Case Of Heat Conduction Research Articles

Heat conduction in a nearly collisionless molecular gas

Transport phenomena in molecular gases and the respective field effects have so far been analyzed in the hydrodynamic and Knudsen regimes. In between there is a rather large gap covered by the transition regime. For the case of heat conduction through a diatomic diamagnetic gas between parallel plates an attempt is made to partially bridge the gap by the beginning of a density expansion derived through iteration of an integral form of the Boltzmann equation. Second-order perturbation in non-sphericity of both intermolecular and gas-wall interactions is applied to determine the effect of a magnetic field. Explicit expressions are derived for a simple model.

Natural Convection Experiment with Liquid NaK under Transverse Magnetic Field

The effect of a magnetic field on laminar natural convection of liquid metal was studied experimentally using NaK as conducting fluid. The magnetic field was imposed horizontally and parallel to a uniformly heated vertical plate, to act perpendicularly across the convective flow. In a low magnetic field, the temperature profile across the layer of flowing fluid acquired an η-shaped profile characterized by a valley close to the wall and a peak further away, which had the effect of raising heat transfer rate above that obtained in the absence of magnetic field. When the magnetic field was intensified, its braking effect on the flow approached the temperature profile to the case of pure heat conduction through solid, to result in dwindling heat transfer rate.

Temperature rise induced by a laser beam II. The nonlinear case

The steady-state solution of the nonlinear heat-conduction equation when the thermal conductivity is strongly temperature dependent is expressed (for arbitrary geometry and heat source) in terms of the corresponding solution for the linear heat-conduction case. This permits a closed-form expression for the maximum temperature rise when a Gaussian laser beam hits the surface of a crystal, as well as integral representations for the spatial distribution of the temperature rise. This solution is essential in understanding laser annealing.

Functional Calculus of a Transferable Scalar in a Turbulent Flow

Abstract The problem of dispersion of a transferable scalar in a turbulent fluid is considered. The Hopf functional equation for the characteristic functional of the velocity vector and transferable scalar is derived and discussed. The special case of heat conduction with random initial condition is studied in detail. It is shown that the Hopf equation accepts a Gaussian characteristic functional solution and several examples are discussed.

Thermal analysis of cyclic cryogenic regenerators

The system of partial differential equations, together with their boundary conditions, has been established to describe the performance of regenerators subjected to cycling flows, including commonly neglected effects such as internal energy changes of the fluid due to pressure cycling and longitudinal matrix conduction. Exact solutions are obtained for the case of an infinitely large matrix heat capacity. For the case of finite matrix heat capacity the method of perturbations is employed, and the solutions can be considered exact throughout the regenerator except for the regions near the boundaries. In general, the results are contained in a set of three coupled ordinary differential equations, which must be solved numerically. For the important case of negligible matrix heat conduction, however, a closed-form solution is presented here.

On Nonlinear Initial Boundary Value Problems of Heat Conduction and Diffusion

Heat conduction and diffusion problems are frequently encountered by physicists and engineers in various fields of applications. This paper treats the nonlinear case of the heat conduction and diffusion equation in both its theoretical and practical aspects. In the theoretical part of the paper, a nonlinear heat conduction equation is derived under quite general assumptions, and it is shown that well-known particular forms of the equation can be derived from the general form under certain simplifying assumptions. The commonly encountered initial boundary value conditions are enunciated and some special forms of the problem formulated. Existence and uniqueness theorems are stated and proved briefly. The practical part of the paper shows the development, which lags that of the theory, of approximate methods suitable for numerical applications and the restrictions to which these methods are subject.

On the “Governing Principle of Dissipative Processes” and its Extension to Non‐linear Problems

AbstractAfter the description of the general structure of non‐equilibrium (irreversible) thermodynamics, the local (or differential) principles developed previously are outlined in all the representations. Hereafter the universal form of the integral principle of thermodynamics is given from which transport equations governing irreversible processes are derived within the framework of a general “Γ” formalism. The derivation of the transport equations is followed by the discussion of the partial and alternative forms of the integral principle previously developed by us and by others. We demonstrate that the “local potential method” can be founded also in an exact manner by means of the partial form of our integral principle. In the following, the types of tasks and theories of non‐linear thermodynamics are classified, further on, the general proof of a supplementary theorem is given, which has already been confirmed for the case of heat conduction. This theorem allows extension of the validity of the universal form of the integral principle to certain types of non‐linear problems. Finally the relation of our integral principle to VOJTA's functional variational principle is discussed and it is stated that the two different formulations are the alternative representations of a single general principle, for which the name: “governing principle of dissipative processes” would be most appropriate.

Kinetic Theory Description of Rarefied Gas Flow

In an early paper with H. Bolza and Max Born, von Karman examined the relation between the “collision-dominated” and “collision-free” regimes in the simple case of steady flow or steady heat conduction through a porous medium. Von Karman also formulated the boundary condition at a gas-solid interface, and showed that the temperature “jump” arises quite naturally from the two-sidedness of the velocity distribution function, independently of any statements about the thermal accommodation coefficient. We reexamine this question of the transition between gas-kinetics and gasdynamics with the aid of the Maxwell moment method, utilizing a two-sided Maxwellian-type distribution as weighting function. The main features deduced from this approach are illustrated by considering two simple examples: (1) steady heat conduction between two concentric cylinders, and between two concentric spheres; (2) the “signaling problem” generated by the sudden motion and heating of an infinite flat plate.

The A.C. Admittance of Temperature-Dependent Circuit Elements

The relation between the isothermal and steady-state current-voltage characteristics of a device in which the current depends on both the voltage and the temperature is deduced quite generally without assuming electrical linearity or Newtonian cooling. The phenomena of voltage turnover in a device with a positive temperature coefficient of d.c. conductance, and of current turnover in a device with a negative coefficient are deduced. The a.c. admittance for a small a.c. voltage superimposed on the d.c. voltage determining the operating point is derived in terms of the thermal admittance of the device, i.e. the complex ratio of the alternating components of power and temperature. For the special case in which the thermal admittance consists of a shunt combination of a fixed thermal capacity and thermal conductance the electrical admittance has been evaluated in detail. Three-element equivalent circuits (of two resistors and a capacitor or inductor) can represent the electrical admittance of such a device and its locus is semicircular; the reactive element represents the effect of thermal inertia and its nature is determined by the sign of the temperature coefficient of d.c. conductance. Where heat loss from the device is by a path which is not `thermally short at the frequency under consideration the thermal admittance no longer has simple form but has to be determined from the diffusion equation. This is done for the case of one-dimensional heat conduction (e.g. along the supporting wires of a thermistor or a lamp) or for radial heat flow (e.g. from the contact in a point-contact rectifier). The general condition for the admittance locus still to be circular, but with the centre not on the conductance axis, is formulated.

Reciprocal Relations in Irreversible Processes. I.

Examples of coupled irreversible processes like the thermoelectric phenomena, the transference phenomena in electrolytes and heat conduction in an anisotropic medium are considered. For certain cases of such interaction reciprocal relations have been deduced by earlier writers, e.g., Thomson's theory of thermoelectric phenomena and Helmholtz' theory for the e.m.f. of electrolytic cells with liquid junction. These earlier derivations may be classed as quasi-thermodynamic; in fact, Thomson himself pointed out that his argument was incomplete, and that his relation ought to be established on an experimental basis. A general class of such relations will be derived by a new theoretical treatment from the principle of microscopic reversibility. (\textsection{}\textsection{}1-2.) The analogy with a chemical monomolecular triangle reaction is discussed; in this case a a simple kinetic consideration assuming microscopic reversibility yields a reciprocal relation that is not necessary for fulfilling the requirements of thermodynamics (\textsection{}3). Reciprocal relations for heat conduction in an anisotropic medium are derived from the assumption of microscopic reversibility, applied to fluctuations. (\textsection{}4.) The reciprocal relations can be expressed in terms of a potential, the dissipation-function. Lord Rayleigh's "principle of the least dissipation of energy" is generalized to include the case of anisotropic heat conduction. A further generalization is announced. (\textsection{}5.) The conditions for stationary flow are formulated; the connection with earlier quasi-thermodynamic theories is discussed. (\textsection{}6.) The principle of dynamical reversibility does not apply when (external) magnetic fields or Coriolis forces are present, and the reciprocal relations break down. (\textsection{}7.)