Nonlinear flow and heat transfer dynamics of impinging jets onto slightly-curved surfaces

The nonlinear flow and heat transfer characteristics for a slot-jet impinging on slightly-curved surfaces are experimentally studied here. The effects of curved surface geometry and jet Reynolds number on the jet velocity distribution and circumferential Nusselt numbers are examined. Two different slightly-curved surface geometries of convex and concave are used as target surfaces. The nozzle geometry is a rectangular slot, and the dimensionless nozzle-to-surface distance equals to L¹ = 8. The constant heat fluxes are accordingly applied to the surfaces to obtain an impingement cooling by the air jet at ambient temperature. The measurements are made for the jet Reynolds numbers of Re = 8617, Re = 13 350 and Re = 15 415 for both curved surfaces. The velocity distributions of issuing jet from the nozzle exit to the target surface are obtained by a highly sensitive hot-wire anemometer. The T-type thermocouples are used to measure local temperatures of both the air jet and the plates. Two-dimensional velocity measurements show that the surfaces are remained out of the potential core region for all Re tested here. New correlations for local, stagnation point, and average Nusselt numbers as a function of jet Reynolds number and dimensionless circumferential distance are reported. The correlations reveal that the impinging cooling rate is higher with the concave surface and increase with increasing Re.     

Unsteady conjugate natural convection in a square enclosure filled with a porous medium

Mathematical simulation of unsteady natural convection modes in a square cavity filled with a porous medium having finite thickness heat-conducting walls with local heat source in conditions of heterogeneous heat exchange with an environment at one of the external boundaries has been carried out. Numerical analysis was based on Darcy–Forchheimer model in dimensionless variables such as a stream function, a vorticity vector and a temperature. The special attention was given to analysis of Rayleigh number effect Ra = 10⁴, 10⁵, 10⁶, of Darcy number effect Da = 10^?5, 10^?4, 10^?3, ∞, of the transient factor effect 0 < τ < 1000 and of the heat conductivity ratio k_2,1 = 3.7 × 10^?2, 5.7 × 10^?4, 6.8 × 10^?5 on the velocity and temperature fields. The influence scales of the defining parameters on the average Nusselt number have been detected.     

Nonlinear flow and heat transfer dynamics of a slot jet impinging on a slightly curved concave surface

The nonlinear flow and heat transfer characteristics for a slot jet impinging on a slightly curved concave surface are experimentally studied here. The effects of jet Reynolds number on the jet velocity distribution and circumferential Nusselt numbers are examined. The nozzle geometry is a rectangular slot and the dimensionless nozzle-to-surface distance equals to L^⁎ = 8. The constant heat fluxes are accordingly applied to the surface to obtain an impingement cooling by the air jet at ambient temperature. The measurements are made for the jet Reynolds numbers of 8617, 13 350 and 15 415. New correlations for local, stagnation point, and average Nusselt numbers as a function of jet Reynolds number and dimensionless circumferential distance are reported.     

Natural convection in a nanofluid-filled eccentric annulus with constant heat flux wall: A lattice Boltzmann study with immersed boundary method

A numerical investigation of natural convection in a Cu–water nanofluid-filled eccentric annulus with constant heat flux wall is presented. The governing equations of the flow and temperature fields are solved by lattice Boltzmann method (LBM), and the Dirichlet and Neumann boundary conditions are treated using the immersed boundary method (IBM). Influences of the Rayleigh number (10³ ≤ Ra ≤ 10⁷), eccentricity (ε = −0.625,0 and 0.625), nanoparticles volume fraction (0 ≤ ϕ ≤ 0.03) and radial ratio (rr = 2.33,2.6 and 3) on the streamlines, isotherms and Nusselt number are studied. It is found that the inclusion of the nanoparticles into pure fluid changes the flow pattern. And the Nusselt number has a positive relationship with nanoparticle volume fraction, Rayleigh number and radial ratio. Also, it can be confirmed that Nusselt number in the case with negative eccentricity (ε = −0.625) is larger than the others.     

Lattice Boltzmann simulation of mixed convection heat transfer in eccentric annulus

Mixed convection heat transfer in eccentric annulus was simulated numerically by lattice Boltzmann model (LBM) based on multi-distribution function double-population approach. The effect of eccentricity on heat transfer at various locations was examined at Ra = 10⁴ and σ = 2. Velocity and temperature distributions as well as Nusselt number are obtained. The results are validated with published results and shown that multi-distribution function approach can evaluate the velocity and temperature fields in curved moving boundaries with a good accuracy in comparison with the previous studies. The results show that the average Nusselt number increases when the inner cylinder moves downward regardless of the radial position.     

Unsteady mixed convection flow over a vertical stretching sheet in a parallel free stream with variable wall temperature

The objective of the paper is to investigate the numerical study of an unsteady two-dimensional mixed convection flow along a vertical semi-infinite power-law stretching sheet in a parallel free stream with a power-law wall temperature distribution of the form T_W(x) = T_∞ + Ax^2m?1. The unsteadiness is caused by the free stream velocity as well as by the stretching sheet velocity. The governing non-linear partial differential equations in the velocity and temperature fields are written in non-dimensional form using suitable transformations. The resulting final set of coupled non-linear partial differential equations is solved using an implicit finite difference scheme in combination with a quasi-linearization technique. The effects of various governing parameters on the velocity and temperature profiles are discussed in the present numerical study. In addition, the numerical results for the local skin friction coefficient and local Nusselt number are also presented. The numerically computed results are compared with previously reported work and are found to be in excellent agreement.     

Laminar natural convection of non-Newtonian power-law fluids between concentric circular cylinders

The two-dimensional steady-state natural convection of power-law fluids is studied numerically between two concentric horizontal cylinders with different constant temperatures. The governing equations are discretized using finite volume technique based on second order upwind and are solved using the SIMPLE algorithm. The effects of Rayleigh number (10³ ≤ Ra ≤ 10⁵) and Prandtl number (10 ≤ Pr ≤ 10³) on the dimensionless velocity and temperature are investigated for both pseudoplastic and dilatant fluids. Also the mean Nusselt number for various values of governing parameters is obtained and discussed. The results indicate that with increasing the power-law index from 0.6 to 1.4, the mean Nusselt number decreases. In the best case among the range of parameters considered here the heat transfer rate for pseudo-plastic fluid (n = 0.6) is 170% higher than the Newtonian one and for dilatant fluid (n = 1.4) the heat transfer rate is 43% lower than the Newtonian fluid. So the pseudoplastic and dilatant fluids are more efficient than Newtonian fluids for cooling and insulating purposes, respectively. It is shown that as the Rayleigh number increases the cooling effect of pseudoplastic fluid and the insulating effect of dilatant fluid become more pronounced.     

Numerical study of heat transfer by natural convection and surface thermal radiation in an open cavity receiver

In this work the numerical results of natural convection and surface thermal radiation in an open cavity receiver considering large temperature differences and variable fluid properties are presented. Numerical calculations were conducted for Rayleigh number (Ra) values in the range of 10⁴–10⁶. The temperature difference between the hot wall and the bulk fluid (ΔT) was varied between 100 and 400 K, and was represented as a dimensionless temperature difference (φ) for the purpose of generalization of the trends observed. Noticeable differences are observed between the streamlines and temperature fields obtained for φ = 1.333 (ΔT = 400 K) and φ = 0.333 (ΔT = 100 K). The total average Nusselt number in the cavity increased by 79.8% (Ra = 10⁶) and 88.0% (Ra = 10⁴) as φ was varied from 0.333 to 1.333. Furthermore the results indicate that for large temperature differences (0.667 ? φ ? 1.333) the radiative heat transfer is more important that convective heat transfer.     

Finite element based heatline approach to study mixed convection in a porous square cavity with various wall thermal boundary conditions

A penalty finite element method based simulation is performed to analyze the influence of various walls thermal boundary conditions on mixed convection lid driven flows in a square cavity filled with porous medium. The relevant parameters in the present study are Darcy number (Da = 10^?5 ? 10^?3), Grashof number (Gr = 10³ ? 10⁵), Prandtl number (Pr = 0.7–7.2), and Reynolds number (Re = 1–10²). Heatline approach of visualizing heat flow is implemented to gain a complete understanding of complex heat flow patterns. Patterns of heatlines and streamlines are qualitatively similar near the core for convection dominant flow for Da = 10^?3. Symmetric distribution in heatlines, similar to streamlines is observed irrespective of Da at higher Gr in natural convection dominant regime corresponding to smaller values of Re. A single circulation cell in heatlines, similar to streamlines is observed at Da = 10^?3 for forced convection dominance and heatlines are found to emanate from a large portion on the bottom wall illustrating enhanced heat flow for Re = 100. Multiple circulation cells in heatlines are observed at higher Da and Gr for Pr = 0.7 and 7.2. The heat transfer rates along the walls are illustrated by the local Nusselt number distribution based on gradients of heatfunctions. Wavy distribution in heat transfer rates is observed with Da ? 10^?4 for non-uniformly heated walls primarily in natural convection dominant regime. In general, exponential variation of average Nusselt numbers with Grashof number is found except the cases where the side walls are linearly heated. Overall, heatlines are found to be a powerful tool to analyze heat transport within the cavity and also a suitable guideline on explaining the Nusselt number variations.     

The temperature statistics of a surfactant-covered air/water interface during mixed convection heat transfer and evaporation

Surface temperature fields were measured of an air/water interface where heat was transferred from the water to the air under mixed convection conditions. The interfacial temperature field was measured using an infrared (IR) camera for mean wind speeds ranging from 0 to 4.0 m/s, in 1.0 m/s increments. Statistics of these surface temperature fields, specifically, the root mean square (rms) and the skewness were obtained. Plots of the rms versus the heat flux showed linear behavior for low wind speeds (U = 0–3 m/s), and the skewness was also found to increase with heat flux for U = 0–3 m/s, although these data exhibited significant scatter. The scaled root mean square temperature was revealed to be governed by the ratio Ra^1/3/(Re^14/5Pr^1/3) where Ra is the Rayleigh number, Re¹ the Reynolds number based on water side friction velocity and Pr is the Prandtl number.     

Heat transfer from a nano-sphere with temperature and velocity discontinuities at the interface

A singular perturbation method has been used to derive a general equation for the rate of heat transfer from a sphere at low Knudsen number. The final expression includes both velocity slip and temperature slip at the interface and applies to a general Stokesian flow regime. The asymptotic analysis was carried up to the order Pe³ln(Pe). By choosing an expression for the drag multiplier, the derived expression for the Nusselt number may be applied to solid, fluid as well as porous spheres, which are special cases of the general solutions. Comparisons with known results for these special cases indicate the accuracy and wide range of applicability of the derived general expression. The inclusion of the temperature slip at the interface makes this equation applicable to particles, bubbles and drops of nanometer sizes, in the continuum or the slip-flow regime, that is for Knudsen number Kn < 0.1. Our results show that the velocity slip at the interface does not affect significantly the overall Nusselt number, Nu. However, the temperature slip affects the heat transfer significantly. If the temperature discontinuity becomes large, the sphere becomes almost adiabatic. This indicates that, if a temperature slip is possible at the interface of nanospheres, it must be taken into account by using the derived expression for Nu. Our results at the limit of Pe = 0 are compared very well with experimental results found in the literature.     

Natural convection effects on heat and mass transfer in a curvilinear triangular cavity

Finite element method is used in this study to analyze the effects of buoyancy ratio and Lewis number on heat and mass transfer in a triangular cavity with zig-zag shaped bottom wall. Buoyancy ratio is defined as the ratio of Grashof number of solutal and thermal. Inclined walls of the cavity have lower temperature and concentration according to zig-zag shaped bottom wall. Enclosed space consists mostly of an absorber plate and two inclined glass covers that form a cavity. Both high temperature and high concentrations are applied to bottom corrugated wall. Computations were done for different values of buoyancy ratio (?10 ? Br ? 10), Lewis number (0.1 ? Le ? 20) and thermal Rayleigh number (10⁴ ? Ra_T ? 10⁶). Streamlines, isotherms, iso-concentration, average Nusselt and Sherwood numbers are obtained. It is found that average Nusselt and Sherwood numbers increase by 89.18% and 101.91% respectively as Br increases from ?10 to 20 at Ra_T = 10⁶. Also, average Nusselt decreases by 16.22% and Sherwood numbers increases by 144.84% as Le increases from 0.1 to 20 at this Rayleigh number.     

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.     

Experimental investigation of mixed convection heat transfer in a rectangular channel with discrete heat sources at the top and at the bottom

Mixed convection heat transfer in a top and bottom heated rectangular channel with discrete heat sources has been investigated experimentally for air. The lower and upper surfaces of the channel were equipped with 8 × 4 flush-mounted heat sources subjected to uniform heat flux. Sidewalls, the lower and upper walls were insulated and adiabatic. The experimental study was made for an aspect ratio of AR = 6, Reynolds numbers 955 ≤ Re_Dh ≤ 2220 and modified Grashof numbers Gr* = 1.7 × 10⁷ to 6.7 × 10⁷. From experimental measurements, surface temperature and Nusselt number distributions of the discrete heat sources were obtained for different Grashof numbers. Furthermore, Nusselt number distributions were calculated for different Reynolds numbers. Results show that surface temperatures increase with increasing Grashof number. The row-averaged Nusselt numbers first decrease with the row number and then, due to the increase in the buoyancy affected secondary flow and the onset of instability, they show an increase towards the exit as a result of heat transfer enhancement.     

Combined convection heat transfer for thermally developing aiding flow in an inclined circular cylinder with constant heat flux

Experiments have been conducted to study the local and average heat transfer by mixed convection for hydrodynamically fully developed, thermally developing and thermally fully developed laminar air flow in an inclined circular cylinder. The experimental setup consists of aluminum cylinder as test section with 30 mm inside diameter and 900 mm heated length (L/D = 30), is subjected to a constant wall heat flux boundary condition. The investigation covers Reynolds number range from 400 to 1600, heat flux is varied from 70 W/m² to 400 W/m² and cylinder angles of inclination including 30°, 45° and 60°. The hydrodynamically fully developed condition has been achieved by using aluminum entrance section pipes (calming sections) having the same inside diameter as test section pipe but with variable lengths. The entrance sections included two long calming sections, one with length of 180 cm (L/D = 60), another one with length of 240 cm (L/D = 80) and two short calming sections with lengths of 60 cm (L/D = 20), 120 cm (L/D = 40). The results present the surface temperature distribution along the cylinder length, the local and average Nusselt number distribution with the dimensionless axial distance Z⁺. For all entrance sections, the results showed an increase in the Nusselt number values as the heat flux increases and as the angle of cylinder inclination moves from θ = 60° inclined cylinder to θ = 0° horizontal cylinder. The mixed convection regime has been bounded by the convenient selection of Re number range and the heat flux range, so that the obtained Richardson numbers (Ri) is varied approximately from 0.13 to 7.125. The average Nusselt numbers have been correlated with the (Rayleigh numbers/Reynolds numbers) in empirical correlations.     

Experimental investigation of mixed convection heat transfer for thermally developing flow in a horizontal circular cylinder

An experimental investigation is presented on mixed (free and forced) convection to study the local and average heat transfer for hydrodynamically fully developed, thermally developing and thermally fully developed laminar air flow in a horizontal circular cylinder. The experimental setup consists of aluminum cylinder as test section with 30 mm inside diameter and 900 mm heated length (L/D = 30), is subjected to a constant wall heat flux boundary condition. The investigation covers Reynolds number range from 400 to 1600, the heat flux varied from 60 W/m² to 400 W/m² and with cylinder inclination angle of θ = 0° (horizontal). The hydrodynamically fully developed condition is achieved by using an aluminum entrance section pipes (calming sections) having the same inside diameter as test section pipe but with variable lengths. The entrance sections included two long calming sections, one with length of 180 cm (L/D = 60), another one with length of 240 cm (L/D = 80) and two short calming sections with lengths 60 cm (L/D = 20), 120 cm (L/D = 40). The surface temperature variation along the cylinder surface, the local and average Nusselt number variation with the dimensionless axial distance Z⁺ were presented. For all entrance sections, it was found an increase in the Nusselt number values as the heat flux increases. It was concluded that the free convection effects tended to decrease the heat transfer results at low Re while to increase the heat transfer results for high Re. The combined convection regime could be bounded by a suitable selection of Re number ranges and the heat flux ranges. The obtained Richardson numbers (Ri) range varied approximately from 0.13 to 7.125. The average Nusselt numbers were correlated with the (Rayleigh numbers/Reynolds numbers). The proposed correlation has been compared with available literature and showed satisfactory agreement.     

Performance evaluation of mixed convection in an inclined square channel with uniform temperature walls

Numerical analyses were performed for the effect of inclined angle on the mixing flow in a square channel with uniform temperature walls (T_w = 30 °C) and inlet temperature (T₀ = 10 °C). Three-dimensional governing equations were solved numerically for Re = 100, Pr = 0.72 and various inclined angles (from ?90° to 90°). Three-dimensional behavior of fluid in a channel was examined for each angle. Thermal performance was evaluated using the relationship between Nusselt number ratio and pressure loss ratio with and without buoyancy induced flow as a parameter of inclined angles. High heat transfer and low pressure loss region was from ?15° to ?60° in thermal performance using mean Nusselt number ratio.     

Lattice Boltzmann simulation of the double diffusive natural convection and oscillation characteristics in an enclosure filled with porous medium

This article adopts lattice Boltzmann method to investigate the double diffusive natural convection around a heated cylinder in an enclosure filled with porous medium. The heated cylinder is located at the center of the enclosure with high temperature and concentration. Four surrounding walls are assumed to be low temperature and concentration. The distributions of velocity, temperature and concentration are solved by three independent lattice Bhatnagar-Gross-Krook (LBGK) equations. The influence of Darcy number Da (10^–4 ≤ Da ≤ 10^− 2), Lewis number Le (0.2 ≤ Le ≤ 10.0) and buoyancy ratio Br (− 10.0 ≤ Br ≤ 10.0) on the double diffusive natural convection are inspected numerically. Results are presented in terms of isotherms, streamlines, isoconcentrations, average Nusselt and Sherwood numbers. At Br = − 50.0, the effect of Darcy number on unsteady flow characteristics is also investigated by the time history and phase space trajectory. It is found that the flow undergoes steady-state, unsteady doubling periodic oscillation, quasi-periodic oscillation and non-periodic oscillation when Darcy number Da is varied from 10^− 4 to 10^− 2.     

Numerical study of the heat sink with un-uniform fin width designs

The thermal performances of the heat sink with un-uniform fin width designs with an impingement cooling were investigated numerically. The governing equations are discretized by using a control-volume-based finite-difference method with a power-law scheme on an orthogonal non-uniform staggered grid. The coupling of the velocity and the pressure terms of momentum equations are solved by the SIMPLEC algorithm. The well-known k ? ε two-equations turbulence model is employed to describe the turbulent structure and behavior. The parameters include the five Reynolds number (Re = 5000–25000), three fin heights (H = 35, 40, 45 mm), and five fin width designs (Type-1–Type-5). The objective of this study is to examine the effects of the fin shape of the heat sink on the thermal performance. The results show that the Nusselt number increases with the Reynolds number. The increment of the Nusselt number decreases gradually with the increasing Reynolds number. Furthermore, the effects of fin dimensions on the Nusselt number at high Reynolds numbers are more significant than that at low Reynolds numbers. It is also found that there is potential for optimizing the un-uniform fin width design.     

Investigation of mixed convection heat transfer in a horizontal channel with discrete heat sources at the top and at the bottom

Mixed convection heat transfer from arrays of discrete heat sources inside a horizontal channel has been investigated experimentally. Each of the lower and upper surfaces of the channel was equipped with 8 × 4 flush mounted heat sources subjected to uniform heat flux. Sidewalls, lower and upper walls are insulated and adiabatic. The experimental parametric study was made for aspect ratios of AR = 2, 4 and 10, at various Reynolds and Grashof numbers. From the experimental measurements, row-average surface temperature and Nusselt number distributions of the discrete heat sources were obtained and effects of Reynolds and Grashof numbers on these numbers were investigated. From these results, the buoyancy affected secondary flow and the onset of instability have been discussed. Results show that top and bottom heater surface temperatures increase with increasing Grashof number. The top heater average-surface temperatures for AR = 2 are greater than those of bottom ones. For high values of Grashof numbers where natural convection is the dominant heat transfer regime (Gr¹/Re² ≫ 1), temperatures of top heaters can have much greater values. The variation of the row-average Nusselt numbers for the aspect ratio of AR = 4, show that with the increase in the buoyancy affected secondary flow and the onset of instability, values of Nusselt number level off and even rise as a result of heat transfer enhancement especially for low Reynolds numbers.