The theory presented here describes the motion of a large gas bubble rising through upward-flowing liquid in a tube. The basis of the theory is that the liquid motion round the bubble is inviscid, with an initial distribution of vorticity which depends on the velocity profile in the liquid above the bubble. Approximate solutions are given for both laminar and turbulent velocity profiles and have the form \begin{equation} U_s = U_c+(gD)^{\frac{1}{2}}\phi(U_c/(gD)^{\frac{1}{2}}), \end{equation}Us being the bubble velocity, Uc the liquid velocity at the tube axis, g the acceleration due to gravity, and D the tube diameter; ϕ indicates a functional relationship the form of which depends upon the shape of the velocity profile. With a turbulent velocity profile, a good approximation to (1) which is suitable for many practical purposes is \begin{equation} U_s = U_s + U_{s0}, \end{equation}Us0 being the bubble velocity in stagnant liquid. Published data for turbulent flow are known to agree with (2), so that in this case the theory supports a well-known empirical result. Our laminar flow experiments confirm the validity of (1) for low liquid velocities.
Using a method due to Davies & Taylor (1950), a simple model is employed to derive the velocity of a two-dimensional gas bubble rising in liquid along the axis of a channel of finite width. The asymptotes of the solution agree well with previous results and an experimental investigation confirms the effect of channel width on bubble velocity. Measured values of velocity are, however, approximately 9% higher than theoretical values due to the three-dimensional nature of the real flow.
The velocity of a large gas bubble rising along the axis of a cylindrical container filled with liquid, is derived to a first approximation from a simple flow model of the system. The resulting expressions involve ratios of certain infinite series, which are evaluated for a few cases, and the theory is then seen to agree with previously accepted results at its asymptotes. Experiments are performed which confirm the first approximation and which allow the theory to be recast as a semi-empirical theory relating bubble volume with velocity. In this form it agrees with the results of Uno & Kintner (1956), but indicates that the relation between volume and curvature which they employed was in error.
There has been a long history of the use of two electromagnetic techniques to measure surface-breaking cracks in metals. Both the alternating current potential drop (ACPD) technique and the eddy current technique have given good agreement with experimental results, even though the theoretical models on which their interpretations are based use contrasting assumptions for the boundary condition on the metal surface. The model for the ACPD technique assumes that the magnetic scalar potential satisfies the 2D Laplace equation, while eddy current modeling assumes an approximation of Born type in which the surface field is unperturbed by the presence of the crack. This paper considers a general model matching the thin-skin electromagnetic field around a surface-breaking crack to that in the free space above and shows that the two contrasting boundary conditions are extremes of a more general one. The Laplace approximation is valid for high permeability materials such as mild steel, while the Born approximation is appropriate for materials of low permeability and high conductivity such as aluminum. Experimental investigations of the magnetic fields near semielliptical cracks in mild steel and aluminum show quantitative agreement with the theory.
An instrument designed to measure the a-c field accurately has been built and fully tested on steel, aluminium, and titanium. These field measurements can be interpreted in terms of crack size, which provides a new technique for nondestructive testing (NDT) that requires no prior calibration. This paper describes the basic electronic measuring system, theoretical derivations of the electrical-field distribution, and application to industrial problems such as crack measurement in threads, shafting, welded connections, etc.
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