Five Assumptions

Up to this point the treatment has been fairly general. The only assumptions involved are  

An additional approximation that may be made is the systematic omission of higher-order terms in the flux expressions, because of truncations of the Taylor series mentioned in connection with Eqs. (5.10) and (5.20). This implies the assumption that v, and T, and their spatial derivatives, do not change significantly over molecular dimensions. The results in Table 1 are therefore frequently appropriate starting points for the study of the rheology, diffusion, and heat conduction in flexible polymer mixtures, both in solutions and melts. However, for the study of cross effects the first three terms in the Taylor series are needed, as discussed in Sects 14-16. 

In this section we discuss five assumptions that have traditionally been introduced in order to continue with the development of the kinetic theory of transport phenomena and get useful results. Some of these assumptions are made because we do not at present know enough about the distribution functions that appear in the expressions in Tables 1 and 2, in particular the pair distribution function and the momentum-space distnbution function. Other assumptions are introduced in order to simplify the subsequent problem-solving for specific molecular models. All the assumptions we present here can be challenged some of them should be modified, and some of them may ultimately be eliminated. 

---

One must note that probability alone can only detect alikeness in special cases, thus cause-effect cannot be directly determined - only estimated. If linear regression is to be used for comparison of X and Y, one must assess whether the five assumptions for use of regression apply. As a refresher, recall that the assumptions required for the application of linear regression for comparisons of X and Y include the following (1) the errors (variations) are independent of the magnitudes of X or Y, (2) the error distributions for both X and Y are known to be normally distributed (Gaussian), (3) the mean and variance of Y depend solely upon the absolute value of X, (4) the mean of each Y distribution is a straight-line function of X, and (5) the variance of X is zero, while the variance of Y is exactly the same for all values of X. 

An algebraically very simple theory can be developed from a set of five assumptions. As will be shown, its predictions are reasonably borne out by a fair body of data. It is now generally accepted as a suitable basis for discussing simple cases. It can be adapted to meet special circumstances and extended to overcome some of the weaknesses of the assumptions. Indeed, exception can be taken (and has been taken) to nearly every one of these assumptions, and the extent of their validity and the need for modification of the theory are further discussed in Section IV. 

A straight-line model is the most used, but also the most misused, model in analytical chemistry. The analytical chemist should check five basic assumptions during method validation before deciding whether to use a straight-line regression model for calibration purposes. These five assumptions are described in detail by MacTaggart and Farwell [6] and basically are linearity, error-free independent variable, random and homogeneous error, uncorrelated errors, and normal distribution of the error. The evaluation of these assumptions and the remedial actions are discussed hereafter. 

To simplify the subsequent treatments, we make the following five assumptions (i) polymer particles contain at most one radical, (ii) a radical with no longer than s monomer units can desorb from and enter into the particles with the same rate irrespective of chain length, (iii) instantaneous termination occurs when another radical enters the particle already containing a radical, (iv) no distinction is made between radicals with or without an initiator fragment on its end, and (v) water-phase reactions such as propagation, termination, and transfer i an be neglected from a kinetic point of view, as shown in Section II,A, Under these assumptions, the rate coefiident for radical desorption from the particles is derived with both deterministic and stochastic approaches,... 

Sklash and Farvolden (1982) listed five assumptions that must hold for reliable hydrograph separations using environmental isotopes ... 

DeWalle et al. (1988) used a three-component model for the special case where the discharge of one component was known. Weis etal. (1991) used a three-component model for the case where the isotopic compositions of new overland and new subsurface flow were identical, and the silica contents of new and old subsurface flow were identical. Ogunkoya and Jenkins (1993) used a model where one component was time varying. Hinton et al. (1994) listed five assumptions required for three-component models using two tracers. 

The universal applicability of the simple two-component mixing model has been frequently challenged (Kennedy et al., 1986 Rodhe, 1987 Sklash, 1990 McDonnell et al., 1991 Bishop, 1991 Kendall and McDonnell, 1993). These challenges address the five assumptions governing the use of the two-component mixing model given in Sklash and Farvolden (1982) and listed above. 

The van der Waals and Platteeuw principal equation (1) has five assumptions, which enable refinement of the theory ... 

Although real gases do not conform exactly to the five assumptions listed, we will see next that these assumptions do indeed explain ideal gas behavior—behavior shown by real gases at high temperatures and/or low pressures. 

In the presentation of this writers experimental results (below) the term apparent distribution coefficient is taken to mean the interface distribution coeflBcient computed according to Equation 5 and the five assumptions listed with Equation 3. The adjective apparent acknowledges the fact that these assumptions may not be satisfied. The k-values shown in Figures 9 to 17 are apparent distribution coeflBcients according to this definition. 

In Sect. 12 five assumptions are introduced in order to carry the development further. In Sect 13 the equation for the singlet distribution is solved for simple models, both for isothermal and nonisothermal conditions the solutions are given in terms of the Finger strain tensor, which describes the kinematics of the fluid motion. 

List five assumptions of traditional hazard control management. 

Additionally, criticality safety of fhe dry storage system includes the following conserva-five assumptions ... 

The first five assumptions are made in most analyses of solids conveying. The last five assumptions have been relaxed by various workers. The general approach to solids conveying analysis is to consider an element of the solid bed in the screw channel and determine all forces that are acting on this solid bed element. The most important forces are the frictional forces at the boundaries and the forces resulting from pressure gradients in the solid bed. 

There are, in general, five assumptions underlying all the mechanistic interface theoretical studies, namely (1) elastic constituent material behavior, (2) zero thickness for the interface, (3) perfect bond, (4) identical constituent bulk and in situ properties, and (5) a regular or repeating array of fibers. It is known that actual composites violate most, if not all, of these assumptions. However, the theoretical predictions still are an invaluable tool in identifying important local geometry and material variables, and in obtaining quantitative estimates of the stress state at the interface. ... 

Name five assumption that are normally made to solve problems in elementary solid mechanics. 

The numeral and small single letters in the circles in Figure 11.2 denote system states. The following five assumptions are associated with this model ... 

The reader may immediately object to some of these assumptions, and is right to do so. Indeed the determination of whether the railroads have a "safety problem" lies in the investigation of whether one or more of these five assumptions do not hold. The next five chapters deal with that determination. 

---