Kirchoff Transformation Research Articles

Nonlinear Heat Transfer From Particles in Supercritical Carbon Dioxide Near the Critical Point

Abstract Particle to fluid heat transfer in supercritical carbon dioxide (sCO2) is encountered in energy technologies and in materials synthesis. Near the critical point, the extreme pressure and temperature sensitivity of sCO2’s thermal conductivity will change the expected heat transfer in these systems. The current work combines the Kirchoff transformation for thermal conductivity with the conduction shape factor for a sphere, allowing prediction of heat transfer in these systems and quantification of the impact of these property changes. Results show that the heat transfer is non-linear for supercritical heat transfer, with the non-linearity particularly significant near the critical point. The results also show that approaches such as an average thermal conductivity based on film temperature are unlikely to accurately predict heat transfer in this region. The methods described in this paper can be applied to fluid–particle heat transfer at low Reynolds number in other fluids with large variations in thermal conductivity.

Pemodelan Matematika Infiltrasi Air pada Saluran Irigasi Alur

Water is one of the main necessity of agricultural activities, because without enough water agricultural crops will not be produced optimally. The way to insufficient water in agricultural crops is irrigation. One of the irrigation methods which is used on agriculture in the world is furrow irrigation method. Water gets into the soil from the bottom of the furrow and furrow’s wall towards the root zone of the plants. The complexity of the water infiltration process in the ground makes infiltration analysis by laboratory experiment difficult to do and needs substantial cost. The alternative way which can do is with mathematical modeling. This paper discusses about mathematical modeling of water infiltration in furrow irrigation channel trapezoidal in shape. This mathematical modeling is shaped boundary condition problem with a cross section of a closed and limited line of irrigation. Governing equation obtanined from Richard equation which then transformed using Kirchoff transformation and non dimensional variable into the modified Helmholtz equation. While, the boundary condition is shaped mixture Neuman and Robin boundary condition.

Suction Potential and Water Absorption from Periodic Channels in Different Types of Homogeneous Soils

<p><span style="font-size: 10.000000pt; font-family: 'CMR10';">In this paper, problems involving infiltration from periodic identical trapezoidal channels into homogeneous soils with root water uptake are considered. The governing equation of infiltration through soil is transformed to a modified Helmholtz equation using Kirchoff transformation with dimensionless variables. A DRBEM with a predictor-corrector scheme is employed to obtain numerical solutions to the modified Helmholtz equation. The proposed method is used to solve infiltration problems in three different types of soil. Results are shown to be physically meaningful. </span></p>

Steady Heat Transfer through a Two‐Dimensional Rectangular Straight Fin

Exact solutions for models describing heat transfer in a two‐dimensional rectangular fin are constructed. Thermal conductivity, internal energy generation function, and heat transfer coefficient are assumed to be dependent on temperature. We apply the Kirchoff transformation on the governing equation. Exact solutions satisfying the realistic boundary conditions are constructed for the resulting linear equation. Symmetry analysis is carried out to classify the internal heat generation function, and some reductions are performed. Furthermore, the effects of physical parameters such as extension factor (the purely geometric fin parameter) and Biot number on temperature are analyzed. Heat flux and fin efficiency are studied.

Model order reduction of linear and nonlinear 3D thermal finite-element description of microwave devices for circuit analysis: Research Articles

Electrothermal models of power devices are necessary for the accurate analysis of their performances. For this reason, this article deals with a methodology to obtain an electrothermal model based on a reduced model of a 3D thermal finite-element (FE) description for its thermal part and on pulsed electrical measurements for its electrical part. The reduced thermal model is based on the Ritz vector approach, which ensures a steady-state solution in every case. An equivalent SPICE subcircuit implementation for circuit simulation is proposed and discussed. An extension of the method to a nonlinear reduced model based on the Kirchoff transformation is also proposed. The complete models have been successfully implemented in circuit simulators for several HBT or PHEMT device structures. Many results concerning devices and circuits are presented, including simulation of both the static and dynamic collector-current collapse in HBTs due to the thermal phenomenon. Moreover, the results in terms of the circuit for an X-band high-power amplifier are also presented. As for the nonlinear approach, results concerning an homogeneous structure is given. © 2005 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2005.

Model order reduction of linear and nonlinear 3D thermal finite-element description of microwave devices for circuit analysis

Electrothermal models of power devices are necessary for the accurate analysis of their performances. For this reason, this article deals with a methodology to obtain an electrothermal model based on a reduced model of a 3D thermal finite-element (FE) description for its thermal part and on pulsed electrical measurements for its electrical part. The reduced thermal model is based on the Ritz vector approach, which ensures a steady-state solution in every case. An equivalent SPICE subcircuit implementation for circuit simulation is proposed and discussed. An extension of the method to a nonlinear reduced model based on the Kirchoff transformation is also proposed. The complete models have been successfully implemented in circuit simulators for several HBT or PHEMT device structures. Many results concerning devices and circuits are presented, including simulation of both the static and dynamic collector-current collapse in HBTs due to the thermal phenomenon. Moreover, the results in terms of the circuit for an X-band high-power amplifier are also presented. As for the nonlinear approach, results concerning an homogeneous structure is given. © 2005 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2005.

Mathematical modelling of heat transfer during electron-beam autocrucible melting by means of the steady-state Stefan problem

A two-dimensional axisymmetric mathematical model of electron-beam autocrucible melting is developed and examined. Here, the hypothesis is used that forced convective heat transfer in the melt may be modelled with the help of the coefficient of effective thermal conductivity, λE. A simplified approach is used in which λE is assumed to be known. In another approach the value of λE depends on a prescribed value of a mean melt-stirring velocity and a mean liquid-pool radius which is determined in the course of solving of the problem. With the help of the Kirchoff transformation and a Green function we may reduce the problem to a nonlinear Hammerstein integral equation. Here, a dependence of the thermal-conductivity coefficient on the temperature, λS(T), at the cooled surfaces is disregarded and constant (mean) values of λS are utilized. In order to solve the problem in the case where this dependence λS(T) is taken into account, an axuiliary Green-function method is proposed which also permits to take into account a change of the heat-exchange coefficients on the autocrucible. This reduces the problem to a system of three integral Hammerstein equations. Numerical solutions of the nonlinear integral equations are obtained with the help of a variational (projective-net) method for the case of circular scanning of an electron beam over the heated surface. The computational results are well consistent with experimental data.

On friction-induced temperatures of rubbing metallic pairs with temperature-dependent thermal properties

This paper investigates the temperature rises for dry sliding systems when the variation in the thermal conductivity with temperature is taken into account. For the purpose of the analysis, it has been assumed that the thermal conductivity of the rubbing materials vary linearly with temperature. Accordingly, materials are classified into three categories based on that variation: materials for which the conductivity drop with temperature elevation (class a): materials for which the conductivity increases with temperature elevation (class b): and materials for which the conductivity-temperature curve has an inflation point (class c). The variable conductivity temperatures are obtained by applying the so called ‘Kirchoff transformation’ to the fundamental solution of the heat equation. The results indicate that the behavior of the conductivity with temperature is significantly influential to the magnitude of the temperatures reached by the rubbing pair. For a variety of sliding pairs analyzed in this work, significant variation between the constant and the variable conductivity predictions were found. For example, the temperature rise for a mild steel (AISI 1020) rubbing pair, sliding at 6 m/s and 30 N nominal load, predicted by the variable conductivity solution is about 30% higher than that predicted using a constant conductivity solution. It is also shown that the estimates of the heat conducted through the surface may be in error (by about 30–40%) if based on a constant conductivity solution. Such behavior has direct effects on the thickness of the thermally affected subsurface layer (the so-called thermal skin), and the thermal distortion of the contact interface. The error introduced in the estimates of the temperature rises for class c materials is shown to be proportional to the ratio between the inflation to the melting temperatures of the moving solid.

A remark on the flash temperature theory

The classical solution of the temperature development in a sliding solid pair treats the thermal conductivity as a non-variant quantity that is not affected by temperature elevation. This may introduce an erroneous estimate of the temperature distribution in the contacting pair. This error may be intensified in the presence of considerable temperature gradients. In this note, we apply the well known Kirchoff transformation to correct the temperature estimates in a practical case of sliding friction. Thus, accounting for the point wise variation of the thermal conductivity in the contacting pair. We compare temperature estimates, obtained by the classical solution, to those obtained by the variable conductivity solution. The results indicate that the behavior of the thermal conductivity with temperature is influential to the magnitude of the peak flash temperatures developed at the tip of the contact asperities.

Steady temperature fields associated with a moving rod in a medium with nonlinear thermal conductivity

The two-dimensional problem is considered of determining the temperature field when a rectangular rod of width h and at constant temperature moves at constant speed V in a medium with a temperature dependent thermal diffusivity D(T). It is assumed that Vh/D(T) ⪡ 1 and the solution is first considered in the far field (“the outer solution”) where the problem is reduced to a nonlinear ordinary differential equation. Properties of this “outer solution” are then used to set a boundary condition at infinity for the “inner problem” of determining the temperature close to the rod. Finally, this inner problem is solved by using the Kirchoff transformation to reduce the nonlinear equation to the Laplace equation valid close to the rod.

Solution of nonlinear elliptic equations with boundary singularities by an integral equation method

A boundary integral equation (BIE) formulation is presented for the numerical solution of certain two-dimensional nonlinear elliptic equations subject to nonlinear boundary conditions. By applying the Kirchoff transformation, all nonlinear aspects are first transferred to the boundary of the solution domain. Then the accurate solution of problems in which there are boundary singularities is demonstrated by including the analytic nature of the singular solution in only those regions nearest the singularity. Because of this, only a minor modification of the classical nonlinear BIE is required and this results in a substantial improvement in the accuracy of the numerical results throughout the entire solution domain. The BIE has previously been applied to either nonlinear or singular problems and so the method presently described constitutes an extension in this field.

Temperature profiles in solid targets irradiated with finely focused beams

The quantitative understanding of the processes involving focused beams, such as recrystallization of semiconductors or thermal annealing and recording, requires a detailed knowledge of the temperature profiles within the target. We have derived first a normalized analytical representation for the distribution of power dissipation in targets bombarded with electron beams. This representation has then been combined with the Green’s function approach and Kirchoff transformation to predict the steady state and transient temperature profiles in targets with linear and nonlinear thermal conductivities. This calculation for the case of transient heating when thermal conductivity changes with temperature is not strictly accurate and, therefore, a numerical technique was developed involving successive over relaxation. Considerable care was needed in choosing the distribution of mesh size because of the great difference between beam radius (down to 2 μm) and target dimensions (250–500 μm). Numerical calculations for silicon (a nonlinear conductor) agreed well with transient analytical calculations using Kirchoff transformation.