Unequal Temperatures Research Articles

Stability analyses of two-temperature radiative shocks: formulation, eigenfunctions, luminosity response and boundary conditions

We present a general formulation for stability analyses of radiative shocks with multiple cooling processes, longitudinal and transverse perturbations, and unequal electron and ion temperatures. Using the accretion shocks of magnetic cataclysmic variables as an illustrative application, we investigate the shock instabilities by examining the eigenfunctions of the perturbed hydrodynamic variables. We also investigate the effects of varying the condition at the lower boundary of the post-shock flow from a zero-velocity fixed wall to several alternative types of boundaries involving the perturbed hydrodynamic variables, and the variations of the emission from the post-shock flow under different modes of oscillations. We found that the stability properties for flow with a stationary-wall lower boundary are not significantly affected by perturbing the lower boundary condition, and they are determined mainly by the energy-transport processes. Moreover, there is no obvious correlation between the amplitude or phase of the luminosity response and the stability properties of the system. Stability of the system can, however, be modified in the presence of transverse perturbation. The luminosity responses are also altered by transverse perturbation.

Measurement of the Quasistatic Component of Microaccelerations Using a Convection Sensor onboard a Satellite

The problem of the interpretation of measurements made by means of a convection sensor is considered. The sensor is a cubic chamber filled by a viscous fluid (gas). Fixed and unequal temperatures are maintained on two opposite sides of the cube; the other sides are perfect heat conductors. Two differential thermocouples are placed inside the chamber to measure the temperature difference at two pairs of fixed points. The sensor is mounted aboard the Earth's satellite. Mathematical models of various degrees of complexity are proposed which describe processes of heat and mass transfer under the action of a quasistatic component of microaccelerations. The results of mathematical simulation of the data of sensor thermocouples presenting a response to the real quasistatic component of microaccelerations which took place aboard the Mirstation are given. It is shown that under usual conditions of an orbital mission the sensor presents a linear low-frequency filter. By combining the data of several identical sensors, tightly arranged and oriented in a certain way, it is possible to measure low-frequency components of the angular acceleration of the satellite and linear microaccelerations at the point of the sensor position.

Kinetic theory of fluidized binary granular mixtures with unequal granular temperature

We derive balance laws and constitutive relations for a binary granular mixture with unequal granular temperature. The complete pair distribution function for two colliding spheres was assumed as the product of Maxwellian velocity distribution for each species. The constitutive relations together with the transport equations form a model for binary granular mixture with unequal granular temperature.

Combined forced and free flow of a power-law fluid in a vertical annular duct

An investigation of mixed convection and flow reversal in a vertical annular duct is presented with reference to laminar and fully-developed flow of a power-law fluid. The boundary surfaces are supposed to be isothermal, with unequal temperatures. The momentum balance and the energy balance equations as well as the viscous stress constitutive equation are solved analytically in order to obtain the velocity field, the viscous stress field and the temperature field. First, two special cases are analyzed: mixed convection of a Newtonian fluid; forced convection of a power-law fluid. Then, in the general case, the evaluation of the friction factors is employed to determine the conditions for the occurrence of flow reversal, for fixed values of the power-law index and of the ratio between the duct inner and outer radii.

Perturbative analysis of two-temperature radiative shocks with multiple cooling processes

The structure of the hot downstream region below a radiative accretion shock, such as that of an accreting compact object, may oscillate due to a global thermal instability. The oscillatory behaviour depends on the functional forms of the cooling processes, the energy exchanges of electrons and ions in the shock-heated matter, and the boundary conditions. We analyse the stability of a shock with unequal electron and ion temperatures, where the cooling consists of thermal bremsstrahlung radiation which promotes instability, plus a competing process which tends to stabilize the shock. The effect of transverse perturbations is considered also. As an illustration, we study the special case in which the stabilizing cooling process is of order 3/20 in density and 5/2 in temperature, which is an approximation for the effects of cyclotron cooling in magnetic cataclysmic variables. We vary the efficiency of the second cooling process, the strength of the electron-ion exchange and the ratio of electron and ion pressures at the shock, to examine particular effects on the stability properties and frequencies of oscillation modes.

On fully-developed mixed convection and flow reversal of a power-law fluid in a vertical channel

The fully-developed mixed convection of power-law fluids is investigated for laminar flow in a parallel-plate vertical channel. The boundary condition of uniform and unequal temperatures prescribed at the channel walls is considered. The velocity field, the viscous stress field and the temperature field are obtained by solving analytically the momentum and energy balance equations as well as the stress-strain constitutive equation. An expression which allows the evaluation of the friction factors is presented. The condition for the occurrence of flow reversal is obtained.

Performance Analysis of Double Stack Cold Plates Covering All Conditions of Asymmetric Heat Loading

Double stack, force-cooled cold plates, which are used as electronic chassis, are investigated for all conditions of unequal temperature and heat loading. A closed form solution that relates the plate temperatures to composite heat loads is derived. This analysis covers all regimes of operation. In particular, this includes the previously ignored regime in which no adiabatic point exists anywhere on either of the fins connecting cover and splitter plates. The heat dissipating advantage of the double stack structure, over a similarly designed single stack structure, is predicted with the analysis presented in the paper.

Electron and ion temperature gradients and suprathermal tail strengths at Parker's solar wind sonic critical point

New constraints are placed on the electron and ion temperature gradients at Parker's sonic point, demonstrated here as the location for an ion where the force of gravity and the electric field parallel to the magnetic field are balanced. The equal‐temperature, one‐fluid versions of these results precisely recover Parker's necessary and sufficient conditions for the supersonic expansion. The two species constraints are largely within the Parker envelope. For all possible critical points, the electron partial pressure must exceed 0.2437 of the plasma pressure at this point. The location of the critical point relative to the isothermal critical point is determined by the partial pressure average of the electron and ion logarithmic derivatives of temperature at the critical point. The bulk acceleration at the critical point is controlled by (1) the ion (as distinct from the electron) temperature gradient at this point and (2) the ratio of the ion to electron pressure at the critical point. The sonic point is a velocity space deLaval nozzle rather than a traditional configuration space nozzle. Critical points are most frequently accompanied by positive radial ion and electron temperature gradients. The electron and ion temperature gradients required for a critical point are coupled and not arbitrary. Parker's one‐fluid discussion requires, by this analysis, both electron and ion temperature gradients to have positive radial exponents β = 3/4 near the critical point and a local effective gas polytrope behavior of γ = 10/13 < 1. For equal electron and ion temperatures, all possible critical points require positive ion temperature gradients at the critical point. Nonthermal electron and ion distributions are demonstrated to be generally required near the sonic critical point of a spherically symmetric solar wind. Effective polytrope exponents near the critical point range from γi(r*) < 1 and 1.5 > γe(r*) > 1/3. The allowed critical points with γ < 1 are used to infer a range of nonthermal κ distributions that permit this unusual γ < 1 behavior; ion nonthermal suprathermal tail indices ranging between 2 < κi < 5 are required, reflective of decidedly nonthermal distributions at the critical point and consistent with previous inferences of their size [Scudder, 1992b]. Even when unequal electron and ion temperatures are considered, the vast preponderance of the critical points delineated also is found with 1/3 < γe < 1, with the attendant range of electron nonthermal indices, κe, ranging from 2 to the Maxwellian limit of ∞. Critical points with maximal acceleration invariably occur for a positive electron temperature power law exponent of βe ≃ 1.12 that corresponds to κe ≃ 3.3, whether or not the ions and electrons have a common temperature at the critical point.

Fin efficiency and mechanisms of heat exchange through fins in multi-stream plate-fin heat exchangers: formulation

The ideal conditions underlying various methods proposed in literature for the thermal design of multi-stream plate-fin heat exchangers are discussed. The basic differential equation for heat transfer from a fin connected to two walls, when the fin bases are at unequal temperatures is solved. Two general situations when T A > T B > T 0 and T A > T 0 > T B are investigated. The possible mechanisms, with respect to fin efficiency, of heat exchange in a plate-fin exchanger passage are examined and the relevant heat transfer equations are derived. The significance of the mechanisms is discussed. It has been shown that the equations for heat transfer across fin bases remain identical for all mechanisms. The relations between various dimensionless factors involved are presented in the form of graphs. The significance of the findings and their use in the development of a general method for rating multi-stream plate-fin heat exchangers are elucidated.

Buoyancy-induced recirculation bubbles and heat convection of developing flow in vertical channels with fin arrays

Developing fluid flow and heat transfer characteristics of the mixed convection in a vertical parallel-plate channel with fin array are analyzed in this study. The channel walls are maintained at uniform but unequal temperatures, and the fins of extremely high thermal conductivity are mounted on the relatively hotter wall. The stream function-vorticity method coupled with the power-law scheme are employed to solve the continuity, momentum and energy equations numerically. Results show that the streamwise periodic variation of the cross-sectional area causes the flow and temperature fields to attain a periodically fully developed character after a number of modules from the inlet. Meanwhile, if wall heating is sufficiently strong, an adverse pressure gradient can be developed downstream in the channel, and a series of buoyancy-induced recirculation bubbles may appear adjacent to the colder wall. The periodically fully developed solutions provided by Cheng et al. ( Int. J. Heat Mass Transfer 1992, 35, 2643–2653) are found to accurately portray the behavior of the present developing flow in the downstream region.

Influence of trapped thermal particles on internal kink modes in high temperature tokamaks

The effects of thermal trapped particles on the n=1 internal kink mode are studied using drift kinetic theory. Strong modifications of the magnetohydrodynamic (MHD) results are found, and marginal stability generally occurs at nonzero rotation frequency. For equal electron and ion temperatures, the trapped particles increase the marginal poloidal beta at q=1 substantially above the MHD value. For unequal electron and ion temperatures, the drift resonance with the hotter species becomes increasingly destabilizing and for sufficiently unequal temperatures, this leads to instability below the ideal-MHD threshold. Treatment of trapped thermal particles requires consideration of the effects of an electrostatic potential. The potential is weakly stabilizing for the internal kink mode. Furthermore, finite beta couples unstable, nearly electrostatic, trapped particle modes to the internal kink mode. At high beta, thermal fluctuations of the trapped particle modes can lead to significant internal kink displacements.

Effects of trapped thermal particles on the n=1 internal kink mode in tokamaks.

The effects of thermal trapped particles on the internal kink mode are studied using drift kinetic theory. For equal electron and ion temperatures, the trapped particles increase the marginal poloidal beta at q=1 substantially. For unequal temperatures, drift resonance with the hotter species is destabilizing and can lead to instability below the ideal magnetohyrodynamics threshold. An electrostatic potential is weakly stabilizing for the internal kink. At high beta, fluctuations of trapped particle modes couple to the internal kink mode and can lead to large displacements of the central region.

Inertia Effects in Externally Pressurized Conical Bearings with Unequal Slider Pad Temperature

This paper investigates the influence of convective inertia on the characteristics of an externally-pressurized conical bearing with unequal slider pad temperature, rotating with a uniform angular velocity. The fluid is assumed to be incompressible and have a constant viscosity. The governing equations of motion using conical coordinates appear as a system of non-linear partial differential equations. This system is simplified by averaging inertia terms across the film thickness, and the resulting system is solved numerically using a finite difference scheme. An energy equation with convection and dissipation terms is used to determine temperature distribution. It is observed that inclusion of the inertia terms causes a decrease in the dimensionless load capacity and an increase in dimensionless torque. Temperature decreases in the lower half of the bearing gap, and increases in the upper half when inertia terms are included. Presented at the 46th Annual Meeting In Montreal, Quebec, Canada April 29–May 2,...

Partition-extrapolation integration applied to radiative transfer in cylindrical media

An integration technique constructed to treat integrands with singularity has been extended to the solutions of the equation of radiative transfer in integral form. To illustrate the technique, radiative transfer is considered in an absorbing and isotropic scattering solid cylinder exposed to diffuse radiation, as well as radiative equilibrium in a participating medium between concentric cylinders with unequal surface temperatures. The integral equations for the two problems are solved by using the collocation method. Since the integral equations have singular kernels, the partition-extrapolation technique is employed. The technique yields estimates of the integration with singularity by extrapolation of the integration results of several subdomains without singularity. Comparisons of the results obtained by the present method with those obtained by the F-N method and the Monte Carlo method show that the partition-extrapolation technique generates accurate results.

Poloidal beta scaling for a bootstrapped tokamak

A new scaling of the poloidal plasma beta with the bootstrap fraction is derived for a steady-state tokamak. In this derivation, the seed current profile is specified, but no assumptions are made regarding the functional form of the total current profile. The scaling, which holds for a wide class of density and temperature profiles, exhibits a quadratic dependence of the poloidal beta on the bootstrap fraction. The effects of the ionic charge and unequal electron and ion temperatures are considered.

Effects of heat transfer on the dynamics and transport of small particles suspended in gases

Cases discussed in the article of recent studies include thermophoretic drift of spherical particles in a gas through which energy is diffusing, photophoretic drift of radiation-absorbing suspended spheres, phoresis and apparent Brownian diffusivity of asymmetric particles, including spheres with nonuniform accomodation coefficient and unequal primary particle size aggregates, scavenging of fine particles by evaporating droplets, and coagulation rate of unequal temperature spherical particles. General expressions for the relevant particle drift velocities, coagulation rate constants, or effective particle diffusivities are used to carry out illustrative calculations for multiphase systems of current and future engineering interest, including fine particle synthesis/materials processing, and combustion gas cleaning

Forced convection in parallel-plate channels with a series of fins mounted on the wall

Fluid flow and heat transfer characteristics are analyzed for the laminar forced-convection flows in a parallel-plate channel with an array of fins mounted on the wall. The two plates are maintained at uniform but unequal temperatures and the mounted fins are assumed to be perfectly conductive. Numerical solutions for the Reynolds number within the range 10 ≤ Re ≤ 300, fin height within 0 ≤ e H ≤ 0·5 , and fin pitch within 0 ≤ d H ≤ 4 are presented. Analytical solutions for the infinite and zero pitch cases, representing the lower-bound and upper-bound values, respectively, are developed and compared with the numerical predictions. Results show that both the heat transfer and friction factor (i) increase with the Reynolds number and (ii) are highly sensitive to the geometric arrangements.

The role of nucleation in the initial phases of a rapid depressurization of a subcooled liquid

A two-fluid model with zero slip between the phases but which allows for unequal phase temperatures, changing interfacial area and the effect of heterogeneous nucleation has been used to evaluate the initial depressurization of a subcooled or saturated light. The results compare favourably with existing water and dichlorodifluoromethane experiments and suggest that the algorithm, in association with well-tabulated data, could be used to evaluate the heterogeneous nucleation factor.

Local chemical and thermal-hydraulic analysis of U-tube steam generators

In order to know how pH distribution affects corrosion in a U-tube steam generator, a study of the combination of water chemistry and thermal-hydraulic conditions is suggested. A two-fluid (unequal velocity and unequal temperature) formulation is proposed to describe the convective transport of volatile species in each phase, and a spherical bubble model is developed on the basis of the penetration theory to describe the interfacial mass transfer. The thermal-hydraulic local conditions are obtained by the U-tube steam generator design analysis code FAUST which is based on the three-dimensional two-fluid model. The results of the present study are compared with dynamic equilibrium model calculations. This study shows that, in contrast with dynamic equilibrium calculations, the pH is lower in the cold-leg side than in the hot-leg side because of liquid recirculation. Just above the tube sheet, however, the lower void fraction in this region than that in the hot-leg region results in higher pH, which agrees with the prediction of the dynamic equilibrium model.

On the stability of degenerate dwarf radiative shocks - New one- and two-temperature calculations

The present numerical hydrodynamic calculations of radiative shock structure and evolution in degenerate dwarf accretion flows give attention to their stability properties; this is, moreover, the first systematic study of the time-dependent behavior of radiative shocks assuming unequal ion and electron temperatures. The results obtained are applied to the observed quasi-periodic oscillations seen in four AM Her stars, and it is concluded that the observed quasi-periodic optical oscillations can be explained by the shock oscillation model.