Reliability Engineering Tutorial Questions (With Solution) (1)

CN3135 PROCESS SAFETY, HEALTH AND ENVIRONMENT (RELIABILITY ENGINEERING TUTORIAL) Question 1: Given the fault tree gates shown in Figure 1 below and the following set of failure probabilities: (i)

(ii)

T

(v)

2

L1

2

L1

L2

OR

OR

3

2

AND

1

1

L1 OR

AND

T AND

1

T

OR

1

2

(iv)

T

OR

AND

1

(iii)

T

3

2

3

4

Figure 1 Component

Failure Probability

1

0.1

2

0.2

3

0.3

4

0.4

(a) Determine an expression for the probability of the top event in terms of the component failure probabilities. (b) Determine the minimal cut sets. (c) Compute a value for the failure probability of the top event. Use both the expression of part (a) and the fault tree itself.

Question 1 (SOLUTION):

Question 2: The storage tank system shown in Figure 2 is used to store process feedstock. Overfilling of storage tanks is a common problem in the process industries. To prevent overfilling, the storage tank is equipped with a high-level alarm and a high-level shutdown system. The high-level shutdown system is connected to a solenoid valve that stops the flow of input stock. System High-level alarm Operator stops flow High-level switch system

Failures / demand 0.01 0.1 0.01

Figure 2 – Level Control System with Alarm (a) Develop an event tree for this system using the “failure of level indicator” as the initiating event. Given that the level indicator fails 4 times / yr, estimate the number of overflows expected per year. Use the data in the table provided above. (b) Develop a fault tree for the top event of “storage tank overflows.” Use the data in Table 12-1 (from Textbook) to estimate the failure probability of the top event and the expected number of occurrences per year. Determine the minimal cut sets. What are the most likely failure modes? Should the design be improved?

Question 2 (SOLUTION):

Reliability (R = e (-mt)) 0.549 0.183 0.803 0.183 0.957 0.657 0.183

Probability (P = 1 - R) 0.451 0.817 0.197 0.817 0.043 0.343 0.817 0.853 0.825 0.704 0.838 0.880 0.737

1 2 3 4 5 6 7

Flow control valve Level measurement Chart recorder Level measurement Alarm Solenold valve Level Switch / measurement

Faults/yr (m = -ln(R)) 0.6 1.7 0.22 1.7 0.044 0.42 1.7

2&3 4&5 E&F 1&C 6&7 B&D

OR gate (F) OR gate (E) AND gate (C) OR gate (B) OR gate (D) AND gate (A)

1.920 1.744 1.218 1.818 2.118 1.335

0.147 0.175 0.296 0.162 0.120 0.263

-ln(R)

R = 1-P

MBTF = 1/m =

0.749

Question 3: Using the system shown in Figure 3 below, draw the fault tree and determine the failure characteristics of the top event (vessel pressure exceeds MAWP)

Figure 3 – A control system to prevent the pressure from exceeding the MAWP

Question 3 (SOLUTION):

1 Level measurement 2 Controller 3 Pressure Switch 4 Valve 5 Pressure Measurement 6 Controller 7 Pressure Switch 8 Valve LIC = PIC = Probability of top event = reliability, R of top event = m of top event = MTBF =

Faults/yr (m=-LN(R)) 1.7 0.29 0.14 0.6 1.41 0.29 0.14 0.6

Reliability (R = e (-mt)) 0.183 0.748 0.869 0.549 0.244 0.748 0.869 0.549

Probability (P = 1 - R) 0.817 0.252 0.131 0.451 0.756 0.252 0.131 0.451

2.73 2.44 0.853 0.147 1.919 0.521

0.065 0.087

0.935 0.913

Question 4: Determine P, R,  and the MTBF for the top event of the system shown in Figure 4. Also list the minimal cut sets.

Figure 4 – Determine the failure characteristics of the top event

Question 4 (SOLUTION):

1 2 3 4 5 1&2 3&4 34 & 5 345 & 12

AND gate OR gate OR gate OR gate

Faults/yr (m = -ln(R)) 0.25 0.4 0.3 0.4 1

Reliability (R = e (-mt)) 0.779 0.670 0.741 0.670 0.368

Probability (P = 1 - R) 0.221 0.330 0.259 0.330 0.632

0.076 0.7 1.7 1.776

0.927 0.497 0.183 0.169

0.073 0.503 0.817 0.831

MBTF = 1/m =

0.563

Question 5: Using the system shown in Figure 5, draw the fault tree and determine the failure characteristics of the top event (vessel overflow). In this problem you have human intervention; that is, when the alarm sounds, someone turns off valve 7.

Figure 5 – Control system to prevent vessel overflow

Question 5 (SOLUTION):

1 2 3 4 5 6 7 12 12 & 3 & 7 456 1237 & 456

Chart Alarm Level measurement Solenoid valve Controller Level measurement Hand Valve

Faults/yr (m = -ln(R)) 0.22 0.044 1.7 0.42 0.29 1.7 0.13

Reliability (R = e (-mt)) 0.803 0.957 0.183 0.657 0.748 0.183 0.878

Probability (P = 1 - R) 0.197 0.043 0.817 0.343 0.252 0.817 0.122

AND gate OR gate OR gate AND gate

0.009 1.839 2.410 1.450

0.991 0.159 0.090 0.235

0.009 0.841 0.910 0.765

MBTF = 1/m =

0.69