Profile homogeneous-function

Homogeneous function profiles 16-6 Separable profiles 16-7 Example Infinite linear profile 16-8 Example Double parabolic profile... 

The power-law profiles satisfy Eq. (16-19b), and are therefore homogeneous function profiles. Consequently, the above expression is consistent with the more general result of Eq. (16-20). In the case of the infinite linear profile ( = 1), Eq. (17-6) reduces to the exact eigenvalue equation of Eq. (16-29), apart from a change in the multiplicative constant on the ri t from 1.018 to 3/(2 t ) S 1.024, a relative error of 0.6%. This excellent agreement is independent of and and demonstrates the accuracy of the Gaussian approximation for arbitrary eccentricity. 

The integral length scale provides a simple measure of the plume width and homogeneity and is calculated based on the area under the correlation function profiles ... 

We consider noncircular fibers with profiles expressible in terms of an arbitrary homogeneous function f (x, y) in the form... 

Figure 19 (a) Peak melting temperature as a function of the branch content in ethylene-octene copolymers (labelled -O, and symbol —B (symbol, ) and -P (symbol, A) are for ethylene-butene and ethylene-propylene copolymers, respectively) and obtained from homogeneous metallocene catalysts show a linear profile, (b) Ziegler-Natta ethylene-octene copolymers do not show a linear relationship between peak melting point and branch content [125]. Reproduced from Kim and Phillips [125]. Reprinted with permission of John Wiley Sons, Inc. 

Figure 2 Relative concentration profiles of oxidation products during thermal degradation as a function of depth for a plaque with thickness 3 mm Nearly homogeneous profile when the degradation depth, a, is 3.0 mm, similar to sample thickness, and heterogeneous profile when the degradation depth, a, is 0.1 mm, smaller than the sample thickness. The profiles were calculated from Equation (4).

In Figure 9.1(c), the opposite extreme case of a very porous solid B is shown. In this case, there is no internal diffusional resistance, all parts of the interior of B are equally accessible to A, and reaction occurs uniformly (but not instantaneously) throughout the particle. The concentration profiles are flat with respect to radial position, but cB decreases with respect to time, as indicated by the arrow. This model may be called a uniform-reaction model (URM). Its use is equivalent to that of a homogeneous model, in which the rate is a function of the intrinsic reactivity of B (Section 9.3), and we do not pursue it fiirther here. 

Such systems of differential equations are called homogeneous. They have as solutions, linear combinations of exponential functions, where the eigenvalues, Xi, of the matrix K are the exponentials. In the first, irreversible example, equation (5.34), the eigenvalues of K are Xi=-ki, X,2=-fe and X3=0. Thus, the concentration profiles are linear combinations of the vectors e-, where t is the vector of times. In matrix notation we can write... 

Henry and Fauske (1975, 1976) have proposed a model to describe the events leading to a large-scale vapor explosion in a free contact mode. Their initial, necessary conditions are that the two liquids, one hot and the other cold, must come into intimate contact, and the interfacial temperature [Eq. (1)] must be greater than the homogeneous nucleation temperature of the colder liquid. Assuming the properties of both liquids are not strong functions of temperature, the interface temperature is then invariant with time. Temperature profiles within the cold liquid may then be computed (Eckert and Drake, 1972) as... 

The concentration profile evolves as a cosine function but the amplitude is decreasing with time exponentially. When Df/(L/m) =0.11665, = 0.01, the sample may be considered homogenized. 

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