid, and the local values of the heat transfer coefficient led to the solution of three types of inverse problems in which: (i) the temperature, (ii) the heat flux, and (iii) the thermal properties of a material were being sought.

Results and Discussion
Numerical calculations were performed for the chosen time interval of the experiment set (from 300 s to 500 s) and the values of the basic experimental parameters are shown in Table 1. In the calculations presented below, the local values of the void fraction (see Figure 5) were approximated either by a step function (for the CTM and the FEMT) or by polynomials (for the HPTM).
The results are presented as the heat transfer coefficient:   Table 1.   Table 1.   Table 1.  Table 1.
When analyzing the heat transfer coefficient dependence versus distance from the mini-channel inlet, achieved for three time moments: 380 s, 420 s, and 460 s (Figures 7a,  8a and 9a), an increase in coefficient values with the distance is noticed. In these cases, a certain amount of vapor bubbles increasing with time (and supplied heat flux) is observed near the channel outlet (Figures 7b, 8b and 9b), resulting in higher void fraction     Table 1.  Table 1.  Table 1. Figure 11. Heat transfer coefficient versus time, the distance from the mini-channel inlet x = 0.0178 m; other experimental parameters are listed in Table 1.  Table 1.  Table 1.
As can be read from Figures 10-13, the heat transfer coefficient increased with time, whereas the highest values were reached at the outlet of the mini-channel (referring to the calculations using the CPM and HPTM). The results for the first selected distance from the mini-channel inlet (0.0114 m) showed a slight variation in the coefficient values ( Figure 10). Furthermore, it was observed that the coefficient was the lowest in comparison to the other presented data. Analyzing the data collected at a distance of 0.0306 m  Table 1.  Table 1.  Table 1.
As can be read from Figures 10-13, the heat transfer coefficient increased with time, whereas the highest values were reached at the outlet of the mini-channel (referring to the calculations using the CPM and HPTM). The results for the first selected distance from the mini-channel inlet (0.0114 m) showed a slight variation in the coefficient values ( Figure 10). Furthermore, it was observed that the coefficient was the lowest in comparison to the other presented data. Analyzing the data collected at a distance of 0.0306 m  Table 1.
When analyzing the heat transfer coefficient dependence versus distance from the mini-channel inlet, achieved for three time moments: 380 s, 420 s, and 460 s (Figures 7a, 8a  and 9a), an increase in coefficient values with the distance is noticed. In these cases, a certain amount of vapor bubbles increasing with time (and supplied heat flux) is observed near the channel outlet (Figures 7b, 8b and 9b), resulting in higher void fraction (up to 8.54%). The local heat transfer coefficients obtained for the first selected moment of time, i.e., 340 s ( Figure 6a), are the lowest in comparison with other results. It should be underlined that, although the temperature data indicate the occurrence of subcooled flow boiling, one can hardly see any bubbles (Figure 6b), and the maximum void fraction does not exceed 0.58%. The curves based on the CTM and the HPTM computations show the same trends, whereas the FEMT curves slightly differ from them, indicating a higher sensitivity to the experimental data including void fraction: reaching minimum in the middle of the channel, and the maximum of the coefficient values at the channel inlet and outlet. Furthermore, all plots of heat transfer coefficient appeared to be similar, which confirms the acceptable range of variation of the coefficient.
As can be read from Figures 10-13, the heat transfer coefficient increased with time, whereas the highest values were reached at the outlet of the mini-channel (referring to the calculations using the CPM and HPTM). The results for the first selected distance from the mini-channel inlet (0.0114 m) showed a slight variation in the coefficient values ( Figure 10). Furthermore, it was observed that the coefficient was the lowest in comparison to the other presented data. Analyzing the data collected at a distance of 0.0306 m from the inlet (Figure 13), it was noticed that the heat conduction coefficient obtained by the FEMT indicated a sudden maximum at the time moment approx. 380 s.   Table 1.   Table 1.
The data illustrated in Figures 14 and 15 concerning the calculation results with the help of the CTM and the HPTM, confirm an increase of the heat transfer coefficient with time and distance from the mini-channel inlet. The highest values of the coefficient were achieved near the channel outlet, for higher values of the heat flux (accompanying increasing distance from the inlet). Local values of the heat transfer coefficient due to the FEMT computations, collected close to the mini-channel inlet, achieve relative high values (see Figure 16) in comparison to the other results.  Table 1.
The data illustrated in Figures 14 and 15 concerning the calculation results with the help of the CTM and the HPTM, confirm an increase of the heat transfer coefficient with time and distance from the mini-channel inlet. The highest values of the coefficient were achieved near the channel outlet, for higher values of the heat flux (accompanying increasing distance from the inlet). Local values of the heat transfer coefficient due to the FEMT computations, collected close to the mini-channel inlet, achieve relative high values (see Figure 16) in comparison to the other results.
The heat transfer coefficient calculated by the CTM achieved the highest values compared to the results from the other two methods. Furthermore, the coefficient increases with increasing distance from the mini-channel inlet and with time ( Figure 14). Similar values of the heat transfer coefficient were obtained by the HPTM, which follows from inspection (see: Figures 6-16). They are additionally confirmed by the maximum relative differences (MRD) which will be discussed in Section 7. While the CTM and the HPTM yield comparable results, the differences between the coefficients from the FEMT computations and the aforementioned methods were higher. It was noticed that the FEMT method reacts faster to dynamic changes in measurements in comparison with the CTM and the HPTM. The growth of vapor phase in the channel (i.e., higher value of void fraction) caused a greater decrease of the heat transfer coefficient calculated by the FEMT than by the other two methods. Moreover, the FEMT looks to be more sensitive to local changes in experimental parameters. A sudden drop in the measured plate temperature, which occurred during the experimental set at the time moment of 400 s, can be observed. It caused an equally sudden jump of the heat transfer coefficient shown in Figure 13. Distinctive features of each calculation method are presented in Table 3.  Table 1.
The heat transfer coefficient calculated by the CTM achieved the highest values compared to the results from the other two methods. Furthermore, the coefficient increases with increasing distance from the mini-channel inlet and with time ( Figure 14). Similar values of the heat transfer coefficient were obtained by the HPTM, which follows from inspection (see: Figures 6-16). They are additionally confirmed by the maximum relative differences (MRD) which will be discussed in Section 7. While the CTM and the HPTM yield comparable results, the differences between the coefficients from the FEMT computations and the aforementioned methods were higher. It was noticed that the FEMT method reacts faster to dynamic changes in measurements in comparison with the CTM and the HPTM. The growth of vapor phase in the channel (i.e., higher value of void fraction) caused a greater decrease of the heat transfer coefficient calculated by the FEMT than by the other two methods. Moreover, the FEMT looks to be more sensitive to local changes in experimental parameters. A sudden drop in the measured plate temperature, which occurred during the experimental set at the time moment of 400 s, can be observed. It caused an equally sudden jump of the heat transfer coefficient shown in Figure 13. Distinctive features of each calculation method are presented in Table 3.  Table 1. Table 3. Distinctive features of the mathematical methods used in calculations.

Distinctive Features
Mathematical Method

CTM HPTM FEMT
Can be used to solve inverse problems + + + Can be used to solve problems described by a nonlinear differential equation -+ - The solution satisfies exactly the governing differential equation

Comparison of the Authors' Results with Theoretical Correlations
The data from the authors' own experimental results were used in calculations by the three mathematical methods described above and three theoretical correlations dedicated for small channels. Three correlations were chosen for comparison, namely those proposed by: (i) Lazarek and Black (1982) [57], (ii) Tran et al. (1996) [58], and (iii) Piasecka (2015) [59]. Short characteristics of these correlations are presented in Table 4. Table 4. Characteristics of selected theoretical heat transfer correlations concerning small channels.

Author/Authors Correlation Characteristics
Lazarek and Black (1982) [57] α TP = 30·Re 0.857 The heat transfer coefficients calculated by the presented mathematical methods as well as derived from the chosen theoretical correlations were compared with one another using the authors' experimental data, namely the data collected at the four time moments: 340 s, 380 s, 420 s, and 460 s. The results are illustrated in Figure 17 as heat transfer coefficient dependence in function of distance from the mini-channel inlet, calculated according to own methods and correlations from the literature. When analysing the results, it was found that the heat transfer coefficient values achieved due to the proposed mathematical methods turned out to be lower in comparison with most of the results computed from theoretical correlations. Moreover, it was noticed that the lowest values of the heat transfer coefficient as the results of FEMT application occurred at the following time moments of the experiment set: 380 s, 420 s, and 460 s, i.e., at higher values of the supplied heat flux.
To thoroughly analyze the results from experiments and calculations according to the proposed mathematical methods and selected theoretical correlations, the maximum relative difference (MRD) between two heat transfer coefficients at the fixed time moment was determined from the following formula: The results of calculations according to Equation (22) are listed in Table 5.
According to the data shown in Table 5, the best compliance was found using the correlation by Lazarek and Black, especially for the initial time moments of the experimental set. For the remaining correlations, MRDs take higher values and reach even 78%. The highest MRD for the results obtained by theoretical correlations was for the HPTM at t = 340 s. However, when the heat flux increases, the FEMT gives the results whose MRD achieves higher values than for other methods.
The MRDs between the results obtained by the CTM and the HPTM and the results obtained by the CTM and the FEMT take higher values. Except for the results obtained for t = 340 s, it was noticed that the MRD increases with heat flux (and the void fraction) for the CTM and the HPTM methods. However, the MRD does not exceed 31%, regardless of the method applied.  Table 1.
To thoroughly analyze the results from experiments and calculations according to the proposed mathematical methods and selected theoretical correlations, the maximum relative difference (MRD) between two heat transfer coefficients at the fixed time moment was determined from the following formula: The results of calculations according to Equation (22) are listed in Table 5.  Table 1.

Conclusions
This work focuses on the mathematical modeling of time-dependent heat transfer process during flow boiling in a mini-channel heat sink. The data from the experimental set conducted on the set-up with the test section comprising five mini-channels of 1 mm depth. During the experiment, the test section was inclined (75 • ) to the horizontal plane. An infrared camera recorded the temperature field on the outer heated surface of the mini-channel wall while a high-speed camera was monitoring two-phase flow patterns, simultaneously. Based on the saved images during the experiment, after their analyses using own software, void fraction in 10 areas along the flow was determined and used in further calculations.
Having proposed a mathematical model of the posed problem, the aim was to identify the heat transfer coefficient on the heated wall-fluid contact surface. In terms of mathematical computations, the coefficient which appears in the Robin boundary condition should be somehow evaluated. Three methods based on Trefftz functions, namely the FEM with Trefftz-type basis functions (FEMT), the Classical Trefftz Method (CTM), and the Hybrid Picard-Trefftz Method (HPTM), were used in the paper for numerical calculations. The methods are effective in solving the inverse heat transfer problems. The proposed methods allowed calculating two-dimensional time-dependent temperature distributions in two contacting areas: the heater plate and the flowing fluid. In particular, the HPTM method is useful in solving time-dependent nonlinear problems. The CTM and the HPTM methods do not require the use of an advanced mathematical apparatus. The FEMT can be employed to solve problems in the domains of very complicated shapes. In this method, the set of nodes in the domain can be adjusted to the temperature measurements for exactly fulfilling the boundary conditions in nodes. Compared with the other methods presented in this article, a smaller number of the Trefftz functions can be used to approximate the solution of the given problem. The numerical results obtained from the proposed mathematical model and three calculation methods were validated by comparing them to the results gained using theoretical correlations. The comparison between the heat transfer coefficients, calculated by the tested mathematical methods and achieved from three correlations selected from the literature, all based on the data from the authors' experiments, gave satisfactory results. The best compliance was found by using the correlation by Lazarek and Black. The analysis proved that the values of temperature and heat transfer coefficient were similar.
In the near future, the research will focus on: (i) adaptation of the mathematical model to heat transfer problems with higher void fraction and including different mini-channel inclinations to the horizontal plane, (ii) improvement of the proposed calculation methods, and (iii) comparing the results from the Trefftz functions to those obtained with the help of commercial software, like ANSYS CFX/Fluent and STAR-CCM+. It is expected that the results of experimental studies will widen the knowledge of heat transfer during flow boiling in mini-and micro-channel heat sinks within a wide range of thermal and flow parameters, for various geometries and spatial orientation of the channels. Such a wide experimental research combined with the application of numerical methods will certainly provide a big amount of data.
The confrontation of the mathematical model and advanced computational methods with the experiment will allow for a detailed understanding of heat transfer processes occurring during flow boiling in mini-channels. It is worth to underline that the flow boiling heat-transfer process during fluid flow in mini-channels may find practical application in modern miniaturization of devices used in electronics, transportation, nuclear technology, medicine, and other energy brands. Compact heat exchangers with minigaps are strongly needed as the elements of computer units, which require intensive cooling. The calculation results could be useful for designing cooling systems in mini-scale-as mini-channel heat exchangers. Weber number We = w l 2 ·d h ·ρ l σ l