Influence of boundary conditions on the optimization of a geothermal heat pump studied using a thermodynamic model

This paper is the logical follow-up to a work [1] whose results were presented at the 28th French Thermal Congress which was to be held in Belfort in 2020. The model developed at that time is completed in this proposal to consider the specificity of the geothermal heat pump. This is a machine operating upon a mechanical vapor compression cycle, the limit of which is an inverse Carnot cycle. Its specificity consists of a cold loop at the source with the geothermal exchanger and the evaporator, then a hot loop at the sink with the condenser and a floor heat exchanger in the application considered here. We are particularly concerned with the optimal sizing of these heat exchangers through their effectiveness. The parametric sensitivity of this distribution to various boundary conditions is studied, especially by focusing on different conditions at the source: (1) imposed soil temperature, corresponding to a Dirichlet condition, (2) imposed heat flux (including adiabatic case), corresponding to a Neumann condition, (3) imposed mechanical power consumed by the heat pump, and (4) imposed coefficient of performance COP, to all cases being associated a finite thermal capacity in thermal contact with the geothermal exchanger operating in steady-state conditions.

Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal

The chemotaxis-Stokes system [Formula: see text] is considered subject to the boundary condition [Formula: see text] with [Formula: see text] and a given nonnegative function [Formula: see text]. In contrast to the well-studied case when the second requirement herein is replaced by a homogeneous Neumann boundary condition for [Formula: see text], the Dirichlet condition imposed here seems to destroy a natural energy-like property that has formed a core ingredient in the literature by providing comprehensive regularity features of the latter problem. This paper attempts to suitably cope with accordingly poor regularity information in order to nevertheless derive a statement on global existence within a generalized framework of solvability which involves appropriately mild requirements on regularity, but which maintains mass conservation in the first component as a key solution property.

Optimal Estimation of Thermal Diffusivity in a Thermal Energy Transfer Problem with Heat Generation, Convection Dissipation and Lateral Heat Flow

This work focuses on determining the coefficient of thermal diffusivity in a one-dimensional heat transfer process along a homogeneous and isotropic bar, embedded in a moving fluid with heat generation. A first type (Dirichlet) condition is imposed on one boundary and a third type (Robin) condition is considered at the other one. The parameter is estimated by minimizing the squared errors where noisy observations are numerically simulated at different positions and instants. The results are evaluated by means of the relative errors for different levels of noise. In order to enhance the estimation performance, an optimal design technique is chosen to select the most informative data. Finally, the improvement of the estimate is discussed when an optimal design is used.

The Meyers Estimates for Domains Perforated along the Boundary

In this paper, we consider an elliptic problem in a domain perforated along the boundary. By setting a homogeneous Dirichlet condition on the boundary of the cavities and a homogeneous Neumann condition on the outer boundary of the domain, we prove higher integrability of the gradient of the solution to the problem.

Application of the vertical model of the seasonally thawed layer for the Tulik Lake area (Alaska)

Практический интерес в районах вечной мерзлоты представляет глубина сезонного оттаивания. Построена одномерная (в вертикальном направлении) упрощенная полуэмпирическая модель динамики вечной мерзлоты в “приближении медленных движений границ фазового перехода”, основанная на задаче Стефана и эмпирических соотношениях. Калибровочные параметры модели выбираются для исследуемого района с использованием натурных измерений глубины оттаивания и температуры воздуха. Проверка работоспособности численной модели проведена для района оз. Тулик (Аляска). Получено согласие рассчитанных значений глубины талого слоя и температуры поверхности почвы с результатами измерений Due to the change in global air temperature, the assessment of permafrost reactions to climate change is of interest. As the climate warms, both the thickness of the thawed soil layer and the period for existence of the talik are increased. The present paper proposes a small-size numerical model of vertical temperature distributions in the thawed and frozen layers when a frozen layer on the soil surface is absent. In the vertical direction, thawed and frozen soils are separated. The theoretical description of the temperature field in soils when they freeze or melt is carried out using the solution of the Stefan problem. The mathematical model is based on thermal conductivity equations for the frozen and melted zones. At the interfacial boundary, the Dirichlet condition for temperature and the Stefan condition are set. The numerical methods for solving of Stefan problems are divided into two classes, namely, methods with explicit division of fronts and methods of end-to-end counting. In the present work, the method with the selection of fronts is implemented. In the one-dimensional Stefan problem, when transformed to new variables, the computational domain in the spatial variable is mapped onto the interval [0 , 1]. In the presented equations, the convective terms characterize the rate of temperature transfer (model 1). A simplified version of the Stefan problem solution is considered without taking into account this rate (“approximation of slow movements of the boundaries of the phase transition”, model 2). The model is tuned to a specific object of research. Model parameter values can vary significantly in different geographic regions. This paper simulates the dynamics of permafrost in the area of Lake Tulik (Alaska) in summer. Test calculations based on the proposed simplified model show its adequacy and consistency with field measurements. The developed model can be used for qualitative studies of the long-term dynamics of permafrost using data of the air temperature, relative air humidity and precipitation

The solution of arbitrary smoothness of the one-dimensional wave equation for the problem with mixed conditions

In this paper, we represented an analytical form of a classical solution of the wave equation in the class of continuously differentiable functions of arbitrary order with mixed boundary conditions in a quarter of the plane. The boundary of the area consists of two perpendicular half-lines. On one of them, the Cauchy conditions are specified. The second half-line is separated into two parts, namely, the limited segment and the remaining part in the form of a half-line. The Dirichlet condition is specified on the segment, as well as the Neumann condition is fulfilled on the second part in the form of a half-line. In a quarter of the plane, the classical solution of the problem under consideration is determined. To construct this solution, a particular solution of the original wave equation is established. For the given functions of the problem, the concordance conditions are written, which are necessary and sufficient for the solution of the problem to be classical of high order of smoothness and unique.

Homogenization and Correctors for Stochastic Hyperbolic Equations in Domains with Periodically Distributed Holes

The goal of this paper is to present new results on homogenization and correctors for stochastic linear hyperbolic equations in periodically perforated domains with homogeneous Neumann conditions on the holes. The main tools are the periodic unfolding method, energy estimates, probabilistic and deterministic compactness results. The findings of this paper are stochastic counterparts of the celebrated work [D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006) 467–496]. The convergence of the solution of the original problem to a homogenized problem with Dirichlet condition has been shown in suitable topologies. Homogenization and convergence of the associated energies results recover the work in [M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal. 97 (2016) 301–327]. In addition to that, we obtain corrector results.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

PurposeIn this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.Design/methodology/approachFrom the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.FindingsAs the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

The Hybrid FEM-DBCI for the Solution of Open-Boundary Low-Frequency Problems

This paper describes a particular use of the hybrid FEM-DBCI, for the computation of low-frequency electromagnetic fields in open-boundary domains. Once the unbounded free space enclosing the system has been truncated, the FEM is applied to the bounded domain thus obtained, assuming an unknown Dirichlet condition on the truncation boundary. An integral equation is used to express this boundary condition in which the integration surface is selected in the middle of the most external layer of finite elements, very close to the truncation boundary, so that the integral equation becomes quasi-singular. The method is described for the computation of electrostatic fields in 3D and of eddy currents in 2D, but it is also applicable to the solution of other kinds of electromagnetic problems. Comparisons are made with other methods, concluding that FEM-DBCI is competitive with the well-known FEM-BEM and coordinate transformations for what concerns accuracy and computing time.

A note on the exact boundary controllability for an imperfect transmission problem

In this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.