Diffusion of macromolecular solutions in the boundary layer

Summary This paper concerns the study of a diffusion layer of macromolecular solutions annularly injected in a cylindric pipe in turbulent flow. The study includes firstly the analysis of the longitudinal and transversal projections of the concentration gradient, secondly the study of the state of turbulence in the diffusion layer.The concentrations are determined by the measurement of the concentration salt (salt was added to the injected solution). A mathematical model of the development of concentration is proposed.The turbulent state of diffusion layer has been studied with Laser-Doppler-Anemometry. An analysis of Eulerian autocorrelation and the power spectral density has been done.

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Zusammenfassung Es werden Ergebnisse über die turbulente Diffusionsschicht von nicht-newtonschen Flüssigkeiten (PÄO) mitgeteilt, die an der Wand eines kreiszylindrischen Rohrs injiziert werden. Diese Untersuchungen betreffen die Analyse der Längs- und Querkomponenten des Konzentrationsgradienten in der Diffusionsschicht und im Turbulenzfeld.Die Konzentrationen werden mit Hilfe von Salz (NaCl) gemessen, das der injizierten Polymerlösung beigefügt worden ist. Ein mathematisches Modell für die Konzentrationsverteilung wird vorgeschlagen.Der Turbulenzzustand der Diffusionsschicht wird mit Hilfe der Laser-Doppler-Anemometrie untersucht. Die Eulersche Autokorrelation und das Intensitätsspektrum werden analysiert.

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Nomenclature c polymer concentration (wppm) - c ₀ wall value ofc - c _i value ofc at the slot - D diameter of the pipe - e slot thickness - L ₀ distance from the slot at which the wall concentration drops toe ^–1of its original value - m distance in meter from the slot - U means freestream velocity - u _i injection velocity - x distance from the slot - y normal distance from the surface - boundary layer thickness - characteristic height of the diffusion, i. e. the value ofy at whichc/c ₀ = 0.5 With 4 figures     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe

A model is presented describing the changes that occur in the diffusion boundary layer upon injection of a macromolecular solution (PEO) into a cylindrical pipe under turbulent flow conditions (Re 40,000). A shape parameter was introduced to describe the shape of the turbulent plume. The value of this parameter was found to be the same for water and various dilute PEO solutions. The proposed model gives a good approximation at low homogeneous concentrations. x downstream distance from the slot - y normal distance from the wall - R radius of the pipe - C concentration - C _w wall concentration - Q _i flow rate injection - Q _t flow rate - C _j =C _i *Q _i /Q _t equivalent homogeneous polymer concentration - L _tf characteristic length of the diffusion plume - characteristic height of the diffusion plume, i.e. the value ofy at whichC/C _w = 0.5 - thickness of the diffusion boundary layer - x ₀ characteristic distance from the slot, i.e. the value ofx at which/R = 1/2 - ⁺ shape parameter of the diffusion boundary layer - ⁺ —/R nondimensionalized variables - x ⁺ —x/L _tf nondimensionalized variables     

Effect of polymeric additives on cavitation and radiated noise in water flowing past a circular cylinder

The present study is concerned with the effect of high molecular weight polymers on the cavitating flow around a cylinder. A decrease of the incipient cavitation number and disappearance of the transient cavitation regime are observed when polymers are added to the flow. Simultaneously, the radiated noise and the drag on the cylinder are substantially reduced for σ values above critical.     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe — Evolution of wall concentration

Summary In this paper, we are presenting a model of the evolution of the wall concentration of a macromolecular solution (PEO) annularly injected in a cylindrical pipe in a turbulent flow. This model valid for all diffusion zones is based on mathematical and physical considerations and proves to be in good agreement with the experimental data.

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Zusammenfassung Es wird ein Modell der Entwicklung der Wandkonzentration einer makromolekularen Lösung (PÄO) vorgestellt, die in einem wandnahen Ringspalt in die turbulente Strömung durch ein zylindrisches Rohr injiziert worden ist. Dieses für alle Diffusionszonen gültige Modell basiert auf mathematischen und physikalischen Betrachtungen und erweist sich für die Beschreibung der experimentellen Daten als gut geeignet.

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C _w wall concentration - C _i initial concentration before injection - L ₀ distance from the slot at which the wall concentration drops toe ^-1 of its original value - L _IT ,L _IF ,L _F characteristic lengths - L _I length scale of the second region - x downstream distance from the source - n _I ,n _T ,n _F characteristic exponents - K ₀,K _I ,K _F characteristic constants - ln natural logarithm - q _i flow rate of injection - Q _T flow rate - C _j =C _{i · q i} /Q _T concentration in homogeneous medium - A, B, C, m constants - p andq annex variables - Re Reynolds number With 7 figures     

Diffusion of macromolecular solutions in a turbulent boundary layer of a cylindrical pipe

This paper presents a study of the turbulent structure of the diffusion boundary layer. A macromolecular solution (PEO WSR 301) is injected into a cylindrical pipe under turbulent flow conditions (Re ≈ 40 000). Laser velocimetry was the experimental technique used. Velocity profiles and turbulence intensities are determined in a boundary layer with injections of newtonian or non-newtonian fluids. We attempt to give an interpretation of these results as a function of the decomposition of the diffusion field.     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe

This paper presents a model for the evolution of a transversal concentration profile of a macromolecular solution (PEO) injected into a cylindrical pipe at turbulent flow conditions (R 40000). This model, based on the diffusion of a scalar quantity emitted by two diametrically opposed point sinks, proves to be in good agreement with the experimental data. C concentration - C _w wall concentration - C _i initial concentration before injection - x downstream distance from the slot - y normal distance from the wall - characteristic height of diffusion, i.e. the value ofg at whichC/C _w = 0.5 - n characteristic exponent - R radius of pipe - D diameter of pipe - a, b constants - L _m mixing length, i.e. the value ofx at whichC _w /C _i =Q _i /Q _T - Q _i flow rate of injection - Q _T flow rate - f, g annex functions - n ₀ maximal value ofn