Forced Convection with Laminar Pulsating Counterflow in a Saturated Porous Circular Tube

An analytical solution is obtained for forced convection in a circular tube occupied by a core–sheath-layered saturated porous medium with counterflow produced by pulsating pressure gradients. The case of the constant heat-flux boundary conditions is considered, and the Brinkman model is employed for the porous medium. A perturbation approach is used to obtain analytical expressions for the velocity, temperature distribution, and transient Nusselt number for convection produced by an applied pressure gradient that fluctuates with small amplitude harmonically in time about a non-zero mean. It is shown that the fluctuating part of the Nusselt number alters in magnitude and phase as the dimensionless frequency increases. The magnitude increases from zero, goes through a peak, and then decreases to zero. The height of the peak depends on the values of various parameters. The phase (relative to that of the steady component) decreases from π/2 to − π/2 as the frequency increases.     

Forced convection of a power-law fluid in a porous channel – numerical solutions

A detailed numerical study of laminar forced convection in a porous channel which contains a fibrous medium saturated with a power-law fluid was performed. Hydrodynamic and heat transfer results are presented for a configuration that has uniform heat flux or uniform temperature heating at the walls. The flow in the porous medium was modeled using the modified Brinkman-Forchheimer-extended Darcy model for power law fluids in which the non-Darcy effects of inertia and boundary were considered. Parametric studies were conducted to examine the effects of Darcy number, power law index, inertia parameter and Prandtl number. The results indicate that when the power law index is decreased, the velocity gradient near the walls increases but these effects are reduced gradually as the Darcy number decreases until the Darcy regime (Da≤10⁻⁶) is reached in which case the effects of power law index become negligible. As the power law index is decreased, the thermal boundary layer thickness decreases significantly only in the non-Darcy regime. Consequently, as the power law index decreases, the fully developed Nusselt number increases considerably in the non-Darcy regime whereas in the Darcy regime the change in Nusselt number is very small. As the Prandtl number increases, the local Nusselt number increases and this effect is more significant for shear thinning fluids (n<1.0). Received on 2 March 1998     

Radiation effect on mixed convection over an isothermal cone in porous media

The radiation effect on the mixed convection flow of an optically dense viscous fluid adjacent to an isothermal cone embedded in a saturated porous medium with Rosseland diffusion approximation is numerically investigated. The entire regime of the mixed convection is included, as the mixed convection parameter of χ varies from 0 (pure free convection) to 1 (pure forced convection). The transformed nonlinear system of equations is solved by using an implicit finite difference method. Numerical results are given for the dimensionless temperature profiles and the local Nusselt number for various values of the mixed convection parameter χ, the cone angle parameter m, the radiation-conduction parameter R _d and the surface temperature parameter H. The local Nusselt number decreases initially, reaches a minimum in the intermediate value of χ and then increases gradually. It is apparent that increasing the cone angle parameter m enhances the local Nusselt number. The local Nusselt number is significantly increased for the large values of the radiation-conduction parameter R _d and the surface temperature parameter H, i.e., radiation effect becomes pronounced. Received on 25 October 1999     

Numerical analysis of a symmetrically heated vertical channel with obstruction

 In this paper, a numerical investigation of laminar natural convection flows in a vertical channel with obstructions is carried out. The main purpose was to analyze the effects of the locations of symmetric obstructions. The computations were performed in a two-dimensional domain and a symmetric uniform wall temperature has been taken as thermal boundary condition. The governing equations were solved using a control volume method and the SIMPLER algorithm for the velocity–pressure coupling was employed. The profiles of the local Nusselt number were given for three different locations of the obstructions. The variation of the average Nusselt number and inlet flow rate versus the modified Rayleigh number were investigated. The results demonstrated that the average Nusselt number decreases as the distance of the obstructions from the inlet increases. Received on 17 January 2000     

Forced convection of gaseous slip-flow in porous micro-channels under Local Thermal Non-Equilibrium conditions

Steady laminar forced convection gaseous slip-flow through parallel-plates micro-channel filled with porous medium under Local Thermal Non-Equilibrium (LTNE) condition is studied numerically. We consider incompressible Newtonian gas flow, which is hydrodynamically fully developed while thermally is developing. The Darcy–Brinkman–Forchheimer model embedded in the Navier–Stokes equations is used to model the flow within the porous domain. The present study reports the effect of several operating parameters on velocity slip and temperature jump at the wall. Mainly, the current study demonstrates the effects of: Knudsen number (Kn), Darcy number (Da), Forchheimer number (Γ), Peclet number (Pe), Biot number (Bi), and effective thermal conductivity ratio (K _R) on velocity slip and temperature jump at the wall. Results are given in terms of skin friction (C _f Re ^*) and Nusselt number (Nu). It is found that the skin friction: (1) increases as Darcy number increases; (2) decreases as Forchheimer number or Knudsen number increases. Heat transfer is found to (1) decreases as the Knudsen number, Forchheimer number, or K _R increases; (2) increases as the Peclet number, Darcy number, or Biot number increases.     

Experimental investigation to study flow and heat transfer characteristics over a NACA0018 aerofoil for different height ratios

Experiments are carried out to study flow and heat transfer characteristics over NACA0018 aerofoil when the body approaches the wall of a wind tunnel. Investigations have been done to study the effect of wall proximity due to flow separation around the body at Reynolds number 2.5 × 10⁵, different height ratios and various angles of attack. The static pressure distribution has been measured on upper and lower surfaces of the aerofoil. The results have been presented in the form of pressure coefficient, drag coefficient for different height ratios. Pressure coefficient values are decreased and then increased on the lower surface of the aerofoil and decreased on the upper surface of the aerofoil at all angles of attack. The negative pressure coefficient and drag coefficient decreases as the body approaches the upper wall of wind tunnel. The maximum value of drag coefficient has been observed at an angle of attack 30° for the aerofoil at all height ratios. The Heat transfer experiments have been carried out under constant heat flux condition. Heat transfer coefficients are determined from the measured wall temperature and ambient temperature and presented in the form of Nusselt number. The variation of local as well as average Nusselt number has been shown with non dimensional distance for different angles of attack and for various height ratios. The local as well as average Nusselt number decreases as the height ratio decreases for all non-dimensional distance and angles of attack respectively. Maximum value of average Nusselt number has been observed at an angle of attack 40°.     

Constitutive Relations for Monatomic Gases¶Based on a¶Generalized Rational Approximation¶to the Sum of the Chapman-Enskog Expansion

. We study the asymptotic behavior as time goes to infinity of solutions to the initial‐boundary‐value problem on the half space for a one‐dimensional model system for the isentropic flow of a compressible viscous gas, the so‐called p‐system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity be zero at the boundary . When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary, then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity. The proof is given by an elementary energy method. (Accepted March 2, 1998)     

Heat transfer due to a round jet impinging normal to a circular cylinder

An experimental investigations of heat transfer for a stationary isothermal circular cylinder exposed normal to an impinging round air-jet has been reported. The circumferential heat transfer distributions as well as axial Nusselt number is measured. The measurements are taken as a function of the Reynolds number ranging from 3.8 × 10³ to 4 × 10⁴, the cylinder separation distance to the nozzle diameter (z/d) varying from 7 to 30, and the nozzle to cylinder diameter ratio (d/D) changing from 0.06 to 0.14. The output results indicated that the axial and radial distributions of the local heat transfer peaked at the impingement point. The heat transfer rate increases as the values of z decreases, for the same d and Re. The drop-off of the Nusselt number with increasing axial distance or radial angle from the impingement point was more pronounced for smaller z and d. The peripheral and surface average Nusselt numbers were determined by integration. The experimental data was used to produce correlations for both average and stagnation point heat transfer. Received on 4 January 1999     

Transient conjugate heat transfer from a hemispherical plate during free liquid jet impingement on the convex surface

This paper considers the analysis of transient heating of a hemispherical solid plate of finite thickness during impingement of a free liquid jet. A constant heat flux was imposed at the inner surface of the hemispherical plate at t = 0 and heat transfer was monitored for the entire duration of the transient until a steady state condition was reached. Calculations were done for Reynolds number (Re) ranging from 500 to 1,500 and dimensionless plate thicknesses to nozzle diameter ratio (b/d _n) from 0.083 to 1.5. Results are presented for local and average Nusselt number using water as the coolant and various solid materials such as silicon, constantan, and copper. It was detected that increasing the Reynolds number decreases the time for the plate to achieve the steady-state condition. Also, a higher Reynolds number increases the Nusselt number. Hemispherical plate materials with higher thermal conductivity maintain lower temperature non-uniformity at the solid–fluid interface. Increasing the plate thickness decreases the maximum temperature in the solid and increases the time to reach the steady-state condition.     

Computational flow and heat transfer of multiple circular jets impinging on a flat surface with effusion

Computational investigations are reported on the local flow and heat transfer characteristics from staggered, multiple circular air jets impinging on a flat surface with effusion holes. The geometrical and flow parameters for the computational study are chosen as per the experimental arrangement of Cho and Rhee J Turbomachinery 123:601–608, (14) so as to explain salient features observed in these experiments. The two peaks in the Nusselt number observed in the case of H/D = 6 and three peaks in the case of H/D = 2 are attributed to the flow characteristics such as primary vortices forming an up-wash region, followed by secondary vortices resulting in a secondary stagnation zone. The magnitude of local peak in heat transfer increases up to 88% with increasing values of D/d from 0.5 to 1.5 at Re = 10,000.     

Correlations for heat transfer coefficient and pressure drop in the annulus of concentric helical coils

Heat transfer and pressure drop characteristics in the annulus of concentric helical coils heat exchangers were experimentally investigated. The effects of coil curvature ratio, flow configuration, number of turns and addition of surfactant were investigated. Five test coils were designed and manufactured to study the effect of different parameters on heat transfer and pressure drop. The liquids used in the present study were water and oleyl-dihydroxy-etheyl-amine-oxide (ODEAO, C₂₂H₄₅NO₃ = 371) non-ionic aqua surfactant solution flowing through the annulus side. The inner side Reynolds number range 11,000–27,000 and the annulus side range 5,000–19,000. The results showed that the annulus Nusselt number decreases as annulus curvature ratio increases and increases when number of turns decrease. Moreover, the friction factor increases with the curvature ratio and also increases as number of turns decreases. Both Nusselt number and friction factor decrease when ODEAO concentration increases.     

Non-Darcian and Anisotropic Effects on Free Convection in a Porous Enclosure

The free convective flow and heat transfer, within the framework of Boussinesq approximation, in an anisotropic fluid filled porous rectangular enclosure subjected to end-to-end temperature difference have been investigated using Brinkman extended non-Darcy flow model. The studies involve simultaneous consideration of hydrodynamic and thermal anisotropy. The flow and temperature fields in general are governed by, Ra, the Rayleigh number, AR, the aspect ratio of the slab, K*, the permeability ratio and k*, the thermal conductivity ratio, and Da, Darcy number. Numerical solutions employing the successive accelerated replacement (SAR) scheme have been obtained for 100 ≤ Ra ≤ 1000, 0.5 ≤ AR ≤ 5, 0.5 ≤ K* ≤ 5, 0.5 ≤ k* ≤ 5, and 0 ≤ Da ≤ 0.1. It has been found that [`(Nu)]{\overline {Nu}}, average Nusselt number increases with increase in K* and decreases as k* increases. However, the magnitude of the change in [`(Nu)]{\overline {Nu}} depends on the parameter Da, characterizing the Brinkman extended non-Darcy flow.     

Numerical solutions of pulsating flow and heat transfer characteristics in a pipe

Numerical studies are made of flow and heat transfer characteristics of a pulsating flow in a pipe. Complete time-dependent laminar boundary-layer equations are solved numerically over broad ranges of the parameter spaces, i.e., the frequency parameter β and the amplitude of oscillation A. Recently developed numerical solution procedures for unsteady boundary-layer equations are utilized. The capabilities of the present numerical model are satisfactorily tested by comparing the instantaenous axial velocities with the existing data in various parameters. The time-mean axial velocity profiles are substantially unaffected by the changes in β and A. For high frequencies, the prominent effect of pulsations is felt principally in a thin layer near the solid wall. Skin friction is generally greateer than that of a steady flow. The influence of oscillation on skin friction is appreciable both in terms of magnitude and phase relation. Numerical results for temperature are analyzed to reveal significant heat transfer characteristics. In the downstream fully established region, the Nusselt number either increases or decreases over the steady-flow value, depending on the frequency parameter, although the deviations from the steady values are rather small in magnitude for the parameter ranges computed. The Nusselt number trend is amplified as A increases and when the Prandtl number is low below unity. These heat transfer characteristics are qualitatively consistent with previous theoretical predictions.     

Effect of viscous dissipation on fully developed combined convection in a horizontal double-passage channel

 Fully developed combined convection in a horizontal double-passage channel has been investigated by taking into account the effect of viscous dissipation. Uniform wall temperatures with asymmetric and symmetric heating have been considered. The results showed that the increase in Brinkman number decreases the Nusselt number on the hot wall and increases that on the cold wall specially when the baffle becomes near the hot wall or the cold wall, respectively. Received on 20 February 2001 / Published online: 29 November 2001     

A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder

Forced convection heat transfer from an unconfined circular cylinder in the steady cross-flow regime has been studied using a finite volume method (FVM) implemented on a Cartesian grid system in the range as 10 ≤ Re ≤ 45 and 0.7 ≤ Pr ≤ 400. The numerical results are used to develop simple correlations for Nusselt number as a function of the pertinent dimensionless variables. In addition to average Nusselt number, the effects of Re, Pr and thermal boundary conditions on the temperature field near the cylinder and on the local Nusselt number distributions have also been presented to provide further physical insights into the nature of the flow. The rate of heat transfer increases with an increase in the Reynolds and/or Prandtl numbers. The uniform heat flux condition always shows higher value of heat transfer coefficient than the constant wall temperature at the surface of the cylinder for the same Reynolds and Prandtl numbers. The maximum difference between the two values is around 15–20%.     

Thermal radiation effects on the flow and heat transfer in a liquid film on an unsteady stretching sheet

The influence of thermal radiation on the flow and heat transfer within Newtonian liquid film over an unsteady stretching sheet with and without thermocapillarity is examined. The governing non‐linear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformation, which is solved numerically for different values of the thermal radiation parameter and the thermocapillarity parameter. The results show that the dimensionless velocity, the film thickness and the local Nusselt number increase as the thermocapillarity parameter increases, while the free surface temperature decreases with increasing the thermocapillarity parameter. Also, both the dimensionless temperature and the free surface temperature increase and the local Nusselt number decreases as the thermal radiation parameter increases. Copyright © 2010 John Wiley & Sons, Ltd.     

Natural Convection in a Spherical Porous Annulus: The Brinkman Extended Darcy Flow Model

The present study reports results of a numerical investigation of natural convection in a spherical porous annulus. The inner and outer surfaces are subjected to constant temperatures. The Brinkman extended Darcy flow model is considered in this study. The problem has been solved numerically employing successive accelerated replacement scheme. The effect of parameters like Rayleigh number, Darcy number and radius ratio on the fluid flow and heat transfer has been examined. There exists a critical radius ratio $(\mathrm{rr} = 3)$ at which the average Nusselt number attains a peak value. The average Nusselt number decreases as the Darcy number increases.     

Slip-flow heat transfer in a microchannel with viscous dissipation

Forced convection flow in a microchannel with constant wall temperature is studied, including viscous dissipation effect. The slip-flow regime is considered by incorporating both the velocity-slip and the temperature-jump conditions at the surface. The energy equation is solved for the developing temperature field using finite integral transform. To increase β_v Kn is to increase the slip velocity at the wall surface, and hence to decrease the friction factor. Effects of the parameters β_v Kn, β, and Br on the heat transfer results are illustrated and discussed in detail. For a fixed Br, the Nusselt number may be either higher or lower than those of the continuum regime, depending on the competition between the effects of β_v Kn and β. At a given β_v Kn the variation of local Nusselt number becomes more even when β becomes larger, accompanied by a shorter thermal entrance length. The fully developed Nusselt number decreases with increasing β irrelevant to β_v Kn. The increase in Nusselt number due to viscous heating is found to be more pronounced at small β_v Kn.     

A Modified Nested-Gridding for Upscaling–Downscaling in Reservoir Simulation

This article reports a numerical study of double-diffusive convection in a fluid-saturated vertical porous annulus subjected to discrete heat and mass fluxes from a portion of the inner wall. The outer wall is maintained at uniform temperature and concentration, while the top and bottom walls are adiabatic and impermeable to mass transfer. The physical model for the momentum equation is formulated using the Darcy law, and the resulting governing equations are solved using an implicit finite difference technique. The influence of physical and geometrical parameters on the streamlines, isotherms, isoconcentrations, average Nusselt and Sherwood numbers has been numerically investigated in detail. The location of heat and solute source has a profound influence on the flow pattern, heat and mass transfer rates in the porous annulus. For the segment located at the bottom portion of inner wall, the flow rate is found to be higher, whereas the heat and mass transfer rates are higher when the source is placed near the middle of the inner wall. Further, the average Sherwood number increases with Lewis number, while for the average Nusselt number the effect is opposite. The average Nusselt number increases with radius ratio (λ); however, the average Sherwood number increases with radius ratio only up to λ = 5, and for λ > 5 , the average Sherwood number does not increase significantly.     

Convective heat transfer characteristics of laminar pulsating pipe air flow

 Heat transfer characteristics to laminar pulsating pipe flow under different conditions of Reynolds number and pulsation frequency were experimentally investigated. The tube wall of uniform heat flux condition was considered. Reynolds number was varied from 780 to 1987 while the frequency of pulsation ranged from 1 to 29.5 Hz. The results showed that the relative mean Nusselt number is strongly affected by pulsation frequency while it is slightly affected by Reynolds number. The results showed enhancements in the relative mean Nusselt number. In the frequency range of 1–4 Hz, an enhancement up to 30% (at Reynolds number of 1366 and pulsation frequency of 1.4 Hz) was obtained. In the frequency range of 17–25 Hz, an enhancement up to 9% (at Reynolds number of 1366 and pulsation frequency of 17.5 Hz) was indicated. The rate of enhancement of the relative mean Nusselt number decreased as pulsation frequency increased or as Reynolds number increased. A reduction in relative mean Nusselt number occurred outside these ranges of pulsation frequencies. A reduction in relative mean Nusselt number up to 40% for pulsation frequency range of 4.1–17 Hz and a reduction up to 20% for pulsation frequency range of 25–29.5 Hz for Reynolds numbers range of 780–1987 were considered. This reduction is directly proportional to the pulsation frequency. Empirical dimensionless equations have been developed for the relative mean Nusselt number that related to Reynolds number (750 < Re < 2000) and the dimensionless frequency (3<Ω<18) with about 10% rms. Received on 16 May 2000 / Published online: 29 November 2001