Radial Equilibrium Research Articles

The Simulation of Axial Compressor Performance Using an Annulus Wall Boundary Layer Theory

This paper summarizes the development of a computer program to simulate axial compressor performance. The program incorporates a well-established technique for cascade performance prediction and a modified radial equilibrium method of calculating the mainstream axial velocity distribution. The program’s most important feature is a new theory of annulus wall boundary layers which predicts annulus boundary layer development and losses. The empirical input to the present annulus wall boundary layer model is very restricted and involves well defined quantities: shape factor, skin friction, and leakage coefficients. Special provision is made for cases where the annulus wall boundary layers are merged; this aspect needs improvement however. Theoretically derivable losses due to the annulus wall, in combination with cascade losses, yield overall compressor efficiency. In the interest of being abstemious with empiricism, no attempt has been made to introduce high Mach number cascade loss corrections at this time and the values of the empirical parameters in the boundary layer theory are held fixed. Considering the very restricted empirical content of the model and the absence of adjustable parameters, the current predictions of compressor performance are quite good.

Radial equilibrium adsorption dynamics for the rectangular isotherm

The problem of radial equilibrium adsorption dynamics, taking into account longitudinal diffusion, was discussed for highly convex (rectangular) isotherms. A rule was obtained for movement of the adsorption wave front in a cylindrical adsorber. The time of the adsorber protective action was estimated.

Some remarks on the string problem treated by Singh and Demin

The problem of the motion of a string attached to a satellite on a circular orbit, as treated by Singh and Demin, is reconsidered. In their paper they discuss problems of uniqueness and stability. In particular the radial equilibrium positions were found to be unstable in a certain sense. In the present paper it is shown that: (i) with the stability definition used by Singh and Demin the equilibrium of a string hanging in a uniform gravity field would also be unstable; (ii) a definition of stability more appropriate for continuous systems would establish the stability of the string both in orbit and in a uniform gravity field.

The formation of plasma equilibrium in a Q-machine with radial potential discontinuities

A potassium-ion plasma forms in a Q-machine with mirror magnetic fields at the hotplates. Discontinuity in the thermoelectric potential at the hotplate edge produces a strong localized radial electric field that both electrostatically confines the ions and creates a high-velocity rotating annulus. Neglecting time-dependent losses, the radial equilibrium in the collisional regime with the column radius comparable to the ion gyro radius is adequately described by an isothermal single-fluid model, including both ion viscosity and rotational inertia. The radial density profiles are consistent with classical diffusion at the column edge.

Rotor Equilibria of Non-Neutral Plasmas

Equilibria are considered within a Vlasov-Maxwell model for an electron gas column rotating about an axis parallel to a uniform external magnetic field B0. A condition is derived which restricts the values of ωR (angular speed of mean rotation), Ωc (electron cyclotron frequency), and ωp (electron plasma frequency on axis) in order that the general class of rigid-rotor distributions f0(H⊥ − ωRL, pz) corresponds to radially confined equilibria in the nonrelativistic, nondiamagnetic case. It is demonstrated that rigid-rotor distributions have the property that a specification of the radial equilibrium density profile n0(r) uniquely determines the distribution ∫ dpzf0(H⊥ − ωRL, pz) of electrons perpendicular to B0. Explicit examples in this regard are discussed in both the nonrelativistic, and relativistic, diamagnetic cases. This can serve as a useful experimental diagnostic whereby a determination of n0(r) yields the perpendicular distribution ∫ dpzf0(H⊥ − ωRL, pz).

Paper 18: Large Compressors: Some Aspects of Applications and Design

The choice between axial and centrifugal compressors is briefly discussed, and cost comparison applicable over a limited range is given. The present design trends and future prospects of both types of machine are considered. Methods of radial equilibrium calculations and compressor characteristic predictions are given as examples of the design tools which are available for effecting improvements in axial compressor aerodynamics, and in the evaluation of performance.

Elastic expulsion from a long tube

The paper presents a theoretical analysis of elastic expulsion from a long pressurized tube following instantaneous severance. The governing equations of motion and continuity are used, together with an equation representing radial equilibrium of the tube wall, to construct a differential equation for the inner radius of the tube as a function of axial position and time. Similarity solutions for this equation are presented for a Newtonian fluid. The effect of the relaxed/fully-extended radius ratio on wall dilatation and expulsion rate is discussed.

Hydromagnetic Stability of Thin Self-Gravitating Disks and Spiral Structure

The hydromagnetio stability of infinitesimally thin, ourrent-oarrying, differentially rotating disks is oonsidered. The perturbation theory desoribed is first order (linear) throughout. It is shown that azimuthal eleotrio ourrent produoes instability for all (radial) wavelengths. The oase of radial equilibrium eleotrio ourrent reduoes to the non-hydromagnetio problem treated by Lin and Shu (1964), although hydromagnetio effeots are expeoted for a. disk of finite thiokness or to second order due to azimuthal perturbation ourrent.

On the Approximated Expression of the Axial Perturbation Velocities in the Actuator Disc Theory

Fluid motion in an annulus is described, in the absence of radial velocity, by the simple radial equilibrium equation. This equation is used to describe the flow in an axial turbomachine some distance ahead and behind the rows of blades, where radial displacements have already decayed.In the neighbourhood of and between the blades the fluid motion is far more complex than the flow in the radial equilibrium condition. Most of the difficulties of the real problem can be overcome by introducing the actuator disc mathematical model. The actuator disc divides the complex flow field into two simpler fields, each free from blades, i.e. a simple annulus. The equations of motion in each field are easily linearised by posing some particular restrictions concerning the velocity profiles and by conceiving the velocity as the sum of the radial equilibrium solution and of small perturbed quantities. These are radial and axial perturbation velocities vanishing far from the actuator disc, and are given by a perturbation velocity potential.

Differences in Normal Stress in Aqueous Solutions of Turbulent Drag Reducing Additives

WHEN a fluid is in motion the normal stresses acting on the faces of a small fluid element which is instantaneously in the form of a cube are not in general equal. For example, in a Newtonian liquid the tensile stresses τxx, τyy, τzz acting on the cube faces perpendicular to orthogonal axes x, y, z are given by where p is pressure, μ is viscosity, and u, v w are the velocity components in the x, y, z directions. In the special case of flow through a long straight tube of uniform cross-section, however, u does not vary with x, and v and w are zero, so that then τxx = τyy = τzz = −p for a Newtonian liquid. For non-Newtonian liquids, on the other hand, such an equality may no longer obtain, even for this restricted type of flow. Thus for axial flow in the annular passage between two concentric tubes, of inner arid outer radius r1 and r2, respectively, the radial normal stress τrr may differ from the circumferential stress τθθ. For radial equilibrium or Thus measurements of the pressure difference can give an indication of the average magnitude of the normal stress difference τrr − τθθ.

Internal Aerodynamics of Turbo-Machines

In this paper it is shown how principles which are fundamental to external aerodynamics have been applied to study the internal aerodynamics of turbo-machines. Mention is made of aerofoil cascade, radial equilibrium, and actuator disc theories. Results are also presented of a number of supporting experimental investigations.

Research on the Axisymmetric Flow through an Axial-Flow Turbo-Machine : 2nd Report, Axisymmetric Flow at Design Point in Multistage Turbo-Machines of Various Vortex Types

In this report, the author has treated the axisymmetric flow at design point in multistage turbo-machines of various vortex types, such as the constant reaction type and the exponential type, using a similar method to that described in 1st Report. Regarding the axisymmetric flow through an axial-flow turbo-machine, the radial equilibrium theory and the actuator disc theory have been expounded, but the effects of impeller breadth and blade shape have not been considered in these theories. The method described in the present paper takes these effects into account. Numerical examples are shown comparing the results calculated on the basis of the above mentioned theories.

Prediction of Axial-Flow Turbomachine Performance by Blade-Element Methods

The blade-element method has been used effectively in the solution of the design problem for axial-flow turbomachines. A considerable body of data has been acquired to support the use of this system. The method has been applied in the investigation reported in this paper to the performance-prediction problem for blade rows in incompressible and compressible flows. Radial distributions of velocity and fluid properties were determined by numerical solution of the radial equilibrium condition, with radial entropy gradients included by using blade-element losses estimated from available experimental information. The effect of a loss gradient on the predicted performance of a typical axial-flow pump configuration is indicated and, for the case of an axial-flow compressor rotor operating in air, performance predicted is compared with experimentally measured performance.

Research on the Axisymmetric Flow through an Axial-Flow Turbo-Machine : 1st Report, On the Flow through a Free Vortex Type Impeller

Regarding the axisymmetric flow through an axial-flow turbo-machine, the radial equilibrium theory and the actuator disc theory have been expounded, in which the effects of impeller breadth and blade shape have not been considered. In this paper a method is presented, in which the above mentioned disadvantages are eliminated. To the axisymmetric flow through a turbo-machine of the free vortex type, the author applies the fundamental partial-differential equations derived from the equations given by Lorenz and Bauersfeld, and the following cases are treated. (1) The flow through an impeller at the off-design point is treated on the assumption that the flow in front of the impeller is a potential one, and numerical examples are shown. (2) The flow through an impeller at the design point is treated on the assumption that the flow in front of the impeller is an inviscid shear flow, and it is shown that the flow in the rear of the impeller has a more uniform axial-velocity distribution.

Research on the Axisymmetric Flow through an Axial-Flow Turbo-Machine : 1st Report, On the Flow through a Free Vortex Type Impeller

Regarding the axisymmetric flow through an axial-flow turbo-machine, the radial equilibrium theory and the actuator disc theory have been expounded, in which the effects of impeller breadth and blade shape have not been considered. In this paper a method is presented, in which the above mentioned disadvantages are eliminated. To the axisymmetric flow through a turbo-machine of the free vortex type, the author applies the fundamental partial-differential equations derived from the equations given by Lorenz and Bauersfeld, and the following cases are treated. (1) The flow through an impeller at the off-design point is treated on the assumption that the flow in front of the impeller is potential one, and numerical examples are shown. (2) The flow through an impeller at the design point is treated on the assumption that the flow in front of the impeller is inviscid shear flow, and it is shown that the flow in the rear of the impeller has a more uniform axial-velocity distribution.

Reconsideration of the Annulus Flow Problem in Axial Turbomachinery

The theory pertaining to the annulus flow problem in axial turbomachinery is reviewed and a simplified analysis which is essentially concerned with the effect of streamline curvature is worked out. Some generalized results are obtained for the case of a single blade row and for the oscillating flow between repeating blade rows of arbitrary spacing. The effect of finite blade width is assessed. The simple radial equilibrium hypothesis which is still attractive to designers is reconsidered and a modified radial equilibrium theory which directly eliminates the oscillatory terms is proposed as being more accurate and simpler than the conventional radial equilibrium theory.

Three-Dimensional Design of Multi-Stage Axial-Flow Compressors

The design of multi-stage axial-flow compressors, particularly where the blade rows are closely spaced, presents a problem to the designers owing to the mutual interference of the separate blade rows. The actuator disc approximation, applied to the design of single stages at a time has been shown to be useful, and this technique can be used for the multistage machine if the mutual interference of the separate stages may be neglected. Carmichael and Horlock (1)† have carried out calculations by including only the mutual interference of adjacent rows (see also Horlock (2) for an introduction to actuator disc theory). Using the theory of simple radial equilibrium of the flow at the trailing edges of the blade rows, it has been demonstrated, Howell (3), that the flow tends to a state which is termed the ‘ultimate steady flow’ after about two stages. The design of the third stage on a machine may be attempted, making use of this assumption, treating this stage as if it were flanked by many stages. The first part of the paper deals with an extremely simple method of design, for the ultimate steady flow, using the actuator disc theory, and a particular example of this method is compared with a more exact solution of the equations, based on a modified radial equilibrium approach. Both results are then compared with a set of experimental results taken from an experimental three-stage axial-flow compressor in the Department of Mechanical Engineering at the University of Birmingham. In the second part of the paper, the validity of the above assumptions regarding the ultimate steady flow state is discussed, along with the problem of the ‘entry stages’. These problems may be resolved by an exact solution for the flow through a series of identical stages in which parameters such as number of stages, blade row spacing and blade row axial length may be systematically varied. Such an approach is described, and the results of a series of calculations are presented. It is shown that the overall behaviour of the entry stages depends upon the ratio of blade row axial length to blade row spacing. Thus, for small values of this ratio, the flow tends slowly to the ultimate state, but for small axial clearance the ultimate state is reached virtually by the third stage. The influence on the entry stages of blade row spacing and of number of stages is also discussed. It is shown that the treatment of the practical multi-stage machine, even for closely spaced blading, reduces to the separate treatment of the first stage only (inlet guide vanes plus one rotor and one stator), plus the ultimate steady flow treatment to obtain the behaviour of the third stage; otherwise the exact approach may be applied to the first three stages taken as a group, although this involves more calculation. On a high-speed digital computer, such as the Ferranti ‘Mercury’ machine which was, in fact, used for the calculations, a 3-stage compressor design would take about 15 minutes.

Closure to “Discussions of ‘Radial Equilibrium in Supersonic Compressors’” (1959, ASME J. Basic Eng., 81, pp. 557–558)

Closure to “Discussions of ‘Radial Equilibrium in Supersonic Compressors’” (1959, ASME J. Basic Eng., 81, pp. 557–558)

Radial Equilibrium in Supersonic Compressors

The radial equilibrium of flow through nearly radial shock waves in supersonic compressors is considered. The most restricted useful cases for which equilibrium is possible are a normal shock in a rotor, an oblique shock in a stator, or a normal shock with variable total head in a stator. The equilibrium requirement restricts the range of solutions but many useful ones are still possible.

29—THE MECHANICS OF TWISTED YARNS: THE INFLUENCE OF TRANSVERSE FORCES ON TENSILE BEHAVIOUR

An idealized theoretical model of a yarn is assumed, and, after considering the stresses acting on a small element of the yarn, the conditions for radial equilibrium in the yarn are worked out. This leads to expressions for the tensile and transverse stresses in the fibres as functions of (i) the position in the yarn, (ii) the stress which would be present in fibres under the same extension as the yarn, (iii) the twist angle of the yarn, and (iv) the Poisson's ratio of the fibres. Integration of the components of these stresses parallel to the yarn axis gives an expression for yarn tension. Finally, the behaviour when fibres start to break is considered.