Chapter 7. Systems 7.1 INTRODUCTION 7.2 MATHEMATICAL MODELING OF LIQUID LEVEL SYSTEMS. Steady State Flow. A. Bazoune

Transcription

1 Chapter 7 Flud Systems and Thermal Systems 7.1 INTODUCTION A. Bazune A flud system uses ne r mre fluds t acheve ts purpse. Dampers and shck absrbers are eamples f flud systems because they depend n the vscus nature f a flud t prvde dampng. In addtn t prvdng dampng, ther applcatns f flud systems nclude actuatrs and prcesses that nvlve mng, heatng, and clng f fluds. Actve vehcle suspensns use hydraulc and pneumatc actuatrs t prvde frces t supplement the passve sprng and dampng elements. Water supply, waste treatment, and ther chemcal prcessng applcatns are eamples f a general categry f flud systems called lqud-level-systems, because they nvlve regulatng the vlumes, and therefre the levels f lquds n cntaners such as tanks. A flud mght be ether a lqud r a gas. A flud s sad t be ncmpressble f the flud s densty remans cnstant despte changes n the flud pressure. If the densty changes wth pressure, the flud s cmpressble. 7.2 MATHEMATICAL MODELING OF LIQUID LEVEL SYSTEMS Lamnar Steady State Flw Turbulent D Fgure 7.1 (a) Velcty prfle fr lamnar flw Fgure 7.1 (b) Velcty prfle fr turbulent flw Flw dmnated by vscsty frces s called When nerta frces dmnate, the flw s 1/9

2 lamnar flw and s characterzed by a smth, parallel lne mtn f the flud and lw ρvd eynlds number e = < 2000 µ where ρ s the mass densty f the flud, µ s the dynamc vscsty f the flud, v s the average velcty f flw, and D s characterstc length. Frctn frce s lnearly prprtnal t velcty, f f = bv called turbulent flw and s characterzed by an rregular and eddylke mtn f the flud and Hgh eynlds number e > Frctn frce vares as a pwer f velcty f f = bv α esstance and Capactance f Lqud-Level Systems. Cnsder the flw thrugh a shrt ppe cnnectng tw tanks as shwn n Fgure 7-2. The resstance fr lqud flw n such a ppe r restrctn s defned as the change n the level dfference (the dfference f the lqud levels f the tw tanks) necessary t cause a unt change n flw rate; that s, Change n level dfference esstance = = 3 Change n flw rate m / ( H H ) 1 2 Q m s H 1 H 2 Q Fgure 7-2 Tw tanks cnnected by a shrt ppe wth a valve Snce the relatnshp between the flw rate and the level dfference dffers frm lamnar flw and turbulent flw, we shall cnsder bth cases n what fallws. esstance n Lamnar Flw. Q + q Cntrl valve Capactance C H + h Lad valve Fgure 7-3 esstance (a) Lqud level system; Q + q ο 2/9

3 Fr lamnar flw, ( e < 2000 ), the relatnshp between the steady-state flw rate and steadystate head at the level f restrctn s gven by Q = l K H Where Q = steady-state lqud flw rate, m 3 / s, K = cnstant, m 2 / s and H = steady-state head, m. Fr lamnar flw, the resstance l s dh 1 H l = = = dq K Q The lamnar-flw resstance s cnstant and s analgus t the electrcal resstance, where. Heght Vltage ( ) Steady state flw rate Q Current ( ) ( H ) e and ( ) esstance n Turbulent Flw. Fr turbulent flw; ( e > 3000 ), the steady-state flw rate s gven by l l Q = K t H (7-1) where Q = steady-state lqud flw rate, m 3 / s, K = cnstant, m 2.5 / s and H = steady-state head, m. The resstance t fr turbulent flw s btaned frm t Then dh t = dq Kt dq Kt dh 2 H 2H dq = dh = = = 2 H dh 2 H d Q ( Q/ H ) Q Thus t 2H = Q (7-2) Head H O H h Q P q 2H h Slpe = = Q q 1 tan ( ) t Flw rate Fgure 7-3 (b) curve f head versus flw rate 3/9

4 Capactance The capactance f a tank s defned t be the change n quantty f stred lqud necessary t cause a unty change n the ptental (head). The ptental (head) s the quantty that ncludes the energy level f the system). r 3 Change n lqud stred m Capactance C = Change n head m r m Capactance C = Crss-Sectnal area (A) f the tank. ate f change f flud vlume n the tank flw n flw ut 2 dv = q n q dh C = qn q ut ut ( ) d A h dh = q q A = q q n ut n ut C dh = q n q ut Mathematcal Mdelng f Lqud-level Systems. Cnsder the system shwn n fgure 7-3(a). If the peratng cndtn as t the head and flw rate vares lttle fr the perd cnsdered, a mathematcal mdel can easly be fund n terms f resstance and capactance. Assume turbulent flw, and defne Q + q Cntrl valve Capactance C H + h Lad valve Fgure 7-3 esstance (a) Lqud level system; Q + q ο H = steady-state head (befre any change has ccurred), m. h = small devatn f head frm ts steady-state value, m. Q = steady-state flw rate (befre any change has ccurred), m 3 /s. q = small devatn f nflw rate frm ts steady-state value, m 3 /s. q = small devatn f utflw rate frm ts steady-state value, m 3 /s. Tank: The rate f change n lqud stred n the tank s equal t the flw n mnus flw ut = C dh q q 4/9

5 r ( ) C dh = q q (7-3) where C s the capactance f the tank. In the present system, we defne h and q as small devatns frm steady state head and steady state utflw rate, respectvely. Thus, dh = h, dq = q esstance : The resstance may be wrtten as dh h h = = q = dq q h Substtute q = nt Equatn (7-3), we btan dh h C = q r dh C + h = q (7-4) Ntce that C s the tme cnstant f the system. Equatn (7-4) s a lnearzed mathematcal mdel fr the system when h s cnsdered the system utput. If q rather than h, s cnsdered the system utput, then substtutng h = q n the abve equatn gves dq C + q = q (7-5) Analgus Systems. The lqud level system cnsdered here s analgus t the electrcal system shwn n Fgure 7-4(a). It s als analgus t the mechancal system shwn n Fgure 7-4(b). Fr the electrcal system, a mathematcal mdel s de C + e = e (7-6) Fr the mechancal system, a mathematcal mdel s b d + = (7-7) k Equatns (7-5), (7-6) and (7-7) are f the same frm; thus they are analgus. (b) (a) Fgure 7-4 Systems analgus t lqud level system shwn n Fgure 7-3(a). (a) Electrcal system. (b) mechancal system 5/9

6 Lqud-Level System wth Interactn. 7.3 LINEAIZATION OF NONLINEA SYSTEMS Lnearzatn f z = f ( ) abut a pnt (, ) z. Cnsder a nnlnear system whse nput s and utput s z, the relatnshp between z and may be wrtten as z ( ) z = f ( ) z = f (7-21) z If the nrmal peratng cndtn crrespnds t a pnt (, ) z, then Equatn (7-21) can be epanded nt a Taylr seres abut ths pnt as fllws: 2 df 1 d f 2 z = f ( ) = f ( ) + ( ) + ( ) + d 2! d 2 = = (7-22) 2 2 d f d are evaluated at the peratng pnt, = where the dervatves df d,, z = z. If the varatn ( ) s small, we can neglect the hgher-rder terms n z = f, Equatn (7-22) can be wrtten ( ). Ntng that ( ) Ntng that z = ( ) df z = f + d ( ) ( ) = f, we can wrte Equatn (7-22) as z z = m( ) (7-23) where df m = d = 6/9

7 Equatn (7-23) ndcates that z z s prprtnal t. The equatn s a lnear mathematcal mdel fr the nnlnear system gven by Equatn (7-21) near the peratng pnt, z z. Equatn (7-23) represents an equatn f the tangent lne t the curve z = f ( ) at the peratng pnt (, z ) wth a slpe m z z ( ) z = f z z = m( ) Fgure 1. Lnearzatn f the functn, z y = f ( ) abut the pnt ( ) Lnearzatn f z = f (, y ) abut a pnt (,, ) y z. Net, cnsder a nnlnear system whse utput z s functn f tw nputs and y such that (, ) z = f y (7-24) T btan a lnear mathematcal mdel fr ths nnlnear system abut an peratng pnt (, y, z ), we epand Equatn (7-24) nt a Taylr seres abut ths pnt as fllws: z = f (, y ) + ( ) + ( y y ) y f 2 f f 2 + ( ) 2 ( )( y y ) ( y y ) ! y y where the partal dervatves are evaluated at the peratng pnt, =, y = y, and z = z. Near ths pnt, the hgher-rder terms may be neglected. Ntng that z = f (, y ), a lnear mathematcal mdel f ths nnlnear system near the peratng pnt =, y = y, and z = z s where z z = m ( ) + n( y y ) m = n = y =, y = y =, y = y Eample 7-3 (Tetbk Page 336) Lnearze the nnlnear equatn z = y n the regn 5 7, 10 y 12. Fnd the errr f the lnearzed equatn s used t calculate the value f z when y. = 5 and = 10 7/9

8 Slutn Snce the regn cnsdered s gven by 5 7, 10 y 12, chse = 6, y = 11. Then z = y = 66. Let us btan a lnearzed equatn fr the nnlnear equatn near a pnt = 6, = 11 and z = 66. Epandng the nnlnear equatn nt a Taylr s seres abut the pnt =, y = y and z = z and neglectng the hgher rder terms, we have z z = m ( ) + n( y y ) where m = = [ y] = y = y = 11 =, y= y =, y= y =, y= y m = = [ y] = = = 6 =, y= y =, y= y y y =, y= y Hence the lnearzed equatn s r z 66 = 11( 6) + 6( y 11) z = y 66 When = 5 and y = 10, the value f z gven by the lnearzed equatn s ( ) ( ) y, z = y 66 = = = 49 z = y = 5 10 = 55. The errr s thus z = 100 = 2 % 49 The eact value s ( ) ( ) terms f percentage, the errr s Eample 7-4 (Tetbk Page 336) =. In Cnsder the lqud level-system shwn n Fgure 7-8. At steady state, the nflw rate s Q = Q, the utflw rate s Q = Q, and the head s H = H. Assume that the flw s turbulent. Then Q = K H Fr ths system, we have dh C = Q Q = Q K H Where C s the capactance f the tank. Let us defne dh 1 K H = Q = C C (, ) f H Q (7-25) 8/9

9 Assume that the system perates near the steady-state cndtn ( H, Q ). That s, H = H + h and Q = Q + q, where h and q are small quanttes (ether pstve r negatve). At steady-state peratn, dh = 0. Hence, f ( H, Q ) = 0. Slutn Snce the regn cnsdered s gven by 9/9