scholarly journals Determining 2D temperature field in flow boiling with the use of Trefftz functions

2019 ◽  
Vol 213 ◽  
pp. 02026 ◽  
Author(s):  
Sylwia Ho ejowska
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Problem ◽  
Flow Boiling ◽  
Temperature Fields ◽  
Glass Pane ◽  
Trefftz Method ◽  
Flowing Liquid ◽  
Mean Square Errors

The paper proposes the use of Trefftz method to solve the triple coupled heat conduction problem in flow boiling of refrigerant in an asymmetrically heated minichannel. A mathematical model of heat transfer in a rectangular minichannel is suggested. Two sets of Trefftz functions were used to determine 2D temperature fields at a fluid flow in the minichannel tilted at a known angle. The procedure for the calculation of the liquid temperature was coupled with the process of determining temperature fields in two adjacent elements of the experimental stand with the minichannel, i.e. in the glass pane and the heating foil. Heat transfer in the glass, foil and liquid is described using various 2D differential equations with an adequate set of boundary conditions. Solving those equations led to the solving of the triple coupled heat conduction problem made up of one direct and two subsequent inverse problems. The results are presented as: (1) 2D temperature of the glass pane, the heating foil, the flowing liquid, (2) mean square errors between temperature approximations and selected boundary conditions, (3) the heat transfer coefficient versus the distance from the minichannel inlet.

scholarly journals The application of Picard method in identification of the heat transfer coefficient in flow boiling in a minichannel

2018 ◽  
Vol 240 ◽  
pp. 01011
Author(s):  
Mirosław Grabowski ◽  
Sylwia Hożejowska ◽  
Anna Pawińska ◽  
Mieczysław E. Poniewski
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Temperature Distribution ◽  
Boundary Conditions ◽  
Transfer Coefficient ◽  
Flow Boiling ◽  
Image Processing Technique ◽  
Two Dimensional ◽  
Trefftz Method ◽  
Flowing Liquid

This paper summarizes the procedure of experimental determination of flow boiling heat transfer coefficient in the rectangular minichannel 2 mm wide, 1.8 mm deep and 193 mm long. In this experiment, the basic thermal and flow parameters were measured and the temperature distribution on the foil insulating the heater was recorded using an infrared camera. A high speed video camera observed the vapor patterns forming during the boiling process. The void fraction was determined with the image processing technique. Stationary heat transfer process in the insulating foil, the heater and the flowing liquid was described using two-dimensional Laplace’s equation (for the foil), the Poisson equation (for the heater) and energy equation (for the liquid) – all complemented with an appropriate system of boundary conditions. The system of the differential equations with adopted boundary conditions gives the three consecutive inverse problems in elements of the test section containing minichanel. Trefftz method was used to determine two dimensional temperature distribution of the foil and the heater. The liquid temperature was calculated using the Picard-Trefftz method. The values of the experimentally obtained heat transfer coefficients with the values of these coefficients calculated on the basis of correlations known from the literature were also compared.

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

2018 ◽  
Vol 180 ◽  
pp. 02099 ◽  
Author(s):  
Kinga Strąk ◽  
Beata Maciejewska ◽  
Magdalena Piasecka
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Boiling Heat Transfer ◽  
Transfer Problem ◽  
Heat Conduction Problem ◽  
Flow Boiling ◽  
Inverse Heat Conduction Problem ◽  
Heated Wall ◽  
Trefftz Method

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.

scholarly journals A spacetime collocation Trefftz method for solving the inverse heat conduction problem

2019 ◽  
Vol 11 (7) ◽  
pp. 168781401986127 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Wei-Po Huang ◽  
Yan Su
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Heat Conduction Problem ◽  
Domain Boundary ◽  
Inverse Heat Conduction Problem ◽  
Test Problems ◽  
Inverse Heat Conduction ◽  
Trefftz Method ◽  
Base Functions

In this article, a novel spacetime collocation Trefftz method for solving the inverse heat conduction problem is presented. This pioneering work is based on the spacetime collocation Trefftz method; the method operates by collocating the boundary points in the spacetime coordinate system. In the spacetime domain, the initial and boundary conditions are both regarded as boundary conditions on the spacetime domain boundary. We may therefore rewrite an initial value problem (such as a heat conduction problem) as a boundary value problem. Hence, the spacetime collocation Trefftz method is adopted to solve the inverse heat conduction problem by approximating numerical solutions using Trefftz base functions satisfying the governing equation. The validity of the proposed method is established for a number of test problems. We compared the accuracy of the proposed method with that of the Trefftz method based on exponential basis functions. Results demonstrate that the proposed method obtains highly accurate numerical solutions and that the boundary data on the inaccessible boundary can be recovered even if the accessible data are specified at only one-fourth of the overall spacetime boundary.

scholarly journals Heat Transfer Coefficient Identification in Mini-Channel Flow Boiling with the Hybrid Picard–Trefftz Method

Energies ◽  
2018 ◽  
Vol 11 (8) ◽  
pp. 2057 ◽  
Author(s):  
Mirosław Grabowski ◽  
Sylwia Hożejowska ◽  
Anna Pawińska ◽  
Mieczysław Poniewski ◽  
Jacek Wernik
Keyword(s):  
Heat Transfer ◽  
Test Section ◽  
Two Phase Flow ◽  
Heat Conduction Problem ◽  
Flow Boiling ◽  
Inverse Heat Conduction Problem ◽  
Phase Flow ◽  
Transfer Coefficients ◽  
Two Phase ◽  
Trefftz Method

This paper summarizes the results of the flow boiling heat transfer study with ethanol in a 1.8 mm deep and 2.0 mm wide horizontal, asymmetrically heated, rectangular mini-channel. The test section with the mini-channel was the main part of the experimental stand. One side of the mini-channel was closed with a transparent sight window allowing for the observation of two-phase flow structures with the use of a fast film camera. The other side of the channel was the foil insulated heater. The infrared camera recorded the 2D temperature distribution of the foil. The 2D temperature distributions in the elements of the test section with two-phase flow boiling were determined using (1) the Trefftz method and (2) the hybrid Picard–Trefftz method. These methods solved the triple inverse heat conduction problem in three consecutive elements of the test section, each with different physical properties. The values of the local heat transfer coefficients calculated on the basis of the Robin boundary condition were compared with the coefficients determined with the simplified approach, where the arrangement of elements in the test section was treated as a system of planar layers.

Characterization of the Heat Transfer Boundary Conditions during Cooling of a Horizontal Disk with a Water Column

2007 ◽  
Vol 539-543 ◽  
pp. 2479-2484
Author(s):  
Bernie Hernández-Morales ◽  
J.S. Téllez-Martínez ◽  
G. Sánchez-Sarmiento
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Water Column ◽  
Heat Conduction Problem ◽  
Radial Symmetry ◽  
Heat Extraction ◽  
Radial Coordinate ◽  
Thermal Boundary Conditions

To model the microstuctural and mechanical responses of quenched metallic components, the evolution of the thermal field must be known precisely; the latter, in turn, depends on accurate values of the thermal boundary conditions. In this work, the heat transfer boundary conditions on both sides of a stainless steel disk, held horizontally while a water column impinged on its lower surface to cool it from 850°C to room temperature, were characterized as heat flux histories which are functions of the radial coordinate. Thermal responses, measured with embedded thermocouples and a computer-controlled data acquisition system, were used to estimate the heat flux histories by solving the corresponding inverse heat conduction problem (IHCP), considering radial symmetry. The optimization problem also included the estimation of sub-areas associated with different heat extraction rates on both the lower and upper surfaces of the disk. The fluctuating interaction between the water column and the cooling disk was captured in the estimated heat flux histories. The estimated thermal boundary conditions were validated by computing the thermal response at the thermocouple locations by solving the direct heat conduction problem (DHCP) with a computer program based on the finite-element method. A good agreement between experimentally determined and computed thermal responses was observed, thus verifying the methodology employed.

Microprocessor Silicon Die Temperature Prediction by Simplified Boundary Conditions

2015 ◽  
Author(s):  
Koji Nishi ◽  
Tomoyuki Hatakeyama ◽  
Shinji Nakagawa ◽  
Masaru Ishizuka
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Thermal Resistance ◽  
Hot Spot ◽  
Three Dimensional ◽  
Thermal Design ◽  
Target Temperature ◽  
One Dimensional ◽  
Thermal Network

The thermal network method has a long history with thermal design of electronic equipment. In particular, a one-dimensional thermal network is useful to know the temperature and heat transfer rate along each heat transfer path. It also saves computation time and/or computation resources to obtain target temperature. However, unlike three-dimensional thermal simulation with fine pitch grids and a three-dimensional thermal network with sufficient numbers of nodes, a traditional one-dimensional thermal network cannot predict the temperature of a microprocessor silicon die hot spot with sufficient accuracy in a three-dimensional domain analysis. Therefore, this paper introduces a one-dimensional thermal network with average temperature nodes. Thermal resistance values need to be obtained to calculate target temperature in a thermal network. For this purpose, thermal resistance calculation methodology with simplified boundary conditions, which calculates thermal resistance values from an analytical solution, is also introduced in this paper. The effectiveness of the methodology is explored with a simple model of the microprocessor system. The calculated result by the methodology is compared to a three-dimensional heat conduction simulation result. It is found that the introduced technique matches the three-dimensional heat conduction simulation result well.

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