Heat and mass transfer in laminar flow between parallel porous plates

Summary The problem of heat transfer in a two-dimensional porous channel has been discussed by Terrill [6] for small suction at the walls. In [6] the heat transfer problem of a discontinuous change in wall temperature was solved. In the present paper the solution of Terrill for small suction at the walls is revised and the whole problem is extended to the cases of large suction and large injection at the walls. It is found that, for all values of the Reynolds number R, the limiting Nusselt number Nu increases with increasing R.Nomenclature stream function - 2h channel width - x, y distances measured parallel and perpendicular to the channel walls respectively - U velocity of fluid at x=0 - V constant velocity of fluid at the wall - =y/h nondimensional distance perpendicular to the channel walls - f() function defined in equation (1) - coefficient of kinematic viscosity - R=Vh/ suction Reynolds number - density of fluid - C _p specific heat at constant pressure - K thermal conductivity - T temperature - x=x ₀ position where temperature of walls changes - T ₀, T ₁ temperature of walls for x<x ₀, x>x ₀ respectively - = (T – T ₁)/T ₀ – T ₁) nondimensional temperature - =x/h nondimensional distance along channel - R ^* = Uh/v channel Reynolds number - Pr = C _p/K Prandtl number - _n eigenvalues - B _n() eigenfunctions - B _n ⁽⁰⁾ , () eigenfunctions for R=0 - B ₀ ⁽ⁱ⁾ , B ₀ ⁽ⁱⁱ⁾ ... change in eigenfunctions when R0 and small - K _n constants given by equation (13) - h heat transfer coefficient - Nu Nusselt number - _m mean temperature - C _n constants given by equation (18) - perturbation parameter - B _0i () perturbation approximations to B ₀() - Q = B ₀/ ₀ derivative of eigenfunction with respect to eigenvalue - z nondimensional distance perpendicular to the channel walls - F(z) function defined by (54)     

Approximate method for calculating turbulent boundary layer with positive pressure gradient

A single-parameter integral method is proposed for calculating the turbulent boundary layer with positive pressure gradient which makes it possible to calculate the friction, thermal flux, and layer thickness both ahead of the separation point and in some region behind the separation point.Notation u velocity - density - ^* displacement thickness - ^** momentum thickness - energy thickness - M Mach number - r radius - dynamic viscosity - c_p specific heat at constant pressure - Reynolds number based on initial boundary layer thickness - P Prandtl number - p₁ static pressure at point of initial interaction - p₂ static pressure at pressureplateau - p₀ stagnation pressure - T₀ stagnation temperature - I enthalpy - T_e recovery temperature - T_w ⁰ temperature factor - H form parameter - r₁ recovery coefficient Indices 0 denotes initial section of boundary layer - 1 parameters taken at edge of boundary layer - w parameters taken at the wall temperature - * parameters referred to flow on a flat plate with =0     

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.     

Oscillatory flow and mass transfer within asymmetric and symmetric channels with sinusoidal wavy walls

Mass transfer for oscillatory flow was studied experimentally in channels with two different geometries, i.e., a periodically converging-diverging channel and a serpentine channel, both having sinusoidal wavy walls. The experiments were carried out under the following conditions: 10<Re<500 and 0.008<St<0.05. The channel geometries were found to have an important effect on the flow patterns and the mass transfer rates. At low Strouhal numbers of less than 0.023, the mass transfer rates for both channels were almost identical, regardless of different flow patterns and wall shear stresses. At high Strouhal numbers, however, the serpentine channel had a smaller mass transfer rate than the converging-diverging channel. The mass transfer characteristics were explained in terms of the vortex dynamics, wall shear stresses and fluid mixing based on numerical analysis and flow visualizations. The serpentine channel yields a better mass transfer and pumping power performance than the converging and diverging channel at low Strouhal numbers.

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Schwingungsströmung und Stofftransport innerhalb asymmetrischer und symmetrischer Kanäle mit sinusförmig gewellten Wandungen

Zusammenfassung Der Stofftransport bei Schwingungsströmungen in Kanälen mit zwei verschiedenen Geometrien experimentell untersucht, d.h. in einem periodisch konvergierenden und divergierenden Kanal und einem schlangenförmig gewundenen Serpentinenkanal, wobei die Kanäle jeweils sinus förmig gewellte Wandungen aufwiesen. Die Versuche wurden unter folgenden Bedingungen, ausgeführt; 10<Re<500 und 0.008<St<0.05. Es zeigte sich, daß die Kanalgeometrien einen erheblichen Einfluß auf die Strömungsmuster und Stofftransportraten haben. Bei niedrigen Strouhal-Zahlen unter 0.023 waren die Stofftransportraten beider Kanäle praktisch identisch, und zwar unabhängig von unterschiedlichen Strömungsmustern und Wand-Scherspannungen. Bei hohen Strouhal-Zahlen dagegen zeigte der Serpentinenkanal eine geringere Stofftransportrate als der Konvergenz-Divergenz-Kanal. Diese Stofftransport-Charakteristik wurde, basierend auf numerischer Analyse und Sichtbarmachung der Strömung, erklärt in Form von Vortexdynamic, Wand-Scherspannungen und Flüssigkeitsmischung. Bei niedrigen Strouhal-Zahlen erbringt der Serpentinenkanal ein besseres Stofftransport- und Pumpleistungsverhalten als der Konvergenz-Divergenz-Kanal.

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Nomenclature A area of mass transfer surface - a wave amplitude of wavy wall - c _b concentration of the ferricyanide ion - D diameter of piston - D molecular diffusivity of the ferricyanide ion - F Faraday constant - f frequency of oscillation - H _min minimum spacing between wavy walls - Re Reynolds number, Eq. (4) - s length of stroke of piston - Sc Schmidt number - Sh Sherwood number, Eq. (5) - St Strouhal number, Eq. (3) - T period of oscillation - U _p peak velocity based onH _min - W width of wavy wall Greek symbols wavelength of wavy wall - kinematic viscosity - _w wall vorticity - _max vortex strength - angle of misalignment of the two channel walls This work was supported in part by a Grant-in-Aid for Science Research (No. 63750889 and No. 03302031) from the ministry of Education, Science and Culture of Japan. The author acknowledge with thanks the assistance of graduate student Shigeki Matsune in the computations.     

Analysis of a method and a system to determine heat transfer and properties of fluids in the absence of temperature measurements and a modified Fourier Equation

The technique to determine by capacitance measurements heat transfer, thermal transport and dielectric properties of fluids introduced recently is now analyzed for a simple system of spherical geometry. The temperature distribution under programmed heat input to a fluid annulus between solid walls is computed by finite difference method for the determination of the capacitance time function of the arrangement. A system of heavy wall structure and heated long enough will produce a capacitance-time curve which is a function of thermal conductivity only. Thermal diffusivity is of influence in thin wall systems. The capacitance change of a heavy wall arrangement is related to the thermal conductivity of the test fluid by a modified Fourier equation. This equation describes the heat flow through the fluid layer but includes the thermal expansion of the solid walls. The change of geometry with T is therefore accounted for. For other multicomposite structures the Fourier equation must be further modified by including the thermal expansion of all materials of the structure and possibly also their compressibilities.

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Zusammenfassung Die kürzlich eingeführte Methode der Bestimmung von Wärmeübergang, thermischen Transport und dielektrischen Größen mittels Kapazitäts-Zeit-Messung wird analysiert für ein einfaches kugeliges System. Die Temperaturverteilung in der Flüssigkeit im Kugelspalt zwischen zwei festen Körpern wird für konstante Wärmezufuhr von außen mittels der Differenzmethode bestimmt und daraus die Kapazitäts-Zeit-Funktion ermittelt. Es wird gezeigt, daß die Kapazitäts-Zeit-Kurve nur eine Funktion der Wärmeleitzahl ist für den Fall dickwandiger Anordnungen. Für dünnwandige Systeme wird sie auch abhängig von der Temperaturleitzahl. Es wird eine modifizierte Fourier-Gleichung eingeführt, die den Wärmetransport durch die Flüssigkeit beschreibt, dabei aber die Änderung der Geometrie der Schicht berücksichtigt, die sich wegen der thermischen Ausdehnung der festen Wände bei der Einstellung der Temperaturdifferenz ergibt. Für andere mehrschichtige Körper muß die Fourier-Gleichung weiterhin modifiziert werden durch Berücksichtigung der thermischen Ausdehnungskoeffizienten aller beteiligten Materialien und möglicherweise auch ihrer Kompressibilitäten.

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Nomenclature A average cross-sectional area of fluid layer - A coefficient matrix - B matrix defined by Eq. (20) - B₀ geometric constant of fluid layer (A/L) at reference temperature - C capacitance of arrangement - C_i, C_r capacitance of layer of fluid i and reference fluid at temperature T - capacitances at reference temperature - C_H, c_l specific heats of outer and inner wall - FA...FE constants defined in Eqs. (13 ... 17) - L thickness of fluid layer - M_H, M_L mass of outer and inner wall - P power input to the system - R constant defined by Eq. (24) - T temperature - T_ref reference temperature - T (O, t), T (L, t) temperatures of outer and inner wall at time t - T _i ⁿ , T _i+0 ^n+m temperatures at location i and time n (m=number of t's; 0=number of x's) - T temperature difference across fluid layer - T apparent temperature difference - t_h, T_l temperature increases of outer and inner wall - T_max temperature change of system from one to another thermal equilibrium condition a thermal diffusivity - k, k_i, k_r thermal qonductivity of fluids and of fluid i and reference fluid - q heat flow through fluid layer - ^rh,^rl inner radius of outer wall and outer radius of inner wall - ^rOH,^rOL radii at reference temperature - t time - t time interval - x coordinate - ¯x vector of unknown T_i ⁿ⁺¹ - x length interval Greek symbols linear thermal expansion coefficient - _H, _L linear thermal expansion coefficient of materials of outer and inner wall - dielectric constant - _i, _ref dielectric constant of fluid i and reference fluid - ₀ permittivity of free space - multiplyer of conduction Eq. (7) in finite difference form - time needed to establish quasi-steady state conditions in the system heated by a constant power input In honor of Prof. Dr. E. Schmidt to his 80th Birthday.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

Laminar flow in the entrance region of a porous-wall channel

Summary The effect of fluid injection at the walls of a two-dimensional channel on the development of flow in the entrance region of the channel has been investigated. The integral forms of the boundary layer equations for flow in the channel were set up for an injection velocity uniformly distributed along the channel walls.With an assumed polynomial of the n-th degree for the one-parameter velocity profile a solution of the above boundary layer equations was obtained by an iteration method. A closed form solution was also obtained for the case when a similar velocity profile was assumed. The agreement between the entrance region velocity profiles of the present analysis for an impermeable-walled channel and of Schlichting¹) and Bodoia and Osterle²) is found to be very good.The results of the analysis show that fluid injection at the channel walls increases the rate of the growth of the boundary layer thickness, and hence reduces considerably the entrance length required for a fully developed flow.Nomenclature h half channel thickness - L entrance length with wall-injection - L ₀ entrance length without wall-injection - p static pressure - p=p/U ₀ ² dimensionless pressure - Re=U ₀ h/ Reynolds number at inlet cross-section - u velocity in the x direction at any point in the channel - =u/U ₀ dimensionless velocity in the x direction at any point in the channel - U _av average velocity at a channel cross-section - U _c center line velocity - U ₀ inlet cross-section velocity - _c =U _c /U ₀ dimensionless center line velocity - v velocity in the y direction at any point in the channel - v ₀ constant injection velocity of fluid at the wall - v=v/v ₀ dimensionless velocity in the y direction at any point in the channel - x distance along the channel wall measured from the inlet cross-section - x=x/hRe dimensionless distance in the x direction - y distance perpendicular to the channel wall - y=y/h dimensionless distance in the y direction - thickness of the boundary layer - =/h dimensionless boundary layer thickness - =/ dimensionless distance within the boundary layer region - =v ₀ h/ injection parameter or injection Reynolds number - kinematic viscosity - 1+ie - mass density of the fluid - parameter defined in (14)     

Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux

Summary The effect of an internal heat source on the heat transfer characteristics for turbulent liquid metal flow between parallel plates is studied analytically. The analysis is carried out for the conditions of uniform internal heat generation, uniform wall heat flux, and fully established temperature and velocity profiles. Consideration is given both to the uniform or slug flow approximation and the power law approximation for the turbulent velocity profile. Allowance is made for turbulent eddying within the liquid metal through the use of an idealized eddy diffusivity function. It is found that the Nusselt number is unaffected by the heat source strength when the velocity profile is assumed to be uniform over the channel cross section. In the case of a 1/7-power velocity expression, the Nusselt numbers are lower than those in the absence of internal heat generation, and decrease with diminishing eddy conduction. Nusselt numbers, in the absence of an internal heat source, are compared with existing calculations, and indications are that the present results are adequate for preliminary design purposes.Nomenclature A hydrodynamic parameter - a half height of channel - a ₁ a constant, 1+0.01 Pr Re ^0.9 - a ₂ a constant, 0.01 Pr Re ^0.9 - C _p specific heat at constant pressure - D _h hydraulic diameter of channel, 4a - h heat transfer coefficient, q _w/(t _w–t _b) - I ₁ integral defined by (17) - I ₂ integral defined by (18) - k diffusivity parameter, (1+0.01 Pr Re ^0.9)^1/2 - m exponent in power velocity expression - Nu Nusselt number, hD _h/ - Nu ₀ Nusselt number in absence of internal heat generation - Pr Prandtl number, / - Q heat generation rate per volume - q _w wall heat flux - Re Reynolds number for channel, 2/ - s ratio of heat generation rate to wall heat flux, Qa/q _w - T dimensionless temperature, (t _w–t)/(t _w–t _b) - t fluid temperature, t _w wall temperature, t _b fluid bulk temperature - u fluid velocity in x direction, , fluid mean velocity - x longitudinal coordinate measured from channel entrance - x ⁺ dimensionless longitudinal coordinate, 2(x/a)/Pr Re - y transverse coordinate measured from channel centerline - z transverse coordinate measured from channel wall, a–y - molecular diffusivity of heat, /C _p - dummy variable of integration - dummy variable of integration - _H eddy diffusivity of heat - _M eddy diffusivity of momentum - dummy variable of integration - fluid thermal conductivity - _T dimensionless diffusivity, Pr ( _H/) - fluid kinematic viscosity - dummy variable of integration - fluid density - dummy variable of integration - ratio of eddy diffusivity for heat transfer to that for momentum transfer, _H/ _M - average value of - dimensionless velocity distribution, u/     

Control of unstable oscillations in a confined and bounded injecting wall channel

The analysis of a confined flow field generated through two separated injecting walls was carried out by studying the effect of a mass flow rate difference between the two walls. Such a parameter has been found to play a major role in unstable flow field behaviour. Concretely speaking, we have identified two vortex shedding phenomena, i.e. the main flow and the wall vortex shedding phenomena. Results clearly show that when the mass flow rate is increased at the second injecting wall, wall vortex coherence is enhanced and impinging of such structures forces a coupling phenomenon to develop between flow field dynamics and acoustics. On the other hand, only a 15% mass flow rate difference of the first injecting block is sufficient to prevent such coupling between acoustics and vortex shedding phenomenon. Consequently, the resonance phenomenon is pronouncedly weakened and significant oscillation reduction is achieved.Nomenclature sound velocity (m/s) - nth longitudinal acoustic mode (Hz) - h_t height of the nozzle throat (m) - h_c channel height (m) - l length between the edge of the second injecting block and the nozzle location (m) - L channel length (m) - P mean pressure at the front-head (Pa) - P fluctuating pressure (Pa) - q_m, q₁, q₂ total, first and second injecting block mass flow rate (kg/s) - S_x normalised power spectral density of the x fluctuations (Hz^-1) - s=w h_c characteristic surface area (m²) - T temperature of the flow (K) - u, v longitudinal and lateral velocity component (m/s) - u_X longitudinal mean velocity at X location (m/s) - u_m maximum longitudinal velocity (m/s) - u, v longitudinal and lateral fluctuating velocity (m/s) - characteristic acoustic velocity (m/s) - v_w wall injection velocity (m/s) - w channel width (m) - X, Y, Z non-dimensional axis normalised respectively by l, h_c and w - dynamic viscosity (kg/ms) - density (kg/m³) - time delay (s)Dimensionless parameters turbulence intensity - Mach number - Re_c= M a h_c/ Reynolds number - Re_w= v_w h_c/ wall injection Reynolds number - correlation coefficient of pressure and velocity fluctuations - normalised longitudinal velocity component - parameter of unbalanced mass flow rate between the two injecting blocks - specific heat ratio - coherence function     

Calculations for a cylindrical electric arc with allowance for energy transfer by radiation with the hydrogen at a pressure of 100 atm

An approximate method is described for the consideration of energy transfer by radiation during the utilization of real properties of a gas (in particular, the frequency-dependent absorption coefficient under conditions of local thermal equilibrium). With increasing pressure, it becomes necessary to take self-absorption into account over almost the entire frequency spectrum.Calculations are carried out for a wall-stabilized cylindrical electric arc in hydrogen as an example for a pressure of 100 atm and channel radii of 0.3, 1, and 3 cm at values of current strength up to the order of 10 A. The strong effect of radiation on the current-voltage characteristic of the arc, the gas temperature, and the nature of its distribution over the arc radius is demonstrated.The process of energy transfer by radiation plays a significant and sometimes predominant role in the thermal balance of electric arcs with high current strengths [1–9]. Calculations have been performed for cylindrical arcs in atmospheres of argon and hydrogen [5, 7] with allowance for energy transfer by radiation and for atmospheric pressure in which case the gas is essentially transparent to radiation. Approximate estimates were obtained for the self-absorbed portion of the radiation.The role played by radiation increases with increasing current strength, arc radius, and pressure, while self-absorption in this process extends over an increasingly large region of the spectrum. Hence, calculations must be carried out for the arc if conditions are such that the gas in the arc does not transmit radiation.In [10–13], an approximate method was developed for taking into account energy transfer by radiation in the presence of intense selfabsorption as applied to heat transfer problems under conditions of local thermal equilibrium with allowance for the variation of the absorption coefficient as a function of the frequency. The conditions for local thermal equilibrium in an arc passing through an argon or hydrogen atmosphere are fulfilled for pressures greater than atmospheric pressure and for current strengths greater than 10 A [14–16], The results of [10–12] were used as the foundation for calculations based on an electric arc in argon at atmospheric pressure, under which conditions, self-absorption affects only the transitions to the ground state. The part played by radiation in the heat transfer process is smaller than the part played in the energy transfer by conduction. Calculations confirmed the results of [5, 7].The role of energy transfer by radiation in the energy balance of the arc increases with increasing pressure, while in turn, the role of the continuous spectrum increases for the radiation. The results of calculations performed for a wall-stabilized arc burning in an atmosphere of hydrogen at a pressure of 100 atm are given in the present paper. In this case, almost the entire energy supply is lost by radiation. The approximate method of accounting for energy transfer by radiation is demonstrated by an example.Notation and T gas density and temperature, respectively - u velocity - c_p heat capacity of the gas at constant pressure - coefficient of thermal conductivity - coefficient of electrical conductivity - x and r cylindrical coordinates - r₀ channel radius - I current strength - E electric field strength - u ° equilibrium value of radiation energy density - u value of radiation energy density - radiation frequency - divergence of energy flux density transported by radiation - k absorption coefficient - c speed of light - _i emissivity of the i-th region of the spectrum     

Thermal wave propagation within a medium with the inert heat source

A heat conduction equation of a new type is derived which takes into account the finite velocity of heat flux propagation and the relaxation of heat source capacity. The equation is solved for a semi-infinite body and a step change in temperature at the surface. The analysis shows that as the time increases the obtained solution moves from the solution of the classical hyperbolic equation without energy generation towards the solution of the classical hyperbolic equation with energy generation.

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Ausbreitung thermischer Wellen in einem Medium mit träger Wärmequelle

Zusammenfassung Es wird eine neuartige Wärmeleitungsgleichung abgeleitet, welche die endliche Geschwindigkeit der Ausbreitung des Wärmestromes und die Relaxation der Kapazität der Wärmequelle berücksichtigt. Die Gleichung wird für einen halbunendlichen Körper und eine schrittweise Temperaturänderung an der Oberfläche gelöst. Die Analyse zeigt, daß mit zunehmender Zeit sich die Lösung der klassischen hyperbolischen Gleichung ohne Wärmeerzeugung in eine solche mit ebenfalls klassischer hyperbolischer Gleichung mit Wärmeerzeugung wandelt.

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Nomenclature a thermal diffusivity,k/( c _p - c _p specific heat at constant pressure - C speed of heat propagation - C ₁,C ₂ constants - k thermal conductivity - q _v steady capacity of internal heat source - q _vd transient capacity of internal heat source - r ₁,r ₂ roots of characterisitc equation - t time - t _k relaxation time of heat flux - t _q relaxation time of internal heat source capacity - T temperature - T ₀ surface temperature - u() unit step function - x, y, z Cartesian coordinates - X dimensionless coordinate - , constant coefficients - dimensionless temperature - density - dimensionless time - _r-t_qt_k ratio of relaxation times - dimensionless steady capacity of internal heat source - _d dimensionless transient capacity of internal heat source     

Injection moulding of thermoplastics: Freezing during mould filling

The injection moulding of thermoplastic polymers involves, during mould filling, flows of hot melts into mould networks, the walls of which are so cold that frozen layers form on them. Theoretical analyses of such flows are presented here. Br Brinkman number - c _L polymer melt specific heat capacity - c _S frozen polymer specific heat capacity - e exponential function - erf() error function - Gz Graetz number in thermal entrance region - Gz ^* modified Graetz number in thermal entrance region - Gz overall Graetz number - h channel half-height - h ^* half-height of polymer melt region - H mean heat transfer coefficient - k _L polymer melt thermal conductivity - k _S frozen polymer thermal conductivity - ln( ) natural logarithm function - L length of thermal entrance region in pipe or channel - m viscosity shear rate exponent - M(,,) Kummer function - Nu Nusselt number - p pressure - P pressure drop in thermal entrance region - P _f pressure drop in melt front region - Pe Péclet number - Pr Prandtl number - Q volumetric flow rate - r radial coordinate in pipe - R pipe radius - R ^* radius of polymer melt region - Re Reynolds number - Sf Stefan number - t time - T temperature - T _i inlet polymer melt temperature - T _m melting temperature of polymer - T _w pipe or channel wall temperature - U(,,) Kummer function - u _r radial velocity in pipe - u _x axial velocity in channel - u _y cross-channel velocity - u _z axial velocity in pipe - V melt front velocity - w channel width - x axial coordinate in channel - x _f melt front position in channel - y cross-channel coordinate - z axial coordinate in pipe - z _f melt front position in pipe - () gamma function - dimensionless thickness of frozen polymer layer - _i i-th term (i = 1,2,3) in power series expansion of - dimensionless axial coordinate in pipe - _f dimensionless melt front position in pipe - dimensionless cross-channel coordinate - ^* dimensionless half-height of polymer melt region - dimensionless temperature - _i i-th term (i = 0, 1, 2, 3) in power series expansion of - _i first derivative of _i with respect toø - _i second derivative of _i with respect toø - ^* dimensionless wall temperature - thermal diffusivity ratio - - latent heat of fusion - µ viscosity - µ ^* unit shear rate viscosity - dimensionless axial coordinate in channel - _f dimensionless melt front position in channel - dimensionless pressure drop in thermal entrance region - _f dimensionless pressure drop in melt front region - _L polymer melt density - _s frozen polymer density - dimensionless radial coordinate in pipe - ^* dimensionless radius of polymer melt region - ø dimensionless similarity variable in thermal entrance region - dummy variable - dimensionless contracted axial coordinate in thermal entrance region - dimensionless similarity variable in melt front region - ^* constant     

Unsteady effects in a thin viscous shock layer near a three-dimensional stagnation point of a body moving with given acceleration and deceleration

An unsteady viscous shock layer near a stagnation point is studied. The Navier-Stokes equations are analyzed in the limit 1, Re₀ , df/dt = ^n-mF(t/^m). The Reynolds number Re₀ is defined in the paper by Eq. (1.3) (df/dt is the velocity of the body with respect to an inertial frame of reference moving with the original steady velocity –V'_t8, ₂ = ( – 1)/( + 1)). Various flow regimes in the case 1, l, n max(2m, m + 1), m 0, where ² = 1/Re₀ are analyzed. Equations are derived that generalize the asymptotic analysis to the case of a viscous unsteady flow of gas in a thin three-dimensional shock layer. The problem of a thin unsteady viscous shock layer near the stagnation point of a body with two curvatures is formulated. Examples of numerical solution are given for different ratios of the principal curvatures of the body, the wall temperature, the parameters of the original steady flow, and the acceleration and deceleration regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–111, March–April, 1981.I thank Yu. D. Shevelev for a fruitful discussion of the work.     

Slip flow in the hydrodynamic entrance region of a tube and a parallel plate channel

Summary The problem of slip flow in the entrance region of a tube and parallel plate channel is considered by solving a linearized momentum equation. The condition is imposed that the pressure drop from momentum considerations and from mechanical energy considerations should be equal. Results are obtained for Kn=0, 0.01, 0.03, 0.05, and 0.1 and the pressure drop in the entrance region is given in detail.Nomenclature A cross-sectional area of duct - c mean value of random molecular speed - d diameter of tube - f _p - f _t - h half height of parallel plate channel - Kn Knudsen number - L molecular mean free path - n directional normal of solid boundary - p pressure - p ₀ pressure at inlet - r radial co-ordinate - r _t radius of tube - R non-dimensional radial co-ordinate - Re _p 4hU/ - Re _t 2r _t U/ - s direction along solid boundary - T absolute temperature - u velocity in x direction - u* non-dimensional velocity - U bulk velocity = (1/A) _A u dA - v velocity in y direction - x axial co-ordinate - x* stretched axial co-ordinate - X non-dimensional axial co-ordinate - X* non-dimensional stretched axial co-ordinate - Y non-dimensional channel co-ordinate - eigenvalue in parallel plate channel - stretching factor - eigenvalue in tube - density - kinematic viscosity - i index - p parallel plate - t tube - v velocity vector - gradient operator - ² Laplacian operator     

Binary laminar boundary layer on a vertical surface in the presence of combined free and forced convection

Heat and mass transfer at a vertical surface is examined in the case of combined free and forced convection. The boundary layer equations, transformed to ordinary differential equations, contain a parameter that determines the effect of free convection on the forced motion. Criteria are offered for differentiating the free-convection, forced-convection, and combined regimes.Notation x, y coordinates - u, v velocity components - g acceleration of gravity - T temperature - kinematic viscosity - coefficient of thermal expansion - a thermal diffusivity - ₁ partial vapor density - D diffusion coefficient - W₂ mass velocity of air - independent variable - _w shear stress at wall - thermal conductivity - r latent heat of phase transition - , dimensionless temperature and partial vapor density - m^* the complex (m ₁–m _1w )/(1–m(_1w ) - c_p specific heat at constant pressure - G Grashof number - R Reynolds number - P Prandtl number - S Schmidt number     

Self-similar problem of the motion of a plane layer of heated material with an arbitrary equation of state

The self-similar problem of the nonstationary motion of a plane layer of material in which energy from an external source is released for values of the flux density q₀ on the boundary which are constant in time is considered. The self-similar variable is = m/t, where m is the Lagrangian mass coordinate and t is the time. The characteristic values of the velocity, density, and pressure do not vary with time. For a self-similar problem the energy flux density q must also depend only on the self-similar variable. In this case q() can be an arbitrary function of its argument and can be given by a table. Examples are presented of actual physical processes in which the mass of the energy-release zone increases linearly with time. The equation of state can have an arbitrary form, including specification by a table. The gaseous state of matter for an arbitrary variable adiabatic exponent, the condensed state, and a two-phase state can be described. A solution of the self-similar problem is presented for the heating of a half-space bounded by a vacuum for a certain specific equation of state and various flux densities q₀ and velocities M of the advance of the energy-release zone.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 5, pp. 136–145, September–October, 1975.     

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     

On the boundary conditions at the macroscopic level

We study the problem of the boundary conditions specified at the boundary of a porous domain in order to solve the macroscopic transfer equations obtained by means of the volume-averaging method. The analysis is limited to the case of conductive transport but the method can be extended to other cases. A numerical study enables us to illustrate the theoretical results in the case of a model porous medium. Roman Letters _sf interfacial area of the s-f interface contained within the macroscopic system m² - A _sf interfacial area of the s-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - d _s , d _f microscopic characteristic length m - g vector that maps to _s, m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l REV characteristic length, m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for the f-phase at the f-s interface - n _e outwardly directed unit normal vector at the dividing surface - T ^* macroscopic temperature field obtained by solving the macroscopic equation (3), K - V averaging volume, m³ - V _s , V _f volume of the considered phase within the averaging volume, m³ - volume of the macroscopic system, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² Greek Letters _s , _f volume fraction - ratio of thermal conductivities - _s , _f thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding to T ^* , K - spatial deviation temperature K - error on the temperature due to the macroscopic boundary conditions, K - spatial average - ^s , ^f intrinsic phase average     

The effect of surface mass transfer on buoyancy induced flow in a variable porosity medium adjacent to a vertical heated plate

The effect of surface mass transfer on buoyancy induced flow in a variable porosity medium adjacent to a heated vertical plate is studied for high Rayleigh numbers. Similarity solutions are obtained within the frame work of boundary layer theory for a power law variation in surface temperature,T _Wx and surface injectionv _Wx(–1/2). The analysis incorporates the expression connecting porosity and permeability and also the expression connecting porosity and effective thermal diffusivity. The influence of thermal dispersion on the flow and heat transfer characteristics are also analysed in detail. The results of the present analysis document the fact that variable porosity enhances heat transfer rate and the magnitude of velocity near the wall. The governing equations are solved using an implicit finite difference scheme for both the Darcy flow model and Forchheimer flow model, the latter analysis being confined to an isothermal surface and an impermeable vertical plate. The influence of the intertial terms in the Forchheimer model is to decrease the heat transfer and flow rates and the influence of thermal dispersion is to increase the heat transfer rate.

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Der Effekt des Oberflächenstoffaustausches bei auftriebsinduzierter Strömung in einem variablen porösen Medium, das an eine vertikale, beheizte Platte angrenzt

Zusammenfassung Es wird der Effekt des Oberflächenstoffaustausches in auftriebsinduzierter Strömung in einem variablen porösen Medium, das an eine vertikale, beheizte Platte angrenzt, für große Reynoldszahlen untersucht. Ähnliche Lösungen werden im Rahmen der Grenzschicht-Theorie, durch Variation des Potenzansatzes der Oberflächentemperatur,T _Wx , und der Oberflächengeschwindigkeit,v _Wx(–1/2), erreicht. Die Analyse vereinigt sowohl den Ausdruck, der Porösität und Permeabilität verbindet, als auch den Ausdruck, der Porösität und Wärmeleitfähigkeit miteinander verbindet. Der Einfluß der Temperaturverteilung auf Strömung und Wärmeübergangskennzahlen wird ebenfalls im Detail analysiert. Als Ergebnis der vorliegenden Untersuchung ergibt sich die Tatsache, daß variable Porösität Wärmeübertragungsrate und Betrag der Geschwindigkeit in Wandnähe steigert. Die bestimmenden Gleichungen, sowohl für das Darcysche Strömungsmodell als auch für das Forchheimersche Strömungsmodell, werden mit Hilfe eines implizierten Differenzenschemas gelöst. Die Berechnung wird für die beiden Fälle, isotherme Oberfläche und undurchlässige vertikale Platte, angewandt. Der Einfluß der Terme für die Trägheitskräfte im Forchheimerschen Modell senkt Wärmeübergangs- und Durchgangsrate, wogegen die Wärmeübergangsrate durch den Einfluß der Temperaturverteilung erhöht wird.

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Nomenclature a constant defined by Eq. (12) - A constant defined by Eq. (12) - B constant defined by Eq. (3) - b _s/_f ratio of thermal conductivities - C constant defined by Eq. (1) - C _P specific heat of the convective fluid - d particle diameter - f dimensionless function defined by Eq. (14) - f _w lateral mass flux parameter - g acceleration due to gravity - k ₀ mean permeability of the mediumk ₀= ₀ ³ d ²/150 (1– ₀)² k ₀=1.75d/(1– ₀) 150 (Inertia parameter) - L length of the source or sink - m mass transfer - n constant defined in Eq. (12) - k (y) permeability of the porous medium - k (y) interial coefficient in the Ergun expression - Gr modified Grashof numberGr=(g k ₀ k ₀ (T _w–))/ ² - R _a Rayleigh number (g k ₀ x T _w–)/ - R _ad modified Rayleigh number (g k ₀ d|T _w–|)/ - N _u Nusselt number - s x/d - Q overall heat transfer rate - T temperature - T _w surface temperature - T ambient fluid temperature - u velocity in vertical direction - v velocity in horizontal direction - x vertical coordinate - y horizontal coordinate Greek symbols ₀ mean thermal diffusivity _f/ C_p - coefficient of thermal expansion - constant defined in Eq. (4) - ratio of particle to bed diameter - _e effective thermal conductivity - f thermal conductivity of fluid - _s thermal conductivity of solid - dimensionless similarity variable in Eq. (13) - value of at the edge of the boundary layer - constant defined in Eq. (1) - _e effective molecular thermal diffusivity - (y) porosity of the medium - ₀ mean porosity of the medium - viscosity of the fluid - ₀ density of the convective fluid - stream function - w condition at the wall - condition at infinity     

An analytical solution for the temperature profiles during injection molding,including dissipation effects

An analytical solution is obtained for the stationary temperature profile in a polymeric melt flowing into a cold cavity, which also takes into account viscous heating effects. The solution is valid for the injection stage of the molding process. Although the analytical solution is only possible after making several (at first sight) rather stringent assumptions, the calculated temperature field turns out to give a fair agreement with a numerical, more realistic approach. Approximate functions were derived for both the dissipation-independent and the dissipation-dependent parts which greatly facilitate the temperature calculations. In particular, a closed-form expression is derived for the position where the maximum temperature occurs and for the thickness of the solidified layer.The expression for the temperature field is a special case of the solution of the diffusion equation with variable coefficients and a source term.Nomenclature a thermal diffusivity [m²/s] - c specific heat [J/kg K] - D channel half-height [m] - L channel length [m] - m 1/ - P pressure [Pa] - T temperature [°C] - T _W wall temperature [°C] - T _i injection temperature [°C] - T _A Br independent part of T - T _B Br dependent part of T - T ^core asymptotic temperature - v _z() axial velocity [m/s] - W channel width [m] - x cross-channel direction [m] - z axial coordinate [m] - (x) gamma function - (a, x) incomplete gamma function - M(a, b, x) Kummer function - small parameter - () temperature function - thermal conductivity [W/mK] - viscosity [Pa · s] - ₀ consistency index - power-law exponent - density [kg/m] - similarity variable Dimensionless variables Br Brinkman number - Gz Graetz number -