bulk velocity through orifice,

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1 150A Review Sessin Other Frictin Lsses Bernulli hf accunts fr all types f drag: is drag due t skin frictin is drag due t fittings (tabulated fractin f the velcity head) is drag due t units (a given r calculated pressure drp) Kf fr sudden expansin/cntractin based n smaller crss sectin velcity Ke = 1.0 fr filling large tank Kc = 0.4 fr emptying large tank Other tabulated Kf values: Flw Measurement and Cntrl Orifice meter Small & cheap t install Measure pressure drp thrugh small crss sectin rifice Create large irrecverable pressure drp: bulk velcity thrugh rifice, Measure rifice cefficient fr fluid, Rtameter Gd fr small vlumes f gasses r liquids Measure balance f flw, gravity and buyancy n flat is the tube crss sectinal area where the tp f the flat sits must be measured fr a specific gemetry & fluid is the crss sectinal area f the tp f the flat is flat density, is fluid density is fluid velcity

2 150A Review Sessin Pumps Incmpressible fluids are transprted by pumps Head describes pump functinality Centrifugal pumps High flw rate, mderate head increase Inlet in center, impeller accelerates fluid, cnverted t utlet Psitive displacement pumps Mderate flw rate, high head increase Admit fixed vlume int inlet, eject at higher pressure at utlet Units f head Frm Bernulli, therefre Head is generally pltted with units f meters r feet (any term in Bernulli divided by g) Pump curves Fr a specific pump manufacturer, mdel and RPM nly!! Head (y-axis) vs. vlumetric flw rate (x-axis) Dwnward curving lines fr discrete values f impeller diameter Steeply dwnward curving lines fr hrsepwer t pump fluid with s.g.=1 U-shaped lines fr cntinues values f pump efficiency Methd fr reading: 1) Lcate system pint = crssing f desired head & flw rate 2) Interplate between the efficiency system pint, that is yur η 3) G up t next highest impeller diameter line, that is the DI yu need 4) The ttal head frm the intersectin f Q and DI is the max pump head 5) If yu are pumping water yu can interplate the pump Hp frm the chart, but it is better t calculate it frm the pump wrk equatin Pump cavitatin Cavitatin ccurs when the Pinlet is < Pvap and yu vaprize yur fluid Pumps designed fr liquids wn t pump gasses effectively, need Pinlet > Pvap NPSH gives safety margin fr a particular pump beynd the Pvap Read NPSH ff the chart (usually a separate axis) The minimum head (pressure + velcity) yu need at pump inlet is: Nte that the equality abve is Bernulli fr the system befre pump

3 150A Review Sessin Flw Arund Immersed Objects Dynamic frces (drag frces) n a particle due t relative mtin f bject and fluid (shear stress) Definitin: Fk = frce n the particle Cd = drag cefficient (analgus t frictin factr in pipe flw) u = apprach velcity (far frm bject) A = prjected area f particle = density f the fluid Reynlds number fr flw arund an bject: Stkes flw fr creeping flw arund a sphere: Re < 1 (r apprximately Re < 10) Newtn s regime fr turbulent flw arund a sphere: 1000 < Re < 200,000 Intermediate regime, use the chart (nte variatin w/ bject shape): Particle reaches terminal velcity when Gravity: Buyancy: Terminal velcity: Settling (terminal) velcity:

4 150A Review Sessin Stkes regime cmmn fr small particles: (Re<1, ), Newtn s regime: (1000<Re<200,000, Cd 0.44), Criterin fr settling better than Reynlds number (Re) because it depends nly n physical parameters K < 2.6 Stkes regime K > 68.9 Newtn s regime Fixed and Fluidized Beds Fixed bed fluid des nt impart enugh drag t vercme gravity, s particles are statinary; bed length Lb cnstant; ΔP increases with increasing u Fluidized bed drag + buyancy vercme gravity, s particles begin mving with flw; bed length Lb increases with increasing u; ΔP is cnstant Ergun equatin describes a fixed/packed bed: ΔPErgun = pressure drp in the bed (type f frictinal lss, ) u = superficial r bulk velcity thrugh bed Lb = bed length = density f the fluid Φs = sphericity f particle (tabulated), Φs = Dp = equivalent spherical particle diameter ε = prsity r vid fractin, 1 st term in Ergun = Blake-Kzeny eqn fr laminar flw (ε<0.5, Re p<10) 2 nd term in Ergun = Burke-Plummer eqn fr turbulent flw (Re p>1,000) Reynlds number: Fr gases, use average gas density between inlet and utlet pressures. Interstitial velcity: u = u/ε between particles in bed (u > u) Minimum fluidizatin velcity: ρp = particle density ρf = fluid density umf = minimum fluidizatin velcity εmf = vid fractin at minimum fluidizatin, ; Mp = mass f all particles ; Ab = crss sectinal area f bed ; Lb,mf bed length at minimum fluidizatin

5 150A Review Sessin Cmpressible Flw Relatins cme frm cmbinatin f mass balance, mechanical energy balance, ttal energy balance and equatin f state! We have used tw types f thermdynamic pathways: Isthermal lng pipes, expsed t cnstant T envirnment, include frictin Isentrpic/reversible shrt units (nzzles & cmpressrs), ignre frictin is the flw descriptr, because G remains cnstant as ρ and u change Often have fully develped turbulent flw -> at the end, must calculate Re and make sure yu are in the flat sectin f the Isthermal flw: Maximum speed is isthermal speed f sund vs. Re chart If flw is subsnic (nrmal), then pressure will drp frm Pinlet t Pexit in pipe Use the general relatin: Check that u2 (end f pipe) < usund t make sure flw is subsnic If flw ges snic (chked), pressure will nt be able t drp t end Use Gmax equatin: Check snic flw [P2 (end f pipe) > Pexit]: Nte that these equatins are nn-linear, s yu need t iterate t slve! Isentrpic flw in nzzles: Isentrpic speed f sund Mach number describes flw, describes fluid prperties With enugh ΔP driving frce, fluid can g cnstrictive thrat Snic thrat can then g supersnic in diverging nzzle (because f mass balance, fluid must accelerate as it expands in diverging nzzle) 4 equatins used fr relevant prperties (* = thrat, = stagnant reservir): These values tabulated fr (air, diatmic gasses) Search table fr given, find Ma,, and there 1) Given, find Ma: 2) 3) 4) Maximum mass flw rate defined by snic flw at thrat

6 150A Review Sessin Nrmal shcks (γ=1.4) Due t imperfectins in a nzzle, flw can drp ff the isentrpic pathway, causing a nrmal shck in a given lcatin as it decelerates t u<usund Nrmal shck tables relate cnditins befre and after a shck Find Ma n isentrpic table fr the given shck lcatin Lk up that Ma as Ma1 n nrmal shck table, recrd ratis f pst shck values Ma2, P2 and T2 relative t P1 and T1 befre shck After nrmal shck, fluid will cntinue t travel n new isentrpic pathway Use isentrpic flw table t find fictinal A *, T and P that wuld have given yu the same cnditins as the pst-shck cnditins Use isentrpic flw table t find cnditins further dwn the nzzle relative t thse A *, T and P values Cmpressrs Cmpressible fluids are transprted using a cmpressr, wrk in increases pressure and als therefre temperature f the fluid Cmpressin rati describes cmpressr functinality Rtating centrifugal cmpressrs (can be multistage) High flw rates, lwer cmpressin ratis Inlet in center, impeller accelerates fluid, cnverted t utlet Reciprcating pistn cmpressrs (i.e. bicycle pump) Lw flw rates, high cmpressin ratis Admit fixed vlume int inlet, eject at higher pressure at utlet Determine number f required centrifugal cmpressr stages: Guideline (1) max cmpressin rati/stage ~3 Guideline (2) max discharge temperature ~350 F Wrk is minimized if each stage has the same cmpressin rati Assumptin that fluid is cled t initial temperature T1 befre each stage Minimal wrk frm cmpressr with infinite stages (isthermal cmpressr) but that is nt realistic Assume cmpressrs are isentrpic (reversible) then incrprate inefficiencies: P = cmpressr pwer n = number f stages with equal cmpressin ratis is wrk/stage

7 150A Review Sessin = cmpressr efficiency Can als d this using a Pressure-Enthalpy chart Pressure (y-axis) vs. Specific Enthalpy (x-axis) Dme n left is liquid-vapr regin; d nt want t frm liquid! Mstly vertical lines are Temperature Gradually psitively slped lines are Specific Vlume Mre steeply psitively slped lines are Specific Entrpy Methd fr using chart: 1) Find starting pint using initial P and T 2) Read ff H at that pint 3) Fllw a line f cnstant S frm starting pint t ending P (f that stage) 4) Read ff T and H at that end pint 5) ; 6) Repeat fr multiple stages, assuming perfect intercling Calculate final temperature is heat generated during cmpressin Need t add extra heat due t inefficiencies (assume all in fluid) Equatin: Chart: start at Pend, Tend, g right by t get T2, actual Plytrpic cmpressin Accunts fr actual path being nt perfectly isentrpic Includes plytrpic efficiency in calculating wrk (j = number f stages)