Boundary Values Of Temperature Research Articles

Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer

We study a dynamic boundary condition problem in heat transfer which represents the interaction between a conducting solid enclosed by a conducting shell. Both the solid and the shell are thermally inhomogeneous and anisotropic. Interaction is modelled by considering the solid as a source of thermal energy to the shell. A constitutive equation proposed by Carslaw and Jaeger establishes a relation between temperature in the shell and the boundary value of temperature in the solid. This gives rise to a dynamic boundary condition problem that has not been studied in the recent literature. The system of equations so obtained is presented as an implicit evolution equation which involves a pair of unbounded linear operators that map between two different spaces. We extend the operators to a jointly closed pair for which the implicit equation makes sense. The solution of the initial value problem is constructed by means of a holomorphic family of solution operators. The class of admissible initial states is surprisingly large.

Time-Fractional Heat Conduction in a Half-Line Domain due to Boundary Value of Temperature Varying Harmonically in Time

The Dirichlet problem for the time-fractional heat conduction equation in a half-line domain is studied with the boundary value of temperature varying harmonically in time. The Caputo fractional derivative is employed. The Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate are used. Different formulations of the considered problem for the classical heat conduction equation and for the wave equation describing ballistic heat conduction are discussed.

Analysis of heat conduction in a rarefied gas at rest with a temperature jump at the boundary by consistent-order extended thermodynamics

In the context of the first- and second-order theories of consistent-order extended thermodynamics, a systematic approach is established for analyzing the temperature jump at the boundary through studying one-dimensional stationary heat conduction in a rarefied gas at rest. Thereby an approach to the free boundary-value problem in general is explored. Boundary values of temperature are assumed to be in the form of power expansion with respect to the Knudsen number, based on which analytical expressions of the temperature jump aswell as entropy production at the boundary are derived explicitly. Dependencies of these two boundary quantities on both the Knudsen number and accommodation factor are also extensively discussed. The present analysis is expected to be the basis for the study of higher-order theories of consistent-order extended thermodynamics.

Freezing processes in porous media: Formation of ice lenses, swelling of the soil

When moist porous soil is freezing, a volume expansion is generally observed. The volume increase is mainly due to a water migration process from the base of the soil up to the freezing front, which separates the lower unfrozen part from the upper frozen one. The coupled heat-mass transfer process is accompanied, under particular conditions, to the formation of pure ice segregated layers. Conversely, if the freezing process is too fast or the overburden pressure acting on the column of soil is relevant, no macroscopic accumulation of ice is observed. It is generally accepted that a thin transition region (frozen fringe), where water and ice coexist in the porous space, separates the unfrozen from the frozen parts of the soil. The strong interest on ground freezing is motivated by at least two reasons: preventing frost damages produced on roads pavements, pipelines, or other structures, and predicting the effects of artificial-freezing techniques for tunneling or underground constructions. We are going to present a mathematical model of frost heave where water is driven through the porous space by the coupling of a pressure and a chemical gradient. In particular, the interest is focused on detecting which are the boundary values for temperature (or thermal flux) that determine the process of lens formation or frost penetration, once the properties of the soil are known.

FREEZING PROCESSES IN SATURATED SOILS

When a soil saturated with water is subject to freezing, a volume expansion can generally be observed. The increase in volume is due not only to the different densities of water and ice, but mainly to a water migration process that is transported from the base of the soil up to an intermediate region where the change of phase occurs. It is generally accepted that a transition region, called frozen fringe, where ice and water coexist in the porous space, separates the unfrozen from the frozen part of the soil. Immediately over the frozen fringe a pure segregated layer of ice (ice lens) can form. If the freezing process is too fast or the weight acting on the soil (overburden pressure) is relevant, the porous matrix does not separate and the shifting of the frozen fringe towards the base of the soil (frost penetration) is observed. Many mathematical models have been proposed for the frost heave process. The main features of the one-dimensional model we are considering are summarized in Sec. 1. In Ref. 9 the case of assignment of the boundary thermal fluxes (at the base and on top of the soil) is studied. In practical cases, sometime the boundary temperatures, instead of the thermal fluxes, can be registered or imposed: in this paper we will investigate the model in such case. The main purpose is to detect which are the boundary values for temperature that determine the process of lens formation or frost penetration, once the properties of the soil are known.

Some tests with a surface energy balance scheme, including a bulk parameterisation for vegetation, in a mesoscale model

The problem of generating realistic boundary values of temperature and humidity in mesoscale models is discussed as well as the importance of allowing for non-linear interactions between the state of the surface and the atmosphere. An energy balance scheme, based on a bulk parameterisation of vegetation and simple prognostic equations for temperature and humidity of the soil interface, is presented. It is implemented in a hydrostatic, higher order closure model for the meso-γ-scale. Some tentative results are presented and critically evaluated. The presented parameterisation is very simple in its description of the interaction between plant and atmosphere, but shows the importance of such effects. Regarded as a building block in a mesoscale model, however, it is concluded that the present scheme is rather complicated. It is concluded that, although the results presented are realistic and show the importance of variable soil and vegetation types, it is highly doubtful if the presented parameterisation is useful in a wider perspective. Some of the parameterisations are less well founded from a physical point of view, and even if it were not so, the modeller will be hard put to specify all specifiable parameters and initialize all variables for any arbitrary area and event. It is thus concluded that this problem may still have to be dealt with in a simpler fashion. The question is raised if it should not be possible to obtain comparatively good results, similar to those in this paper, by simpler means.

Fog formation in boundary value problems

AbstractFor a Lewis number one gas, the equations of energy and diffusion state that if local equilibrium exists in a fog region, fog may form in a continuous manner, but it may be vaporized only discontinuously at the boundary between fog and nonfog regions. The local rate of formation of fog is proportional to the square of the local gradient of temperature or concentration. A class of solutions to the transport equations has been obtained for problems in which no vaporization occurs at the fog boundary. Under these conditions, the temperature and concentration at the boundary of the fog region are both shown to be constant at values that depend only upon the initial and/or boundary values of temperature and concentration and upon the equilibrium curve.

Calculations of temperature regime and heat budget of sea ice in the central Arctic

The equation of heat conduction, including variable thermal conductivity and specific heat, an internal heat source diminishing with depth, and an advective term, is integrated numerically for sea ice of equilibrium thickness. The annual cycle of thickness (ablation-accretion) is imposed as an external parameter. The boundary values for temperature and the vertical distribution of ice salinity are taken from empirical data. The computed temperature field is in good agreement with observations. The thermal history of individual particles of ice, the relative effect of the internal heat source (penetrating solar radiation), heat storage, and the annual cycle of heat flux by conduction at various depths are described. The observed maximum of brine volume at 40- to 70-cm depth is explained as the combined effect of salinity profile and internal absorption of radiation. The requirement that heat flux in the ice plus the heat equivalent of surface ablation equal the heat flux in the atmospheric boundary layer is well met by Badgley's values of radiative and turbulent heat transfer. During the melting season, June 15 to August 20, the surface of the ice receives about 4.5 kcal/cm2 and loses, during the freezing season, August 21 to June 14, an only slightly greater amount of heat to the atmosphere. The annual sum of heat conduction at the base of the ice is 3.6 kcal/cm2. Of this, 2.0 kcal/cm2 originates from ice accretion and 1.6 kcal/cm2 is drawn from the ocean. The atmosphere over the central Arctic receives an annual total of 2.5 kcal/cm2, which is mainly the heat of fusion of exported ice.