Entrance Region Problem Research Articles

Analytical solution and numerical simulation of the thermal entrance region problem for laminar flow through a circular pipe

In this paper, the assumptions implicited in Leveque’s approximation are re-examined, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This report gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form, and presents a similarity solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth order (RK4) method. Finally, other important thermal results obtained from this analysis, such as; approximate Nusselt number in the thermal entrance region was discussed in detail. A comparison with the previous study available in literature has been done and found an excellent agreement with the published data.

Heat transfer characteristics of a partially insulated tube with an impulsive temperature rise

The heat transfer for a laminar forced convection inside a two dimensional planar symmetric duct is analyzed. The fluid passage is formed by two parallel plates, and flow is fully developed and incompressible. Flow is isothermal to a position x o = 0, where the wall temperature jumps impulsively to T 1 > T 0 and remains at this value up to the position x 1, where it jumps back T 0. The problem is considerably simplified by introducing a transformation to reduce the heat transfer problem to the standard thermal entrance region problem for flow between parallel plates. Various heat transfer characteristics for different values of Prandtl and Nusselt numbers are analyzed and found to be physically realistic.

SIMULTANEOUS DEVELOPMENT OF VELOCITY AND TEMPERATURE PROFILES IN THE ENTRANCE REGION OF A PARALLEL PLATE CHANNEL: LAMINAR FLOW WITH UNIFORM WALL HEAT FLUX

Available boundary layer type solutions to the combined hydrodynamic and thermal entrance region problem are known to exhibit a discontinuity in the gradients of the velocity and temperature distributions in the entrance region. A new solution is presented which alleviates this shortcoming. The new solution is based on the hydrodynamic inlet-filled region concept originally proposed by Ishizawa (1966) and later adopted by Mohanty and Das (1982) to hydrodynamically developing flow in a channel. This concept is extended to the combined entry length problem by dividing the thermal entrance length into two lengthwise regions, a thermal inlet region and a thermally filled region. In the former, the effect of heat transfer between fluid and wall is confined within the thermal boundary layer developing along the wall. At the end of the thermal inlet region, the thermal boundary layers meet at the duct axis but the temperature profile is not yet developed. In the thermally filled region, the heat effects propagat...

Historical Development of the Theory of Heat and Thermodynamics: Review and Some Observations

Abstract This article traces the historical development of the basic concepts in thermodynamics from the time of Galileo (1592) to the time of Planck (1900) to gain historical perspectives. The review of the theory of heat as a branch in physics and thermodynamics is based on secondary sources. Some observations are made regarding the analogous physical phenomena in conduction and convection heat transfer problems. It is pointed out that Newton's cooling problem (1701) is closely related to Graetz's thermal entrance region problem (1885), and Prandtl's boundary-layer concept (1904) can be traced to transient heat conduction problems involving semiinfinite solids. It is also noted that the heat pipe (1964) represents a limiting case of the Carnot cycle (1824) with zero power output.

Small-disturbance stability of a non-isothermal tube flow of power law fluids

This paper, consisting of two parts, is intended to be a continuation of the previous works by the same authors [1], [2], [3] in which the thermal entrance region problem for steady, laminar tube flow of power law fluids with temperature dependent rheological properties and various thermal wall conditions has been numerically solved. In the present paper a small-disturbance stability of the same flow is considered. As the first approach to the problem an attempt is made, to study the flow stability just at the end of thermal entrance region where the temperature and velocity profiles are already fully developed and do not change in flow direction. Only the most simple thermal wall condition of constant temperature is assumed. In Part 1 a basic equation is derived that makes it possible to predict those critical Reynolds numbers over which the laminar tube flow becomes unstable and, hence, the equations of motion, continuity and energy, governing it and discussed in aforementioned papers lose their physical meaning. The equation to be derived in this paper is an extension of the wellknown Orr-Sommerfeld equation on purely viscous power law fluids with temperature dependent fluid consistency. Also the effects of viscous dissipation are included. In Part 2 some particular numerical solutions to the problem will be presented and discussed.

Laminar forced convection in converging or diverging planar symmetric ducts

Approximate velocity profiles are developed and a new transformation is introduced to reduce the heat transfer problem for laminar forced convection inside a diverging or converging planar symmetric duct subjected to uniform wall temperature to the standard thermal entrance region problem of Graetz type for flow between two parallel plates.

Heat and mass transport in turbulent liquid jets

Available experimental data and theory for heat and mass transport in turbulent liquid jets are critically reviewed. It is shown that mass transfer in turbulent jets is an entrance region problem and that experimental data must be analyzed accordingly. Finite difference numerical solution of the entrance region problem shows that turbulent transport has a minor effect on mass transfer for the parameter range investigated. New experimental data for evaporation from turbulent jets are presented for a 4 mm dia. jet. Comparisons are made with data from various previous condensation studies, and an attempt made to explain the trends and apparent discrepancies.

Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area

The concepts of fully developed flow and heat transfer have been generalized to accommodate ducts whose cross-sectional area varies periodically in the streamwise direction. The identification of the periodicity characteristics of the velocity components and of a reduced pressure function enables the flow field analysis to be confined to a single isolated module, without involvement with the entrance region problem. A similar modular analysis can be made for the temperature field, but the periodicity conditions are of a different nature depending on the thermal boundary conditions. For uniform wall temperature, profiles of similar shape recur periodically. On the other band, for prescribed wall heat flux which is the same for all modules, the temperature field itself is periodic provided that a linear term related to the bulk temperature change is subtracted. The concepts and solution procedure for the periodic fully developed regime were applied to a heat exchanger configuration consisting of successive ranks of isothermal plate segments placed transverse to the mainflow direction. The computed laminar flow field was found to be characterized by strong blockage effects and massive recirculation zones. The fully developed Nusselt numbers are much higher than those for conventional laminar duct flows and show a marked dependence on the Reynolds number.

Buoyancy effects on heat transfer and liquid solidification-free zone in a convectively-cooled horizontal square channel

The effects of free convection on thermal entrance region problem and liquid solidification-free zone in a convectively-cooled horizontal square channel are studied by numerical method for steady-state fully developed laminar conditions using the large Prandtl number assumption. The Rayleigh number effects on the heat transfer results and the length of liquid solidification-free zone are investigated for Biot number of 2 and 20. It is concluded that the free convection effects should be included in the analysis of ice formation in a convectively-cooled pipe or channel when the Rayleigh number is greater than say 104.

Magnetohydrodynamic entrance region problem with heat transfer

The entrance region problem for a viscous, electrically and thermally conducting fluid inside a parallel plate channel is considered. A one-parameter Pohlhausen-type method with two integral equations momentum integral and the energy integral-has been employed. This method was earlier proposed by Campbell and Slattery[1] who used it to solve the entrance region problem for simple viscous flow through a circular tube. The governing equations are reduced to a system of non-linear ordinary differential equations in the pressure, the velocity boundary layer thickness, and the thermal boundary layer thickness. That system is solved numerically for quadratic velocity and temperature profiles. These profiles, of course, are also dependent on the unknown thickness of the velocity and thermal boundary layers. Graphs have been sketched to show the effects of heat transfer and imposed magnetic field. Also, a new definition for the entrance length has been given, and a proposal is put forward for work leading toward further clarifications of the Pohlhausen-type techniques.

Measurements of velocity profile in developing liquid flows

AbstractAn optical technique for the quantitative determination of point velocities in rapidly developing laminar flows of transparent liquids is useful as an experimental tool in problems of complex geometries and fluids. The accuracy of the technique has been confirmed by comparison of experimental data with the numerical solution of the equation of motion for the entrance region problem. A distributor capable of generating a flat velocity profile has been developed for this purpose.

Comparison of Turbulent Heat-Transfer Results for Uniform Wall Heat Flux and Uniform Wall Temperature

The purpose of this note is to examine in a more precise way how the Nusselt numbers for turbulent heat transfer in both the fully developed and thermal entrance regions of a circular tube are affected by two different wall boundary conditions. The comparisons are made for: (a) Uniform wall temperature (UWT); and (b) uniform wall heat flux (UHF). Several papers which have been concerned with the turbulent thermal entrance region problem are given. 1 Although these analyses have all utilized an eigenvalue formulation for the thermal entrance region there were differences in the choices of eddy diffusivity expressions, velocity distributions, and methods for carrying out the numerical solutions. These differences were also found in the fully developed analyses. Hence when making a comparison of the analytical results for uniform wall temperature and uniform wall heat flux, it was not known if differences in the Nusselt numbers could be wholly attributed to the difference in wall boundary conditions, since all the analytical results were not obtained in a consistent way. To have results which could be directly compared, computations were carried out for the uniform wall temperature case, using the same eddy diffusivity, velocity distribution, and digital computer program employed for uniform wall heat flux. In addition, the previous work was extended to a lower Reynolds number range so that comparisons could be made over a wide range of both Reynolds and Prandtl numbers.