Szyszkowski equation

For equations such as (A2.1.13) involving more than two variables the problem is no longer trivial. Most such equations are not integrable.  [c.334]

It is possible to write two such equations for the initial state,(corresponding to the reduced aquo-complex  [c.605]

The terms Po, Pa, Pt, Pat, Paa, and Pt,t, are adjustable parameters whose values are determined by using linear regression to fit the data to the equation. Such equations are empirical models of the response surface because they have no basis in a theoretical understanding of the relationship between the response and its factors. An empirical model may provide an excellent description of the response surface over a wide range of factor levels. It is more common, however, to find that an empirical model only applies to the range of factor levels for which data have been collected.  [c.676]

Published Cost Correla.tions. Purchased cost of an equipment item, ie, fob at seller s site or other base point, is correlated as a function of one or more equipment—size parameters. A size parameter is some elementary measure of the size or capacity, such as the heat-transfer area for a heat exchanger (see HeaT-EXCHANGETECHNOLOGy). Historically the cost—size correlations were graphical log—log plots, but the use of arbitrary equation forms for correlation has become quite common. If cost—size equations are used in computer databases, some limit logic must be included so that the equation is not used outside of the appHcable size range.  [c.441]

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful.  [c.454]

If data on several furnaces of a single class are available, a similar treatment can lead to a partially empirical equation based on simphfied rules for obtaining (GS )r or an effective A. Because Eq. (5-178) has a structure which covers a wide range of furnace types and has a sound theoretical basis, it provides safer structures of empirical design equations than many such equations available in the engineering hterature.  [c.588]

Packed Red Reactors The commonest vessels are cylindrical. They will have gradients of composition and temperature in the radial and axial directions. The partial differential equations of the material and energy balances are summarized in Table 7-10. Example 4 of Modeling of Chemical Reactions in Sec. 23 is an apphcation of such equations.  [c.702]

Methods for estimating the height of the active sec tion of counterflow differential contactors such as packed towers, spray towers, and falling-film absorbers are based on rate expressions representing mass transfer at a point on the gas-hquid interface and on material balances representing the changes in bulk composition in the two phases that flow past each other. The rate expressions are based on the interphase mass-transfer principles described in Sec. 5. Combination of such expressions leads to an integral expression for the number of transfer units or to equations related closely to the number of theoretical plates. The paragraphs which follow set forth convenient methods for using such equations, first in a general case and then for cases in which simplifying assumptions are vahd.  [c.1354]

Particle-Size Equations. 20-5  [c.1819]

Particle-Size Equations It is common practice to plot size-distribution data in such a way that a straight line results, with all the advantages that follow from such a reduction. This can be done if the cui ve fits a standard law such as the normal probability law. According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. In order to maintain a symmetrical beU-shaped cui ve for the frequency distribution it is necessary to plot the population density (e.g., percentage per micron) against size.  [c.1823]

The terms Po, Pa, Pt, Pat, Paa, and are adjustable parameters whose values are determined by using linear regression to fit the data to the equation. Such equations are empirical models of the response surface because they have no basis in a theoretical understanding of the relationship between the response and its factors. An empirical model may provide an excellent description of the response surface over a wide range of factor levels. It is more common, however, to find that an empirical model only applies to the range of factor levels for which data have been collected.  [c.676]

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system.  [c.15]

The theory of all three programming methods will now be discussed and the simplest form, flow programming, will be considered first. However, the procedures that are used to theoretically calculate the effect of the different programming techniques needs some prior discussion. It is difficult to derive explicit equations to simulate the change in retention time with programming conditions and, even if such equations are derived, they are, with few exceptions, exceedingly complex and difficult to relate to the actual physical processes that are taking place. Today, with the extremely rapid calculating capabilities of the modern computer, the development of explicit equations to describe programming effects is unnecessary. The actual process can be imitated in small steps and the ultimate effect obtained by summing the effect of all the steps. This method of treatment will be applied to three forms of programming.  [c.144]

Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases.  [c.452]

Such equations are generally successful only for the case of apolar liquids and solids, for which = 1, ys = y and yc = y, giving  [c.23]

Ca plug flow)- the case where the effluent composition is fixed instead of the reactor size. Equations 8-152 and 8-154 can be manipulated to show that for small Dg /uL,  [c.745]

It is possible for numerical integration to fail, in the sense of yielding physically meaningless results, for certain combinations of coupled equations. This can occur when different reactions in a kinetic scheme have greatly different time scales. An important example is a reaction system to which the steady-state approximation is applicable, the rate of change of an intermediate concentration being essentially zero as a consequence of two very large opposing fluxes. Then very small changes in initial conditions may result in grossly different numerical results. Such equations are said to be stiff. Curtiss and Hirschfelder called attention to the problem of stiffness in numerical integration, and several computer programs capable of integrating stiff equations are available.  [c.109]

After expanding the products, we can equate the coefficients on each side of the equation for each power of X, leading to a series of relations representing successively higher orders of perturbation. Here are the first three such equations (after some rearranging), corresponding to powers of 0,1, and 2 of X  [c.268]

Table 4-156 gives the maximum water-to-cement ratio for Class A cement as 5.5 gal/sack. Thus using the absolute volume for Class A cement from Table 4-154 as 3.59 gal/sack, Equation 4-324 is  [c.1185]

Ib/sack Equation 4-324 becomes  [c.1208]

As A will be a function of current density, T will be a function of electrode area, and comparisons should therefore be made with cells of standard size. Equation 12.12 shows that high throwing indices will result when polarisation rises steeply with current (AE, AEj) and cathode efficiency falls steeply (cj >> f i)- The primary current ratio, P = affects the result because  [c.366]

In Equation (11), V is the total volume containing n moles of component i, n moles of component j, etc. The differentiation is carried out such that, in addition to temperature and pressure, all mole numbers (except n ) are held constant.  [c.16]

Equation (12), applicable at low or moderate pressures, is used in this monograph for typical vapor mixtures. However, when the vapor phase contains a strongly dimerizing component such as carboxylic acid. Equation (7) is not applicable and  [c.16]

For such components, as the composition of the solution approaches that of the pure liquid, the fugacity becomes equal to the mole fraction multiplied by the standard-state fugacity. In this case,the standard-state fugacity for component i is the fugacity of pure liquid i at system temperature T. In many cases all the components in a liquid mixture are condensable and Equation (13) is therefore used for all components in this case, since all components are treated alike, the normalization of activity coefficients is said to follow the symmetric convention.  [c.18]

When the pressure is low and mixture conditions are far from critical, activity coefficients are essentially independent of pressure. For such conditions it is common practice to set P = P in Equations (18) and (19). Coupled with the assumption that v = v, substitution gives the familiar equation  [c.22]

This chapter presents a general method for estimating nonidealities in a vapor mixture containing any number of components this method is based on the virial equation of state for ordinary substances and on the chemical theory for strongly associating species such as carboxylic acids. The method is limited to moderate pressures, as commonly encountered in typical chemical engineering equipment, and should only be used for conditions remote from the critical of the mixture.  [c.26]

The fugacity coefficient is a function of temperature, total pressure, and composition of the vapor phase it can be calculated from volumetric data for the vapor mixture. For a mixture containing m components, such data are often expressed in the form of an equation of state explicit in the pressure  [c.26]

Unfortunately, good binary data are often not available, and no model, including the modified UNIQUAC equation, is entirely adequate. Therefore, we require a calculation method which allows utilization of some ternary data in the parameter estimation such that the ternary system is well represented. A method toward that end is described in the next section.  [c.66]

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation.  [c.105]

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965).  [c.108]

Bianco and Marmur [143] have developed a means to measure the surface elasticity of soap bubbles. Their results are well modeled by the von Szyszkowski equation (Eq. III-57) and Eq. Ill-118. They find that the elasticity increases with the size of the bubble for small bubbles but that it may go through a maximum for larger bubbles. Li and Neumann [144] have shown the effects of surface elasticity on wetting and capillary rise phenomena, with important implications for measurement of surface tension.  [c.90]

There are three forms of the Langmuir-Szyszkowski equation, Eq. III-57, Eq. Ill-107, and a third form that expresses ir as a function of F. (n) Derive Eq. III-57 from Eq. Ill-107 and (b) derive the third form.  [c.93]

It was further shown that such equations could be extended to a wide variety of amides and related compounds and even allowed the prediction of the degradation products of agrochemicals of the benzoylphenylurea type [13]  [c.183]

When the pore structure is viewed as an assembly of interconnected channels, the flux vectors are formed by adding contributions from the separate channels, so before a model of this type can be constructed it is necessary to have equations which will predict the fluxes in a single chann< throughout the range from very small to very large diameters. But we have already seen that no such equations are presently available. The classic-Knudsen theory gives the fluxes for very small pores or very low pressures, and the continuum theory developed in Chapter 4 gives the fluxes for large pores or high pressures, but in intermediate cases there is no adequate theory.  [c.67]

Algorithm and Solution of the Linearized Equations. The procedure is to solve the finite difference equation sets for each dependent variable (inner iteration) sequentially and repeatedly. Each cycle of calculation (outer iteration) consists of computing the velocity components from the momentum equations using the most recently calculated pressure field (a guessed pressure field is used during the first cycle of iteration). The velocities and the pressure fields are then corrected to satisfy the continuity equation. This is followed by the calculation of the other field variables, eg, turbulence quantities, temperature, and concentration, if these influence the flow field. The procedure is repeated until the solution converges, ie, until the so-called residual source (the sum of the absolute values of the residuals of all the equations) is less than a predetermined value. This procedure is followed in the semi-imphcit method for pressure-linked equation (SIMPLE) algorithm (17). Other variants of the SIMPLE algorithms, SIMPLER, SIMPLEC, PISO, PISOC, etc, differ primarily in the procedure used in correcting for continuity. A large proportion of the computer time is used to solve the individual transport equations. Each of the transport equations consists of a single diagonally dominant equation. Good solvers for handling such equations are available. The alternating direction method, line relaxation method, the Stone algorithm, and the incomplete Cholesky conjugate gradient method are but a few examples. To avoid numerical instabiUties, relaxation parameters are used in that only a fraction of the change of the dependent variables calculated at the current iteration step are appHed to the next iteration step. The relaxation parameters can be different for each of the dependent variables.  [c.101]

Cetane number is difficult to measure experimentally. Therefore, various correlation equations have been developed to predict cetane number from fuel properties. One such equation may be found in ASTM D4737 to calculate a cetane index (Cl). ASTM D975 allows use of Cl as an approximation if cetane numbers are not available.  [c.192]

Upper Bound for the Real Roots Any number that exceeds all the roots is called an upper bound to the real roots. If the coefficients of a polynomial equation are all of hke sign, there is no positive root. Such equations are excluded here since zero is the upper oound to the real roots. If the coefficient of the highest power of P x) = 0 is negative, replace the equation by —P x) = 0.  [c.433]

These equations are integrated from some initial conditions. For a specifiea value of. s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and h depend just on x and y (and not u), and the equation is linear if a, h, and/all depend on X and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks. Ref. 245. For further information, see Refs. 79, 159, 192, and 245.  [c.457]

Wallis One-dimensional Two-phase Flow, McGraw-Hill, New York, 1969) and Govier and Aziz present mass, momentum, mechanical energy, and total energy balance equations for two-phase flows. These equations are based on one-dimensional behavior for each phase. Such equations, for the most part, are used as a framework in which to interpret experimental data. Reliable prediction of multiphase flow behavior generally requires use of data or correlations. Two-fluid modeling, in which the full three-dimensional microscopic (partial differential) equations of motion are written for each phase, treating each as a continuum, occupying a volume frac tiou which is a continuous function of position, is a rapidly developing technique made possible by improved computational methods. For some relatively simple examples not requiring numerical computation, see Pearson Chem. Engr Sci., 49, 727-732 [1994]). Constitutive equations for two-fluid models are not yet sufficiently robust for accurate general-purpose two-phase flow computation, but may be quite good for particular classes of flows.  [c.652]

See pages that mention the term Szyszkowski equation : [c.2676]    [c.607]    [c.397]    [c.456]    [c.638]    [c.1255]    [c.1281]    [c.400]    [c.29]    [c.29]   
Physical chemistry of surfaces (0) -- [ c.67 ]