Bashforth

The general case has been solved by Bashforth and Adams [14], using an iterative method, and extended by Sugden [15], Lane [16], and Paddy [17]. See also Refs. 11 and 12. In the case of a figure of revolution, the two radii of curvature must be equal at the apex (i.e., at the bottom of the meniscus in the case of capillary rise). If this radius of curvature is denoted by b, and the elevation of a general point on the surface is denoted by z, where z = y - h, then Eq. II-7 can be written... 

As in the case of capillary rise, Sugden [27] has made use of Bashforth s and Adams tables to calculate correction factors for this method. Because the figure is again one of revolution, the equation h = a lb + z is exact, where b is the value of / i = R2 at the origin and z is the distance of OC. The equation simply states that AP, expressed as height of a column of liquid, equals the sum of the hydrostatic head and the pressure... 

F. Bashforth and J. C. Adams, An Attempt to Test the Theories of Capillary Action, University Press, Cambridge, England, 1883. 

A. K. Biswas and G. R. Bashforth, The Physical Chemistry of Metallurgical Processes, Chapman, London,... 

Bl. Bashforth, F., and Adams, H., Capillary Action. Cambridge Univ. Press, London and New York, 1883. 

Recently Bashforth et al. (B15) have made a study of the special region in which the transition from countercurrent to cocurrent liquid flow occurs with vertical upward gas flow. At this point, the liquid film is suspended, and the net liquid flow rate is zero corresponding to the limiting flooding case for wetted wall columns. 

For intermediate values of rja, or for tubes of intermediate size, no general formula has been given. Bashforth and Adams have however published tables from which the form of any capillary surface may be calculated, and with the aid of these Sugden has further calculated a table of values of rjh for all values of rja, between 0 and 6. h is here the radius of curvature at the crown... 

In Equation (83) R is given by Equation (36). Expression (83) is known as the Bashforth-Adams equation. It is conventional to express this equation in dimensionless form by expressing all distances relative to the radius at the apex b ... 

The Bashforth-Adams equation —the composite of Equations (36) and (84) —is a differential equation that may be solved numerically with /5 and 4> as parameters. Bashforth and Adams solved this equation for a large number of values between 0.125 and 100 by compiling values of x/b and z/b for 0° < 4> < 180°. Their tabular results, calculated by hand before the days of computers, were published in 1883. Other workers subsequently extended these... 

The Bashforth-Adams tables provide an alternate way of evaluating 7 by observing the profile of a sessile drop of the liquid under investigation. If, after all, the drop profiles of Figure 6.15 can be drawn using 0 as a parameter, then it should also be possible to match an experimental drop profile with the (3 value that characterizes it. Equation (85) then relates 7 to 0 and other measurable quantities. This method is claimed to have an error of only 0.1%, but it is slow and tedious and hence not often the method of choice in practice. 

Once (3 is known for a particular profile, the Bashforth-Adams tables may be used further to evaluate b ... 

Several additional points might be noted about the use of the Bashforth-Adams tables to evaluate 7. If interpolation is necessary to arrive at the proper (3 value, then interpolation will also be necessary to determine (x/bl. . This results in some loss of accuracy. With pendant drops or sessile bubbles (i.e., negative /3 values), it is difficult to measure the maximum radius since the curvature is least along the equator of such drops (see Figure 6.15b). The Bashforth-Adams tables have been rearranged to facilitate their use for pendant drops. The interested reader will find tables adapted for pendant drops in the material by Padday (1969). The pendant drop method utilizes an equilibrium drop attached to a support and should not be confused with the drop weight method, which involves drop detachment. 

Several other methods for determining 7 —notably, the maximum bubble pressure, the drop weight, and the DuNouy ring methods (see Section 6.2) —all involve measurements on surfaces with axial symmetry. Although the Bashforth-Adams tables are pertinent to all of these, the data are generally tabulated in more practical forms that deemphasize the surface profile. 

What is the Bashforth-Adams equation How would you use it to determine the surface tension of a liquid and the contact angle of a liquid with a surface ... 

BASF see Badische aniline-und sodafabrik 2 B4 Bashforth chronograph see Chronographs 3 C308 Basic cupric azide 1 A533 Basic lead acetate 1 A28 Basic lead azide 1 A555... 

An instrument invented in England ca 1866 by Bashforth, was probably more accurate than... 

Francis Bashforth, English astronomer, mathematician, and ballastician, 1819-1912 Forest Ray Moulton, US astronomer, 1872-1951 14Rudolf Lipschitz, German mathematician, 1832-1903... 

There are several methods that we can use to increase the order of approximation of the integral in eqn. (8.72). Two of the most common higher order explicit methods are the Adams-Bashforth (AB2) and the Runge-Kutta of second and fourth order. The Adams-Bashforth is a second order method that uses a combination of the past value of the function, as in the explicit method depicted in Fig. 8.19, and an average of the past two values, similar to the Crank-Nicholson method depicted in Fig. 8.21, and written as... 

At z=0 AP = 2y/b.) Defining /3=Apgb2/y° and substituting into the Young-Laplace equation yields one form of the Bashforth-Adams equation ... 

One can measure the maximum pressure that can be applied to a gas bubble at the end of a vertical capillary, of radius r and depth t in a liquid, before it breaks away (Figure 3.13) [143], Before break-away the bubble has the shape of a sessile drop and is described by the equation of Bashforth and Adams. The pressure in the tube is the sum of the hydrostatic pressure (Apgt) and the pressure due to surface tension. Equations have been published which allow calculation of surface tension using Bashforth-Adams and density and depth data. 

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