Points turning

The production phase commences with the first commercial quantities of hydrocarbons ( first oil ) flowing through the wellhead. This marks the turning point from a cash flow point of view, since from now on cash is generated and can be used to pay back the prior investments, or may be made available for new projects. Minimising the time between the start of an exploration campaign and first oil is one of the most important goals in any new venture. 

Figure Al.1.1. Wavefimctions for the four lowest states of the hamronie oseillator, ordered from the n = Q ground state (at the bottom) to tire u = 3 state (at the top). The vertieal displaeement of the plots is ehosen so that the loeation of the elassieal turning points are those that eoineide with the superimposed potential fimetion (dotted line). Note that the number of nodes in eaeh state eorresponds to the assoeiated quantum number.

Figure Al.1.2. Probability density (v[/ vt/) for the n = 29 state of the hamionic oscillator. The vertical state is chosen as in figure A1.1.1. so that the locations of the turning points comcide with the superimposed potential fiinction.

Of course the real projectile-surface interaction potential is not infinitely hard (cf figure A3,9,2. As E increases, the projectile can penetrate deeper into the surface, so that at its turning point (where it momentarily stops before reversing direction to return to the gas phase), an energetic projectile interacts with fewer surface atoms, thus making the effective cube mass smaller. Thus, we expect bE/E to increase with E (and also with W since the well accelerates the projectile towards the surface). 

The physical situation of interest m a scattering problem is pictured in figure A3.11.3. We assume that the initial particle velocity v is comcident with the z axis and that the particle starts at z = -co, witli x = b = impact parameter, andy = 0. In this case, L = pvh. Subsequently, the particle moves in the v, z plane in a trajectory that might be as pictured in figure A3.11.4 (liere shown for a hard sphere potential). There is a point of closest approach, i.e., r = (iimer turning point for r motions) where... 

To integrate this expression, we note that 0 starts at n when r = co, then it decreases while r decreases to its turning point, then r retraces back to oo while 0 continues to evolve back to n. The total change in 0 is then twice the integral... 

Figure B2.5.2. Schematic relaxation kinetics in a J-jump experiment, c measures the progress of the reaction, for example the concentration of a reaction product as a fiinction of time t (abscissa with a logaritlnnic time scale). The reaction starts at (q. (a) Simple relaxation kinetics with a single relaxation time, (b) Complex reaction mechanism with several relaxation times x.. The different relaxation times x. are given by the turning points of e as a fiinction of ln((). Adapted from [110].

Figure C3.5.4. Ensemble-averaged loss of energy from vibrationally excited I2 created by photodissociation and subsequent recombination in solid Kr, from 1811. The inset shows calculated transient absorjDtion (pump-probe) signals for inner turning points at 3.5, 3.4 or 3.3 A.

There are both bound and eontinuum solutions to the radial Sehrodinger equation for the attraetive eoulomb potential beeause, at energies below the asymptote the potential eonfmes the partiele between mO and an outer turning point, whereas at energies above the asymptote, the partiele is no longer eonfmed by an outer turning point (see the figure below). 

Figure 1.13 Plot of potential energy, V(r), against bond length, r, for the harmonic oscillator model for vibration is the equilibrium bond length. A few energy levels (for v = 0, 1, 2, 3 and 28) and the corresponding wave functions are shown A and B are the classical turning points on the wave function for w = 28...

Each point of intersection of an energy level with the curve corresponds to a classical turning point of a vibration where the velocity of the nuclei is zero and all the energy is in the form of potential energy. This is in contrast to the mid-point of each energy level where all the energy is kinetic energy. 

As V increases, the two points where j/l, the vibrational probability, has a maximum value occur nearer to the classical turning points. This is illustrated for u = 28 for which A and B are the classical turning points, in contrast to the situation for u = 0, for which the maximum probability is at the mid-point of the level. 

The wavelength of the ripples in ij/ increases away from the classical turning points. This is more apparent as v increases and is pronounced for u = 28. 

The classical turning point of a vibration, where nuclear velocities are zero, is replaced in quantum mechanics by a maximum, or minimum, in ij/ near to this turning point. As is illustrated in Figure 1.13 the larger is v the closer is the maximum, or minimum, in ij/ to the classical turning point. 

Figure 7.21 illustrates a particular case where the maximum of the v = 4 wave function near to the classical turning point is vertically above that of the v" = 0 wave function. The maximum contribution to the vibrational overlap integral is indicated by the solid line, but appreciable contributions extend to values of r within the dashed lines. Clearly, overlap integrals for A close to four are also appreciable and give an intensity distribution in the v" = 0 progression like that in Figure 7.22(b). 

Hydrocarbons and carbon monoxide emissions can be minimised by lean air/fuel mixtures (Fig. 2), but lean air/fuel mixtures maximize NO emissions. Very lean mixtures (>20 air/fuel) result in reduced CO and NO, but in increased HC emissions owing to unstable combustion. The turning point is known as the lean limit. Improvements in lean-bum engines extend the lean limit. Rich mixtures, which contain excess fuel and insufficient air, produce high HC and CO concentrations in the exhaust. Very rich mixtures are typically used for small air-cooled engines, needed because of the cooling effect of the gasoline as it vaporizes in the cylinder, where CO exhaust concentrations are 4 to 5% or more. 

Fan horsepower is obtained from Fig. 12-15. Connecting the point representing 100 percent of standard tower performance with the turning point and extending this straight line to the horsepower scale show that it will require 0.041 hp/fr of actual effective tower area. For a tower area of 1000 ft" 41.0 fan hp is required to perform the necessary cooling. 

Fig. 3. One-dimensional barrier along the coordinate of an exoergic reaction. Qi(E), Q i(E), QiiE), Q liE) are the turning points, coo and CO initial well and upside-down barrier frequencies, Vo the barrier height, — AE the reaction heat. Classically accessible regions are 1, 3, tunneling region 2.

Fig. 19. Cubic parabola potential. Turning points are shown. The dashed line indicates the stable potential with the same well frequency.

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