Distribution, equation for

Equation 2.40 is an empirical equation known as the one-seventh power velocity distribution equation for turbulent flow. It fits the experimentally determined velocity distribution data with a fair degree of accuracy. In fact the value of the power decreases with increasing Re and at very high values of Re it falls as low as 1/10 [Schlichting (1968)]. Equation 2.40 is not valid in the viscous sublayer or in the buffer zone of the turbulent boundary layer and does not give the required zero velocity gradient at the centre-line. The l/7th power law is commonly written in the form... 

The distribution equations for this particular case—corresponding, for instance, to a peritectic reaction taking place during cooling of a magma chamber—were evaluated by Neumann et al. (1954) starting from a modified form of general equation 10.86 ... 

When the ideas of symmetry and of microscopic reversibility are combined with those of probability, statistical mechanics can deal with many stationary state nonequilibrium problems as well as with equilibrium distributions. Equations for such properties as viscosity, thermal conductivity, diffusion, and others are derived in this way. 

Write a distribution equation for each enzyme form—for example, E/E0 = Ne/D, where Ne and D are the numerator (for E) and denominator terms, respectively. 

Note that we now have Boltzmann distribution equations for each type of defect., and that the energy, E, to form the defect is like an activation energy. The fraction of defects present, either Ny / N or Ni / N, is an exponential function of this activation energy. 

Equations for absorber, stripper, enricher, and exhauster cascades plus the feed stage are summarized in Table 12.3. The equations are readily combined to obtain product distribution equations for the types of separators in Fig. 12.24. For a fractionator, we combine (12-98), (12-100), and (12-103) to eliminate lp and Vp, noting that r +i = vp and lu+i = If- The result is... 

Tablb 34. Pboouct Distribution Equations for Parallei., Series, anu Compubx-sbribs Reactions... 

The derivation of the product distribution equations for the complex-series reaction of the type... 

Each component separated will have a different value for K, reflecting their relative affinities for the stationary phase the generalised form of the distribution equation for each component is... 

In addition, to these binding parameters the corresponding affinity distribution can likewise be generated from the corresponding affinity distribution equation (Eq. (15)). This expression is more complex than the affinity distribution equation for the Freundlich isotherm (Eq. (8)) [26]. However, it is still a simple algebraic expression that calculates the number of binding site (Ni) having an association... 

Enzyme distribution equations for the Ordered Bi Bi system (Eq. (4.31)) are written in terms of kinetic constants and substrate concentrations. They aU have the same denominators as the rate equation and thus, if they are multiplied by the same factors given in Eq. (4.38), used to convert the rate equation into kinetic constant form, their denominators will also be expressed in terms of kinetic constants. The terms in the numerators of these equations represent entire denominator terms, which are easily expressed in terms of kinetic constants, except when some denominator terms are split between the numerators of two or more equations. There are two split denominator terms in Eqs. (4.31). The rate constants multiplying A5 and PQ in the numerator of the fEAB]/[Eo] distribution are only parts of (coefAB) and (coefPQ), respectively the other part of (coefAB) comes from the [EQ]/tEo] distribution, and the other part of (coefPQ) comes from the [ A]/[Eo] distribution. 

However, in steady-state systems, the relative concentration of a particular enzyme form may be expressed by several denominator terms, and, often, a particular denominator term represents more than one enzyme form. Generally, in steady-state systems, the denominator terms multiplied by the factor F are those which appear in the numerator of the distribution equation for the enzyme form combining with the inhibitor. If a distribution equation in terms of kinetic constants cannot be written for the enzyme form combining with the inhibitor, than Ki cannot be calculated kinetically. 

The rate equation is obtained in the following manner. In the rate equation in the absence of products (Eq. (9.15)), those terms in the denominator representing free enzyme are multiplied by ii+B/KO, where Jf is the dissociation constant of the EB complex. The terms which represent the free enzyme are found in distribution equations for this system in the absence of products, these terms are JCiAlifB and KaB (found in Eq. (9.13)). Thus, the velocity equation becomes... 

Table a. Numerator terms of enzyme distribution equations for e mechanism in c ion (i4<34)> bach numerator term is multiplied by a corresponding Michaelis pH biD( k>D. 

In order to introduce the terms for a dead-end inhibitor into the velocity equation, each enzyme distribution equation is multiplied by an appropriate Michaelis pH function. In order to do so, the corresponding distribution equations for the Ordered Bi Bi mechanism, found in Chapter 9 (Eq. (9.13)), were divided by Va, followed by a partial elimination ofiTeq- The entire procedure is shown in Table 2. 

Distribution equations for bisubstrate reactions in the steady state are often very complex expressions (Chapter 9). However, in the chemical equilibrium, the distribution equations for all enzyme forms are usually less complex. Consider an Ordered Bi Bi mechanism in reaction (16.12) with a single central complex ... 

The last equation in Table 4 for the Bi Bi Uni Uni Ping Pong mechanism (in the absence of C and R) is relatively long and complex. The right-hand portion of the denominator in this equation represents the reciprocal concentration of free enzyme in chemical equilibrium multiplied by [EJ. If the products C and R are present, the distribution equation for [E]/[Eo] would be more complex (Eq. (12.73)) consequently, the resulting full rate equation for the A - Q isotope exchange becomes rather cumbersome. 

Establishing Mean Life to Failure. The mean cycles are determined from fitting the data at each stress level to a two-parameter Weibull distribution and computing the mean from the cumulative distribution equation for a probability of 50%. The two-parameter Weibull probability density function and cumulative distribution function as given by Madayag (1969) are... 

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