Uniform Heat Generation Research Articles

Flow instability in a large aspect ratio porous medium with an internal heat source

The problem of convective heat transfer in a fluid saturated porous medium is of great importance in various applications ranging from geothermal engineering to underground nuclear waste disposal. Previous studies in this regard were mainly done for a low range of Rayleigh number and small aspect ratio. In this paper, natural convection in a horizontal as well as inclined porous channel with uniform internal heat generation is investigated for a large aspect ratio. Numerical solutions are obtained using the Arakawa-Dufort-Frankel scheme and are mainly governed by two parameters, namely, the aspect ratio of the duct, γ = b a and the Rayleigh number Ra = KβgaA′ Q g / ανk. For a duct with an aspect ratio of 10, a complicated solution structure, i.e. the presence of stable, periodic, quasi-periodic and chaotic behavior is observed upon increasing the Rayleigh number. The results show that when the angle of inclination of a porous channel increases, chaotic behavior appears for a smaller Ra. Results also show that the flow stability increases with increasing angle of inclination.

Step-function current in a metallic foil as a step-function heat-flux source

An exact solution is derived to the one-dimensional, time-dependent, heat-conduction equation for a two-layer, semi-infinite, composite solid with uniform heat generation in the surface layer, no heat transfer through the surface plane, and uniform initial temperature. The interface between the two layers is assumed to have no thermal contact resistance. This solution enables a discussion of the ideality with which a step-function electric current in a metallic foil can generate a step-function heat flux into a contacting semi-infinite solid. Previous measurements of thermal diffusivity (based on the above conditions) have relied on the idealized constant-flux solution for data reduction. It is shown here that the temperature errors in the substrate arising from nonideality of the constant-flux boundary condition increase with depth into the substrate, foil thickness, and decreasing thermal conductivity/diffusivity of the substrate.

NATURAL CONVECTION IN A DIFFERENTIALLY HEATED SQUARE CAVITY WITH INTERNAL HEAT GENERATION

A high-resolution, finite-difference numerical study is reported on natural convection in a square cavity. The vertical sidewatts of the cavity are differentially heated, and a uniform internal heat generation is also present. Two principal parameters are considered, the internal Rayleigh number RaI, which represents the strength of the internal heat generation, and the external Rayleigh number Rag, which denotes the effect due to the differential heating of the side walls. The internal Rayleigh number varies in the range 1010 RaI ≤ 107, while the external Rayleigh number is set at RaE = 5 x 107 for most computations. As the relative strength of the internal heat generation increases, the flows near the tap portion of the heated sidewall are directed downward. When the effect of the internal heat generation is dominant, the thermal energy leaves the system for the surroundings over the top portion of the heated wall. Only in the bottom pari of the heated wall is heat transfer directed into the system. The...

Boundary Element Heat Conduction and Thermoelastic Analyses of Interface Crack in LSI Package in Consideration of Contact Between Crack Surfaces.

The interface crack initiated between the die pad (lead frame) and the resin encapsulant in the LSI package sometimes leads to package cracking and becomes critical to package fracture design. On the other hand, the property of thermal diffusion has become a main concern for recent package design of power semiconductor devices. To simulate these problems, we developed the boundary element heat conduction and thermoelastic analysis code, which can deal with thermoelastic contact of the interface crack. The stress intensity factors for an interface crack are calculated by the extrapolation method which is proposed by one of the authors. Using the above technique, the interface crack initiated in a plastic-molded LSI package is analyzed under conditions of uniform temperature loading and heat generation in the Si-chip. Calculated interfacial stress intensity factor Ki shows a decreasing trend against the interface crack length in all the above cases. Taking into account the contact of the crack surface, the results of the disturbance in temperature distribution are outstanding. However, the normalized stress intensity factor Ki/|ΔTmax| Shows almost identical behavior against with the crack length and is less dependent on the contact region length and the amount of heat generation.

The effect of water density extremum on heat transfer within a cylinder containing a heat-generating porous media

A numerical model was used to investigate the effect of the density extremum of water in a cylinder containing a heat-generating porous media. The modeled vertical cylinder had an adiabatic bottom surface with isothermal sides and top. The density of water was varied with temperature, as was the heat generation rate, aspect ratio, and boundary temperature to reveal the affects upon heat transfer and maximum temperatures. Results of the explicit finite difference model using Darcy's formulation for the momentum equations indicate that the uniform internal heat generation rate and the surface area to volume ratio determine maximum temperatures and steady-state flow patterns. At very low heat generation rates, the boundary temperature does not significantly affect the location of the maximum velocity unless the boundary temperature is at the extremum temperature of 4°C. When larger internal heat generation rates are considered, and as the surface area to volume ratio decreases, the effect of boundary temperature becomes less significant. Because the density extremum of water is so near its melting point of 0°C, the effect of the maximum density is essentially negligible for boundary temperatures above the melting point of water.

Experimental Determination of Stator Endwall Heat Transfer

Local Stanton numbers were experimentally determined for the endwall surface of a turbine vane passage. A six-vane linear cascade having vanes with an axial chord of 13.81 cm was used. Results were obtained for Reynolds numbers based on inlet velocity and axial chord between 73,000 and 495,000. The test section was connected to a low-pressure exhaust system. Ambient air was drawn into the test section; inlet velocity was controlled up to a maximum of 59.4 m/s. The effect of the inlet boundary layer thickness on the endwall heat transfer was determined for a range of test section flow rates. The liquid crystal measurement technique was used to measure heat transfer. Endwall heat transfer was determined by applying electrical power to a foil heater attached to the cascade endwall. The temperature at which the liquid crystal exhibited a specific color was known from a calibration test. Lines showing this specific color were isotherms, and because of uniform heat generation they were also lines of nearly constant heat transfer. Endwall static pressures were measured, along with surveys of total pressure and flow angles at the inlet and exit of the cascade.

Transient convection in saturated porous layers with internal heat sources

The present investigation is focused on the two-dimensional, transient behavior of convective heat transfer in porous, rectangular ducts saturated with a fluid and in which there is uniform internal heat generation. In earlier works the steady-state multiplicity features of this flow have been studied. In the present work the evolutionary path to such steady states is examined. In several cases, a sustained oscillatory behavior has been observed. The solution structure is governed by two parameters, namely the aspect ratio of the duct, γ = b/ a and the Rayleigh number, R = KβgaA′ Q g αvk. For a duct with an aspect ratio of unity, a complicated solution structure is observed upon increasing the dynamical parameter. A steady, symmetric two-cell pattern observed for R of up to 4400 gives way to a periodic regime for R of up to 5400, then to a chaotic regime over a narrow range of R and a return to a steady-state solution at R = 5800. Upon increasing γ to 8, several multiple steady-state solutions are observed. The transition to oscillatory convection occurs at an earlier value of R with increasing y. None of the oscillatory solutions are symmetric about the center line.

Energy transfer in an elliptic annulus with uniform heat generation

Energy transfer in an elliptic annulus with uniform heat generation

Heat flow provinces reconsidered

The concept of a heat flow province, an area in which there is apparently a linear relationship between near-surface heat flow and radiogenic heat generation, is considered. The relationship has been suggested for a wide variety of tectonic terrains, ranging in age from Archaean to Cenozoic. It is expressed as q = q 0 + A 0D in which q and A o are the measured heat flow and heat generation and q o and D are interpreted to be the uniform reduced heat flow from the mantle and lower crust, and D is a scale depth for the distribution of radiogenic elements. Reported values for D are in a remarkably narrow range, and are typically ∼ 10 km. However, there can be substantial uncertainties associated with the data. In particular, heat generation measurements from a limited number of sites may not adequately represent even a relatively uniform pluton. When the data are re-analysed with reasonable estimates of uncertainty associated with them, the calculated values of q o and D are, in most cases, significantly different from those calculated without regard to uncertainties in the data. Only the data from the Sierra Nevada batholith, a deep-seated plutonic complex, appear to satisfy the relationship even when their uncertainties are taken into account. With the re-analysis, D generally falls in the approximate range 10–20 km. This is also the range of depths to a commonly-observed discontinuity in the mid- to lower-crust, sometimes called the Conrad discontinuity, which suggests a possible correlation between the two. A model in which blocks of crust 10–20 km thick, depths being randomly arranged, are each assigned a uniform heat generation, randomly generated from a reasonable range, produces an apparent relationship between heat flow and heat generation that is at least as good as most of those deduced from real data. The model produces crustal temperatures that are generally considerably lower in the lower crust than those commonly calculated; this is consistent with the observations from seismic refraction that the depth to the Moho can vary widely even in stable shield environments. It is concluded that the concept of a heat flow province as originally expounded is invalid.

Thermal Convection in a Rectangular Cavity Filled With a Heat-Generating, Darcy Porous Medium

Two-dimensional, steady natural convection in a rectangular cavity filled with a heat-generating, saturated porous medium has been studied numerically for the case when the vertical walls of the cavity are isothermal and the horizontal walls are either adiabatic or cold. Results are presented in terms of the streamlines and isotherms, the maximum temperature in the cavity, and the local and overall Nusselt numbers. The buoyant flow together with the uniform heat generation produces a highly stratified medium at high Rayleigh numbers. Although the maximum temperature in the cavity θmax invariably increases with the Rayleigh number Ra and aspect ratio A, the rate of increase diminishes with this enhancement in Ra and A. However, the change in the horizontal wall boundary condition from adiabatic to cold reduces θmax. The local heat flux on the bounding walls is a strong function of the Rayleigh number, the aspect ratio, and the wall boundary conditions. The variation in overall Nusselt number is qualitatively similar to that observed in the case of a differentially heated cavity, and the present heat transfer rates are close to that for the cavity heated by applying a uniform heat flux. Several correlations are presented for maximum temperature and overall Nusselt number.

Temperature distributions in fins with uniform and non-uniform heat generation and non-uniform heat transfer coefficient

Temperature distributions in a straight fin of rectangular profile (or in a cylindrical fin) with uniform and non-uniform internal heat-generation characteristics and non-uniform heat transfer coefficients are derived analytically. The heat transfer coefficient is assumed to be a power function of the difference between the temperature of the fin and that of the fluid surrounding it. The power of this function is taken as being equal to — 1, 0, 1 and 2. Non-uniform internal heat generation is assumed to depend on the fin temperature and this dependency is expressed in a polynomial equation up to the third degree. The results obtained for uniform internal heat generation are also presented in tabular and graphical forms and an interpolation method is proposed to determine the fin temperature and the fin effectiveness if the foregoing power is a fraction between — 1 and 2.

A simplified model of heat generation during the uniaxial tensile test

The temperature rise in a sheet tensile specimen has been calculated by the finite difference method for a plain-carbon steel at various strain rates and in several environments. Prior to necking, a uniform heat generation function is used with the governing flow equation while during the post-uniform strain, an empirical heat generation function is used. The empirical function is based on a strain distribution equation generated by curve fitting of experimental data. The effect of heat transfer conditions on the temperature increase has been discussed. The maximum temperature rise in air may reach 42 K at the center of an I.F. steel specimen at a strain rate of 10-2/s. The instability strain during tensile testing has been predicted by taking into account strain hardening, strain-rate hardening, and deformationinduced heating. The results show that significant deformation heating can occur during tensile testing in air at “normal” strain rates near 10-2/s, and that the uniform elongation can be affected markedly. Predictions for other alloys based on tabulated data are also presented.

Heat transfer from exothermically reacting fluid in vertical unstirred vessels—II. Free-convection heat transfer correlations and reactor safety

The maximum temperature in exothermically reacting fluid undergoing free convection occurs just under the liquid surface. In order to avoid temperature run-away it may not exceed a critical value. Experiments at two scales show that under identical conditions the maximum and mean temperatures increase with vessel size. Based upon a heat balance and a correlation between Nusselt and Rayleigh numbers, a method is worked out to predict the maximum temperature from the exothermicity of the reaction, the physical properties of the fluid and the vessel size. Nu-Ra correlations applicable to a vertical unstirred vessel with a dished base are given for laminar and also for turbulent free convection. They were obtained from active and passive experiments and incorporate a modified Ra, suitable for uniform internal heat generation. They are similar to those for a vertical plate. An outline of some remaining problems on heat transfer in exothermically reacting fluids is given.

Multiple Solutions for Buoyancy-Induced Flow in Saturated Porous Media for Large Peclet Numbers

The problem of buoyancy-induced secondary flow in fluid-saturated porous media is examined using a numerical model. The natural convection is coupled either with a forced axial flow or uniform internal heat generation. In both cases the model equations are shown to exhibit dual solutions over certain ranges of flow parameter. In the two-parameter space of aspect ratio and Grashof number, the flow transition between the two-vortex and four-vortex pattern can be explained in terms of a tilted cusp.

Pseudo-Steady-State Natural Convection Heat Transfer Inside a Vertical Cylinder

The SIMPLER numerical method was used to calculate the pseudo-steady-state natural convection heat transfer to a fluid inside a closed vertical cylinder for which the boundary temperature was spatially uniform and the temperatures throughout the entire system were increasing at the same rate. (Pseudo-steady state is comparable to the steady-state problem for a fluid with uniform heat generation and constant wall temperature.) Stream functions, temperature contours, axial velocities, and temperature profiles are presented. The range of calculation was 0.25 < H/D < 2, Ra < 107, and Pr = 7. This range includes conduction to weak turbulence. A characteristic length defined as 6 × (volume)/(surface area) was used since it seemed to produce good regression results. The overall heat transfer for the convection-dominated range was found to be correlated by Nu = 0.519 Ra0.255, where the temperature difference for both the Nusselt and Rayleigh numbers was the center temperature minus the wall temperature. Correlations using other temperature differences are also presented for estimating the volumetric mean and minimum temperatures.

TRANSIENT BEHAVIOR OF NATURAL CONVECTION INSIDE A VERTICAL CYLINDER

The transient behavior for natural convection heat transfer within a vertical cylindrical enclosure was studied using the SIMPLER numerical algorithm. Initially, the temperature of the internal fluid was uniform. The unsteady state heat transfer was initiated by a temperature step change applied to all boundaries. The boundary temperature was then increased at the same rate as the center temperature until pseudosteady state was achieved. (Pseudosteady state is comparable to steady state convection for a fluid with uniform heat generation.) The transient heat transfer coefficient and the intensity of flow, along with the magnitude and the location of the minimum temperature, mean temperature, and maximum stream function are presented. Temperature and stream function contours are used to show the penetration of heat from the boundary and the development of the flow patterns. The local Nusselt number is also provided as supplementary information. A cylinder with HID = 1 was considered, with the Prandtl numbe...

The optimum dimensions of convective pin fins with internal heat generation

The solution of the optimization problem for convective pins with uniform internal heat generation is presented. Pins with constant thickness, triangular and parabolic profile are treated, and the cases considered are those of a given heat generation density, or total heat generation, when either the heat dissipation or the width of the pin is prescribed. The results are generalized, in a nondimensional form, and presented graphically. The effect of the pin's thermal conductivity upon the optimum dimensions is discussed, and limiting values of the heat generation and the heat dissipation, that may be imposed on the pin for a feasible optimization, are also obtained.

Convective Heat Transfer in a Circular Annulus With Various Wall Heat Flux Distributions and Heat Generation

Convective heat transfer for steady laminar flow between two concentric circular pipes with walls heated and/or cooled independently and subjected to uniform heat generation is presented in analytical closed form utilizing the linearized Navier-Stokes and energy equations. The flow field is hydrodynamically and thermally fully developed. The effect of heat generation is depicted in Fig. 1 where the ratio of the Nusselt number with heat generation to without heat generation is plotted against the radius ratio, the core size ω. It is seen that heat generation may have positive as well as negative effect on the Nusselt number.

Natural convection in a vertical porous enclosure with internal heat generation

A theoretical study is performed on steady natural convection inside a rectangular porous enclosure with uniform internal heat generation and cooling from the side walls. For low Rayleigh numbers, the problem is tackled by regular expansion in the Rayleigh number and the resulting Poisson equations are sequentially solved by the modified Galerkin method. For large Rayleigh numbers, the boundary layer equations valid near the vertical walls along with the core region equations are examined using the Gill linearization method. The analytical predictions agree well with the numerical solutions based on the power-law difference representation of the full governing equations.

Natural convection in a heat generating fluid bounded by two horizontal concentric cylinders

AbstractConvective flow and heat transfer of a Boussinesq fluid contained between two horizontal concentric cylinders is investigated under the effects of two driving mechanisms – an externally‐imposed temperature gradient across the annulus, and a uniform internal heat generation. Numerical results for flow field and temperature distribution are obtained in terms of four dimensionless parameters, namely the radius ratio, R, the Prandtl number, Pr, the Rayleigh number, Ra*, and the ratio, S, between the characteristic temperature induced by internal heating and the applied temperature difference between the boundaries. Depending on the value of S, the flow pattern is made up of either one or two vortices in each half cavity, and heat is transferred into or out of the cavity through the hot wall. In particular, for a certain value of the applied temperature difference, the hot wall apparently acts as a thermally‐insulated boundary, the internal heat is completely lost through the cold wall, and the fluid undergoes a transition from a bicellular to a unicellular flow regime.