One-dimensional Heat Diffusion Research Articles

Stability of superconductors cooled by supercritical helium in rapidly changing magnetic fields

The stability behavior of superconductors with respect to disturbances extended over a large magnet volume, e.g., by fast magnetic field changes, is discussed. Cooling by supercritical helium is considered, which is the usual cooling medium in cable-in-conduit conductors. On a time scale of a few milliseconds the heat transfer is mainly determined by heat diffusion. The solution of the one-dimensional heat diffusion equation allows the calculation of the stability limit as a function of the conductor parameters and of the external field change. Calculations for exponential and sinusoidal field variations were performed and compared with available experimental data. An easy-to-use stability criterion is given for exponential heat load.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Pulsed heating of surfaces: Comparison between numerical simulation, analytical models, and experiments

A general numerical solution and two approximate analytical solutions of the one-dimensional heat diffusion equation are used to calculate the temperature of a surface exposed to pulsed heating. The calculations simulate the case of pulsed CO 2 laser heating of fused quartz. The numerical solution (based on a finite difference model) includes explicitly the temperature dependence of thermal parameters and is in excellent agreement with experimental measurements on the same system. The analytical solutions are unable to reproduce the temperature history of the sample, and their limitations are discussed.

Determination of the thermal diffusivity and specific heat using an exponential heat pulse, including heat-loss effects

The one-dimensional heat diffusion equation has been solved analytically for the case of a heat pulse of the form F(t) = exp(−t/τ)/τ applied to the front face of a homogeneous body including the effects of heat loss from the front and back faces. Approximate expressions are presented which yield a simple, accurate technique for the determination of the thermal diffusivity and specific heat, suitable to a wide range of heat-pulse time constant and heat-loss parameters, without recourse to graphical techniques or requiring further computer analysis. A procedure is described for the determination of an effective time constant to allow application of the present results to the case of a nonexponential heat pulse. Experimental results supporting the theoretical analysis are presented for five samples of silicon germanium alloys of various thicknesses, determined using a xenon flash tube heat-pulse exhibiting an exponential dependence. Proper consideration of the experimental heat pulse shape is shown to lead to reliable corrections to the apparent thermal diffusivity, even for relatively long heat-pulse times.

Approximate solution of nonlinear transient heat conduction in one dimension

Three approximate methods, viz. the Heat Balance Integral method (HBI), the Modified Heat Balance Integral method (MHBI) and the Double Integral Method (DIM) in combination with hybrid profiles (HP - consisting of an exponential and polynomial) are evaluated for relative merits in solving the onedimensional heat diffusion equation. Applications to two linear test problems (TP1 - semi-infinite solid with constant heat flux at its boundary, TP 2 - semi-infinite solid with step change in its surface temperature) yield the HP-MHBI as the most desirable combination. Applications to nonlinear problems show that these combinations are ideally suited for obtaing reliable and accurate appoximate solutions.

Measurement of thermal diffusivity for polymer film by flash radiometry

A new method to determine the thermal diffusivity of polymer film by flash radiometry is presented. The transient infrared radiation from the sample surface was measured after the sample film was heated by a short-duration pulsed light. The thermal diffusivity was determined by fitting the transient infrared signal to the theoretical curve of a one-dimensional heat diffusion model. The transient infrared signal of polymethylmethacrylate (PMMA) film could be well fitted by the theoretical curve using appropriate thermal diffusivity in the wide temperature range between 100 K and near the softening point of PMMA. The temperature dependence of thermal diffusivity of PMMA film showed that the thermal diffusivity sensitively reflects the molecular motion of polymer chain.

Approximate solution of one-dimensional heat diffusion problems via hybrid profiles

This paper presents hybrid temperature profiles, a combination of an exponential and a polynomial, capable of providing highly accurate but approximate solutions to one-dimensional heat diffusion problems by the use of the heat balance integral method. After establishing their capability in the case of two test problems, namely, an initially isothermal semi-infinite medium subject to either a specified heat flux or temperature at its surface, two typical applications are considered. These show that the hybrid profile-heat balance integral combination is an excellent approximation that may be used as a first step before launching on a numerical solution in cases that do not have available analytical solutions.

Numerical simulation of cometary nuclei: III. Internal temperatures of cometary nuclei

A one-dimensional heat-diffusion model was used to calculate internal temperatures in cometary nuclei composed of either crystalline or amorphous ice, and for a range of orbits. It was found that the final central temperature, T c, was a complex function of the comet's orbital semimajor axis, a, and eccentricity, e, as well as the functional form of the thermal conductivity. For cometary nuclei with identical thermal properties, T c was found to decrease with eccentricity for a short-period orbit with a = 3 AU. For an intermediate-period orbit with a = 20 AU, T c initially increased with eccentricity but then declined at large values of e for a crystalline ice nucleus, while for amorphous ice T c increased monotonically. In addition, it was found that for conductivities of similar magnitude, crystalline ice (for which the conductivity varies inversely proportional to temperature) reached the final central temperature twice as fast as amorphouslike ice (for which the conductivity is proportional to temperature). T c also depended on the magnitude of the conductivity. A four- to fivefold decrease in the conductivity resulted in a 3–4°K decrease in T c at large eccentricities, while at small eccentricities T c was only weakly dependent on the conductivity. Finally, the numerical results are compared to the analytical solutions of J. Klinger (1981, Icarus 47, 320–324) and C. P. McKay, S. W. Squyres, and R. T. Reynolds (1986, Icarus, 66, 625–629), and a numerical correction factor is derived for the McKay et al. expression for the central temperature.

Ultrasonic determination of thermodynamic threshold parameters for irreversible cutaneous burns.

In vivo ultrasonic measurements of the depth of conductive cutaneous burns experimentally induced in anesthetized Yorkshire pigs are reported as a function of burn time for the case in which the skin surface temperature is maintained at 100 degrees C. The data are used in the solution of the one-dimensional heat diffusion equation with time-dependent boundary conditions to obtain the threshold temperature (65.3 degrees C) and the energy of transformation per unit mass (408 J . g-1) associated with the transition of the tissue from the state of viability to the state of necrosis. The simplicity of the mathematical model and the expediency of the ultrasonic measurements in studies of thermal injury are emphasized.

Photoacoustic study of a UV initiated solid-state polymerization

We report a detailed photoacoustic study of a highly exothermic solid-state reaction, the photopolymerization of diacetylene crystals. Both the amplitude and the phase shift of the photoacoustic signal are monitored during the course of the polymerization at various UV excitation wavelengths and various chopping frequencies. A one-dimensional heat-diffusion model employing impedance boundary conditions is shown to fit the experimental data quite adequately. We conclude from the theoretical interpretation of the results that the quantum yield for photopolymerization is very high (∼100) and that the polymer acoustically couples much more effectively to the gas than does the monomer.

Numerical solution of a time-optimal parabolic boundary-value control problem

A special time-optimal parabolic boundary-value control problem describing a one-dimensional heat-diffusion process is solved numerically. Using a bang-bang principle recently proved by Lempio, this problem can be transformed in such a way that the variables are jumps of bang-bang controls. A discretization is performed in two steps, and the convergence of the approximate solutions is proved. Finally, an algorithm to solve the discrete problem is developed and some numerical results are discussed.

Cryogenic stability of composite conductors taking into account transient heat transfer

To predict stability behaviour of various conductors under different cooling conditions, a computer program was developed to solve the one-dimensional heat diffusion equation along the conductor using temperature dependent material parameters. Transient heat transfer is included basing on results of earlier heat transfer measurements. To check the validity of these calculations experiments were performed using a copper stabilized multifilamentary NbTi-superconductor under pool boiling conditions. The normal transition was initiated by an ohmic heater. The minimum energy needed to induce quenching as well as the propagation velocity of the normal zone are then compared with numerical results and show good agreement.

Maximum likelihood identification of a heat diffusion process—a pole clustering effect

Parametric models of a one-dimensional heat diffusion process are determined using the maximum likelihood method. The process is a linear, infinite dimensional system. Statistical teats indicate that the appropriate orders of the models obtained are relatively low. It is found empirically that successive terms in the modal expansion of the transfer function of the process, having gain factors of the same sign, are identified as a single term.

Numerical treatment of a parabolic boundary-value control problem

The purpose of this paper is to present a method to compute optimal controls for a class of one-dimensional heat-diffusion processes. The approach used is in the spirit of the Ritz method and approximates the given problem with simpler tasks which are solved by means of algorithms based on the principles of semi-infinite programming. General convergence properties of the procedures are shown. Some illustrative numerical examples are also given.

On the treatment of radiative transfer in the lunar diurnal heat flow

The earlier attempts to explain infrared observations of the temperature variation of the lunar surface in terms of heat flow analysis [Wesselink, 1948; Jaeger, 1953] consisted of the solution of the one-dimensional heat diffusion equation expressible as (1) with the specific heat, density, and thermal conductivity as constants, subject to the boundary condition at the surface imposed by solar insolation (2) in which an infrared emissivity of unity and a constant specific insolation is assumed. The unit step function, used to represent zero insolation between sunset and sunrise, is H(X), and A is the Bond albedo of the moon. The sun is treated as a point source during lunation heating and cooling. The manner of treating the sun's altitude, wt, in the equation implies that the solution is for a lunar point at which the sun passes through the zenith during the course of the given lunation. Jaeger's treatment was an extension of Wesselink's method in that two-layer cases of multiple values ofKρc were considered.