Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 2: Thermally developing flow

In this part of the study, consideration is given to thermally developing laminar forced convection in a pipe including viscous dissipation. The axial heat conduction in the fluid is neglected. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Both the wall heating (the fluid is heated) case and the wall cooling (the fluid is cooled) case are considered. The distributions for the developing temperature and local Nusselt number in the entrance region are obtained. Results show that the temperature profiles and local Nusselt number are influenced by the Brinkman number (Br) and the thermal boundary condition used for the wall. Significant viscous dissipation effects have been observed for large Br.     

Influences of Boundary Condition on Laminar Natural Convection of Bingham Fluids in Square Cross-Sectioned Cylindrical Annular Enclosures with Differentially Heated Vertical Walls

Numerical simulations have been carried out to analyze steady-state laminar natural convection of yield stress fluids obeying Bingham model in square cross-sectioned cylindrical annular enclosures with differentially heated vertical walls for both constant wall temperature and constant wall heat flux boundary conditions for active walls. The simulations have been performed under the assumption of axisymmetry for a nominal Rayleigh number range of 10³ to 10⁶ and nominal Prandtl number range of 10 to 10³ for different ratio of internal cylinder radius to cylinder height range of 0.125 to 16. The mean Nusselt number on the inner periphery for the constant wall heat flux configuration has been found to be smaller than that in the case of constant wall temperature configuration for a given set of values of nominal Rayleigh and Prandtl numbers for both Newtonian and Bingham fluid cases. The mean Nusselt number normalized by the corresponding value obtained for pure conductive transport increases with increasing internal radius before approaching the corresponding mean Nusselt number for square enclosures regardless of the boundary conditions. Detailed physical explanations have been provided for the effects of the aforementioned parameters on the mean Nusselt number on the inner periphery. Finally, the new Nusselt number correlations have been proposed for laminar natural convection of both Newtonian and Bingham fluids in square cross-sectioned cylindrical annular enclosures for both constant wall temperature and constant wall heat flux boundary conditions.     

Analysis of the effect of viscous dissipation for laminar flow in stadium-shaped ducts

The fully-developed laminar forced convection of a Newtonian fluid in a duct with stadium-shaped cross section has been analyzed. The effect of viscous dissipation has been taken into account. Three different thermal boundary conditions have been considered: (T) uniform wall temperature distribution; (H1) axially uniform wall heat flux distribution with peripherally uniform wall temperature distribution; (H2) axially and peripherally uniform wall heat flux distribution. The adiabatic-wall boundary condition has also been analyzed as a special case of the H2 boundary condition. The velocity and temperature distributions in the fluid, as well as the Fanning friction factor and the Nusselt number, have been evaluated numerically, by employing a Galerkin finite element method. As expected, the numerical evaluation of the dimensionless temperature distribution and of the Nusselt number reveals that increasing discrepancies between the H1 and H2 boundary conditions exist if the stadium-shaped duct is gradually flattened.     

Wall coupling effect in channel forced convection with streamwise periodic boundary heat flux variation

The present paper deals with the laminar forced convection in a parallel-plane channel, and is aimed to investigate the effect of conducting walls. On the external boundaries of the duct walls a thermal boundary condition is prescribed, such that the wall heat flux longitudinally varies with sinusoidal law. The local energy balance equation is written separately for the fluid and the solid regions, with reference to the fully developed regime, and then solved both analytically and numerically. Moreover, the local and average Nusselt numbers in a longitudinal period are evaluated. The average Nusselt number, if regarded as a function of the dimensionless pulsation, displays an interesting feature. In fact, for all the considered cases, it has a minimum, so that there exists a value of the dimensionless pulsation such that the heat exchange between the fluid and the solid wall is considerably inhibited.     

Thermally developing flow and heat transfer in rectangular microchannels of different aspect ratios

Laminar convective heat transfer in the entrance region of microchannels of rectangular cross-section is investigated under circumferentially uniform wall temperature and axially uniform wall heat flux thermal boundary conditions. Three-dimensional numerical simulations were performed for laminar thermally developing flow in microchannels of different aspect ratios. Based on the temperature and heat flux distributions obtained, both the local and average Nusselt numbers are presented graphically as a function of the dimensionless axial distance and channel aspect ratio. Generalized correlations, useful for the design and optimization of microchannel heat sinks and other microfluidic devices, are proposed for predicting Nusselt numbers. The proposed correlations are compared with other conventional correlations and with available experimental data, and show very good agreement.     

An analytical study of pulsating laminar heat convection in a circular tube with constant heat flux

Pulsating laminar convection heat transfer in a circular tube with constant wall heat flux is investigated analytically. The results show that both the temperature profile and the Nusselt number fluctuate periodically about the solution for steady laminar convection, with the fluctuation amplitude depending on the dimensionless pulsation frequency, ω*, the amplitude, γ, and the Prandtl number, Pr. It is also shown that pulsation has no effect on the time-average Nusselt numbers for pulsating convection heat transfer in a circular tube with constant wall heat flux.     

The effect of uniform lateral mass flux on free convection about a vertical cone embedded in a saturated porous medium

The effect of uniform lateral mass flux on natural convection about a cone embedded in a saturated porous medium is numerically analyzed. The surface is maintained at a uniform wall temperature (UWT) or uniform heat flux (UHF). The transformed governing equations are solved by Keller box method. Numerical data for the dimensionless temperature profiles and the local Nusselt number are presented for a wide range of the mass flux parameter. In general, it has been found that the local surface heat transfer rate increases owing to suction of fluid. This trend reversed for blowing of fluid. The mass flux parameter is found to have a more pronounced effect on the local Nusselt number for the case of UWT than it does for the case of UHF.     

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.     

Analytical Solution of Viscoelastic Fluid Flow and Heat Transfer through an Annulus

Abstract

Forced convection heat transfer to a fully developed laminar flow of simplified Phan-Thien-Tanner fluid in an annulus was studied analytically. Inner and outer cylinders were stationary and subjected to constant heat flux. The results of the velocity and temperature distributions, dimensionless pressure gradient, and Nusselt number were obtained for a wide range of aspect ratios and dimensionless viscoelastic groups. Results indicated the gap between two cylinders as well as viscoelastic behavior of the fluid have noticeable effects on flow and heat transfer.     

Laminar mixed convection heat and mass transfer in vertical rectangular ducts with film evaporation and condensation

The investigation of mixed convection heat and mass transfer in vertical ducts with film evaporation and condensation has been numerically examined in detail. This work is primarily focused on the effect of film evaporation and condensation along the wetted wall with constant temperature and concentration on the heat and mass transfer in rectangular vertical ducts. The numerical results, including the distributions of dimensionless axial velocity, temperature and concentration distributions, Nusselt number as well as Sherwood number are presented for moist air mixture system with different wall temperatures and aspect ratios of the rectangular ducts. The results show that the latent heat transport with film evaporation and condensation augments tremendously the heat transfer rate. Better heat transfer enhancement related with film evaporation is found for a system with a higher wall temperature.     

Uniform lateral mass flux effect on natural convection of non-Newtonian fluids over a cone in porous media

In this paper, we have investigated a boundary layer analysis for uniform lateral mass flux effect on natural convection of non-Newtonian power-law fluids along an isothermal or isoflux vertical cone embedded in a porous medium. Numerical results for the dimensionless temperature profiles as well as the local Nusselt number are presented for the mass flux parameter, viscosity index n and geometry shape parameter λ. The local surface heat transfer increases for the case withdrawal of fluid, the increase of the value of λ. The local Nusselt number is found to be significantly affected by the surface mass flux than the viscosity index.     

Fully developed electro-osmotic heat transfer in microchannels

Thermally fully developed, electro-osmotically generated convective transport has been analyzed for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Such a flow is established not by an imposed pressure gradient, but by a voltage potential gradient along the length of the tube. The result is a combination of unique electro-osmotic velocity profiles and volumetric heating in the fluid due to the imposed voltage gradient. The exact solution for the fully developed, dimensionless temperature profile and corresponding Nusselt number have been determined analytically for both geometries and both thermal boundary conditions. The fully developed temperature profiles and Nusselt number are found to depend on the relative duct radius (ratio of the Debye length to duct radius or plate gap half-width) and the magnitude of the dimensionless volumetric source.     

Numerical Analysis Of Mixed Convection In Inclined Tubes With External Longitudinal Fins

Laminar mixed convection in the entrance region of an inclined tube with longitudinal external fins (corresponding to the basic element of a flat plate solar collector) has been studied numerically. The system is subjected to a uniform solar flux and convective losses on its top surface; it is insulated on its bottom surface. The results show that the secondary flow induced by buoyancy has a very significant effect on the axial flow and on the isotherms in both the fluid and the solid. Furthermore, due to circumferential conduction in the tube wall, more heat reaches the fluid from the bottom half of the fluid–solid interface than from its top half. The circumferentially average Nusselt number shows the usual decrease with axial distance from the tube inlet towards a constant value in the fully developed region. This constant value is considerably higher for the finned tube than the corresponding value for a bare tube. Both these values are significantly higher than one for fully developed forced convection. A parametric study shows the effects of the fin and tube materials, of the solar flux and the fin’s width on the thermal performance of the collector.     

The effect of viscous dissipation in thermally fully-developed electro-osmotic heat transfer in microchannels

The influence of viscous dissipation on thermally fully-developed, electro-osmotically generated flow has been analyzed for a parallel plate microchannel and circular microtube under imposed constant wall heat flux and constant wall temperature boundary conditions. Such a flow is established not by an imposed pressure gradient, but by a voltage potential gradient along the length of the tube. The result is a combination of unique electro-osmotic velocity profiles and volumetric heating in the fluid due to the imposed voltage gradient. For large ratio of the microtube radius (or microchannel half-width) to Debye length, the wall-normal fluid velocity gradients can be extremely high, which has the potential for significant viscous heating. The solution for the fully-developed, dimensionless temperature profile and corresponding Nusselt number have been determined for both geometries and for both thermal boundary conditions. It is shown that three dimensionless parameters govern the thermal transport: the relative duct radius (ratio of the duct radius or plate gap half-width to Debye length), the dimensionless volumetric source (ratio of Joule heating to wall heat flux), and a dimensionless parameter that relates the magnitude of the viscous heating to the Joule heating. Surprisingly, it is shown that the influence of viscous dissipation is only important at low values of the relative duct radius. For magnitudes of the dimensionless parameters which characterize most practical electro-osmotic flow applications, the effect of viscous dissipation is negligible.     

Convective-diffusive heat transfer in a packed-bed with constant heat flux at the wall: An asymptotic solution for ξ → large

Packed-bed systems with axial convective, radial diffusive heat transfer and with a constant heat flux at the wall are studied. Asymptotic solutions valid for axial distances sufficiently away from the inlet position of the packed-bed are presented and their behavior investigated. Two geometrical configurations (i.e., the parallel plate and the cylindrical tube) are analyzed. Also, lower bounds for the axial coordinate are derived and their effect on the validity of the asymptotic solution are investigated. The asymptotic temperature profile is shown to yield the correct bulk temperature profile and to predict the correct Nusselt number for both types of geometries.     

Natural convection heat transfer of water in a horizontal gap with downward-facing circular heated surface

An experimental study of natural convection heat transfer from a downward-facing horizontal circular heated surface in a water gap has been carried out. The results were correlated in different forms of Nusselt number vs Rayleigh number according to different independent variables. The effects of different characteristic length and temperature were discussed and the gap size is the preferred characteristic length, the average fluid temperature between bulk temperature and the saturated temperature is the preferred film temperature. For the estimation of the natural convection heat transfer under the present conditions, empirical correlations in which Nusselt number is expressed as a function of Rayleigh number, or Rayleigh and Prandtl numbers both, may be used. However, the best accuracy is provided by an empirical correlation which expresses the Nusselt number as a function of the Rayleigh and Prandtl numbers, as well as the gap width-to-heated surface diameter ratio, the dimensionless temperature. Artificial neural networks have been trained successfully for analyzing the influences of the gap width-to-heated surface diameter ratio and the wall temperature difference between the temperature of wall and ambient fluid on natural convection heat transfer based on the experimental data in the present study. The results show that the Nusselt number will increase by increasing the gap ratio and decrease by increasing the wall temperature difference.     

Physical Effects of Variable Thermophysical Fluid Properties on Flow and Thermal Development in Micro-Channel

Micro-scale cooling is an efficient and effective cooling technique to achieve the goal of higher heat removal capabilities. The present research focuses to find the physical effects of fluid property variations on flow and thermal development in micro-channel. The effects of temperature-dependent density, viscosity, and thermal conductivity variations on single-phase laminar forced convection are numerically investigated. The problem is especially simulated for hydrodynamically and thermally developing water flow in micro-channel with no-slip, no-temperature jump, and constant wall heat flux boundary conditions. It is observed that the density variation induces radially inward flow due to continuity, which sharpens the axial velocity profile and decreases Nusselt number compared to constant property solution. The axial velocity profile significantly alters due to viscosity variation. This alteration varies along the micro-flow and it induces radially flow due to flow continuity. The reducing rate of Nusselt number for viscosity variation is substantially lower than constant property solution due to a significant flattening effect of the axial velocity profile, which augments the Nusselt number. Thermal-conductivity variation across the flow induces radial conduction, which enhances convection compared to constant property solution. Additionally, the effects of thermophysical fluid property variations on static gauge pressure drop are also investigated.     

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.     

Natural convection heat and mass transfer of non-Newtonian power law fluids with yield stress in porous media from a vertical plate with variable wall heat and mass fluxes

This work studies the heat and mass transfer by natural convection from a vertical plate with variable wall heat and mass fluxes in a porous medium saturated with a non-Newtonian power law fluid with yield stress for the general case of power law variations in wall heat and mass fluxes. The governing equations are transformed into a dimensionless form by the similarity transformation and then solved by a cubic spline collocation method. Results are presented for velocity, temperature, and concentration profiles, as well as the Nusselt and Sherwood numbers for various parameters of the power law fluid with yield stress in porous media. The existence of threshold pressure gradient in the power law fluids tends to decrease the fluid velocity and the local Nusselt and Sherwood numbers. An increase in the power law exponent increases the local Nusselt and Sherwood numbers.     

Numerical study of conjugated heat transfer in metal foam filled double-pipe

The two equation numerical model has been applied for parallel flow double-pipe heat exchanger filled with open cell metal foams. The model fully considered solid–fluid conjugated heat transfer process coupling heat conduction and convection in open cell metal foam solid matrix, interface wall and fluid in both inner and annular space in heat exchanger. The non-Darcy effect and the wall thickness are also taken into account. The interface wall heat flux distribution along the axial direction is predicted. The numerical model is firstly verified and then the influences of solid heat conductivity, metal foam porosity, pore density, relative heat conductivity and inner tube radius of the heat exchanger on dimensionless temperature distribution and heat transfer performance of heat exchanger are numerically studied. It is revealed that the proposed numerical model can effectively display the real physical heat transfer process in the double pipe heat exchanger. It is expected to provide useful information for the design of metal foam filled heat exchanger.