Numerical study of simultaneous natural convection heat transfer from both surfaces of a uniformly heated thin plate with arbitrary inclination

 Numerical studies were conducted to investigate the natural convection heat transfer around a uniformly heated thin plate with arbitrary inclination in an infinite space. The numerical approach was based on the finite volume technique with a nonstaggered grid arrangement. For handling the pressure–velocity coupling the SIMPLE-algorithm was used. QUICK scheme and first order upwind scheme were employed for discretization of the momentum and energy convective terms respectively. Plate width and heating rate were used to vary the modified Rayleigh number over the range of 4.8×10⁶ to 1.87×10⁸. Local and average heat transfer characteristics were compared with regarding to the inclination angle. The empirical expressions for local and average Nusselt number were correlated. It has been found that for inclination angle less than 10^∘, the flow and heat transfer characteristics are complicated and the average Nusselt number can not be correlated by one equation while for inclination angle larger than 10^∘, the average Nusselt number can be correlated into an elegant correlation. Received on 18 April 2001 / Published online: 29 November 2001     

An experimental study on the natural convection heat transfer between inclined plates (Lower plate isothermally heated and the upper plate thermally insulated as well as unheated)

Presented in this paper are the results of natural convection heat transfer between inclined parallel plates. Lower plate is heated isothermally while the upper plate is both unheated and insulated. Plate inclinations were 0^°, 30^°, 45^° measured from vertical position. Experiments were carried out for different temperature differences in air to determine the effect of plate spacing and plate inclination on heat transfer. It was found that heat transfer rate depends on plate spacing and inclination.     

Turbulent free convection in a vertical converging channel

This paper presents the results of experimental and numerical investigations of the problem of turbulent natural convection in a converging-plate vertical channel. The channel has two isothermally heated inclined walls and two adiabatic vertical side walls. The parameters involved in this study are the channel geometry represented by the channel width at exit, the inclination of the heated walls and the temperature difference between the heated walls and the ambient. The investigation covered modified Rayleigh numbers up to 10⁸ in the computational study and up to 9.3 × 10⁶ in the experimental work. The experimental measurements focused on the velocity field and were carried out using a PIV system and included measurements of the mean velocity profiles as well as the root-mean-square velocity and shear stress profiles. The experiments were conducted for an inclination angle of 30°, a gap width of 10 mm and two temperature differences (∆T=25.4°C and 49.8°C). The velocity profiles in the lower part of the channel indicated the presence of two distinct layers. The first layer is adjacent to the heated plate and driven by buoyancy forces while the second layer extends from the point of maximum velocity to the channel center plane and driven mainly by shear forces. The velocity profile at the upper portion of the channel has shown the merging of the two boundary layers growing over the two heated walls. The measured values of the Reynolds shear stress and root mean square of the horizontal and vertical velocity fluctuation components have reached their maximum near the wall while having smaller values in the core region. The computational results have shown that the average Nusselt number increases approximately linearly with the increase of the modified Rayleigh number when plotted on log–log scale. The variation of the local Nusselt number indicated infinite values at the channel inlet (leading edge effect) and high values at the channel exit (trailing edge effect). For a fixed value of the top channel opening, the increase of the inclination angle tended to reduce flow velocity at the inlet section while changing the flow structure near the heated plates in such a way to create boundary-layer type flow. The maximum value of the average Nusselt number occurs when θ = 0 and decreases with the increase of the inclination angle. On the other hand, the increase of the channel width at exit for the same inclination angle caused a monotonic increase in the flow velocity at the channel inlet.     

Experimental investigation of mixed convection in a rectangular duct with a heated plate in the middle of cross section

The flow and heat transfer in an inclined and horizontal rectangular duct with a heated plate longitudinally mounted in the middle of cross section was experimentally investigated. The heated plate and rectangular duct were both made of highly conductive materials, and the heated plate was subjected to a uniform heat flux. The heat transfer processes through the test section were under various operating conditions: Pr ≈ 0.7, inclination angle ϕ = −60° to +60°, Reynolds number Re = 334–1,911, Grashof number Gr = 5.26 × 10²–5.78 × 10⁶. The experimental results showed that the average Nusselt number in the entrance region was 1.6–2 times as large as that in the fully developed region. The average Nusselt numbers and pressure drops increased with the Reynolds number. The average Nusselt numbers and pressure drops decreased with an increase in the inclination angle from −60° to +60° when the Reynolds number was less than 1,500. But when the Reynolds number increased to over about 1,800, the heat transfer coefficients and pressure drops were independent of inclination angles.     

Forced convection of turbulent flow in triangular ducts with different angles and surface roughnesses

The experimental investigations were consisting of two parts. The first part was carried out to study the effect of corner geometry on the steady-state forced convection inside horizontal isosceles triangular ducts with sharp corners. The electrically-heated triangular duct was used to simulate the triangular passage of a plate-fin compact heat exchanger. The isosceles triangular ducts were manufactured with duralumin, and fabricated with the same length of 2.4 m and hydraulic diameter of 0.44 m, but five different apex angles (i.e. θ _a =15^∘,30^∘, 40^∘,60^∘, and 90^∘) respectively. The investigation was performed under turbulent flow condition covering a wide range of Reynolds number (i.e. 7000≤Re _D ≤20000). It was found that the best thermal performance is achieved with the apex angle of 60^∘. The second part was performed to investigate the effect of surface roughness on the forced convection of the same system. Horizontal equilateral triangular ducts with an apex angle of 60^∘ were fabricated with the same length and hydraulic diameter, but different average surface roughnesses of 1.2 m,3.0 m and 11.5 m respectively. It was concluded that the duct with a higher surface roughness will have a better heat transfer performance. Non-dimensional expressions for the determination of the heat transfer coefficient of the triangular ducts with different apex angles and surface roughnesses were also developed. Received on 15 December 1997     

Forced convection from a straight elliptical tube

The paper deals with the problem of two-dimensional laminar forced convection heat transfer from a straight isothermal tube of elliptic cross-section placed in a uniform stream. The study is based on numerical solutions of the conservation equations of mass, momentum, and energy which covers the entire flow domain including the wake region. The parameters influencing the heat transfer process are essentially the Reynolds number, Re, the tube geometry represented by its minor to major axis ratio, Ar, and the angle of inclination, λ. The study focuses on the effect of Re, Ar, and λ on the heat transfer process in the range of Re from 20 to 500. The study reveals that the rate of heat transfer reaches its maximum when λ=0^∘ while the minimum occurs when λ=90^∘. The results also show that smaller axis ratio gives higher heat transfer rate when λ=0^∘. The local Nusselt number and surface vorticity distributions are plotted for a number of cases and the effect of vortex shedding on the overall rate of heat transfer is briefly discussed. Received on 20 September, 1997     

Numerical and experimental studies of natural convective heat transfer from vertical and inclined narrow isothermal flat plates

Natural convective heat transfer from an isothermal narrow flat plate embedded in a plane adiabatic surface and inclined at moderate positive and negative angles to the vertical has been numerically and experimentally studied. The solution has the Rayleigh number, the dimensionless plate width, the angle of inclination, and the Prandtl number as parameters. Attention was restricted to a Prandtl number of 0.7. The numerical results have been obtained for Rayleigh numbers between 10³ and 10⁷ for dimensionless plate widths of between 0.3 and 1.2 and for angles of inclination between +45° and −45°. In the experimental study, results have been obtained for Rayleigh numbers between 4 × 10² and 10⁵ for dimensionless plate widths of 0.4 and 2.5 and for angles of inclination between +45° and −45° to the vertical. Empirical equations for the heat transfer rate have been derived.     

Oscillatory Couette flow in the presence of an inclined magnetic field

Oscillatory MHD Couette flow of electrically conducting fluid between two parallel plates in a rotating system in the presence of an inclined magnetic field is considered when the upper plate is held at rest and the lower plate oscillates non-torsionally . An exact solution of the governing equations has been obtained by using Laplace transform technique. Asymptotic behavior of the solution is analyzed for M ² ≪1, K ² ≪1 and ω ≪1 and for large M ², K ² and ω. Numerical results of velocities are depicted graphically and the frictional shearing stresses are presented in tables. It is found that a thin boundary layer is formed near the lower plate, for large values of rotation parameter K ², Hartman number M ² and frequency parameter ω. The thickness of this boundary layer increases with increase in inclination of the magnetic field with the axis of rotation.     

Interaction of turbulent natural convection and surface thermal radiation in inclined square enclosures

Two-dimensional numerical studies of flow and temperature fields for turbulent natural convection and surface radiation in inclined differentially heated enclosures are performed. Investigations are carried out over a wide range of Rayleigh numbers from 10⁸ to 10¹², with the angle of inclination varying between 0° and 90°. Turbulence is modeled with a novel variant of the k–ε closure model. The predicted results are validated against experimental and numerical results reported in literature. The effect of the inclination of the enclosure on pure turbulent natural convection and the latter’s interaction with surface radiation are brought out. Profiles of turbulent kinetic energy and effective viscosity are studied to observe the net effect on the intensity of turbulence caused by the interaction of natural convection and surface radiation. The variations of local Nusselt number and average Nusselt number are presented for various inclination angles. Marked change in the convective Nusselt number is found with the orientation of enclosure. Also analyzed is the influence of change in emissivity on the flow and heat transfer. A correlation relevant to practical applications in the form of average Nusselt number, as a function of Rayleigh number, Ra, radiation convection parameter, N _RC and inclination angle of the enclosure, φ is proposed.     

Laminar natural convection of pure and saline water along a vertical isothermal cylinder

 In most studies concerning laminar natural convection along a vertical isothermal cylinder a linear relationship between fluid density and temperature has been used and kinematic viscosity and thermal diffusivity have been considered constant calculated at ambient temperature. However, it is known that the density–temperature relationship for water is non-linear at low temperatures and kinematic viscosity and thermal diffusivity are functions of temperature. In this study the problem of laminar natural convection of pure and saline water along a vertical isothermal cylinder has been investigated in the temperature range between 20 and 0 ^∘C taking into account the temperature dependence of ν, α and ρ. The results are obtained with the numerical solution of the boundary layer equations. The variation of ν, α and ρ with temperature has a strong influence on free convection characteristics. Received on 17 May 1999     

The crumbling of a viscous prism with an inclined free surface

Summary We study the two-dimensional instantaneous Stokes flow driven by gravity in a viscous triangular prism supported by a horizontal rigid substrate and a vertical wall. The oblique side of the prism, inclined at an angle α with respect to the substrate, is a fluid-air interface, where the stresses are zero and surface tension is neglected. We develop the stream function ψ in polar coordinates (r,θ) centered at the vertex of α and split it into an inhomogeneous part, which accounts for gravity effects, and a homogeneous part, which is expressed as a series expansion. The inhomogeneous part and the first term of the expansion may be envisioned, respectively, as self-similar solutions of the first kind and of the second kind for r→0, each one holding in complementary α domains with a crossover at α _c =21.47^∘, which we study in some detail. The coefficients of the series are calculated by truncating the expansion and using the method of direct collocation with a suitable set of points at the wall. The solution strictly holds for t=0, because later the free surface ceases to be a plane; nevertheless, it provides a good approximation for the early time evolution of the fluid profile, as shown by the comparison with numerical simulations. For 0<α<45^∘, the vertex angle remains constant and the edge remains strictly at rest; the transition at α _c manifests itself through a change in the rate of growth of the curvature. The time at which the edge starts to move (waiting time) cannot be calculated since the instantaneous solution ceases to be valid. For α>45^∘, the instantaneous local solution indicates that the surface near the vertex is launched against the substrate so that the edge starts to move immediately with a power law dependence on time t. However, due to the high value of the exponent, the vertex may seem to be at rest for some finite time even in this case. Received 29 August 1997; accepted for publication 21 January 1998     

The effect of the VOF–CSF static contact angle boundary condition on the dynamics of sliding and bouncing ellipsoidal bubbles

The static contact angle is the only empiricism introduced in a Volume of Fluid–Continuum Surface Force (VOF–CSF) model of bubbly flow. Although it has previously been shown to have a relatively limited effect on the accuracy of velocity and shape predictions in the case of large gas bubbles sliding under inclined walls (e.g. Cook and Behnia, 2001), it may have a more determining influence on the numerical prediction of the dynamics of smaller ellipsoidal bubbles which were shown by Tsao and Koch (1997) to bounce repeatedly when sliding under inclined walls at certain wall inclinations. The present paper reports on the influence of surface tension and the static contact angle on the dynamics of an ellipsoidal air bubble of equivalent diameter D_e = 3.4 mm. The bubble Eötvös and Morton numbers are Eo = 1.56 and Mo = 2 × 10⁻¹¹ respectively. The computational results are achieved with a Piecewise Linear Construction (PLIC) of the interface and are reviewed with reference to experimental measurements of bubble velocity and interface shape oscillations recorded using a high speed digital camera. Tests are performed at plate inclination angles θ ∈ {10°, 20°, 30°, 45°} to the horizontal and computational models consider three static contact angles θ_c ∈ {10°, 20°, 30°}. The static contact angle has been found to have a significant effect on the bubble dynamics but to varying degree depending on the plate inclination. It is shown to promote lift off and bouncing when the plate inclination angle reaches 30° in a way that does not necessarily reflect experimental observations.     

Variation of the stress intensity factor along the crack front of interacting semi-elliptical surface cracks

Summary  In this study, the interaction between two semi-elliptical co-planar surface cracks is considered when Poisson's ratio ν = 0.3. The problem is formulated as a system of singular integral equations, based on the idea of the body force method. In the numerical calculation, the unknown density of body force density is approximated by the product of a fundamental density function and a polynomial. The results show that the present method yields smooth variations of stress intensity factors along the crack front very accurately, for various geometrical conditions. When the size of crack 1 is larger than the size of crack 2, the maximum stress intensity factor appears at a certain point, β₁=177^∘, of crack 1. Along the outside of crack 1, that is at β₁=0∼90^∘, the interaction can be negligible even if the two cracks are very close. The interaction can be negligible when the two cracks are spaced in such a manner that their two closest points are separated by a distance exceeding the small crack's major diameter. The variations of stress intensity factor of a semi-elliptical crack are tabulated and charted. Received 30 August 1999; accepted for publication 22 February 2000     

A new approach on MHD natural convection boundary layer flow past a flat plate of finite dimensions

A new approach on MHD natural convection boundary layer flow from a finite flat plate of arbitrary inclination in a rotating environment, is presented. This problem plays a significant role on boundary layer flow control. It is shown that taking into account the pressure rise region at the leading edge of the plate leads to avoid separation and the back flow is reduced by the strong magnetic field. It is also shown that the frictional drag at the leading edge of the plate is reduced when the inclination angle α=π/4. In the case of isothermal flat plate, the bulk temperature becomes identical for any value of Gr (Grashof number) when the value of M ² (Hartmann number) and K ² (rotation parameter) are kept fixed.     

Drag Reduction of a Circular Cylinder Using an Upstream Rod

Experimental studies on the drag reduction of the circular cylinder were conducted by pressure measurement at a Reynolds number of 82 000 (based on the cylinder diameter). A rod was placed upstream of and parallel to the cylinder to control the flow around the cylinder. The upstream rod can reduce the resultant force of the cylinder at various spacing between the rod and the cylinder for α < 5^∘(α defined as the staggered angle of the rod and the cylinder). For α > 10^∘, the resultant force coefficient has a large value, so the upstream rod cannot reduce the force on the cylinder any more. For α = 0^∘ and d/D = 0.5 (where d and D are the diameter of the rod and the cylinder, respectively), the maximum drag of the cylinder reduces to 2.34% that of the single cylinder. The mechanism of the drag reduction of the cylinder with an upstream rod in tandem was presented by estimating the local contributions to the drag reduction of the pressure variation. In the staggered arrangement, the flow structures have five flow patterns (they are the cavity mode, the wake splitting mode, the wake merge mode, the weak boundary layer interaction mode and the negligible interaction mode) according to the pressure distribution and the hydrogen bubble flow visualization. The half plane upwind of the cylinder can be divided to four regions, from which one can easily estimates the force acting on the circular cylinder with an upstream rod in staggered arrangement.     

Numerical simulation of fully developed turbulent flow and heat transfer in annular-sector ducts

 The mixing length theory is employed to simulate the fully developed turbulent heat transfer in annular-sector ducts with five apex angles (θ₀=18,20,24,30,40^∘) and four radius ratios (R _o/R _i=2,3,4,5). The Reynolds number range is 10⁴10⁵. The numerical results agree well with an available correlation which was obtained in following parameter range: θ₀=18,20,24,30,40^∘, R _o/R _i=4 and Re=10⁴5×10⁴. The present work demonstrates that the application range of the correlation can be much extended. Apart from the mixing length theory, the kɛ model with wall function and the Reynolds stress model are also employed. None of the friction factor results predicted by the three models agrees well with the test data. For the heat transfer prediction the mixing length theory seems the best for the cases studied. Received on 17 July 2000 / Published online: 29 November 2001     

Three-dimensional channel flow of second grade fluid in rotating frame

An analysis is performed for the hydromagnetic second grade fluid flow between two horizontal plates in a rotating system in the presence of a magnetic field.The lower sheet is considered to be a stret...     

Laminar mixed convection in horizontal concentric annuli with non-uniform circumferential heating

 Steady, laminar, mixed convection in the fully developed region of horizontal concentric annuli has been investigated numerically for the case of non-uniform circumferential heating. Two heating conditions were studied, one in which a 180^∘ arc encompassing the top half of inner surface of the inner cylinder is uniformly heated while the bottom half is kept insulated, and the other in which the heated and the insulated surfaces were reversed. The fluid flow and heat transfer characteristics were found to be affected by the heating conditions. For the investigated range of the governing buoyancy parameter, the modified Grashof number (Gr^*), it was found that bottom heating arrangement gives rise to a vigorous secondary flow, with the result that the average Nusselt numbers are much higher than those for pure forced convection. On the other hand, the local Nusselt numbers are nearly circumferentially uniform. In the case of top heating arrangement, a less vigorous secondary flow is induced because of temperature stratification, with average Nusselt numbers that are substantially lower than those for bottom heating and with large circumferential variation of the local Nusselt number. Received on 15 March 2000     

Experimental verification of space marching methods for solving inverse heat conduction problems

Two space marching methods for solving one-dimensional nonlinear inverse heat conduction problems developed earlier are validated using data obtained during emergency cooling simulation of PWR pressure vessels. The model of pressure vessel was preheated at 350 ^∘C and then cooled by injection of cold water at 20 ^∘C. Based on the vessel temperature histories measured at the distance of 50 mm from the inner surface and at the outside surface the transient temperature distribution in the pressure vessel was obtained. The comparison of the results obtained by both space marching methods with experimental data shows high accuracy of the space marching methods. Received on 3 May 1999     

Experimental study of air natural convection on metallic foam-sintered plate

The natural convection on metallic foam-sintered plate at different inclination angles was experimentally studied. Seven copper foam samples with different pore densities (10–40 pore per inch), porosities (0.90–0.95), and aspect ratios (the ratio of foam thickness to sample length, 0.1–0.5) were measured at inclination angles of 0° (vertical orientation), 15°, 30°, 45°, 60°, 75°, 90° (horizontal orientation). The heat conduction and natural convection inside the foam both contributed to the total heat transfer. Although, the form and viscous drag, which are influenced by permeability and viscous friction in the thermal boundary layer respectively, tend to suppress the natural convection, the heat transfer was finally enhanced by the foam sintered surface due to large surface area extension. Optimum inclination range 60–75° corresponding to maximum average Nu number was found in the heat flux range of 600–1800 W/m². The sintered foam surface with lower porosity and pore density was recommended for heat transfer enhancement. Particularly, the sample with porosity 0.9, pore density of 10 PPI, aspect ratio of 0.5 offered the highest average Nu number among the studied samples. An empirical correlation for modified Nusselt number at isoflux boundary condition considering the foam morphology parameter and inclination angle was proposed within deviation ±15% between the correlation and the experimental data.