Greater than unity

The formation volume factor for water (B, reservoir volume per stock tank volume), is close to unity (typically between 1.00 and 1.07 rb/stb, depending on amount of dissolved gas, and reservoir conditions), and is greater than unity due to the thermal contraction and evolution of gas from reservoir to stock tank conditions. 

Reactions in porous catalyst pellets are Invariably accompanied by thermal effects associated with the heat of reaction. Particularly In the case of exothermic reactions these may have a marked influence on the solutions, and hence on the effectiveness factor, leading to effectiveness factors greater than unity and, In certain circumstances, multiple steady state solutions with given boundary conditions [78]. These phenomena have attracted a great deal of interest and attention in recent years, and an excellent account of our present state of knowledge has been given by Arls [45]. 

This dissipative force is proportional to the relative velocity of the two beads and acts so as tc reduce their relative momentum, v is tire difference between the two velocities (Vy = v, — v ) and vP rjj) is a weight function that depends upon the distemce and disappears for interbead distances greater than unity (i.e. r ). 

Nitro compounds, when hquid, have characteristic odours, are insoluble in water, highly refractive and with a density greater than unity. Many are crystalline sohds. Most nitro compounds are slightly coloured, generally yellow the intensity of the colour increases with the number of nitro groups. The following reactions will assist in their detection. 

In all other cases the observed result will depend upon both the speed of mixing and the speed of nitration. The relative rate will be greater than unity by an amount peculiar to the conditions of the experiment. Again, if the alkylbenzene is sufficiently reactive to be nitrated upon encounter, whilst benzene is not, the relative rate will be greater than unity and, for the experimental conditions, will be a limiting upper value no matter what aromatic is used. 

It has been necessary to comment upon these various studies because Olah and his co-workers have suggested that whilst nitrations, like those with nitronium salts, which give a relative rate of reaction of toluene with respect to benzene not much greater than unity involve the nitronium ion as the electrophile, this is not so in other cases. It is important to consider these opinions closely. In the earlier of the two relevant papers it is agreed that since nitrations of toluene with nitronium tetrafluoroborate in sulpholan show no abnormal o -ratio there... 

A similar study of the nitration of 2,5-dichloro- and 2,5-dibromo-nitrobenxene under a variety of conditions has been made. At the very high acidities in oleum the o /)-ratio for nitration was less than unity. It increased with decreasing acidity of the medium and became greater than unity at roughly the acidity represented by 89-90 % sulphuric acid. The results were interpreted in terms of the interaction between the nitronium ion and the nitro group, but the results are complicated and the interpretation not compelling. 

Despite the considerable amount of work which has been reported, our knowledge of the nitration of biphenyl is not in a satisfactory state. The 0 p-T3.tw varies considerably with the conditions of nitration, and the cause of the variation is not fully understood. Nitrations with solutions prepared from nitric acid and acetic anhydride have generally given o -ratios greater than unity, the most consistent value being 2-2, obtained at o °C. The corresponding partial rate factors are reported later. 

This reaction is carried out under base-catalyzed conditions and with a formaldehyde/phenol ratio greater than unity. The resulting product is called a C state resin or resite. 

If Fj = F2, with both values greater than unity, an S-shaped curve passing through the point (0.5, 0.5) would also result, but in this case reflected across the 45° line compared to item (3). 

Although Table 7.1 is rather arbitrarily assembled, note that it contains no systems for which rj and r2 are both greater than unity. Indeed, such systems are very rare. We can understand this by recognizing that, at least in the extreme case of very large r s, these monomers would tend to simultaneously homopolymerize. Because of this preference toward homopolymerization, any copolymer that does form in systems with rj and r2 both greater than unity will... 

For the case of separating a binary mixture, the following conventions are used. The concentrations of the streams are specified by the mol fraction of the desired component. The purpose of the separation process is usually to obtain one component of the mixture in an enriched form. If both components are desired, the choice of the desired component is an arbitrary one. The upflowing stream from the separation stage is the one in which the desired component is enriched, and by virtue of this convention, a is defined as a quantity the value of which is greater than unity. However, for the processes considered here, a exceeds unity by only a very small fraction, and the relationship between the concentrations leaving the stage can be written, without appreciable error, in the form... 

The relative volatility, a, is a direct measure of the ease of separation by distillation. If a = 1, then component separation is impossible, because the hquid-and vapor-phase compositions are identical. Separation by distillation becomes easier as the value of the relative volatihty becomes increasingly greater than unity. Distillation separations having a values less than 1.2 ate relatively difficult those which have values above 2 are relatively easy. 

Ertl and DuUien [ibid.] found that Hildebrand s equation could not fit their data with B as a constant. They modified it by applying an empirical exponent n (a constant greater than unity) to the volumetric ratio. The new equation is not generally useful, however, since there is no means for predicting /i. The theory does identify the free volume as an important physical variable, since n > for most hquids implies that diffusion is more stronglv dependent on free volume than is viscosity. 

Component reliability will vary as a function of the power of a dimensional variable in a stress function. Powers of dimensional variables greater than unity magnify the effect. For example, the equation for the polar moment of area for a circular shaft varies as the fourth power of the diameter. Other similar cases liable to dimensional variation effects include the radius of gyration, cross-sectional area and moment of inertia properties. Such variations affect stability, deflection, strains and angular twists as well as stresses levels (Haugen, 1980). It can be seen that variations in tolerance may be of importance for critical components which need to be designed to a high reliability (Bury, 1974). 

As a complication some sources define a flow index as the reciprocal of that defined above so that some care has to be taken in interpretation. In such cases the values are greater than unity for polymer melts and the greater the value the greater the divergence from Newtonian behaviour.)... 

Thus as (y) will always be greater than unity, the resistance to mass transfer term in the mobile phase will be, at a minimum, about forty times greater than that in the stationary phase. Consequently, the contribution from the resistance to mass transfer in the stationary phase to the overall variance per unit length of the column, relative to that in the mobile phase, can be ignored. It is now possible to obtain a new expression for the optimum particle diameter (dp(opt)) by eliminating the resistance to mass transfer function for the liquid phase from equation (14). 

The benefit-to-cost (B/C) ratio is a benchmark that is determined by taking the total present value of all of the financial benefits of an air pollution control project and dividing it by the total present value of all the costs of the project. If the ratio is greater than unity, then the benefits outweigh the costs, and we may conclude that the project is economically worthwhile. 

The present values of the benefits and costs are kept separate, and expressed in one of two ways. First, there is the pure B/C ratio, which implies that if the ratio is greater than unity, the benefits outweigh the costs and the project is viable. Second, there is the net B/C ratio, which is the net benefit (benefits minus costs) divided by the costs. In this latter case, the decision criterion is that the benefits must outweigh the costs, which means that the net ratio must be greater than zero (if the benefits exactly equaled the costs, the net B/C ratio would be zero). In both cases, the highest B/C ratios are considered as the best projects. 

To begin with, we reeall that, for nonassoeiating fluids at T = = 1.35, Plisehke and Henderson [58] have observed a depletion of the fluid density at a hard wall, starting from the bulk fluid density p = 0.4 up to p = 0.65. For higher bulk fluid densities the eontaet values of the redueed density profile, p z)/p, beeome greater than unity and the profiles... 

Table 4-1 lists some rate constants for acid-base reactions. A very simple yet powerful generalization can be made For normal acids, proton transfer in the thermodynamically favored direction is diffusion controlled. Normal acids are predominantly oxygen and nitrogen acids carbon acids do not fit this pattern. The thermodynamicEilly favored direction is that in which the conventionally written equilibrium constant is greater than unity this is readily established from the pK of the conjugate acid. Approximate values of rate constants in both directions can thus be estimated by assuming a typical diffusion-limited value in the favored direction (most reasonably by inspection of experimental results for closely related... 

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