Axial Heat Conduction Research Articles

Computational modelling of thermal transport phenomena in continuous casting process based on non-orthogonal control volume approach

An efficient computational simulation scheme based on non-orthogonal control volume discretization and co-ordinate transformation techniques has been developed for solving the thermal transport phenomena, which involves tracking of the interface between solid and liquid phases (solidification front) and evaluation of the temperature profile during continuous casting operation. Conservation equations are reformulated in differential-integral form in terms of the transformed co-ordinates. All the terms arising from the non-orthogonality of the control volume have been retained in the numerical solution methodology, and a front tracking procedure has been formulated based on an iterative solution scheme. The formulation has been applied to solve the thermal transport phenomena in solidification processing of an Al-Mg alloy cylindrical ingot during continuous casting, which also includes axial conduction of heat. Theoretical evaluation of the solidification front and temperature distribution in the ingot are in good agreement with the experimentally measured data

Time constants in thermal laser medicine: II. Distributions of time constants and thermal relaxation of tissue

The thermal response of a semi-infinite medium in air, irradiated by laser light in a cylindrical geometry, cannot accurately be approximated by single radial and axial time constants for heat conduction. This report presents an analytical analysis of heat conduction where the thermal response is expressed in terms of distributions over radial and axial time constants. The source term for heat production is written as the product of a Gaussian shaped radial term and an exponentially shaped axial term. The two terms are expanded in integrals over eigenfunctions of the radial and axial parts of the Laplace heat conduction operator. The result is a double integral over the coupled distributions of the two time constants to compute the temperature rise as a function of time and of axial and radial positions. The distribution of axial time constants is a homogeneous slowly decreasing function of spatial frequency indicating that one single axial time constant cannot reasonably characterize axial heat conduction. The distribution of radial time constants is a function centred around a distinguished maximum in the spatial frequency close to the single radial time constant value used previously. This suggests that one radial time constant to characterize radial heat conduction may be a useful concept. Special cases have been evaluated analytically, such as short and long irradiation times, axial or radial heat conduction (shallow or deep penetrating laser beams) and, especially, thermal relaxation (cooling) of the tissue. For shallow penetrating laser beams the asymptotic cooling rate is confirmed to be proportional to which approaches for , where t is the time and is the laser pulse duration. For deep penetrating beams this is proportional to . For intermediate penetration, i.e. penetration depths about equal to spot size diameters, this is proportional to . The double integral has been evaluated numerically and the results have been compared with the various approximations available including the new results and the single time constant model. The present analysis completes our previous work, presents a closed-form formulation for the non-ablative thermal response of laser irradiated tissue and provides insight into the practical value of using time constants for representing heat conduction effects, in particular for the rate of cooling of the tissue surface.

The Conjugate Graetz Problem With Axial Conduction

Our study is motivated by recent investigations of heat transfer in microheat exchangers fabricated on silicon wafers. The objective is to assess the importance of axial heat conduction in both the solid and the liquid. To this end, we studied slug and Poiseuille flows between parallel plates and in circular cylinders

An extract analytical solution for the extended turbulent Graetz problem with Dirichlet wall boundary conditions for pipe and channel flows

For turbulent flows in ducts axial heat conduction effects within the fluid can be important for low Prandtl number fluids (liquid metals). The paper presents an entirely analytical solution to the extended turbulent Graetz problem with Dirichlet wall boundary conditions. The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation for turbulent flow into a pair of first-order partial differential equations. The present approach, which is based on the solution method of Papoutsakis et al. for laminar pipe flow, is not plagued by any uncertainties arising from expansions in terms of eigenfunctions belonging to a nonselfadjoint operator. The obtained analytical results are compared with measurements of Gilliland et al. and Sleicher et al. showing good agreement between measured and predicted values.

Laminar forced convection with sinusoidal wall heat flux distribution: axially periodic regime

Stationary and laminar forced convection in a circular tube with a sinusoidal axial distribution of wall heat flux is studied under the hypothesis that both axial heat conduction and viscous dissipation in the fluid are negligible. Two cases are considered: a sinusoidal wall heat flux distribution with a vanishing mean value; a sinusoidal wall heat flux distribution which does not change its sign. In both cases, the temperature field and the local Nusselt number are evaluated analytically in the fully developed region, i.e. where the local Nusselt number depends periodically on the axial coordinate. It is shown that, in the first case, the fully developed region presents an infinite sequence of axial positions where the local Nusselt number is singular. In these positions, the wall heat flux has a non-vanishing value even if the wall temperature equals the bulk temperature.

A transient-state technique for the heat transfer coefficient measurement in a corrugated plate heat exchanger channel based on frequency response and residence time distribution

Transient techniques are widely used for the measurement of heat transfer coefficient in heat exchangers: using such methods generally leads to simplified experimental procedures. The measured coefficient is obtained by minimizing the distance between a model of the system and experimental data. In this paper, we use such a transient-state technique to measure the global heat exchange coefficient between a liquid and corrugated plates. The model usually employed in the heat exchangers area is based on the hypothesis that the fluid flow is similar to a plug flow. Axial dispersion is then considered by defining an apparent axial dispersion coefficient in the fluid and by considering axial or transversal heat conduction in the solid. We propose here to model the fluid flow by an equivalent flow pattern obtained by inert tracer experiments. Such a representation, similar to those used to model flows through chemical reactors, is then completed by the heat transfer model in the solid. In the latter, only transversal conduction is considered. The model that we derive is in fact similar to those used to model percolation processes through porous media. We give some experimental evidences based on pressure drop measurements that such a comparison is correct. The frequency response is then used to estimate the heat transfer coefficient between the fluid and the solid. The values obtained by this method are very close to that already known for such corrugated plates.

Catalytic honeycomb reactors with electrical heating

Experimental data on the ignition and time course of temperatures and conversion for CO oxidation both on an electrically heated ceramic monolith reactor and the system metallic monolith (preheater)—ceramic monolith (afterburner) and presented. Three types of nonstationary two-phase models are described, numerical algorithms for their solution are discussed and their applicability to description of experimental data is evaluated. The spatially one-dimensional AD model, including axial dispersion in the gas phase and axial heat conduction in the solid phase, is recommended particularly for its robust computational properties.

Transient lumped heat pipe analyses

A transient lumped heat pipe formulation for conventional heat pipes is presented and the lumped analytical solutions for different boundary conditions at the evaporator and condenser are given. For high temperature heat pipes with a radiative boundary condition at the condenser, a nonlinear ordinary differential equation is solved. In an attempt to reduce computer demands, a transient lumped conductive model has been developed for noncondensible gas-loaded heat pipes. The lumped flat-front transient model was extended by accounting for axial heat conduction across the sharp vapor-gas interface. The analytical solutions for conventional and gas-loaded heat pipes were compared with the corresponding numerical results of the full two-dimensional conservation equations and experimental data, with good agreement.

Effects of Axial Heat Conduction in the Metal Wall of a Detonator Delay Element

Abstract We investigate the effect of axial heat conduction in the metal wall of a detonator delay element on the speed of a deflagration wave. An axisymmetric time dependent model is formulated in which the reaction rate is described by a single step kinetic equation. Numerical solutions are obtained and it is found that the flame speed is reduced below its one-dimensional adiabatic value at early times due to heat loss into the cold wall, but exceeds the adiabatic value at later times because the unreacted pyrotechnic ahead of the flame is preheated by radial conduction. These effects happen to almost cancel out for a typical 8s delay element. It is also shown that the wall temperature can exceed the decomposition temperature of the pyrotechnic leading to the onset of a radial flame. A simplified analytical model is presented which allows the wall temperature to be estimated.

Conjugated Periodic Turbulent Forced Convection in a Parallel Plate Channel

The transient conjugated turbulent heat transfer with axial conduction in the wall and convection boundary conditions is solved with the generalized integral transform technique for the flow of a Newtonian fluid in a parallel-plate duct subjected to periodically varying inlet temperature. A lumped model that neglects transverse temperature gradients in the solid, but takes into account the axial heat conduction along the wall, is adopted. Accurate numerical results are presented for the fluid bulk temperature, wall temperature, and wall heat flux. The effects of the conjugation parameter, fluid-to-solid heat capacitance ratio, and Biot number on the behavior of the periodic responses are investigated.

Measurement of heat transfer coefficient and axial dispersion coefficient using temperature oscillations

A periodic transient test technique based on the axial dispersion model is proposed for the determination of both heat transfer coefficients and axial dispersion coefficients in heat exchangers. The model uses a parameter called the axial dispersive Peclet number to account for the deviation of the flow pattern from ideal plug flow. It takes both axial dispersion in the fluid and axial heat conduction in the wall into account and is solved analytically by means of a complex Fourier transform. Experiments conducted on dented copper tubes show that axial dispersion has a significant effect on the dynamic temperature response of a heat exchanger.

Frequency response of a supported thermocouple wire: Effects of axial conduction

Theoretical expressions are derived for the steady-state frequency response of a supported thermocouple wire. In particular, the effects of axial heat conduction are demonstrated for both a supported one material wire and a two material wire with unequal material properties across the junction. For the case of a one material supported wire, an exact solution is derived which compares favorably with an approximate expression that only matches temperatures at the support junction. Moreover, for the case of a two material supported wire, an analytical expression is derived that closely correlates numerical results.

Temperature Distribution Inside a Solar Collector Storage Tank of Finite Wall Thickness

A mathematical model for the transient conjugated behavior of a hot-water storage tank having finite wall thickness is presented. An analytical closed-form solution for the temperature field within the tank is obtained. This solution takes into consideration the axial conduction of heat in both fluid and solid wall and the heat capacity of the solid wall. Plots of the solution show that finite wall thickness tends, as expected, to decrease the thermal stratification within the tank. This effect becomes less apparent at high Peclet numbers.

Measurement of frequency response in short thermocouple wires

Experimental measurements are made for the steady-state frequency response of a supported thermocouple wire. In particular, the effects of axial heat conduction are demonstrated for both a supported one material wire (type K) and a two material wire (type T) with unequal material properties across the junction. The data for the amplitude ratio and phase angle are correlated to within 10% with the theoretical predictions of Fralick and Forney (1992) for a laser-heated junction with no bead. This is accomplished by choosing a natural frequency ωn for the wire data to correlate the first-order response at large gas temperature frequencies. Moreover, it is found that a large bead size (roughly twice the wire diameter) will increase the amplitude ratio at low frequencies and distort the phase angle. In the latter case, it is necessary to increase the thermocouple wire length by 20% in the theory to correlate the data when the junctions have beads (silver soldered).

Effect of axial solid heat conduction and mass diffusion in a rotary heat and mass regenerator

This research analyzes the effect of axial heat conduction and mass diffusion on the performance of a solid desiccant wheel. A one-dimensional transient heat and mass transfer model which contains four nonlinear partial differential equations is developed. The equations are solved using a full-implicit finite-difference thod. A limiting case with a zero solid diffusion resistance is considered. Three different regeneration temperatures, namely 26, 50 and 85 C, are considered. The parameter, ( λ 4 2 Ntu/Bi), is found to be the most important factor governing the axial heat conduction and mass diffusion effect.

Combined Free and Forced Convection of Low-Prandtl-Number Fluids in a Vertical Pipe.

Numerical analyses have been conducted for combined free and forced laminar convection of low-Prandtl-number fluids in a vertical pipe. The effects of both buoyancy force and axial heat conduction on the hydrodynamic and thermal characteristics are systematically investigated for the fluid heating and cooling with upward flow at Pe=0.7 to 71(Pγ=0.007 to 0.71 with Re=50 and 100) and |Gγ/Re|=10 to 500. At relatively high |Gγ/Re|, flow reversals are produced at the center of a pipe and near the wall in the heating and cooling cases, respectively. The regime of reverse flow is identified for both heating and cooling cases in Pe vs |Gγ/Re| coordinates. The axial conduction effect becomes significant and even dominant at very low Pe numbers. Numerical results are presented for velocity profiles, streamlines showing the occurrence of flow recirculation, temperature profiles, and distributions of the Nusselt number and wall shear stress with |Gγ/Re| as a parameter.

Hot spot locations and temperature distributions in a forced convection photochemical reactor

We have computed steady axisymmetric temperature distributions in a tubular photoreactor in which fluid in laminar flow absorbs light from an azimuthally uniform, radially incident light source. In addition to the primary on-axis temperature maximum, a second hot spot forms off the centerline as the optical density γ increases. This results from a shift of light absorption toward the wall, which is held at a fixed temperature (e.g. by forced convection cooling on the outside). At still larger γ the centerline hot spot disappears, leading to a decrease in the maximum temperature and accompanied by an increase in the temperature near the wall. There is a critical value of γ, depending on the Peclet number Pe, below which the hot spot is located on the centerline and above which a second hot spot forms and moves off the centerline. The reduction in maximum centerline temperature may be advantageous for product selectivity and yield in the central core of the reactor, but may be disadvantageous with respect to the formation of light-absorbing deposits on the interior of the tube wall. Axial heat conduction has significant effects for Pe < 50, including displacement of the hot spot upstream from the exit plane of the illuminated section, as well as an increase in the maximum temperature.

Conjugated turbulent heat transfer with axial conduction in wall and convection boundary conditions in a parallel-plate channel

The steady-state conjugated turbulent heat transfer with axial conduction in the wall and convection boundary conditions is solved with the generalized integral transform technique for the flow of Newtonian fluid in parallel-plate duct. A lumped wall model that neglects transverse temperature gradients in the solid but that takes into account the axial heat conduction along the wall is adopted. Highly accurate results are presented for the fluid bulk and wall temperatures and Nusselt number. The effects of the conjugation parameter, Biot number, and the dimensionless channel length on Nusselt number and fluid bulk and wall temperatures are systematically investigated.

Analytical solutions for transient conjugated heat transfer in parallel plate and circular ducts

Taking into consideration the axial conduction of heat in both fluid and solid wall and the heat capacity of the solid wall, an exact analytical solution has been obtained for the unsteady conjugated forced convection heat transfer in a steady slug flow inside circular and parallel plate ducts. This analytical exact solution considers transients initiated by time-variations of inlet fluid temperature.

Thermal radiation and laminar forced convection in the entrace region of a pipe with axial conduction and radiation

Combined forced convection and thermal radiation in the thermal entrance region of a circular pipe with axial heat conduction and thermal radiation are numerically studied in this article. The fluid is treated as a gray, absorbing and emitting medium with a fully developed velocity entering an isothermal semi-infinite long pipe. The method of moments is applied to approximately model the radiative heat transport process. The governing equations are solved by applying the SIMPLE algorithm. The effects of the Peclet number and its interaction with the conduction-radiation parameter and optical thickness on heat transfer behavior in the thermal entrance region are also investigated. The results obtained indicate that radiation plays more important role at a low Peclet number than at a high Peclet number and that axial radition is negligible when the Peclet number N>20. The numerical results for limited cases of no axial heat conduction and radiation are also compared with the available data published in open literature. In all cases, good agreement with these studies was obtained.