Rate logarithmic

Critical heat production rates (i.e., heat production rates that still do not lead to a runaway), are often determined by small scale experiments. However, the effect of scale-up on these rates, as discussed in [161], must be taken into account. An indication of the effect of scaling in an unstirred system is shown in Figure 3.2. In this figure, the heat production rate (logarithmic scale) is shown as a function of the reciprocal temperature. Point A in the figure represents critical conditions (equivalent heat generation and heat removal) obtained in a 200 cm3 Dewar vessel set-up. It can be calculated from the Frank-Kamenetskii theory on heat accumulation [157, 162] that the critical conditions are lowered by a factor of about 12 for a 200 liter insulated drum. These conditions are represented by... 

Styrene. Figure 6 shows the over-all reaction rate (logarithmic) plotted vs. time in the case of the y-EP of styrene. Ubf in the beginning goes up steeply after about 40 minutes (at a dose rate of 200 rad per hour) it becomes constant for some time a period of zero order with respect to monomer concentration. After about 140 minutes and 40 to 50% conversion, v j. decreases linearly a period of first order with respect to monomer concentration. 

Fig. 14. Variations in the rate logarithm (10J for isotopic oxygen exchange vs variations in the electron work function (J in ev). Exchange on CuO additions made at 412°.

In these systems of equations, jci represents coded values of strain rate logarithm log V, X2 is the temperature in Kelvins, JC3 is the lo/W ratio, JC4 is the IJ W... 

Fig. 6.7 a Creep curve at a temperature T = 1412 °C and under a resolved shear stress <7 = 5 kg/mm. b Creep rate (logarithmic scale) versus 1/kT, for the constant resolved shear stress <7 = 5 kg/mm. The slope gives the creep activation energy [45]. With kind permission, Permissions Dept., EDP Sciences by Dr. Corinne Griffon and Professor Escaig... 

As can be noted, the interaction energy remains positive at short separations and tends to infinity at a much slower rate (logarithmically) in comparison to the linear model. It ean be easily estimated that for h 0.01, the differenees between the linear and nonlinear models increase to an order of magnitude. It is interesting to mention, however, that in the ease of the e.p. boundary conditions, the asymptotic expression for the interaetion energy at short separations remains the same for the linear and nonlinear models provided that CTj a°. On the other hand, for larger separations, the expressions for the interaetion energy reduee to the same asymptotic form for the two boundary conditions... 

Figure 8.32 Stress (logarithmic scale) versus steady-state creep rate (logarithmic scale) for an S-590 alloy at four temperatures. (Reprinted with permission of ASM International. All rights reserved, www.asminternational.org)...

Once the production potential of the producing wells is insufficient to maintain the plateau rate, the decline periodbegins. For an individual well in depletion drive, this commences as soon as production starts, and a plateau for the field can only be maintained by drilling more wells. Well performance during the decline period can be estimated by decline curve analysis which assumes that the decline can be described by a mathematical formula. Examples of this would be to assume an exponential decline with 10% decline per annum, or a straight line relationship between the cumulative oil production and the logarithm of the water cut. These assumptions become more robust when based on a fit to measured production data. 

The ultrasound system should have more independent channels and allow the transmitter pulse to be individually adjustable in width and amplitude, and an increased frequency range for the logarithmic amplifier was desired. The digitization should be improved both with respect to sampling rate and resolution. 

The kinetics of reactions in which a new phase is formed may be complicated by the interference of that phase with the ease of access of the reactants to each other. This is the situation in corrosion and tarnishing reactions. Thus in the corrosion of a metal by oxygen the increasingly thick coating of oxide that builds up may offer more and more impedance to the reaction. Typical rate expressions are the logarithmic law,... 

We have seen various kinds of explanations of why may vary with 6. The subject may, in a sense, be bypassed and an energy distribution function obtained much as in Section XVII-14A. In doing this, Cerefolini and Re [149] used a rate law in which the amount desorbed is linear in the logarithm of time (the Elovich equation). 

The above rate law has been observed for many metals and alloys either anodically oxidized or exposed to oxidizing atmospheres at low to moderate temperatures—see e.g. [60]. It should be noted that a variety of different mechanisms of growth have been proposed (see e.g. [61, 62]) but they have in common that they result in either the inverse logaritlnnic or the direct logarithmic growth law. For many systems, the experimental data obtained up to now fit both growth laws equally well, and, hence, it is difficult to distinguish between them. 

Figure C2.8.7. Principal oxide growth rate laws for low- and high-temperature oxidation inverse logarithmic, linear, paralinear and parabolic.

In order to obtain more insight into the local environment for the catalysed reaction, we investigated the influence of substituents on the rate of this process in micellar solution and compared this influence to the correspondirg effect in different aqueous and organic solvents. Plots of the logarithms of the rate constants versus the Hammett -value show good linear dependences for all... 

Fig. 2.4, illustrates the variation with the concentration of sulphuric acid of the logarithm of the second-order rate coefficients for the nitration of a series of compounds for which the concentration of effective... 

Rates of nitration in perchloric acid of mesitylene, luphthalene and phenol (57 I-6i-i %), and benzene (57 i-64 4%) have been deter-mined. The activated compounds are considered below ( 2.5). A plot of the logarithms of the second-order rate coefficients for the nitration of benzene against — ( f + log over the range of acidity... 

The relative basicities of aromatic hydrocarbons, as represented by the equilibrium constants for their protonation in mixtures of hydrogen fluoride and boron trifluoride, have been measured. The effects of substituents upon these basicities resemble their effects upon the rates of electrophilic substitutions a linear relationship exists between the logarithms of the relative basicities and the logarithms of the relative rate constants for various substitutions, such as chlorination and... 

Relative electrophilic localization energies vs. logarithms of partial rate factors for nitration (a) Hiickel, (6) PPP with fixed /3. (From Dewar Thompson. ) (iv) Plot of log K vs. AB c. (From Dewar. )... 

A plot against Hammett s cr-constants of the logarithms of the rate constants for the solvolysis of a series of Mz-substituted dimethylphenylcarbinyl chlorides, in which compounds direct resonance interaction with the substituent is not possible, yielded a reasonably straight line and gave a value for the reaction constant (p) of — 4 54. Using this value of the reaction constant, and with the data for the rates of solvolysis, a new set of substituent parameters (cr+) was defined. The procedure described above for the definition of cr+, was adopted for... 

The applicability of the two-parameter equation and the constants devised by Brown to electrophilic aromatic substitutions was tested by plotting values of the partial rate factors for a reaction against the appropriate substituent constants. It was maintained that such comparisons yielded satisfactory linear correlations for the results of many electrophilic substitutions, the slopes of the correlations giving the values of the reaction constants. If the existence of linear free energy relationships in electrophilic aromatic substitutions were not in dispute, the above procedure would suffice, and the precision of the correlation would measure the usefulness of the p+cr+ equation. However, a point at issue was whether the effect of a substituent could be represented by a constant, or whether its nature depended on the specific reaction. To investigate the effect of a particular substituent in different reactions, the values for the various reactions of the logarithms of the partial rate factors for the substituent were plotted against the p+ values of the reactions. This procedure should show more readily whether the effect of a substituent depends on the reaction, in which case deviations from a hnear relationship would occur. It was concluded that any variation in substituent effects was random, and not a function of electron demand by the electrophile. ... 

Log arithmic-Mean Driving Force. As noted eadier, linear operating lines occur if all concentrations involved stay low. Where it is possible to assume that the equiUbrium line is linear, it can be shown that use of the logarithmic mean of the terminal driving forces is theoretically correct. When the overall gas-film coefficient is used to express the rate of absorption, the calculation reduces to solution of the equation... 

Another aspect of plasticity is the time dependent progressive deformation under constant load, known as creep. This process occurs when a fiber is loaded above the yield value and continues over several logarithmic decades of time. The extension under fixed load, or creep, is analogous to the relaxation of stress under fixed extension. Stress relaxation is the process whereby the stress that is generated as a result of a deformation is dissipated as a function of time. Both of these time dependent processes are reflections of plastic flow resulting from various molecular motions in the fiber. As a direct consequence of creep and stress relaxation, the shape of a stress—strain curve is in many cases strongly dependent on the rate of deformation, as is illustrated in Figure 6. 

Effect of Temperature. In addition to being often dependent on parameters such as shear stress, shear rate, and time, viscosity is highly sensitive to changes in temperature. Most materials decrease in viscosity as temperature increases. The dependence is logarithmic and can be substantial, up to 10% change/°C. This has important implications for processing and handling of materials and for viscosity measurement. 

An overwhelming body of evidence, starting with the earliest investigations (2), supports the contention that the rate of destmction of microorganisms is logarithmic, ie, first order with respect to the concentration of microorganisms. The process can be described by the following expression ... 

Table 32 Logarithms of the Partial Rate Factors Calculated According to the Standard Conditions...

The death rate coefficient is usually relatively small unless inhibitoiy substances accumulate, so Eq. (24-10) shows an exponential rise until S becomes depleted to reduce [L. This explains the usual growth curve (Fig. 24-21) with its lag phase, logarithmic phase, resting phase, and declining phase as the effect of takes over. 

Fig. 27. Logarithm of normalized rate constant ln(fc/nto) versus dimensionless coupling strength C /Q for PES (4.28) with Q = 0.1, n = 1, F /a>o = 3. Separate points and dashed line correspond to instanton result and numerical data [Hontscha et al. 1990].

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