Weibull curve

Construction of a Weibull curve when no failures have occurred. 

Taking 100% overtopping for zero freeboard (the actual data are only a little lower), a Weibull curve can be fitted through the data. Equation (15.5) can be used to predict the number or percentage of overtopping waves or to establish the armor crest level for an allowable percentage of overtopping waves. 

Equation (8.42) shows that the relation between the fine powder ratio and external force P can be expressed by the Weibull equation. The broken curve (the relationship curve between the fine powder ratio and external force) is a Weibull curve. This is closely related to the agreement between SPCS and the Weibull distribution. Experimental results in the literature show that the parameter M of the WSCS has a value very close to the Weibull modulus in the SPCS data. Both relationships are reflections of the distribution of defects in the material. In the measurement of WSCS, the disabled probability of [E(P)] is the fine powder ratio (dm/m). 

Sometimes the data do not fit a single straight line, and may give rise to two straight lines. Such a distribution is called a bimodal one. If that happens, we can think of eliminating the distribution that gives a low value of m. What we should do is observe the fracture surface and identify the two types of cracks and the phases in which they occur. Once the identification is done, then we arrive at the possible causes for their occurrence. Once we know the causes, we can modify the fabrication steps to eliminate the cracks giving rise to the lower value of m. Thus, Weibull curves and fracture analysis can be used to improve a ceramic material. 

Explain why a plant accident is more likely to happen during startup of a new plant or a retro-fit process. Refer to Chapter 20 and careful review the presentation or tlie bathtub curve tliat is represented by the Weibull distribution. 

The Weibull distribution provides a iiiatliematical model of all tliree stages of the batlitub curve. Tliis is now discussed. An assumption about failure rate tliat reflects all tliree stages of tlie batlitub curve is... 

To illustrate probability calculations involving tlie exponential and Weibull distributions introduced in conjunction willi llie batlitub curve of failure rate, consider first llie case of a mansistor having a constant rate of failure of 0.01 per tliousand hours. To find the probability tliat llie transistor will operate for at least 25,000 hours, substitute tlie failure rate... 

The procedure is to fit the population frequency curve as a straight line using the sample moments and parameters of the proposed probability function. The data are then plotted by ordering the data from the largest event to the smallest and using the rank (i) of the event to obtain a probability plotting position. Two of the more common formulas are Weibull... 

Although the Noyes-Whitney equation has been used widely, it has been shown to be inadequate in modeling either S-shape experimental data or data with a steeper initial slope. Therefore, a more general function, based on the Weibull distribution [8], was proposed [9] and applied empirically and successfully to all types of dissolution curves [10] ... 

F. Langenbucher. Linearization of dissolution rate curves by the Weibull distribution. J. Pharm. Pharmacol. 

Alternative methods and algorithms may be used, such as the model-independent approach to compare similarity limits derived from multi-variate statistical differences (MSD) combined with a 90% confidence interval approach for test and reference batches (21). Model-dependent approaches such as the Weibull function use the comparison of parameters obtained after curve fitting of dissolution profiles. See Chapters 8 and 9 for further discussion of these methods. 

Fig. 8.6 Estimated risk of liver cancer, P(d), in relation to dose of aflatoxin, d, as determined with different dose-incidence models. The models for the different curves. are as follows OH. one-hit model MS, multi-stage model W, Weibull model MH, multihit model MB, Mantel-Bryan (log-probit model) (from Krewski and Van Ryzin, 1981).

Validation was carried out using the area under the dissolution curve, this being the major response to be optimized. The dissolution curves fitted the Weibull distribution. 

Drug release from controlled release matrix tablets has been described by many kinetic theories [20,21]. Fig. 4 illustrates the release profiles of the validated dissolution (experiments 12, 12b and 12c) and the release profiles obtained from fits to the Weibull, Higuchi and Hixson-Crowell models. Figs 5 and 6 show the curves after linear transformation. 

The analysis of covariance between a continuous variable (P is the curve shape parameter from the Weibull function) and a discrete variable (process) was determined using the general linear model (GLM) procedure from the Statistical Analysis System (SAS). The technique of the heterogeneity of slopes showed that there was no significant difference (Tables 5 and 6). 

A comparison of the simulation results and fittings with the Weibull and the power-law model is presented in Figure 4.7. Obviously, the Weibull model describes quite well all release data, while the power law diverges after some time. Of course both models can describe equally well experimental data for the first 60% of the release curve. 

Figure 4.7 Number of particles inside a cylinder as a function of time with initial number of drug molecules n0 = 2657. Simulation for cylinder with height 21 sites and diameter 21 sites (dotted line). Plot of curve n (t) = 2657exp (—0.04910 72), Weibull model fitting (solid line). Plot of curve n(t) = 2657 (l — 0.0941° 45), power-law fitting (dashed line).

In 1951, Weibull [116] described a more general function that can be applied to all common types of dissolution curves. This function was introduced in the pharmaceutical field by Langenbucher in 1972 [117] to describe the accumulated fraction of the drug in solution at time t, and it has the following form 1... 

Figure 9.3 depicts state probability curves for the Erlang and the Weibull distributions. The hazard rates as functions of time are also illustrated. For v 1 and 1, we obtain the behavior corresponding to an exponential... 

Following Huttinger (1990), we can correlate the modulus and strength of carbon fiber to its diameter. We make use of Weibull statistics to describe the mechanical properties of brittle materials (see Chapter 10). Brittle materials show a size effect, i.e. the experimental strength decreases with increasing sample size. This is demonstrated in Fig.8.10 which shows a log-log plot of Young s modulus as a function of carbon fiber diameter for three different commercially available carbon fibers. The curves in Fig. 8.10 are based on the following expression ... 

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