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:

AND

T

L1

1 2

OR

L2

3 4

OR

T

1

OR

AND

L1

32

AND

T

1L1

2 3

OR

T

1 2

ORAND

T

21

(i) (ii) (iii) (iv)

(v)

Figure 1

(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.

Component Failure Probability

1 0.1

2 0.2

3 0.3

4 0.4

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.

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?

System Failures / demand

High-level alarm 0.01

Operator stops flow 0.1

High-level switch system 0.01

Question 2 (SOLUTION):

Faults/yr (m = -ln(R)) Reliability (R = e(-mt)) Probability (P = 1 - R)

1 Flow control valve 0.6 0.549 0.451

2 Level measurement 1.7 0.183 0.817

3 Chart recorder 0.22 0.803 0.197

4 Level measurement 1.7 0.183 0.817

5 Alarm 0.044 0.957 0.043

6 Solenold valve 0.42 0.657 0.343

7 Level Switch / measurement 1.7 0.183 0.817

2 & 3 OR gate (F) 1.920 0.147 0.853

4 & 5 OR gate (E) 1.744 0.175 0.825

E & F AND gate (C) 1.218 0.296 0.704

1 & C OR gate (B) 1.818 0.162 0.838

6 & 7 OR gate (D) 2.118 0.120 0.880

B & D AND gate (A) 1.335 0.263 0.737

-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):

Faults/yr (m=-LN(R)) Reliability (R = e(-mt)) Probability (P = 1 - R)

1 Level measurement 1.7 0.183 0.817

2 Controller 0.29 0.748 0.252

3 Pressure Switch 0.14 0.869 0.131

4 Valve 0.6 0.549 0.451

5 Pressure Measurement 1.41 0.244 0.756

6 Controller 0.29 0.748 0.252

7 Pressure Switch 0.14 0.869 0.131

8 Valve 0.6 0.549 0.451

LIC = 2.73 0.065 0.935

PIC = 2.44 0.087 0.913

Probability of top event = 0.853

reliability, R of top event = 0.147

m of top event = 1.919

MTBF = 0.521

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):

Faults/yr (m = -ln(R)) Reliability (R = e(-mt)) Probability (P = 1 - R)

1 0.25 0.779 0.221

2 0.4 0.670 0.330

3 0.3 0.741 0.259

4 0.4 0.670 0.330

5 1 0.368 0.632

1 & 2 AND gate 0.076 0.927 0.073

3 & 4 OR gate 0.7 0.497 0.503

34 & 5 OR gate 1.7 0.183 0.817

345 & 12 OR gate 1.776 0.169 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):

Faults/yr (m = -ln(R)) Reliability (R = e(-mt)) Probability (P = 1 - R)

1 Chart 0.22 0.803 0.197

2 Alarm 0.044 0.957 0.043

3 Level measurement 1.7 0.183 0.817

4 Solenoid valve 0.42 0.657 0.343

5 Controller 0.29 0.748 0.252

6 Level measurement 1.7 0.183 0.817

7 Hand Valve 0.13 0.878 0.122

12 AND gate 0.009 0.991 0.009

12 & 3 & 7 OR gate 1.839 0.159 0.841

456 OR gate 2.410 0.090 0.910

1237 & 456 AND gate 1.450 0.235 0.765

MBTF = 1/m = 0.69