# Risk management, Fragility, Uncertainties, Fault Tolerant Systems, Explosion

1. A propane tank is located 100 m away from the nearest buildings. The fragility to explosion can be expressed by

p = 0 for I < 0.7 MPa ? s

p = aI for 0.7 < I < 1.3 MPa ? s

p = 1 for I > 1.3 MPa ? s

Accounting for various release and ignition scenarios, including uncertainties, the estimated probability distribution of impulses resulting from such explosions is given by the following set of values:

I (MPa ? s) 0.1 0.3 0.5 0.7 0.9 1.2

p 0.06 0.1 0.4 0.3 0.1 0.04

Supposing that you can improve the building so that you can affect the threshold of damage, increasing it above 0.7 MPa ? s, and supposing the high end at 1.3 MPa ? s remains the same, and so is the linear in-between behavior:

(a) Where would you place the lower threshold if you wanted the failure probability not to exceed 5%?

(b) What would you do if your answer in (a) was the best you could do to improve the fragility, and you wanted a failure probability of less than 0.1% (moral certainty) for no failures?

2. A fault-tolerant nuclear reactor protection system consists of 3 processors and 6 memories. The system will fail if (a) any 2 of the 3 processors fail, or (b) any 2 of the 6 memories fail. The nominal failure rates of these processors and memories, provided by the manufacturers, are 1 per 10,000 hours and 1 per 2,000 hours respectively.

(a) Calculate the probability that the system will fail in 3,000 hours.

(b) Supposing no failure of any component occurs in 5,000 hours of operation. What would be your revised estimate of failure rates of the components in your system?

3. In a failure tolerant system you have a choice between the following failure logics:

(a) 1 out of 4

(b) 2 out of 8

(c) 2 out of 5

(a). Assuming that the probability of failure of each component is the same (p), which one of the three options would you prefer?

(b). What if the price of a component was $50,000 , and the loss associated with system failure was $100,000 ?

4. A team of consultants is asked to evaluate the safety merits of a new nuclear power plant design. The designer company is very experienced, and one can assure that the design is good with high probability (say 98%). Within the time limitation of the review, we can assume that there is a 5% chance that the consultant team will miss important flaws.

(a). Given a positive report by the consultant, what is the probability of a faulty design?

(b). In light of your result, if the consultants were paid $200,000 and considering the loss from a faulty design could be as high as $10,000,000, was this inspection a good investment?

#### Solution Preview

1. A propane tank is located 100 m away from the nearest buildings. The fragility to explosion can be expressed by

p = 0 for I < 0.7 MPa ? s

p = aI for 0.7 < I < 1.3 MPa ? s

p = 1 for I > 1.3 MPa ? s

Accounting for various release and ignition scenarios, including uncertainties, the estimated probability distribution of impulses resulting from such explosions is given by the following set of values:

I (MPa ? s) 0.1 0.3 0.5 0.7 0.9 1.2

pload 0.06 0.1 0.4 0.3 0.1 0.04

Supposing that you can improve the building so that you can affect the threshold of damage, increasing it above 0.7 MPa ? s, and supposing the high end at 1.3 MPa ? s remains the same, and so is the linear in-between behavior:

(a) Where would you place the lower threshold if you wanted the failure probability not to exceed 5%?

At I = 1.3 MPa.s, the cumulative probability of fragility to explosion is equal to 1, so a = 1/1.3 = 0.7692.

If the lower threshold is 0.7 MPa.s, the discrete cumulative probability of fragility to explosion is given as follows:

I (MPa ? s) 0.1 0.3 0.5 0.7 0.9 1.2

pfragility 0 0 0 0 0.6923 0.9231

The failure probability is

Pfailure =  pload x pfragility = 0.06 x 0 + 0 x 0.1 + 0 x 0.4 + 0 x 0.3 + 0.6923 x 0.1 + 0.04 x 0.9231 = 0.1062

This value exceeds 5%. To lower the failure probability to 5%, we need to raise the threshold to 0.9 MPa.s. In that case, the discrete cumulative probability of fragility to explosion is given as follows:

I (MPa ? s) 0.1 0.3 0.5 0.7 0.9 1.2

pfragility 0 0 0 0 0 0.9231

The failure probability with the threshold of 0.9 MPa.s is

Pfailure =  pload x pfragility = 0.06 x 0 + 0 x 0.1 + 0 x 0.4 + 0 x 0.3 + 0.0 x 0.1 + 0.04 x 0.9231 = 0.0369 < 5%

(b) What would you do if your answer in (a) was the best you could do to improve the fragility, and you wanted a failure probability of less than 0.1% (moral certainty for no failures)?

Since we can not improve the fragility, the only method to lower the failure probability is to reduce the load probability. This can be done through installation of safety systems and shileds. If we keep the threshold at 0.9 MPa.s , we need to reduce the probability of load I = 1.2 MPa.s to be less than 0.001/0.9231 ~0.0011.

2. A fault-tolerant nuclear reactor protection ...

#### Solution Summary

Problems involving fragility to explosion, failure of fault tolerant systems and Bayesian probability are solved in detail.