The heat transfer coefficient determination with the use of the Beck-Trefftz method in flow boiling in a minichannel

In this paper, the solution of the two-dimensional inverse heat transfer problem with the use of the Beck method coupled with the Trefftz method is proposed. This method was applied for solving an inverse heat conduction problem. The aim of the calculation was to determine the boiling heat transfer coefficient on the basis of temperature measurements taken by infrared thermography. The experimental data of flow boiling heat transfer in a single vertical minichannel of 1.7 mm depth, heated asymmetrically, were used in calculations. The heating element for two refrigerants (FC-72 and HFE-7100, 3M) flowing in the minichannel was the plate enhanced on the side contacting with the fluid. The analysis of the results was performed on the basis of experimental series obtained for the same heat flux and two different mass flow velocities. The results were presented as infrared thermographs, heated wall temperature and heat transfer coefficient as a function of the distance from the minichannel inlet. The results was discussed for the subcooled and saturated boiling regions separately.


Introduction
Boiling is an effective heat transfer process, which due to the phase change provides high heat transfer capacity. Flow boiling heat transfer is used in many applications utilizing the heat-removal processes. Variations in the process intensity may be, however, a limitation to the efficiency of boiling heat transfer, characterized by heat transfer coefficient values and can be estimated due to the heat transfer coefficient values.
Inverse heat conduction problems [1,2], such as heat transfer coefficient identification, are solved with a number of numerical methods. One of these methods is the Beck method [3 ,4], which introduces sensitivity coefficients as derivatives of the measured quantity with respect to the identified quantity and transforms an inverse problem into several direct problems. In this paper, the Trefftz method [5] was used to solve direct problems by approximating the unknown solution to a differential equation with a linear combination of functions strictly satisfying the governing differential equation. The Trefftz method was explored in more detail in [6][7][8][9][10][11][12][13][14].

Experimental setup and methodology
The major elements of the experimental stand are presented in Fig. 1a. The essential part of the stand is the test section (Fig. 1b) with a single rectangular minichannel, 1.7 mm in depth, 16 mm in width and 180 mm in length. The heated plate (12) made of Haynes-230 alloy, 0.45 mm thick, was enhanced by vibration-assisted laser surface on the side contacting refrigerant.
It was possible to measure temperature on the outer side of the plate due to infrared camera ( Fig. 1, 2). The E60 FLIR infrared camera (2) had an accuracy of ± 1 °C or ± 1% in the temperature range 0 ÷ 120 °C [23]. This smooth plate surface was coated with a black paint of known emissivity (of 0.83) [24]. The enhanced surface of the heated plate was observed through a glass panel (14). The flow patterns were recorded using a quick shot camera (3). K-type thermocouples (17) and pressure sensors measured temperature and pressure at the inlet and outlet of the minichannel, respectively.
During experiment, there was a laminar flow of refrigerant along a minichannel and a gradual increase in the electric power supplied to the plate followed by an increase in the heat flux transferred to the fluid. Experimental parameters for several settings of the heat flux supplied to the heated plate during subcooled and saturated boiling region were collected. The experimental parameters are presented in Table 1 and physical properties of refrigerants in Table 2.

3
Heat transfer coefficient determination

Main assumptions
It was assumed that the heat flow in the minichannel was stationary and two-dimensional. The variation in temperature along the channel width was neglected.
Local values of the heat transfer coefficient were calculated from Newton's law: where: x -the flow direction, y -the direction perpendicular to the flow direction, related to the thickness of the plate, q -density of the heat flux transferred from the plate to the fluid, T -plate temperature,  -thickness of the plate, T f -temperature of the fluid, in the subcooled boiling region it equals to T l (x) and in the saturated nucleate boiling region it equals to T sat (x), T l (x) -liquid temperature calculated from the assumption of the linear distribution of the liquid temperature along the channel and T sat -saturation temperature determined on the basis of the linear distribution of the fluid pressure along the channel. The heat flux density q was calculated from Fourier's law.

Methodology of calculations, boundary conditions
The plate temperature T was determined by solving the inverse heat conduction problem in the heated plate: , and the boundary conditions (see Fig. 2): where: I  current supplied to the heated plate, The inverse problem (Eqs. 2-6) was solved using the Beck method.

The Beck method
The Substituting Eq. (7) into Eq. (2) leads to 1+K direct problems which were solved using the Trefftz method [8].
The values of m T for K m  , 2 , 1  in expression Eq. (7) were calculated by minimizing the following functional:

Results and discussion
The study focused on heat transfer coefficient identification in the regions of subcooled and saturated boiling. The coefficient values were determined by solving the inverse heat conduction problem using the Beck method in combination with the Trefftz method.  Table 1. Fig. 4. Plate temperature vs. distance from the minichannel length obtained for the subcooled boiling region, experimental parameters shown in Table 1.  Table 1.
The results are presented graphically as : -infrared thermograms (Fig. 3a) and two-phase flow structures images (Fig. 3b), for q w = 120 kW•m -2 , two mass flow velocities, working fluids: HFE-7100 and FC-72, the saturated boiling region; -the temperature of the outer heated plate surface determined through infrared camera measurements, for all anayzed settings of the heat flux versus the distance from the minichannel inlet, obtained for the subcooled boiling region (Fig. 4a-d) and the saturated boiling region (Fig. 5a,b); -the heat transfer coefficient for all anayzed settings of the heat flux versus the distance from the minichannel inlet obtained for the subcooled boiling region (Fig. 6a-d) and the saturated boiling region, Fig.(7a,b). The highest temperatures of the heated plate surface were recorded for HFE-7100 at mass flow velocity of Q m = 12•10 -3 kg•s -1 in the subcooled and saturated boiling regions (except for the inlet part of the channel at subcooled boiling region). Lower temperatures were recorded for FC-72 at mass flow velocity of Q m = 18•10 -3 kg•s -1 (except for the inlet part of the channel).
For HFE-7100, at the subcooled boiling region, the temperature rose uniformly with increasing distance from the minichannel inlet, whereas for FC-72, the temperature changes were minor.  Table 1.  Table 1.
The heated plate temperatures recorded for HFE-7100 at the saturated boiling region were higher (irrespective of the mass flow velocity) than those provided by FC-72. At this region, no substantial plate temperature increase was observed.
In the subcooled boiling region, heat transfer coefficients were in the range 1.5 to 7.0 kW•m -2 •K -1 , that is, much lower than those obtained for the saturated boiling region. The lowest coefficients for both liquids were at the inlet to the minichannel, and the highest -at the outlet. The heat transfer coefficient calculated when HFE-7100 was used, started rising with the increasing distance from the minichannel inlet, reaching the highest values at a 2/3 length of the channel from the inlet. In the saturated boiling region, the heat transfer coefficient reached substantially higher values compared to the subcooled boiling region (similarly as observed in [26][27][28]) with the highest being from 10 kW•m -2 K -1 to 120 kW•m -2 K -1 . No data were collected for HFE-7100 for Q m = 18•10 -3 kg•s -1 at heat fluxes of q w = 120 kW•m -2 and q w = 125 kW•m -2 for both fluids for Q m = 18•10 -3 kg•s -1 in the central part of the minichannel (Fig. 7a-b). It was assumed that the measurement error, T p -T sat , was about 0.5 K, and the values T p -T sat < 1 K were not considered in the calculations. At the saturated boiling of FC-72 and at higher mass flow velocity, the heat transfer coefficient reached the lowest values of all tested liquid and mass flow velocities up to 65 kW•m -2 •K -1 . At lower mass flow velocity for both liquids, the coefficient had much higher values (2-3 times higher) up to about 120 kW•m -2 •K -1 . At the given mass flow velocity, heat transfer coefficients provided by HFE-7100 near the minichannel inlet were higher compared to FC-72.

Conclusion
In this work, the solution of the two-dimensional inverse heat transfer problem with the use of the Beck method coupled with the Trefftz method was proposed. The experimental data of flow boiling heat transfer in a single vertical minichannel of 1.7 mm depth, heated asymmetrically, were used in calculations. The heating element for two refrigerants (FC-72 and HFE-7100) flowing in the minichannel was the plate enhanced on the side contacting with the fluid. The results were presented as infrared thermographs, heated wall temperature and heat transfer coefficient as a function of the distance from the channel inlet.
The highest temperatures of the heated plate surface were recorded for HFE-7100 at mass flow velocity of Q m = 12•10 -3 kg•s -1 in the subcooled and saturated boiling regions. Lower temperatures were recorded for FC-72 at mass flow velocity of Q m = 18•10 -3 kg•s -1 .
In the subcooled boiling region, local heat transfer coefficients were in the range 1.5 to 7.0 kW•m -2 K -1 . The lowest values of the coefficients were obtained with HFE-7100 at mass flow velocity of Q m = 18•10 -3 kg•s -1 at all the heat fluxes were recorded along the same outlet section of the minichannel.
In the saturated boiling region, the heat transfer coefficient reached substantially higher values compared to the subcooled boiling region, with the highest being from 10 to 120 kW•m -2 K -1 . For FC-72 and at higher mass flow velocity, the heat transfer coefficient reached the lowest values of all tested refrigerants and mass flow velocities.