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
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 pressure @ 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 lines @ 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
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:
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
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 Pexit @ 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 snic @ cnstrictive thrat Snic fluid @ 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
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 pressure @ 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
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)