RELIABILITY ANALYSIS OF REVERSE OSMOSIS FILTRATION SYSTEM USING COPULA Текст научной статьи по специальности «Медицинские технологии»

RELIABILITY ANALYSIS OF REVERSE OSMOSIS FILTRATION SYSTEM USING COPULA

Anas Sani Maihulla

Department of Mathematics, Sokoto State University, Sokoto- Nigeria anas.maihulla@ssu.edu.ng

Ibrahim Yusuf

Department of Mathematical sciences, Bayero University, Kano- Nigeria

iyusuf@buk.edu.ng

Abstract

In this study, the reliability metrics used to assess the strength of a three-subsystem reverse osmosis filtering system. The subsystems include sand filter, carbonated filter, and precision filter. Each subsystem is composed of active components that can operate in series parallel. The system of partial differential equations was built using the mnemonic rule and analytically solved. Other reliability variables that were investigated for determining system strength included availability, reliability, mean time to failure (MTTF), profit analysis, and sensitivity analysis. The Maple software was used to obtain numerical solutions. In addition, a graphical representation of the numerical results was provided to demonstrate the behaviors of reliability characteristics with regard to time and failure rate. The study could assist water treatment firms and their repairers in overcoming some of the challenges faced by repairers of specialized manufacturing and industrial systems working in harsh settings or contaminated environments unfit for human consumption.

Keywords: mnemonic, profit, sensitivity, reliability, repair, system, copula

I. Introduction

The removal of pollutants from drinking water caused by humans is a modern-day technical challenge. You'll learn about the detectable contamination of drinking water caused by anthropogenic (human-made) contaminants that is still present after a quick review of the treatment steps that municipal water goes through before it reaches your tap. Prescription medications, herbicides, and hormones have all been discovered in the drinking water systems of our respective countries. The engineering design method could be used to find solutions to a real-world problem (contaminated water) that could be dangerous to people's health. Water is perhaps the most important nutrient in our diets. In reality, a normal adult needs to drink around 2 liters (8 glasses) of water per day to replace the water lost through the epidermis, respiratory system, and urine. The Reliability, Availability, Maintainability, and Dependability Analysis of a Complex Reverse Osmosis Machine System in Water Purification was studied by Maihulla A. S. et al. [1] Al-Ghouti, M.A et al. [2] evaluated the recent developments and applications for municipal solid waste bottom and fly ashes. The Gumbel-Hougaard family copula was used to forecast the

Anas Sani Maihulla and Ibrahim Yusuf

RELIABILITY ANALYSIS OF REVERSE OSMOSIS FILTRATION RT&A, No 2 (68)

SYSTEM USING COPULA_Volume 17, June 2022

reliability and performance of a small serial solar photovoltaic system for rural use was carried out by Maihulla A.S. and Yusuf I. [3]. Evaluation of Ozone Pretreatment on Reverse Osmosis Flux Parameters for Surface Water Treatment, Ozone by Shalana L. et al. [4]. The study pertaining the Availability and reliability analysis of integrated reverse osmosis was carried out by safder U. et al. [5]. Calixto, Eduardo. [6] Discussed about Reliability, Availability, and Maintainability (RAM Analysis). The RAM analysis and availability optimization of thermal power plant water circulation system using PSO was carried out by Hanumant P. et al. [7]. The study that tackled the Reliability Assessment for Hybrid Systems of Advanced Treatment Units of Industrial Wastewater Reuse Using Combined Event Tree and Fuzzy Fault Tree Analyses. Was conducted by Farzad P. et al. [8]. Hajeeh, M. [11] & Chaudhuri, D. [9] conducted a research titled reliability and availability assessment of reverse osmosis Desalination. The feasibility and reliability of the Life Cycle Assessment for desalination, is a study carried out by Zhou, J. et al. [10]. Revas P. et al. [12] study the Real-Time Implementation of an Expert Model Predictive Controller in a Pilot-Scale Reverse Osmosis Plant for Brackish and Seawater Desalination. Goyal et al. [13] defined the most vulnerable aspect of serial processes such as evaporation systems in the sugar industry and water treatment plants. Using STP, this research deconstructs the efficiency indices of the power generation system. Simple probability theory concepts and the Markovian birth-death process were used to investigate the power system. As a Markov process progresses from one stage to the next, it becomes more complex.

M.F. Idrees [14] also works with the Performance Analysis and Treatment Technologies of a Reverse Osmosis Plan. C. Li, S. Besarati, and colleagues [15] A few years ago, I did research on reverse osmosis desalination using a low temperature supercritical organic Rankine cycle. In order to meet environmental and economic standards, any sophisticated reverse osmosis plant must have automation and reliability. S. Srivastava. [16]. S. Sadri [17] Created a computational model based on diffusion and convection transport mechanisms, as well as the concentration polarization concept, to predict RO membrane performance using a variety of feed water concentrations, feed flow rates, feed water pressures, membrane specifications, and feed water properties. Y. Li et al. [18] conducted a study of the concepts and categorization of membrane distillation, with an emphasis on the variables influencing it and ways to improving its efficacy. Experiments were conducted by E. O. Ezugbe et al. [19] Pure water and NaCl solutions ranging from 15 g/L to 300 g/L, as well as two distinct fiber types and architectures, were used. Vacuum membrane distillation (VMD) is a saltwater desalination technique. The pure water permeability and global heat transfer coefficient of the two systems were compared. The effects of hydrodynamics on global heat and mass transport coefficients are discussed. R. Tundis et al. [20] The chemical profile, antioxidant and anti-obesity properties of concentrated fractions obtained from micro-filtered OMW treated by direct contact membrane distillation were studied (DCMD). Using ultrahigh performance liquid chromatography, several phenols chosen as phytochemical markers were measured (UHPLC). To treat the pollutants found in olive mill wastewater (total organic carbon (TOC), dissolved organic carbon (DOC), total phosphorus (TP), total nitrogen (TN), and total polyphenols), a sequential Direct Contact Membrane Distillation (DCMD) and a Reverse Osmosis (RO) hybrid membrane system were used. The study was conducted by D. Teresa [21]. The influence of permeate flow and pressure on pollutant parameter removals was also studied. One of the biggest challenges that humanity must address in the twenty-first century is the scarcity of freshwater. P. Biniaz et al. [22] explored how an ecologically acceptable, cost-effective, and energy-efficient membrane distillation process might be developed (MD) process can reduce pollution caused by industrial and domestic wastes. Garud R. M. et al. [23] conducted a Short Review on Process and Applications of Reverse Osmosis. The Gumbel-Hougaard family copula was used to model the reliability and performance of a solar photovoltaic system was analyzed by Maihulla A. S. et al. [24]. Y.G. Lee et al. [25] created a model with five input factors (feed temperature, feed total dissolved solids (TDS), transmembrane pressure (TMP), feed flow rate, and time) and two output parameters (permeate TDS

Anas Sani Maihulla and Ibrahim Yusuf

RELIABILITY ANALYSIS OF REVERSE OSMOSIS FILTRATION RT&A, No 2 (68)

SYSTEM USING COPULA_Volume 17, June 2022

and flow rate) to estimate the performance of a saltwater reverse osmosis (SWRO) desalination plant It was then used to simulate feed water temperature. D. Wirth [26] compared two hollow fiber module designs (inside/out and outside/in).

Our motivation for investigating the reverse osmosis filtration system derives from a severe problem that the water purification businesses are experiencing as a result of purification filter failure. And the slow progress in technological advancement in water purification as a result, as well as its relevance in the lives of people all over the world. Industry is working hard to keep up with the growing complexity of filtration systems. According to the paper's findings, Gumbel-Haugaard Family Copula analysis was performed to examine the filtration system's strength, efficiency, and performance improvement. Users will be able to serve the expense of medical care owing to un-pure water if the strength, efficiency, and performance of the filtration system are assessed. Protect yourself from aquatic pollution. The research is broken into five sections, one of which being the present introduction. Filtration modeling is discussed. Later in the second portion Section 4 contains a discussion and explanation of the results from the third section, which involves an analytical analysis of the system. Section 5 is devoted to the conclusion and ramifications of the findings.

II. Methods

I. Filtration

One of the most common methods for removing these materials is gravity filtration. Water containing solid impurities (e.g., precipitates after water softening) is passed through a porous material, usually sand and gravel layers, in this procedure. The force of gravity pushes the water through the medium. The gaps between the sand and gravel grains allow small water molecules to flow through. Precipitation-derived solids, on the other hand, become trapped in the pores and thus remain in the porous medium. The solid contaminants have been removed from the water that passes through the bottom of the filter.

II. Description of the model

A model with three series-parallel subsystems, A, B, and C, is shown in Figure 1. Two identical units work as 1-out-of-2 in subsystem A (sand filter), three identical units work as 1-out-of-3 in subsystem B (carbonated filter), and two parallel units work as 2-out-of-2 in subsystem C (precision filter). The two types of system failures are partial and complete failures. When a unit in a subsystem fails but the system continues to function, it is called partial failure, whereas total failure occurs when all of the subsystems fail. Copula is used to repair a system that has completely failed. In the system, there are eleven states: eight that are operational and three that are total failure states (see Figure2). The states are described briefly in Table 1.

III. Results

I. Formulation and Solution of Mathematical Model

The following set of difference-differential equations is related to the aforementioned mathematical model based on consideration likelihood and argument continuity.

Q

{— + a1 + a2 + 2a3) P0(t) = J p1P1(x, t)dx + J p2P2(y, t)dy + J #(y)P6(y, t)dy +

f0°° (p(x)PB (x, t)dx + f0°° (p (y)P9(y, t)dy +

/0" V(z)Pi0(z, t)dz

(1)

(¿ + ¿ + «1 + ^+ Pi) PiM = 0 (2)

RELIABILITY ANALYSIS OF REVERSE OSMOSIS FILTRATION SYSTEM USING COPULA RT&A, No 2 (68) Volume 17, June 2022

(i+h+ai + a2 + P2(y,t)=0 (3)

(jt + h + ai + a2+ p2) p3(y-t) = 0 (4)

(Tt + h + a2+ &) P4(y,t) = 0 (5)

(Tt + i; + a2+ P*) Ps(y.t) = 0 (6)

(i + Tx + ai+ Pi) p6(x-t) = 0 (7)

£t+lX + al+ Pi) P7(X't) = 0 (8)

(£ + £ + *W) P8(x,t) = 0 (9)

(yt + i; + v(y)) p9(y,t) = 0 (10)

(irt + i + v(z)) pioM = 0 (11)

Boundary conditions P1(0,t) = aiP0(t) P2(0,t) = U2P0(t) P3(0,t) = a2P0(t) P4(0,t) = aia2P0(t) Ps(0,t) = aia2P0(t) P6(0,t) = aiU2P0(t) P7(0,t) = aia2P0(t) PB(0,t) = a2(l + ai + a2)Po(t) P9(0,t) = ai(l + ai)Po(t) Pw(0,t) = 2a3P0(t) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)

Initial condition P0 (t) = 1 and other transition probability at t=0 are zero Laplace transformation of (1) - (21)

f œ f œ f œ

(s + ai + a2 + 2a3)P0(s) = I ßiPi(x,s)dx + I ß2P2(y,s)dy + I ^(y)P6(y,s)dy +

J0 J0 J0

Ç 9(x)f}8 0, s)dx + f0œ <p{y)Pg(y, s)dy+ f™ <p(z)Pw(z, s)dz (22)

(s + ai + a2+ pi ) ï'i(x,s) = 0 (23)

(s + ^+a1 + V2+ P2 ) P2(y,s) = 0 (24)

(s + ^+ai + a2+ P2 ) Ps(y,s) = 0 (25)

(s + -^+a2+ M P*(y,s) = 0 (26)

(s + -^+a2+ M Ps(y,s) = 0 (27)

(s + -^+ai+ pi) P6(x,s) = 0 (28)

(s + -^+ai+ pi) P7(x,s) = 0 (29)

(s + yx + y(x)) P8(x,s) = 0 (30)

(s + -^+cp(y)) Pg(y,s) = 0 (31)

(s + ^+cp(z)) Pi0(z,s) = 0 (32)

Laplace of the Boundary conditions

Pi(0,s) = aiP0(s) (33)

P2(0,s) = a2P0(s) (34)

P3(0,s) = a2P0(s) (35)

P4(0,s) = a^Pote) (36)

Ps(0,s) = aia2Po(s) (37)

P6(0,s) = aia2Po(s) (38)

P7(0,s) = aia2Po(s) (39)

PB(0,s) = a2(l + ai + a2)Po(s) (40)

Anas Sani Maihulla and Ibrahim Yusuf

RELIABILITY ANALYSIS OF REVERSE OSMOSIS FILTRATION RT&A, No 2 (68)

SYSTEM USING COPULA_Volume 17, June 2022

P9(0,s) = aUl + ai)Po(s) (41)

Plo(0,s) = 2a3Po(s) (42) Solving equation (22) to (32) with the help of boundary condition (33) to (42) and applying the below shifting properties of Laplace:

J0>—. e-%f(*)dx]dx = L f-^} =(43)

S™[e-sxJ(x) e-iof№]dx = L[Sf(x)}=Sf(s) (44)

And the identity; P1 (s) = J™ P1 (x, s)dx (45)

P^^Pi^M1-^^} (46)

P2(S) = P2(0,S){1-^^+^} (47)

P3(s) = Ps(0,s){1-f^^} (48)

P4(s)=P4(0,s){i-^s^} (49) P5(s) = P5(0,s){.(50)

P6(s) = P6(0Mld^} (51)

P7(S) = P7Q,S){1-%^} (52)

P8(s) = P8(0,s){i-f^} (53)

P9(S)=P9(0,S){1-^^} (54)

PiO(s) = PiO(0,s){1-f^} (55) Substituting the Laplace of the Boundary conditions i.e (33) to (42) into (43) to (55)

Pi(*)="i{1-^^}P°(s) (56)

P2^) = ^2{1-^j:^}P0(s) (57)

= (58)

P4(s) = aia2{i-Slfa+2a2)}P0(s) (59)

P5(s) = aia22{i-^s2+fiai)}P0(s) (60)

P6(s) = aia2{i-l1+(;^+2ai)}P0(s) (61)

P7is) = aiai{i^^}P0is) (62)

P8(s) = ai(1 + ai + a2) p^®} P0(s) (63)

P9(s) =ai(1 + ai){i^f^}Po(s) (64)

P^io(5) = 2a3{^^(^}Po(5) (65)

(s + ai + a2 + 2a3)Po(s) =1+ piS^S + ai + ai)Po(s) + fcS^S + ai + ai)Po(s) + [<p(x)Sv(x)(S)

+<p(y)SV(y)(S) + <p(z)Sviz)(S)] Po(s) (66) s + ai + a2 + 2a3 - [PiSPl(S + ai + a2) + PiSp2(S + ai + a2) + [ptoS^C) + p(y)S^(y)(S) +

p(z)S^(z)(S)]]!^o(s) = 1 (67) D(s) = s + ai + a2 + 2a3- [PiSPl(S + ai + a2) + PiSp2(S + ai + ai) + [pCOS^C^ +

p(y)Sv(y)(S) + p(z)Sviz)(S)]] (68)

Since D(s)x Po(s)=1 (69)

Pup(S) = Po(S) + Pi(S) + P3(S) + P4(S) + Ps(S) + P6(S) + P7(S) (70)

P,® =1^ Ffe^} + a2 + aia2 a1ai {i-^} +

RELIABILITY ANALYSIS OF REVERSE OSMOSIS FILTRATION RT&A, No 2 (68) SYSTEM USING COPULA_Volume 17, June 2022

(i-Sp1(S+g1)} +a a2 (i-Sp1{S+a1)^ -S+a1 } i 2 { S+a1

^^J + ^Fg^M (71)

II. Availability Analysis

Setting all repairs to 1. i.e. ^(x) = ^(y) = y(z) - pi - p2 - p3 - 1 (72)

S.(s)=^n8L ,

J S+2.7i83 S S+$

Taking the values of different parameters as ai - 0.01, ai - 0.02, a3 - 0.03, a4 - 0.04 in

(48) Then taking the inverse Laplace transform, we can obtain, the expression for availability as:

D(s) = S + 0.12- [-003- + + 0.06 . ( 2 7W3 ) ] (73)

v ' S+i.04 S+i.03 KS+2.7iB3' J v '

p (S)=[1+ + + ^ (74)

L S+i.04 S+i.03 S+i.03J v '

Taki"g•rw=,^, but* =1 and 1 =

0.001, a2 - 0.002, a3 - 0.003

And repair rates ^(x) - ^(y) - y(z) — pi — p2 — p3 — 1 in equation (69), and applying the inverse Laplace transform to (69), the expression for system availability is

Pup(t) - ^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^099809e"Z722649460f + (-0.006078301855 } (75)

- 0.0014178109921) e( -1-014501644 - 0.00099381113161) ,+ (-0.00607830 1 855 + 0.0014178109921) e( -1-014501644 + 0.00099381113161) i + , ,0i 0602552 e"002864725083'

Taking t = 0, 10,...,100, availability of the system is obtained and presented in Table 1 below:

Table 1: Availability variance with respect to time

Time in days Availability

0 1.000000

10 0.997679

20 0.997531

30 0.997383

40 0.997235

50 0.997086

60 0.996939

70 0.996791

80 0.996643

90 0.996495

100 0.996347

1,001 1

0,999

JT 0,998

S

CS

H 0,997

a

0,996 0,995 0,994

10

Availability

20

30

40

50

60 70

Time

80

90

100

Figure 1: Variation of availability with time

III. Reliability Analysis Letting all repair rates, ^(x) = ^(y) = ^(z) = p1 = p2 = p3 = 0 in equation (70), and taking the values of failure rates and employing inverse Laplace transformation, the expression is reliability relation.

R(t) = { 0.002909090909 e 001100000000' + 0.2352941176 e 001600000000 f } (76)

+ 0.3617967914e-003300000000f + 0.4000000000 e"001300000000'

Using t = 0, 10...100 as time units in equation (72), reliability is determined and shown in Table 2 below.

Table 2: Variation in Reliability as a Function of Time

Time in Days Reliability

0 1.000000

10 0.941981

20 0.887595

30 0.836601

40 0.788773

50 0.743903

60 0.701796

70 0.662272

80 0.625161

90 0.590308

100 0.557565

1,2

1

0,8

.■ÎT

s ia 0,6

"s K

0,4

0,2

0

Reliability

10 20

30

40

50

60

Time

70

80

90

100

Figure 2. Variation of reliability with time

IV. Cost Analysis

The expression for the expected profit incurred in [0, t)

t

Ep (t) == Pp (t )dt-K 2

0

Taking fixed values of parameters of equation (62), the subsequent equation (69) follows; -0.018340e-87072t+0.008479e-20122t

Ev(t)

Ep (t ) =

+0.000094e-15618t - 598.072776e-00016t +0.000183eU3000t + 0.000237^™ +5985.0821

-— )

(77)

(78)

Table 3: Expected profit as a function of time

Time in days Ev(t)

0 0 0 0 0 0 0

10 3.97831 4.97831 5.97831 6.97831 7.97831 8.97831

20 7.95435 9.95435 11.95435 13.95435 15.95435 17.95435

30 11.92892 14.92892 17.92892 20.92892 23.92892 26.92892

40 15.90200 19.90200 23.90200 27.90200 31.90200 35.90200

50 19.87361 24.87361 29.87361 34.87361 39.87361 44.87361

60 23.84373 29.84373 35.84373 41.84373 47.84373 53.84373

70 27.81238 34.81238 41.81238 48.81238 55.81238 62.81238

80 31.77954 39.77954 47.77954 55.77954 63.77954 71.77954

90 35.74523 44.74523 53.74523 62.74523 71.74523 80.74523

100 39.70944 49.70944 59.70944 69.70944 79.70944 89.70944

100

90

80

70

tsi 60

fio 50

O- 40

30

20

10

0

Expected profits against time

10 20 30 40

50

60 70

Time

80 90 100

K1=0.01 K2=0.02 K3=0.03 K4=0.04 K5=0.05

Figure 3: Variation of profit with time

K6=0.06

V. Formulation and Analysis Mean Time to Failure Setting repairs to zero in equation (70), the expression for MTTF is defined as follows: Fixing a1 = 0.001, a2 = 0.002, a3 = 0.003 varying, failure rate in equation (66), MTTF is computed with respect to failure rate as presented in Table 3 below.

Table 4: Variation of MTTF with failure rates

Failure rate MTTF

(a) (b) (c)

0.001 222.8152 250.2918 401.6008

0.002 200.5337 222.8153 286.5720

0.003 182.3034 200.9009 222.8153

0.004 167.1114 183.0318 182.2731

0.005 154.2567 168.1965 154.2157

0.006 143.2384 155.6951 133.6447

0.007 133.6892 145.0273 117.9162

0.008 125.3336 135.8265 105.5002

0.009 117.9610 127.8176 95.44993

MTTF Against failure rate

450 400 350 300

fc 250 £ 200

150 100 50 0

0,001 0,002 0,003 0,004 0,005 0,006 0,007 0,008 0,009

Failure rate

-Subsystem 1

Subsystem 2

-•—Subsystem 3

Figure 4: MTTF as function of Failure rate

VI. Sensitivity analysis corresponding to MTTF The partial differentiation of MTTF with respect to the failure rate of the system can be used to investigate the system's sensitivity in MTTF. Using the set of parameters as ai = 0.001, a2 = 0.002 a3 = 0.003, The MTTF sensitivity can be calculated in the partial differentiation of MTTF, as shown in the Table below and associated graphs in the Figure.

Table 5. MTTF sensitivity as function of failure rate

Failure rate d (MTTF) a1 d(MTTF) a2 d(MTTF) as

0.001 -24757.25 -30952.89 -100000.61

0.002 -20053.37 -24386.44 -82020.57

0.003 -16573.03 -19689.49 -49563.88

0.004 -13925.95 -16214.28 -33163.29

0.005 -11865.90 -13571.10 -23737.80

0.006 -10231.31 -11514.08 -17826.70

0.007 -8912.61 -9881.903 -13877.30

0.008 -7833.35 -8565.144 -11108.57

0.009 -6938.88 -7487.473 -9092.820

ity

ivi ti

£

n

e

c-n

0

-20000 -40000 -60000 -80000 -100000 -120000

Sensitivity Analysis 0,001 0,002 0,003 0,004 0,<

Failure rate

-•—Sand filter

Carbonated Filter

Precision Filter

Figure 5: Sensitivity analysis corresponding to mean time to failure (MTTF)

Figure 6: Filtration system's transition diagram

Perfect state / / Partial failure state

complete failure state

1 out of 2 1 out of 3 2 out of 2

Figure 7: Reliability block diagram of the filtration system

VII. Description of the system

an: Failure rate for the subsystems. Where n=1,2,3.

Repair rate for the subsystems with incomplete failure Where q =1,2,3. ^(k) : Repair rate for the subsystems with complete failure. Where k= x, y, z. Po: Denote initial state where the system is working perfectly.

P1: Denote state with an incomplete failure in subsystem 1 due to failure of first unit and copula repair is busy in repairing the failed unit.

P2: Denote state with a complete failure in subsystem 2 due to failure of first unit and repair

machine is busy in repairing the failed unit.

P3: Denote state with a incomplete failure in subsystem 2

P4: Denote state with an incomplete failure state due to failure of first unit in subsystem 1 and one unit from subsystem 2.

P5: Denote state with an incomplete failure state due to failure of first unit in subsystem 1, first and second units from subsystem 2.

P6: Denote state with an incomplete failure due to the failure of first and second units from subsystem 2, and one unit from subsystem 1.

P7: Denote incomplete state of system due to failure of first unit from each of subsystems one and two

P8: Denote complete failure state in subsystem 1. P9: Denote complete failure state in subsystem 2. P10: Denote complete failure state in subsystem 3.

IV. Discussion

I. Interpretation of the Result and Conclusion

Table1 and Figure 1 provide information how availability and time changes when failure rates are fixed at different values. When failure rates are fixed at lower values a1 = 0.001, a2= 0.002 and a3 = 0.003. Table 1 shows the results, and Figure 1 shows the corresponding result, the system's availability decreases over time. And the decreases from the first point are greater than those from the second. This is due to the units in subsystem 1 having less redundancy. Figure 2 shows that the system is more reliable than it is available. However, this is due to a copula repair for a component that had completely failed. And the system will be assumed to be operational during the repair. The system's availability reduces with time and eventually stabilizes at zero over a sufficiently extended length of time. As a result, the graphical representation of the model shows that one may confidently describe the future behavior of a complex system at any time for any given set of parametric parameters. Figure 2 and table 2 indicates that incorporating copula considerably decreases system reliability with time. The graphical representation of the model shows that one may confidently forecast the future behavior of a complex system at any time for any given set of parametric variables. When the values of availability and reliability in Tables 1 and 2 are compared, it is clear that when repair is offered, the system performs far better than replacement.

Tables 4 and figure 4 yield the mean-time-to-failure (MTTF) of the system with respect to variation in the failure rates, a1,a2, and a3 respectively when all other parameters are held constant Shown by color graphs (green, Pink, and ash) respectively. The system is also examined using the Gumbel-Hougaard family copula. According to the findings, incorporating copula considerably enhances system reliability.

Cost analysis of the system is done in the analytic part of the paper Figure 3 and table 3 shows the variation of cost-profit with variation in the values of parameters. Table 5 and figure 5 displayed the result of sensitivity analysis corresponding to mean time to failure (MTTF) with respect to the failure rates afc from the table, MTTF sensitivity decreases with increase in afc.

For the analysis of profit the following are used:

(a) Fixing K2 = 0.01 varying time t from 0 to 100

(b) Fixing K2 = 0.02 varying time t from 0 to 100

(c) Fixing K2 = 0.03 varying time t from 0 to 100

(d) Fixing K2 = 0.04 varying time t from 0 to 100

(e) Fixing K2 = 0.05 varying time t from 0 to 100

Table 4 displayed the result of expected profit Ep (t) with respect to K2. From the table, it i evident that expected profit increases as K decreases.

II. Conclusion

Due to a lack of data on the filtration system, the current paper developed a reliability modeling approach to investigate the filtration system's overall strength, efficiency, and performance. The reliability, availability, MTTF, and profit function of this paper can all be evaluated. We present a new filtration system model that includes three subsystems: a sand filter, an activated carbon filter,

Anas Sani Maihulla and Ibrahim Yusuf

RELIABILITY ANALYSIS OF REVERSE OSMOSIS FILTRATION RT&A, No 2 (68)

SYSTEM USING COPULA_Volume 17,June 2022

and a precision filter in this study.

The findings of the study suggest that reliability modeling can be used to investigate the strength, efficiency, and performance enhancement of a reverse osmosis (RO) filtering system. The system's strength, efficiency, and performance improvement are determined at this point. This study could be enhanced to incorporate a system with numerous subsystems and several repair machines to reduce repair facility congestion and handle problems using supplemental variable techniques. Among other things, the current effort will benefit water manufacturing and industrial uses that are hazardous to humans.

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