Iteratioal Joural o Miig Scie IJMS Volume, Issue, 6, PP -4 ISSN 454-946 Olie wwwarcjouralsorg Compariso o Mathematical Approimatio Methods or Mie Vetilatio Network Aalysis Farhag Sereshki Associate Proessor Faculty o Miig Egieerig Petroleum ad Geophysics Shahrood Uiversity o Techology Shahrood, Ira arhag@shahroodutacir Ebrahim Elahi Faculty o Egieerig Departmet o Miig Egieerig Uiversity o Sista & Baluchesta, aheda, Ira elahi@egusbacir Amir Saari PhD Cadidate Faculty o Miig Egieerig Petroleum ad Geophysics Shahrood Uiversity o Techology Shahrood, Ira amirsaari57@yahoocom Abstract: Various maual ad computerized methods or aalysis o mie vetilatio etwork are preseted The selectio o a suitable method depeds o the target o etwork aalysis I the purpose o aalysis is ivestigatio o the eect o as o mie etwork, usig computerized methods based o mathematical approimatio methods is better The most otable eature has bee the developmet o computerised mie vetilatio etwork aalysis This has made predictio o a requiremets ad airlow distributios i comple mie vetilatio circuits easible usig desktop computers A umber o methods o mathematical approimatios are preseted such as the ardy Cross method ad its modiied versios, Newto-Raphso techique, critical path, liear aalysis, o-liear programmig ad optimizatio techiques Usig o ardy Cross method i vetilatio sotware has become commo I ret years, the use o Newto-Raphso techique i mie vetilatio etwork aalysis has bee growig Accordigly, i this paper, a model is developed or aster coverge to the ial solutio based o the ardy Cross method that is amed colatio model o ardy Cross method Also, a more precise use o the Newto-Raphso techique i mie vetilatio etwork aalysis is also perormed Keywords: Network Aalysis, Mies Vetilatio, Fas, optimizatio, Mathematical Approimatios Methods INTRODUCTION The basic requiremet or the mie vetilatio system is to provide air or people to breath ad i a coditio that will ot cause ay immediate or uture ill eects Because o the prosses o miig, i positive airlow through the workigs was ot provided the air would very quickly become stale, cotamiated ad uit or huma cosumptio The vetilatio system must thereore be suiciet to deal with the cotamiats released durig miig I they are ot adequately dealt with as they are idetiied, they may become at best a discomort to mie workers, ad at worst cause serious or eve atal illess Although the mie eviromet by tuel, wize ad shat is coected to the sura, this coectio aloe is ot adequate or providig good airlow i the etworks It is essary to esure the air is clea by usig various istrumets ad artiicial methods Accordigly, or dilutio o harmul gases i the mie vetilatio etworks requires kowledge o the atural ad artiicial vetilatio circuits Elahi, 4; Madai, 6 Vetilatio desig methods i udergroud mies are based o priciples such as mappig, idetiicatio o vetilatio braches ad odes, calculatio o vetilatio resista o the each braches, the airlow itesity o the each brach, modiied low itesity, pressure loss or each brach, atural vetilatio, etwork regularizatio ad ially selectio o the regulator doors or udergroud as, alog with selectio mai a ARC Page
Farhag Sereshki et al Maual ad computerized methods or the aalysis o mie vetilatio etwork are preseted The selectio o the type o method depeds o the target o etwork aalysis I the brach low itesity ad directio i a mie etwork have bee ied ad determied i this state, usig o maual methods or the aalysis o the etwork is better The purpose o the mie vetilatio etwork aalysis i maual methods is oly or the selectio o mai ad auiliary as ad the air regulators But i the purpose o mie vetilatio etwork aalysis is the ivestigatio o the eect o oe or more as o a mie etwork, use o mathematical approimatio methods is better, because the maual method caot calculate low itesity i each o braches i comple etworks The mathematical approimatio methods begi with a assumptio o low itesity ad directio i each o the braches o the mie etwork The calculatio error or each loop is determied usig mathematical approimatio equatios, ad the the assumed airlow itesity will be corrected Airlow itesity correctios based o the provided mathematical equatio cotiue to be coducted util calculatio error is less tha or equal to a required accuracy Use o mathematical approimatio methods or solvig huge ad comple etworks by huma is impossible; thereore utilisatio o computers is essary to solve them A umber o computer sotware's is available or aalysis o mies vetilatio etworks Oe o the most popular o them is Vetsim sotware that is based o ardy Cross equatio Elahi, 4; Madai, The irst mathematical equatios or estimatig the low itesity error i each loop, was preseted by ardy Cross i 96 Although this equatio was proposed or the aalysis o water etworks, this equatio was later used ad improved or mie vetilatio etwork aalysis by Wag 98a ad El-Nagdy 8 I additio to the metioed methods, other techiques such as, Newto-Raphso techique Madai & Maleki, 7; Wag, 989, critical path Wag, 98b, liear aalysis Bhamidipati & Procarioe, 985; Kamba et al, 995, o-liear programmig u & ogso, 99; Wag, 984 ad optimizatio techiques Collis, 978 are also preseted A REVIEW OF APPROXIMATE MATEMATICS METODS ardy- Cross Method ardy Cross's amous equatio is as ollows ad accordig to equatio, ca be used to estimate the low itesity error i each loop Cross, 96; Elahi, 4 P R R R R R R R R R R R R R R R R R P i R i i R i i R i i Where: : The actual low itesity m /sec : The primary or assumed low itesity m /sec : The low itesity error i loop m /sec Iteratioal Joural o Miig Scie IJMS Page
Compariso o Mathematical Approimatio Methods or Mie Vetilatio Network Aalysis R : The miig work resista i each brach K Murgue P : The pressure loss or each brach mm water The ardy- Cross equatio solutio steps are as ollows: First Stage Accordig to the juctio law o equal low ito ad out o every juctio, assumed low itesity with assumed directio or each o the braches o the mie vetilatio etwork is established Secod Stage Idetiy closed loops i mie vetilatio etwork accordig to equatio ad select a assumed directio or them N R Where: N N R B N J N B N J : Number o useul loops with a sura ode : Number o braches i mie vetilatio etwork : Number o juctios i mie vetilatio etwork Third Stage Calculatio o air pressure loss or each o braches i loop accordig to equatio : P R 4 Fourth Stage The calculatio o low itesity error or each loop accordig to equatio : It should be oted or the umerator part o the equatio, i the airlow directio i brach was aliged with the loop low directio, the pressure loss is a positive sig, otherwise it is egative For the deomiator part o the equatio it is always summed as a positive sig 5 Fith Stage The calculatio o the ew low or each brach o mie vetilatio etwork: The error correctio or each brach is determied by the algebraic summatio o low itesity error correctios o the loops cotaied the brach It should be oted, i the loop directio was aliged with the brach low directio, the error is applied as positio; otherwise it is subtracted rom the assumed low Also, i the value o ew low itesity i the brach is egative, the airlow directio i the brach is reversed 6 Sith Stage Repeat the above operatios rom third to ith stages util the calculatio error is less tha or equal to the required accuracy Wag Modiied Method Wag's correctio is relatig to ourth stage o the ardy-cross method At this stage, the amout o low itesity error o each loop is estimated accordig to equatio 4 The purpose o this modiicatio is that it elimiates the positive or egative sig o the pressure loss i equatio, ad also removes the decisio to chage the airlow directio i a low is egative Accordigly, i the low directio chages, the amout o low itesity chages to a positive sig, otherwise a egative sig should be appear i equatio 4 I act, the equatios ad 4 or the low itesity with positive value are quite similar i i b R P P ki i i i b ki i R i i i Fi 4 Iteratioal Joural o Miig Scie IJMS Page
Farhag Sereshki et al Iteratioal Joural o Miig Scie IJMS Page 4 Where: ki b : Fudametal matri elemet o the loop I the airlow directio i brach was aliged with the loop low directio, i this case, this elemet is equivalet to i otherwise equivalet to - which appeara i equatio 4 i P : Icreasig pressure by reaso o atural vetilatio i the brach Fi P : Icreasig pressure by reaso o istalled a i the brach Newto Raphso techique Oe o methods to solvig umerical computatio is usig the Newto- Raphso techique This techique is based o the deiitio o the derivative ad the correctio o it I this techique, the iitial guess o error value or the solutio o the equatio is estimated ad the iteratively corrected The mathematical equatio ca be epressed as ollows: Where: : Iitial guess : Aswer o the et step : The value o the uctio o the basis o iitial guess : The value o the uctio o the basis o ially aswer is equivalet to : The value o the derivative o uctio This techique or the irst time was preseted or the aalysis o mie vetilatio etworks by Wag ad i i 985 Also, this techique was used by Madai ad Maleki i 8 or P, equatios aalysis i mie vetilatio etworks Based o P, equatios aalysis i mie vetilatio etworks by usig Newto-Raphso techique is epressed as ollows:
Compariso o Mathematical Approimatio Methods or Mie Vetilatio Network Aalysis Iteratioal Joural o Miig Scie IJMS Page 5 The Newto-Raphso techique solutio steps based o equatio are as ollows: First stage Accordig to braches' law, assumptio low itesity with assumptio directio i each o the braches o the mie vetilatio etwork is cosidered Secod stage: Idetiicatio o useul loops i mie vetilatio etwork accordig to equatio ad select a assumed directio or them Third stage: Calculatio o equatio based o air pressure loss or each brach i the etwork P R 4 Fourth stage: Repeat the above operatios rom third stage util calculatio error is less tha or equal to the required accuracy
Farhag Sereshki et al CONFATION MODE OF ARDY- CROSS METOD This correctio is relatig to ourth ad ith stages o the ardy-cross method I the primary ad modiied methods, these two stages were calculated separately Meaig that ater completig the ourth stage calculatios or all o the loop errors, the ith stage o the operatio to apply the loops errors to the braches begis But i these two stages are applied to each loop together, the ilue o each o each loop correctio ca be cosidered o others, ad the ivestigatio has led to more rapid acss to the ial aswer I other words, ater the irst loop error calculatio is doe, the relevat braches are corrected or low itesity ad oly the are the secod loop calculatio error perormed Thereore, solvig o ourth ad ith stages step by step together o ardy-cross method the ivestigatio has led to more rapid acss to the ial aswer To prove this claim, cosider the ollowig two models First Model A eample mie s vetilatio etwork, accordig to Figure is cosidered by a a with a pressure o mm water The miig resista ad low itesity o each brach i this etwork is show i Table Iitially, this etwork was simulated with Vetsim sotware that so ar is the most complete mie vetilatio etwork aalysis sotware kow, ad the the results are compared with maual method to solvig the ardy- Cross equatios, the Wag modiied method ad the colatio model The result o simulatio with Vetsim sotware or low itesity distributio o each brach i the metioed etwork is show i Figure Figure Mie vetilatio hypothetical etwork Table Mie vetilatio hypothetical etwork properties Brach Miig resista k Murgue Flow itesity m /sec ab 5 4 4 4 d 6 8 Figure Distributio o low itesity with Vetsim sotware For aalysis o low itesity o each braches by ardy- Cross method, requires to assumptio low itesity The assumed lows itesity with required loops selectio is show i Figure Iteratioal Joural o Miig Scie IJMS Page 6
Compariso o Mathematical Approimatio Methods or Mie Vetilatio Network Aalysis Figure ypothetical lows itesity with loop selectio The irst stage o calculatio o ardy- Cross method ad Wag modiied method chage the airlow directio where low itesity has a egative value is accordig to equatio ad which is as ollows Results o other stages are show i Table 6 4 4 54 6 4 4 54 4 4 8 668 4 4 8 6 8 45 6 4 4 ab ab 4 668 7 6 45 869 d d 8 668 58 668 45 68 45 55 Accordig to Table ad ater iteratios, the method reached coclusio with the results obtaied rom Vetsim sotware beig quite similar Thereore, the Wag modiied method i this model is ot able to coverge aster to reach the ial solutio Table The results o ardy Cross Method ad its corrected stages Stage ab d 4 7 58 55 869 68 944 454 895 4697 56 8 6 6 88 69 877 75 4 77 679 86 7 9985 96 5 544 5 5766 48 98 856 5 68 79 567 945 8776 5 45 58 674 444 99 877 4 58 669 49 98 877 4 58 668 49 98 877 The irst stage o calculatio o colatio model o ardy-cross method is as ollows ad results o other stages are show i Table Iteratioal Joural o Miig Scie IJMS Page 7
Farhag Sereshki et al 6 44 54 6 44 54 4 4 ab ab 4 8869 6 58869 d d 4 8869 8 69 4 8869 8 8869 69 75 69 469 8 69 588 469 58869 8 4984 469 58869 d d 58869 4984 9585 469 4984 9685 4984 576 Accordig to Table ad ater times iteratios doig the method reached the coclusio with the results obtaied rom the Vetsim sotware beig quite similar Thereore, the colatio model o ardy-cross method is able to coverge aster to reach the solutio the ial aswer Table The results o colatio model o ardy- Cross Method stages Stage ab 4 75 588 576 9585 9685 765 888 44 59 6 757 7 7944 4456 688 65 4 994 49 44 8 88 5 874 5 4 69 889 857 4 58 669 49 98 877 4 58 668 49 98 877 Secod Model A assumed mie's vetilatio etwork, accordig to Figure 4 is cosidered with two as with the ad 8 mm water pressures respectively The miig resista ad low itesity o each brach i this etwork is show i Table Iitially, this etwork was simulated with Vetsim sotware, ad the the results are compared with the maual method o solvig the ardy Cross equatios, the Wag modiied method ad the colatio model The result o simulatio with Vetsim sotware or low itesity distributio o each brach is show i Figure 5 d Figure4 Mie vetilatio hypothetical etwork Iteratioal Joural o Miig Scie IJMS Page 8
Compariso o Mathematical Approimatio Methods or Mie Vetilatio Network Aalysis Figure5 Distributio o low itesity with Vetsim sotware For aalysis o low itesity o each braches by ardy- Cross method, requires to assumptio low itesity These assumptio lows itesity with required loops selectio is show i Figure 6 Figure6 ypothetical lows itesity with loop selectio The irst stage calculatio o the ardy- Cross method ad the Wag modiied method chage the airlow directio where low itesity is a egative value is accordig to Table 4 Ater 54 times iteratios the results compared with the Vetsim sotware are quite similar Thereore, the Wag modiied method i this model is ot able to coverge aster to reach the ial aswer solutio Table4 The results o ardy Cross Method ad its corrected stages Stage ab -855 884 58 78 9975 699 47 698 5545 8 469 567 8874 6649 77975 98 874 85 4 9875 454 4748 6994 5969 746 7 466 4574-47 96 6597 99 674 645 67 9487 685 64 846 4 5 7456 5995 687 8589 6 5767 7 5899 6876 4 859 45 5865 7 589 6877 5 8595 4 587 75 589 6877 54 8596 4 587 74 589 6877 The irst stage o calculatio o the colatio model o ardy-cross method is as ollows ad results o other stages are show i Table 5 6 54 44 5 6855 6 54 44 4 6855 855 ab ab 4 6855 855 6 6855 855 d d Iteratioal Joural o Miig Scie IJMS Page 9 d
Farhag Sereshki et al 4 855 8 65 4 855 8 855 65 4757 65 85 8 65 8965 85 855 8486 85 855 d d 855 8486 4466 85 8486 96649 8486 564 As show i Table 5, ater 9 times iteratios the coclusio reached is similar to Vetsim sotware Thereore, the colatio model o ardy-cross method is able to coverge aster to reach the solutio Table5 The results o colatio model o ardy Cross Method stages Stage ab -855 4757 8965 564 4466 96649 79459 454 6475 667 4659 65 756 46 799 4 577 4 4 4798 685-4947 4 54 75 5 5877 55 5 974 5558 7 887 5749 7479 5866 659 5 8577 49 57 7 589 685 8596 44 586 7 5899 6875 5 8595 4 587 75 589 6876 9 8596 4 587 74 589 6877 4 COMPARISON BETWEEN CONFATION MODE OF ARDY-CROSS METOD AND NEWTON-RAPSON TECNIUE So ar, the compariso o the Newto- Raphso techique i the aalysis o the mies vetilatio etwork is ot ully accurate Thereore, or a more precise evaluatio o this techique, a compariso was doe betwee this model ad colatio model o ardy- Cross method, resultig i two dieret evaluatio models 4 First Model A eample mie vetilatio etwork is cosidered accordig to Figure ad Table Accordig to Table, the colatio model o ardy-cross method is able to solve ater times iteratios ad the low itesity is calculated as show i Figure The solvig stages o calculatio o Newto- Raphso techique i mies vetilatio etworks aalysis accordig to Figure ad Table 6 are show i 4 stages Compariso o these results with the results obtaied rom the Vetsim sotware which as show i Figure is quite similar d Iteratioal Joural o Miig Scie IJMS Page
Compariso o Mathematical Approimatio Methods or Mie Vetilatio Network Aalysis ab d 8 d 5 4 ab d 8 8 6 6 d d 6 d 4 8 6 4 8 884 - - 856-5 - - 5-64 - 4 5688 897 556 677644-897 - 556-67764 646475-49494 - 757-49494 6847-7944 - 757-7944 8894-448 44896 6947 47 6 48-7547 - 66784-7846 599954-8 - 6458-8 58447-8 - 6458-8 988858-76 74 4864 497 5469 98-759 - 67-78566 59787-877 - 6865-877 58955-88 - 6865-88 987544-4 66 49 Table6 The results o Newto- Raphso Techique stages Stage ab -4E- 5 5E- 5-75896 - 676-7856 76 7677 487 9 578 74 459 756 6 758 999 854 4 6 67 49 99 877 4 4 58 668 49 98 877 Iteratioal Joural o Miig Scie IJMS Page d 4
Farhag Sereshki et al 4 Secod Model A eample mie vetilatio etwork is cosidered accordig to Figure 4 ad Table Accordig to Table 5, the colatio model o ardy-cross method is able to solve ater 9 times iteratios, the low itesity is calculated as show i Figure 5 The solvig stages o calculatio o Newto-Raphso techique i mies vetilatio etworks aalysis or this model is similar to irst model, but the diere is the +5 value added to uctio Thereore, i accorda with Figure 6, the irst to ourth stages o the calculatios are show as ollows i Table 7 Ater the ourth stage the results o this techique will diverge ad its results ad results obtaied rom Vetsim sotware which has bee preseted i Figure 5 is ot the same 884 - - 856-5 - - 5-64 49 4 688 75768 9854478 68774-7577 - 985448-6877 86979-479 - 479 97-59559 - 645-59559 - 645 69 4898 455 7577 7954 598 5448-96 - 688676-65 - 47898-866 -6-866 79-4976 -6-4976 868895 8964 4795 665-44 -68-7454 - 685-8858 - 7746 8475-7949 - 86987-7949 - 86987 58 578-47 46-47 4766 96777 Table7 The results o Newto- Raphso Techique stages Stage ab -98-7868 -555-75565 - 689-95 -87577 69 8455 8 7699 667-87967 795-58676 -5 587 5775-4685 6446-858 -7746 958 9 4-5565 9976-689 7795 47 494 5-495 455 957 868 49 998-756 789-94646 -46447 47689 4899 5-6788 79-79 -7676 449 48 4949 669 687 676 8 9474 5-9 746 6855 4559 495 9774-64876 847-49 -979 45446 49 5-8 689 657 674 6865 476 4-5 -4778 68 84 578 45-8884 6597 7755 75644 64486 889 5 974-588 94987 9695-988 8 55-968 88944-774 4 4 97 5 CONCUSIONS Various methods o maual ad computerized aalysis o mie vetilatio etwork are preseted The selectio o type o method depeds o target o etwork aalysis I the purpose o mie vetilatio etwork aalysis is ivestigatio o the eect o oe or more as o mie etwork, use o computerized methods based o mathematical approimatio methods is better The ardy Cross method or mathematical approimatio methods or mies vetilatio etwork aalysis has become more commo The coverge o this method depeds o values, iitial assumed lows ad directio ad loops selectio I this method, i the value o ew low i the brach is egative, it must reverse the directio o the brach Iteratioal Joural o Miig Scie IJMS Page d 4
Compariso o Mathematical Approimatio Methods or Mie Vetilatio Network Aalysis Wag's correctio is relatig to ourth stage o the ardy-cross method I this correctio model, i the value o ew low itesity i brach is egative, i the directio o the brach stays the same, ad the low itesity becomes egative Colatio model o ardy- Cross method is relatig to ourth ad ith stages o the ardy-cross method Accordigly, i these two stages are perormed sequetially step by step together or each brach, ad the ilue o each o the other to be cosidered, i this case, the ivestigatio has led to more rapid acss to the ial aswer I other words, ater the irst loop error calculatio must be doe to correct the relevat braches o low itesity ad the calculate the error o the secod loop Accordig to the eamples preseted, colatio model is able to decrease the repeat o calculatio o the ardy- Cross method, approimately 5 pert Newto- Raphso techique is oe o the mathematical approimatio methods which, is usig i mies vetilatio etwork aalysis i ret years, however, so ar the eact validatio is ot perormed Based o the results o this study, it ca be cocluded that i some o the models, this techique is ot able to calculate the real value o low itesity i mies vetilatio etworks aalysis I i a etwork, the assumed iitial lows directio is doe i accorda with the real lows directio, i this case, with ull coide it ca be claimed that the most rapid way to reachig the ial aswer i the aalysis o mies vetilatio etworks is usig o Newto- Raphso techique I i a etwork, some o assumed lows directios are ot be i accorda with the real lows directio, usig o Newto- Raphso techique i the aalysis o mies vetilatio etworks or reachig the ial aswer i some o models may result i diverge ad icorrect aswers REFERENCES Bhamidipati, S S, Procarioe, J A, 985 iear Aalysis or the Solutio o Flow Distributio Problems, Proedigs o the d US Mie Vetilatio Symposium, Mousset_Joes, P Ed, Rotterdam, Netherlads, 645-654 Collis, M, Cooper,, elgaso, R, Keigto, J, eblac,, 978 Solvig the Pipe Network Aalysis Problem Usig Optimizatio Techiques, Maagemet Scie, Vol 4, 747-76 Cross,, 96 Aalysis o Flow i Networks o Coduits or Coductors, Bulleti 86, Egieerig Eperimet Statio, Uiversity o Illiois, Urbae Elahi, E, 4 The Priciples o Desigig Vetilatio i Mies, Publicatio o Jihad Amikabir Uiversity, Ira I Persia El-Nagdy, K A, 8 Aalysis o Comple Vetilatio Networks i Multiple Fa Coal Mie, PhD thesis, West Virgiia Uiversity u, W, ogso, I, 99 The Optimizatio o Airlow Distributio i Vetilatio Networks Usig A Noliear Programmig Method, Miig Scie ad Techology, Vol, No, 9-9 Kamba, G M, Jacques, E, Patigy, J, 995 Applicatio o the Simple Method to the Optimal Adjustmet o the Parameters o A Vetilatio Network, Proedigss o the 7th US Mie Vetilatio Symposium, Wala, A M Ed, SME, ittleto, Co, 46-465 Madai,, Mies Vetilatio, Vol, Tehra: Amirkabir Uiversity o Techology Tehra Polytechic Press, Ira I Persia Madai,, 6 Mies Vetilatio, Vol, Prit 5, Tehra: Uiversity Ceter Pub, Ira I Persia Madai,, Maleki, B, 7 Mies vetilatio etwork aalysis usig Newto Raphso method based o D equatios, Amirkabir Joural o Scie & Research, Vol 66, 97-, Ira I Persia Madai,, Maleki, B, 8 Mies vetilatio etworks aalysis based o equatios i Newto Raphso method, Iraia Joural o Miig Egieerig IJME, Vol, No 5, 7-77 Wag YJ, i S 985 The Newto Method o Calculatig the Vetilatio Network, Proedigs, d Coere o the Use o Computers i the Coal Idustry, AIME, New York, pp 88-9 Wag, Y J, 98a Vetilatio Network Theory, Mie Vetilatio ad Air Coditioig, d ed, artma Ed, Wiley-Iterscie, NY, 67-95 Wag, Y J, 98b Critical Path Approach to Mie Vetilatio Networks with Cotrolled Flow, Tras SME-AIME,Vol 7, 86-87 Iteratioal Joural o Miig Scie IJMS Page
Farhag Sereshki et al Wag, Y J, 984 A No-iear Programmig Formulatio or Mie Vetilatio Networks with Natural Splittig, Iteratioal Joural o Rock Mechaics ad Miig Scie, Vol, No, 4-45 Wag, Y J, 989 A Produre or Solvig A More Geeralized System o Mie Vetilatio Network Equatios, Proedigs o the 4th US Mie Vetilatio Symposium, SME, ittleto, Co, 49-44 Iteratioal Joural o Miig Scie IJMS Page 4