On the transport properties of fluid-particle flow

The hydrodynamic forces acting on a solid particle in a viscous, incompressible fluid medium at low Reynolds number flow is investigated mathematically as a prerequisite to the understanding of transport processes in two-phase flow involving solid particles and fluid. Viscous interaction between a small number of spherical particles and continuous solid boundaries as well as fluid interface are analyzed under a “point-force” approximation. Non-spherical and elastic spherical particles in a simple shear flow area are then considered. Non-steady motion of a spherical particle is briefly touched upon to illustrate the transient effect of particle motion.

A macroscopic continuum description of particle-fluid flow is formulated in terms of spatial averages yielding a set of particle continuum and bulk fluid equations. Phenomenological formulas describing the transport processes in a fluid medium are extended to cases where the volume concentration of solid particles is sufficiently high to exert an important influence. Hydrodynamic forces acting on a spherical solid particle in such a system, e.g. drag, torque, rotational coupling force, and viscous collision force between streams of different sized particles moving relative to each other are obtained. Phenomenological constants, such as the shear viscosity coefficient, and the diffusion coefficient of the bulk fluid, are found as a function of the material properties of the constituents of the two-phase system and the volume concentration of solid. For transient heat conduction phenomena, it is found that the introduction of a complex conductivity for the bulk fluid permits a simple mathematical description of this otherwise complicated process. The rate of heat transfer between particle continuum and bulk fluid is also investigated by means of an Oseen-type approximation to the energy equation.

High pressure states in condensed matter: I. High pressure behavior of the iron-sulphur system with applications to the earth's core. II. Empirical equation of state for organic compounds at high pressures

Part I:

The earth's core is generally accepted to be composed primarily of iron, with an admixture of other elements. Because the outer core is observed not to transmit shear waves at seismic frequencies, it is known to be liquid or primarily liquid. A new equation of state is presented for liquid iron, in the form of parameters for the 4th order Birch-Murnaghan and Mie-Grüneisen equations of state. The parameters were constrained by a set of values for numerous properties compiled from the literature. A detailed theoretical model is used to constrain the P-T behavior of the heat capacity, based on recent advances in the understanding of the interatomic potentials for transition metals. At the reference pressure of 10⁵ Pa and temperature of 1811 K (the normal melting point of Fe), the parameters are: ρ = 7037 kg/m³, K_S0 = 110 GPa, K_S' = 4.53, K_S" = -.0337 GPa-1, and γ = 2.8, with γ α ρ^-1.17. Comparison of the properties predicted by this model with the earth model PREM indicates that the outer core is 8 to 10 % less dense than pure liquid Fe at the same conditions. The inner core is also found to be 3 to 5% less dense than pure liquid Fe, supporting the idea of a partially molten inner core. The density deficit of the outer core implies that the elements dissolved in the liquid Fe are predominantly of lower atomic weight than Fe. Of the candidate light elements favored by researchers, only sulfur readily dissolves into Fe at low pressure, which means that this element was almost certainly concentrated in the core at early times. New melting data are presented for FeS and FeS₂ which indicate that the FeS₂ is the S-hearing liquidus solid phase at inner core pressures. Consideration of the requirement that the inner core boundary be observable by seismological means and the freezing behavior of solutions leads to the possibility that the outer core may contain a significant fraction of solid material. It is found that convection in the outer core is not hindered if the solid particles are entrained in the fluid flow. This model for a core of Fe and S admits temperatures in the range 3450K to 4200K at the top of the core. An all liquid Fe-S outer core would require a temperature of about 4900 K at the top of the core.

Part II.

The abundance of uses for organic compounds in the modern world results in many applications in which these materials are subjected to high pressures. This leads to the desire to be able to describe the behavior of these materials under such conditions. Unfortunately, the number of compounds is much greater than the number of experimental data available for many of the important properties. In the past, one approach that has worked well is the calculation of appropriate properties by summing the contributions from the organic functional groups making up molecules of the compounds in question. A new set of group contributions for the molar volume, volume thermal expansivity, heat capacity, and the Rao function is presented for functional groups containing C, H, and O. This set is, in most cases, limited in application to low molecular liquids. A new technique for the calculation of the pressure derivative of the bulk modulus is also presented. Comparison with data indicates that the presented technique works very well for most low molecular hydrocarbon liquids and somewhat less well for oxygen-bearing compounds. A similar comparison of previous results for polymers indicates that the existing tabulations of group contributions for this class of materials is in need of revision. There is also evidence that the Rao function contributions for polymers and low molecular compounds are somewhat different.

On the added mass of sphere in a circular cylinder considering real fluid effects

An experimental method combined with boundary layer theory is given for evaluating the added mass of a sphere moving along the axis of a circular cylinder filled with water or oil. The real fluid effects are separated from ideal fluid effects.

The experimental method consists essentially of a magnetic steel sphere propelled from rest by an electromagnetic coil in which the current is accurately controlled so that it only supplies force for a short time interval which is within the laminar flow regime of the fluid. The motion of the sphere as a function of time is recorded on single frame photographs using a short-arc multiple flash lamp with accurately controlled time intervals between flashes.

A concept of the effect of boundary layer displacement on the fluid flow around a sphere is introduced to evaluate the real fluid effects on the added mass. Surprisingly accurate agreement between experiment and theory is achieved.

Topics in vortex motion

Six topics in incompressible, inviscid fluid flow involving vortex motion are presented. The stability of the unsteady flow field due to the vortex filament expanding under the influence of an axial compression is examined in the first chapter as a possible model of the vortex bursting observed in aircraft contrails. The filament with a stagnant core is found to be unstable to axisymmetric disturbances. For initial disturbances with the form of axisymmetric Kelvin waves, the filament with a uniformly rotating core is neutrally stable, but the compression causes the disturbance to undergo a rapid increase in amplitude. The time at which the increase occurs is, however, later than the observed bursting times, indicating the bursting phenomenon is not caused by this type of instability.

In the second and third chapters the stability of a steady vortex filament deformed by two-dimensional strain and shear flows, respectively, is examined. The steady deformations are in the plane of the vortex cross-section. Disturbances which deform the filament centerline into a wave which does not propagate along the filament are shown to be unstable and a method is described to calculate the wave number and corresponding growth rate of the amplified waves for a general distribution of vorticity in the vortex core.

In Chapter Four exact solutions are constructed for two-dimensional potential flow over a wing with a free ideal vortex standing over the wing. The loci of positions of the free vortex are found and the lift is calculated. It is found that the lift on the wing can be significantly increased by the free vortex.

The two-dimensional trajectories of an ideal vortex pair near an orifice are calculated in Chapter Five. Three geometries are examined, and the criteria for the vortices to travel away from the orifice are determined.

Finally, Chapter Six reproduces completely the paper, "Structure of a linear array of hollow vortices of finite cross-section," co-authored with G. R. Baker and P. G. Saffman. Free streamline theory is employed to construct an exact steady solution for a linear array of hollow, or stagnant cored vortices. If each vortex has area A and the separation is L, then there are two possible shapes if A^(1/2)/L is less than 0.38 and none if it is larger. The stability of the shapes to two-dimensional, periodic and symmetric disturbances is considered for hollow vortices. The more deformed of the two possible shapes is found to be unstable, while the less deformed shape is stable.

Particle fluid mechanics in shear flows, acoustic waves, and shock waves

Three different categories of flow problems of a fluid containing small particles are being considered here. They are: (i) a fluid containing small, non-reacting particles (Parts I and II); (ii) a fluid containing reacting particles (Parts III and IV); and (iii) a fluid containing particles of two distinct sizes with collisions between two groups of particles (Part V).

Part I

A numerical solution is obtained for a fluid containing small particles flowing over an infinite disc rotating at a constant angular velocity. It is a boundary layer type flow, and the boundary layer thickness for the mixture is estimated. For large Reynolds number, the solution suggests the boundary layer approximation of a fluid-particle mixture by assuming W = W_p. The error introduced is consistent with the Prandtl’s boundary layer approximation. Outside the boundary layer, the flow field has to satisfy the “inviscid equation” in which the viscous stress terms are absent while the drag force between the particle cloud and the fluid is still important. Increase of particle concentration reduces the boundary layer thickness and the amount of mixture being transported outwardly is reduced. A new parameter, β = 1/Ω τ_v, is introduced which is also proportional to μ. The secondary flow of the particle cloud depends very much on β. For small values of β, the particle cloud velocity attains its maximum value on the surface of the disc, and for infinitely large values of β, both the radial and axial particle velocity components vanish on the surface of the disc.

Part II

The “inviscid” equation for a gas-particle mixture is linearized to describe the flow over a wavy wall. Corresponding to the Prandtl-Glauert equation for pure gas, a fourth order partial differential equation in terms of the velocity potential ϕ is obtained for the mixture. The solution is obtained for the flow over a periodic wavy wall. For equilibrium flows where λ_v and λ_T approach zero and frozen flows in which λ_v and λ_T become infinitely large, the flow problem is basically similar to that obtained by Ackeret for a pure gas. For finite values of λ_v and λ_T, all quantities except v are not in phase with the wavy wall. Thus the drag coefficient C_D is present even in the subsonic case, and similarly, all quantities decay exponentially for supersonic flows. The phase shift and the attenuation factor increase for increasing particle concentration.

Part III

Using the boundary layer approximation, the initial development of the combustion zone between the laminar mixing of two parallel streams of oxidizing agent and small, solid, combustible particles suspended in an inert gas is investigated. For the special case when the two streams are moving at the same speed, a Green’s function exists for the differential equations describing first order gas temperature and oxidizer concentration. Solutions in terms of error functions and exponential integrals are obtained. Reactions occur within a relatively thin region of the order of λ_D. Thus, it seems advantageous in the general study of two-dimensional laminar flame problems to introduce a chemical boundary layer of thickness λ_D within which reactions take place. Outside this chemical boundary layer, the flow field corresponds to the ordinary fluid dynamics without chemical reaction.

Part IV

The shock wave structure in a condensing medium of small liquid droplets suspended in a homogeneous gas-vapor mixture consists of the conventional compressive wave followed by a relaxation region in which the particle cloud and gas mixture attain momentum and thermal equilibrium. Immediately following the compressive wave, the partial pressure corresponding to the vapor concentration in the gas mixture is higher than the vapor pressure of the liquid droplets and condensation sets in. Farther downstream of the shock, evaporation appears when the particle temperature is raised by the hot surrounding gas mixture. The thickness of the condensation region depends very much on the latent heat. For relatively high latent heat, the condensation zone is small compared with Ʌ_D.

For solid particles suspended initially in an inert gas, the relaxation zone immediately following the compression wave consists of a region where the particle temperature is first being raised to its melting point. When the particles are totally melted as the particle temperature is further increased, evaporation of the particles also plays a role.

The equilibrium condition downstream of the shock can be calculated and is independent of the model of the particle-gas mixture interaction.

Part V

For a gas containing particles of two distinct sizes and satisfying certain conditions, momentum transfer due to collisions between the two groups of particles can be taken into consideration using the classical elastic spherical ball model. Both in the relatively simple problem of normal shock wave and the perturbation solutions for the nozzle flow, the transfer of momentum due to collisions which decreases the velocity difference between the two groups of particles is clearly demonstrated. The difference in temperature as compared with the collisionless case is quite negligible.

I. The effect of droplet solidification upon two-phase flow in a rocket nozzle. II. A kinetic theory investigation of some condensation-evaporation phenomena by a moment method

Part I:

The perturbation technique developed by Rannie and Marble is used to study the effect of droplet solidification upon two-phase flow in a rocket nozzle. It is shown that under certain conditions an equilibrium flow exists, where the gas and particle phases have the same velocity and temperature at each section of the nozzle. The flow is divided into three regions: the first region, where the particles are all in the form of liquid droplets; a second region, over which the droplets solidify at constant freezing temperature; and a third region, where the particles are all solid. By a perturbation about the equilibrium flow, a solution is obtained for small particle slip velocities using the Stokes drag law and the corresponding approximation for heat transfer between the particle and gas phases. Singular perturbation procedure is required to handle the problem at points where solidification first starts and where it is complete. The effects of solidification are noticeable.

Part II:

When a liquid surface, in contact with only its pure vapor, is not in the thermodynamic equilibrium with it, a net condensation or evaporation of fluid occurs. This phenomenon is studied from a kinetic theory viewpoint by means of moment method developed by Lees. The evaporation-condensation rate is calculated for a spherical droplet and for a liquid sheet, when the temperatures and pressures are not too far removed from their equilibrium values. The solutions are valid for the whole range of Knudsen numbers from the free molecule to the continuum limit. In the continuum limit, the mass flux rate is proportional to the pressure difference alone.

Friction and heat transfer coefficients in smooth and rough pipes with dilute polymer solutions

Measurements of friction and heat transfer coefficients were obtained with dilute polymer solutions flowing through electrically heated smooth and rough tubes. The polymer used was "Polyox WSR-301", and tests were performed at concentrations of 10 and 50 parts per million. The rough tubes contained a close-packed, granular type of surface with roughness-height-to-diameter ratios of 0.0138 and 0.0488 respectively. A Prandtl number range of 4.38 to 10.3 was investigated which was obtained by adjusting the bulk temperature of the solution. The Reynolds numbers in the experiments were varied from =10,000 (Pr= 10.3) to 250,000 (Pr= 4.38).

Friction reductions as high as 73% in smooth tubes and 83% in rough tubes were observed, accompanied by an even more drastic heat transfer reduction (as high as 84% in smooth tubes and 93% in rough tubes). The heat transfer coefficients with Polyox can be lower for a rough tube than for a smooth one.

The similarity rules previously developed for heat transfer with a Newtonian fluid were extended to dilute polymer solution pipe flows. A velocity profile similar to the one proposed by Deissler was taken as a model to interpret the friction and heat transfer data in smooth tubes. It was found that the observed results could be explained by assuming that the turbulent diffusivities are reduced in smooth tubes in the vicinity of the wall, which brings about a thickening of the viscous layer. A possible mechanism describing the effect of the polymer additive on rough pipe flow is also discussed.