Equations power function

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

Horvath and Lin s equation is very similar to that of Huber and Hulsman, only differing in the magnitude of the power function of (u) in their (A) and (D) terms. These workers were also trying to address the problem of a zero (A) term at zero velocity and the fact that some form of turbulence between particles aided in the solute transfer across the voids between the particles. 

The data in Figure 3.6-3 were fitted to a simple power function to give the resulting equation, k = 390q , = 0.83. A potentially more informative way to look... 

In all cases, (4.1) - (4.10), the power series associated with the equation defines a function element with center at the origin. In the sequel, we call it the function element associated with the equation. The function elements associated with the equations (4.1), (4.3), (4.4), (4.7) are the power series (x), r(x), s(x), t(x). 

The computer-optimized y values obtained for a number of conditions are given in Table VI. It can be seen that the first condition assumes simple power functions only and a value for B strictly in compliance with Eq. (18). The rms error achieved is good, but marked improvements are obtained by relaxing the equations for A and B in stages, as shown, the final result giving a much better rms error. It was not necessary in the analysis to separate the data into low- and high-velocity regimes, as was the case for round-tube data, since the lowest mass velocity is not so low as to cause difficulty. 

Equations 18 and 22 may be solved simultaneously, yielding the power function... 

If a power curve is available for a particular system geometry, it can be used to calculate the power consumed by an agitator at various rotational speeds, liquid viscosities and densities. The procedure is as follows calculate the Reynolds number for mixing ReM , read the power number Po or the power function from the appropriate power curve and calculate the power PA from either equation 5.13 rewritten in the form... 

For power-law functions the (scaled) elasticities do not depend on the substrate concentration, that is, unlike Michaelis Menten rate equations, power-law functions will not saturate for increasing substrate concentration. 

From a consideration of either Eqs. (113) or (114) (K3), it is evident that a saddle point is predicted from the fitted rate equation. This could eliminate from consideration any kinetic models not capable of exhibiting such a saddle point, such as the generalized power function model of Eq. (1) and the several Hougen-Watson models so denoted in Table XVI. 

While first-order models have been used widely to describe the kinetics of solid phase sorption/desorption processes, a number of other models have been employed. These include various ordered equations such as zero-order, second-order, fractional-order, Elovich, power function or fractional power, and parabolic diffusion models. A brief discussion of these models will be provided the final forms of the equations are given in Table 2. 

For comparison purposes, regression parameters were computed for the model defined by Equations 6, 7, 8, and 10 and the model obtained by replacing In (1/R) in those equations by R. The dependent variable (y) is particulate concentration because it is desired to predict particulate content from reflectance values. Data from Tables I and II were also fitted to exponential and power functions where the independent variable (x) was reflectance but the fits were found to be inferior to that of the linear relationship. 

The correlation coefficients generated for mono-, bi- and triexponential fits obtained by nonlinear regression analyses are summarized in Table 1. Wilson el al. [8] reported that the rate of tobramycin release from Simplex PMMA beads could be fitted to monoexponential and power functions however, they obtained r2 values<0.9 for both fits. Our results show that, although the monoexponential fit is poor, both biexponential and triexponential fits provided r2 values>0.9. Since the biexponential relationship in equation (2) is proposed to fit our physical model, this approach was adopted in analysis of computer fits to release data. The rate constants, a and P, represent an initial, rapid surface release and a prolonged matrix diffusion-controlled release respectively. 

It seems almost inconceivable that earlier investigators would not have made the same recommendation, as these power-function equations have been in use for approximately thirty years. Nevertheless, no early reference was found which explicitly suggested the use of n or ra for such purposes, although it is highly possible that such references may exist. 

It can be shown (V2) that Eq. (51) is identical to an equation presented earlier by Alves, Boucher, and Pigford (A3) for power-function non-Newtonians. This is due to the fact that neglecting the derivative in Eq. (48) implies assumption of this type of fluid behavior. Eq. (51) is to be slightly preferred to that of Alves et al. on the basis that its readily possible to take the slope of a semilogarithmic plot of the term (1/n") — 1 versus the torque t in order to determine the error... 

Incidentally, plotting data on a log-log paper (i.e., log-log plot) is often used for determining the value of an exponent, if the data can be represented by an empirical power function, such as Equation 2.6. 

Numerical solution of equations (13)—(19) for polypropylene extrusion was made in 29,34> using approximation of the flow function (flow curve) by a piecewise power function. To find the root of b(f, M) of Eq. (13), the authors used a formal search algorithm compiled as a standard program for computer M-20 (USSR). Figure 2 gives dependency of b/f upon M (M is the specific moment of a core s rotation, i.e., the moment related to the length of the channel). It can be seen in Fig. 2 that (b/f) is a strictly decreasing function. 

Allometric scaling (allometry) is the discipline that predicts human PK using pre-clinical data (Ritschel et al., 1992). This approach is based on empirical observations that various physiological parameters are functions of body size. The most widely used equation in allometry is a one-term power function ... 

The intrinsic kinetics was measured in an isothermal integrated reactor and the reaction rate equations in terms of power function have been established... 

Differential Rate Laws 5 Mechanistic Rate Laws 6 Apparent Rate Laws 11 Transport with Apparent Rate Law 11 Transport with Mechanistic Rate Laws 12 Equations to Describe Kinetics of Reactions on Soil Constituents 12 Introduction 12 First-Order Reactions 12 Other Reaction-Order Equations 17 Two-Constant Rate Equation 21 Elovich Equation 22 Parabolic Diffusion Equation 26 Power-Function Equation 28 Comparison of Kinetic Equations 28 Temperature Effects on Rates of Reaction 31 Arrhenius and van t Hoff Equations 31 Specific Studies 32 Transition-State Theory 33 Theory 33... 

Havlin and Westfall (1985) and Havlin et al. (1985) used a power-function equation to describe potassium release from soils. The integrated form of the power-function equation can be expressed (Havlin and West-fall, 1985) as... 

Havlin and Westfall (1985) found that the power-function equation described potassium release from soils well. The a and k values were highly correlated with nonexchangeable potassium release. They found that k from Eq. (2.53) was highly correlated with potassium uptake and the relative yield of alfalfa (Medicago sativa L.) as shown in Fig. 2.8. 

Sparks and Jardine (1984) found that the first-order equation best described potassium adsorption kinetics on clay minerals and soils. However, Havlin and Westfall (1985) reported that the power-function equation described nonexchangeable potassium release kinetics better than first-order or a number of other models. 

Equation 5 expresses the variation in the concentration of monomer A in the feed stream entering the reactor as a function of time. Since this variation is a power function of time, the process has been named "power feed."... 

In the computer program MINTEQA2 in addition to using the Van Hoff equation there is the option of using a power function of the form... 

Leiva et al. [65] have reported for poly(itaconates) monolayers the surface behavior at the air - water interface at different surface concentrations. They have found that for these type of polymers, the air - water interface at 298 K, is a bad solvent, very close to the theta solvent. At the semidilute region concentration, the surface pressure variation was expressed in terms of the scaling laws as a power function of the surface concentration. According to equation (3.3), the log it vs log T plot shows a linear variation with slope 2 v/(2 u-1). 

The Freundlich equation, empirical in origin, relates positive adsorption to a power function of c, as follows ... 

The solution of the differential equations above is a power function of time, namely c(t) = f3ta with parameters ft and a satisfying the initial condition c (to) = co. Usually P and a are estimated by curve fitting on experimental data, and the parameters of (2.22) and (2.23) are obtained by... 

Noncompartmental models were introduced as models that allow for transport of material through regions of the body that are not necessarily well mixed or of uniform concentration [248]. For substances that are transported relatively slowly to their site of degradation, transformation, or excretion, so that the rate of diffusion limits their rate of removal from the system, the noncompartmental model may involve diffusion or other random walk processes, leading to the solution in terms of the partial differential equation of diffusion or in terms of probability distributions. A number of noncompartmental models deal with plasma time-concentration curves that are best described by power functions of time. 

The other example is the power function equation, or, by another name, fractional power or modified Freundlich equation (Wahba and Zaghloul 2007) ... 

---