Distribution functions Weibull

Fig. 18.3. (a) The Weibull distribution function, (b) When the modulus, m, changes, the survival probability changes os shown. 

Dorko et al. [442] have used the Weibull distribution function for the consideration of reactions in which decomposition is accompanied by melting. Following a procedure described by Kao [446], they used a mixed Weibull function, written as a linear combination of separate functions, viz. 

Figure 5.3 Accumulated fraction of drug dissolved, q(t) /q as a function of At according to the Weibull distribution function (5.11).

The Kawakita compaction equation is another equation which is often used for ceramic powder pressing. It can be derived by considering that compaction is similar to packing by tapping, where the compaction pressure, P, is directly substituted for the number of taps, N, in the analysis in Section 13.5.1. The Kawakita equation is a special case, where the value of m in the Weibul distribution function for tapping is 1. In the Kawakita equation, the compaction, C, defined as the relative reduction in volume is given by [72]... 

Mean values in particle dimension, thickness, length and slenderness ratio (length/thickness), used in this test are shown in Table 1. These mean values were based on two hundred measurements of each furnish type and calculated using the Weibull distribution function. 

Use of Eq. [23] automatically implies a power law distribution of pore sizes. A variety of other distribution functions, including the log-normal, incomplete Gamma, and Weibull distribution functions, have also been used to characterize natural pore-size distributions (e.g., Brutsaert, 1966). Furthermore, it is possible to parameterize the pore size distribution without resorting to a particular distribution function model using moment analysis (Brutsaert, 1966 Powers et al., 1992). 

The Normal (or Gaussian) and Weibull distribution functions are most commonly used in engineering design. The normal distribution is usually expressed in one of the following two functional forms ... 

Rodrigo, D., Barbosa-Canovas, G.V., Martinez, A., and Rodrigo, M. 2003. Weibull distribution function based on empirical mathematical model for inactivation of Fscherichia coli by pulsed electric fields. Journal of Food Protection 66 1007-1012. 

M. Sugiura, M. Kurobe, S. Tamura, and S. Ikeda, Kinetics of digestive enzyme stability in solid state I Application of Weibull distribution function to solid-state enzyme inactivation, J. Pharm. Sci. 68, 1381-1383 (1979). 

The Weibull distribution function is shown in Figure 16.19 and gives the probability of survival (Ps), or, alternatively, the probability of failure (Fp), of a stressed volume V. 

The Weibull distribution function contains three parameters ... 

Until now, the Weibull distribution function and the ideas sketched above have formed the basis of the state-of-the-art mechanical design process for ceramic components [2-4]. Consequently, the strength testing [82-84] of ceramics and the determination of Weibull distributions have become standardized [6, 85-87]. 

The Weibull distributions functions for the large and small specimens are also shown in Figure 12.8. It can be recognized that the Weibull line is shifted to lower strength values in the first case, and to higher strength values in the latter case. 

Figure 12.11 Strength distribution of a silicon nitride ceramic (same data as in Figure 12.8). A Weibull distribution function (solid curve) and a Gaussian distribution function (dashed curve) are fitted to the data. The data are plotted (a) in...

The tensile strength of alumina based fibers is governed by the population of flaws. It displays a statistical character, which Is depicted with a Weibull distribution function. 

Weibull distribution function A probabiiity distribution function which can be used to describe the scatter in tensiie strength data for a given lot of fibers having the same cross section area and test gauge iength. 

Confidence and tolerance bounds on predictions that use the Weibull distribution function. 

Table 2 Wetting dynamic contact angle values of white beeswax coated potassium chloride microcapsules with different core-to-wall ratios and dissolution parameters of the Weibull distribution function (the medium for the wetting and dissolution measurements was distilled water)...

The wall material has lipophilic character, so the release curves are S-shaped, with an expressed lag-time period. To describe the release patterns of the nondisintegrating lipophilic matrices the Rosin-Rammler-Ben-nett-Sperling-Weibull distribution function (Eq. 3) [10] was used ... 

The mean dissolution time (t ) determined by the Weibull distribution function was shortened as the surface tension of the surfactant aqueous solution decreased (Table 3). In accordance with the contact angle decreasing, greater water uptake can be observed, parallel to the drug release rate increasing. 

Weibull distribution function is one of the widely used cumulative probability functions for predicting lifetime in reliability test [34]. This is because it can easily approximate the normal distribution, logarithmic normal distribution and exponential distribution functions. In addition, it is also possible to analyze data even when two or more failure modes are present at the same time. The cumulative probability F(t) of a failure system can be introduced just as Weibull distribution function based upon a weakest-link model [34], which is expressed as ... 

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