Flow characteristics in flow networks

A flow network is a system of mutually intersecting holes in a plate or an assembly of plates. The flow at each intersection is characterized by a collision of two flow streams, resulting in complex flow patterns through the downstream holes. In the case of multiple intersections, the flow is periodically disrupted at each succeeding intersection, thus preventing the formation of a fully-developed flow through the holes.An experimental study is presented in this paper to determine flow characteristics in flow networks with various geometry. The intersecting pressure loss coefficient which represents the performance of flow networks is defined and its magnitude empirically determined as functions of geometric and flow conditions. A method is developed to measure the ramming loss in an intersection tube. Flow visualization by means of hydrogen bubble method is applied to observe flow patterns and mixing behavior in the flow network. A physical model is developed to predict the intersection pressure loss in flow networks.List of symbols A total section area of the flow network holes - a section are a of one hole in the flow network - a _t throat area of the orifice - b semi-minor axis of the intersection throat ellipse (Fig. 8) - C _d overall flow discharge coefficient with intersection - C _do overall flow discharge coefficient in the absence of intersection - D _h hydraulic diameter of the flow channel - d hole diameter - f flow friction coefficient - FF compressible flow function - H major axis of the intersection ellipse (Fig. 8) - K _b, K₀ pressure loss coefficients for the miter bend, and quadrant-edged orifice, respectively - K _c, K_e, K_x flow contraction, expansion, and intersection coefficients, respectively - L length of the hole in the flow network, i.e. flow length inside holes - L _e equivalent length of a pipe for the miter bend pressure loss - N _h number of holes in the flow network - N _x number of intersections for each hole - p pitch distance between holes - P _a, P_s, P_t total pressure in the plenum, the ambient pressure, and absolute total pressure in the plenum, respectively - _Pb, p₀ pressure losses in the miter bend and through the quadrant-edged orifice, respectively - p _T, p_H pressure drops in the flow network and its half unit, respectively - Q, Q flow rates passing through the test section equivalent to standard condition and in operating conditions, respectively - R univeral gas constant - s test plate thickness - T, T _t air temperature in the plenum and the absolute temperature of air, respectively - V fluid flow velocity - W mass flow rate of air - diameter ratio in the quadrant-edged orifice - dynamic viscosity of fluid - kinematic viscosity of fluid - intersection angle between holes - fluid density     

The effects of span position of winglet vortex generator on local heat/mass transfer over a three-row flat tube bank fin

The naphthalene sublimation method was used to study the effects of span position of vortex generators (VGs) on local heat transfer on three-row flat tube bank fin. A dimensionless factor of the larger the better characteristics, JF, is used to screen the optimum span position of VGs. In order to get JF, the local heat transfer coefficient obtained in experiments and numerical method are used to obtain the heat transferred from the fin. A new parameter, named as staggered ratio, is introduced to consider the interactions of vortices generated by partial or full periodically staggered arrangement of VGs. The present results reveal that: VGs should be mounted as near as possible to the tube wall; the vortices generated by the upstream VGs converge at wake region of flat tube; the interactions of vortices with counter rotating direction do not effect Nusselt number (Nu) greatly on fin surface mounted with VGs, but reduce Nu greatly on the other fin surface; the real staggered ratio should include the effect of flow convergence; with increasing real staggered ratio, these interactions are intensified, and heat transfer performance decreases; for average Nu and friction factor (f), the effects of interactions of vortices are not significant, f has slightly smaller value when real staggered ratio is about 0.6 than that when VGs are in no staggered arrangement. A cross section area of flow passage [m²] - A _mim minimum cross section area of flow passage [m²] - a width of flat tube [m] - b length of flat tube [m] - B _pT lateral pitch of flat tube: B _pT = S ₁/T _p - d _h hydraulic diameter of flow channel [m] - D _naph diffusion of naphthalene [m²/s] - f friction factor: f = pd _h/(Lu ² _max/2) - h mass transfer coefficient [m/s] - H height of winglet type vortex generators [m] - j Colburn factor [–] - JF a dimensionless ratio, defined in Eq. (23) [–] - L streamwise length of fin [m] - L _PVG longitudinal pitch of vortex generators divided by fin spacing: L _pVG = l _VG/T _p - l _VG pitch of in-line vortex generators [m] - m mass [kg] - m mass sublimation rate of naphthalene [kg/m²·s] - Nu Nusselt number: Nu = d _h/ - P pressure of naphthalene vapor [Pa] - p non-dimensional pitch of in-line vortex generators: p = l _VG/S ₂ - Pr Prandtl number [–] - Q heat transfer rate [W] - R universal gas constant [m²/s²·K] - Re Reynolds number: Re = ·u _max·d _h/ - S ₁ transversal pitch between flat tubes [m] - S ₂ longitudinal pitch between flat tubes [m] - Sc Schmidt number [–] - Sh Sherwood number [–]: Sh = hd _h/D _naph - Sr staggered ratio [–]: Sr = (2Hsin – C)/(2Hsin) - T _p fin spacing [m] - T temperature [K] - u _max maximum velocity [m/s] - u average velocity of air [m/s] - V volume flow rate of air [m³/s] - x,y,z coordinates [m] - z sublimation depth[m] - heat transfer coefficient [W/m²·K] - heat conductivity [W/m·K] - viscosity [kg/m²·s] - density [kg/m³] - attack angle of vortex generator [°] - time interval for naphthalene sublimation [s] - fin thickness, distance between two VGs around the tube [m] - small interval - C distance between the stream direction centerlines of VGs - p pressure drop [Pa] - 0 without VG enhancement - 1, 2, I, II fin surface I, fin surface II, respectively - atm atmosphere - f fluid - fin fin - local local value - m average - naph naphthalene - n,b naphthalene at bulk flow - n,w naphthalene at wall - VG with VG enhancement - w wall or fin surface     

An equation of motion for an incompressible Newtonian fluid in a packed bed

The application of a volume average Navier-Stokes equation for the prediction of pressure drop in packed beds consisting of uniform spherical particles is presented. The development of the bed permeability from an assumed porous microstructure model is given. The final model is quasi-empirical in nature, and is able to correlate a wide variety of literature data over a large Reynolds number range. In beds with wall effects present the model correlates experimental data with an error of less than 10%. Numerical solutions of the volume averaged equation are obtained using a penalty finite element method.Nomenclatures d length of a representative unit cell - d _e flow length in Representative Unit Cell - d _p characteristic pore size - D _T column diameter - D _P equivalent particle diameter - e _v energy loss coefficient for elbow - f _app apparent friction factor - f _v packed bed friction factor, defined by Equation (30) - F term representing impermeability of the porous medium - I integral defined by Equation (3) - L length of packed column - N Number of RUC in model microstructure - P pressure - P interstitial pressure - P pressure deviation - Re_p Reynolds number,v _p d _p/ - Re_s Reynolds number,v _s d/gm - Re_b Reynolds number,v _s D _p/ - S _fs fluid solid contact area - T tortuosity - v fluid velocity - v velocity deviation - v _p velocity in a pore - v _s superficial velocity in the medium - v interstitial velocity - V _o total volume of representative unit cell - V pore volume of representative unit cell - change in indicated property - u normal vector onS _fs - porosity - viscosity - density - coefficient in unconsolidated permeability model     

A experimental investigation of turbulent flow and heat transfer in elliptical duets

Fully developed turbulent flow and heat transfer to air and water in ducts of elliptical cross section have been investigated experimentally. For the ducts of aspect ratio 2.5 1 and larger, a reduction in the overall heat transfer rate was found in the lower turbulent Reynold's number range (Re<25,000). Similar effects have been noted by investigators of narrow triangular cross sections where flow measurements indicated the possible co-existence of laminar and turbulent flow resulting in localised increases in thermal resistance. It was found that the analogy between momentum and heat transfer could not be applied directly to the larger aspect ratio ducts where significant circumferential variations of wall temperature occurred.

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Zusammenfassung Voll entwickelte turbulente Strömung und Wärmeübertragung an Luft und Wasser in elliptischen Kanälen wurden experimentell untersucht. Für Kanäle mit Achsenverhältnissen von 2,5 1 und größer fand man eine Verringerung des Wärmedurchgangs im Bereich geringer Reynolds-Zahlen (Re < 25 000). Ähnliche Effekte waren von anderen Autoren in engen Dreieckskanälen gefunden worden, wobei man aus Strömungsmessungen das gleichzeitige Auftreten von laminarer und turbulenter Strömung mit örtlicher Zunahme des thermischen Widerstandes folgern konnte. Die Analogie zwischen Impuls- und Wärmeübertragung konnte nicht unmittelbar auf Kanäle mit großem Achsenverhältnis, bei denen die Umfangstemperatur beträchtlich variierte, angewendet werden.

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Nomenclature A cross-sectional area - b duct wall thickness - C_p specific heat at constant pressure - d_e equivalent diameter of noncircular cross-section (=4A/p) - f Fanning friction coefficient - h local heat transfer coefficient (=q_w/(T_w-T_b)) - ¯h average circumferential heat transfer coefficient - k thermal conductivity of fluid - k_w thermal conductivity of wall material - K^* wall conductivity parameter (= k_wb/kd_e) - p wetted perimeter - q_w wall heat flux - T_b bulk fluid temperature - T_w local wall temperature - absolute viscosity - kinematic viscosity (=/) - mass density - Nu Nusselt number (= h d_e/k) - Nu average circumferential Nusselt number (= ¯h d_e/k) - Pr Prandtl number (= C_p/k) - Re Reynolds number (= d_e/) - St Stanton number (= Nu/Re · Pr)     

Transients in flow and local heat transfer due to a pressure wave in pipe flow

The effect of a pressure wave on the turbulent flow and heat transfer in a rectangular air flow channel has been experimentally studied for fast transients, occurring due to a sudden increase of the main flow by an injection of air through the wall. A fast response measuring technique using a hot film sensor for the heat flux, a hot wire for the velocities and a pressure transducer have been developed. It was found that in the initial part of the transient the heat transfer change is independent of the Reynolds number. For the second part the change in heat transfer depends on thermal boundary layer thickness and thus on the Reynolds number. Results have been compared with a simple numerical turbulent flow and heat transfer model. The main effect on the flow could be well predicted. For the heat transfer a deviation in the initial part of the transient heat transfer has been found. From the turbulence measurements it has been found that a pressure wave does not influence the absolute value of the local turbulent velocity fluctuations. They could be considered to be frozen.Nomenclature A surface area (m²) - D diameter (m) - h heat transfer coefficient (Wm^–2 K^–1) - p pressure drop (Pa) - P pressure (Pa) - Q heat flow (W) - R tube radius (m) - T bulk temperature (K) - T _s surface temperature (K) - t time (s) - u velocity (m/s) - V voltage (V) - y distance from wall (m) - viscosity (N s m^–2) - kinematic viscosity (m^–2 s^–1) - density (kg m^–3) - _w wall shear stress (N m^–2) - Nu Nusselt number - Re Reynolds number     

Heat transfer of gas-particle flow in a supersonic convergent-divergent nozzle

Summary Heat flux, wall heat transfer coefficients, and wall pressures are determined for high velocity flow of gas-solid mixtures in a converging-diverging nozzle. Flow separation accompanied with oblique shock formation occurs in the diverging section of the nozzle. The shock strength is reduced upon the addition of solid particles. The wall pressure in the convergent section of the nozzle appears unaffected by the presence of solid particles. In the divergent section, however, the wall pressure is slightly lowered. At the maximum ratio of solid to air flow used in the experiments (3.7) increases in the heat transfer rate of up to 20 and 50 percent are obtained in the convergent and separated (divergent) regions of the nozzle, respectively. Slightly larger increases in the wall heat transfer coefficients are also obtained. It is concluded that the wall heat flux and heat transfer coefficients are influenced strongly by the presence of disturbances upstream of the nozzle inlet.Nomenclature W _a air flow rate - W _s solids flow rate - x axial distance from nozzle entrance - L axial length of nozzle - specific heat ratio of fluid - A _e exit cross section of flow - A _* throat cross section of flow - P ₀ inlet pressure - P _s wall separation pressure - P _a ambient exhaust pressure - shock wave angle - shock wave deflection angle - M ₁ Mach number upstream of shock wave - Mach number normal to shock wave - q heat flux - k _f thermal conductivity of fluid - T _wi inside wall temperature - T _wo outside wall temperature - T _ad adiabatic wall temperature - h wall heat transfer coefficient - C nozzle constant - A local cross section of flow - c _p specific heat of fluid - Pr Prandtl number - viscosity of fluid - r _c throat radius of curvature - factor accounting for variation of and Units absolute temperature °R(ankine) °F+459.7 - conductivity 1 BTU (hr ft °F)^–1 4.137×10^–3 cal (s cm °C)^–1 - specific heat 1 BTU (1b °F)^–1 1 cal (g °C)^–1 - absolute pressure 1 psia 0.0680 atm Supported in part by aid provided by the UCLA Space Science Center (Grant NsG 236-62 Libby).Listed for readers not familiar with the units adopted in this paper (editor).     

Laminar thermally developing flow inside right-angularly triangular ducts

An analytical approach based on the generalized integral transform technique is presented, for the solution of laminar forced convection within the thermal entry region of ducts with arbitrarily shaped cross-sections. The analysis is illustrated through consideration of a right triangular duct subjected to constant wall temperature boundary condition. Critical comparisons are made with results available in the literature, from direct numerical approaches. Numerical results for dimensionless average temperature and Nusselt numbers are presented for different apex angles.Nomenclature a,b sides of right triangular duct - A _c cross-sectional area of duct - c _p specific heat of fluid - D _h =4A _c /p hydraulic diameter, with P the wet perimeter - h(z) heat transfer coefficient at duct wall - k thermal conductivity - Pe=c _p D _h /k Peclet number - T(x, y, z) temperature distribution - T _o inlet temperature - T _w prescribed wall temperature - u(x, y); U(X, Y) dimensional and dimensionless velocity profile - average flow velocity - x; X dimensional and dimensionless normal coordinate (Fig. 1) - x ₁(y); X ₁(Y) dimensional and dimensionless position at irregular boundary (Fig. 1) - y; Y dimensional and dimensionless normal coordinate (Fig. 1) - z; Z dimensional and dimensionless axial coordinate Greek letters side of right triangular duct in X direction (dimensionless) - side of right triangular duct in Y direction (dimensionless) - density of fluid - (X, Y, Z) dimensionless temperature distribution - * apex angle of triangular duct (Fig. 1) - ** apex angle of triangular duct (Fig. 1)     

Dynamics of compressible air flow with friction in a variable-area duct

An adiabatic compressible air flow with friction in a variable-area duct is investigated. A nozzle-flow apparatus is built in which compressed air flows in a frictional diverging conical duct. According to the experimental investigation together with the numerical solution of the governing differential equations it is colligated that the dominance of either factor, friction or variation of area effect, depends upon the product of the inequality .

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Dynamik einer kompressiblen Luftströmung mit Reibung in einem Kanal veränderlichen Querschnitts

Zusammenfassung Es wurde eine adiabate kompressible Luftströmung mit Reibung in einem Kanal veränderlichen Querschnitts untersucht. Dazu wurde ein düsenförmiger Kanal gebaut, in dem die komprimierte Luft im erweiterten Teil strömt. Sowohl die Versuche als auch die numerische Lösung der beschreibenden Differentialgleichungen ergaben, daß entweder der Reibungsbeiwert oder die Änderung des Strömungsquerschnittes von dem Produkt abhängen.

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Nomenclature A cross-sectional area of the duct - A _in inlet cross-sectional area of the duct - D diameter of a cross-sectional area - f friction coefficient - M Mach number - /D relative surface roughness of the duct wall - P local static pressure at any axial location - P₀ stagnation pressure at any location - P_{0 in} inlet stagnation pressure - mass flow rate - T static temperature - T_o stagnation temperature - R air gas constant - Re Reynolds number - ratio of specific heats - x measured from the duct inlet     

Heat transfer and equilibrium temperature of a sphere in a supersonic rarefied gas flow

Some results are presented of experimental studies of the equilibrium temperature and heat transfer of a sphere in a supersonic rarefied air flow.The notations D sphere diameter - u, , T,,l, freestream parameters (u is velocity, density, T the thermodynamic temperature,l the molecular mean free path, the viscosity coefficient, the thermal conductivity) - T₀ temperature of the adiabatically stagnated stream - T_e mean equilibrium temperature of the sphere - T_w surface temperature of the cold sphere (T_we) - mean heat transfer coefficient - _e air thermal conductivity at the temperature T_e - P Prandtl number - M Mach number     

Reynolds number dependence of the drag coefficient for laminar flow through electrically-heated photoetched screens

Experiments are performed to measure the drag coefficient of electrically-heated screens. Square-pattern 80 mesh and 100 mesh screens of 50.8 m-wide wires photoetched from 50.8 m thick Inconel sheets are examined. Ambient air is passed through these screens at upstream velocities yielding wire-width Reynolds numbers from 2 to 35, and electrical current is passed through the screens to generate heat fluxes from o to 0.17 MW/m², based on the total screen area. The dependence of the drag coefficient on Reynolds number and heat flux is determined for these two screens by measuring pressure drops across the screens for a variety of conditions in these ranges. In all cases, heating is found to increase the drag coefficient above the unheated value. A correlation relating the heated drag coefficient to the unheated drag coefficient is developed based on the idea that the main effect of heating at these levels is to modify the Reynolds number through modifying the viscosity. This correlation is seen to reproduce the experimental results closely.List of Symbols A total screen cross sectional area - C fitting coefficient, near unity - c _D heated drag coefficient - c _{D, 0} unheated drag coefficient - C _p air specific heat at constant pressure - D photoetched wire width, sheet thickness - h _s stagnation point heat-transfer coefficient - k air thermal conductivity - M distance between adjacent wires - O open area fraction - p air pressure - p air pressure drop across screen - Pr Prandtl number for air, c _p/k - Q total electrical power to screen - R radius of curvature at stagnation point - Re _D wire width Reynolds number, UD/ - T air temperature - U air speed upstream of screen - air specific heat ratio - air density - air viscosity - exponent in temperature power law for viscosity - () quantity () evaluated at heated screen temperature The authors thank John Lewin and Bob Meyer for their assistance in the design and fabrication of the heated screen test facility and Tom Grasser for his help in performing the experiments. This work was performed at Sandia National laboratories, supported by the U.S. Department of Energy under contract number DE-AC04-94AL85000.     

Treatment of moisture condensation on fins

An analysis of natural convection from a vertical plate fin when the fin base temperature is below the dew point of the surrounding air is presented in this paper. The analytical solution derived is based upon a constant heat and mass transfer coefficient and is also valid for forced convection. The results of this simplified theory are compared with a numerical solution where the coupling of convection and conduction is taken into account. An experimental verification of the results is also shown.

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Aus Kondensation von Feuchtigkeit an Rippen

Zusammenfassung Es wird eine Analyse der freien Konvektion an einer vertikalen plattenförmigen Rippe dargestellt, bei der die Temperatur im Anfangsbereich der Rippe unterhalb des Taupunktes der umgebenden Luft liegt. Die abgeleitete analytische Lösung beruht auf einem konstanten Wärme- und Stoffübergangskoeffizienten und gilt auch für die erzwungene Konvektion. Die Resultate dieser vereinfachten Theorie werden mit einer numerischen Lösung verglichen, in der die Verbindung von Konvektion und Wärmeleitung in Betracht gezogen wird. Angeführt wird auch eine experimentelle Bestätigung der Resultate.

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Nomenclature a _f thermal diffusivity of air - A, B constants in Eq. (7) - c constant defined in Eq. (3) - D diffusion coefficient - f an arbitrary function ofT andx in Eq. (12) - F ₁,F ₂ coefficients in differential Eq. (13) - g gravitational acceleration - h heat transfer coefficient - h _m mass transfer coefficient - k thermal conductivity of fin - k _f thermal conductivity of air - l latent heat of moisture condensation - L total length of fin - L _w length of wet fin - m parameter, (h/kt)^1/2 - m _l dimensionless parameter, 1+ B/T _r - m _y parameter,m m _l ^1/2 - p pressure of surrounding air - p _ws saturation pressure of water vapor - p _w partial pressure of water vapor in air - Pr Prandtl number,/a _f - q total heat fluxl - q _c convective heat flux - q _m heat flux - q _r radiative heat flux - R parameter in Eq. (14) - R _w specific gas constant of water vapor - t half thickness of fin - T temperature - T _b base temperature of wet fin - T _c base temperature of dry fin=saturation temp. of vapor - T _r reference temperature defined in Eq. (15) - T temperature of surrounding air - T temp, difference between fin surface and surroundings - v initial temperature for quasilinearization - x vertical coordinate, see Fig. 1 - y horizontal coordinate, see Fig. 1 - coefficient of thermal expansion - emissivity - dimensionless parameter in Eq. (14) - ø _d heat flux of dry fin - ø _tot total heat flux of dry-wet fin - kinematic viscosity - Stefan-Boltzman coefficient - relative humidity of air     

Optimum rib size to enhance forced convection in a horizontal triangular duct with ribbed internal surfaces

The optimum rib size to enhance heat transfer had been proposed through an experimental investigation on the forced convection of a fully developed turbulent flow in an air-cooled horizontal equilateral triangular duct fabricated on its internal surfaces with uniformly spaced square ribs. Five different rib sizes (B) of 5 mm, 6 mm, 7 mm, 7.9 mm and 9 mm, respectively, were used in the present investigation, while the separation (S) between the center lines of two adjacent ribs was kept at a constant of 57 mm. The experimental triangular ducts were of the same axial length (L) of 1050 mm and the same hydraulic diameter (D) of 44 mm. Both the ducts and the ribs were fabricated with duralumin. For every experimental set-up, the entire inner wall of the duct was heated uniformly while the outer wall was thermally insulated. From the experimental results, a maximum average Nusselt number of the triangular duct was observed at the rib size of 7.9 mm (i.e. relative rib size ). Considering the pressure drop along the triangular duct, it was found to increase almost linearly with the rib size. Non-dimensional expressions had been developed for the determination of the average Nusselt number and the average friction factor of the equilateral triangular ducts with ribbed internal surfaces. The developed equations were valid for a wide range of Reynolds numbers of 4,000 < Re _D < 23,000 and relative rib sizes of under steady-state condition. A Inner surface area of the triangular duct [m²] - A _C Cross-sectional area of the triangular duct [m²] - B Side length of the square rib [mm] - C _P Specific heat at constant pressure [kJ·kg^–1·K^–1] - C ₁, C ₂, C ₃ Constant coefficients in Equations (10), (12) and (13), respectively - D Hydraulic diameter of the triangular duct [mm] - Electric power supplied to heat the triangular duct [W] - f Average friction factor - F View factor for thermal radiation from the duct ends to its surroundings - h Average convection heat transfer coefficient at the air/duct interface [W·m^–2 ·K^–1] - k Thermal conductivity of the air [W·m^–1 ·K^–1] - L Axial length of the triangular duct [mm] - Mass flow rate [kg·s^–1] - n ₁, n ₂, n ₃ Power indices in Equations (10), (12) and (13), respectively - Nu _D Average Nusselt number based on hydraulic diameter - P Fluid pressure [Pa] - Pr Prandtl number of the airflow - _c Steady-state forced convection from the triangular duct to the airflow [W] - _l Heat loss from external surfaces of the triangular duct assembly to the surroundings [W] - _r Radiation heat loss from both ends of the triangular duct to the surroundings [W] - Re _D Reynolds number of the airflow based on hydraulic diameter - S Uniform separation between the centre lines of two consecutive ribs [mm] - T Fluid temperature [K] - T _a Mean temperature of the airflow [K] - T _ai Inlet mean temperature of the airflow [K] - T _ao Outlet mean temperature of the airflow [K] - T _s Mean surface temperature of the triangular duct [K] - T Ambient temperature [K] - U Mean air velocity in the triangular duct [m·s^–1] - _r Mean surface-emissivity with respect to thermal radiation - Dynamic viscosity of the fluid [kg·m^–1·s^–1] - Kinematic viscosity of the airflow [m²·s^–1] - Density of the airflow [kg·m^–3] - Stefan-Boltzmann constant [W·m^–2·K^–4]     

An investigation of laminar radial flow between two parallel discs

Summary Experiments have been carried out to test recent theoretical predictions of the pressure distribution for laminar flow between parallel discs, including inertia effects. The experimental investigation covered the condition where the inertia effects were always completely dominant over the central region of the discs in contrast to other recent experimental work on the problem where the central injection diameter was considerably larger. The present experiments subject the theories to a stringent test, due to the dominance of the inertia effects, and it is found that the inertia effects predicted by the various theoretical analyses are significantly smaller than those shown by the experimental results. It is suggested that the theoretical approach requires further development before it will cover the conditions where the central injection diameter is small.Nomenclature r, y, cylindrical co-ordinates - u velocity in r direction - U _m mean velocity in r direction at radius r - density - coefficient of viscosity - Q volume flow per unit time - 2h gap between parallel discs - p static pressure - R r/h - P h ³ p/Q - R _e Q/h     

Heat transfer in laminar flow in a uniformly porous channel with an applied transverse magnetic field

Summary The steady laminar flow of an incompressible, viscous, and electrically conducting fluid between two parallel porous plates with equal permeability has been discussed by Terrill and Shrestha [6]. In this paper, using the solution of [6] for the velocity field, the heat transfer problems of (i) uniform wall temperature and (ii) uniform heat flux at wall are solved.For small suction Reynolds numbers we find that the Nusselt number, with increasing Reynolds number, increases for case (i) and decreases for (ii).Nomenclature stream function - 2h channel width - x, y distances measured parallel, perpendicular to the channel walls - U velocity of fluid in the x direction at x=0 - V constant velocity of suction at the wall - nondimensional distance, y/h - nondimensional distance, x/h - f() function defined in (1) - density - coefficient of kinematic viscosity - R suction Reynolds number, V h/ - Re channel Reynolds number, 4U h/ - B ₀ magnetic induction - electrical conductivity - M Hartmann number, B ₀ h(/)^1/2 - K constant defined in (3) - A constant defined in (5) - 4R/Re - q local heat flux per unit area at the wall - k thermal conductivity - T temperature of the fluid - X –1/ ln(1–) - C _p specific heat at constant pressure - j current density - Pr Prandtl number, C _p/k - P mass transfer Péclet number, R Pr - Pe mass transfer Péclet number, P/ - T ₀ temperature at x=0 - T _H() temperature in the fully developed region - T _h(X, ) temperature in the entrance region - Y _n () eigenfunctions, uniform wall temperature - _n eigenvalues - e() function defined by (24) - B _n 2/3 _n ² - A _n constants defined by (28) - a _2m constants defined by (30) - F _n () eigenfunctions, uniform wall heat flux - a _n , b _n , c _n , d _n , e _n constants defined by (45) and (48) - S a parameter, U ²/q - h ₁ heat transfer coefficient - T _m mean temperature - Nu Nusselt number - Nu _T Nusselt number, uniform wall temperature - Nu _q Nusselt number, uniform wall heat flux     

Momentum and heat transfer in the entrance region of a parallel plate channel: developing laminar flow with constant wall temperature

A new integral method of solution is presented for developing laminar flow and heat transfer in the entrance region of a parallel plate channel with uniform surface temperature. Unlike earlier Karman-Pohlhausen analyses, the new analysis provides solutions which are free from jump discontinuities in the gradients of the velocity and temperature distributions throughout and at the end of the entrance region. The hydrodynamic and thermal results from the present analysis therefore join smoothly and asymptotically to their fully-developed values. The heat transfer results obtained are further found to agree well with previously published numerical solutions.Nomenclature a half width of the channel, m - D _h hydraulic diameter (=4a), m - h local heat transfer coefficient,W/(m²·K) - h _m mean heat transfer coefficient defined by Eq- (9),W/(m²·K) - k thermal conductivity, W/(m·K) - L _H axial length of the hydrodynamic entrance region, m - L _T axial length of the thermal entrance region, m - L _in,H axial length of the hydrodynamic inlet region, m - L _in,T axial length of the thermal inlet region, m - Nu _x local Nusselt number,hD _h /k, dimensionless - Nu _m mean Nusselt number defined by Eq. (9),h _mD_h/k, dimensionless - P pressure, N/m² - P _O pressure at the entrance, N/m² - Pr Prandtl number,c _p /k, dimensionless - Re Reynolds number, 4aU _o /v, dimensionless - T absolute temperature, K - T _b fluid bulk temperature, K - T _c centerline temperature, K - T _w wall temperature, K - U _c centerline velocity, m/s - U ₀ velocity of the fluid at entrance, m/s - U core velocity, m/s - u velocity component inx direction, m/s - v velocity component iny direction, m/s - x spatial coordinate, axial distance, m - y spacial coordinate measured from channel wall, m Greek Letters molecular thermal diffusivity, m²/s - hydrodynamic shape factor, dimensionless - _T thermal shape factor, dimensionless - hydrodynamic boundary layer thickness, m - * /a, dimensionless - _T thermal boundary layer thickness, m - * _{T T} /a, dimensionless - dimensionless distance,y/ ory/a - Pohlhausen's shape factor, dimensionless - dynamic viscosity coefficient, kg/(m·s) - v kinematic viscosity,/, m²/s - dimensionless axial distance,x/(a·Re) - _H dimensionless axial length of the hydrodynamic entrance region (=L _H /(a·Re)) - _T dimensionless axial length of the thermal entrance region (=L _T /(a·Re)) - _in,H dimensionless axial length of the hydrodynamic inlet region (=L _in,H/(a·Re)) - _in,T dimensionless axial length of the thermal inlet region (L _in,T /(a·Re)) - fluid density, kg/m³     

Optimization of a pyrotechnic igniter with the release of reactive particles

This paper deals with the effects of reactive particles on the performance of a pyrotechnic igniter. These particles are placed on the inner surface of a flash tube, released into the main flow of the gas and ignited by the passage of one of the two discontinuities (the shock wave or the contact surface). Two particle sizes have been studied (3m and 10m). It is shown that the best performance is achieved with small particles released into the flow by the shock wave. Another focal point of this study is the combining of two fundamentally different methods to calculate the two phase flow.Nomenclature a₀ sound speed in region 0 - a₂ sound speed in region 2 - C _D drag coefficient - d average particle diameter - d rate change of the particle diameter - e _g total internal energy of the gas - e _s particle internal energy=C ₃ T _s - F drag force - rn mass flow rate - Mo _c shock wave Mach number - N particle number desity - N _u Nusselt number - P pressure - P ₀ pressure in region 0 - P ₂ pressure in region 2 - P _r Prandtl number - Q heat convection - R _e Reynolds number - T _g gas temperature - T _s particle temperature - u ₂ velocity in region 2 - u _g gas velocity - u _s barycentric velocity of the particles - ratio of specific heats - _g thermal conductivity of the gas - _g gas dynamic viscosity - _g gas density - _s apparent density of the particles - _s true density of the particles - defined by (8) This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

Temperature separation and friction losses in vortex tube

The process of energy separation and friction losses in a vortex tube is studied in detail. The hot and cold exit air temperatures were measured. Experiments have been conducted at inlet pressure of 3.5, 5, 7.5 and 9 bar, at inlet temperature of 292.15 and 298.15 K and at cold air mass ratio from 0 to1. The results demonstrate that the hot air temperature reaches its maximum value at a cold air mass ratio of nearly 0.82, while the minimum value of cold air temperature is found at a cold air mass ratio of 0.3. Based on energy and mass balances as well as on the definition of internal energy and on experimental results a new model for the determination of hot and cold exit gas temperature has been developed. The model includes the relevant primary parameters and predicts the experimental results as well as the data published in the literature sufficiently accurate for engineering purposes.A cross-section area m³ - D diameter of the pipe m - F model parameter - f friction factor - L length of the tube m - m mass flow rate kg/s - y cold air mass ratio - P static pressure Pa - T temperature K - t thickness of the orifice m - R gas constant J/kg K - v velocity of fluid m/s - density of the fluid kg/m³ - friction factor for pipe - friction factor for orifice and tee junction - 1 inlet of compressed gas - 2 exit of hot gas - 3 exit of cold gas - atm atmospheric pressure - c cold exit gas - f friction - h hot exit gas - o orifice plate - T tee junction     

Understanding the Piche evaporimeter as a simple integrating mass transfer meter

Results are presented of a comparison of measured and calculated evaporation rates of the Piche evaporimeter under indoor and outdoor (within a meteorological screen) conditions. In both cases, application of mass transfer formulae in use for horizontal (turbulent) flow to the evaporating blotting paper of the instrument yield very good results under pure forced convection conditions. For mixed convection regimes, comparisons using either pure free (combined heat and mass transfer) or pure forced convection equations give as expected too low calculated values. Reasons for such differences with measured values are reviewed. Our forced convection results confirm that main stream turbulence is only of influence on mass transfer to a zero incidence flow in combination with pressure gradient (bluff body) effects, which under our conditions appear to be absent around the Piche surfaces. The same results prove absence of any influence of the particular temperature distribution over the blotting paper on the mass transfer. The understanding and importance of these conclusions in relation to the use of the Piche evaporimeter as a simple integrating mass transfer meter under actual farming conditions are discussed. The importance to obtain such mass transfer data is explained in the introduction.Nomenclature A Numerical constant in free convection Sherwood number - Coefficient of thermal expansion (K^–1) - C _{(s, b)} Water vapour concentration average at the evaporating surface (s) and in the bulk air (b) (g m^–3) - D Coefficient of molecular diffusion of water vapour in air (m² s^–1) - d Characteristic dimension of the paper disc in the direction of flow (m) - E _{(c, m)} Evaporation rates of the Piche evaporimeter, calculated (c) and measured (m) (units in text) - e _{(s, b)} Partial water vapour pressure average at the evaporating surface (s) and in the bulk air (b) (mbar) - Gr Grashof number - g Acceleration of gravity (m s^–2) - m Number of measuring periods - n Numerical constant in free convection Sherwood number - Coefficient of kinematic viscosity of air (m² s^–1) - P Atmospheric pressure (mbar) - Re Reynolds number - _{(s, b)} Air density average at the evaporating surface (s) and of the bulk air (b) (g m^–3) - Sh Sherwood number - T _{(s, b)} Temperature average of the evaporating surface (s) and in the bulk air (b) (K) - T _{(vs, vb)} Virtual temperature average at the evaporating surface (s) and in the bulk air (b) (K) - U Wind speed (air movement) average of the bulk air (m s^–1)     

Thermal and inertia effects in externally pressurized conical thrust oil bearings

This paper presents theoretical and experimental investigations of thermal and inertia effects on the performance of externally pressurized conical thrust bearings. The analysis, as well as the experimental results, revealed that the increase in oil temperature due to pad rotation has a detrimental effect on the load carrying capacity, while it increases the flow rate. Increasing the speed of rotation, will increase or decrease the bearing load carrying capacity depending on the recess dimensions.Nomenclature c lubricant specific heat - F frictional torque - h film thickness - L load carrying capacity - P pressure - P pressure ratio (P/P ₁) - P ₁ inlet pressure - Q volume flow rate - r radius measured on cone surface - r radius ratio (R/R ₃) - R ₁ supply hole radius - R ₂ recess radius - R ₃ outside radius of bearing - S inertia parameter (0.15 ² R ₃ ² /P ₁) - T temperature - u, v, w velocity components (see Fig. 2) - z coordinate normal to cone surface - lubricant density - lubricant viscosity - 2 cone apex angle - rotational speed - recess depth