Non-linear couette flow with heat transfer using the B-G-K model

In recent years a large number of investigators have devoted their efforts to the study of flow and heat transfer in rarefied gases, using the BGK [1] model or the Boltzmann kinetic equation. The velocity moment method which is based on an expansion of the distribution function as a series of orthogonal polynomials in velocity space, has been applied to the linearized problem of shear flow and heat transfer by Mott-Smith [2] and Wang Chang and Uhlenbeck [3]. Gross, Jackson and Ziering [4] have improved greatly upon this technique by expressing the distribution function in terms of half-range functions and it is this feature which leads to the rapid convergence of the method. The full-range moments method [4] has been modified by Bhatnagar [5] and then applied to plane Couette flow using the B-G-K model. Bhatnagar and Srivastava [6] have also studied the heat transfer in plane Couette flow using the linearized B-G-K equation. On the other hand, the half-range moments method has been applied by Gross and Ziering [7] to heat transfer between parallel plates using Boltzmann equation for hard sphere molecules and by Ziering [83 to shear and heat flow using Maxwell molecular model. Along different lines, a moment method has been applied by Lees and Liu [9] to heat transfer in Couette flow using Maxwell's transfer equation rather than the Boltzmann equation for distribution function. An iteration method has been developed by Willis [10] to apply it to non-linear heat transfer problems using the B-G-K model, with the zeroth iteration being taken as the solution of the collisionless kinetic equation. Krook [11] has also used the moment method to formulate the equivalent continuum equations and has pointed out that if the effects of molecular collisions are described by the B-G-K model, exact numerical solutions of many rarefied gas-dynamic problems can be obtained. Recently, these numerical solutions have been obtained by Anderson [12] for the non-linear heat transfer in Couette flow,

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

Unsteady flow and heat transfer on a semi-infinite flat plate with an aligned magnetic field

The unsteady laminar boundary layer flow of an electrically conducting fluid past a semi-infinite flat plate with an aligned magnetic field has been studied when at time t > 0 the plate is impulsively moved with a constant velocity which is in the same or opposite direction to that of free stream velocity. The effect of the induced magnetic field has been included in the analysis. The non-linear partial differential equations have been solved numerically using an implicit finite-difference method. The effect of the impulsive motion of the surface is found to be more pronounced on the skin friction but its effect on the x-component of the induced magnetic field and heat transfer is small. Velocity defect occurs near the surface when the plate is impulsively moved in the same direction as that of the free stream velocity. The surface shear stress, x-component of the induced magnetic field on the surface and the surface heat transfer decrease with an increasing magnetic field, but they increase with the reciprocal of the magnetic Prandtl number. However, the effect of the reciprocal of the magnetic Prandtl number is more pronounced on the x-component of the induced magnetic field. (C) 1999 Elsevier Science Ltd. All rights reserved.

Boundary-layer convective heat transfer in a convergent channel

In this paper the numerical solution of the heat transfer problem in a convergent channel with uniform and non-uniform wall temperatures under boundary-layer approximations has been presented. Also, a semi-analytical solution for uniform wall temperature has been obtained.

A non-linear analogue-to-digital converter for linearization of transducer input-output relationships

Analogue and digital techniques for linearization of non-linear input-output relationship of transducers are briefly reviewed. The condition required for linearizing a non-linear function y = f(x) using a non-linear analogue-to-digital converter, is explained. A simple technique to construct a non-linear digital-to-analogue converter, based on ' segments of equal digital interval ' is described. The technique was used to build an N-DAC which can be employed in a successive approximation or counter-ramp type ADC to linearize the non-linear transfer function of a thermistor-resistor combination. The possibility of achieving an order of magnitude higher accuracy in the measurement of temperature is shown.

Heat-Transfer During Dropwise Condensation

A comprehensive model is developed for previous termheat transfernext term during previous termdropwise condensationnext term based on the assumption that previous termheat transfernext term takes place through the bare surface in between drops to form nuclei at nucleation sites during the waiting period required for nucleation. The dynamics of drop formation and surface renewal, and the presence of non-condensable gases in the vapour have been considered. The resulting equation expresses the dependence of the vapour-side previous termheat transfernext term coefficient on the previous termheatnext term flux, properties of the vapour, previous termcondensationnext term coefficient, mole fraction of non-condensable gases in the vapour, free area available for previous termcondensation,next term surface roughness and surface thermal properties. The equation is tested with the available data and the agreement is found to be satisfactory.

Free convection heat transfer to mercury in vertical annuli

Data on free convection heat transfer to water and mercury are collected using a test rig in vertical annuli of three radii ratios, the walls of which are maintained at uniform temperatures. A theoretical analysis of the boundary layer equations has been attempted using local similarity transformation and double boundary layer approach. Correlations derived from the present theoretical analysis are compared with the analysis and the experimental data available in literature for non-metallic fluids and also with the present experimental data on water and mercury. Generalised correlations are set up for expressing the ratio of heat transferred by convection to the heat transferred by pure conduction and Nusselt's number, in terms of Grashof, Rayleigh and Prandtl numbers, based on the theoretical analysis and the present data on mercury and water. The present generalised correlations agree with the reported and present data for non-metallic fluids and liquid metals with an average deviation of 9% and maximum deviation of ± 13.7%.

A simple non-linear D/A converter

A simple generalized technique for realizing a non-linear digital to analogue converter (N-DAC), based on the principles of ' segment of equal digital interval ' is described. The simplicity of the proposed technique is demonstrated by realizing an N-DAC having a square law transfer function.

Heat transfer to power-law fluids in thermal entrance region with viscous dissipation for constant heat flux conditions

An analytical solution of the heat transfer problem with viscous dissipation for non-Newtonian fluids with power-law model in the thermal entrance region of a circular pipe and two parallel plates under constant heat flux conditions is obtained using eigenvalue approach by suitably replacing one of the boundary conditions by total energy balance equation. Analytical expressions for the wall and the bulk temperatures and the local Nusselt number are presented. The results are in close agreement with those obtained by implicit finite-difference scheme. It is found that the role of viscous dissipation on heat transfer is completely different for heating and cooling conditions at the wall. The results for the case of cooling at the wall are of interest in the design of the oil pipe line.

Steady laminar flow with suction or injection and heat transfer through porous pipe

Following the method due to Bhatnagar (P. L.) [Jour. Ind. Inst. Sic., 1968, 1, 50, 1], we have discussed in this paper the problem of suction and injection and that of heat transfer for a viscous, incompressible fluid through a porous pipe of uniform circular cross-section, the wall of the pipe being maintained at constant temperature. The method utilises some important properties of differential equations and some transformations that enable the solution of the two-point boundary value and eigenvalue problems without using trial and error method. In fact, each integration provides us with a solution for a suction parameter and a Reynolds number without imposing the conditions of smallness on them. Investigations on non-Newtonian fluids and on other bounding geometries will be published elsewhere.

Flow and heat transfer in steady laminar compressible boundary layer swirling flow in a nozzle

The axisymmetric steady laminar compressible boundary layer swirling flow of a gas with variable properties in a nozzle has been investigated. The partial differential equations governing the non-similar flow have been transformed into new co-ordinates having finite ranges by means of a transformation which maps an infinite range into a finite one. The resulting equations have been solved numerically using an implicit finite-difference scheme. The computations have been carried out for compressible swirling flow through a convergent conical nozzle. The results indicate that the swirl exerts a strong influence on the longitudinal skin friction, but its effect on the tangential skin friction and heat transfer is comparatively small. The effect of the variation of the density-viscosity product across the boundary layer is appreciable only at low-wall temperature. The results are in good agreement with those of the local-similarity method for small values of the longitudinal distance.

Flow structure and heat transfer characteristics behind a diaphragm in the presence of a diffusion flame

The characteristics of the separated flow behind a diaphragm over a burning surface are investigated experimentally. This complex problem of practical significance involving recirculation, blowing and combustion reactions is studied in a two-dimensional combustion tunnel. The flame structure, recirculation patterns and heat transfer to the surface are presented for a range of values of free stream and fuel injection velocities as well as for different heights of the diaphragm. The trends of heat transfer vs axial distance are shown to be similar to those resulting from a non-reactive heated stream with a diaphragm. Treating the case of a boundary layer diffusion flame as that corresponding to the zero height of the diaphragm, the heat transfer augmentation due to recirculation is estimated. It is found that at considerable downstream distances (xfh > 3), the heat transfer rates with diaphragm overtake the rates from a developing boundary layer case. Flow visualization studies with particle track photography show that there are many similarities between the reactive and the non-reactive cases.

Heat transfer correlations for PCM-based heat sinks with plate fins

Characterization of melting process in a Phase Change Material (PCM)-based heat sink with plate fin type thermal conductivity enhancers (TCEs) is numerically studied in this paper. Detailed parametric investigations are performed to find the effect of aspect ratio of enclosure and the applied heat flux on the thermal performance of the heat sinks. Various non-dimensional numbers, such as Nusselt number (Nu), Rayleigh number (Ra), Stefan number (Ste) and Fourier number (Fo) based on a characteristic length scale, are identified as important parameters. The half fin thickness and the fin height are varied to obtain a wide range of aspect ratios of an enclosure. It is found that a single correlation of Nu with Ra is not applicable for all aspect ratios of enclosure with melt convection taken into account. To find appropriate length scales, enclosures with different aspect ratios are divided into three categories, viz. (a) shallow enclosure, (b) rectangular enclosure and (c) tall enclosure. Accordingly, an appropriate characteristic length scale is identified for each type of enclosure and correlation of Nu with Ra based on that characteristic length scale is developed. (C) 2010 Elsevier Ltd. All rights reserved.

Flow And Heat-Transfer Over An Upstream Moving Wall With A Magnetic-Field And A Parallel Free Stream

The flow and heat transfer over an upstream moving non-isothermal wall with a parallel free stream have been considered. The magnetic field has been applied in the free stream parallel to the wall and the effect of induced magnetic field has been included in the analysis. The boundary layer equations governing the steady incompressible electrically conducting fluid flow have been solved numerically using a shooting method. This problem is interesting because a solution exists only when the ratio of the wall velocity does not exceed a certain critical value and this critical value depends on the magnetic field and magnetic Prandtl number. Also dual solutions exist for a certain range of wall velocity.

A model for heat transfer in a honey bee swarm

A swarm is a temporary structure formed when several thousand honey bees leave their hive and settle on some object such as the branch of a tree. They remain in this position until a suitable site for a new home is located by the scout bees. A continuum model based on heat conduction and heat generation is used to predict temperature profiles in swarms. Since internal convection is neglected, the model is applicable only at low values of the ambient temperature T-a. Guided by the experimental observations of Heinrich (1981a-c, J. Exp. Biol. 91, 25-55; Science 212, 565-566; Sci. Am. 244, 147-160), the analysis is carried out mainly for non-spherical swarms. The effective thermal conductivity is estimated using the data of Heinrich (1981a, J. Exp. Biol. 91, 25-55) for dead bees. For T-a = 5 and 9 degrees C, results based on a modified version of the heat generation function due to Southwick (1991, The Behaviour and Physiology of Bees, PP 28-47. C.A.B. International, London) are in reasonable agreement with measurements. Results obtained with the heat generation function of Myerscough (1993, J. Theor. Biol. 162, 381-393) are qualitatively similar to those obtained with Southwick's function, but the error is more in the former case. The results suggest that the bees near the periphery generate more heat than those near the core, in accord with the conjecture of Heinrich (1981c, Sci. Am. 244, 147-160). On the other hand, for T-a = 5 degrees C, the heat generation function of Omholt and Lonvik (1986, J. Theor. Biol. 120, 447-456) leads to a trivial steady state where the entire swarm is at the ambient temperature. Therefore an acceptable heat generation function must result in a steady state which is both non-trivial and stable with respect to small perturbations. Omholt and Lonvik's function satisfies the first requirement, but not the second. For T-a = 15 degrees C, there is a considerable difference between predicted and measured values, probably due to the neglect of internal convection in the model.