3. Determine the expected failure rates, f, and mean time between failures (MTBF), 1/f, for control systems with SIL1, SIL2, and SIL3 ratings with PFDs of 10-2, 10-3, and 10-4, respectively.

SIL1, SIL2, and SIL3 ratings with PFDs of 10-2, 10-3, and 10-4, respectively.

PFD is the failure probability denoted as P.

Let's define R as its inverse.

Therefore, R = 1 - P

The failure Rate f is given by

f = - In R.

The mean time between failures (MTBF), 1/f

For SIL1,

P = 2/10 = 0.2

R = 1 - 0.2 = 0.8

failure rate f = - In (0.8) = 0.2231

MTBF = 1/f = 4.481

For SIL2,

P = 3/10 = 0.3

R = 1 - 0.3 = 0.7

failure rate f = - In (0.7) = 0.3367

MTBF = 1/f = 2.804

For SIL3,

P = 4/10 = 0.4

R = 1 - 0.4 = 0.6

failure rate f = - In (0.6) = 0.5108

MTBF = 1/f = 1.9576