Unequal Heat Research Articles

Analytical Solution for Transient Laminar Fully Developed Free Convection in Open-Ended Vertical Concentric Porous Annuli

Free and mixed convection problems in a vertical porous annulus have been extensively studied. These studies use numerical or experimental techniques to investigate the thermal behavior of the mental porous annulus. Parang and Keyhani have obtained closed-form solutions for the special case where the inner and outer walls are heated by uniform but unequal heat fluxes. The lack of analytical solutions for transient fully developed laminar natural convection in vertical concentric porous annuli, with different fundamental combinations of isothermal and isoflux thermal boundary conditions, motivated the present work. The purpose of this paper is to present, in closed forms, transient fully developed free convection solutions, corresponding to four fundamental thermal boundary conditions in vertical concentric annuli. The solutions obtained, which take both Darcian and Brinkman effects into consideration, may be used as a preliminary check for numerical solutions of more practical situations. 4 refs.

Local coefficients for forced convection in curved rectangular channels of large aspect ratio with unequal uniform heating

A finite-element representation and the FIDAP™ code were used to solve the equations of conservation for mass, momentum and energy for fully developed angular flow and fully developed convection in the annulus between two concentric cylinders. Local Nusselt numbers were determined for a complete range of heat flux density ratios, both positive and negative, on the inner and outer surfaces, for Dean numbers up to 2 1 2 times the critical value for the onset of vortical motion. The Nusselt numbers are found to depend strongly on the heat flux density ratio as well as on the Dean number. The applicability of the results to double-spiral heat exchangers is discussed.

ELF harmonic radiation from the Tromsø Heating Facility

Using a new experimental technique the response of the auroral ionosphere to a very powerful HF wave amplitude modulated by an ELF square wave has been determined independent of the obfuscating effects of earth‐ionosphere‐waveguide resonances. It has been found that when resonance effects are compensated for, the odd harmonic amplitude ratios are predominantly those one would expect for a square wave, slightly modified by the non‐linear relationship between generated ELF signal and HF radiated power. The even harmonic amplitude ratios are found to be much lower than those observed in earlier experiments and have been modelled by assuming unequal electron heating and cooling time constants and hence different rise and fall times for the ionospheric currents.

The Effects of Internal Heat Generation, Anisotropy, and Base Temperature Nonuniformity on Heat Transfer from a Two-Dimensional Rectangular Fin

Abstract A finite-element approach is used to study the two-dimensional performance of a straight fin of rectangular profile operating in a convecting environment. The analysis includes the effects of internal heat generation, anisotropy of fin material, base temperature elevation (depression), and unequal heat transfer coefficients for tip and main convection surfaces. Numerical results are presented showing the impact of these effects on the heat removal capability of the fin. The graphical information provided is believed to be useful for predicting the performance of a two-dimensional rectangular fin under more general and realistic conditions.

Heat exchange between unequal countercurrent vessels asymmetrically embedded in a cylinder with surface convection

A three-dimensional approximate analytic solution is presented for the unequal countercurrent heat transfer between parallel paired vessels asymmetrically embedded in a long cylinder with surface convection. The analysis assumes that the velocity profile in the vessels is parabolic, the Peclet number is large compared to unity and the radius of the cylinder is several times larger than the distances from the center of the cylinder to the vessels. A perturbation method is employed by taking the reciprocal of the Peclet number as a small parameter. The new solution approach takes into consideration the axial thermal interaction and describes the three-dimensional asymmetric thermal field due to the vessels, the wall temperatures of which are nonuniform in the cross-sectional plane. Application of the model to whole limb heat transfer is discussed.

Area and capital cost targets for heat exchanger network synthesis with constrained matches and unequal heat transfer coefficients

A pair of “transshipment” nonlinear programs (NLP) are formulated to calculate the area and capital cost targets for heat exchanger network (HEN) synthesis with unequal heat transfer coefficients and different capital cost laws (for different materials of construction, pressure ratings, etc.) when there are constraints on the number of matches (exchangers), forbidden matches and required matches with specified areas (for revamp synthesis). With these NLPs, the trade-off between area and number of units can be evaluated before synthesis. In addition to the targets themselves, solution of the area and capital cost targeting NLPs yield: (1) temperature profiles (much like the composite temperature—enthalpy curves) providing a model for the “ideal” profiles in a HEN achieving the area or capital cost target and corresponding utility target; and (2) a proper selection of stream matches and their heat loads to minimize area (or capital cost). These matches and heat loads provide an excellent starting point for synthesis of HENs achieving (within a few percent) the area and capital cost targets with the same number of exchangers in the HEN as matches in the NLP solution (e.g. corresponding to the units target).

On the Generalization of the Weinbaum-Jiji Bioheat Equation to Microvessels of Unequal Size; The Relation Between the Near Field and Local Average Tissue Temperatures

The extensive series of experiments reported in Lemons et al. [1] show that measureable local tissue temperature fluctuations are observed primarily in the vicinity of the 100-500 micron countercurrent vessels of the microcirculation and thus strongly support the basic hypothesis in the new bioheat equation of Weinbaum and Jiji [2] that these countercurrent microvessels are the principal determinants of local blood-tissue heat transfer. However, the detailed temperature profiles in the vicinity of these vessels indicate that large asymmetries in the local temperature field can result from the significant differences in size between the countercurrent artery and vein. Using the superposition techniques of Baish et al. [9], the paper first presents a solution to the classic problem of an unequal countercurrent heat exchanger with heat loss to the far field. This solution is then used to generalize the Weinbaum-Jiji bioheat equation and the conductivity tensor that appears in this equation to vessels of unequal size. An asymptotic analysis has also been developed to elucidate the relationship between the near field temperature of the artery-vein pair and the local average tissue temperature. This analysis is used to rigorously prove the closure approximation relating the local arterial-venous temperature difference and the mean tissue temperature gradient which had been derived in [2] using a more heuristic approach.

Forced convection — axial conduction between parallel walls with unequal heat fluxes

Laminar flow heat transfer is computed for a situation in which a fluid moves along a parallel plate channel with unequal wall heat fluxes (one wall is insulated). The fluid enters the heating section through an upstream region which is perfectly insulated. This situation serves to describe an upper bound for the commonly encountered case of double pipe heat exchangers with identical thermal conditions. A control volume approach has been employed for the numerical work enabling a fast calculation for the thermally developing regime in the parallel plate channel. The merits of the adopted procedure are assessed by comparison with other results available in the literature for the one-region and for the two-region problem.

The photophoretic force on nonspherical particles

Previous calculations of the photophoretic force due to unequal heating of gas-suspended, illuminated particles have been restricted to spheres. An analytical formula for the photophoretic force acting on an arbitrary body in the free molecule regime is presented. Provided the source function is known, the force vector can be calculated by integration without solving the heat transfer equation. In general, the photophoretic force deviates from the direction of light. Three extreme, idealized cases are discussed: a particle with uniform source function, a perfectly opaque particle, and a perfectly transparent particle with an adsorbing core.

ASYMMETRY IN FREEZE‐DRYING

ABSTRACTThis paper describes what is meant by asymmetric freeze‐drying and reports experimental observations of the phenomenon. Current mathematical models of freeze‐drying cannot account for asymmetry. This theoretical deficiency is corrected by the development and solution of a freeze‐drying theory termed the asymmetrically‐retreating‐ice‐front (ARIF) model. An extension of the well‐known uniformly‐re‐treating‐ice‐front (URIF) model, the ARIF theory accounts for heat flux through the frozen region and for unequal heat fluxes to different sample faces. The ARIF theory is able to predict temperatures and moisture fractions during drying, total drying times, and degrees of ice‐core asymmetry which agree reasonably well with experimental results.

An Efficient Algorithm for Evaluating Arrays of Extended Surface

In the consideration of single longitudinal fins of rectangular, trapezoidal, and triangular profile and in arrays of extended surface composed of these fins, it is shown that conditions of heat flow and temperature excess at the fin or array tip induce conditions of heat flow and temperature excess at the fin or array base. In particular, there is a linear transformation between the aforementioned data at the fin tip and the fin base. The conventional fin efficiency is abandoned and single fins or arrays of extended surface are instead characterized by a single important parameter, the heat flow to temperature excess ratio, which is a function only of fin geometry and heat transfer parameters. Algorithms are provided for combining the effects of individual fins in arrays of extended surface. A modification of the procedures developed leads to an exact solution for the double stack with unequal heat distribution at opposite ends, a problem which has heretofore required iterative solution procedures.

Transient boiling flow instabilities in a multi-channel upflow system

In this study, the transient boiling flow instabilities in a four parallel channel upflow system with and without cross-connections have been experimentally investigated using Freon-11 as the test fluid. Several series of transient experiments have been run using a step increase in power input with a constant flow control valve setting and a step decrease in flow rate with a constant heat input. Two major modes of oscillations, namely density-wave type and pressure-drop type oscillations, have been observed. Total system pressure drop has been plotted as a function of mass flow rate for various heat inputs and flow control valve settings, and the stable and unstable regions for transient density-wave type and pressure-drop type oscillations have been determined and the boundaries of sustained instabilities have also been indicated. Experiments have been conducted with equal and unequal heat inputs and the results have been compared with each other. A comparison of the transient instabilities observed in four parallel channel system with and without cross connections has also been made.

Combined influence of wall conductance, pressure work, viscous dissipation and joule heating on MHD channel flow heat transfer

To investigate the combined influence of viscous dissipation, pressure work, Joule heating, arbitrary voltage ratio, unequal wall conductances and wall heat fluxes on the fully developed laminar MHD channel flow heat transfer, the exact solution of the energy equations for fluid and channel walls are derived assuming the Hartmann velocity profile. It is concluded that there can be a substantial difference, depending upon Hartmann number, electric field and Brinkman number, between the Nusselt number considering the wall conductance and that neglecting it. Representative results are presented in diagrams.

Experimental models for the evaluation of microwave biological effects

The theoretical and empirical approaches to biological effects of microwaves are discussed. The bioeffects should be considered as a chain of events: primary interaction analyzed in terms of biophysics and early direct, early indirect, late (delayed) effects analyzed in terms of physiology. Thermal balance experiments are discussed pointing out the necessity to take unequal deep body heating by microwaves into account. The use of physiologic rhythms in microwave bioeffects experimentation is presented. Possible use of pharmacodynamic approaches (simultaneous use of drugs and microwave exposure) is outlined. Possible ways of developing biological response microwave dosimetry are indicated. Unexplained effects are pointed out and the necessity of further experiments to clarify these stressed. Some possible experimental models are presented.

Combined Free and Forced Convection Through Vertical Rectangular Channels With Unequal Heating From Sides

Fully developed laminar combined free and forced convection through vertical rectangular channels is studied. Uniform heat input per unit length is considered. All fluid properties are considered invariant with temperature except for the variation of density in the body-force term of the equation of motion. Broad sides of the duct are assumed at uniform temperature. The short sides have been treated for adiabatic condition; or under nonzero finite and even unequal heat fluxes. An exact solution of the problem has been presented. Nusselt numbers have been evaluated and their variation with respect to aspect ratios, buoyancy parameter, and short side heat fluxes have been studied.

Effect of heating of blood on the quantitative determination of carbon monoxide haemoglobin.

Routine determination of the HbCO content of blood from cases burned alive is carried out byWolff's andMaehly's spectrophotometric methods. There are sometimes appreciable differences between samples obtained from various parts of the body. As it could be assumed that the variations of HbCO might be explained by unequal heating of the blood, a simple in vitro investigation was carried out, on blood exposed to coal-gas and heat. Judging from the results obtained,Wolff's andMaehly's methods may not always give acceptable values, when carried out on heated blood.

Bilateral Heat Transfer to Liquid Metals Flowing Turbulently Through Annuli

The results of an analytical study of bilateral heat transfer to liquid metals flowing turbulently through annuli are reported. The study was based on the assumptions of 1) uniform, though not necessarily equal, heat fluxes from the walls, 2) fully-established temperature and velocity profiles, and 3) no effect of transverse temperature variations on the physical properties of the liquid metal. The fraction, ξ, of the total heat delivered to the flowing stream which comes from the inner wall was varied from zero to unity. Also two special situations were treated: (A) equal heat fluxes from both walls, and (B) uniform but unequal heat fluxes at the walls, with equal wall temperatures at a given axial position along the channel.The results are presented in the form of Nusselt numbers for r2/r1 values from 1 to 7, Peclet numbers for 102 to 104, and ξ values from 0 to 1.The coefficient for the heat transferred from the inner wall goes to infinity at a certain value of ξ, which depends only on the ratio r2/r1. The same thing occurs for the coefficient for the outer walls, except at a different values of ξ.The interesting observation was made that at Pe = 6700, the Nusselt numbers for the following situations are all equal to about 35: 1) flow in a pipe, 2) flow between parallel plates with equal heat fluxes from both plates, 3) flow in any annulus with equal heat fluxes from the walls, with reference to the heat transferred from the outer wall, 4) flow in any annulus with a uniform heat flux from each wall and having equal wall temperatures at a given axial position, with reference to the heat transferred from the outer wall.

Laminar forced convection in rectangular channels with unequal heat addition on adjacent sides

Temperature distributions are determined analytically for fully developed laminar heat transfer in channels with aspect ratios from 1 to ∞. The channel walls are uniformly heated, but the heat flux on the short sides is an arbitrary fraction between 0 and 1 of the heat flux on the broad sides. For all cases, the wall temperatures are compared on the basis that the total heat transferred per unit channel length is maintained at a fixed value. The poor convection due to the low velocities in the corners and along the narrow walls always caused the peak temperatures to occur at the corners. The lowest peak temperatures were found when all the heating took place at only the broad walls rather than when heating was partly distributed to the short sides. This results from the fact that, when four sides are heated, more energy is being supplied to the low velocity comer regions. For heating at only the broad walls, the corner temperature decreases rapidly as the aspect ratio is increased to about 10 and insignificantly thereafter. In the limit of infinite aspect ratio, the wall temperature distribution does not approach a constant as is the case for infinite parallel plates.

The effect of axially varying and unsymmetrical boundary conditions on heat transfer with turbulent flow between parallel plates

The solution is presented to the heat-transfer problem with turbulent flow between parallel plates with heating on one side only. The configuration has some practical significance when considered as a limiting case of the annulus with a heated core. The velocity and eddy diffusivity variations due to Deissler are used but are modified in the central region of the passage by assuming a constant eddy diffusivity. In solving the energy equation the eddy diffusivities for heat and momentum are taken equal. The eigenvalues and functions necessary to calculate the variation of Nusselt number with distance along the passages are tabulated for three Reynolds and three Prandtl numbers. From the two basic solutions of uniform temperature and uniform heat flux the analysis is extended to certain axial variations which are of practical interest and finally, by using superposition, the result is obtained for unequal uniform heat fluxes on each side of the passage.

Extended and Long‐Range Weather Forecasting

This article discusses extended and long‐range weather forecasting. The article begins by briefly explaining how weather conditions are the result of atmospheric motions of various types and their interactions with one another. These motions are ultimately produced and maintained by the unequal heating of the earth by the sun. The spin of the earth about its axis transforms these motions into curved wind systems. The migrations of these curved wind systems around the earth produce frequent changes in weather. Thus, the earth's spin is responsible for short‐period weather changes, whereas the earth's orbital rotation around the sun produces the longer‐period changes.