Steady-state Heat Flow Equation Research Articles

Geotherms from the temperature-depth–constrained solutions of 1-D steady-state heat-flow equation

We use the Curie depth derived from spectral analysis of near-surface magnetic anomaly data to constrain the solution of a one-dimensional (1-D) steady-state heat-flow equation. The method, in addition to anchoring the geotherm at deep levels in the crust, yields the ratio of radiogenic heat production to thermal conductivity. We evaluate the utility of this constraint in two granitic regions in the United States where radiogenic heat production, thermal conductivity, and heat flow are well determined. We also examine the results of this method in the context of previous research in the central Red Sea and coastal Saudi Arabia. In a test case using data from New Hampshire, USA, the steady-state approximation applies, and we calculate heat-production values (∼8 μW/m3) that agree with measurements on samples of Conway granite in the region. In the second example, our Wyoming geotherm is hotter by ∼200 °C at the Curie depth, possibly reflecting the 15 km thicker crust (as observed from the dense EarthScope USArray stations in comparison to only two earlier refraction profiles) and the consequent greater radiogenic heat production in the lower crust. In the Colorado Front Range, our radiogenic heat-production value is somewhat higher than predicted from previous heat-flow studies and may reflect heat advected by intrusions. Our results indicate that basal crustal temperatures in northern Colorado may be close to solidus of rocks of felsic composition, a scenario that is consistent with the geological history of the region and has important implications for crustal rheology. Our estimates of depth-integrated heat production from the crustal columns (25 and 55 mW/m2 Wyoming and Colorado, respectively) also agree well with estimates from previous studies.

Using the collage method to solve one-dimensional two-point boundary value problems at steady-state

We consider an inverse problem for two-point boundary value problems (BVPs) D ( u ) = f ( x ) − q ( x ) u ( x ) , u ( A ) = u A , u ( B ) = u B , where the goal is to recover an approximation of the physically meaningful functions of x and parameters in the differential operator D . We rewrite the BVP inverse problem as an initial value problem (IVP) inverse problem and recast this inverse problem in terms of approximating a “target” function by the fixed point of a contractive operator on a complete metric space. We formulate different collage coding frameworks to solve such problems. Although the approach has broad applicability, we focus on two model problems: approximating the variable diffusivity in a steady-state heat-flow equation, and approximating the flexural rigidity in a steady-state vibrational beam. We present two examples each treated with three different frameworks.

On the thermal impact of gas venting and hydrate crystallization

At the gas venting rates observed in some seafloor localities, heat advection and the latent heat of hydrate crystallization could significantly warm the subsurface, reduce the hydrate stability zone thickness, and decompose previously crystallized hydrate. Effects of heat advection and the latent heat of hydrate crystallization are limited by the duration of time over which venting occurs and ultimately by the time it takes to plug vents with hydrate, however. We examine the connection between gas venting, hydrate crystallization, and subsurface temperature using a new one‐dimensional analytical solution to the steady state heat flow equation with constant hydrate crystallization, and we examine the effects of the duration of venting employing one‐dimensional finite element solutions to the transient heat flow equation that include space‐ and time‐dependent hydrate dissolution and crystallization. We show that if lateral losses of heat are negligible, hydrate crystallization and the advection of heat by gas flow at the more vigorous rates of gas discharge observed on the seafloor could increase the temperature gradient near the surface by more than an order of magnitude and decrease the feed gas hydrate stability zone thickness from ∼570 to <200 m before the vents plug with hydrate. We compute how the chemistry of the venting gas and crystallizing hydrate evolve as a vent plugs with hydrate.

Thermal behavior using analytical and numerical model in multi-finger power heterojuction Bipolar transistors

The Transmission Line Matrix (TLM) explicit method has been used to model the transient thermal proprieties of various microwave HBT’s power structure. Control of the self-heating is of a paramount importance and changes the electronic performance of the device. The thermal effect is important to the AlGaAs / GaAs multiple emitter heterojunction bipolar transistors (HBT’s) due to the high current handling capability of such devices. Aimed toward predicting the HBT behavior and physical trends, we developed an analytical model incorporating the high current as well as thermal effects. The temperature distribution along the depth direction of the substrate is obtained from a three dimensional analysis by solving the steady-state heat flow equation. In this paper, an investigation of the thermal behavior of single emitter and multiemitter finger HBT microwave power device is exposed.

Thermal Modelling of Packaged Silicon Millimeterwave IMPATT Diode—CW Case

Temperature distribution in millimeterwave IMPATT diode structure and the conventional package assembly comprising of gold and diamond heatsinks, is computed numerically in quasi-three dimensions, for CW mode of operation. Steady state heat flow equation is descretised using finite difference code and the resulting equations are solved by Successive Over Relaxation (SCR) technique with optimised parameters. Effect of intermediate layer of gold, commonly used in the packaging of these diodes, is studied for optimal thermal resistance of the packaged diode.

An exact recursion relation solution for the steady-state surface temperature of a general multilayer structure

A recursion relation technique has been used in the past to determine the surface potential from the multilayer electrical Laplace equation. This has provided for a vastly simplified evaluation of the electrical spreading resistance and four-probe resistance. The isomorphism of the multilayer Laplace equation and the multilayer steady-state heat flow equation suggests the possibility of developing a recursion relation applicable to the multilayer thermal problem. This recursive technique is developed and is shown to provide the surface temperature of the multilayer steady-state heat flow equation. For the three-layer case, the thermal recursion relation readily yields the surface results,which are identical with those presented by Kokkas and the TXYZ thermal code. This recursive technique can be used with any number of layers while incurring only a small increase in computation time for each added layer. For the case of complete, uniform top surface coverage by a heat source, the technique gives rise to the generalized one-dimensional thermal resistance result. An example of the use of the new recursive method is provided by the preliminary calculations of the surface temperature of a buried oxide (SOI, SIMOX) structure containing several thicknesses of the surface silicon layers. This new technique should prove useful in the investigation and understanding of the steady-state thermal response of modern multilayer microelectronic structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerical model calculations of the effects of grain sizes and orientations on the thermal conductivities of composites

A numerical method is used to investigate the effects of particle sizes and orientations on the thermal conductivities of two-component composites. The method is based on a finite-difference simulation of the divided-bar thermal conductivity measuring device, and solves the steady-state heatflow equation in two dimensions throughout a region representing the divided bar with included sample. Calculations are made for two-component particle sizes for two orientations and for random distributions. The results are compared with those from a geometric model in which the conductivity of the composite depends on the volume fractions of the different components and with conductivities calculated using realspace renormalization group theory (RSRG) and the effective medium approximation (EMA). The results show substantial differences between samples with different particle orientation and sizes. For the particular distributions examined here, differences of up to a factor of one and one-half occur in some instances.

Prediction and measurement of the thermal conductivity of composite materials

The transverse thermal conductivity of continuous fibre composite materials has been described in terms of a number of models which yield analytic expressions that are simple to evaluate but limited in application. A more general solution to the two-dimensional, steady-state heat flow equation is obtained by using the finite difference method. This involves extensive computing but this method has the advantage that a rigorous solution is obtained. The results of some experimental measurements are considered and a solution to a measurement problem for composites with fibres at any angle to the imposed temperature gradient is discussed. Heat flow in anisotropic composite materials in directions at any angle to the axes of symmetry is discussed.

Electron beam melting in microfocus X-ray tubes

Using a general expression for the spatial dependence of the temperature determined by a steady-state heat flow equation for a cylindrical X-ray target anode with dimensions much larger than the incident electron beam radius, the radius of the melt region on the anode surface has been calculated as a function of the total incident electron beam power. These calculations are shown to be in excellent agreement with experimentally measured values of the melt radius for tungsten, copper, and molybdenum target anode materials which indicates that electron beam melting is well described using the steady-state heat flow equation. A novel method is also presented for the unambiguous determination of the incident electron beam radius, and therefore the X-ray source size, which makes use of the dependence of the onset of target anode melting on the electron beam radius. This approach is shown to yield consistent values for the electron beam radius on tungsten, copper, and molybdenum target anodes.

Inter-diffusion coefficients parallel to the C-Axis in iron-rich clinopyroxenes calculated from microstructures

Subsolidus marginal zoning in calcium-poor clinopyroxenes and intermediate zoning in discontinuously zoned subcalcic- to calcium-rich clinopyroxenes from ironrich igneous rocks is used to calculate the interdiffusion coefficient, DCa−(Fe,Mg), parallel to the crystallographic caxis. Wagner's mathematical models describing the displacement of interfaces in solids as the result of isothermal diffusion are adopted. The steady-state heat flow equation is used to approximate the diffusion times. The calculated interdiffusion coefficients are of a reasonable order of magnitude, viz. 6.0×10−20−2.0×10−17cm2· sec−1 at about 900° C.