The Simple Power Law Expression

1 The Simple Power Law Expression (SPLE). One the principal objectives of catalyst deactivation research is to measure deactivation rates under conditions rej[M esentative d catalytic pocesses as a functimi of inqxMtant M oeess variables such as reactant ctmcentratimis and tenqrerature and to correlate these data in the form of a deactivatimi rale expression that xedicts 

Sintering rates have been generally conelated by an empirical rate equation involving either surface area S or dispersion D in a SPUE of the form  

2 The General Power Law Expression. A prranising solution to this dilemma is the api cation of the GPLE proposed by Puentes,  

2 Quantitative Comparisons of Thermal Stability of Model-Supported Metals Using GPLE Kinetic Parameters and Comparisons with Real Supported Metals. - 

1 EflFects of Reaction ConditiOTis and Time on Thermal Stability. 

---

Comparison and/or correlation of previous sintering rate data has been historically further complicated by the simplistic, widespread use of the simple power law expression (SPLE) from which are derived reaction orders and activation energies that vary with time, temperature, and metal concentration. Most of these experimental and theoretical complications are overcome by use of the general power law expression (GPLE) from which more physically reasonable reaction orders (of one or two) and activation energies are obtained. This result has important mechanistic implications since a number of fundamental processes such as emission of atoms from crystallites, diffusion of adatoms on a support, collision of crystallites, or recombination of metal atoms may involve second order processes. 

The main advantage of the power-law distribution is its mathematical simplicity compared to other distribution functions. It is much easier to perform calculations with the simple power-law expression compared to the lognormal distribution given by (8.33). Figure 8.9 indicates that the power-law distribution can provide a reasonable approximation to atmospheric aerosol number distributions... 

Equation 3.72 is also found to be applicable to polyatomic gases. Viscosity of gases at low density increases with temperature in a power law with power index in the range of 0.6-10. The simple power law expression is given as... 

Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

The five reactions were analyzed using simple power law expression of the type of Eqs. (44) and (45) ... 

The data will be used to formulate a tentative model to be used in the design of the experiment. It will consist of a simple power-law expression for each step with each of the parameters estimated from the reference data. For the reaction scheme ... 

The rate expression for Fiseher-Tropseh (FT) synthesis has been obtained using a 25 wt.% C0/AI2O3 eatalyst in a 1 liter continuously stirred tank reactor (CSTR) operated at 493K, 1.99 MPa (19.7 atm), H2/CO feed ratios of 1.0-2.4 with varying space velocities to produce 14-63% CO eonversion. Adjusting the ratios of inert gas and added water permitted the impact of added water to be made at the same total flow rate and H2 and CO partial pressures. The addition of water at low levels during FT sjmthesis did not impact CO conversion but at higher levels it decreased CO conversion relative to the same conditions without water addition. The catalytic activity recovered after water addition was terminated. The temporary reversible decline in CO conversion when water was added may be due to the kinetic effect of water by inhibition of CO and/or H2 adsorption. The data of this study are fitted fairly well by a simple power law expression of the form ... 

The addition of water at higher levels in FT synthesis decreased the CO conversion but the activity recovered after water addition was terminated. A rate expression has been obtained for a 25 wt.% C0/AI2O3 catalyst operated in a 1 liter continuous stirred tank reactor (CSTR) at 493K, 19.7 atm. (1.99 MPa), over a range of reactant partial pressures. The data of this study are fitted by a simple power law expression of the form ... 

Much of the recent data reviewed by Heywood et a/. are in agreement in these terms. However, as discussed above, all of the parameters used (i.e. z, Ne Re, Re) have some small dependence on Reynolds number. This can be accommodated by a simple power law expression ... 

Are any other simple power law expressions (i.e., those for which aU orders are either zero or positive or negative integers) consistent with the data Consider only forms in which Pco/ co ... 

In the developments already presented, the rate equations used were simple power law expressions. But, as discussed previously, catalytic rate equations are much more complex and often require the use of LHHW models. Many attempts have been made to incorporate these models in the analysis (e.g., Chu and Hougen, 1962 Krasuk and Smith, 1965 Roberts and Satterfield, 1965, 1966 Hutchings and Carberry, 1966 Schneider and Mitschka, I966a,b Kao and Satterfield, 1968 Rajadhyaksha et al., 1976 see in particular Aris, 1975 Luss, 1977). Clearly, graphical representation becomes cumbersome when a large number of adsorbed species is involved. However, the problem is quite tractable where only one species is adsorbed. 

The rate of duplex film spreading of PDMS oils on either water or surfactant solution is given by a simple power law expression provided that, in the case of surfactant solutions, the air-water and oil-water surface tensions are maintained constant by rapid transport of surfactant to the relevant surfaces. Thus, Fay [86], and later Hoult [87], derived by dimensional analysis that the time dependence of the distance y from the origin to the spreading front, in the case of linear spreading as a duplex film, is given by... 

The relaxation data for most propellants obeys a simple power law expression as indicated by these data when plotted logarithmically in figure 6.9. From Farris discussion dealing with material characterization [1], a logical choice for the functional is... 

This rate, measured the previous way, must be correlated with the temperature and concentration as in the following simple power law rate expression ... 

In practice this relationship is only approximately correct because most plastics are not linearly viscoelastic, nor do they obey completely the power law expressed by equation (2.62). However this does not detract from the considerable value of this simple relationship in expressing the approximate solution to a complex problem. For the purposes of engineering design the expression provides results which are sufficiently accurate for most purposes. In addition. 

For simple power law rate equations the effectiveness can be expressed in terms of the Thiele modulus, Eq 7.28. In those cases restriction is to irreversible, isothermal reactions without volume change. Other cases can be solved, but then the Thiele modulus alone is not sufficient for a correlation. 

We have thus far written unimolecular surface reaction rates as r" = kCAs assuming that rates are simply first order in the reactant concentration. This is the simplest form, and we used it to introduce the complexities of external mass transfer and pore diffusion on surface reactions. In fact there are many situations where surface reactions do not obey simple rate expressions, and they frequently give rate expressions that do not obey simple power-law dependences on concentrations or simple Arrhenius temperatures dependences. 

Figure 1.3 shows a plot of 0 versus partial pressure for various values of the adsorption equilibrium constant. These show that as the equilibrium constant increases for a given pressure, we increase the surface fraction covered, up to a value of 1. As the pressure increases, we increase the fraction of the surface covered with A. But we have only a finite amount of catalyst surface area, which means that we will eventually reach a point where increasing the partial pressure of A will have little effect on the amount that can be adsorbed and hence on the rate of any reaction taking place. This is a kind of behavior fundamentally different from that of simple power-law kinetics, where increasing the reactant concentration always leads to an increase in reaction rate proportional to the order in the kinetic expression. 

Compared to the HT shift reaction fewer publications exists on the reaction kinetics of LT shift reaction. Studies made before 1979 may be found in [602]. A rather simple power law (Eq. 85), for example [624] fitted well measurements between 200 and 250 °C, but the weak point is that the exponents are temperature dependent. An expression of the Langmuir -Hinshelwood-type, published in [625], includes additionally the influence of H2 and C02 concentration on the reaction rate. 

The Nusselt number can be expressed by a simple power-law relation of the form... 

Preliminary kinetic studies have been performed. The Langmuir-Hinshelwood rate expression was used to correlate results of experiments as it was indicated by the shape of kinetic curves (see Fig. 6). However, the reaction order with respect to hydrogen appeared to be dependent on temperature, while activation energy depends on pressure (9.6 kJ/mol at 11 bar and 35.5 kJ/mol at 21 bar). Therefore the rate of benzaldehyde consumption was approximated using the following simple power law equation ... 

If surface reaction is assumed to be rate limiting and irreversible (and no adsorbed inerts are involved), the overall rate expression for consumption of A becomes -rA = A aCa/(1 + KaCa + KbCb), where k is the surface reaction rate constant and Ka and A b are adsorption equilibrium constants. If the surface is only sparsely covered, i.e., KaCa + KbCb 1, this can be approximated as simply va kKACA = k CA-This illustrates how a simple power law rate expression can apply, under some circumstances, for what is actually a relatively complex mechanism. 

Equation 5.109 is rather complex under reaction conditions several terms in the denominator are expected to be kinetlcally unimportant leading to a simple power-law reaction-rate expression. o... 

The form of the kinetic expression can be of importance, however. For example, we can consider the CSTR system with a rate equation of more complex form than simple power law, such as... 

Nonetheless, rate expressions more complex than a simple power law are sometimes useful. For example, a power law expression does not provide any insight into the reasons for changing reactant order (i.e., a changing value of a ) with temperature or organic reactant concentration. However, such effects are frequently observed in oxidation reactions and are often consistent with more fundamentally based rate expressions. Consider, for example, what one would suppose to be the simple oxidation of methane. Golodets (p. 445) states that methane oxidation over metal oxide catalysts may be interpreted by the following mechanism ... 

---