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Numerical solution of the heat equation with non-linear, time derivative-dependent source term
(2010)
The mathematical modeling of heat conduction with adsorption effects in coated metal structures yields the heat equation with piecewise smooth coefficients and a new kind of source term. This term is special, because it is non-linear and furthermore depends on a time derivative. In our approach we reformulated this as a new problem for the usual heat equation, without source term but with a new non-linear coefficient. We gave an existence and uniqueness proof for the weak solution of the reformulated problem. To obtain a numerical solution, we developed a semi-implicit and a fully implicit finite volume method. We compared these two methods theoretically as well as numerically. Finally, as practical application, we simulated the heat conduction in coated aluminum fibers with adsorption in the zeolite coating. Copyright © 2010 John Wiley & Sons, Ltd.
Lightning safety guidelines
(2010)
Quantifying and minimizing uncertainty is vital for simulating technically and economically successful geothermal reservoirs. To this end, we apply a stochastic modelling sequence, a Monte Carlo study, based on (i) creating an ensemble of possible realizations of a reservoir model, (ii) forward simulation of fluid flow and heat transport, and (iii) constraining post-processing using observed state variables. To generate the ensemble, we use the stochastic algorithm of Sequential Gaussian Simulation and test its potential fitting rock properties, such as thermal conductivity and permeability, of a synthetic reference model and—performing a corresponding forward simulation—state variables such as temperature. The ensemble yields probability distributions of rock properties and state variables at any location inside the reservoir. In addition, we perform a constraining post-processing in order to minimize the uncertainty of the obtained distributions by conditioning the ensemble to observed state variables, in this case temperature. This constraining post-processing works particularly well on systems dominated by fluid flow. The stochastic modelling sequence is applied to a large, steady-state 3-D heat flow model of a reservoir in The Hague, Netherlands. The spatial thermal conductivity distribution is simulated stochastically based on available logging data. Errors of bottom-hole temperatures provide thresholds for the constraining technique performed afterwards. This reduce the temperature uncertainty for the proposed target location significantly from 25 to 12 K (full distribution width) in a depth of 2300 m. Assuming a Gaussian shape of the temperature distribution, the standard deviation is 1.8 K. To allow a more comprehensive approach to quantify uncertainty, we also implement the stochastic simulation of boundary conditions and demonstrate this for the basal specific heat flow in the reservoir of The Hague. As expected, this results in a larger distribution width and hence, a larger, but more realistic uncertainty estimate. However, applying the constraining post-processing the uncertainty is again reduced to the level of the post-processing without stochastic boundary simulation. Thus, constraining post-processing is a suitable tool for reducing uncertainty estimates by observed state variables.
Objective: As high-field cardiac MRI (CMR) becomes more widespread the propensity of ECG to interference from electromagnetic fields (EMF) and to magneto-hydrodynamic (MHD) effects increases and with it the motivation for a CMR triggering alternative. This study explores the suitability of acoustic cardiac triggering (ACT) for left ventricular (LV) function assessment in healthy subjects (n=14). Methods: Quantitative analysis of 2D CINE steady-state free precession (SSFP) images was conducted to compare ACT’s performance with vector ECG (VCG). Endocardial border sharpness (EBS) was examined paralleled by quantitative LV function assessment. Results: Unlike VCG, ACT provided signal traces free of interference from EMF or MHD effects. In the case of correct Rwave recognition, VCG-triggered 2D CINE SSFP was immune to cardiac motion effects—even at 3.0 T. However, VCG-triggered 2D SSFP CINE imaging was prone to cardiac motion and EBS degradation if R-wave misregistration occurred. ACT-triggered acquisitions yielded LV parameters (end-diastolic volume (EDV), endsystolic volume (ESV), stroke volume (SV), ejection fraction (EF) and left ventricular mass (LVM)) comparable with those derived fromVCG-triggered acquisitions (1.5 T: ESVVCG=(56± 17) ml, EDVVCG=(151±32)ml, LVMVCG=(97±27) g, SVVCG=(94± 19)ml, EFVCG=(63±5)% cf. ESVACT= (56±18) ml, EDVACT=(147±36) ml, LVMACT=(102±29) g, SVACT=(91± 22) ml, EFACT=(62±6)%; 3.0 T: ESVVCG=(55±21) ml, EDVVCG=(151±32) ml, LVMVCG=(101±27) g, SVVCG=(96±15) ml, EFVCG=(65±7)% cf. ESVACT=(54±20) ml, EDVACT=(146±35) ml, LVMACT= (101±30) g, SVACT=(92±17) ml, EFACT=(64±6)%). Conclusions: ACT’s intrinsic insensitivity to interference from electromagnetic fields renders
We analyze the influence of dipole-dipole interactions in an electromagnetically induced transparency set up for a density at the onset of cooperative effects. To this end, we include mean-field models for the influence of local-field corrections and radiation trapping into our calculation. We show both analytically and numerically that the polarization contribution to the local field strongly modulates the phase of a weak pulse. We give an intuitive explanation for this local-field-induced phase modulation and demonstrate that it distinctively differs from the nonlinear self-phase-modulation that a strong pulse experiences in a Kerr medium.