HEAT TRANSFER IN A LAMINAR CYLINDRICAL WALL JET WITH UNIFORM SURFACE HEAT FLUX

The steady state flow and heat transfer characteristics of a laminar cylindrical wall jet are obtained for uniform surface heat flux conditions. Local nonsimilarity solutions as well as series solutions are presented for the velocity and thermal fields. Numerical results are given for the wall shear stress, surface temperature variation and temperature field for a Prandtl number of 0.73.     

STEADY LAMINAR FLOW THROUGH TWISTED PIPES: FLUID FLOW AND HEAT TRANSFER IN RECTANGULAR TUBES

Fluid flow and heat transfer characteristics in a twisted rectangular tube having an aspect ratio of two were studied using a numerical solution to the momentum and energy equations. Fluid flow solutions are presented for a fully developed laminar flow of a Newtonian fluid. Heat transfer results are presented for the case of axially uniform wall temperature. For the case of peripherally uniform wall temperature, the overall Nusselt number in a twisted rectangular tube was found to be higher than a straight tube by up to 30 percent over certain ranges of twist lengths. However, for the case of non-uniform wall temperature, the overall absolute Nusselt number increased very rapidly with decrease in the twist length     

An analysis of thermally generated compressive waves

The rapid heating of a wall bounding a semi-infinite region of gas generates a slightly supersonic compressive wave and thereby increases the rate of heat transfer to and through the gas. Generalized correlating expressions are presented for the wave velocity and for the amplitude in pressure, temperature, and gas velocity as a function of distance and time for any specified exponential rate and extent of increase of the temperature of the wall. These expressions are based on idealized analytical solutions modified by virtue of finite-difference solutions and experimental data. The combination of all three of these elements proved to be essential in order to understand and correlate the behavior. Although a compressive wave is generated for any rate and extent of heating of the wall, significant effects are observed only for very rapid heating and very large temperature differences. A critical time constant for exponential heating is identified such that the strength of the wave is equivalent to that for an instantaneous step in wall temperature.     

NATURAL CONVECTION INSIDE A HORIZONTAL CYLINDER

Natural convection of a fluid contained in an infinitely long horizontal cylinder at large Prandtl number and unit-order Grashof number is analyzed. The motion is generated by an imposed cosine wall temperature distribution which includes an arbitrary phase angle. The phase angle is a measure of the location of the wall temperature extrema.

From an asymptotic ordering of the energy and vorticity transport equations for large Prandtl number it is shown that the core region, which contains fluid surrounded completely by a boundary-layer flow along the cylinder wall, may assume either of two configurations.

For heating angles near the heating-from-the-side case (wall temperature extrema at the ends of the horizontal diameter) linearized forms of the boundary-layer equations are developed which yield solutions that match the core configuration not considered previously. The form of the results agrees generally with experimental evidence for heating-from-the-side.     

NATURAL CONVECTION INSIDE A HORIZONTAL CYLINDER

Natural convection of a fluid contained in an infinitely long horizontal cylinder at large Prandtl number and unit-order Grashof number is analyzed. The motion is generated by an imposed cosine wall temperature distribution which includes an arbitrary phase angle. The phase angle is a measure of the location of the wall temperature extrema.

From an asymptotic ordering of the energy and vorticity transport equations for large Prandtl number it is shown that the core region, which contains fluid surrounded completely by a boundary-layer flow along the cylinder wall, may assume either of two configurations.

For heating angles near the heating-from-the-side case (wall temperature extrema at the ends of the horizontal diameter) linearized forms of the boundary-layer equations are developed which yield solutions that match the core configuration not considered previously. The form of the results agrees generally with experimental evidence for heating-from-the-side.     

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 propagate throughout the entire cross section and the temperature profile undergoes adjustment in a fully thermal region to finally attain the fully developed form. A thermal shape factor is also introduced in the thermally filled region which ensures that all thermal quantities attain their fully developed values asymptotically. The new model is used to obtain solutions to the combined entry length problem for laminar flow through a parallel plate channel under the constant wall heat flux boundary condition. The analysis gives considerably better results for the local Nusselt number and thermal entrance length than previously available.     

ANALYTICAL TEMPERATURE DISTRIBUTION FOR MULTIDIMENSIONAL BODIES EXPOSED TO A CONSTANT SURFACE HEAT FLUX

For many decades, solutions for transient temperature distributions in multidimensional objects were determined by combining as a product the solutions of one-dimensional objects necessary to delimit the contour of these multidimensional objects. These product solutions are usually restricted to two types of boundary conditions: a constant wall temperature and a constant heat transfer coefficient

This paper considers the case of an object exposed to a constant surface heat flux. It is shown that when the surface of multidimensional object is submitted to a constant heat flux density, its temperature distribution can be obtained by the simple addition of one-dimensional temperature distributions. Only three one-dimensional solutions are necessary to solve all possible multidimensional problems. These are the solutions for a semi-infinite slab, an infinite plate and an infinite cylinder. The equations describing the temperature profile within each of these one-dimensional objects are presented as well as their graphical representations in a generalized form for rapid determination of temperatures.     

Finite element solution of low peclet number fluid flow in a round pipe with the cauchy boundary condition

This paper presents a finite element solution to the problem of low Peclet number fluid flow in the thermal entrance region of a round pipe. The velocity is assumed to be laminar and fully developed throughout the pipe and the fluid temperature is kept uniform atX = —∞. The pipe wall is adiabatic at X ≤ 0 and cooled convectively at X ≥ 0. The solutions include temperature distributions and Nusselt numbers for the parameters, Bi = 0.04, 0.4, 4, 20 and Pe = 1, 3, 5, 10, 20, 30, which are in excellent agreement with the existing analytic solution except in the region near the singular point Δ A temperature discrepancy in the analytic solution at this point is physically impossible. The finite element method overcomes this mathematical difficulty and shows a greater value in the Nusselt number due to a higher wall temperature at X ≥ 0.     

Laminar mixed convection in the thermal entrance region of horizontal rectangular channels with uniform heat input axially and uniform wall temperature circumferentially

This paper presents a numerical study on laminar mixed convection in the thermal entrance region of horizontal rectangular channels with uniform heat input axially and uniform wall temperature circumferentially. A relatively novel numerical method of solution is developed to obtain the developing velocity and temperature fields. The values of Prandtl number are 0.7 and 7.2, corresponding to air and water, respectively. The values of Rayleigh number are 0, 10⁴, 3 × 10⁴ and 10⁵. The channel aspect ratios considered are 0.2, 0.5, 1, 2 and 5. Variations in local friction factor ratios and local Nusselt numbers are presented. It is found that the circumferential boundary condition of uniform wall temperature significantly increases the value of local Nusselt number as compared to that found in earlier works under the boundary condition of uniform wall heat flux. But the boundary condition effect on the friction factor is shown to be comparatively minor. The asymptotic solutions at z → are compared to the existing numerical data with good agreement.     

NONSIMILAR SOLUTIONS FOR HEAT TRANSFER IN WALL JET FLOWS

An analysis is presented to investigate the flow and heat transfer characteristics of a laminar plane wall jet with non-isothermal wall as well as uniform suction and blowing at the surface and a laminar cylindrical wall jet. The approach used is local nonsimilarity method, wherein, the nonsimilanty terms appearing in the momentum and energy equations are retained and simplifications are introduced only in the auxiliary system of equations. To insure the accuracy of the results, solutions are obtained for three levels of truncation of the governing equations. For the case of plane wall jet problem, both a series solution as well as local nonsimilarity solution have been given and the agreement between the two is found to be very good. For the cylindrical wall jet problem, the results obtained by the local nonsimilarity approach in the present paper have been compared with the series solution results. Numerical results for the wall shear stress, velocity distribution, wall heat transfer rate and temperature field are presented.     

Lumped-differential analysis of concurrent flow double-pipe heat exchanger

A mixed lumped-differential formulation for double-pipe heat exchangers is presented that radially lumps the temperature distribution in the outer channel, providing a differential problem for the inner channel that involves a more general type of boundary condition for the wall temperature. The generalized integral transform technique is extended to allow for the analytical solution of this class of problems. Applicability limits for the simplified model are then established in terms of related parameters, based on numerical results obtained for bulk temperatures and Nusselt numbers, as compared to more involved approaches for the complete differential model.     

FINITE ELEMENT ANALYSIS FOR LAMINAR FLOW AND HEAT TRANSFER IN FINITE ROD BUNDLES

This investigation deals with the application of finite element method to solve the thermohydraulic problem of laminar fully developed flow in the interior and wall sub-channels of finite fuel rod bundles. A variational principle has been used for the solution of the momentum and energy equations. Wall shear stress and temperature distributions, ƒRe and Nusselt numbers are obtained for the sub-channels of different configurations. The results are compared with solutions generated by collocation and finite difference methods.     

Squeezing flow of viscoplastic fluids subject to wall slip

Polymer processing operations such as compression molding, sheet forming and injection molding can be modeled by squeezing flows between two approaching parallel surfaces in relative motion. Squeezing flows also find applications in the modeling of lubrication systems, and in the determination of rheological properties. Here, analytical solutions are developed for the constant-speed squeezing flow of viscoplastic fluids. It is assumed that the fluid is purely viscous, and hence viscoelastic effects unimportant. The rheological behavior of the viscoplastic fluids is represented by the Herschel-Bulkley viscosity function. The deformation behavior of commonly encountered viscoplastic fluids is generally complicated by the presence of wall slip at solid walls, which is a function of the wall shear stress. The slip coefficient that relates the slip velocity to the shear stress is affected by the material of construction and also the roughness of the solid surfaces, leading to the possibility of different slip coefficients at various solid surfaces. The model developed in this study accommodates the use of different slip coefficients at different solid surfaces. The accuracy of the solutions is established, and the effects of various parameters such as slip coefficient and apparent yield stress are examined. The solutions provide useful design expressions that can be utilized for squeezing flows of viscoplastic fluids, with or without wall slip at the solid boundaries.     

Influence of solute type and concentration on ice scaling in fluidized bed ice crystallizers

During crystallization of ice from aqueous solutions, ice crystals exhibit a marked tendency to adhere to the cooled heat exchanger wall resulting in the formation of an insulating ice layer, often referred to as ice scaling. A promising method to avoid ice scaling is the application of a solid-liquid fluidized bed heat exchanger in which fluidized steel particles remove the ice crystals from the walls. This paper presents experiments with a single-tube fluidized bed heat exchanger in which ice crystals were produced from aqueous solutions of various solutes with varying concentrations. The experiments reveal that ice scaling is only prevented when a certain temperature difference between wall and solution is not exceeded. This transition temperature difference appears to increase approximately linearly with the solute concentration and is higher in aqueous solutions with low diffusion coefficients. The observed phenomena are explained by the hypothesis that ice scaling is only prevented when the mass transfer controlled growth rate of ice crystals on the wall does not exceed the scale removal rate induced by the fluidized steel particles. In conclusion, a model based on these physical phenomena is proposed to predict ice scaling in fluidized bed heat exchangers.     

Diffusion-Controlled Growth or Dissolution of Spheres with Variable Temperature

Theoretical calculation of growth or dissolution of spherical particles is a challenging problem even under isothermal conditions. This paper examines the diffusion-controlled behavior of spherical particles for time-dependent and spatially uniform temperature. Analytical solutions are possible for growth from zero size even when temperature changes, making diffusivity a variable. Numerical methods for cases not capable of analytical solution are also developed; these include the variation of interfacial concentration with temperature. Analytical solutions are restricted to growth from zero but it is shown that these solutions represent the asymptotic behavior for growth from finite initial size. Nearly all solutions for dissolving spheres must be computed numerically but quasi-steady-state analytical solutions applicable to both growth and dissolution, in suitable cases, are also developed.     

FINITE ELEMENT ANALYSIS FOR LAMINAR FLOW AND HEAT TRANSFER IN FINITE ROD BUNDLES

This investigation deals with the application of finite element method to solve the thermohydraulic problem of laminar fully developed flow in the interior and wall sub-channels of finite fuel rod bundles. A variational principle has been used for the solution of the momentum and energy equations. Wall shear stress and temperature distributions, ?Re and Nusselt numbers are obtained for the sub-channels of different configurations. The results are compared with solutions generated by collocation and finite difference methods.     

Dependence of Heat Transfer in a Circular Tube with Prescribed Wall Flux on Peclet Number and on Heating Length

Solutions for the arithmetic average of the temperatures over the cross-section of a circular tube at each axial position are presented for elliptic heat transfer problems with a Neumann wall boundary condition and a finite heating section. Using the Green's function method, solutions in the form of simple expressions are derived. Although these solutions are only approximations to the available exact solutions, they do not require the solution of an eigenvalue problem and are reasonably accurate under conditions for which the temperature field is approximately radially uniform. This new approach and the straightforward way of determining how the average temperature field depends on the Peclet number and on the length of the heating section allow for computation of temperature fields for lower values of the Peclet number and for higher values of heating length than have been reported using analytical solutions. Peclet number bounds are also given for the importance of axial conduction.     

Characterization of polymer adsorption in tube flow of drag reducing fluids

Elucidation of the polymer adsorption and flow characteristics at the tube wall is essential for an understanding of turbulent drag reduction. The polymer adsorbed onto the tube wall, in the flow of dilute solutions of linear random coiling macromolecules, also produces a concentrated fluid layer at the surface of the adsorption zone, as a result of the flow of the solvent micromolecules in the porous network comprising the adsorption zone.Velocity profiles are developed and used to determine the radial variation in the adsorption zone of porosity, as well as fractional surface coverages and mean separation or interpenetration distances between macromolecules in the various adsorption layers. The polymer concentration build-up in the concentrated fluid layer is also evaluated. Predictions of the latter for aqueous Polyox WSR-301 solutions are in qualitative agreement with experimental measurements and suggest that turbulent drag reduction is related to the level of polymer build-up in the concentrated fluid layer.     

Heat transfer in laminar flow of non-Newtonian fluids

The main objective of this work is the consideration of local heat transfer coefficients for non-Newtonian power-law pseudoplastic liquid in laminar flow in circular conduits. The wall boundary conditions chosen are cases involving uniformly constant heat flux and step change in heat flux.Analytical solutions are developed for the wall temperature profile and compared with experimental data. Additionally, the experimental data have been correlated for comparison with existing relationships, hitherto not verified adequately. The limits of experimental data are:

     

Heat transfer in tubes and ducts with wall mass transfer

This study is concerned with the developing temperature field for flow in circular tubes and parallel plate channels with mass transfer through the wall. Some unique features of the developing temperature field are outlined for both constant temperature and constant heat flux boundary conditions. Solutions to the heat transfer problem are extended to include arbitrary prescribed temperature or heat flux boundary conditions. A practical problem is solved in two ways, utilizing respectively these solutions, and a comparison of the results is made.