Heat and fluid flow characteristics of gases in micropipes

In this study, laminar forced convective heat transfer of a Newtonian fluid in a micropipe is analyzed by taking the viscous dissipation effect, the velocity slip and the temperature jump at the wall into account. Hydrodynamically and thermally fully developed flow case is examined. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Either wall heating (the fluid is heated) case or wall cooling (the fluid is cooled) case is examined. The Nusselt numbers are analytically determined as a function of the Brinkman number and the Knudsen number. Different definitions of the Brinkman number based on the definition of the dimensionless temperature are discussed. It is disclosed that for the cases studied here, singularities for the Brinkman number-dependence of the Nusselt number are observed and they are discussed in view of the energy balance.     

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer rates, as given by the Nusselt number, is numerically evaluated subject to constant wall heat flux (H2) and constant wall temperature (T) thermal boundary conditions. Numerical results are obtained using a continuum based, three-dimensional, compressible, unsteady computational fluid dynamics algorithm with slip velocity and temperature jump boundary conditions applied to the momentum and energy equations, respectively. For the limiting case of parallel plate channels, analytic solutions for the thermally and hydrodynamically fully developed momentum and energy equations are derived, subject to both first- and second-order slip velocity and temperature jump boundary conditions, from which analytic Nusselt number solutions are then obtained. Excellent agreement between the analytical and numerical results verifies the accuracy of the numerical algorithm, which is then employed to obtain three-dimensional rectangular channel and thermally/hydrodynamically developing Nusselt numbers. Nusselt number data are presented as functions of Knudsen number, Brinkman number, Peclet number, momentum and thermal accommodation coefficients, and aspect ratio. Rarefaction and viscous dissipation effects are shown to significantly affect the convective heat transfer rate in the slip flow regime.     

Analysis of laminar dissipative flow and heat transfer in a porous saturated circular tube with constant wall heat flux

A theoretical study of the thermal development of forced convection was performed using a circular tube filled with a saturated porous medium, with constant wall heat flux, and with the effect of viscous dissipation. The solution was obtained using the method of separation of variables. The Sturm–Liouville system was solved for the eigenvalues. Ordinary differential equations for the eigenfunctions were solved numerically by the fourth-order Runge–Kutta method. Results show that, in the presence of the viscous dissipation, both the level and distribution of temperature are altered remarkably, even for small values of the Brinkman number, Br, which is the ratio of heat generation caused by viscous dissipation to the value of heat flux at the wall. The value of the local Nusselt number, Nu, is demonstrably independent of Br, unlike the situation in which the wall temperature remains constant.     

Analysis of laminar heat transfer in micro-Poiseuille flow

The present study examines laminar forced convective heat transfer of a Newtonian fluid in a microchannel between two parallel plates analytically. The viscous dissipation effect, the velocity slip and the temperature jump at the wall are included in the analysis. Both hydrodynamically and thermally fully developed flow case is examined. Either the hot wall or the cold wall case is considered for the two different thermal boundary conditions, namely the constant heat flux (CHF) and the constant wall temperature (CWT). The interactive effects of the Brinkman number and the Knudsen number on the Nusselt numbers are analytically determined. Different definitions of the Brinkman number based on the definition of the dimensionless temperature are discussed. It is disclosed that for the cases studied here, singularities for the Brinkman number-dependence of the Nusselt number are observed and they are discussed in view of the energy balance.     

Similarity solutions for flows and heat transfer in microchannels between two parallel plates

In this paper we give analytical similarity solutions of the Navier–Stokes equations coupled with energy equation of Newtonian fluid in a microchannel between two parallel plates taking into account the effects of viscous dissipation, the velocity slip and the temperature jump at the wall. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). We provide new similarity transformations for the governing equations and derive the expressions of Poiseuille number (Po) and Nusselt number (Nu). Then, the homotopy analysis method (HAM) is employed to solve the nonlinear differential equations with related boundary conditions. Both the dimensionless analytical expressions of velocity and temperature are obtained. The rarefaction effects on velocity distribution and flow friction are exhibited. The interactive effects of the Brinkman number (Br) and the Knudsen number (Kn) on Nu are analytically studied for both the CHF and CWT cases.     

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.     

Thermal entry flow for a viscoelastic fluid: the Graetz problem for the PTT model

A theoretical study of the entrance thermal flow problem is presented for the case of a fluid obeying the Phan-Thien and Tanner (PTT) constitutive equation. This appears to be the first study of the Graetz problem with a viscoelastic fluid. The solution was obtained with the method of separation of variables and the ensuing Sturm-Liouville system was solved for the eigenvalues by means of a freely available solver, while the ordinary differential equations for the eigenfunctions and their derivatives were calculated numerically with a Runge-Kutta method.The scope of the present study was quite wide: it encompassed both the plane and axisymmetric geometries for channel and tube flows; two types of thermal boundary conditions with either an imposed wall temperature or an applied heat flux; inclusion of viscous dissipation; and elastic (through the Weissenberg number) and elongational (through the PTT parameter ?) effects. The main underlying assumptions were those of constant physical properties, negligible axial heat conduction, and fully developed hydrodynamic conditions. The results are discussed in terms of the main effects brought about by viscoelasticity and viscous dissipation on the Nusselt number variation and the bulk temperature.     

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.     

Numerical solution of power law fluids flow and heat transfer with a magnetic field in a rectangular duct

The steady laminar flow and heat transfer of an incompressible, electrically conducting, power law non-Newtonian fluids in a rectangular duct are studied in the presence of an external uniform magnetic field. The momentum and energy equations are solved iteratively using a finite difference method. Two cases of the thermal boundary conditions are considered; (1) T thermal boundary condition “constant temperature at the wall” and (2) H2 thermal boundary condition “constant heat flux at the wall”. The viscous and Joule dissipations are taken into consideration in the energy equation. A numerical solution for the governing partial differential equations is developed and the influence of the magnetic field on the velocity distribution, the friction factor and the average Nusselt number are discussed.     

Effects of viscous dissipation and fluid axial heat conduction on heat transfer for non-Newtonian fluids in ducts with uniform wall temperature: Part I: Parallel plates and circular ducts

Laminar heat transfer in parallel plates and circular ducts subject to uniform wall temperature is studied by taking into account both viscous dissipation and fluid axial heat conduction in an infinite region. Developing temperature fields are evaluated numerically by a finite-difference method for various Brinkman numbers (Br) and Peclet numbers (Pe). Nusselt numbers are presented graphically for Pe = 10 and Pe → ∞, and Br = 0, ± 0.5 and ± 1 for non-Newtonian fluids described by the power-law model with the flow index of n = 0.5, 1.0 and 1.5. It is shown that Nusselt number has a single fixed value independent of Br in the thermally developing region and its numerical value is equal to that at the fully developed region for non-zero Br, when the preheating of incoming fluid due to both viscous dissipation and fluid axial heat conduction is considered.     

Viscous-dissipation effects on the heat transfer in a Poiseuille flow

In this study, laminar heat-convection in a Poiseuille flow of a Newtonian fluid with constant properties is analyzed by taking the viscous dissipation into account. At first, both hydrodynamically and thermally fully-developed flow case is investigated. Then, consideration is given to thermally-developed laminar forced-convection. The axial heat-conduction in the fluid is neglected. Two different thermal boundary-conditions are considered: the constant heat-flux and the constant wall-temperature. Both the hot-wall and the cold-wall cases are considered. In the literature, the viscous-dissipation effect is commonly represented by the Brinkman number. Several different definitions of the Brinkman number arise depending on the thermal boundary conditions. Either for the thermally fully-developed case or the thermally-developing case (the Graetz problem), temperature distributions and the Nusselt numbers are analytically determined as functions of the Brinkman number.     

Forced convective heat transfer in a microtube including rarefaction,viscous dissipation and axial conduction effects

Subsonic gas convective heat transfer in a microtube with a constant cross-sectional area and uniform wall temperature is investigated both analytically and numerically. First, the effect of rarefaction on heat transfer characteristics, at a distance from the inlet where Nu becomes constant, is analytically investigated for two cases: (i) including and (ii) neglecting the viscous dissipation effect. An exact solution for Nu in fully developed flow is presented for the case without viscous dissipation, while a closed-form solution for the asymptotic Nu is also provided for the case with viscous dissipation. Next, a numerical model is employed to investigate the simultaneous effects of rarefaction, viscous dissipation, and axial conduction for developing hydrodynamic and temperature conditions. The Nusselt number is substantially affected by viscous dissipation, rarefaction and axial conduction.     

Effects of viscous dissipation during forced convection of power-law fluids in microchannels

The effect of viscous dissipation in forced convection of power-law fluids through microchannels of different cross-sectional geometries is studied numerically over the ranges of power-law index, 0.8 ≤ n ≤ 1.2 and Brinkman number, 0.001 ≤ Br ≤ 0.1 while keeping Péclet number constant at Pe = 10. Two types of thermal boundary conditions, namely, uniform wall temperature (T₂) and uniform heat flux (H₂), have been employed at the microchannel wall and the results of the temperature fields are expressed in terms of Nusselt number. The interplay between the fluid rheology and viscous dissipation effect gives rise to significant alteration in the net convective transport and thus can be beneficial in the thermal design of biofluidic devices.     

Heat transfer by laminar Hartmann flow in thermal entrance region with a step change in wall temperatures: the Graetz problem extended

Thermally developing laminar Hartmann flow through a parallel-plate channel, including both viscous dissipation, Joule heating and axial heat conduction with a step change in wall temperatures, has been studied analytically. Expressions for the developing temperature and local Nusselt number in the entrance region are obtained in terms of Peclet number Pe, Hartmann number M, Brinkman number Br, under electrically insulating wall conditions, χ=−1 and perfectly conducting wall conditions, χ=0. The associated eigenvalue problem is solved by obtaining explicit forms of eigenfunctions and related expansion coefficients. We show that the nonorthogonal eigenfunctions correspond to Mathieu's functions. We propose a new asymptotic solution for the modified Mathieu's differential equation. The asymptotic eigenfunctions for large eigenvalues are also obtained in terms of Pe and M. Results show that the heat transfer characteristics in the entrance region are strongly influenced by Pe, M, Br and χ.     

Unsteady MHD flow and heat transfer with viscous dissipation past a stretching sheet

In this paper a study is carried out to analyze the unsteady heat transfer effects of viscous dissipation on the steady boundary layer flow past a stretching sheet with prescribed constant surface temperature in the presence of a transverse magnetic field. The sheet is assumed to stretch linearly along the direction of the fluid flow. The assumed initial steady flow and temperature field neglecting dissipation effects becomes transient by accounting dissipation effects when time t′ > 0. The temperature and the Nusselt number are computed numerically using an implicit finite difference method. The obtained steady temperature field with dissipation is of practical importance.     

A numerical study of single-phase convective heat transfer in microtubes for slip flow

The steady-state convective heat transfer for laminar, two-dimensional, incompressible rarefied gas flow in the thermal entrance region of a tube under constant wall temperature, constant wall heat flux, and linear variation of wall temperature boundary conditions are investigated by the finite-volume finite difference scheme with slip flow and temperature jump conditions. Viscous heating is also included, and the solutions are compared with theoretical results where viscous heating has been neglected. For these three boundary conditions for a given Brinkman number, viscous effects are presented in the thermal entrance region along the channel. The effects of Knudsen and Brinkman numbers on Nusselt number are presented in graphical and tabular forms in the thermal entrance region and under fully developed conditions.     

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.     

Viscous dissipation effects on the asymptotic behaviour of laminar forced convection for Bingham plastics in circular ducts

The present study concentrates on the effects of viscous dissipation and the yield shear stress on the asymptotic behaviour of the laminar forced convection in a circular duct for a Bingham fluid. It is supposed that the physical properties are constant and the axial conduction is negligible. The asymptotic temperature profile and the asymptotic Nusselt number are determined for various axial distributions of wall heat flux which yield a thermally developed region. It is shown that if the asymptotic value of wall heat flux distribution is vanishes, the asymptotic value of the Nusselt number is zero. The case of the asymptotic wall heat flux distribution non-vanishing giving a value of the Nusselt number dependent on the Brinkman number and on the dimensionless radius of the plug flow region was also analysed. For an infinite asymptotic value of wall heat flux distributions, the asymptotic value of the Nusselt number depends on the dimensionless radius of the plug flow region and on the dimensionless parameter which depends on the asymptotic behaviour of the wall heat flux. The condition of uniform wall temperature and convection with an external isothermal fluid were also considered. The comparison with other existing solutions in the literature in the Newtonian case is analysed.     

Laminar forced convection slip-flow in a micro-annulus between two concentric cylinders

Forced convection heat transfer in hydrodynamically and thermally fully developed flows of viscous dissipating gases in annular microducts between two concentric micro cylinders is analyzed analytically. The viscous dissipation effect, the velocity slip and the temperature jump at the wall are taken into consideration. Two different cases of the thermal boundary conditions are considered: uniform heat flux at the outer wall and adiabatic inner wall (Case A) and uniform heat flux at the inner wall and adiabatic outer wall (Case B). Solutions for the velocity and temperature distributions and the Nusselt number are obtained for different values of the aspect ratio, the Knudsen number and the Brinkman number. The analytical results obtained are compared with those available in the literature and an excellent agreement is observed.     

Analysis of temperature gradient bifurcation in porous media – An exact solution

The phenomenon of temperature gradient bifurcation in a porous medium is analyzed by studying the convective heat transfer process within a channel filled with a porous medium, with internal heat generation. A local thermal non-equilibrium (LTNE) model is used to represent the energy transport within the porous medium. Exact solutions are derived for both the fluid and solid temperature distributions for two primary approaches (Models A and B) for the constant wall heat flux boundary condition. The Nusselt number for the fluid at the channel wall is also obtained. The effects of the pertinent parameters such as fluid and solid internal heat generations, Biot number and fluid to solid thermal conductivity ratio are discussed. It is shown that the internal heat generation in the solid phase is significant for the heat transfer characteristics. The validity of the one equation model is investigated by comparing the Nusselt number obtained from the LTNE model with that from the local thermal equilibrium (LTE) model. The results demonstrate the importance of utilizing the LTNE model in the present study. The phenomenon of temperature gradient bifurcation for the fluid and solid phases at the wall for Model A is established and demonstrated. In addition, the temperature distributions for Models A and B are compared. A numerical study for the constant temperature boundary condition was also carried out. It was established that the phenomenon of temperature gradient bifurcation for the fluid and solid phases for the constant temperature boundary condition can occur over a given axial region.