Isothermal Microtube Heat Transfer With Second-Order Slip Flow and Temperature Jump Boundary Conditions

2006 ◽  
Author(s):  
Nian Xiao ◽  
John Elsnab ◽  
Susan Thomas ◽  
Tim Ameel
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Nusselt Number ◽  
Boundary Conditions ◽  
Temperature Jump ◽  
Slip Flow ◽  
Second Order ◽  
Order Model ◽  
First Order ◽  
Second Order Model

Two analytical models are presented in which the continuum momentum and energy equations, coupled with second-order slip flow and temperature jump boundary conditions, are solved. An isothermal boundary condition is applied to a microchannel with a circular cross section. The flow is assumed to be hydrodynamically fully developed and thermal field is either fully developed or thermally developing from the tube entrance. A traditional first-order slip boundary condition is found to over predict the slip velocity compared to the second-order model. Heat transfer increases at the upper limit of the slip regime for the second-order model. The maximum second-order correction to the first-order Nusselt number is on the order of 18% for air. The second-order effect is also more significant in the entrance region of the tube. The Nusselt number decreases relative to the no-slip value when slip and temperature jump effects are of the same order or when temperature jump effects dominate. When temperature jump effects are small, the Nusselt number increases relative to the no-slip value. Comparisons to a previously reported model for an isoflux boundary condition indicate that the Nusselt number for the isoflux boundary condition exceeds that for the isothermal case at all axial locations.

Microtube Gas Flows With Second-Order Slip Flow and Temperature Jump Boundary Conditions

2006 ◽  
Author(s):  
Nian Xiao ◽  
John Elsnab ◽  
Tim Ameel
Keyword(s):  
Nusselt Number ◽  
Boundary Conditions ◽  
Temperature Jump ◽  
Order Approximation ◽  
Slip Flow ◽  
Order Effect ◽  
Second Order ◽  
First Order ◽  
Slip Regime ◽  
Energy Equations

Second-order slip flow and temperature jump boundary conditions are applied to solve the momentum and energy equations in a microtube for an isoflux thermal boundary condition. The flow is assumed to be hydrodynamically fully developed, and the thermal field is either fully developed or developing from the tube entrance. In general, first-order boundary conditions are found to over predict the effects of slip and temperature jump, while the effect of the second-order terms is most significant at the upper limit of the slip regime. The second-order terms are found to provide a correction to the first-order approximation. For airflows, the maximum second-order correction to the Nusselt number is on the order of 50%. The second-order effect is also more significant in the entrance region of the tube. Nusselt numbers are found to increase relative to their no-slip values when temperature jump effects are small. In cases where slip and temperature jump effects are of the same order, or where temperature jump effects dominate, the Nusselt number decreases when compared to traditional no-slip conditions.

scholarly journals FEATURES OF HEAT TRANSFER IN A FLAT POROUS MICROCHANNEL WITH THE SECOND ORDER SLIP BOUNDARY CONDITIONS

2021 ◽  
Vol 43 (2) ◽  
pp. 5-12
Author(s):  
A.A. Avramenko ◽  
N.P. Dmitrenko ◽  
Yu.Yu. Kovetska ◽  
O.I. Skitsko
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Prandtl Number ◽  
Boundary Conditions ◽  
Second Order ◽  
Hydrodynamic Resistance ◽  
First Order ◽  
Slip Boundary ◽  
Velocity Jump ◽  
The Heat Transfer Coefficient

The results of the study of heat transfer under forced convection in a flat porous microchannel taking into account the boundary conditions of slippage of the first and second order are considered. The simulation showed that with decreasing porosity the flow velocity in the central part of the microchannel and the slipping velocity on the wall decrease due to the increase in hydrodynamic resistance. Taking into account the influence of the boundary conditions of the second order shows that the magnitude of the velocity jump on the wall varies depending on the value of the parameter A2. The jump decreases with a positive value of A2, with a negative value - increases in comparison with the case A2 = 0 (first order boundary conditions). Qualitatively similar effects of porosity and second-order boundary conditions were also observed with respect to temperature profiles. The results of the calculation of the relative Nusselt number showed that the decrease in porosity contributes to the intensification of heat transfer. The dynamics of the change in the heat transfer coefficient with an increase in the Knudsen number indicates that an increase in the Prandtl number also leads to an improvement in the thermal interaction of the flow with the channel wall. The analysis of taking into account the boundary conditions of the second order showed that at small values of the Prandtl number (Pr ≤ 1) the influence of the parameter A2 was not observed. At A2 < 0 the effects of the boundary conditions of the second order lead to an increase in the relative Nusselt number, whereas at A2> 0 the value of the normalized Nusselt number decreases in comparison with the case A2 = 0 (boundary conditions of the first order).

Slip-Flow and Conjugate Heat Transfer in Rectangular Microchannels

2008 ◽  
Author(s):  
H. D. Madhawa Hettiarachchi ◽  
Mihajlo Golubovic ◽  
William M. Worek ◽  
W. J. Minkowycz
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Nusselt Number ◽  
Conjugate Heat Transfer ◽  
Temperature Jump ◽  
Slip Flow ◽  
Numerical Code ◽  
Published Data ◽  
Rectangular Microchannels ◽  
Knudsen Numbers

Slip-flow and conjugate heat transfer in rectangular microchannels are studied numerically for thermally developing laminar flow subjected to constant wall temperature (T) and constant wall heat flux (H2) boundary conditions. A three-dimensional numerical code based on finite volume method is developed to solve the coupled energy equations in the wall and fluid regions together with temperature jump at the wall-fluid boundary. A modified convection-diffusion coefficient at the wall-fluid interface is defined to incorporate the temperature-jump boundary condition. The numerical code is validated by comparing the present results with the published data. The effect of rarefaction and wall conduction on the heat transfer in the entrance region is analyzed in detail. Results show that the wall conduction has a considerable influence on the developing Nusselt number along the channel for the H2 boundary condition, particularly at low Knudsen numbers. In the case of the T thermal boundary condition, negligible influence of wall conduction on the Nusselt number is observed for all Knudsen numbers considered.

Evaluation of Nusselt Number for a Flow in a Parallel Plates Using MHD Second Order Slip Model

2021 ◽  
Author(s):  
Hatice Simsek
Keyword(s):  
Magnetic Field ◽  
Boundary Condition ◽  
Nusselt Number ◽  
Slip Flow ◽  
Mhd Flow ◽  
Second Order ◽  
Parallel Plates ◽  
First Order ◽  
Condition Model ◽  
Boundary Condition Model

Abstract In this study, two separate boundary condition models, as proposed by Beskok and Karniadakis [1] and Deissler [2], widely preferred for the second order boundary condition, were used. These two proposed boundary condition models were solved in the presence of a magnetic field moving normal to the plate surface in magneto-hydrodynamic (MHD) flow between micro-parallel plates with constant wall heat flux. The energy equation for the second-order temperature jump boundary condition, taking into account the momentum and viscous dissipation, as well as the corresponding Nusselt value were solved analytically in slip flow regime.The flow of an incompressible viscous flow between fixed micro-parallel plates with electrical conductivity is assumed to be constant, laminar, hydrodynamically and thermally developed. The closed form solutions for the temperature field and the fully developed Nusselt number are derived as a function of the Magnetic parameter (MHD), Knudsen number and Brinkman number and shown graphically and in a tabular form. The second order boundary condition model proposed by Deissler [2] predicts the Nusselt number to be at lower values when compared to the first order boundary condition model, and the second order boundary condition model proposed by Beskok and Karniadakis [1] predicts the Nusselt number to be at higher values than that of the first order boundary condition model. Moreover, increasing the magnetic field parameter M, led to higher Nusselt values in the slip flow model proposed by both Deissler [2] and Beskok and Karniadakis [1] compared to that when M = 0.

scholarly journals Evaluation of Nusselt Number for a Flow in a Microtube With Second-Order Model Including Thermal Creep

2012 ◽  
Author(s):  
Barbaros Çetin
Keyword(s):  
Nusselt Number ◽  
Temperature Jump ◽  
Slip Flow ◽  
Second Order ◽  
Thermal Creep ◽  
Brinkman Number ◽  
Order Model ◽  
Closed Form Solutions ◽  
Constant Wall Heat Flux ◽  
Thermo Physical Properties

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

Second-Order Correlation of Klinkenberg-Corrected Permeability and its Experimental Verification on Heterogeneously-Stressed Gas Shale

2020 ◽  
Author(s):  
Yufei Chen ◽  
Changbao Jiang ◽  
Juliana Y. Leung ◽  
Andrew K. Wojtanowicz ◽  
Dongming Zhang ◽  
...  
Keyword(s):  
Experimental Data ◽  
Second Order ◽  
Gas Shale ◽  
Order Model ◽  
First Order ◽  
Gas Slippage ◽  
Steady State Method ◽  
Slippage Effect ◽  
Stress Heterogeneity ◽  
Second Order Model

Abstract Shale is an extremely tight and fine-grained sedimentary rock with nanometer-scale pore sizes. The nanopore structure within a shale system contributes not only to the low to ultra-low permeability coefficients (10−18 to 10−22 m2), but also to the significant gas slippage effect. The Klinkenberg equation, a first-order correlation, offers a satisfying solution to describe this particular phenomenon for decades. However, in recent years, several scholars and engineers have found that the linear relation from the Klinkenberg equation is invalid for most gas shale reservoirs, and a need for a second-order model is, therefore, proceeding apace. In this regard, the purpose of this study was to develop a second-order approach with experimental verifications. The study involved a derivation of a second-order correlation of the Klinkenberg-corrected permeability, followed by experimental verifications on a cubic shale sample sourced from the Sichuan Basin in southwestern China. We utilized a newly developed multi-functional true triaxial geophysical (TTG) apparatus to carry out permeability measurements with the steady-state method in the presence of heterogeneous stresses. Also discussed were the effects of two gas slippage factors, Klinkenberg-corrected permeability, and heterogeneous stress. Finally, based on the second-order slip theory, we analyzed the deviation of permeability from Darcy flux. The results showed that the apparent permeability increased more rapidly as the pore pressure declined when the pore pressures are relatively low, which is a strong evidence of the gas slippage effect. The second-order model could reasonably match the experimental data, resulting in a lower Klinkenberg-corrected permeability compared with that from the linear Klinkenberg equation. That is, the second-order approach improves the intrinsic permeability estimation of gas shales with the result being closer to the liquid permeability compared with the Klinkenberg approach. Analysis of the experimental data reported that both the first-order slippage factor A and the second-order slippage factor B increased with increasing stress heterogeneity, and that A was likely to be more sensitive to stress heterogeneity compared with B. Interestingly, both A and B first slightly increased and then significantly as the permeability declined. It is recommended that when the shale permeability is below 10−18 m2, the second-order approach should be taken into account. Darcy’s law starts to deviate when Kn &gt; 0.01 and is invalid at high Knudsen numbers. The second-order approach seems to alleviate the problem of overestimation compared with the Klinkenberg approach and is more accurate in permeability evolution.

Three Dimensional Numerical Analysis of Laminar Slip-Flow Heat Transfer in Rectangular Microchannels

2008 ◽  
Author(s):  
H. D. Madhawa Hettiarachchi ◽  
Mihajlo Golubovic ◽  
William M. Worek
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Temperature Jump ◽  
Slip Flow ◽  
Three Dimensional ◽  
Velocity Slip ◽  
Thermal Boundary Condition ◽  
Navier Stokes ◽  
Constant Wall Temperature ◽  
Rectangular Microchannels

Slip-flow and heat transfer in rectangular microchannels are studied numerically for constant wall temperature (T) and constant wall heat flux (H2) boundary conditions under thermally developing flow. Navier-Stokes and energy equations with velocity slip and temperature jump at the boundary are solved using finite volume method in a three dimensional cartesian coordinate system. A modified convection-diffusion coefficient at the wall-fluid interface is defined to incorporate the temperature-jump boundary condition. Validity of the numerical simulation procedure is stabilized. The effect of rarefaction on heat transfer in the entrance region is analyzed in detail. The velocity slip has an increasing effect on the Nusselt (Nu) number whereas temperature jump has a decreasing effect, and the combined effect could result increase or decrease in the Nu number. For the range of parameters considered, there could be high as 15% increase or low as 50% decrease in fully developed Nu is plausible for T thermal boundary condition while it could be high as 20% or low as 35% for H2 thermal boundary condition.

Convective Heat Transfer in Elliptical Microchannels Under Slip Flow Regime and H1 Boundary Conditions

2015 ◽  
Vol 138 (4) ◽  
Author(s):  
Pamela Vocale ◽  
Gian Luca Morini ◽  
Marco Spiga
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Aspect Ratio ◽  
Boundary Conditions ◽  
Convective Heat Transfer ◽  
Cross Section ◽  
Convective Heat ◽  
Slip Flow ◽  
The Cross ◽  
Cross Section Geometry

In this work, hydrodynamically and thermally fully developed gas flow through elliptical microchannels is numerically investigated. The Navier–Stokes and energy equations are solved by considering the first-order slip flow boundary conditions and by assuming that the wall heat flux is uniform in the axial direction, and the wall temperature is uniform in the peripheral direction (i.e., H1 boundary conditions). To take into account the microfabrication of the elliptical microchannels, different heated perimeter lengths are analyzed along the microchannel wetted perimeter. The influence of the cross section geometry on the convective heat transfer coefficient is also investigated by considering the most common values of the elliptic aspect ratio, from a practical point of view. The numerical results put in evidence that the Nusselt number is a decreasing function of the Knudsen number for all the considered configurations. On the contrary, the role of the cross section geometry in the convective heat transfer depends on the thermal boundary condition and on the rarefaction degree. With the aim to provide a useful tool for the designer, a correlation that allows evaluating the Nusselt number for any value of aspect ratio and for different working gases is proposed.

Numerical Modeling of Micro Fin Arrays Using Slip Flow and Temperature Jump Boundary Conditions

2008 ◽  
Author(s):  
C. B. Sobhan ◽  
Muhsin M. Ameen ◽  
Praveen P. Abraham
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Temperature Jump ◽  
Slip Flow ◽  
Convection Heat Transfer ◽  
Transient State ◽  
Velocity Slip ◽  
Operating Pressure ◽  
Parametric Studies ◽  
Explicit Finite Difference

A numerical investigation of natural convection heat transfer from a rectangular fin array of microscale dimensions, where a “down and up” flow pattern occurs, is carried out. The stream function vorticity formulation is used in the analysis and the governing equations of the transient two dimensional field are solved using an explicit finite difference scheme. The dimensions of the domain are such that the problem falls under the slip flow regime. The non continuum effects are modeled through Maxwell’s velocity slip and Smoluchowski’s temperature jump boundary conditions. The steady state velocity and temperature distributions in the field are obtained by marching through the transient state. The average heat transfer coefficient and the Nusselt Number are calculated. The influence of the fin spacing, fin height and operating pressure on the performance of the fin array is studied through parametric studies and some conclusions are drawn regarding the significance of non continuum effects in the micro scale dimensions considered.

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