Mapping Frost-Sensitive Areas with a Three-Dimensional Local-Scale Numerical Model. Part I. Physical and Numerical Aspects

Numerical models of heat and moisture diffusion in the soil-vegetation-atmosphere continuum are linked through the moisture flux from the surface to the atmosphere. This mass flux represents a heat exchange as latent heat flux, coupling water, and energy balance equations. In this paper, a new approach for estimating key parameters governing moisture and heat diffusion equation and the closure function which links these equations, is introduced. Parameters of the system are estimated by developing objective functions that link atmospheric forcing, surface states, and unknown parameters. This approach is based on conditional averaging of heat and moisture diffusion equations on land surface temperature and moisture states, respectively. A single objective function is expressed that measures moisture and temperature-dependent errors solely in terms of observed forcings and surface states. This objective function is minimized with respect to the parameters to identify evaporation and drainage models and estimate water and energy balance flux components. The approach is calibration free (surface flux observations are not required), it is not hampered by missing data and does not require continuous records. Uncertainty of parameter estimates is obtained from the inverse of Hessian of the objective function, which is an approximation of the error covariance matrix. Uncertainty analysis and analysis of the covariance approximation, guides the formulation of a well-posed estimation problem. Accuracy of this method is examined through its application over three different field sites. This approach can be applied to diverse climates and land surface conditions with different spatial scales, using remotely sensed measurements.

<abstract> <bold><sc>Abstract.</sc></bold> Soil solarization is a method of non-chemical disease control that employs solar radiation to heat the soil to temperatures that are lethal for soilborne pathogens. Application of this technique in closed greenhouses has allowed it to spread in areas characterized by temperate climate, such as the Mediterranean basin. Optimization of the technique requires knowledge of the physical processes inside the soil-mulch-greenhouse system that determine the thermal regimes in the soil. In this work, a one-dimensional physical model is developed for simulation of the temperature and moisture content in mulched soil during solarization treatment in a closed greenhouse. The model also calculates the air temperature and relative humidity inside the greenhouse. The model includes heat transfer in the soil by both apparent conduction (which incorporates transfer of latent heat by vapor movement under the influence of a temperature gradient) and enthalpy transport by liquid and vapor flow. The temperature and water content in the soil are calculated using coupled partial differential equations of heat and moisture diffusion. Furthermore, in the calculations of radiative fluxes, infinite reflections are considered. The input variables required are outside air temperature and relative humidity, outside solar radiation flux, wind velocity, soil texture and bulk density, and radiometric properties of the soil as well as of the covering and mulching films. The model is validated using data collected during a soil solarization treatment carried out in a full-scale commercial greenhouse. The results show good agreement between measured and modeled data, especially concerning soil temperature. A sensitivity analysis is also carried out in order to investigate soil temperature changes induced by variations in the radiometric properties of the films. The results show that the highest temperature changes in the soil are determined by the shortwave transmittance of the cover.

Solutions of heat or diffusion equations with the boundary conditions which is a dynamic random field are discussed. This kind of method can be used to obtain the description of heat equations or diffusion equations based on observed physical reality, ie ordinary differential equations, representing heat or diffusion propagation, with a boundary condition that satisfies stochastic differential equations. The heat or diffusion equations obtained from the method are the compared to the heat equation or the stochastic diffusion. The comparison is emphasized on the existence and properties of Green functions.

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

Development of a numerical model for the evaluation of the urban thermal environment

A three-dimensional numerical model was developed to predict the microclimate near the ground surface of local-scale domains during radiative frost events. Its performances are compared with an observational topo-climatological survey of minimum temperatures at a height of 0.5 m above the soil surface which was carried out, during radiative float events, in the Hefer Valley, Israel. Considering only topography and soil type in the numerical simulation, relatively good agreement is obtained between predicted and observed minimum temperature. A more realistic picture is given when vegetation is incorporated in the model although larger discrepancies with observations are obtained. This is mainly explained by the fact that measurements were always carried out above bare surfaces, even when dense vegetation was present and, therefore do not provide a representative minimum temperature of many areas. This assumption is validated by field measurements of nighttime temperatures in an orchard and above a bare soil in its immediate vicinity.

The transfer coefficients of a laminar boundary layer with variable fluid properties

Optimized electrothermal design of integrated devices through the solution to the non-linear 3-D heat flow equation

Based on data collected over five years of monitoring the Lower Tarim River, we analyzed the variability of soil moisture content (SMC) and the relationship between SMC, groundwater table depth (GWD) and vegetation by using the methods of coefficient of variation (Cv), Pearson correlation and regression. The results of the variability of SMC indicate that it rose with increase in depth of soil layer — SMC in the soil layer of 0–60 cm was relatively small compared to SMC in the soil layer of 100–260 cm which showed a significant increase in variability. SMC and GWD before and after ecological water diversions exhibited significant differences at the site of the Yingsu transect and its vicinity of the watercourse, especially SMC in the soil layer of 100–260 cm increased significantly with a significant rise of GWD and reached maximum values at a GWD of about 4 m. Plant coverage and species diversity significantly improved with increases in SMC in the soil layer of 100–260 cm, both of them approached the maximum values and 92.3% of major plant species were able to grow when SMC was > 10%. To restore the ecosystem of desert riparian forest along the Lower Tarim River, the GWD must be maintained at < 4 m in the vicinity of the watercourse and at about 4 m for the rest of this arid region.

A physically-based image processing approach, based on a single-source surface energy balance framework, is developed here to model the land surface temperature (LST) over complex/rugged geologic terrains at medium to high spatial resolution (&lt;102 m). This approach combines atmospheric parameters with a bulk-layer soil model and remote-sensing-based parameterization schemes to simulate surface temperature over bare surfaces. The model’s inputs comprise a digital elevation model, surface temperature data, and a set of land surface parameters including albedo, emissivity, roughness length, thermal conductivity, soil porosity, and soil moisture content, which are adjusted for elevation, solar time, and moisture contents when necessary. High-quality weather data were acquired from a nearby weather station. By solving the energy balance, heat, and water flow equations per pixel and subsequently integrating the surface and subsurface energy fluxes over time, a model-simulated temperature map/dataset is generated. The resulting map can then be contrasted with concurrent remote sensing LST (typically nighttime) data aiming to remove the diurnal effects and constrain the contribution of the subsurface heating component. The model’s performance and sensitivity were assessed across two distinct test sites in China and Iran, using point-scale observational data and regional-scale ASTER imagery, respectively. The model, known as the Surface Kinetic Temperature Simulator (SkinTES), has direct applications in resource exploration and geological studies in arid to semi-arid regions of the world.

In the solve for the equation for heat transfer and wave movement, using Fourier Transform to simplify the differential equation and solve it. With the periodic property of the wave including standing wave and vibrating string and heat equation and its stable solution, it can provide many new properties of the equation. Then using the same method to solve the heat equation in space, it gives a similar equation like the plane heat transfer equation. The passage mainly discussed about using Fourier transform to solve standing waves equation heat equation in steady state and in the time dependent state. The general way of solving wave equation both standing wave and traveling wave) is finding differential equation for a half period, expending it into the R, using Fourier transform to solve the equation.

The question of how to map the 3D indoor temperature by infrared thermography is solved by a hybrid method which is a combination of infrared thermography and the well known heat diffusion equation. The idea is to use infrared thermography to get the surface temperature of each frontier of the 3D domain of interest. A suitable procedure is devoted to this, allowing an automatic scanning of the whole frontier, the registration of data and computation. These surface temperatures constitute the boundary conditions of the heat equation solved in the domain of interest. The solution of the heat equation allows analyzing and controlling the temperature of every point belonging to the considered domain. This temperature distribution is controlled over the time with a period of the same order than the necessary time to obtain the frontier temperatures and at the end to contribute to the analysis of the thermal comfort. The study is done for the steady-state conditions under various weather situations. In this case the temperature depends only on space coordinates. With such procedure, we can have an idea about the time necessary to reach thermal equilibrium; time which has a great impact on the thermal comfort sensation. The results yielded by this method are compared with those given by others techniques used for temperature measurement. Finally, the method is used to access 3D temperature distribution for various geometric shapes.

Heat and moisture transfer and shrinkage simulation of deep-fat tofu frying

Equivalent heat source approach in a 3D transient heat transfer simulation of full-penetration high power laser beam welding of thick metal plates

view Abstract Citations (225) References (25) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Two-Fluid Model of the Solar Wind Hartle, R. E. ; Sturrock, P. A. Abstract This theoretical study of the solar wind takes account of the fact that the collisional energy-exchange rate between electrons and protons is sufficiently slow that the electrons and protons can have quite dif- ferent temperatures. The basic equations comprise a continuity equation, an equation of motion, and two heat equations, one for each species-electrons and protons. Each heat equation takes account of thermal conductivity in that species and energy exchange with the other species. In order to determine the region of applicability of these fluid-type equations, the electron-electron, and proton-proton energy/ momentum relaxation rates are compared with the local expansion rate. These comparisons indicate that, for the model investigated, the use of fluid equations is justifiable out to about 1O~ Ro. The conditions imposed on the solution of these coupled equations are that the electron (or proton) density and the electron and proton temperatures should have specified values at the "base" (inner boundary) of the model, that there should be a subsonic-supersonic flow transition, and that the electron and proton temperatures should tend to zero as the heliocentric radius tends to infinity. The effects of solar rotation, viscosity, non-thermal heat sources (e.g., wave dissipation), and magnetic field are neglect- ed, and the flow pattern is required to be steady and spherically symmetrical These requirements appear to determine the model uniquely. The equations are solved numerically using iterative procedures It is found that a good fit to the Blackwell model of the electron density of the outer corona is ob- tained, over the range 2-20 Ro, for the following choice of number density n (electrons or protons), electron temperature Te, and proton temperature T~ at the base, the radius of which is taken to be Ro: no = 3 X 1O~ cm3, T60 = T~0 = 2 X 106 ° K. If the base is regarded as being 2 Ro (the radius at which agreement with the Blackwell model begins), the values would be no = 1.5 X 106 cm3; Te0 1 5 X 106 ° K, T~0 = 1.2 X 106 ° K. This model has the following values for these variables and for the velocity v at Earth's orbit: ~E = 15 cm3, VE 250 km sec', TeE = 3 5 X 10~ °K, T~E = 4 4 X 1O~ ° K. These results for ~B, VE, TPE fit most nearly the observed characteristics of the solar wind at geomagneti- cally quiet times, although the density is somewhat high, the velocity is somewhat low, and the proton temperature is quite low. It is believed that the discrepancy between this model and Blackwell data between the base of the corona and 2 Ro is to be attributed to coronal heating, and that departures of the solar-wind charac- teristics near Earth from those of this model are also to be attributed to heating by a flux of non-thermal energy; additional heating inside the radius at which the flow becomes supersonic, which is at 7.1 Ro, will produce primarily an increase in solar-wind velocity, whereas additional heating outside 7.1 Ro will produce primarily an increase in solar-wind temperature. Data are presented which show the effect upon the characteristics of this model at Earth's orbit of a change in density or in electron or proton temperature at the base. The density increases sharply with the base electron temperature, even more sharply with the base proton temperature, and approximately linearly with the base density. The velocity varies almost linearly with the base electron temperature, and is insensitive to the other base quantities The proton temperature varies approximately linearly with the base density and electron temperature, but increases sharply with the base proton temperature The electron temperature increases with the base electron temperature but varies inversely with the base density and proton temperature; these variations are not stron Publication: The Astrophysical Journal Pub Date: March 1968 DOI: 10.1086/149513 Bibcode: 1968ApJ...151.1155H full text sources ADS |