Distinct Boundary Conditions Research Articles

45 deg Round-Corner Rib Heat Transfer Coefficient Measurements in a Square Channel

Cooling channels, roughened with repeated ribs, are commonly employed as a means of cooling turbine blades. The increased level of mixing induced by these ribs enhances the convective heat transfer in the blade cooling cavities. Many previous investigations have focused on the heat transfer coefficient on the surfaces between these ribs and only a few studies report the heat transfer coefficient on the rib surfaces themselves. The present study investigated the heat transfer coefficient on the surfaces of 45 deg, round-corner ribs. Three staggered rib geometries corresponding to blockage ratios of 0.133, 0.167, and 0.25 were tested in a square channel for pitch-to-height ratios of 5, 8.5, and 10, and for two distinct thermal boundary conditions of heated and unheated channel wall. Comparisons were made between the surface-averaged heat transfer coefficients and channel friction factors for sharp-and round-corner ribs and 45 versus 90 deg ribs, reported previously. Heat transfer coefficients of the furthest upstream rib and that of a typical rib located in the middle of the ribroughened region were also compared. It was concluded that: (a) For the geometries tested, the rib average heat transfer coefficient was much higher than that for the area between the ribs. (b) The general effect of rounding the rib corners was a decrease in both rib heat transfer coefficient and channel pressure drop. (c) For the highest blockage ratio ribs (e/Dh = 0.25), 90 deg ribs performed superior to 45 deg ribs. However, this trend reversed for smaller rib blockage ratios. (d) Heat transfer coefficients for the two smaller rib geometries (e/Dh = 0.133 and 0.167) did not vary significantly with the pitch-to-height ratio in the range tested. However, the heat transfer coefficient for the high blockage rib geometry increased significantly as the ribs were brought closer to each other. (e) Under otherwise identical conditions, ribs in the furthest upstream position produced lower heat transfer coefficients than those in the midstream position. (f) Rib thermal performance decreased with the rib blockage ratio. The smallest rib geometry (e/Dh = 0.133) at a pitch-to-height ratio of 10 and the largest rib geometry (e/Dh = 0.25) at a pitch-to-height ratio of 5, both in midstream position, produced the highest and the lowest thermal performances, respectively.

Low-Aspect-Ratio Rib Heat Transfer Coefficient Measurements in a Square Channel

Cooling channels, roughened with repeated ribs, are commonly employed as a means of cooling turbine blades. The increased level of mixing induced by these ribs enhances the convective heat transfer in the blade cooling cavities. Many previous investigations have focused on the heat transfer coefficient on the surfaces between these ribs and only a few studies report the heat transfer coefficient on the rib surfaces themselves. The present study investigated the heat transfer coefficient on the surfaces of round-corner, low-aspect-ratio (ARrib = 0.667) ribs. Twelve rib geometries, comprising three rib height-to-channel hydraulic diameters (blockage ratios) of 0.133, 0.167, and 0.25 as well as three rib spacings (pitch-to-height ratios) of 5, 8.5, and 10 were investigated for two distinct thermal boundary conditions of heated and unheated channel walls. A square channel, roughened with low-aspect-ratio ribs on two opposite walls in a staggered manner and perpendicular to the flow direction, was tested. An instrumented copper rib was positioned either in the middle of the rib arrangements or in the furthest upstream location. Both rib heat transfer coefficient and channel friction factor for these low-aspect-ratio ribs were also compared with those of square ribs, reported previously by the authors. Heat transfer coefficients of the furthest upstream rib and that of a typical rib located in the middle of the rib-roughened region of the passage wall were also compared.

45 deg Staggered Rib Heat Transfer Coefficient Measurements in a Square Channel

For high-blockage ribs with large heat transfer areas, commonly used in small gas turbine blades, the rib heat transfer is a significant portion of the overall heat transfer in the cooling passages. Three staggered 45 deg rib geometries corresponding to blockage ratios of 0.133, 0.167, and 0.25 were tested in a square channel for pitch-to-height ratios of 5, 8.5, and 10, and for two distinct thermal boundary conditions of heated and unheated channel walls. Comparisons were made between the surface-averaged heat transfer coefficients and friction factors for 45 deg ribs, and 90 deg ribs reported previously. Heat transfer coefficients of the furthest upstream rib and that of a typical rib located in the middle of the rib-roughened region were also compared. It was concluded that: (a) For the geometries tested, the rib average heat transfer coefficient was much higher than that for the area between the ribs. (b) Except for two cases corresponding to the highest blockage ribs mounted at pitch-to-height ratios of 8.5 and 10 for which the heat transfer results of 45 deg ribs were very close to those of 90 deg ribs, 45 deg ribs produced higher heat transfer coefficients than 90 deg ribs. (c) At pitch-to-height ratios of 8.5 and 10, all 45 deg ribs produced lower friction factors than 90 deg ribs. However, when they were brought closer to each other (S/e = 5), they produced higher friction factors than 90 deg ribs. (d) Heat transfer coefficients for the two smaller rib geometries (e/Dh = 0.133 and 0.167) did not vary significantly with the pitch-to-height ratio in the range tested. However, the heat transfer coefficient for the high blockage rib geometry increased significantly as the ribs were brought closer to each other. (e) Under otherwise identical conditions, ribs in the furthest upstream position produced lower heat transfer coefficients than those in the midstream position. (f) Rib thermal performance decreased with the rib blockage ratio. For both angles of attack, the smallest rib geometry in the midstream position and at a pitch-to-height ratio of 10 had the highest thermal performance, and the highest blockage rib in the furthest upstream position produced the lowest thermal performance.

Rib Heat Transfer Coefficient Measurements in a Rib-Roughened Square Passage

Three staggered 90 deg rib geometries corresponding to blockage ratios of 0.133, 0.167, and 0.25 were tested for pitch-to-height ratios of 5, 8.5, and 10, and for two distinct thermal boundary conditions of heated and unheated channel walls. Comparisons were made between the surface-averaged heat transfer coefficients and friction factors for ribs with rounded corners and those with sharp corners, reported previously. Heat transfer coefficients of the furthest upstream rib and that of a typical rib located in the middle of the rib-roughened region of the passage wall were also compared. It was concluded that: (a) For the geometries tested, the rib average heat transfer coefficient was much higher than that for the area between the ribs. For the sharp-corner ribs, the rib average heat transfer coefficient increased with blockage ratio. However, when the corners were rounded, the trend depended on the level of roundness. (b) High-blockage-ratio (e/Dh = 0.25) ribs were insensitive to the pitch-to-height ratio. For the other two blockage ratios, the pitch-to-height ratio of 5 produced the lowest heat transfer coefficient. Results of the other two pitch-to-height ratios were very close, with the results of S/e = 10 slightly higher than those of S/e = 8.5. (c) Under otherwise identical conditions, ribs in the furthest upstream position produced lower heat transfer coefficients for all cases except that of the smallest blockage ratio with S/e of 5. In that position, for the rib geometries tested, while the sharp-corner rib average heat transfer coefficients increased with the blockage ratio, the trend of the round-corner ribs depended on the level of roundness, r/e. (d) Thermal performance decreased with the blockage ratio. While the smallest rib geometry at a pitch-to-height ratio of 10 had the highest thermal performance, thermal performance of high blockage ribs at a pitch-to-height ratio of 5 was the lowest. (e) The general effects of rounding were a decrease in heat transfer coefficient for the midstream ribs and an increase in heat transfer coefficient for ribs in the furthest upstream position.

Dynamic Stability of Spinning Beams with an Unsymmetrical Cross-Section and Distinct Boundary Conditions Subjected to Time-Dependent Spin Speed*

ABSTRACT The equations of motion of a spinning beam with a rectangular cross-section are formulated using the Euler beam theory and the assumed mode method. The spin speed consists of steady-state and time-dependent portions. The resulting equations of motion are not in standard Mathieu-Hill's equation form, due to the time-dependent coefficient of the gyroscopic term. These equations of motion are then reduced to a set of first-order differential equations with time-dependent coefficients. The regions of instability due to parametric excitations are determined using the multiple scale method. Numerical results are presented for a spinning beam subjected to combinations of end conditions in the two orthogonal planes of transverse vibration. Widths of the unstable regions are found to decrease as the aspect ratio of the rectangular cross-section approaches unity for spinning beams with an identical set of end conditions in both transverse vibration planes.These regions vanish when the aspect ratio becomes one. However, this is not the case when the beam is subjected to distinct end conditions in the two planes. For a given aspect ratio, interesting changes in the unstable regions are observed as the spin speed varies within, as well as across, critical spin speed zones.

Microphysical irreversibility in quantum mechanics

Irreversibility on the microphysical level is described by a semigroup which is generated by the Hamiltonian and is distinguished from the irreversibility of open systems. The mathematical tool of this quantum mechanical irreversibility is the rigged Hilbert space. The physical basis is found in the distinct initial and boundary conditions for the states and the observables, which can be formulated in terms of an arrow of time. The principal objects of this theory are resonances which are defined by resonance poles of the S-matrix and described by generalized eigenvectors which are Jordan vectors of higher degree. Though there is no physical evidence for these, they have an intriguing semigroup time evolution.

Rifting of Africa and pattern of mantle convection beneath the African plate

Abstract The Mesozoic and Cenozoic tectonic setting of the African continent is characterized by an extensional stress regime, as is evident from the evolution of the West, Central and East African rift systems, the opening of the Indian and Atlantic oceans and the development of Tethys domain, as well as from the results of geophysical investigations. The most prominent feature of the Bouguer gravity anomaly map of Africa is a great belt of negative Bouguer anomalies which extends across the entire African continent from Ethiopia in the East to Benguela in the West. In conjunction with the Red Sea spreading zone this belt appears to encircle a “centre” in Cameroon. It probably corresponds to a distinct zone of lithospheric thinning and extension. The entire African continent is characterized by higher-than-normal elevations. It is located inside the African residual geoid high. Earthquake fault-plane solutions confirm the extensional character of the present tectonic regime of eastern and southern Africa where normal faulting predominates. The average orientation of T axes is N-S in South Africa, NW-SE in Zambia, Mozambique and Madagascar, E-W in Kenya and Ethiopia, and NE-SW in the Red Sea region. From north to south, the orientation of T axes rotates systematically in a clockwise sense by 130° and describes a radial pattern centred on equatorial West Africa. This indicates that the African lithosphere has been subject to continent-wide radial extension around a hypothetical West African “spreading centre”, referred to as the African Spreading Centre A (ASC-A), which coincides with the geographical centre of the African plate. Shear traction exerted by the convecting mantle on the base of the lithosphere is considered to be a major driving force of plate movements. In the case of Africa, the continent-wide regularities in the pattern of lithospheric extension, the elevated position of the continent, as well as the evolution of the mid-oceanic ridge system surrounding Africa, provide distinct boundary conditions for the pattern of large-scale mantle flow beneath the African plate. It is proposed that relatively warm and less dense mantle material rises below Africa, forming a single mega-plume or a number of plumes. At the base of the African lithosphere, the ascending material diverges and flows radially away from the centre of ascent, corresponding to the ASC-A. This ascending flow forms part of a very large mantle convection cell which occupies the Eurafrican hemisphere. The descending branch of this convection cell is located at a distance of 80–90° from the ASC-A. The mid-oceanic ridges surrounding the African continent are located between the upwelling and the downwelling branches of this convection cell, at a distance of 50–60° from the ASC-A. Upper mantle flow below the sea-floor spreading axes is uni-directional and horizontal and is directed away from the ASC-A.

Comprehensive correlations for laminar mixed convection on vertical and horizontal flat plates

This paper presents comprehensive correlation equations of the local Nusselt numbers and surface shear stresses for laminar forced convection, natural convection, and mixed convection on vertical and horizontal flat plates which are maintained with uniform wall temperature or uniform surface heat flux. The correlation for pure forced convection and pure natural convection are very accurate for any Prandtl number between 0.001 and infinity. The correlations for mixed convection coincide very well with the numerical results over the entire regimes of mixed convection intensity and Prandtl number for the eight cases of the two plates with distinct thermal boundary conditions and buoyancy-assisting and-opposing flow conditions.

Thermal convection of micropolar fluids past two-dimensional or axisymmetric bodies with suction/injection

Steady laminar forced convection of micropolar fluids past two-dimensional or axisymmetric bodies with porous walls and different thermal boundary conditions has been analyzed. New coordinate transformations are employed to reduce the streamwise-dependence in the coupled nonlinear boundary-layer equations. The analysis is applied to cylinders and spheres with fluid injection/suction at the surface. Of interest are the effects of material parameters, wall mass transfer, the Prandtl number and the distinct thermal boundary conditions on the local skin friction coefficient and heat transfer coefficient.

Negative resistance and distributed gain in submicron semiconductor diodes using analytical solutions of the Boltzmann transport equation

A perturbational, moments-of-the-Boltzmann-equation approach (PMBE) is used to obtain the impedance Z and propagation constant k of n + nn + diode structures in an analytical fashion. Re (Z) or Im (k) can be examined to determine if negative resistance or gain occurs in the millimetre wave, submillimetre wave, or infrared spectrums where transverse device sizes for semiconductor materials of interest are submicron. Evaluation of Z and k is relatively inexpensive so that hundreds of distinct boundary conditions (BC), corresponding to a single set of interior device parameters, may be studied. Based on such PMBE analytically obtained BC results, a few of the more interesting BC cases could be simulated enlisting a large signal, time domain, spatially varying numerical code. For a single BC cas-e in GaAs, the CPU time ratio between a Monte Carlo code (run on a Cray XM-P) and the analytical PMBE technique (run on a VAX) is on the order of 103 when the Monte Carlo run time is about 1 hour. Examples of PMBE computer runs are provided for 0.4 μm thick devices with donor doping densities of 1015/cc at 100–300 GHz frequencies.

Perturbational solutions of the boltzmann transport equation for n+ nn+ structures

A perturbational, moments-of-the-Boltzmann-equation approach (PMBE) is used to obtain the impedance Z and propagation constant k of n+nn+ diode structures in an analytical fashion. Evaluation of Z and k is relatively inexpensive so that hundreds of distinct boundary conditions (BC), corresponding to a single set of interior device parameters, may be studied. Based on such PMBE analytically obtained BC results, a few of the more interesting BC cases could be simulated enlisting a large signal, time domain, spatially varying numerical code. An example of a PMBE computer run is provided for a 0.4 μm thick device with a donor doping density of 1015/cm3 in the 100–300 GHz frequency spectrum.

Transient thermal stresses in a rectangular plate due to variation of heat-transfer coefficients on upper and lower surfaces

In this paper, a method is presented for the determination of the transient temperature and thermal stresses in a thin finite rectangular plate subjected to the distinct thermal boundary conditions of the third kind at the four edges and allowing the variation of heat-transfer coefficients on the upper and lower surfaces of the plate.

Viscosity dependence of the rotational reorientation of rhodamine B in mono- and polyalcohols. Picosecond transient grating experiments

Rotational reorientation times (Tau/sub ROT/) were obtained for rhodamine B in a series of n-alcohols and polyalcohols of varying viscosities by a transient grating technique. The general trend of the Debye-Stokes-Einstein (DSE) theory was shown to extend to much higher viscosities than previously reported. Detailed analysis of the results suggests that the n-alcohols and polyalcohols provide two distinct hydrodynamic boundary conditions for the rotating species. In the n-alcohols, stick boundary conditions are observed, while the polyalcohols provide approximately slip boundary conditions. The question of a difference in the rotation times for the ground and excited electronic states is addressed and an upper bound is placed on this difference for rhodamine B. 7 figures, 1 table.

Boundary conditions and point defect computer simulation

Abstract Three physically distinct boundary conditions have been previously used in the real space computer simulation of point defects. The difference between them is discussed and their effect on vacancy formation energy is investigated.

On the effect of boundary conditions on finite deflections of heated annuli exhibiting temperature-dependent properties

The present study concerns nonlinear deflection analyses of axisymmetrically heated annular plates for several boundary conditions at the outer radius. Temperature-dependent mechanical and thermal properties of plates are considered within the frame of orthotropic elasticity theory. It is pointed out by numerical calculations that the temperature-dependent effect on finite deflections differs markedly for distinct boundary conditions.

The Effect of Principal Bending Curvature on the Lateral Buckling of Uniform Slender Beams

The general lateral buckling equation is developed for a uniform, slender, simply supported beam fixed in torsion and with a load applied at the shear center of the midspan cross section. In this general equation, the effect of principal bending curvature (i.e., beam deflection prior to buckling) is completely accounted for. Therefore, a distinction is made between beams fixed in torsion about the deformed or undeformed elastic axis, and distinct boundary conditions are derived for each case. The equations for each of the two support conditions are then specialized to include only the first-order effect of principal bending curvature and these equations are compared with similar equations for cantilever beams and beams in pure bending. Finally, simplified buckling load formulas are derived and compared with numerical solutions of the general equations for each of the lateral buckling configurations. The comparison shows that the approximate formulas provide good estimates for the buckling load and that the classical buckling load formulas that neglect principal bending curvature are not always conservative for infinitely slender beams.

Iterative solution of the inverse Sturm-Liouville problem

An iterative construction of the potential function entering in a regular Sturm-Liouville problem is discussed. The given data is in the form of two spectra associated with distinct boundary conditions at one end point.

On maintained convective motion in a fluid heated from below

This paper examines the stability in viscous liquid of a steady regime in which the temperature decreases with uniform gradient between a lower horizontal surface which is heated and an upper horizontal surface which is cooled. The problem has been treated both experimentally and theoretically by Bénard , Brunt, Jeffreys, Low and Rayleigh , and it is known that in stability will occur at some critical value of gh 3 Ap/pkv, h denoting the thickness of the fluid layer, Ap/p the fractional excess of density in the fluid at the top as compared with the fluid at the bottom surface, k the diffusivity and v the kinematic viscosity. The critical value depends upon the conditions at the top and bottom surfaces, which may be either ‘free ’or constrained by rigid conducting surfaces. The theoretical problem is solved here under three distinct boundary conditions, and greater generality than before is maintained in regard to the ‘cell pattern’ which occurs in plan. In addition an approximate method is described and illustrated, depending on a stationary property akin to that of which Lord Rayleigh made wide application in vibration theory. Within the assumptions of the approximate theory (i.e. with neglect of terms of the second order in respect of the velocities) a particular size is associated with every shape of cell (such that ‘ a 2 ’ takes a preferred value), but no particular shape is more likely than another to occur in a layer of indefinite extent (§ 31'). The explanation of the apparent preference for a hexagonal cell pattern (§5) must presumably be sought in a theory which takes account of second-order terms. This conjecture if correct goes some way towards explaining the rather indefinite nature of observed cell-formations (cf. Low 1930, figure 10).