12.209. Before we introduce some PRA principles, it is desirable to clarify the difference between deterministic and probabilistic safety anal­ysis. The term deterministic has a philosophical basis which refers to the mechanical correspondence between causes and effects. For example, we could consider a small-break LOCA in a PWR as a “cause,” and by suitable analytical modeling determine the maximum fuel clad temperature as a function of the break area. The clad temperature would be the “effect” and when related to prescribed limits provides us with a “safety margin.” An evaluation of numerous safety margins is required in licensing appli­cations (§12.132). In contrast, probabilistic safety analysis utilizes statistical methods to evaluate failure probabilities resulting from various initiating events. Here we are concerned with binary states; i. e., an initiating event might be the transition of a given component from an operating state to
a nonoperating state. Then this state could affect the condition of related components, as we shall see.

Elementary Binary State Concepts

12.210. As an introduction to failure concepts, let us consider a com­ponent that is either functioning normally or failed. We define the relia­bility, R(t), as the probability of survival to age t. Then,

number surviving at t ‘ total sample (population)’

We can define unreliability, F(t), as the probability of failure up to age t (t not included):

, 4 number of failures before t

Fit) = ————- —————

7 population

Now, R(t) = 1 — F(t). If we consider the proportion of the population, or sample, that will fail between tx and t2, we can introduce the failure probability density, f(t), where

F(t2) ~ F(t0 = [‘7(0 dt,

J’ і

where f(t) dt is the probability of failure in dt about t:

We are also interested in the rate of failure, r(t), which is sometimes called the hazard rate. This is the probability of failure per unit time at age t i. e., the device must have survived to time t:

№ m

1 — Fit) R(ty

Here R(t) is the number of survivals at t divided by the initial population.

12.211.The behavior of r(t) for many devices is described by the classic “bathtub curve” shown in Fig. 12.15. Characteristically, there are signif­icant early failures during a burn-in period arising from poor manufacturing quality control. Subsequently, there is a flat period of random failures followed by a rising rate in the wear-out range. This concept provides only

BURN-IN

PERIOD

TIME, t

a “taste” of a discipline known as reliability engineering, in which com­ponent and system performance are analyzed.