Asymptotic analysis of vertical geothermal boreholes in the limit of slowly varyng heat injection rates

Theoretical models for the thermal response of vertical geothermal boreholes often assume that the characteristic time of variation of the heat injection rate is much larger than the characteristic diffusion time across the borehole. In this case, heat transfer inside the borehole and in its immediate surroundings is quasi-steady in the first approximation, while unsteady effects enter only in the far field. Previous studies have exploited this disparity of time scales, incorporating approximate matching conditions to couple the near-borehole region with the outer unsteady temperatura field. In the present work matched asymptotic expansion techniques are used to analyze the heat transfer problem, delivering a rigorous derivation of the true matching condition between the two regions and of the correct definition of the network of thermal resistances that represents the quasi-steady solution near the borehole. Additionally, an apparent temperature due to the unsteady far field is identified that needs to be taken into account by the near-borehole region for the correct computation of the heat injection rate. This temperature differs from the usual mean borehole temperature employed in the literatura

Stable high-order finite-difference methods based on non-uniform grid point distributions

It is well known that high-order finite-difference methods may become unstable due to the presence of boundaries and the imposition of boundary conditions. For uniform grids, Gustafsson, Kreiss, and Sundstr¨om theory and the summation-by-parts method provide sufficient conditions for stability. For non-uniform grids, clustering of nodes close to the boundaries improves the stability of the resulting finite-difference operator. Several heuristic explanations exist for the goodness of the clustering, and attempts have been made to link it to the Runge phenomenon present in polynomial interpolations of high degree. By following the philosophy behind the Chebyshev polynomials, a non-uniform grid for piecewise polynomial interpolations of degree q_N is introduced in this paper, where N + 1 is the total number of grid nodes. It is shown that when q = N, this polynomial interpolation coincides with the Chebyshev interpolation, and the resulting finite-difference schemes are equivalent to Chebyshev collocation methods. Finally, test cases are run showing how stability and correct transient behaviours are achieved for any degree q<N through the use of the proposed non-uniform grids. Discussions are complemented by spectra and pseudospectra of the finite-difference operators

On the dynamics of flame edges in diffusion-flame/vortex interactions

We analyze the local flame extinction and reignition of a counterflow diffusion flame perturbed by a laminar vortex ring. Local flame extinction leads to the appearance of flame edges separating the burning and extinguished regions of the distorted mixing layer. The dynamics of these edges is modeled based on previous numerical results, with heat release effects fully taken into account, which provide the propagation velocity of triple and edge flames in terms of the upstream unperturbed value of the scalar dissipation. The temporal evolution of the mixing layer is determined using the classical mixture fraction approach, with both unsteady and curvature effects taken into account. Although variable density effects play an important role in exothermic reacting mixing layers, in this paper the description of the mixing layer is carried out using the constant density approximation, leading to a simplified analytical description of the flow field. The mathematical model reveals the relevant nondimensional parameters governing diffusion-flame/vortex interactions and provides the parameter range for the more relevant regime of local flame extinction followed by reignition via flame edges. Despite the simplicity of the model, the results show very good agreement with previously published experimental results

Analytical solution to the one-dimensional non-uniform absorption of solar radiation in uncoated and coated single glass panes

The analytical solution to the one-dimensional absorption–conduction heat transfer problem inside a single glass pane is presented, which correctly takes into account all the relevant physical phenomena: the appearance of multiple reflections, the spectral distribution of solar radiation, the spectral dependence of optical properties, the presence of possible coatings, the non-uniform nature of radiation absorption, and the diffusion of heat by conduction across the glass pane. Additionally to the well established and known direct absorptance αe, the derived solution introduces a new spectral quantity called direct absorptance moment βe, that indicates where in the glass pane is the absorption of radiation actually taking place. The theoretical and numerical comparison of the derived solution with existing approximate thermal models for the absorption–conduction problem reveals that the latter ones work best for low-absorbing uncoated single glass panes, something not necessarily fulfilled by modern glazings

Application of matched asymptotic expansion techniques to the analysis of geothermal heat exchangers

Most theoretical models for the thermal response of geothermal heat exchangers assume the mean azimuthal borehole wall temperature to be uniform along the boreholes. This simplifying assumption, closely related to the g-functions introduced by Eskilson in 1987, has dominated the research field for the past 30 years, allowing the analysis of large geothermal heat exchangers in reasonable amounts of time. The assumption, however, is not physically correct, which hinders the attainable accuracy. By using matched asymptotic expansion techniques, analytical models for the thermal response of geothermal heat exchangers are derived, which do not require the aforementioned simplification. The resulting expressions, applicable to geothermal heat exchangers with irregularly placed heterogeneous boreholes, show accuracy and flexibility levels comparable to SBM, but with a computational cost in line with the use of g-functions

Four Decades of Studying Global Linear Instability: Progress and Challenges

Global linear instability theory is concerned with the temporal or spatial development of small-amplitude perturbations superposed upon laminar steady or time-periodic three-dimensional flows, which are inhomogeneous in two(and periodic in one)or all three spatial directions.After a brief exposition of the theory,some recent advances are reported. First, results are presented on the implementation of a Jacobian-free Newton–Krylov time-stepping method into a standard finite-volume aerodynamic code to obtain global linear instability results in flows of industrial interest. Second, connections are sought between established and more-modern approaches for structure identification in flows, such as proper orthogonal decomposition and Koopman modes analysis (dynamic mode decomposition), and the possibility to connect solutions of the eigenvalue problem obtained by matrix formation or time-stepping with those delivered by dynamic mode decomposition, residual algorithm, and proper orthogonal decomposition analysis is highlighted in the laminar regime; turbulent and three-dimensional flows are identified as open areas for future research. Finally, a new stable very-high-order finite-difference method is implemented for the spatial discretization of the operators describing the spatial biglobal eigenvalue problem, parabolized stability equation three-dimensional analysis, and the triglobal eigenvalue problem; it is shown that, combined with sparse matrix treatment, all these problems may now be solved on standard desktop computer

Analysis of the Mechanical Properties of Multicrystalline and Monocrystalline Silicon Wafers Manufactured by Casting Methods

Quasi-monocrystalline silicon wafers have appeared as a critical innovation in the PV industry, joining the most favourable characteristics of the conventional substrates: the higher solar cell efficiencies of monocrystalline Czochralski-Si (Cz-Si) wafers and the lower cost and the full square-shape of the multicrystalline ones. However, the quasi-mono ingot growth can lead to a different defect structure than the typical Cz-Si process. Thus, the properties of the brand-new quasi-mono wafers, from a mechanical point of view, have been for the first time studied, comparing their strength with that of both Cz-Si mono and typical multicrystalline materials. The study has been carried out employing the four line bending test and simulating them by means of FE models. For the analysis, failure stresses were fitted to a three-parameter Weibull distribution. High mechanical strength was found in all the cases. The low quality quasi-mono wafers, interestingly, did not exhibit critical strength values for the PV industry, despite their noticeable density of extended defects

Comparison of Different Finite Element Models for the Transient Dynamic Analysis of Laminated Glass for Structural Applications

Commercial laminated glass is usually composed of two glass layers and an interlayer of PVB. The viscoelastic behaviour of the PVB layer has to be taken into account when dealing with dynamic loads. This paper shows different Finite Element (FE) models developed to characterize laminated glass. Results are contrasted with data from different reference test cases. First, a quasi-static test is reproduced with a 2d model through a transient analysis. In addition, flexural modes of vibration in a free-free test configuration have been analysed using 2d as well as 3d FE models. Apart from using transient analysis in order to simulate the dynamic behaviour of laminated glass, an iterative procedure has been employed which permits to identify the correct value of the shear modulus of the PVB layer for each mode in an eigenvalue analysis

Iron porphyrin molecules on Cu(001): Influence of adlayers and ligands on the magnetic properties

The structural and magnetic properties of Fe octaethylporphyrin (OEP) molecules on Cu(001) have been investigated by means of density functional theory (DFT) methods and X-ray absorption spectroscopy. The molecules have been adsorbed on the bare metal surface and on an oxygen-covered surface, which shows a $\sqrt{2}\times2\sqrt{2}R45^{\circ}$ reconstruction. In order to allow for a direct comparison between magnetic moments obtained from sum-rule analysis and DFT we calculate the dipolar term $7$, which is also important in view of the magnetic anisotropy of the molecule. The measured X-ray magnetic circular dichroism shows a strong dependence on the photon incidence angle, which we could relate to a huge value of $7$, e.g. on Cu(001) $7$ amounts to -2.07\,\mbo{} for normal incidence leading to a reduction of the effective spin moment $m_s + 7$. Calculations have also been performed to study the influence of possible ligands such as Cl and O atoms on the magnetic properties of the molecule and the interaction between molecule and surface, because the experimental spectra display a clear dependence on the ligand, which is used to stabilize the molecule in the gas phase. Both types of ligands weaken the hybridization between surface and porphyrin molecule and change the magnetic spin state of the molecule, but the changes in the X-ray absorption are clearly related to residual Cl ligands.Comment: 17 figures, full articl

Order 10 4 speedup in global linear instability analysis using matrix formation

A unified solution framework is presented for one-, two- or three-dimensional complex non-symmetric eigenvalue problems, respectively governing linear modal instability of incompressible fluid flows in rectangular domains having two, one or no homogeneous spatial directions. The solution algorithm is based on subspace iteration in which the spatial discretization matrix is formed, stored and inverted serially. Results delivered by spectral collocation based on the Chebyshev-Gauss-Lobatto (CGL) points and a suite of high-order finite-difference methods comprising the previously employed for this type of work Dispersion-Relation-Preserving (DRP) and Padé finite-difference schemes, as well as the Summationby- parts (SBP) and the new high-order finite-difference scheme of order q (FD-q) have been compared from the point of view of accuracy and efficiency in standard validation cases of temporal local and BiGlobal linear instability. The FD-q method has been found to significantly outperform all other finite difference schemes in solving classic linear local, BiGlobal, and TriGlobal eigenvalue problems, as regards both memory and CPU time requirements. Results shown in the present study disprove the paradigm that spectral methods are superior to finite difference methods in terms of computational cost, at equal accuracy, FD-q spatial discretization delivering a speedup of ð (10 4). Consequently, accurate solutions of the three-dimensional (TriGlobal) eigenvalue problems may be solved on typical desktop computers with modest computational effort