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SENSITIVITY AND PROFITABILITY ANALYSIS OF TWO-UNITS AMMONIA/UREA PLANT
Sara Salim Al Oraimi •
Department of Mechanical and Industrial Engineering, National University of Science and Technology , Sultanate of Oman
S yed M ohd Rlzwan •
Department of Applied Mathematics and Science, National University of Science and Technology , Sultanate of Oman
Kajal Sachdeva •
Department of Mathematics, Maharshi Dayanand University, Rohtak, 124001, Haryana, India
Abstract
This paper presents a reliability modelling of a two-unit ammonia/urea plant. Real maintenance data of the production plant have been used for this purpose. Four types of failure were noted: process, electrical, mechanical and instrumental failures. Both ammonia/urea formation units work in parallel and do not fail simultaneously. Various reliability indices of the plant, such as availability, busy period for repair, and expected number of repairs for each type of failure, have been obtained. Markov processes and regenerative point techniques are used for analysis. Profit analysis for the plant is also done, along with a graphical representation of various parameters. Finally, sensitivity analysis is carried out to see the impact of varied parameters on the profit function of the plant.
Keywords:Ammonia plant; Markov process; regenerativ e point techniques; repairs; failures
1. Introduction
Many resear chers have studied complex industrial systems under various operating conditions and presumptions, contributing to the discipline of reliability modelling and analysis. Rizw an et al. [1,2] presented a reliability modelling strategy and its application to industries, specificall focusing on a biscuit manufacturing factory controlled by a single-unit and two-unit hot standby PLC system. Mathe w et al. [3,4] discussed the reliability modelling of single-unit and two-unit systems in a continuous casting plant. They utilized real maintenance data from a steel production plant to analyze the system's performance and identify different types of failures. Also, Mathe w et al. [5] did a comparativ e analysis of the two models of the CC plant. Mathe w et al. [6,7] presented reliability modelling in an actual CC plant with different installed and full installed capacities, where two EOT cranes operate in parallel. Padma vati et al. [8-13] evaluated the impact of prioritizing repair over maintenance on the overall reliability and availability of the desalination plant. Also, they discussed the implications of the resear ch for the design and operation of desalination plants in terms of cost-effectiveness and efficienc by taking different assumptions for failures and repairs.Rizw an et al. [14-16] analyzed the reliability of a wastewater treatment plant. They estimated various reliability indices associated with the plant. Also, they
highlighted the importance of regular monitoring, repairs, and replacements in maintaining the reliability of systems.
Al Rahbi et al. [17-23] analyzed the reliability of single unit/ multiple units of a rodding anode plant in the aluminium industr y with single/numer ous repair ers and optimized maintenance strategies, impr oved system reliability , and reduced downtime, leading to increased productivity and cost savings. Taj et al. [24-30,33] evaluated reliability analysis conducted on the cable plant's single- or three-unit machine subsystem with repair priority over maintenance. Rizw an et al. [31] examined the three pumps' performance in distributing desalinated water. The study included maintenance data collected over fi e years, encompassing various failur e reasons, restoration times, and waiting times. Rizw an et al. [32] explor ed the reliability and sensitivity analysis of Membrane Biofil Fuel Cells. Thus, the literatur e has widely discussed the reliability modelling and analysis of comp lex industrial systems in various failur e/maintenance circumstances. But the concept of reliability analysis for ammonia/ur ea manufacturing plants has yet to be discussed.
The demand for fertilizers is growing day by day around the world to meet agricultural requir ements. The most used or consumed fertilizer is UREA, which is manufactur ed from ammonia, from different industrial chemical reactions. The UREA fertilize r manufacturing facilities consist ammonia manufacturing plant along with a urea plant. To meet the growing demand for urea in the current market, these facilities must keep production continuously with maximum capacity to meet the market's growing demand. For continuous operation, these facilities must function the plant and equipment efficiently throughout the year without significan technical or maintenance issues. Any unexpected operational failur e, breakdo wn, or downtime may cause plant productivity and efficienc . For that, the operation and maintenance strategies are critical as they help maintain the life and smooth operation of the equipment. These strategies also help reduce plant do wntime.Also, further analysis and resear ch techniques for plant performance, productivity , reliability , availability , maintainability , sensitivity [34, 35], etc., may be carried out to ensur e continuous and smooth plant operations. This paper provides sensitivity and profitabilit analysis, along with reliability analysis of parallel ammonia and urea plants worldwide that have operated for more than 15 years. The research is based on the actual plant data, with some assumptions, failur e rate, or probability.
2. N otations
The following are the notations used in the analysis: Au = failur e rate of ammonia plant
p1/p2/ p3/ p4= probability of process failure/ electrical failure/ mechanical failure/ instrumental failur e in unit 1.
p5/ pe/ p7/ p8=probability of process/ electrical/ mechanical/ instrumental failur e in unit 2. a1/a2/ a3/ a4=repair rate of process/ electrical/ mechanical/ instrumental failur e in unit 1. a5/ a^/ a7/ a7=repair rate of process/ electrical/ mechanical/ instrumental failur e in unit 2. fu(t) = p.d.f. of failur e time.
g1 (t)/ g2 (t)/ g3 (t)/ g4(t)=p.d.f. of repair time due to process/ electrical/ mechanical/ instrumental failur e in unit 1.
g5(t)/ g6(t)/ g7(t)/ g8(t)=p.d.f. of repair time due to process/ electrical/ mechanical/ instrumental failur e in unit 2.
3. Data Summary
The real data from a urea manufacturing company is summarized as follows:
Probability of process failur e in unit 1, p1 = 0.2088
Probability of electrical failur e in unit 1, p2 = 0.0220
Probability of mechanical failur e in unit 1, p3 = 0.1758
Probability of instrumental failur e in unit 1, p4 = 0.1978
Probability of process failur e in unit 2, p5 = 0.1648
Probability of electrical failur e in unit 2, p6 = 0.0220 Probability of mechanical failur e in unit 2, p7 = 0.1868 Probability of instrumental failur e in unit 2, p8 = 0.0220 Failur e rate of urea plant, Au= 0.00031 per hour Repair rate of process failur e in unit 1, a1 = 0.0249 per hour Repair rate of electrical failur e in unit 1, a2 = 0.2 per hour Repair rate of mechanical failur e in unit 1, a3 = 0.0081 per hour Repair rate of instrumental failure in unit 1, a4= 0.01833 per hour Repair rate of process failur e in unit 2, a5 = 0.0150 per hour Repair rate of electrical failur e in unit 2, «6 = 0.0175 per hour Repair rate of mechanical failur e in unit 2, a7 = 0.0057 per hour Repair rate of instrumental failur e in unit 2, a8 = 0.0392 per hour
4. Model Description and Assumptions
• Initially, we have an operativ e ammonia manufacturing plant composed of two parallel units: Unit 1 and Unit 2.
• The four types of failures are observed in both units, i.e., process, electrical, mechanical, and instrumental.
• Both units cannot fail simultaneously .
• Repair is carried out upon failur es.
• Failure rate and repair rates all are taken as general.
5. Stochastic Model
Table 1 shows the rates of transition from state i (Si) to state j (Sj).The set of states {0,1,2,3...,8} all are operativ e and regenerativ e. Table 1
State Transition Table
Si / Sj So S1 S2 S3 S4 S4 S6 S7 SS
So 0 Pif" (t) P2f" (t) P3f" (t) P4f" (t) P5f" (t) P6f" (t) P7f" (t) Psf" (t)
Si gi (t) 0 0 0 0 0 0 0 0
S2 g2 (t) 0 0 0 0 0 0 0 0
S3 g3 (t) 0 0 0 0 0 0 0 0
S4 g4 (t) 0 0 0 0 0 0 0 0
S5 g5 (t) 0 0 0 0 0 0 0 0
S6 g6 (t) 0 0 0 0 0 0 0 0
S7 g7 (t) 0 0 0 0 0 0 0 0
SS gs (t) 0 0 0 0 0 0 0 0
wher e,
State 0 (So) - Both urea processing machines unit 1 and 2 operativ es.
State 1 (Si) - Unit 1 failed due to process failur e, nd Unit 2 is still operativ e.
State 2 (S2) - Unit 1 failed due to electrical failur e and Unit 2 is still operativ e.
State 3 (S3) - Unit 1 failed due to mechanical failur e, and Unit 2 is still operativ e.
State 4 (S4) - Unit 1 failed due to instrumental failur e, and Unit 2 is still operativ e.
State 5 (S5) - Unit 1 is operativ e, and Unit 2 failed due to process failur e.
State 6 (S6) - Unit 1 is operativ e, and Unit 2 failed due to electrical failur e.
State 7 (S7) - Unit 1 is operativ e, and Unit 2 failed due to mechanical failur e.
State 8 (Ss) - Unit 1 is operativ e, and Unit 2 failed due to instrumental failur e.
The transition probability from state i (S,) to state j (Sj), (t) is given by
qoi ( ) = pi/" (t), qio( ) = gi(t
qo2( ) = p2/" (t), q20( ) = g2 (t
qo3( ) = p3/" (t), q30( ) = g3(t
qo4 ( )= P4/" (t), q40 ( ) = g4 (t
qo5( ) = p5/" (t), q50( ) = g5 (t
qo6 ( ) = p6/" (t), q60 ( ) = g6 (t
qo7( ) = p7/" (t), q70( ) = g7 (t
qo8 ( )= p8/" (t), q80 ( ) = g8 (t
The steady-state probability , as
Poi = Pi, P02 = P2, P03 = P3, P04 = P4, P05 = P5, Роб = Рб, P07 = P7, P08 = P8
P10 = P20 = P30 = P40 = P50 = P60 = P70 = P80 = 1 (1)
Sojourn time (^)„ i.e., mean stay time in particular state i, is given as
Ho pTO =0 ./" (t) dt, H5 = JT t. g5 (t) dt,
Hi r> TO =0 .gi (t) dt, H6 = /oTO t.g6(t) dt,
Ц2 r> TO =0 .g2 (t) dt, H7 = /oTO t.g7(t) dt,
H3 r> TO =0 .g3 (t) dt, H8 = /oTO t.g8 (t) dt,
H4 Г TO =0 .g4 (t) dt,
The contribution to mean sojourn time, My,is given by = J0° (t) dt.It can be verifie that
m0i + m02 + m03 + m04 + m05 + m06 + m07 + m0s = ^o,
mio = ^i, m20 = ^2, m30 = m"3, №40 = ^4,
№50 = ^5, №60 = ^6, №70 = mM7, m80 = ^
6. System Performance Measures 6.1. Availability of the System
Defin
A"(t) = probability that it is operativ e at time t, given that the system is in state i at time t = 0.
Sara Salim Al Oraimi, Syed Mohd Rizw an, Kajal Sachde va
SENSITIVITY AND PROFITABILITY ANALYSIS OF RT&A, No 1 (77)
TWO-UNITS AMMONIA / UREA PLANT Volume 19, March 2024
Using the state transitions, we get the following equations:
AU (t) = Mo (t) + qoi (t)© AU (t) + qo2 (t)© AU (t) + qo3 (t)© AU (t) + qo4 (t)© AU (t) + qo5 (t)© AU (t)
+ qo6 (t)© AU (t) + qo7 (t)© AU (t) + qo8 (t)© AU(t) AU (t) = Mi (t) + q 10 (t)© AU (t) A2" (t) = M2 (t) + q2o (t)© AU (t) AU (t) = M3 (t)+ q30 (t)© AU (t) AU (t) = M4 (t)+ q40 (t)© AU (t) AU (t) = M5 (t)+ q50 (t)© AU (t) AU (t) = M6 (t)+ q6o (t)© AU (t) AU (t) = M7 (t)+ q7o (t)© AU (t) AU (t) = M8 (t)+ q8o (t)© AU (t)
(2)
wher e
Mi (t)= probability that the system stays instate i while operating rather than transferring to any other state.
Taking Laplace transfor m of equations (28)-(36) and solving for AU (s), we get
au (s) = NUM
Ao (S) = DU(s)
wher e
N (s) = MQ (s) + q0i (s) M^ (s) + q02 (s) MQ (s) + q03 (s) M3 (s) + q04 (s) M4 (s)
+qo5 (s) M2 (s) + q06 (s) M2 (s) + q07 (s) M2 (s) + q08 (s) MQ (s) (3)
DU (s) =1 - q0i (s)q1o(s) - q02 (s)q2o(s) - q03 (s)q33o(s) - q04 (s)q4o(s) - q05(s)
qs5o (s) - qo6 (s)q6o (s) - qo7 (s)q7o (s) - q08 (s)q8o (s) (4)
The steady-state availability of the system is given by :
AU = lim s AU2 (s) = lim s NUM = NM = N1 (say) (5)
Ao=simos.Ao (s)=simos. Du (s) = Du (0) = Du (say) (5)
wher e
N = p-o + pi Pi + P2 U2 + P3P3 + P4P4 + P5P5 + P6P6 + P7P7 + P8 P8 (6)
DU = Pi Pi + P2P2 + P3P3 + P4P4 + P5 P5 + P6P6 + P7P7 + P8P8 + Po
6.2. Busy Period for Repair
The expected time for which the repair man is busy for the repair of unit 1 and unit 2 due to process failure in steady state is given by:
PBUi = PNf/ DU and PBU2 = PNf/ DU
wher e
PN2u1 = P01 Ui = Pi Ui
PN2u2 = P05 U5 = P5 U5
The expected time for which the repair man is busy for the repair of unit 1 and unit 2 due to electrical failur e in steady state is giv en by:
EB«1 = EN«1/ D« and EB«2 = EN«2/ D«
wher e
EN«1 = P02 «2 = P2 «2 EN«2 = P06 «6 = P6 «6
The expected time for which the repair man is busy for the repair of unit 1 and unit 2 due to mechanical failur e in steady state is giv en by:
MB«1 = MN«1/ D« and MB«1 = MN«1/ D«
wher e
MN«1 = p03 «3 = P3 «3 MN«2 = p07 «7 = p7 «7
The expected time for which the repair man is busy for the repair of unit 1 and unit 2 due to instrumental failur e in steady state is giv en by:
JB"1 = IN«1/ D« and IB«1 = IN«1/ D« wher e
IN«1 = p04 «4 = p4 «4
IN«2 = P08 «8 = p8 «8
6.3. Expected Number of Repairs
The expected number of repairs in unit 1 and unit 2 due to process failure in steady state is given by:
PR«1 = PN«1/ D« and PR«2 = PN3«2/ D«
wher e
PN«1 = P01P10 = P1 PN«2 = P05 P50 = P5
The expected number of repairs in unit 1 and unit 2 due to electrical failur e in steady state is giv en by:
ER«1 = EN«1/ D« and ER«2 = EN«2/ D«
wher e
EN«1 = P02 P20 = P2 EN«2 = p06 P60 = P6
The expected number of repairs in unit 1 and unit 2 due to mechanical failur e in steady state is giv en by:
MR«1 = MN«1/ D« and MR«2 = MN«2/ D«
wher e
MN«1 = P03 P30 = P3 MN«2 = P07 P70 = P7
The expected number of repairs in unit 1 and unit 2 due to instrumental failur e in steady state is giv en by:
JR«1 = IN«1/ D« and JR«2 = JN«2/ D«
Sara Salim Al Oraimi, Syed Mohd Rizw an, Kajal Sachde va
SENSITIVITY AND PROFITABILITY ANALYSIS OF RT&A, No 1 (77)
TWO-UNITS AMMONIA / UREA PLANT Volume 19, March 2024
wher e
IN"1 = po4 P40 = P4 IN"2 = po8 P80 = P8
7. Profit Analysis of the System
The profi equation of the system is as follows:
P" = C0A" - C1 (PB"1 + PR"1 ) - C2 (PB"2 + PR"2) - C3 (EB"1 + ER"1 ) - C4(EB"2 + ER"2)
-C5 (MB"1 + MR"1 ) - C6 (MB"2 + MR"2 ) - C7 (IB"1 + IR"1 ) - C8 (IB"2 + IR"2 )
wher e
C0= Revenue generated by the system
C1 (C2)/ C3(C4)/ C5(C6)/ C7(C8): - Cost per unit time for engaging the repairman and cost for repair due to process/electrical/mechanical/instrumental failur e in unit 1 (unit 2).
8. N umerical A nalysis
In this section, interpretation from graphs and tables has been made for the above-obtained system measur es in Section 4 and Section 5. Let us assume all the failures and repair times follow exponential distribution along with their p.d.f. as:
f" (t) = A"e-À"f,
gi (t) = a,e-a'f, i = 1,2,3, ..8.
Using the values as written in Section 3 that is calculated from real data from a manufacturing company , we get system effectiv eness measur es as:
Availability of Ammonia Plant, A" = 1
Busy Period for Repair of Unit 1 due to Process Failur e, PB"1 = 0.0025
Busy Period for Repair of Unit 2 due to Process Failur e, PBu2 = 0.0033
Busy Period for Repair of Unit 1 due to Electrical Failure, EB"1 = 3.3209 * 10-5
Busy Period for Repair of Unit 2 due to Electrical Failure, EB"2 = 3.7953 * 10-4
Busy Period for Repair of Unit 1 due to Mechanical Failur e, MB"1 = 0.0066
Busy Period for Repair of Unit 2 due to Mechanical Failur e, MB"2 = 0.0099
Busy Period for Repair of Unit 1 due to Instrumental Failur e, IB"1 = 0.0033
Busy Period for Repair of Unit 2 due to Instrumental Failure, IB"2 = 1.6943 * 10-4
Excepted no. of Repair of Unit 1 due to Process Failur e, PR"1 = 6.3036 * 10 5
Excepted no. of Repair of Unit 2 due to Process Failur e, PR"2 = 4.9753 10-5
Excepted no. of Repair of Unit 1 due to Electrical Failur e, ERu1 = 6.6418 * 10-6
Excepted no. of Repair of Unit 2 due to Electrical Failur e, ER"2 = 6.6418 * 10-6
• Excepted no. of Repair of Unit 1 due to Mechanical Failure, MRu1 = 5.3074 * 10-5
• Excepted no. of Repair of Unit 2 due to Mechanical Failure, MRu2 = 5.6395 * 10-5
• Excepted no. of Repair of Unit 1 due to Instrumental Failure, IRu1 = 5.9715 * 10-5
• Excepted no. of Repair of Unit 2 due to Instrumental Failur e, IRu2 = 5.8712 * 10-5
The graph of profi Function (Pu) w.r.t. revenue (C0) for different values of repair rate («1) has been shown in Figure 1.
Figure 1: Change in Profit w.r.t. Revenue and Repair Rate
It shows that the increase in revenue and repair rate increases profit Also, the cut-off points for the system to be profitabl can be observed in Fig. 1.:
• For Co > 108.2567 and a2 = 0.05, Pu > 0.
• For Co > 112.6834 and a2 = 0.01, Pu > 0.
• For Co > 115.8273 and a2 = 0.005, Pu > 0
Similarly , we can draw graphs of the profi function with other parameters to see its effect and cut-off points when the system is profitable
Table 2. Sensitivity and Relativ e Sensitivity Analysis of Profi Function
Parameter Sensitivity Analysis Relative Sensitivity Analysis
Au -3.3476 * 105 -0.0212
a1 59.2593 3.0154 * 10-4
a2 1.0047 4.1063 * 10-5
«3 3.8409 * 103 0.0064
a4 3.5789 * 103 0.0134
a5 17.3446 5.3167 * 10-5
a6 5.0857 1.8188 * 10-5
a7 578.7666 6.7416 * 10-4
as 3.7234 2.9827 * 10-5
C0 1 1.0218
C1 -0.0026 -3.6631 * 10-4
C2 -0.0034 ] - 1.2854 * 10-4
C3 -3.9851 * 10-5 -5.0143 * 10-5
C4 -3.8617 * 10-4 -2.6915 * 10-5
C5 -0.0066 -0.0065
C6 -0.01 -8.9916 * 10-4
C7 -0.0033 -0.0137
Cs -1.7607 * 10-4 -3.4829 * 10-5
Table 2 shows the sensitivity and relativ e sensitivity analysis [34] of the profi function concerning different parameters that affect the system's profit It shows that the profi function decreases rapidly with the change in C3 and increases with the shift in a3.
Also, the decreasing order in which parameters affect the profi function from Table 2 as:
C0 > AM > C7 > a4 > C5 > a3 > C6 > a7 > C1 > a1 > C2 > a5 > C3 > alpha2 > C8 > a8 > C4 >
9. Conclusion
In this paper, the parallel functioning of two units of an ammonia/ur ea plant reliability modelling has been examined. The availability of the plant, the busy period for repairs, and the anticipated number of repairs for each type of failur e have all been obtained as reliability indices. Profi analysis for the plant is also carried out along with the graphical representation with respect to various parameters. Profi increases when revenue and repair rates both increase. The cut-off point is also drawn to determine when a system is profitable Finally, sensitivity analysis is perfor med to assess the effect of various parameters on the plant's profi function. It demonstrates that, in comparison to other factors, revenue and system failur e rate have the most significan impact on the profi function. The model forecasts the failure and repair conditions based on the optimized reliability and profitabilit results.
Acknowledgement
This resear ch is funded under a student's resear ch grant from the College of Engineering, National University of Science and Technology , Sultanate of Oman.
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