Numerical Solution of Axisymmetric Inverse Heat Conduction Problem by the Trefftz Method

In this paper, the issue of flow boiling heat transfer in an annular minigap was discussed. The main aim of the paper was determining the boiling heat transfer coefficient at the HFE-649 fluid–heater contact during flow along an annular minigap. The essential element of the experimental stand was a test section vertically oriented with the minigap 2 mm wide. Thermocouples were used to measure the temperature of the heater and fluid at the inlet and the outlet to the minigap. The mathematical model assumed that the fluid flow was laminar and the steady–state heat transfer process was axisymmetric. The temperatures of the heated surface and of the flowing fluid were assumed to fulfill energy equations with adequate boundary conditions. The problem was solved by the Trefftz method. The local heat transfer coefficients at the fluid–test surface interface were calculated due to the third kind boundary condition at the saturated boiling. Graphs were used to illustrate: the measurement of the heater surface temperature, 2D temperature distributions in the pipe and fluid, and the heat transfer coefficient as a function of the distance from the minigap inlet. The measurement uncertainties and accuracy of the heat transfer coefficient determination were estimated.


Introduction
Stricter energy performance requirements combined with the continuing trend towards miniaturization have made thermal management a key issue in engineering research. As a result, increasingly advanced methods are employed to intensify heat transfer processes. Phase change heat transfer is known to raise the efficacy of the process. With the growth of micro-electromechanical systems (MEMS), the development of compact evaporators is becoming crucial. Compact heat exchangers of different constructions and minigap geometries can be used in many applications. The miniaturized systems are being progressively applied in the electronics, medical, automotive, aerospace, and spacecraft industries, as well as in the military sector, power, and nuclear industries and chemical engineering. The development of mini-and microgaps compact heat exchangers, miniature pumps, minicompressors, mini-and micro-turbines and heat pipes, residential air conditioning systems, refrigeration systems, and other thermal systems have become essentialin heat transfer applications. Some proposals for applications of compact heat exchangers with rectangular minichannels in engineering devices were presented in [1]. The prototype heat exchanger used in our research, as well as the heat exchangers with rectangular minichannels tested in our laboratory, are planned to be used for cooling elements of solar systems. A review and future applications for a new generation of high-temperature solar receivers, including well-established devices used in the market and designs tested in laboratories, were described in [2]. first approach, the measurement module was presented as a planar multilayer wall and, in the second approach, as a multilayer cylindrical wall. The values of the local heat transfer coefficients determined from two approaches were compared. The first results from an experiment with distilled water as the working fluid were shown. Both approaches gave similar results. The results from experiments with HFE-649 flowing in an annular minigap were illustrated in [9]. This gap of a 1 mm width was created between the metal pipe with an enhanced surface contacting fluid and the external glass pipe.
The local values of the heat transfer coefficient for stationary state conditions were calculated using one-dimensional methods in which the multilayer cylindrical wall was assumed to be planar. The results were presented as a function of the heat transfer coefficient along the minigap length and as boiling curves, prepared for selected values of mass flow rates and five types of the enhanced heated surfaces and a smooth one. The observations indicated that the highest local values of the heat transfer coefficient were obtained with using the enhanced surface produced by the electromachining process (spark erosion) at the saturated boiling region. The boiling curves generated for two distances from the minigap inlet have similar plots without a drop in the temperature of the heated surface characteristics for nucleation hysteresis. One of the key parameters affecting boiling heat transfer is development of a heated surface that can be produced by different technologies, as described in [1,[17][18][19]. Reference [10] described a two-dimensional mathematical model of heat transfer in flow boiling of HFE-7100 in a minigap. The Fourier transform and the Trefftz method were used for identification of temperature distributions of heated surfaces and, furthermore, the local heat transfer coefficient. It was underlined that its values obtained from the Fourier transform and from the Trefftz method were similar. This paper reports the results of research and discusses the mathematical method used in the calculations. The test stand includes a test section with a minigap with annular cross-section. The experiments were carried out in stationary conditions. The temperature of the metal surface near the heater was measured with thermocouples. The aim of the experiments was to collect data which, with the proposed mathematical model and computational method, enabled determination of local heat transfer coefficients at the interface between the heated surface and the boiling liquid flowing through a minigap. The specificity of the test section construction as well as the physics of the problem under consideration make it impossible to directly measure the required quantities (temperature of the heated wall and its gradient, fluid temperature) in order to determine the local heat transfer coefficients.
The proposed system of two energy equations in the heater and flowing fluid together with an adequate set of boundary conditions leads to the solution of the conjugated inverse and direct heat conduction problems [20]. Inverse heat conduction problems (IHCPs) like those presented, e.g., in [21][22][23][24][25][26][27], belong to ill-posed problems [28] and require effective and stable solution methods. The Trefftz method [29] complies with this requirement, even when not all boundary conditions are fully known [30][31][32][33][34][35]. In the Trefftz method, the unknown solution of a partial differential equation is approximated by a linear combination of the Trefftz functions (called T-functions) that satisfy the governing equation exactly. A novel idea, presented in this paper, was to apply the Trefftz method to the simultaneous determination of the two-dimensional temperature fields: in the heated surface and fluid flowing through the minigap. It is also worth noting that both the mathematical model and numerical calculations are based on real experimental data. Two sets of the T-functions (for the Laplace equation and the energy equation for fluid) were used to compute the temperature distributions in the heater and the boiling fluid. The computation yields a continuous function representing the heater temperature in the whole domain. It allows to determine its gradient on the boundary of the domain and, consequently, based on the third kind boundary condition, also the values of the local heat transfer coefficients at the fluid-heater contact.

Experimental Data
We would like to emphasize that the results presented in the next sections are based on the data collected from the research conducted on our own experimental stand. Providing calculations on the Energies 2020, 13, 705 4 of 14 basis of experimental data is essential for the reliable prediction of heat transfer in complicated heat transfer systems, in which the sizes of bubbles can be larger than the size of a channel.
The experimental stand comprises: the flow loop with a test section in which the working fluid circulates (HFE-649, 3M), the data and image acquisition system, and the power supply and control system ( Figure 1). The test section with an annular minigap vertically oriented, Figure 2, is the main element of the circulated flow loop.

Experimental Data
We would like to emphasize that the results presented in the next sections are based on the data collected from the research conducted on our own experimental stand. Providing calculations on the basis of experimental data is essential for the reliable prediction of heat transfer in complicated heat transfer systems,in which the sizes of bubbles can be larger than the size of a channel.
The experimental stand comprises: the flow loop with a test section in which the working fluid circulates (HFE-649, 3M), the data and image acquisition system, and the power supply and control system ( Figure 1). The test section with an annular minigap vertically oriented, Figure 2, is the main element of the circulated flow loop. Figure 1. The schematic diagram of main loops realized in the experimental stand: 1-a test section, 2-a gear pump, 3-a compensating tank, 4-a tube-type heat exchanger, 5-a filter, 6-a mass flow meter, 7-a deaerator, 8-a data acquisition station, 9-a pc computer, 10-an ammeter, 11-a voltmeter, 12-an inverter welder, and 13-a shunt. The gap of 2-mm-wide and 180-mm-long was formed between the external glass pipe and the enhanced surface of the copper pipe positioned along the same axis, similarly as to the cartridge heater located inside this pipe. The heater is powered by an autotransformer with adjusted current intensity. The fluid temperature and pressure are measured at the inlet and the outlet to the minigap. K-type thermocouples are used for temperature measurement. Thermocouples were also placed in the gap between the copper pipe and the heater in the thermal paste. These 18 thermocouples were fixed evenly, with a spacing of 10 mm in the flow line according to the pitch.
During the experimental series, there was a gradual increase in the electric power supplied to the cartridge heater at steady-state experiment conditions.

Mathematical Model
In the mathematical model, the test section was modeled as concentric cylindrical layers ( Figure  3) with stationary and temperature-independent thermal properties. Neglecting both the thin thermal conductive filler (~10 −4 m) and the thermocouple size, it was assumed that the surface of the copper pipe surface has direct contact with the heater surface. Furthermore, it was assumed that the heat Figure 1. The schematic diagram of main loops realized in the experimental stand: 1-a test section, 2-a gear pump, 3-a compensating tank, 4-a tube-type heat exchanger, 5-a filter, 6-a mass flow meter, 7-a deaerator, 8-a data acquisition station, 9-a pc computer, 10-an ammeter, 11-a voltmeter, 12-an inverter welder, and 13-a shunt.

Experimental Data
We would like to emphasize that the results presented in the next sections are based on the data collected from the research conducted on our own experimental stand. Providing calculations on the basis of experimental data is essential for the reliable prediction of heat transfer in complicated heat transfer systems,in which the sizes of bubbles can be larger than the size of a channel.
The experimental stand comprises: the flow loop with a test section in which the working fluid circulates (HFE-649, 3M), the data and image acquisition system, and the power supply and control system ( Figure 1). The test section with an annular minigap vertically oriented, Figure 2, is the main element of the circulated flow loop. Figure 1. The schematic diagram of main loops realized in the experimental stand: 1-a test section, 2-a gear pump, 3-a compensating tank, 4-a tube-type heat exchanger, 5-a filter, 6-a mass flow meter, 7-a deaerator, 8-a data acquisition station, 9-a pc computer, 10-an ammeter, 11-a voltmeter, 12-an inverter welder, and 13-a shunt. The gap of 2-mm-wide and 180-mm-long was formed between the external glass pipe and the enhanced surface of the copper pipe positioned along the same axis, similarly as to the cartridge heater located inside this pipe. The heater is powered by an autotransformer with adjusted current intensity. The fluid temperature and pressure are measured at the inlet and the outlet to the minigap. K-type thermocouples are used for temperature measurement. Thermocouples were also placed in the gap between the copper pipe and the heater in the thermal paste. These 18 thermocouples were fixed evenly, with a spacing of 10 mm in the flow line according to the pitch.
During the experimental series, there was a gradual increase in the electric power supplied to the cartridge heater at steady-state experiment conditions.

Mathematical Model
In the mathematical model, the test section was modeled as concentric cylindrical layers ( Figure  3) with stationary and temperature-independent thermal properties. Neglecting both the thin thermal conductive filler (~10 −4 m) and the thermocouple size, it was assumed that the surface of the copper pipe surface has direct contact with the heater surface. Furthermore, it was assumed that the heat The gap of 2-mm-wide and 180-mm-long was formed between the external glass pipe and the enhanced surface of the copper pipe positioned along the same axis, similarly as to the cartridge heater located inside this pipe. The heater is powered by an autotransformer with adjusted current intensity. The fluid temperature and pressure are measured at the inlet and the outlet to the minigap. K-type thermocouples are used for temperature measurement. Thermocouples were also placed in the gap between the copper pipe and the heater in the thermal paste. These 18 thermocouples were fixed evenly, with a spacing of 10 mm in the flow line according to the pitch.
During the experimental series, there was a gradual increase in the electric power supplied to the cartridge heater at steady-state experiment conditions.

Mathematical Model
In the mathematical model, the test section was modeled as concentric cylindrical layers (Figure 3) with stationary and temperature-independent thermal properties. Neglecting both the thin thermal conductive filler (~10 −4 m) and the thermocouple size, it was assumed that the surface of the copper pipe surface has direct contact with the heater surface. Furthermore, it was assumed that the heat transfer in the test section and the minigap is axisymmetric; i.e., the temperature distributions in the Energies 2020, 13, 705 5 of 14 elements of the module depend on only two variables: r, referring to the thickness of the cartridge heater (r 1 ) and the copper pipe (r 2 − r 1 ) and the width of the minigap (r 3 − r 2 ) and z (along the flow direction) [7,10].
transfer in the test section and the minigap is axisymmetric; i.e., the temperature distributions in the elements of the module depend on only two variables: r, referring to the thickness of the cartridge heater (r1) and the copper pipe (r2 − r1) and the width of the minigap (r3 − r2) and z (along the flow direction) [7,10].
The following additional assumptions considering the fluid flow were made: (i) the flow is in steady-state and laminar (Reynolds number < 2100) with only one nonzero velocity component vave parallel to the flow direction, (ii) the flow is nonadiabatic and has a constant cross-section,(iii) the temperature of the fluid at the inlet Tf,in and outlet Tf,out of the minigap is known, and (iv) the gravitational force is out of consideration.
and vave is equal to the average fluid velocity in the minigap.
Boundary conditions (3) and (4) were derived from the knowledge of measurements of the heating surface temperature and current I and voltage drop U  supplied to the heater: -for the copper pipe: -for the fluid: where a = k f c p, f ρ f and v ave is equal to the average fluid velocity in the minigap. Boundary conditions (3) and (4) were derived from the knowledge of measurements of the heating surface temperature and current I and voltage drop ∆U supplied to the heater: where the measurements of copper pipe temperature were approximated by a polynomial T approx (z).
Since the length of the pipe significantly exceeds its width, we can assume that its walls for z = 0 and z = L are isolated: For Equation (2), we assumed that the temperature of the fluid both at the minigap inlet (T f,in ) and outlet (T f,out ) is known, i.e., and for 0 ≤ z ≤ L at the fluid-copper pipe interface, the fluid temperature fulfills the condition In condition (8), the saturation temperature of the fluid T sat is a function of a pressure that changes linearly from the inlet to the outlet of the minigap.
Additionally, neglecting the heat exchange with the environment at the fluid-external glass pipe interfaces, it was assumed that the outer wall ofthe minigap was isolated: Knowledge of functions describing the temperature distribution of the copper pipe and flowing fluid allows to determine the local value of the heat transfer coefficient at the copper pipe-fluid contact from the third kind boundary condition: The reference fluid temperature T f ,ave is calculated as an average fluid temperature in the minigap, i.e.,

Numerical Method
The solution of the presented IHCP leads to solving three inverse problems of determining: (i) the temperature; (ii) the heat flux; and (iii) the thermal properties of a material (heat transfer coefficient, in this case). These are ill-posed problems which require effective and stable methods of solution. In this paper, the Trefftz method was applied to determine the two-dimensional temperature distributions, both of the copper pipe and the flowing fluid. The unknown temperature T cp was approximated by a linear combination of two types of the T-functions f n,0 (r,z) and g n (r,z) appropriate for Laplace Equation (1), i.e., The method for deriving T-functions for Equation (1) was described in detail in [10]. The recursive formulas of T-functions f n,0 (r,z) and g n (r,z),as well as their properties, are listed in Table 1. Table 1. Properties of T-functions f n,0 (r,z) and g n (r,z).

Functions f n,0 (r,z) Functions g n (r,z)
satisfy the Laplace Equation (1) recursive formulas for functions: f 0,0 = 1, f 1,0 = z, and for n ≥ 2 f n,0 = z 2n−1 n 2 f n−1,0 − (r 2 +z 2 ) n 2 f n−2,0 g 0 = lnr, g 1 = zlnr, and for n ≥ 2 g n = z 2n−1 n 2 g n−1 − r 2 +z 2 n 2 g n−2 + 2 n 2 r ∂ fn,0 ∂r recursive formulas for derivatives:  To calculate the temperature distribution of fluid T f flowing in the annular gap between two fixed concentric cylinders, the T-functions u n (r,z) for Equation (2) were used [7]. The T-functions u n (r,z) are the sums of harmonic polynomials f n,0 (r,z) for the Laplace Equation (1) and polynomials f n,i (r,z): The polynomials f n,i (r,z), i = 1, 2, ..., n, are defined by formula [7]: where the inverse operator ∆ −1 for a monomial in the form r n z k is determined as follows [7]: The fluid temperature T f was approximated in the form as a linear combination of the T-functions u n (r,z). The unknown coefficients a n , b n in (12) and c n in (16) are determined by minimizing the adequate functionals. For example, the functional allowing to determine coefficients c n has the form The approximate temperatures T cp and T f obtained by the Trefftz method are the functions which: (i) are continuous, (ii) satisfy Equations (1) and (2)  with the conditions (6)-(9) define the IHCP in the copper pipe and a direct heat conduction problem in the fluid, respectively.

Results and Discussion
Based on the model described in the previous section, the Trefftz method was adopted to determine a two-dimensional temperature field for the copper pipe and the flowing fluid. The calculations were performed for five heat flux densities q in the range of 9.24 ÷ 18.33 kW/m 2 . The main experimental parameters from the series taken into account for calculations were those listed in Table 2. Additionally, Table 2 includes the measurement uncertainties. The uncertainty of temperature measurement by K-type thermocouples was assumed, like in [11]. Other uncertainties were estimated according to the data from the manufacturer of the measuring devices.  The approximate temperatures Tcp and Tf obtained by the Trefftz method are the functions which: (i) are continuous, (ii) satisfy Equations (1) and (2) exactly, and (iii) satisfy the adopted boundary conditions approximately. Equation (1)

subject to the boundary conditions (3)-(5) and
Equation (2) with the conditions (6)-(9) define the IHCP in the copper pipe and a direct heat conduction problem in the fluid, respectively.

Results and Discussion
Based on the model described in the previous section, the Trefftz method was adopted to determine a two-dimensional temperature field for the copper pipe and the flowing fluid. The calculations were performed for five heat flux densities q   in the range of 9.24 ÷ 18.33 kW/m 2 . The main experimental parameters from the series taken into account for calculations were those listed in Table 2. Additionally, Table 2 includes the measurement uncertainties. The uncertainty of temperature measurement by K-type thermocouples was assumed, like in [11]. Other uncertainties were estimated according to the data from the manufacturer of the measuring devices.   In the first step, the temperature distribution of the copper pipe was determined using the Trefftz method utilizing 18 T-functions for Equation (1). Next,five T-functions defined for Equation (2) were used to calculate the fluid temperature. The number of T-functions employed were chosen so that the approximations of the copper pipe temperature Tcp and the fluid temperature Tf would be polynomials of the same degree. To verify the accuracy with which functions Tcp and Tf satisfy the adequate boundary conditions, the mean square error (MSE) defined as in [35] was used. The In the first step, the temperature distribution of the copper pipe was determined using the Trefftz method utilizing 18 T-functions for Equation (1). Next, five T-functions defined for Equation (2) were used to calculate the fluid temperature. The number of T-functions employed were chosen so that the approximations of the copper pipe temperature T cp and the fluid temperature T f would be polynomials of the same degree. To verify the accuracy with which functions T cp and T f satisfy the adequate boundary conditions, the mean square error (MSE) defined as in [35] was used. The temperature T cp and T f fulfill the boundary conditions with the MSEs, whose average values are listed in Table 3 (only representative results). The boundary conditions for which the MSEs have not been shown in Table 3 were satisfied with similar accuracy. Table 3. Average mean square errors (MSEs) for boundary conditions (6), (8), and (9).

Boundary Conditions
Average MSE condition (6) 0.75 K condition (8) 167.58 K condition (9) 2.5 × 10 −3 W/m 2 The 2D temperature distributions of the copper pipe and the flowing fluid are presented in Figure 5. To verify the results obtained from the proposed mathematical model, the local heat transfer coefficients were computed using a simplified approach where the measurement module was treated as a multilayer cylindrical wall and the heat transfer in the direction perpendicular to the flow was neglected [8]: To verify the results obtained from the proposed mathematical model, the local heat transfer coefficients were computed using a simplified approach where the measurement module was treated as a multilayer cylindrical wall and the heat transfer in the direction perpendicular to the flow was neglected [8]: Figure 6 compares the heat transfer coefficients (as a function of the distance from the minigap inlet) computed with the Trefftz method, Equation (10), and the simplified approach, Equation (18).  When analyzing the results presented in Figure 6, it can be noticed that the heat transfer coefficient decreased along the entire heater length of the minigap, which resulted in an increase in temperature in the heating surface (Figures 4 and 5). Increase of the vapor phase in the flow causes a decrease in values of the heat transfer coefficient. The heat transfer coefficient calculated from Formula (10) takes on slightly higher values than the one calculated from Formula (18). Larger differences between the values of heat transfer coefficients h1D and h2D are for lower values of heat fluxes. The maximum relative differences (MRD), determined similarly as in [11], between heat transfer coefficient values computed with Equations (10) and (18) range from to 12.8% to 36.0%, which indicates that the results obtained by both approaches are consistent.
The mean relative errors (MREs) were calculated for the local heat transfer coefficients, as in [11,31]. Table 4 shows the dependence of the MRE on the heat fluxes q   for one-and two-dimensional approaches.  When analyzing the results presented in Figure 6, it can be noticed that the heat transfer coefficient decreased along the entire heater length of the minigap, which resulted in an increase in temperature in the heating surface (Figures 4 and 5). Increase of the vapor phase in the flow causes a decrease in values of the heat transfer coefficient. The heat transfer coefficient calculated from Formula (10) takes on slightly higher values than the one calculated from Formula (18). Larger differences between the values of heat transfer coefficients h 1D and h 2D are for lower values of heat fluxes. The maximum relative differences (MRD), determined similarly as in [11], between heat transfer coefficient values computed with Equations (10) and (18) range from to 12.8% to 36.0%, which indicates that the results obtained by both approaches are consistent.
The mean relative errors (MREs) were calculated for the local heat transfer coefficients, as in [11,31]. Table 4 shows the dependence of the MRE on the heat fluxes q for one-and two-dimensional approaches. Based on the data shown in Table 4, it was observed that the MREs took lower values for the one-dimensional approach. Generally, the MRE decreased when the heat flux q supplied to the cartridge heater increased.

Conclusions
The results of flow boiling heat transfer in an annular minigap were presented. The working fluid circulated in a flow loop was HFE-649. The essential element of the experimental stand was a test section with a minigap of 2 mm width, vertically oriented. During the experimental series, heat flux supplied to the heater was increased. It was assumed the laminar fluid flow and the steady-state heat transfer process. Thermocouples were used to measure the temperature of the heater and fluid at the inlet and the outlet to the minigap.
The paper proposed a mathematical model of heat transfer in flow boiling in a minigap. The use of known harmonic functions and T-functions for energy equations with constant velocity allowed to compute two-dimensional temperature distributions in the copper pipe and flowing liquid and then, based on the third kind boundary condition, the heat transfer coefficient at the heater-fluid interface. The results were presented as two-dimensional fields of heater and flowing fluid temperatures and heat transfer coefficients versus the distance from the minigap inlet.
Based on the results from the experiments carried out and the numerical analysis, the following conclusions can be drawn:

•
The heat transfer coefficient values decreased along the entire heater length of the minigap in the analyzed saturated boiling region, which resulted from the increase of the vapor phase in the flow; • The heat transfer coefficient calculated from both mathematical approaches were similar, as evidenced by the MRD. Larger MRD values were obtained for lower values of the heat flux; • The uncertainty analysis indicated that MRE values determined from the two-dimensional approach were higher than MREs for the one-dimensional approach. MREs decreased with the increase of the heat flux, regardless of the method for determining the transfer coefficient; and • It should be noted that using the Trefftz method is convenient because it yields solutions thatdo satisfy the governing equation exactly. This method allows solving both direct and inverse heat conduction problems.
Further extended research on an improved experimental stand is planned. Having more results of wider ranges of thermal-flow parameters will allow for their reliable comparison with the correlations reported in the literature.

Conflicts of Interest:
The authors declare no conflict of interest.