Fully-developed Nusselt Number Research Articles

An Analytical Study of the Impact of an Inclined Magnetic Field on Couette Flow with Jeffrey Fluid under Local Thermal Non-Equilibrium (LTNE) and Viscous Dissipation

In this work, we examine the effects of viscous dissipation and local thermal non-equilibrium (LTNE) on Couette flow in a duct filled with a porous media under the influence of an angled magnetic field. The bottom plate of the duct is in motion and subjected to a constant heat flux, while the top plate remains stationary and adiabatic. The Jeffrey fluid flow model is consistent with the unidirectional flow in the porous zone. The studies provide more precise measurements of the effects of the Jeffrey parameter (λ), inclined angle (ϕ), Hartmann number (MW), thermal conductivity ratio (ν), Brinkman number (BrW), and Biot number (BiW) on improving heat transmission. The governing equations are solved analytically. The present investigation gives dimensionless temperatures for fluid-solid phases and fully developed Nusselt number (FDNN) profiles. Variation of Jeffrey parameter, inclined angle, Brinkman number, and Hartman number in the temperature field in both phases and FDNN. Furthermore, the temperature in the solid phase is higher than the temperature in the fluid phase for the Jeffrey parameter and Hartman number in the Couette flow, which supports LTNE validation.

The thermally-developing region in metal foam with open pores and high porosity

The thermal development phenomenon in foam-like conductive porous media, e.g. open-cell metal foam, is investigated via simulation. A typical cell of commercial metal foam was geometrically idealized while maintaining its actual structural features as much as possible. Then, clones of the idealized cell were virtually connected to each other to form a sizable vertical sheet representing a slice of metal foam. This one-cell thick sheet had a length of 33 cells (flow direction) and a height of 15 cells. Because of symmetry, this sheet could represent flow and heat transfer in a semi-infinite metal-foam channel sandwiched between two heated parallel plates separated by a distance twice the sheet’s height. The pertinent mass, momentum and heat equations were solved directly at the microscopic level; and the temperature fields were obtained for various approach velocities using a commercial numerical package. The results provided insight into the thermal development region in general, and the thermal entry lengths were determined for air flow at various speeds. The thermally fully-developed Nusselt number was correlated with Reynolds number. To substantiate the results, comparisons were made to analytical predictions; and qualitative contrasts were made to experimental data from the literature.

Heat transfer characterization of rhombic microchannel for H1 and H2 boundary conditions

In this paper, the characterization of laminar convective heat transfer in rhombic shaped microchannel with H1 (axially constant heat flux and circumferentially constant temperature) and H2 (constant axial and circumferential wall heat flux) boundary conditions is presented. A three-dimensional numerical model is developed for hydrodynamically and thermally developing flow in different rhombic geometries with side-angles varying from 10° to 90°. The numerical model is first validated with the other model developed in the literature for rectangular geometry and subsequently the results are compared with rhombic channel with different side-angles. This model is then extended to determine the temperature and heat flux distributions in rhombic microchannel, which is used to find the local and hydrodynamically and thermally fully developed Nusselt numbers. It is found that the fully developed Nusselt number varies only with the side-angles of rhombic channel and generalized correlations are developed as a function of side-angles for H1 and H2 boundary conditions. Correlations for developing length, local and fully-developed Nusselt numbers for rhombic channels are currently not available. These results will be useful in design and optimization of rhombic microchannel for electronic cooling applications.

Optimal unsteady convection over a duty cycle for arbitrary unsteady flow under dynamic thermal load

Developing next generation transient heat exchangers is a transformative technology for efficient thermal management of advanced power electronics and electric machines (APEEM) inside hybrid electric, electric, and fuel cell vehicles as well as renewable energies (wind, solar, tidal). Optimal design criteria for such dynamic heat exchangers should be achieved through addressing internal forced convection with arbitrary flow unsteadiness under dynamic time dependent thermal loads. Exact analytical solutions were obtained for laminar forced-convective heat transfer under arbitrary time-dependent heat flux for steady flow inside a tube (Fakoor-Pakdaman et al., 2014). In this study, the energy equation is solved analytically for arbitrary unsteady flow between parallel plates under dynamic time and position dependent heat flux. As such, laminar pulsating flow between two parallel plates is considered under a harmonic wall heat flux. Exact relationships are obtained to find; (i) temperature distribution for the fluid; (ii) fluid bulk temperature; and (iii) the local and time averaged Nusselt number. New compact relationships are proposed to find the thermal entrance length and the cyclic fully-developed Nusselt number. It is shown that the period of the Nusselt number oscillations is the least common multiple of the periods of the imposed harmonic heat flux and the pulsating flow. We obtained a relationship for the ‘cut-off’ angular frequency of pulsating flow beyond which the heat transfer does not feel the pulsation. This study also shows that for a given harmonic wall heat flux, there is an optimal pulsating flow velocity, with optimum frequency, which enhances the time averaged Nusselt number by up to 27%.

Slip-Flow and Heat Transfer in a Porous Microchannel Saturated with Power-Law Fluid

This study aims to numerically examine the fluid flow and heat transfer in a porous microchannel saturated with power-law fluid. The governing momentum and energy equations are solved by using the finite difference technique. The present study focuses on the slip flow regime, and the flow in porous media is modeled using the modified Darcy-Brinkman-Forchheimer model for power-law fluids. Parametric studies are conducted to examine the effects of Knudsen number, Darcy number, power law index, and inertia parameter. Results are given in terms of skin friction and Nusselt number. It is found that when the Knudsen number and the power law index decrease, the skin friction on the walls decreases. This effect is reduced slowly while the Darcy number decreases until it reaches the Darcy regime. Consequently, with a very low permeability the effect of power law index vanishes. The numerical results indicated also that when the power law index decreases the fully-developed Nusselt number increases considerably especially, in the limit of high permeability, that is, nonDarcy regime. As far as Darcy regime is concerned the effects of the Knudsen number and the power law index of the fully-developed Nusselt number is very little.

Electroosmotic Flow and Heat Transfer in Microchannels: A Closed Form Solution

In this paper, an analysis has been conducted to explore the momentun and thermal transport characterastics of electroosmotic liquid flow in a microchannel under imposed constant wall heat flux boundary condition. The present formulation shows that the problem is governed by three parameters, namely, the length scale ratio (ratio of Debye length to half channel height), the Joule heating parameter (ratio of Joule heating to surface heat flux), and the Brinkman number. A closed form solution of the problem was obtained and the impact of viscous dissipation on the heat transfer behavior was investigated. Analytical exact solutions of dimensionless velocity and temperature profiles, normalized local velocity, volume flow rate, friction coefficient, mean fluid temperature, and the fully-developed Nusselt number were obtained as functions of the governing parameters. Especially, the effects of length scale ratio on major flow parameters (including the normalized local velocity, friction coefficient, and volumetric flow rate) were examined. Also, the viscous dissipation effect on thermal transport characteristics was discussed in depth.

Electrokinetic-Driven Flow and Heat Transfer of a Non-Newtonian Fluid in a Circular Microchannel

An analytical analysis is presented to explore the transport characteristics of electroosmotic flow and associated heat transfer of non-Newtonian power-law fluids in a circular microchannel. The approach selected here is based on the linearized Poisson–Boltzmann distribution equation to get analytical expressions for velocity and temperature profiles, the friction coefficient, and the fully-developed Nusselt number. The key parameters governing the problem include the flow behavior index, the length scale ratio (ratio of half channel diameter to Debye length), and the thermal scale ratio. The results reveal that increasing the length scale ratio tends to increase the friction coefficient. For surface heating, increasing the flow behavior index amplifies the temperature difference between the wall and the fluid, and thus the temperature distribution broadens; while the opposite trend is observed for surface cooling. Depending on the value of the thermal scale ratio, the fully-developed Nusselt number can be either increased or decreased by increasing the flow behavior index and/or the length scale ratio. The effect of flow behavior index on the Nusselt number vanishes as the length scale ratio approaches infinity.

Non-Newtonian Fluid Flow and Heat Transfer in Microchannels

Convective heat transfer of non-Newtonian power-law fluids in a microchannel is investigated. The governing parameters include the flow behavior index, the length scale ratio (ratio of Debye length to half channel height), the Joule heating parameter (ratio of Joule heating to surface heat flux), and the Brinkman number. Analytical expressions are presented for velocity and temperature profiles, as well as the Nusselt number. The flow and heat transfer parameters can be obtained by numerical integrations of the analytical expressions. The dimensionless temperature distribution across the microchannel and the fully-developed Nusselt number are illustrated for a wide range of governing parameters.

Evaluation of Nusselt number for a flow in a microtube using second-order slip model

In this paper, the fully-developed temperature profile and corresponding Nusselt value is determined analytically for a gaseous flow in a microtube with a thermal boundary condition of constant wall heat flux. The flow assumed to be laminar, and hydrodynamically and thermally fully developed. The fluid is assumed to be constant property and incompressible. The effect of rarefaction, viscous dissipation and axial conduction, which are important at the microscale, are included in the analysis. Second-order slip model is used for the slip-flow and temperature jump boundary conditions for the implementation of the rarefaction effect. Closed form solutions for the temperature field and the fully-developed Nusselt number is derived as a function of Knudsen number, Brinkman number and Peclet number.

Investigation of turbulent mixed convection of air flow in vertical tubes using a zonal turbulence model

Fluid flow and heat transfer in a vertical tube under constant heat flux have been studied and the effect of buoyancy force on the heat transfer coefficient is investigated. The finite volume method is used to study turbulent flow in both upward and downward directions. For the turbulence modeling, a zonal k– ε model is employed and the numerical results are compared with available experimental data. The results of the simulation show that for the downward flow, heat transfer is enhanced and for strong buoyancy force, flow reversal is observed. In contrast, for the heated upward flow, heat transfer can be either impaired or enhanced by the buoyancy force depending on its strength. Partial laminarization is caused by the buoyancy in the case of modest buoyancy force. For the condition of stronger buoyancy force, a sudden decrease in the fully-developed Nusselt number is evident in the experimental data and well predicted by the numerical solution. In general, the quantitative agreement between the numerical results and the experimental data is satisfactory.

An experimental and numerical study of forced convection in a microchannel with negligible axial heat conduction

Experiments are conducted for laminar forced convection of water in a microchannel under partially-heated and fully-heated conditions on one wall with negligible axial heat conduction. The microchannel had a trapezoidal cross-sectional shape, with a hydraulic diameter of 155 μm and a heating length of 30 mm. Three-dimensional numerical simulations, based on the Navier–Stokes equations and energy equation, are obtained for forced convection of water in this microchannel under the same experimental conditions. It is found that the numerical predictions of wall temperatures and local Nusselt numbers are in good agreement with experimental data. This confirms that classical Navier–Stokes and energy equations are valid for the modeling of convection in a microchannel having a hydraulic diameter as small as 155 μm. For a microchannel with the same cross-sectional shape with one-wall heated and a heating length of 100 mm, numerical results show that the thermal entrance length is given by z = 0.15RePrD h, with the fully-developed Nusselt number approaching a constant value of 4.00.

Effect of Prandtl number on heat transfer characteristics in an axisymmetric sudden expansion: A numerical study

Using the standard k-e turbulence model, an incompressible, axisymmetric turbulent flow with a sudden expansion was simulated. Effect of Prandtl number on heat transfer characteristics downstream of the expansion was investigated. The simulation revealed circulation downstream of the expansion. A secondary circulation (corner eddy) was also predicted. Reattachment was predicted at approximately 10 step heights. Corresponding to Prandtl number of 7.0, a peak Nusselt number 13 times the fully-developed value was predicted. The ratio of peak to fully-developed Nusselt number was shown to decrease with decreasing Prandtl number. Location of maximum Nusselt number was insensitive to Prandtl number.

Fully-Developed Thermal Transport in Combined Pressure and Electro-Osmotically Driven Flow in Microchannels

Thermally fully-developed heat transfer has been analyzed for combined electro-osmotic and pressure driven flow in a circular microtube. The two classical thermal boundary conditions of constant wall heat flux and constant wall temperature were considered. Such a flow is established by the combination of an imposed pressure gradient and voltage potential gradient along the length of the tube. The induced flow rate and velocity profile are functions of the imposed potential gradient, electro-osmotic mobility of the fluid, the ratio of the duct radius to the Debye length, the established streamwise pressure gradient, and the fluid viscosity. The imposed voltage gradient results in Joule heating in the fluid, with an associated distributed volumetric source of energy. For this scenario, the solution for the fully developed, dimensionless temperature profile and corresponding Nusselt number have been determined. The fully-developed Nusselt number is found to depend on the duct radius/Debye length ratio (termed the relative duct radius), the dimensionless volumetric source, and a dimensionless parameter that characterizes the relative strengths of the two driving mechanisms. This parameter can take on both positive and negative values, depending on the signs of the streamwise voltage and pressure gradients imposed. Analytical results are presented and discussed for a range of the governing dimensionless parameters.

Experimental investigation of laminar mixed convection in an inclined semicircular duct under buoyancy assisted and opposed conditions

Combined free and forced convection is experimentally investigated for laminar water flow in the entrance region of a semicircular duct with upward and downward inclinations within ±20° using the boundary condition of uniform heat input axially. The experiment was designed for determining the effect of inclination (particularly the downward) on the wall temperature, and the local and fully-developed Nusselt numbers at three Reynolds numbers (500, 1000, and 1500) and a wide range of Grashof numbers. For the upward inclinations, Nusselt number was found to increase with Grashof number and the inclination angle (up to 20°), while the effect of Reynolds number was found to be small. For the downward inclination, Reynolds number has a strong effect on Nusselt number and the manner by which it varies with Grashof number. The fully-developed values of Nusselt number agree well in magnitude and trend with theoretical results recently obtained by the authors.

Analysis and experiment of non-Darcian convection in horizontal square packed-sphere channels—1. Forced convection

A numerical and experimental investigation of non-Darcian forced convection in horizontal square packed-sphere channels is presented. The theoretical results are found to be in agreement with the experimental results. The values of the fully-developed Nusselt number are influenced mainly by the channelling effect when the Peclet number is small, but the thermal dispersion effect becomes dominant when the Peclet number is high. The effect of thermal dispersion on heat transfer is rather weak in the near entrance region but becomes relatively significant in the thermally fully-developed region. The ratio of equivalent hydraulic diameter to sphere diameter, D e / d, significantly affects the Nusselt numbers when the Peclet number is high; it is caused again mainly by the thermal dispersion effect.

Forced convection in parallel and tapered passages: the importance of the heat flux boundary condition

It has been established previously that, for heat transfer to a constant property fluid in a long pipe, an exponential wall heat flux ( q ∝ e yx ) leads to a fully-developed Nusselt number. Attention is drawn to the rapid decrease of Nu as the parameter γ is made increasingly negative. For wedge flow there is a corresponding family of thermal boundary conditions for which Nu is constant in the flow direction. If the familiar uniform wall heat flux were to be adopted without reflection for experiments with converging passages, significant reductions in Nu could well be observed and might be attributed to other effects in the accelerated flow.