The extended Graetz problem with piecewise constant wall heat flux for laminar and turbulent flows inside concentric annuli

In the present paper, the heat transfer characteristics in the thermal entrance region of concentric annuli have been analysed for laminar and turbulent internal flow. Axial heat conduction effects in the fluid have been taken into account. The present paper shows an exact analytical solution for the problem of a piecewise uniform wall heat flux. The obtained analytical solution for the extended Graetz problem is as simple and efficient to compute as the related solution of the parabolic problem. The obtained results show the effect of axial heat conduction in the fluid for a semi-infinite heated section as well as for a finite length of the heated section. It is shown, that for a finite length of the heated section, axial heat conduction effects might be important even for higher Peclet number.     

Conjugate forced convection and heat conduction in circular microchannels

The effects of axial heat conduction in the solid walls of microchannels of circular cross-sections are analyzed here. A systematic approach is adopted, with the aim of pointing out the influence of geometrical parameters and of solid wall thermal conductivity on microchannel heat transfer. The reliability of a commonly adopted criterium, based on the so-called axial conduction number, to assess the relevance of axial heat conduction is also discussed. Numerical simulations concern the simultaneously developing laminar flow of a constant property fluid in microchannels of different length, wall thickness and wall material, heated with a uniform heat flux at the outer surface, for different values of the Reynolds number. Moreover, since often in experimental tests the two end sections of the microchannel wall are not perfectly insulated, the effects of heat losses through these sections are also considered. A hybrid finite element procedure, which implies the step-by-step solution of the parabolized momentum equations in the fluid domain, followed by the solution of the energy equation in the entire domain, corresponding to both the solid and the fluid parts, is used for the numerical simulations.     

Laminar heat transfer characteristics of internally finned tube with sinusoidal wavy fin

Comparative numerical study of laminar heat transfer characteristics of annular tubes with sinusoidal wavy fins has been conducted both experimentally and numerically with Re = 299–1,475. The uniform heat flux is imposed on the tube outside wall surface. Two tube materials (copper and stainless steel) are considered. It is found that the fluid temperature profile is not linear but convex along the flow direction due to the axial heat conduction in tube wall, and the effects of axial heat conduction on the heat transfer decreases with an increase in Reynolds number or decrease in tube wall thermal conductivity. The axial distributions of local Nusselt number could reach periodically fully developed after 3–5 cycles. The convectional data reduction method based on the traditional method should be improved for tube with high thermal conductivity or low Reynolds numbers, Otherwise, the heat transfer performance of internally finned tube may be underestimated.     

On the second fundamental problem of combined free and forced convection through vertical noncircular ducts

Analysis of combined free and forced convection through vertical noncircular ducts is carried out using a variational technique. Fully developed flow with uniform axial heat input and uniform peripheral heat flux is assumed. All fluid properties are considered invariant with temperature except the variation of density in the buoyancy term of the equation of motion. The condition of uniform peripheral heat flux is utilized in deriving the variational expression. This procedure releases the thermal boundary condition from satisfying exactly the condition at the wall. A finite-difference procedure is carried out. For pure forced convection case, a particularly simple variational expression is presented. Nusselt numbers for combined free and forced convection are computed for rectangular, rhombic and elliptical ducts. An exact solution is presented for laminar forced convection through elliptic ducts. Variational results are in agreement with this exact solution. The present results are compared with those in the published literature wherever possible, and good agreement is obtained.     

Effects of viscous dissipation on laminar forced convection with axially periodic wall heat flux

The laminar forced convection in a circular duct is investigated in the case of a sinusoidal axial variation of the wall heat flux. The axial heat conduction in the fluid is neglected, while the effect of viscous dissipation is taken into account. The heat transfer in the thermally developed region, where the temperature is the sum of a linear function and a periodic function of the axial coordinate, is analysed. Both the temperature field and the local Nusselt number are evaluated analytically. Comparisons with the solution in the absence of viscous heating are performed. It is shown that the effect of viscous dissipation on the temperature field may be relevant especially in the case of a sinusoidal wall heat flux distribution with a vanishing mean value. Received on 24 July 1998     

Laminar forced convection with sinusoidal wall heat flux distribution: axially periodic regime

Stationary and laminar forced convection in a circular tube with a sinusoidal axial distribution of wall heat flux is studied under the hypothesis that both axial heat conduction and viscous dissipation in the fluid are negligible. Two cases are considered: a sinusoidal wall heat flux distribution with a vanishing mean value; a sinusoidal wall heat flux distribution which does not change its sign. In both cases, the temperature field and the local Nusselt number are evaluated analytically in the fully developed region, i.e. where the local Nusselt number depends periodically on the axial coordinate. It is shown that, in the first case, the fully developed region presents an infinite sequence of axial positions where the local Nusselt number is singular. In these positions, the wall heat flux has a non-vanishing value even if the wall temperature equals the bulk temperature.     

Axial conduction effect in flow through circular porous passages with prescribed wall heat flux

This study presents a series solution for computation of the steady state temperature field in circular ducts with prescribed wall heat flux. The ducts are filled with fluid saturated porous materials. The developed methodology includes a simple transformation that improves the convergence of this series solution. The acquired solution includes the contribution of axial conduction that leads to a modified Graetz-type solution for these fluid passages. Finally, this solution is augmented by the contribution of frictional heating.     

Numerical nonlinear inverse problem of determining wall heat flux

The inverse problem of determining time-variable surface heat flux in a plane wall, with constant or temperature dependent thermal properties, is numerically studied. Different kinds of incident heat flux, including rectangular waveform, are assumed. The solution is numerically solved as a function estimation problem, so that no a priori information for the functional waveforms of the unknown heat flux is needed. In all cases, a solution in the form of a piece-wise function is used to approach the incident flux. Transient temperature measurements at the boundary, from the solution of the direct problem, served as the simulated experimental data needed as input for the inverse analysis. Both direct and inverse heat conduction problems are solved using the network simulation method. The solution is obtained step-by-step by minimising the classical functional that compares the above input data with those obtained from the solution of the inverse problem. A straight line of variable slope and length is used for each one of the stretches of the desired solution. The influence of random error, number of functional terms and the effect of sensor location are studied. In all cases, the results closely agree with the solution.     

Forced convection heat transfer of Giesekus viscoelastic fluid in pipes and channels

A theoretical solution is presented for the convective heat transfer of Giesekus viscoelastic fluid in pipes and channels, under fully developed thermal and hydrodynamic flow conditions, for an imposed constant heat flux at the wall. The fluid properties are taken as constant and axial conduction is negligible. The effect of Weissenberg number (We), mobility parameter (α) and Brinkman number (Br) on the temperature profile and Nusselt number are investigated. The results emphasize the significant effect of viscous dissipation and fluid elasticity on the Nusselt number in all circumstances. For wall cooling and the Brinkman number exceeds a critical value (Br ₁), the heat generated by viscous dissipation overcomes the heat removed at the wall and fluid heats up longitudinally. Fluid elasticity shifts this critical Brinkman number to higher values.     

Heat transfer in laminar and turbulent flows in the thermal entrance region of concentric annuli: Axial heat conduction effects in the fluid

The present analytical study investigates the influence of axial heat conduction within the flow on the heat transfer in the thermal entrance region of a concentric annular duct with laminar and turbulent internal flow. The solution is based on a decomposition of the elliptic energy equation into a pair of first order partial differential equations. By using a new defined vector norm it is possible to obtain a selfadjoint eigenvalue problem for the extended Graetz problem even though the original convective diffusion operator is non-selfadjoint. The obtained exact analytical solutions for the Graetz problem with axial heat conduction are as simple to compute as the related solutions of the parabolic problem. Received on 28 October 1996     

Exact solution of conductive heat transfer in cylindrical composite laminate

This paper presents an exact solution for steady-state conduction heat transfer in cylindrical composite laminates. This laminate is cylindrical shape and in each lamina, fibers have been wound around the cylinder. In this article heat transfer in composite laminates is being investigated, by using separation of variables method and an analytical relation for temperature distribution in these laminates has been obtained under specific boundary conditions. Also Fourier coefficients in each layer obtain by solving set of equations that related to thermal boundary layer conditions at inside and outside of the cylinder also thermal continuity and heat flux continuity between each layer is considered. In this research LU factorization method has been used to solve the set of equations.     

Thermo-mechanical modeling of turbulent heat transfer in gas–solid flows including particle collisions

A thermo-mechanical turbulence model is developed and used for predicting heat transfer in a gas–solid flow through a vertical pipe with constant wall heat flux. The new four-way interaction model makes use of the thermal k_θ–τ_θ equations, in addition to the hydrodynamic k–τ transport, and accounts for the particle–particle and particle–wall collisions through a Eulerian/Lagrangian formulation. The simulation results indicate that the level of thermal turbulence intensity and the heat transfer are strongly affected by the particle collisions. Inter-particle collisions attenuate the thermal turbulence intensity near the wall but somewhat amplify the temperature fluctuations in the pipe core region. The hydrodynamic-to-thermal times-scale ratio and the turbulent Prandtl number in the region near the wall increase due to the inter-particle collisions. The results also show that the use of a constant or the single-phase gas turbulent Prandtl number produces error in the thermal eddy diffusivity and thermal turbulent intensity fields. Simulation results also indicate that the inter-particle contact heat conduction during collision has no significant effect in the range of Reynolds number and particle diameter studied.     

Hydrothermal analysis of non-Newtonian fluid flow (blood) through the circular tube under prescribed non-uniform wall heat flux

The present article aims to investigate the Graetz-Nusselt problem for blood as a non-Newtonian fluid obeying the power-law constitutive equation and flowing inside the axisymmetric tube subjected to non-uniform surface heat flux. After the flow field is determined by solving the continuity and the momentum equations, the energy equation is handled by employing the separation of variables method. The resulting Eigen functions and Eigen values are numerically calculated using MATLAB built-in solver BVP4C. The analysis is first conducted for the situation of constant heat flux and subsequently generalized to apply to the case of sinusoidal variation of wall heat flux along the tube length, using Duhamel's Theorem. Furthermore, an approximate analytic solution is determined, employing an integral approach to solve the boundary layer equations. With respect to the comparison, the results of approximate solution display acceptable congruence with those of exact solution with an average error of 7.4%. Interestingly, with decreasing the power-law index, the discrepancy between the two presented methods significantly reduces. Eventually, the influences of the controlling parameters such as surface heat flux and power-law index on the non-Newtonian fluid flow's thermal characteristics and structure are elaborately discussed. It is found that switching from constant wall heat flux to non-uniform wall heat flux that sinusoidally varies along the tube length significantly improves the simulation's accuracy due to the better characterization of the heat transport phenomenon in non-Newtonian fluid flow through the tube. In the presence of sinusoidally varying wall heat flux with an amplitude of 200 W/m²and when the power-law index is 0.25, the maximum arterial wall temperature is found to be about 311.56 K.     

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer, which is also an exact solution to the unsteady Navier-Stokes(NS) equations. The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions. The wall temperature and heat flux have power dependence on both time and spatial distance. The solution domain, the velocity distribution, the flow field, ...     

Thermal Resonance in Hyperbolic Heat Conduction in Porous Media due to Periodic Ohm’s Heating

The effect of periodic Ohm??s heating on hyperbolic heat conduction in porous media is studied analytically with the objective of identifying the thermal resonance conditions. Local thermal equilibrium conditions are assumed to apply. The paper focuses initially on the temperature solution and looks at the conditions required for resonating the temperature signal. The heat flux solution is then evaluated. While a discrete infinite set of modes can be resonated, it is shown that in practice the resonance in the temperature signal is felt starting from moderately small values of Fourier numbers and becomes too small to be noticed if the Fourier number is extremely small. The temperature solution is shown to represent a standing wave the amplitude of which is strongly affected by the Fourier number. While the heat flux solution is shown to differ from the one obtained for the temperature, it also shows similar features such as the standing wave behavior the amplitude of which is strongly affected by the Fourier number.     

Approche analytique de la convection forcée des fluides de Bingham dans un tubeAnalytical approach for forced convection of Bingham fluids in a tube

The asymptotic behaviour of laminar forced convection for Bingham fluid in a circular tube subjected to an axially varying wall heat flux is studied analytically. The effect of viscous dissipation is taken into account while the axial heat conduction in the fluid is considered as negligible. The asymptotic temperature profile and the asymptotic Nusselt number are determined for various axial wall heat flux distributions which yield a thermally developed region. The results obtained show a diminution in asymptotic Nusselt number when the Brinkman number and the dimensionless radius of the plug flow region increase. Comparisons with the results in the literature for Newtonian fluids show the validity of the present analysis. To cite this article: R. Khatyr et al., C. R. Mecanique 330 (2002) 69–75.     

Laminar forced convection conjugate heat transfer over a flat plate

Coupled heat transfer between laminar forced convection along and conduction inside a flat plate wall is theoretically studied. The laminar convective boundary layer is analyzed by employing the integral technique. The energy equations for the fluid and the plate wall are combined under the condition of the continuity in the temperature and heat flux at the fluid-solid interface. The analysis results in a simple formal solution. Expressions have been obtained for calculating local Nusselt number, wall heat flux and temperature along the plate, all are functions of the local Brun number, Br _x , which is a measure of the ratio of the thermal resistance of the plate to that of the convective boundary layer. The results indicate that for Br _x ≥0.15, neglecting the plate resistance will results in an error of more than 5% in Nusselt number. Comparison of the present solution with other previous studies has been made. The solution may be of a considerable theoretical and practical interest. Received on 19 August 1998     

Non-fourier heat conduction in a plane slab with prescribed boundary heat flux

The non-stationary heat conduction in an infinitely wide plane slab with a prescribed boundary heat flux is studied. An arbitrary time dependent boundary heat flux is considered and a non-vanishing thermal relaxation time is assumed. The temperature and the heat flux density distributions are determined analytically by employing Cattaneo-Vernotte's constitutive equation for the heat flux density. It is proved that the temperature and the heat flux density distributions can be incompatible with the hypothesis of local thermodynamic equilibrium. A comparison with the solution which would be obtained by means of Fourier's law is performed by considering the limit of a vanishing thermal relaxation time.     

Wall Heat Flux and Thermal Stratification Investigations during the Compression Stroke of an engine-like Geometry: A comparison between LES and DNS

The near wall regions in internal combustion engines contain a significant amount of the gaseous mass in the cylinder and thus have a high relevance for the amount of unburned hydrocarbons, the wall heat transfer and the thermal stratification in the cylinder. In this context in the following study the predictive capability of Large Eddy Simulation (LES) with respect to wall heat flux and thermal stratification during the compression stroke i.e. under non-reactive conditions in an Internal Combustion Engine (ICE) are investigated based on a comparison with Direct Numerical Simulations (DNS). Two different modeling approaches for the near wall region, the low Reynolds damping approach and the LES adapted model from Plengsaard and Rutland, have been tested. During the first half of the compression stroke the low Reynolds damping approach agreed well with the DNS data, but increasing deviations were observed after 270° CA (piston halfway up). The underprediction of the wall heat flux at later stages was found to stem from the underestimation of the y ₊ values of the first cell centroid, compared to values obtained by evaluating the DNS data at the same location, and originates from the model used to determine the friction velocity. As a consequence of the underpredicted y ₊ value, the cell is not located in the viscous sublayer as expected, and the temperature gradient which is needed for the heat flux calculation is underpredicted. The results of the LES wall heat transfer model from Plengsaard and Rutland on the other hand showed overall reasonable agreement with the DNS data, but the model strongly depended on the modeling constants. With respect to the increasing thermal stratification during the compression both methods were found to significantly under predict the DNS results. These findings are especially relevant for LES of auto ignition phenomena in engines, since ignition timing and location are known to strongly depend on the temperature distribution.     

Heat dispersion in stationary mixed convection flow about horizontal surfaces in porous media

An analysis has been performed to study the influence of velocity dependent dispersion on transverse heat transfer in mixed convection flow above a horizontal wall of prescribed temperature in a saturated porous medium. The Boussinesq approximation and boundary layer analysis were used to numerically obtain gravity affected temperature and velocity distributions within the frames of Darcy's law and a total thermal diffusivity tensor comprising both of constant coefficient heat conduction and velocity proportional mechanical heat dispersion. Dependending on Pe_∞, the molecular Peclét number basing on the effective thermal diffusivity and the velocity of the oncoming flow, density coupling has distinct influences on heat transfer rates between the wall surface and the porous medium flow region. For small Peclét numbers, when heat conduction is the prevailing mechanism, wall heat fluxes are the higher the larger the density difference between the oncoming and the near wall fluid is. The opposite is true for larger Peclét numbers, when mechanical heat dispersion is the main cause of heat spreading. For Pe_∞ tending to infinity these wall heat fluxes approach finite maximum values in the total heat diffusivity model, they grow beyond any limit if only constant coefficient heat conduction is considered. Thus, the inclusion of mechanical heat dispersion effects yields physically more realistic predictions. Received on 18 September 1996