Adam Fic

Solving Transient Nonlinear Heat Conduction Problems by Proper Orthogonal Decomposition and the Finite-Element Method

ABSTRACT A method of reducing the number of degrees of freedom and the overall computing time by combining proper orthogonal decomposition (POD) with the finite-element method (FEM) has been devised. The POD-FEM technique can be applied both to linear and nonlinear problems. At the first stage of the method a standard FEM time-stepping procedure is invoked. The temperature fields obtained for the first few time steps undergo statistical analysis, yielding an optimal set of globally defined trial and weighting functions for the Galerkin solution of the problem at hand. The resulting set of ordinary differential equations (ODEs) is of greatly reduced dimensionality when compared with the original FEM formulation. For linear problems, the set can be solved either analytically, resorting to the modal analysis technique, or by time stepping. In the case of nonlinear problems, only time stepping can be applied. The focus of this article is on the time-stepping approach, in which the generation of the FEM-POD matrices, requiring some additional matrix manipulations, can be embedded in the assembly of standard FEM matrices. The gain in execution times comes from the significantly shorter time of solution of the set of algebraic equations at each time step. Numerical results are presented for both linear and nonlinear problems. In the case of linear problems, the derived time-stepping technique is compared with the standard FEM and the modal analysis. For nonlinear problems the proposed POD-FEM approach is compared with the standard FEM. Good accuracy of the POD-FEM solver has been observed. Controlling the error introduced by the reduction of the degrees of freedom in POD is also discussed.

Thermal Analysis of Vertical Ground Exchangers of Heat Pumps

The heat rate absorbed from the ground by a vertical ground exchanger of a heat pump unit is considered. The aim is to investigate the time variation of this energy rate for a set of parameters. The analyzed set of alternatives encompasses arrangements of the exchanger tubes, values of the temperature of the heat carrier, thermal parameters of the ground, periodic operation of the compressor, and, when appropriate, different values of seepage velocity. To achieve this goal, the transient temperature distributions in the soil surrounding the ground exchanger are evaluated. The calculations are carried out using both PATRAN-THERMAL, a commercial finite volume code, and FEMCONV, an in-home FEM package. Characteristic features of the latter are discussed briefly along with some results of simulations for a ground exchanger with tube-in-tube (Field-type) elements. It is shown that in every case, the heat rate absorbed from the ground depends on the season and reaches the minimum value in the second part of winter. As expected, a strong influence of the arrangement of the exchanger tubes and the motion of moisture is observed. It is shown that if the prices of the electric energy are variable during a day, it may be profitable to operate an HP unit compressor in a periodic regime. The approximate values of the heat pump unit coefficient of performance, defined as the ratio of heat output of HP and compressor driving power, are evaluated. It is pointed out that this coefficient depends on the heat carrier temperature, and therefore this temperature may also be a subject of optimization calculations.

Reduction of the Dimensionality of Transient FEM Solutions Using Proper Orthogonal Decomposition

A method of reducing the number of degrees of freedom in FEM analysis has been devised. As in the case of FEM, Galerkin weighted residuals is based on weak formulation. The distinct feature of the method is the usage of a set of globally defined trial (and weighting) functions defined as a linear combination of the shape functions. The coefficients of this combination are evaluated employing the Proper Orthogonal Decomposition method ensuring optimal approximation properties of the trial functions. The resulting set of ODEs of much lower dimensionality than the standard FEM is then solved analytically using the modal analysis technique. In the included numerical examples, the number of unknowns is reduced by several orders of magnitude while the maximum error is of the order of 1%. The technique leads also to a significant reduction of the execution times. A method of controlling the error introduced by the POD is proposed.

Inverse Convection-Diffusion Problem of Estimating Boundary Velocity Based on Internal Temperature Measurements

Possibility and algorithm of identification of boundary velocity based on the measurements of internal temperature is presented and studied in the paper. Procedure for solving the steady state inverse convection-diffusion heat transfer problem with potential fluid flow in 2D was worked out. The presented procedure employs sensitivity coefficient technique and finite element method. Due to the nonlinearity, the iterative algorithm was used for solving the inverse problem. The direct and inverse procedures and their solutions are discussed in the paper. The sensitivity of the estimation with respect to changes of selected parameters is also investigated. The presented analysis is directed at heat transfer processes in the ground with groundwater flow.

A comparison of heterogenous and homogenous models of two-phase transonic compressible CO2 flow through a heat pump ejector

This paper presents mathematical model of a two-phase transonic flow occurring in a CO2 ejector which replaces a throttling valve typically used in heat pump systems. It combines functions of the expander and compressor and it recovers the expansion energy lost by a throttling valve in the classical heat pump cycle. Two modelling approaches were applied for this problem, namely a heterogenous and homogenous. In the heterogenous model an additional differential transport equation for the mass fraction of the gas phase is solved. The evaporation and condensation process in this model is described with use of the Rayleigh-Plesset equation. In the homogenous model, phases are traced based on the thermodynamic parameters. Hence the heterogenous model is capable to predict non-equilibrium conditions. Results obtained with both models were compared with the experimental measurements.

Coupling of conductive, convective and radiative heat transfer in Czochralski crystal growth process

Abstract This paper studies the conjugate problems of fluid flow and energy transport (involving conduction, convection and radiation heat transfer) within a material changing its phase. The analysis focuses on the Czochralski crystal growth process. The solidifying material is treated as a pure substance with constant material properties. The solution of the resulting 3-D, axisymmetric, non-linear problem is obtained iteratively using the commercial CFD package Fluent. The algorithm employed here treats each subdomain of the system separately, i.e. the liquid and solid phases of the solidified material, as well as the inertial gas surrounding both phases. Results of a test case shows the velocity field and temperature distribution within a simple system employed for the growth of a single silicon crystal.

3-D periodic CFD model of the heating system in a coke oven battery

Purpose – The purpose of this paper is to develop a 3-D fully transient numerical model of the heat and fluid flow associated with the chemical reactions that occur in the heating system of the coke oven battery. As a result, the model can be used to provide data for the control system of the battery to reduce energy consumption and emissions and to obtain a product of the desired quality. Design/methodology/approach – In the proposed model, an accurate representation of the heating flue geometry, the volumetric heat sources as a result of the coke oven gas combustion, the temperature- and mole fraction-dependent properties of the gases were taken into account. The most important part of the model was the unsteady boundary condition definition that allowed the modeling of the periodic heat delivery to the two oven heating walls, both in the coking and the reversion cycles. Findings – The temperatures obtained using the computational fluid dynamics (CFD) model showed the same pattern of temperature variati...

Analysis of fluid flow and energy transport in Czochralski's process

Abstract In this paper the conjugate fluid flow and energy transport problem (involving conduction–convection–radiation heat transfer) resulting from the Czochralski crystal growth process is analysed. The solidifying material is treated as a pure and semitransparent substance with material properties depending neither on temperature nor on the wavelength. The solution of the problem is obtained iteratively using two computer codes: FLUENT, a commercial CFD package, and BEM-based in-house code capable of analysing the radiative heat transfer in the entire computational domain. Obtained results not only show velocity field and temperature distribution within the bodies under consideration but also demonstrate the influence of thermal radiation on these quantities.

CFD simulations of transport phenomena during transcritical flow of real fluid (CO2) within ejector

Purpose – The purpose of this paper is to overview successful approaches to the computational simulation of real fluid (R744 – carbon dioxide (CO2)) flow within an ejector is presented. Important issues such as the ejector geometry and its optimisation, the adapted equations of state and the proposed models of the process, fluid parameters, etc., are examined and critically discussed. Whenever possible, the discussed models are experimentally validated. In the conclusion, some trends in future research are pointed out. Design/methodology/approach – Flow within CO2 ejector is generally transcritical and compressible. Models existing in the literature are shortly described and critically compared. Whenever possible, those models were validated against the experimental data. In a model validation process, the primary and secondary mass flow rates as well as the pressures at the selected points in the mixing section and diffuser were compared, showing a satisfactory agreement between experimental and computat...

Mathematical Modelling of Two‐Phase Transonic Compressible CO2 Flow Through Heat Pump Ejector

In this paper mathematical modelling of a two‐phase flow occurring in an ejector which replaces a throttling valve typically used in a heat pump cycle is numerically considered. The geometry of the model is three dimensional taking one cross‐sectional symmetry plane into account. In such model different discretizations were generated to study an influence of the rapidly changing quantities on the system operation. The density variations are defined using real gas model i.e. Redlich‐Kwong for CO2 and REFPROP databases were employed. Both models produced similar character of the results. In addition, it was observed that the effects of wall roughness value defined as in planned experiments in such high speed systems can be neglected.