Similarity solution for a cylindrical shock wave in a rotational axisymmetric dusty gas with heat conduction and radiation heat flux

The propagation of shock waves in a rotational axisymmetric dusty gas with heat conduction and radiation heat flux, which has a variable azimuthally fluid velocity together with a variable axial fluid velocity, is investigated. The dusty gas is assumed to be a mixture of non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston (or inner expanding surface). The fluid velocities in the ambient medium are assume to be vary and obey power laws. The density of the ambient medium is assumed to be constant, the heat conduction is express in terms of Fourier’s law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity K and the absorption coefficient α_R are assumed to vary with temperature and density. In order to obtain the similarity solutions the angular velocity of the ambient medium is assume to be decreasing as the distance from the axis increases. The effects of the variation of the heat transfer parameter and non-idealness of the gas in the mixture are investigated. The effects of an increase in (i) the mass concentration of solid particles in the mixture and (ii) the ratio of the density of solid particles to the initial density of the gas on the flow variables are also investigated.     

Nonisothermic flow of gas mixture in a channel at intermediate Knudsen numbers

Solution of the problem of gas mixture flow in a plane channel at intermediate Knudsen numbers is considered on the basis of the 20-moment approximation as a function of distribution. The applied method consists of averaging moment equations valid throughout the flow region (including the Knudsen layers) with the determination of boundary values of macroscopic parameters on the wall using the approximate Loyalka method /1,2/. Expressions are obtained for a binary mixture for the mean molar velocity averaged over the channel cross section, difference of component velocities, and the relative heat flux in the presence of longitudinal gradients of partial pressures, and for the temperature gradients. Respective kinetic coefficients of the Onsager matrix are calculated. Dependence of these coefficients on the Knudsen number, and the properties of molecule scatter on the channel wall are analyzed in detail in the case of one-component gas and of a binary mixture with small relative difference of mass and diameters of molecule scatter.     

CFD predictions for cement kilns including flame modelling,heat transfer and clinker chemistry

Clinker formation in coal-fired rotary cement kilns under realistic operation conditions has been modelled with a commercial axisymmetric CFD code for the gaseous phase including a Monte Carlo method for radiation, a finite-volume code for the energy equation in the kiln walls, and a novel code for the species and energy conservation equations, including chemical reactions, for the clinker. An iterative procedure between the predictions for the temperature field of the gaseous phase, the radiative heat flux to the walls, and the kiln and clinker temperature is used to predict the distribution of the inner wall temperature explicitly, including the calculation of heat flow to the clinker. It was found that the dominant mode of heat transfer between the gas and the kiln walls is by radiation and that the heat lost through the refractories to the environment is about 10% of the heat input and a further 40% is used for charge heating and clinker formation. The predictions are consistent with trends based on experience and limited measurements in a full-scale cement kiln.     

Numerical modelling of interaction process of low-temperature plasma

The problem of the interaction of high-temperature reactive gas mixture with semi-infinite solid propellant plate is investigated. The ignition conditions of the plate are determined in the framework of the well known model of burning powder with constant surface temperature when the heat flux into plate is limited by some critical value. The gas medium is generated due to the fluid (water or methane) evaporation under the action of the impulse electric high-powered discharge in the semi-closed volume. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Axisymmetric boundary layers on non-uniformly heated horizontal surfaces

Similarity solutions for the velocity and temperature induced by axisymmetrically heated horizontal surfaces with a power law temperature distribution are derived and investigated. Two physical situations, a stationary and a radially moving temperature distribution are considered; both above and below the surface. Apart from a perturbation solution for small temperature differences for the latter, the equations have to be numerically integrated. The results include the non-existence of a solution below the surface for particularly large heat inputs. A major difference from previous work on cartesian geometries is that there must be a non-zero heat flux between the surface and fluid, which is related to the total heat flux through the fluid.     

Approximate formulas for heat flows towards an ideally catalytic surface near a plane of symmetry

The three-dimensional flow of a chemically unstable viscous gas near a plane of symmetry of blunt bodies streamlined at the angle of attack, is considered. The investigation is carried out using a model of a thin, viscous shock layer. To a first approximation of the method of successive approximations for a uniform gas simple formulas are obtained for the distribution of the heat flux over the surface, referred to its value at the stagnation point. It is shown that for a chemically unstable gas the distribution of the heat flux along an ideally catalytic surface depends only slightly on the conditions prevailing within the incident flow, is determined mainly by the geometrical characteristics of the body, and is described quite satisfactorily by the formulas obtained. The accuracy of these formulas is determined by comparison with numerical computations carried out for bodies of various shapes, moving at different angles of attack along a planing trajectory of re-entry into the Earth's atmosphere, and during re-entry into the atmosphere at a constant velocity.     

Problem with nonequilibrium boundary conditions in the kinetic theory of gases

The Boltzmann kinetic equation is considered in a new formulation with nonequilibrium distribution functions on free boundaries, which makes it possible to simulate nonequilibrium superand subsonic flows. Transport processes for such flows are analyzed. The possibility of anomalous transport is determined, in which case the heat flux, temperature gradient, and the corresponding components of the nonequilibrium stress tensor and the velocity gradient have the same sign.     

Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration

Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed.     

Heat conduction through a rarefied gas between two rotating cylinders at small temperature difference

Summary Numerical calculations of heat transfer between two coaxial rotating cylinders at a small temperature difference are carried out over wide ranges of the Knudsen number and the angular velocity. The calculations have been performed based on the S-model of the Boltzmann equation by the discrete velocity method. It has been confirmed that in a rotating gas a radial temperature gradient causes both radial and tangential heat fluxes. Also, it has been found that the radial heat flux is affected by the rotation.On temporary leave from Department of Physics, Urals State University, 620083 Ekaterinburg, Russia.     

Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems

In this paper, the homotopy analysis method (HAM) is compared with the homotopy-perturbation method (HPM) and the Adomian decomposition method (ADM) to determine the temperature distribution of a straight rectangular fin with power-law temperature dependent surface heat flux. Comparisons of the results obtained by the HAM with that obtained by the ADM and HPM suggest that both the HPM and ADM are special case of the HAM.     

Dynamic adaptation method in gasdynamic simulations with nonlinear heat conduction

A dynamic adaptation method is applied to gas dynamics problems with nonlinear heat conduction. The adaptation function is determined by the condition that the energy equation is quasi-stationary and the grid point distribution is quasi-uniform. The dynamic adaptation method with the adaptation function thus determined and a front-tracking technique are used to solve the model problem of a piston moving in a heat-conducting gas. It is shown that the results significantly depend on the thermal conductivity chosen. The numerical results obtained on a 40-node grid are compared with self-similar solutions to this problem.     

Spatial decay estimates for the Maxwell-Cattaneo equations with mixed boundary conditions

In this paper the authors derive spatial decay bounds for the temperature and heat flux as defined by the Generalized Maxwell-Cattaneo equations for heat conduction in a semi-infinite cylinder when the temperature and the tangential components of the heat flux vector vanish on the lateral surface of the cylinder. The results here supplement those previously found by the authors [5] when the heat flux vector was assumed to be zero on the lateral surface but no condition was imposed on the temperature.     

Thermal effects in the description of a multicomponent mixture using the density functional method

The use of a model, based on an expression for the total entropy in the form of a functional with the temperature and density gradients of the components, is proposed to describe a multicomponent, multiphase system using continuous hydrodynamics (that is, within the framework of the approach of the continuum mechanics without discontinuities in the hydrodynamic quantities). It is proved that this model is consistent with the zeroth law of thermodynamics. Expressions for the stress tensor, the diffusion fluxes and the heat flux are found from the condition that the entropy production is non-negative. Compared with the classical Newton, Fick and Fourier laws, these expressions contain third-order spatial derivatives, The problem of a mixture between two parallel and impermeable walls at different temperatures is analysed. In this case, the system of dynamic equations reduces to a system of ordinary differential equations. It is shown that the number of free parameters, on which the solution depends, corresponds to the number of boundary and general integral conditions.     

Transient flow in natural gas pipeline – The effect of pipeline thermal model

One-dimensional, nonisothermal gas flow model was solved to simulate the slow and fast fluid transients, such as those typically found in high-pressure gas transmission pipelines. The results of the simulation were used to understand the effect of different pipeline thermal models on the flow rate, pressure and temperature in the pipeline. It was found that simplified flow model with steady-state heat transfer term overestimates the amplitude of the temperature fluctuations in the pipeline. This result indicated that unsteady heat transfer model with the effect of heat accumulation in the surroundings of the pipeline should be used to calculate the gas parameters at locations of interest within high-pressure gas transmission pipelines.     

Spatial decay estimates for the Maxwell-Cattaneo equations with mixed boundary conditions

In this paper the authors derive spatial decay bounds for the temperature and heat flux as defined by the Generalized Maxwell-Cattaneo equations for heat conduction in a semi-infinite cylinder when the temperature and the tangential components of the heat flux vector vanish on the lateral surface of the cylinder. The results here supplement those previously found by the authors [5] when the heat flux vector was assumed to be zero on the lateral surface but no condition was imposed on the temperature.Received: February 7, 2002; revised: June 3, 2002     

Thermal stresses in a plate with transversely isotropic material

In this paper the thermal stresses in a plate with transversely isotropic material have been obtained by the method of Hankel transforms. Three cases of surface temperature over a circular region of exposure with flux of heat, paraboloidal distribution and constant temperature with surface radiation have been considered. Numerical results are presented for the case of surface radiation.     

Numerical study of a mathematical model of a catalytic fuel processor with cocurrent oxidation and conversion flows

The results are presented of the numerical study of a mathematical model in the form of a nonlinear boundary value problem describing the stationary regimes in a catalytic fuel processor. We study a two-dimensional model for the endoblock, with the longitudinal heat and mass transfer by the gas and the transversal heat conductivity along the catalyst in the two-temperature approximation. For the exochannel, a model is considered with the longitudinal heat and mass transfer by the gas flow and the longitudinal heat transfer along the catalytic wall. These two blocks are related to each other through the equality of the temperature and heat flux on the boundary. The results obtained are in good agreement with experimental data.     

On the approximation method of solving integral equations in diffusion problems

In this paper we consider the one-dimensional problem of heat or mass transport in the system with moving ends. We show that without solving the heat transfer equation, the heat flux flowing out from the system can be found when temperature on the boundary of this system is known. We make use of the Banach contraction theorem for appropriate integral equations. Our method also enables us to find the distribution of temperature in the whole domain that forms the physical system.     

Transient modeling of non-isothermal, dispersed two-phase flow in natural gas pipelines

Unsteady-state or transient two-phase flow, caused by any change in rates, pressures or temperature at any location in a two-phase flow line, may last from a few seconds to several hours. In general, these changes are an order of magnitude longer than the transient encountered during single-phase flow. The primary reason for this phenomenon is that the velocity of wave propagation in a two-phase mixture is significantly slower. Interfacial transfer of mass, momentum and energy further complicate the problem. It is primarily due to the numerical difficulties anticipated in accurately modeling transient two-phase flow that the state of the art in this important area is restricted to a handful of studies with direct applicability to petroleum and gas engineering. A limited amount of information on the subject of two-phase transport phenomena is available in the petroleum engineering literature. Most of the publications for two-phase flow of gas assume that temperature is constant over the entire length of the pipeline.This study is the first effort to simulate the non-isothermal, one-dimensional, transient homogenous two-phase flow gas pipeline system using two-fluid conservation equations. The modified Peng–Robinson equation of state is used to calculate the vapor–liquid equilibrium in multi-component natural gas to find the vapor and liquid compressibility factors. Mass transfer between the gas and the liquid phases is treated rigorously through flash calculation, making the algorithm capable of handling retrograde condensation. The liquid droplets are assumed to be spheres of uniform size, evenly dispersed throughout the gas phase.The method of solution is the fully implicit finite difference method. This method is stable for gas pipeline simulations when using a large time step and therefore minimizes the computation time. The algorithm used to solve the non-linear finite difference thermo-fluid equations for two-phase flow through a pipe is based on the Newton–Raphson method.The results show that the liquid condensate holdup is a strong function of temperature, pressure, mass flow rate, and mixture composition. Also, the fully implicit method has advantages, such as the guaranteed stability for large time step, which is very useful for simulating long-term transients in natural gas pipeline systems.