Fluid flow in microtubes with axial conduction including rarefaction and viscous dissipation

Graetz problem inside the microtube is revisited considering rarefaction effect, viscous dissipation term and axial conduction in the fluid for uniform wall temperature boundary condition in the slip flow regime. The flow is assumed to be hydrodynamically fully developed, thermally developing, and the velocity profile is solved analytically. The temperature field is determined by the numerical solution of the energy equation. The rarefaction effect is imposed to the problem via velocity-slip and temperature jump boundary conditions. The local and fully developed Nu numbers are obtained in terms of dimensionless parameters; Pe, Kn, Br, κ. Fully developed Nu numbers and the thermal entrance length are found to increase by the presence of the finite axial conduction.     

Laminar Flow and Heat Transfer to a Non-Newtonian Fluid in an Entrance Region of a Square Duct with Prescribed Constant Axial Wall Heat Flux

Abstract

Forced-convection heat transfer information as a function of the pertinent nondimensional numbers is obtained numerically for laminar incompressible non-Newtonian fluid flow in the entrance region of a square duct with simultaneously developing temperature and velocity profiles for constant axial wall heat flux with uniform peripheral wall temperature. The power-law model characterizes the non-Newtonian behavior.

Finite-difference representations are developed for the equations of the mathematical model, and numerical solutions are obtained assuming uniform inlet velocity and temperature distributions. Results are presented for local and mean Nusselt numbers as functions of the Graetz number and the Prandtl number in the entrance region. Comparisons are made with previous analytical work for Newtonian fluids. The results show a strong effect of the Prandtl number on the Nusselt numbers with fully developed and uniform velocity profiles representing the lower and upper limits, respectively. The results provide a new insight into the true three-dimensional character of the pseudoplastlc fluid flow in the entrance region of a square duct and are accurate.     

Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux

Combined forced and free flow in a vertical rectangular duct is investigated for laminar and fully developed regime. The velocity field, the temperature field, the friction factor and the Nusselt number are evaluated analytically by employing finite Fourier transforms. The thermal boundary condition considered is an axially uniform wall heat flux and a peripherally uniform wall temperature, i.e. an H1 boundary condition. The necessary and sufficient condition for the onset of flow reversal is determined either in the case of upward flow in a cooled duct or in the case of downward flow in a heated duct. The special case of free convection, i.e. the case of a purely buoyancy-driven flow, is discussed. The occurrence of effects of pre-heating or pre-cooling in the fluid is analysed. It is pointed out that although these effects occur in rectangular ducts, they are not present either in circular ducts or in parallel-plate channels.     

Thermally developing flow in elliptic ducts with axially variable wall temperature distribution

The laminar, incompressible, hydrodynamically fully developed and thermally developed and developing flow is studied in straight elliptic ducts with aspect ratio a^* varying from 0.25 to 0.99 (which is an almost circular duct). The duct wall is subjected successively to constant temperature, to circumferential uniform and axially linearly or exponentially varying temperature. Numerical results obtained with the ADI scheme indicate that the friction factor increases as aspect ratio a^* decreases. In the thermally developing flow a high Nusselt number decreases as a^* decreases. In the thermally developed limit as a^* decreases, the Nusselt number increases for small axial wall temperature distributions, while decreases for large axial wall temperature values.     

Forced convection with viscous dissipation in the thermal entrance region of a circular duct with prescribed wall heat flux

The thermal entrance forced convection in a circular duct with a prescribed wall heat flux distribution is studied under the assumptions of a fully developed laminar flow and of a negligible axial heat conduction in the fluid, by taking into account the effect of viscous dissipation. The solution of the local energy balance equation is obtained analytically by employing the Laplace transform method. The effect of viscous dissipation is taken into account also in the region upstream of the entrance cross-section, by assuming an adiabatic preparation of the fluid. The latter hypothesis implies that the initial condition in the entrance cross-section is a non-uniform radial temperature distribution. Two special cases are investigated in detail: an axially uniform wall heat flux, a wall heat flux varying linearly in the axial direction.     

FULLY DEVELOPED LAMINAR HEAT TRANSFER IN CIRCULAR-SEGMENT DUCTS WITH UNIFORM WALL TEMPERATURE

Heat transfer to constant-property, fully developed, laminar flows in circular-segment ducts with uniform wall temperature (T) has been analyzed. Besides representing a compact surface, the segment duct geometry models the flow cross section of a circular tube with a straight-tape insert. Two variations in the T thermal boundary condition are considered: constant axial and circumferential wall temperature, and constant temperature on the curved surface but an adiabatic flat wall. These two conditions model the extremes of the fin effects of a straight-tape insert, i.e., 100% and zero fin efficiencies, respectively. Numerical solutions, obtained by using finite difference techniques, are presented for both the velocity and temperature fields. The isothermal friction factors are in excellent agreement with analytical solutions reported in the literature. The Nusselt number results for the two thermal boundary conditions are presented for different segment shapes, 0° ≤, 6 ≤, 90°, and they represent the lower limits of the heat transfer enhancement due to twisted-tape inserts.     

Entrance heat transfer in rectangular ducts with constant axial energy input

The study of heat transfer in the entrance region of ducts with different cross-sections is important in engineering practice. This paper considers laminar, hydrodynamically fully developed flow in the thermal entrance regions of rectangular passages, emphasizing heat transfer aspects. By having a prescribed heating or cooling rate and considering the wall temperature to depend on the axial coordinate alone, the temperature solution leads to an integral equation. Solution of this equation is found using an inverse technique to determine the temperature at the walls. For verification purposes, an asymptotic solution is developed which produces results that agree very well with those from the inverse analysis. The results include a correlation and computed values of the Nusselt number at entrance locations, for rectangular ducts with different aspect ratios.     

Asymptotic behaviors of heat transfer in porous passages with axial conduction

As reported in the literature, a sufficiently small Peclet number requires the inclusion of axial conduction within a fluid flowing in a duct. In fluid saturated porous ducts, this phenomenon greatly increases the heat transfer rate within the thermal entrance region. Axial conduction effects near the thermal entrance regions in parallel-plate ducts and in circular ducts are emphasized in this study. Having metallic foams as porous materials can cause the effective thermal conductivity to increase and this decreases the Peclet number. Here, a simple solution is being used for determination wall heat flux near the thermal entrance location and the result leads to a relatively simple correlation for determination of the bulk temperature.     

NUMERICAL PREDICTION OF LAMINAR FORCED CONVECTION IN TRIANGULAR DUCTS WITH UNSTRUCTURED TRIANGULAR GRID METHOD

The flow and heat transfer characteristics of smooth triangular ducts with different apex angles of 15, 30, 60, and 90 under the fully developed laminar flow condition were predicted numerically using a finite volume method. The SIMPLE-like algorithm was employed together with an unstructured triangular grid method, where the grid was generated by a Delaunay method. The triangular grid was adopted instead of the traditional rectangular grid to fit better into the triangular cross section of the duct. Two kinds of boundary condition (uniform wall temperature and uniform wall heat flux) were considered. Comparison of the predictions with previous computational results indicated a very good agreement. Both the friction factor and Nusselt number (Nu) showed a strong dependence on apex angle of the triangular duct. When the apex angle was 60, the duct provided the highest steady-state forced convection from its inner surface to the airflow under the laminar flow condition.     

COMPUTATION OF HEAT AND FLUID FLOW IN DUCTS OF ARBITRARY CROSS SECTION

This paper reports on connective heat transfer in irregular ducts maintained under a constant wall temperature. In particular, due to the complexity of the geometry, the paper investigates in detail the fluid flow and convective heat transfer in right-triangular and semicircular ducts. The hydrodynamically fully developed flow and the developing temperature in these geometries are obtained analytically/numerically from the solution of the energy equation employing the method of lines. The energy equation is reformulated by a system of a first-order differential equation controlling the temperature along each line. It was found that reliable closed-form solutions for the temperature distribution in the thermal entrance region can be obtained utilizing 21 lines, or less, displayed in the cross-stream direction of the duct. The grid pattern chosen provides: drastic savings in computing time. Results for the thermal entry region flow heat transfer are presented in tabular and graphical forms. The representative curves illustrating the variation of bulk temperature and Nusselt numbers with pertinent parameters in the entire thermal entry region are plotted. The computed results are compared against some analytical/numerical findings reported in the literature. In all cases, satisfactory comparison is obtained. The asymptotic Nusselt numbers are 1.90, 2.25, and 2.29 for 15°, 30°, and 45° right-triangular ducts, respectively, and 6.030 for semicircular ducts     

Combined forced and free flow in a vertical circular duct subjected to non-axisymmetric wall heating conditions

The fully developed mixed convection flow in a vertical circular duct is investigated analytically, under the assumption of laminar parallel flow. A wall heat flux uniform in the axial direction and dependent on the angular coordinate is considered. As a consequence, the fluid temperature is three dimensional, since it changes in the radial, axial and angular directions. An analytical method based on Fourier series expansions of temperature and velocity fields is adopted to determine the velocity and the temperature distributions as well as the friction factor and the average Nusselt number. The general solution, expressed in terms of Bessel functions, is applied to study a case that has a special importance in technical applications: a duct whose wall is half subject to a uniform heat flux and half adiabatic. The positive and negative threshold values of the ratio between the Grashof number Gr and the Reynolds number Re for the onset of the flow reversal phenomenon are determined. A comparison between the average Nusselt number for the considered non-axisymmetric case and that for the case of a duct subject to a uniform wall heat flux is performed.     

Numerical and experimental analysis of unsteady heat transfer with periodic variation of inlet temperature in circular ducts

This work focusses on a numerical and experimental analysis of unsteady forced convection in hydrodynamically developed and thermally developing laminar air flow in a circular duct, subjected to a periodic variation of the inlet temperature. The experiments were conducted over a wide range of Reynolds number (281.2 ≤ Re ≤ 1024.3) and inlet frequency (0.01 ≤ β ≤ 0.20 Hz) of the periodic heat input. In the numerical study, the non-uniform inlet temperature amplitude profile derived from the experiments, was included in the numerical model. A fully explicit, second-order accurate finite difference scheme was developed and used for the solution of the unsteady energy equation. Numerical results are obtained with the fully developed parabolic velocity profile under the boundary condition of the first kind, which was verified by the experiments. Temperature variations along the centerline of the circular duct are observed to be thermal oscillations with the same frequency as the inlet periodic heat input and amplitudes that decayed exponentially with distance along the duct. Thermal response along the wall exhibits negligible amplitude variation with changes in Reynolds number and inlet frequency. The variation in the periods and amplitudes of the thermal oscillations are observed to be a function of spacial system variables only. Satisfactory agreement between the numerical and experimental results are obtained.     

Cartesian graetz problems with air property variation

Heat transfer and wall friction results have been obtained numerically to demonstrate the effects of gas transport property variation on three classical, laminar flow situations: (a) heating and cooling at constant wall temperature with a fully developed velocity profile at entry, (b) heating and cooling at constant wall temperature with a uniform velocity profile at entry, and (c) heating with constant wall heat flux and a uniform velocity profile at the entrance.     

A forced convection subcooled boiling model for nonuniform axial heat fluxes

The modeling of convective subcooled boiling of water flowing in round tubes subjected to nonuniform axial heat fluxes is described. The effects of different axial heat flux profiles are modeled using a local hypothesis; i.e. flow and thermal development are assumed to occur very rapidly in the subcooled boiling (SCB) flow regime. A computer code has been developed to predict the pressure drop, heat transfer coefficient and wall temperature for nonuniform axial heat fluxes, starting with a well-validated code for uniform axial heat fluxes. The predictions for some common nonuniform axial heat profiles are compared to the uniform heat flux case.     

Temperature dependent viscosity effects on laminar forced convection in the entrance region of straight ducts

A parametric investigation is carried out on the effects of temperature dependent viscosity in simultaneously developing laminar flow of a liquid in straight ducts of arbitrary but constant cross-sections. Viscosity is assumed to vary with temperature according to an exponential relation, while the other fluid properties are held constant. Different cross-sectional geometries are considered, corresponding both to three-dimensional (rectangular, trapezoidal and hexagonal) and to axisymmetric (circular and concentric annular) duct geometries. Uniform wall temperature boundary conditions are imposed on the heated/cooled walls of the ducts. A finite element procedure is employed for the solution of the parabolized momentum and energy equations. Computed axial distributions of the local Nusselt number and of the apparent Fanning friction factor for ducts of the considered cross-sections are presented with reference to both fluid heating and fluid cooling. Numerical results confirm that, in the laminar forced convection in the entrance region of straight ducts, the effects of temperature dependent viscosity cannot be neglected in a wide range of operative conditions.     

Thermally Developing Forced Convection of Non-Newtonian Fluids Inside Elliptical Ducts

Laminar-forced convection inside tubes of various cross-section shapes is of interest in the design of a low Reynolds number heat exchanger apparatus. Heat transfer to thermally developing, hydrodynamically developed forced convection inside tubes of simple geometries such as a circular tube, parallel plate, or annular duct has been well studied in the literature and documented in various books, but for elliptical duct there are not much work done. The main assumptions used in this work are a non-Newtonian fluid, laminar flow, constant physical properties, and negligible axial heat diffusion (high Peclet number). Most of the previous research in elliptical ducts deal mainly with aspects of fully developed laminar flow forced convection, such as velocity profile, maximum velocity, pressure drop, and heat transfer quantities. In this work, we examine heat transfer in a hydrodynamically developed, thermally developing laminar forced convection flow of fluid inside an elliptical tube under a second kind of a boundary condition. To solve the thermally developing problem, we use the generalized integral transform technique (GITT), also known as Sturm-Liouville transform. Actually, such an integral transform is a generalization of the finite Fourier transform, where the sine and cosine functions are replaced by more general sets of orthogonal functions. The axes are algebraically transformed from the Cartesian coordinate system to the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall. The GITT is then applied to transform and solve the problem and to obtain the once unknown temperature field. Afterward, it is possible to compute and present the quantities of practical interest, such as the bulk fluid temperature, the local Nusselt number, and the average Nusselt number for various cross-section aspect ratios.     

Numerical study of developing laminar forced convection of a nanofluid in an annulus

Laminar forced convection of a nanofluid consisting of Al₂O₃ and water has been studied numerically. Two dimensional elliptical governing equations have been solved to investigate the hydrodynamics and thermal behaviors of the fluid flow throughout an annulus. Single phase approach is used for the nanofluid modeling. The velocity and temperature profiles are presented in the fully developed region. The axial evolution of temperature, convective heat transfer coefficient and the friction coefficient at the inner and outer walls' region are shown and discussed. It is shown that the dimensionless axial velocity profile does not significantly change with the nanoparticle volume fraction. But, the temperature profiles are affected by the nanoparticle concentration. In general convective heat transfer coefficient increases with nanoparticle concentration. However, when the order of magnitude of heating energy is much higher than the momentum energy the friction coefficient depends on the nanoparticle concentration. At higher Reynolds numbers for which the momentum energy increases, this dependency on the nanoparticle volume fraction decreases.     

Experimental studies of laminar flow and heat transfer in a one-porous-wall square duct with wall injection

Experiments were conducted to investigate the effect of fluid injection on laminar flow and heat transfer characteristics in a one-porous-wall square duct. Uniform air flow at Re₀ = 400−2000 entered the duct with a cross section of 20 × 20 mm² and a ratio of the active injection length to the hydraulic diameter of 40. Pressurized air was injected through a thick layer of porous material for flow uniformity and a heated porous duct wall at injection rates Re_w = 5−20. All of the measured and deduced data, including the axial velocity profiles, the pressure drops, the friction factors, the porous wall temperatures, the outlet fluid temperatures and the Nusselt numbers, were presented and compared with the previous theoretical results. The deduced friction factors and Nusselt numbers from the experimental data were correlated within differences of ±10% and ±15% respectively.     

Laminar flow between parallel flat plates, with heat transfer, of water with variable physical properties

The laminar fully developed flow of water in a vertical channel is examined, using the complete equations of motion with experimental values (in the range 0–100°C) for the viscosity, conductivity, specific heat, and density. Although the local effects of variations in these quantities are, of course, small, the cumulative effect is significant for moderate temperature differences. Accordingly, emphasis has been placed on the evaluation of integrated properties such as mass flow and heat transfer.

Poiseuille, Couette and mixed Poiseuille-Couette flows are investigated for a range of wall temperature differences and the effects of the temperature dependent properties on the velocity and thermal profiles are discussed in detail. Wall Nusselt numbers, flow rates, skin friction coefficients, friction factors and Reynolds analogy factors are evaluated for all these regimes.     

Convection in laminar three-dimensional separated flow

Distributions of the wall temperature, the Nusselt number, and the friction coefficient on all of the bounding walls of laminar three-dimensional forced convection flow adjacent to backward-facing step in a rectangular duct are reported. A uniform heat flux is imposed on the bounding walls (stepped wall, sidewalls, and flat wall) downstream from the step, while the walls of the duct upstream from the step and the step are treated as adiabatic surfaces. The flow upstream of the step is treated as hydrodynamically fully developed and isothermal, and the outlet flow downstream from the step is treated as being hydrodynamically and thermally fully developed. Local and average results are presented for a Reynolds numbers range of 150-450, and some results are compared with their equivalent from the two-dimensional case.