‘Pop’ safety valves: a compressible flow analysis

Pressure versus lift characteristics of Pop-type safety valves have been predicted from a model in which pressures on various parts of the valve disc can be deduced from compressible flow analysis of three passages in series. In particular, it has been shown that the characteristic snap-open and snap-shut phenomena are inherent in the solutions to the sets of equations, and are critically dependent on the setting of the adjusting ring, bearing out what is known in practice. The model has shown satisfactory agreement with experimental data from a $\text{1}\text{2}\text{″}$ (12.55 mm) pop valve operating in air.     

Two-phase flow in a vertical pipe and the phenomenon of choking: Homogeneous diffusion model—I. Homogeneous flow models

The paper examines the topological structure of all possible solutions which can exist in flows through adiabatic constant-area ducts for which the homogeneous diffusion model has been assumed. The conservation equations are one-dimensional with the single space variable $\text{z.}$ but gravity effects are included. The conservation equations are coupled with three equations of state: a pure substance, a perfect gas with constant specific heats, and a homogeneous two-phase system in thermodynamic equilibrium. The preferred state variables are pressure $\text{P.}$ enthalpy $\text{h.}$ and mass flux $\text{G}{}^2$.The three conservation equations are first-order but nonlinear. They induce a family of solutions which are interpreted as curves in a four-dimensional phase space conceived as a union of three-dimensional spaces ($\text{P, h, G}{}^2\text{, z}$) with $\text{G}{}^2\text{=}\text{const}$ treated as a parameter. It is shown that all points in these spaces are regular, so that no singular solutions need to be considered. The existence and uniqueness theorem leads to the conclusion that through every point in phase space there passes one and only one solution-curve.The set of differential equations, treated as a system of algebraic equations of each point of the phase space, determines the components of a rate-of-change vector which are obtained explicitly by Cramer's rule. This vector is tangent to the solution curve. Each solution curve turns downward in $\text{z}$ at some specific elevation $\text{z}{}^1$, and this determines the condition for choking. Choking occurs always when the exit flow velocity at $\text{L = z}{}^1$ is equal to the local velocity of propagation of small plane disturbances of sufficiently large wavelength, that is when the flow rate $\text{G}$ becomes equal to a specified, critical flow rate, $\text{G}{}^1$. (The possible dependence of the sonic velocity on frequency in a real flow is ignored, because it has not been allowed for in the equations of the model under study.) A criterion, analogous to the Mach number, which indicates the presence or absence of choking in a cross section is the ratio $\text{K = G/G}{}^7$ of the mass-flow rate $\text{G}$ to the local critical mass flow rate. $\text{G}{}^7$, $\text{K = 1}$ denoting choking. The critical parameters depend only on the thermodynamic properties of the fluid and are independent of the gravitational acceleration and shearing stress at the wall.The topological characteristics of the solutions allow us to study all flow patterns which can, and which cannot, occur in a pipe of given length $\text{L}$ into which fluid is discharged through a rounded entrance from a stagnation reservoir and whose back-pressure is slowly lowered. The set of flow patterns is analogous to that which occurs with a perfect gas, except that the characteristic numerical values are different. They must be obtained by numerical integration and the influence of gravity must be allowed for.The preceding conclusions are valid for all assumptions concerning the shearing stress at the wall which make if dependent on the state parameters only, but not on their derivatives with respect to $\text{z}$. However, the study is limited to upward flows for which the shearing stress at the wall and the gravitational acceleration are codirectional.     

Investigation of droplet deposition from a turbulent gas stream

Turbulent deposition of particles from two-phase flow onto the smooth wall of a tube has been studied theoretically and experimentally. A model is proposed for the deposition motion of large particles based on turbulent diffusion in the core followed by a free flight towards the wall. The theory shows that within the Stokes regime, the dimensionless deposition velocity $\text{k-}{}_{\mathrm{<mtext>d</mtext>}}\text{/}\text{u}$* depends on Re and $\text{τ}{}^{\mathrm{+}}$ only, where u* is the friction velocity, Re is the tube Reynolds number and $\text{τ}{}^{\mathrm{+}}$ is the dimensionless particle relaxation time. Deposition data are obtained for air-water droplet flow through a 12.7-mm i.d. acrylic tubing at $\text{Re}\text{= 52,500}$ and 94,600. The proposed theory satisfactorily describes the existing deposition data as well as present measurements, covering a wide range of Re and $\text{τ}{}^{\mathrm{+}}$.     

The role of solvent viscosity in dilute-solution polymer rheology

Capillary viscometry was performed on dilute non-Newtonian solutions of monodisperse polystyrene in theta solvents. The solvents, blends of low-molecular-weight polystyrene with styrene, had viscosities (η_s) that were varied from 0.22–27 Pa s. Data reduction of the dilute limit, [η]/[η₀] vs. β = [η₀]η_sM$\text{γ}\text{?}$/RT (where $\text{γ}\text{?}$ is shear rate) revealed a parametric dependence on η_s that has not before been reported and is not predicted by most molecular theories of polymer dynamics. It is suggested that an internal viscosity model can explain such a phenomenon.     

Plastic flow and dislocation structure at small strains in OFHC copper deformed in tension,torsion and combined tension-torsion

OFHC copper specimens of 39 μm grain size were deformed to small strains (up to 8%) in tension, torsion and combined tension-torsion at 300 K and the resulting dislocation structures, distributions and densities were determined using transmission electron microscopy. Employing the von Mises yield criterion and the plastic-work hypothesis good agreement was obtained for the three testing conditions for (i) equivalent stress $\text{}\text{?}$gs vs equivalent strain $\text{}\text{?}$g3^p curves, (ii) the dislocation structure, distribution and density ρ as a function of $\text{}\text{?}$g3^p, and (iii) $\text{}\text{?}$gs as a function of $\text{ρ}\text{1}\text{2}$. Furthermore, upon comparing the $\text{}\text{?}$gs vs $\text{ρ}\text{1}\text{2}$ curve for polycrystalline copper with the τ_RSS vs $\text{ρ}\text{1}\text{2}$ curve for single crystals, an average Taylor factor M= (σ/τ_RSS) of approximately 3.2 was obtained, which is in good accord with that predicted theoretically for FCC metals. Almost equally good correlations for the stressstrain curves and for the dislocation density were obtained on the basis of maximum shear stress τ_max and maximum shear strain γ^p_max as on the basis of $\text{}\text{?}$gs and $\text{}\text{?}$g3^P. Therefore, the present results do not permit a positive decision on the question whether the dislocation density correlates better with $\text{}\text{?}$gs and $\text{}\text{?}$g3^P or with τ_max and γ^P_max.A single test in which the direction of straining in torsion was reversed yielded a density and distribution of dislocations (and a corresponding value of $\text{}\text{?}$gs) equivalent to those that developed at a smaller strain in unidirectional straining.     

Viscous dissipation effects on thermophoretically augmented aerosol particle transport across laminar boundary layers

The sensitivity of aerosol particle motion to local temperature gradients has motivated this investigation of viscous dissipation effects on mass transport rates across nonisthermal, low mass-loading ‘dusty gas’ laminar boundary layers (lbl). From numerical lbl transfer calculations, including ‘ash’ particle thermophoresis and variable thermophysical properties, it has been found that for a specified wall temperature, $\text{T}{}_{\mathrm{<mtext>w</mtext>}}$, and mainstream static temperature, $\text{T}{}_{\mathrm{<mtext>e</mtext>}}$ cous dissipation within the boundary layer increases total particle deposition rates, its relative importance being dependent on $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{/T}{}_{\mathrm{<mtext>e</mtext>}}$. For combustion turbine blades which operate at near-unity Mach number, neglect of viscous dissipation is found to cause about a 25% underestimate of the fouling rate at $\text{T}{}_{\mathrm{<mtext>w</mtext>}}\text{/T}{}_{\mathrm{<mtext>e</mtext>}}\text{= 0.8}$ for particle diameters between $\text{0.6 × 10}{}^{\mathrm{?2}}\text{μ}\text{m}$ and 0.3 μm. Alternatively, for conditions of fixed adiabatic wall temperature, $\text{T}{}_{\mathrm{<mtext>aw</mtext>}}$, or fixed stagnation (reservoir) temperature, $\text{T}{}_0$, dusty gas acceleration to appreciable Mach numbers is associated with reduced particle arrival rates due, in part, to the associated reduction in mainstream gas temperature. Recently developed mass transfer rate correlations are extended and found to be successful when tested against the present numerical calculations.     

Axisymmetric stagnation point flow and mass transfer for power-law fluids II. Moderate schmidt number correction

The theory for axisymmetric stagnation point flow of power-law fluids has been extended to include the correction terms for convective diffusion at moderate Schmidt numbers. The dimensionless mass transfer rate is expressed as an asymptotic series that is valid for Re^{(1 ? n)/3(1 + n)}Sc^{?$\text{1}\text{3}$} < 1. The result can be used to predict accurate diffusion coefficients for dilute species in fluids with specified power-law characteristics.     

Rheology of suspensions of spherical particles in a newtonian and a non-newtonian fluid

Properties of suspensions of spherical glass beads (25–38 μm dia.) in a Newtonian fluid and a non-Newtonian (NBS Fluid 40) fluid were measured at volume fractions, φ, of 0%, 10%, 20% and 30%. Measurements were made using a modified and computerized Weissenberg Rheogoniometer. Properties measured included steady shear viscosity, η($\text{γ}\text{.}$), first normal stress difference, N₁($\text{γ}\text{.}$), linear viscoelastic properties, η′(ω) and G′(ω), shear stress relaxation, σ^? ($\text{γ}\text{.}$, t), and growth, σ⁺($\text{γ}\text{.}$, t) and normal stress relaxation, N₁^?($\text{γ}\text{.}$, t).For a the Newtonian fluid, increasing φ causes both η and η′ to increase, with η′ showing a slight frequency dependence. Both N₁ and G′ are zero and stress relaxation and growth occur essentially instantaneously. For the NBS fluid, both η and η′ increse with φ at all $\text{γ}\text{.}$ and ω, respectively, the increase being greater as $\text{γ}\text{.}$ and ω approach zero. N₁ and G′ are less affected by the presence of the particles than η and η′ with the effect on G′ being more pronounced than on N₁. For fixed $\text{γ}\text{.}$, stress relaxation and growth exhibit greater non-linear effects as φ is increased. A model for predicting a priori the linear viscoelastic properties for suspensions was found to yeild reasonable estimates up to φ = 20%.     

Two new methods for the approximate determination of the initial coefficient of the first normal stress difference

Two methods for determining the initial coefficient of the first normal stress difference are presented. They are based on the evaluation of the steady viscosity function η($\text{γ}\text{.}$) and the viscosity function η⁺($\text{γ}\text{.}$, t) at the start-up of a flow with a very small rate of deformation $\text{γ}\text{.}$ < $\text{γ}\text{.}$₀. For the functions η($\text{γ}\text{.}$) and η⁺($\text{γ}\text{.}$), equations are given which can be used for a simple evaluation of the integral relationships obtaiend for ψ₁₀. The values for ψ₁₀ calculated by the two methods are compared with values obtained by the well-known methods via measurement of the ψ₁($\text{γ}\text{.}$) or η″(ω)/ω functions and extrapolation to zero). Both methods give values which are in satisfactory agreement with the experimental values.     

Critical J-integral and tearing energies for fracture of reinforced natural rubber

The critical tearing energy and $\text{J-}\text{integral}$ for initiation and rapid crack propagation at 22°C were determined for carbon black reinforced natural rubber. Tearing energies were 20 per cent and up to 100 per cent greater for initiation and rupture, respectively, and more scattered than the $\text{J}$ values. Specimen geometry affects both fracture characterizing parameters because of the energy dissipated during deformation. However, the $\text{J-}\text{integral}$ analysis can partition this energy between the crack tip region and the bulk of the specimen. Therefore, application of the $\text{J-}\text{integral}$ concept for characterizing fracture of elastomers is promising.     

Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous mediaNon unicité de solutions des équations différentielles pour un problème de couches limites en milieux poreux

The free convection, along a vertical flat plate embedded in a porous medium, can be described in terms of solutions to $\text{f?+}\text{α+1}\text{2}\text{ff″?αf′}{}^2\text{=0,}$ for all t∈(0,+∞). The purpose of this Note is to study the nonuniqueness of solutions to this problem, with the initial conditions, $\text{f(0)=a∈}\text{R}$ and f′(0)∈{0,1}, where $\text{α∈(?}\text{1}\text{3}\text{,0).}$ No assumption at infinity is imposed. We show that this problem has an infinite number of unbounded global solutions. Moreover, we prove that the first and the second derivative of solutions tend to 0 as t approaches infinity. To cite this article: M. Guedda, C. R. Mecanique 330 (2002) 279–283.     

The Einstein viscosity correction in n dimensions

The effective viscosity of a dilute suspension of rigid $\text{n-}\text{dimensional}$ hyperspheres in a viscous fluid at small particle Reynolds numbers is determined; the result being

$\text{μ}{}_{\mathrm{eff}}\text{=μ}\text{1+}\text{n+2}\text{2}\text{φ}$

. Expressions are also given for the $\text{n-}\text{dimensional}$ Stokes velocity and pressure fields for a hypersphere in a pure straining flow.     

Modelling the transverse viscoelasticity of green wood using a combination of two parabolic elementsModélisation de la viscoélasticité transverse du bois vert à l'aide d'une combinaison de deux éléments paraboliques

The rigidity in radial compression at different levels of temperature and strain rate have been measured on green Boco, with a drastic softening around $\text{60}{}^{\mathrm{<mspace\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>°</mtext>}}$C attributed to the glassy transition of lignin. The representation of experimental results in an approximated complex diagram revealed a secondary viscoelastic process occurring at lower temperature. A multiparabolic model was used for the analysis. For convenience, each parabolic element was replaced by a generalised Maxwell model with a modified-Gaussian relaxation spectrum. This model fitted correctly the observed behaviour of wood in the time range of 0.05 to 50 sec and temperature between 10 to $\text{90}{}^{\mathrm{<mspace\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>°</mtext>}}$C. To cite this article: S. Bardet, J. Gril, C. R. Mecanique 330 (2002) 549–556.     

An experimental study of choked foam flows in a convergent-divergent' nozzle

Choked flow of a foam in a convergent-divergent nozzle has been investigated. The foam consisted of air and a solution of a surface active agent in water. The upstream gas-liquid volume ratio $\text{δ}{}_0$ was in the range 0.053–1.57. The experimental results are in very good agreement with a homogeneous frictionless nozzle flow theory, assuming isothermal behaviour of the gas and no relative motion between the phases, for throat gas-liquid volume ratios $\text{δ}{}_1$ as high as 0.8; for ratios in the range $\text{0.8 < δ}{}_{\mathrm{t}}\text{< 2.98}$ the agreement, while only approximate, is still quite close. Departures from the homogeneous theory are explained in terms of (a) the failure of the assumption of the isothermal behaviour and (b) the existence of relative velocity between the phases. The latter effect predominates at low values of $\text{δ}{}_1$ but at large values, it appears that both contribute to errors in the predictions.     

Burn-out,circumferential film flow distribution and pressure drop for an eccentric annulus with heated rod

Measurements of (1) burn-out, (2) circumferential film flow distribution, and (3) pressure drop in a $\text{17 × 27.2 × 3500}\text{mm}$ concentric and eccentric annulus geometry are presented. The eccentric displacement was varied between 0 and 3 mm. The working fluid was water. Burn-out curves at 70 bar are presented for mass velocities between 500 and 1500 kg/m²s and for inlet subcoolings of 10°C and 100°C. The film flow measurements correspond to the steam qualities $\text{χ = 19 \%}$ and 24 % for the mass velocity $\text{G = 602}$ kg/m²s and $\text{χ = 20 \%}$ and 23 % for $\text{G = 1200}$ kg/m²s. The influence of the circumferential rod film flow variation on burn-out is discussed.     

Hydromagnetic flow due to torsional oscillations of an infinite disk about a steady non-zero mean

The present investigation is the study of the laminar hydromagnetic flow due to torsional oscillations of an infinite disk about a steady non-zero mean in an electrically conducting fluid. Separate solutions have been obtained for the limiting cases of low and high frequency oscillations. The low frequency solution is obtained by expressing the flow functions in powers (ik) while for high frequencies, the flow functions are expressed in powers of $\text{(ik)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$, where $\text{k =}\text{ω}\text{Ω}$ being the ratio of the frequency of oscillations ω to the mean disk-angular velocity Ω and i² = ?1. It is found that the oscillating part of the transverse shearing stress has a phase-lead while that of the radial shearing stress has a phase lag behind the disk oscillations. The phase-lead in the former case and phase-lag in the latter case decrease with the increase in the strength of the applied magnetic field.     

Complexification phenomenon in an example of sensitive singular perturbationPhénomène de complexification dans un exemple de perturbation singulière sensitive

We consider singular perturbation problems depending on a parameter ε?0 such that for ε>0 the solution u^ε belongs to a Sobolev space on a domain $\text{Ω}$, but the limit u⁰ is not a distribution on $\text{Ω}$. A very simple model problem, solvable by Fourier transform allows us to study the complexification process of u^ε as $\text{ε↘0}$. The limit holds in the topology of a space of analytical functionals. To cite this article: C.A. De Souza, É. Sanchez-Palencia, C. R. Mecanique 332 (2004).