Thermal Entrance Region Research Articles

Thermal Nonequilibrium and Viscous Dissipation in the Thermal Entrance Region of a Darcy Flow With Streamwise Periodic Boundary Conditions

The thermal entrance region in a plane-parallel channel filled by a fluid saturated porous medium is investigated with reference to steady forced convection and to a thermal boundary condition given by a wall temperature longitudinally varying with a sinusoidal law. The effect of viscous dissipation in the fluid is taken into account, and a two-temperature model is employed in order to evaluate separately the local fluid and solid matrix temperatures. The asymptotic temperature distributions are determined analytically. The governing equations in the thermal entrance region are solved numerically by a finite element method and by the method of lines.

Magnetic field effect on laminar heat transfer in a pipe for thermal entry region

The influence of a transverse magnetic field on the local heat transfer of an electrically-conducting laminar fluid flow with high Prandtl number through a pipe was studied experimentally. Experiments indicated an increase in the local Nusselt number. The coupled set of the equations of motion and the energy equation including the viscous and Joule dissipation terms becomes non-linear and is solved numerically using a finite difference scheme. Favorable comparisons with experimental results confirm the correctness of the numerical results. It is found that the influence of magnetic field can be diminished by reducing the angle between the flow direction and the direction of magnetic field. The wall temperature reduces as the value of Hartmann number increases and the reduction rate of the wall temperature decreases as the value of Hartmann number exceeds a certain value. The average Nusselt number asymptotically approaches its limit as the Hartmann number becomes larger. Also, curve fitting can be employed to derive an equation for the average Nusselt number as a function of the Hartmann number. This equation is similar to the Sieder-Tate equation. It is observed that increasing the Hartmann number has no considerable effect on the thermal boundary layer thickness but decreases the temperature of fluid layers inside the boundary layer.

Thermally developing flow in finned double‐pipe heat exchanger

AbstractA numerical solution of the convective heat transfer in the thermal entry region of the finned double‐pipe is carried out for the case of hydro‐dynamically fully developed flow when subjected to uniform wall temperature boundary condition. Adaptive axial grid size is used in order to cater for the variation of large solution gradients in the axial direction. It has been observed that the thermal entrance region is highly effective and there is a substantial enhancement in the heat transfer coefficient. A maximum of 76.4877% increase has been observed in the thermal entrance region as compared with the fully developed region for 24 fins and H*=0.6 when R̂=0.25, whereas for R̂=0.5 the maximum increase is 75.0308% for the same number of fins of same height. It has been observed that no geometry consistently perform better throughout the entrance region. However, the geometries that have optimal performance in the fully developed region perform better in the developing region on average terms. Results show that the Nusselt number and the thermal entrance length are dependent upon various geometrical parameters such as ratio of radii of the inner and the outer pipe, fin height and the number of fins. The limiting case results match well with the literature results. This validates our numerical procedure and computer code. Copyright © 2010 John Wiley & Sons, Ltd.

Exact solution for the conjugate fluid–fluid problem in the thermal entrance region of laminar counterflow heat exchangers

A generalized Lévêque solution is presented for the conjugate fluid–fluid problem that arises in the thermal entrance region of laminar counterflow heat exchangers. The analysis, carried out for constant property fluids, assumes that the Prandtl and Peclet numbers are both large compared to unity, and neglects axial conduction both in the fluids and in the plate, assumed to be thermally thin. Under these conditions, the thermal entrance region admits an asymptotic self-similar description where the temperature varies as a power j of the axial distance, with the particularity that the self-similarity exponent must be determined as an eigenvalue by solving a transcendental equation arising from the requirement of continuity of heat fluxes at the heat conducting wall. Specifically, the analysis reveals that j depends only on the lumped parameter k ˆ = ( A 2 / A 1 ) 1 / 3 ( α 1 / α 2 ) 1 / 3 ( k 2 / k 1 ) , defined in terms of the ratios of the wall velocity gradients, A i , thermal diffusivities, α i , and thermal conductivities, k i , of the fluids entering, 1, and exiting, 2, the heat exchanger. Moreover, it is shown that for large (small) values of k ˆ the solution reduces to the classical first (second) Lévêque solution. Closed-form analytical expressions for the asymptotic temperature distributions and local heat-transfer rate in the thermal entrance region are given and compared with numerical results in the counterflow parallel-plate configuration, showing very good agreement in all cases.

Experimental study of forced and free convective heat transfer in the thermal entry region of horizontal concentric annuli

Forced and free convective heat transfer for thermally developing and thermally fully developed laminar air flow inside horizontal concentric annuli in the thermal entrance length has been experimentally investigated. The experimental setup consists of a stainless steel annulus having a radius ratio of 2 and an inner tube with a heated length of 900 mm subjected to a constant wall heat flux boundary condition and an adiabatic outer annulus. The investigation covers Reynolds number range from 200 to 1000, the Grashof number was ranged from 6.2 × 10 5 to 1.2 × 10 7. The entrance sections used were long tube with length of 2520 mm ( L/ D h = 63) and short tube with length of 504 mm ( L/ D h = 12.6). The surface temperature distribution along the inner tube surface, and the local Nusselt number distribution versus dimensionless axial distance Z t were presented and discussed. It is inferred that the free convection effects tended to decrease the heat transfer at low Re number while to increase the heat transfer for high Re number. This investigation reveals that the Nusselt number values were considerably greater than the corresponding values for fully developed combined convection over a significant portion of the annulus. The average heat transfer results were correlated in terms of the relevant dimensionless variables with an empirical correlation. The local Nusselt number results were compared with available literature and show similar trend and satisfactory agreement.

Laminar flow and heat transfer characteristics of nanoparticle colloidal dispersions in water

Numerical analysis is performed to examine axisymmetric laminar flow and heat transfer characteristics of colloidal dispersions of nanoparticles in water (nanofluids). Effect of volume fraction on flow and heat transfer characteristics is investigated. Eight different materials, alumina, copper, copper oxide, diamond, gold, graphite, silver, and zirconia are considered. Heat transfer and property measurements were conducted previously for Alumina nanofluid and the results have shown that nanofluids behave as homogeneous mixtures. It is found that oxide-based nanofluids offer the least heat transfer enhancement compared to elements-based nanofluids. For a given volume flow rate, all nanofluids exhibited linear increase in heat transfer enhancement with increasing colloids volume fraction, up to 0.05. Furthermore, it is found that in the thermal entrance region, a hydrodynamically developing flow exhibits significantly higher heat transfer enhancement than fully-developed conditions.

Experimental Investigation of Turbulent Heat Transfer in the Entrance Region of Microchannels

Within the entrance region of a closed channel, the effects on flowfield and heat transfer mechanisms are significant. Where microchannels and microdevices are concerned, there are limited heat transfer studies into this region, and those available are for laminar flows. In the turbulent regime, where the hydraulic resistance is conventionally increased, higher pumping powers are required. However, future microscale applications may have a need for turbulent flow data in microchannels. There is currently no heat transfer data available on the turbulent entrance region of microchannels. An experimental investigation has been carried out to explore turbulent convection heat transfer in the entrance region of uniformly heated microtubes. The measurement of local wall temperatures is achieved through the use of unencapsulated thermochromic liquid crystals, a state-of-the-art, nonintrusive thermal-measurement technique. Heat transfer data was obtained for two stainless steel microtubes, with nominal inner diameters of 1.067 and 0.508 mm, over a Reynolds number range of 4000 to 9000. The working fluid is FC-72, and adequate tube entry length is provided for hydrodynamic flow development before heating. Local temperature data and Nusselt values are obtained in the thermal entrance region of the microchannels. The thermal turbulent entrance length is found to remain relatively constant for the Reynolds number range considered for both microtube diameters. This is in good agreement with conventional thermal turbulent entrance studies for pipes.

Non-intrusive measurements of convective heat transfer in smooth- and rough-wall microchannels: laminar flow

The convective heat transfer behavior of laminar flow through a smooth- and two rough-wall microchannels is investigated by performing non-intrusive and spatially resolved measurements of fluid temperature via two-color fluorescent thermometry under constant heat flux conditions at three of the four microchannel walls. Pressure-drop measurements reveal that the apparent friction factors for all surfaces agree well with established macroscale predictions for laminar flow through rectangular ducts with the onset of transition at Re > Recr = 1,800 for smooth-wall flow and deviation from laminar behavior at progressively lower Re with increasing surface roughness. The local Nu for smooth-wall flow agrees well with macroscale predictions in both the thermally developing and developed regimes. With increasing roughness, while an enhancement in local Nu is noted for flow in the thermally developing regime, no measurable influence is noted upon attainment of a thermally developed state. These observations are supported by the examination of temperature profiles across the microchannel at various axial positions and Re, which suggest that the thermal boundary layer may be regenerated locally by roughness in the thermal entrance region of the flow resulting in an increased axial distance (compared to smooth-wall behavior) at which thermally developed flow is attained in the presence of roughness. Finally, estimates of the bulk Nu indicate enhancement in convective heat transfer over the smooth-wall case for laminar flow at higher Re while the smooth-wall bulk Nu data are found to agree well with macroscale predictions.

Effects of viscous dissipation on slip-flow heat transfer in a micro annulus

This study deals numerically with the laminar slip-flow forced convection in a micro annulus with constant wall temperature. The solution takes the effects of viscous dissipation, velocity-slip and temperature-jump conditions at the surface into considerations. A hybrid application of the Laplace transformation technique and the local adaptive differential quadrature method (La-DQM) is used to solve the energy equation for the developing temperature field. Of interest are the effects of the Brinkman number Br, the Knudsen number Kn, and the radius ratio β of inner to outer cylinders on the temperature distribution and the Nusselt number. The results indicate that fully developed Nusselt number increases with an increase in the radius ratio β or a decrease in the Knudsen number Kn, but it is rather insensitive to the Brinkman number. The local Nusselt number in the thermal entrance region increases as Br increases. Comparisons are also made of the dimensionless temperature gradient at the outer cylinder between the present results and published data for the circular tube case β = 0, and the agreement is found out to be generally good.

Transient and steady-state forced convection to power-law fluids in the thermal entrance region of circular ducts: Effects of viscous dissipation, variable viscosity, and axial conduction

A numerical study was conducted on the transient behavior of a hydrodynamically fully developed, laminar flow of power-law fluids in the thermally developing entrance region of circular ducts with taking into account the effects of viscous dissipation, axial conduction, and variations of viscosity with temperature. In this regard, the unsteady-state thermal energy and momentum equations were solved numerically using a finite-difference method, whereas the steady-state thermal energy equation with constant wall heat flux as the boundary condition was solved analytically as the initial condition of the former. The numerical procedure used in the present work was validated with an analytical solution for the special case of Newtonian fluids. The effects of the power-law index, axial conduction, wall heat flux, and variations of fluid viscosity on the local Nusselt number and thermal entrance length were investigated. Moreover, the local Nusselt number values of steady-state conditions were correlated as a function of the power-law index and wall heat flux. Furthermore, a correlation was derived for the thermal entrance length as a function of the power-law index and wall heat flux.

Heat transfer in trapezoidal microchannels of various aspect ratios

Heat transfer in the thermal entrance region of trapezoidal microchannels is investigated for hydrodynamically fully developed, single-phase, laminar flow with no-slip conditions. Three-dimensional numerical simulations were performed using a finite-volume approach for trapezoidal channels with a wide range of aspect ratios. The sidewall angles of 54.7° and 45° are chosen to correspond to etch-resistant planes in the crystal structure of silicon. Local and average Nusselt numbers are reported as a function of dimensionless length and aspect ratio. The effect of Prandtl number upon the thermal entrance condition is explored. The fully developed friction factors are computed and correlated as a function of channel aspect ratio. Correlations are also developed for the local and average Nusselt numbers in the thermal entrance region as a function of a dimensionless axial length variable.

Slip flow heat transfer in an infinite microtube with axial conduction

This paper deals with analytical solution of steady-state heat transfer for laminar, two-dimensional and rarefied gas flow in an infinite microtube subjected to mixed boundary conditions. To account for the slip-flow characteristics of microscale heat transfer, temperature jump condition at the wall has been incorporated in the model while the fluid velocity is assumed to be constant (slug flow). The energy equation in the thermal entrance region has been solved by the method of separation of variables. The solution yields closed form expressions for bulk-mean temperature and Nusselt number in terms of Knudsen number and Peclet number.

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.

Transient forced convection with viscous dissipation to power-law fluids in thermal entrance region of circular ducts with constant wall heat flux

A numerical investigation was conducted on the transient behavior of a hydrodynamically, fully developed, laminar flow of power-law fluids in the thermally developing entrance region of circular ducts taking into account the effect of viscous dissipation but neglecting the effect of axial conduction. In this regard, the unsteady state thermal energy equation was solved by using a finite difference method, whereas the steady state thermal energy equation without wall heat flux was solved analytically as the initial condition of the former. The effects of the power-law index and wall heat flux on the local Nusselt number and thermal entrance length were investigated. Moreover, the local Nusselt number of steady state conditions was correlated in terms of the power-law index and wall heat flux and compared with literature data, which were obtained by an analytic solution for Newtonian fluids. Furthermore, a relationship was proposed for the thermal entrance length.

Effects of non-gray thermal radiation on the heating of a methane laminar flow at high temperature

The effects of the non-gray thermal radiation on the heating of a methane/argon laminar flow at high temperature are investigated. The preheating zone of a tubular reactor is studied, where the thermal decomposition of methane does not occur. The laminar flow is simulated with the commercial Fluent code in an axisymmetric geometry. The discrete ordinates method is applied to the numerical simulation of radiative heat transfer. The non-gray gas radiative model used is the absorption distribution function (ADF) using high temperature methane radiative properties which were recently published. Several thermal entrance regions of tubular reactors are compared and the influence of methane participation in radiative heat transfer is studied. The results show that the temperature field is significantly influenced by radiation due to methane absorption. Furthermore, the average flow temperature increases when the wall temperature, the tube diameter or the methane mole fraction increases. Due to intense absorption bands of methane, it is shown that the influence of methane non-gray thermal radiation should be included in simulations of such reactors.

Thermal Analysis of a Novel Compact Packaged Passively Cooled Laser Diode Array

The temperature of a laser diode array chip must be maintained under a safe level during operation in order to achieve satisfactory performance and lifetime. In this paper, 3-D thermal analysis on diode heatsink is presented. The boundary condition at the bottom of heatsink is constant convective heat transfer coefficient. In addition, heat transfer in thermal-entry region of tubes is accounted for to calculate the convective heat transfer coefficient. Moreover, a 12 mm times 37 mm times 7 mm passively cooled laser diode array cooled by a base heatsink is demonstrated. Up to 62 W output power with an electrical-to-optical conversion efficiency of 49.2% are achieved. System thermal resistance of 0.77 K/W is obtained when the flow rate through the base heatsink with 0.8-mm-wide, 2-mm-high and 20-mm-long channels is 25 cm3/s.

Semi-analytical solution of the extended Graetz problem for combined electroosmotically and pressure-driven microchannel flows with step-change in wall temperature

Analytical solutions are obtained for the temperature and Nusselt number distribution in the thermal entrance region of a parallel plate microchannel under the combined action of pressure-driven and electroosmotic transport mechanisms, by taking into account the effects of viscous dissipation, Joule heating and axial conduction simultaneously, in the framework of an extended Graetz problem. Step changes in wall temperature are considered to represent physically conceivable thermal entrance conditions. The solution of the temperature distributions at the various channel sections essentially involves the determination of a set of eigenvalues and eigenfunctions corresponding to a Sturm Liouiville problem with non self-adjoint operators. The resultant eigenfunctions are non orthogonal in nature, and are obtained in the forms of hypergeometric functions. Parametric variations on the effects of the relative strengths of the pressure gradients and the electric field, ratio of the rate of heat generation to the rate of wall heat transfer, and the Peclet number are analyzed in details, in terms of their influences on the temperature field as well as the Nusselt number distribution.

Effects of duct inclination angle on thermal entrance region of laminar and transition mixed convection

Experimental investigation was performed on the mixed convection heat transfer of thermal entrance region in an inclined rectangular duct for laminar and transition flow. Air flowed upwardly and downwardly with inclination angles from −90° to 90°. The duct was made of duralumin plate and heated with uniform heat flux axially. The experiment was designed for determining the effects of inclination angles on the heat transfer coefficients and friction factors at seven orientations (θ=−90°, −60°, −30°, 0°, 30°, 60° and 90°), six Reynolds numbers (Re≈420, 840, 1290, 1720, 2190 and 2630) within the range of Grashof numbers from 6.8×103 to 4.1×104. The optimum inclination angles that yielded the maximum heat transfer coefficients decreased from 30° to −30° with the increase of Reynolds numbers from 420 to 1720. The heat transfer coefficients first increased with inclination angles up to a maximum value and then decreased. With further increase in Reynolds numbers, the heat transfer coefficients were nearly independent of inclination angles. The friction factors decreased with the increase of inclination angles from −90° to 90° when Reynolds numbers ranged from 420 to 1290, and independent of inclination angles with higher Reynolds numbers.

Thermally developing electroosmotic transport of nanofluids in microchannels

A theoretical analysis is presented in this work to assess the influence of nanofluids on thermally developing and hydrodynamically developed electroosmotic transport in parallel plate microchannels (Graetz problem). The hydraulic diameters of the microchannels are assumed to be beyond a certain threshold limit, so that the electric double layers formed adjacent to the plates do not overlap with each other. The volumetric heating arising from the conduction currents in the flow is modeled using Ohm’s law. The viscous generation terms in the energy equation are neglected, based on the earlier findings that the consequent effects are negligible as compared to the Joule heating effects in electroosmotically driven microchannel flows. Closed form expressions for the pertinent temperature distributions and the Nusselt number variations are obtained by employing the method of separation of variables in conjunction with an eigen value formulation, in order to assess the influence of volume fraction of the dispersed nano-particles on the overall rates of convective transport. It is revealed that the effects of nano-particles in the fluid turn out to be significant in the thermal entrance region only, especially for higher Peclet number values. The implications of the incorporation of nanofluids are demonstrated to be somewhat non-trivial in nature, and are strongly determined by the effective Peclet number values obtained on the basis of the phase-integral values of the thermo-physical properties and the pertinent flow parameters.

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.