Beta factor

This method as superseded by the beta factor method which also lacks a physical basis. 

s method assumes that X, the total constant failure rate for each unit, can be expanded into independent and dependent failure contributions (equation 

For systems with more than two subsystems the beta-factor model, as presented, does not a I - tween different numbers of multiple failures. This siraplificat i 

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Table 3.5 6-1 (from NUREG-1150) provides generic beta factors. 

Example of the Beta-Factor Method Emergency Electric Power... 

This, more physical model that visualizes failure to result from random "shocks," was specialized from the more general model of Marshall and Olkin (1967) by Vesely (1977) for sparse data for the ATWS problem. It treats these shocks as binomially distributed with parameters m and p (equation 2.4-9). The BFR model like the MGL and BPM models distinguish the number of multiple unit failures in a system with more than two units, from the Beta Factor model,... 

Bochkarev AV, Trefilova AN, Tsurkov NA, Klinskii GD (2003) Calculations of beta-factors by ab initio quantum-chemical methods. Russian J Phys Chem 77 622-626 Bode BM, Gordon MS (1998) MacMolPlt a graphical user interface for GAMESS. J Mol Graphics and Modeling 16 133-138... 

Wittenberg, G.K., Haun, D.V., and Parsons, M.L., The use of free-energy minimization for calculating beta factors and equilibrium compositions in flame spectroscopy, Appl. Spectrosc., 33, 626, 1979. 

Before a total sieve tray pressure drop can be summed, the froth pressure in inches of clear liquid over the active area must be calculated. This froth height actually reduces the HHDS value by a factor called the aeration beta correction. This has been done by Smith, who plotted the aeration factor beta vs. FGA (see Eq. (3.120) for FGA). Equation (3.121) is a curve-fit of Smith s beta curve plot [16]. Generally a beta factor of 0.7 to 0.8 is calculated using Eq. (3.121). 

Correct the HHDS sieve tray hydraulic gradient with the beta factor (froth correction) ... 

In Figure 12-6 the third proposed design is shown. This case consists of two conventional transmitters (loo2 voting) with comparison diagnostics, a safety PLC, and two separate shutdown valves (loo2) with individual solenoids. The calculation results are shown in Table 12-7 using a common cause beta factor of 5% for the sensors and 10% for the final elements. 

Common Cause Beta Factor for Pressure Transmitters tcpt 0.02 ... 

Gate 2 represents the entire sensor subsystem. In addition to the Gate 5 results, common cause failures of the two sensors and the safety PLC input circuits are included. Given that the two switches are likely to be similar technology, a common cause beta factor of 5% was chosen. The simplified approximation equation for gate 2, PFD is ... 

Two different types of solenoid valves are used to block fuel flow to a burner in a SIS. The valves are piped in series. Both valves should close when a dangerous condition is detected. Both valves have one failure mode, fail-danger, with a failure rate of 0.0008 failures per year. Both valves are tested once every year. Based on the differences between the valves, a common cause beta factor of 0.01 is assigned. What is the PFD of the valve subsystem including common cause ... 

In the loo2 architecture, redundant units are used. This means that common cause must be included in the modeling. Using a Beta factor of 0.02, the model parameters are shown in Table F-4. 

Using the four total failure rates from Table F-3, common cause must be added to the model because redundant units are present. Using the same Beta factor as before (Beta = 0.02), the failure rates are calculated in Table F-11. 

In what follows only the Beta Factor Model is treated. As to the remaining models the reader is referred to the literature, e.g. [48]. 

The Beta Factor Model is a one parameter model in which the total failure rate of a component is split into an independent part and one due to common cause, i.e. 

The Beta Factor Model was originally developed for treating CCFs in twofold redundant systems of U.S. nuclear power reactors. A factor of 6 = 0.1 resulted. An evaluation of data of the collection in process plants described in [40] gave 6 = 0.084, which insinuates that B = 0.1 is a conservative value for analyses of process plants from the class investigated in [40]. 

However, the application of the model to systems with higher degrees of redundancy is problematic. This was reason for extending the Beta Factor Model to the Multiple Greek Letter (MGL) Model [48]. 

The fault tree for the undesired event filling level too high is shown in Fig. 9.50. High levels can be initiated by the faUure of the activation of control valve RV by LICAl and LICA2, the failure of the control valve RV itself or the failure of instrument air. Further potential causes are that LICAl and L1CA2 remain in their positions for functional tests after inspection or a CCF of both level measurements. The latter is treated using the Beta Factor Method with 6 = 0.1. 

There are many different models that can be used to apply common causes, but the most common (and the one preferred by EC 61508) is the Beta factor (j8) model. This model applies a /J factor between 0 and 1 representing the fraction of the failure of all affected inputs resulting from the common cause. For instance, a fl value of 0.1 implies that 10% of failures where aU inputs fail were in fact the result of a common cause. There are some specialised resources for appropriate CCF values that can be apphed, but fundamentally a sensitivity analysis should be performed to determine how much an effect the CCF has on the top event probability. A large influence would indicate the need for further analysis [see NASA Fault Tree Handbook paragraph 7.2]. 

The Beta factor model allows for the probability of occurrence to be calculated by adjusting the probability of the basic events using the Beta factor. These adjustments are shown in the fault tree above for the basic events. 

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