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The thermal conductivity of gases has been the subject of experimental investigations over a period of several decades (54). The thermal conductivity (hereafter simply referred to as the "conductivity"), being, from a physical point of view, one of the basic properties of a gas, much of the classical kinetic theory was concerned with its interpretation. Its measurement was to examine the validity of the theory. The limits of this validity were similar to those imposed on the theory by other gaseous properities, which lead to the concept of the "ideal gas". As to the conductivity, the behaviour of the majority of naturally occuring gases approximated closely that of the "ideal gas" under normal conditions, i.e. at moderate or room temperatures and a pressure range up to atmospheric pressure. Serious deviations from experimental data occured, however, when it was attempted to apply the theory to gases at higher temperatures and pressures.[...]

Abstract This thesis summarizes certain boundary element methods applied to some initial and boundary value problems. Our model problem is the two-dimensional homogeneous heat conduction problem with vanishing initial data. We use the heat potential representation of the solution. The given boundary conditions, as well as the choice of the representation formula, yield various boundary integral equations. For the sake of simplicity, we use the direct boundary integral approach, where the unknown boundary density appearing in the boundary integral equation is a quantity of physical meaning. We consider two different sets of boundary conditions, the Dirichlet problem, where the boundary temperature is given and the Neumann problem, where the heat flux across the boundary is given. Even a nonlinear Neumann condition satisfying certain monotonicity and growth conditions is possible. The approach yields a nonlinear boundary integral equation of the second kind. In the stationary case, the model problem reduces to a potential problem with a nonlinear Neumann condition. We use the spaces of smoothest splines as trial functions. The nonlinearity is approximated by using the L2-orthogonal projection. The resulting collocation scheme retains the optimal L2-convergence. Numerical experiments are in agreement with this result. This approach generalizes to the time dependent case. The trial functions are tensor products of piecewise linear and piecewise constant splines. The proposed projection method uses interpolation with respect to the space variable and the orthogonal projection with respect to the time variable. Compared to the Galerkin method, this approach simplifies the realization of the discrete matrix equations. In addition, the rate of the convergence is of optimal order. On the other hand, the Dirichlet problem, where the boundary temperature is given, leads to a single layer heat operator equation of the first kind. In the first approach, we use tensor products of piecewise linear splines as trial functions with collocation at the nodal points. Stability and suboptimal L2-convergence of the method were proved in the case of a circular domain. Numerical experiments indicate the expected quadratic L2-convergence. Later, a Petrov-Galerkin approach was proposed, where the trial functions were tensor products of piecewise linear and piecewise constant splines. The resulting approximative scheme is stable and convergent. The analysis has been carried out in the cases of the single layer heat operator and the hypersingular heat operator. The rate of the convergence with respect to the L2-norm is also here of suboptimal order.

Continuous wave thermal propagation experiments were made with two single crystals of bismuth at frequencies up to 7 kHz. The experiments were performed at temperatures close to 4 K (i. e. close to the dielectric-like thermal conductivity peak). Accurate phase shift measurements were made in order to permit the detection of small departures from diffusive propagation. Attenuation measurements were also made. A summary of some microscopic theories of time-dependent thermal propagation in dielectric crystals is given. It is concluded that, for dielectric crystals in both the "hydrodynamic" and "ballistic" phonon gas regimes, the initial deviations from diffusive propagation will be described by a modified heat equation of the Vernotte type: [formula omitted] with appropriate identifications of the relaxation time. The possibility that the small numbers of charge carriers present in bismuth might lead to different forms of deviation is explored. Several types of thin-film insulating layers and superconducting alloy thermometers were investigated. Kodak Photo-Resist was found to be the most useful insulating material. This was used in conjunction with constantan heater films and Pb-In alloy thermometer films. The heat wave detection system employed a radio frequency thermometer bias current, a radio frequency tuned circuit, an envelope detector and phase-sensitive detection of the audio frequency heat wave signals. Heat wave phase lags were measured with a precision of 1°, using the phase-sensitive detector as a null detector. The measurements were analyzed in terms of a thermal transmission line model based on the modified heat equation given above. The electrical analogue of τ in such a model is L/R. A thermal leakage conductance term ⩋(electrical analogue G/C) was included in the model. The results at low frequencies were in excellent agreement with those expected on the basis of the transmission line model under conditions of diffusive propagation at high attenuations. Values of the apparent diffusivity obtained from these measurements were in reasonable agreement with the results of D. C. experiments made by other workers on comparable specimens. The quantity ⩋/ω was shown to be small at all frequencies used. Phase lag measurements at higher frequencies indicated significant departures from diffusive propagation in both crystals. (The crystals had different orientations.) The measurements in this range suggested a harmonic-wave-like mode of propagation. This mode appeared to break down at the highest frequencies examined. Evidence is presented to show that the observed deviations reflected thermal properties of the bismuth crystals rather than properties of the thin films, or spurious electrical effects. The apparent wave velocities were lower, and the corresponding relaxation times were longer than those predicted on the basis of the microscopic theories and from the diffusivity values obtained at low frequencies. In view of these numerical discrepancies, it is suggested that the wave-like mode could be a mode peculiar to the bismuth system, rather than the "second sound" mode predicted for ideal dielectrics. Some further experiments are suggested. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The thermal conductivity of three crystals of cesium iodide ranging in size from three to eight millimeters diameter was measured in the temperature range 1.15°K to 5.40°K. Thermal conductivity measurements were made using the thermal potentiometer method. Differences in the thermal conductivity of the three samples were interpreted in terms of phonon scattering from the boundaries of the crystals, and from internal structure defects. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

Heat pulse experiments have been made on Al₂O₃ single crystals in the temperature range 3.8° K to 35°K with the aim of gaining further insight into the nature of heat transport in solids at low temperatures. Short heat pulses were produced by heating a thin metal film evaporated on to one end of the crystal. The thermal pulse arriving at the other end of the crystal was detected by an indium film thermometer placed in a coil connected to a sensitive radio-frequency bridge, so that the variation of resistance was finally displayed on an oscilloscope. The pulses received at low temperatures (3.8°K to 8°K) show two quite separate parts, an initial sharp rise followed by a slow rise, starting at a definite delay time corresponding to the phonon velocity in the medium. The results up to 18°K do not show appreciable variation in delay time, showing that the heat pulse propagation has not entered a second sound region. As the temperature increases, the amplitude of the initial phonon pulse decreases very much compared with the amplitude of the slow rise. Above 18°K, the small sharp rise can no longer be seen clearly so that the delay time is no longer well defined, and at 30°K only the slow rise is observed. It is found that the conventional theory of heat conduction is inadequate to interpret our results at low temperatures, as it fails to predict the finite delay of the initial rise of the received pulse. A phenomenological approach is taken, using a modified heat equation which has an electrical transmission line analogy. Using Laplace transforms, a solution is obtained and the results calculated with a computer are compared with the experimental curves. It is found that the pulse shape can be interpreted quite satisfactorily, especially at the lowest temperatures. The thermal diffusivity, D, for different temperatures is found, and the apparent thermal conductivity, K, is calculated and compared with Herman's (1955) results. The solution of the modified heat equation is also calculated for liquid He II at 0.25°K and compared with the heat pulses observed by Kramers et al (1954); very good agreement is obtained. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

A one-dimensional, single phase Stefan Problem is considered. This problem is shown to have a unique solution which depends continuously on the boundary data. In addition two algorithms are formulated for its approximate numerical solution. The first algorithm (the Similarity Algorithm), which is based on Similarity, is shown to converge with order of convergence between one half and one. Moreover, numerical examples illustrating various aspects of this algorithm are presented. In particular, modifications to the algorithm which are suggested by the proof of convergence are shown to improve the numerical results significantly. Furthermore, a brief comparison is made between the algorithm and a well-known difference scheme. The second algorithm (a Collocation Scheme) results from an attempt to reduce the problem to a set of ordinary differential equations. It is observed that this set of ordinary differential equations is stiff. Moreover, numerical examples indicate that this is a high order scheme capable of achieving very accurate approximations. It is observed that the apparent stiffness of the system of ordinary differential equations renders this second algorithm relatively inefficient. / Science, Faculty of / Statistics, Department of / Graduate

A research thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfillment of the requirement for the degree of Doctor of Philosophy. August 7, 2013. / In this thesis we consider the construction of exact solutions for models describing heat transfer through extended surfaces (fins). The interest in the solutions of the heat transfer in extended surfaces is never ending. Perhaps this is because of the vast application of these surfaces in engineering and industrial processes. Throughout this thesis, we assume that both thermal conductivity and heat transfer are temperature dependent. As such the resulting energy balance equations are nonlinear. We attempt to construct exact solutions for these nonlinear models using the theory of Lie symmetry analysis of differential equations. Firstly, we perform preliminary group classification of the steady state problem to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended by one element. Some reductions are performed and invariant solutions that satisfy the Dirichlet boundary condition at one end and the Neumann boundary condition at the other, are constructed. Secondly, we consider the transient state heat transfer in longitudinal rectangular fins. Here the imposed boundary conditions are the step change in the base temperature and the step change in base heat flow. We employ the local and nonlocal symmetry techniques to analyze the problem at hand. In one case the reduced equation transforms to the tractable Ermakov-Pinney equation. Nonlocal symmetries are admitted when some arbitrary constants appearing in the governing equations are specified. The exact steady state solutions which satisfy the prescribed boundary conditions are constructed. Since the obtained exact solutions for the transient state satisfy only the zero initial temperature and adiabatic boundary condition at the fin tip, we sought numerical solutions. Lastly, we considered the one dimensional steady state heat transfer in fins of different profiles. Some transformation linearizes the problem when the thermal conductivity is a differential consequence of the heat transfer coefficient, and exact solutions are determined. Classical Lie point symmetry methods are employed for the problem which is not linearizable. Some reductions are performed and invariant solutions are constructed. The effects of the thermo-geometric fin parameter and the power law exponent on temperature distribution are studied in all these problems. Furthermore, the fin efficiency and heat flux are analyzed.

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