Lab7_Lecture - Lab 7: Scaling Laws and ME395, Fall...

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Unformatted text preview: Lab 7: Scaling Laws and ME395, Fall 2011 Prof. Kenn Oldham 11/14/11 Scaling Laws Virtually all physical phenomena depend on dimensional scaling, with respect to length, mass, and Kme. Boeing 747 F ­16 Beech Bonanza quetzalcoatlus When results follow clear trends over a wide range, we oNen consider there to be a major scaling law in effect. Canada goose swallows dragonfly gnat Flight speed versus weight, Tennekes, 1996 bumblebee Another Examples These scaling laws oNen reflect very basic physical principles, reflected into potenKally very complex derived quanKKes. As such, they tend to be become more precise as their circumstances become more specific. Scaling Laws The previous examples come from the world of “allometric scaling”, which specifically concerns itself with physiological relaKonships in terms of size, anatomy, and even behavior. In engineering, where we oNen have clear equaKons for behavior, scaling can be very easy to perform. Example: Beam bending ω =β 2 EI ρA However, what do we do when those equaKons are not available to us? € One powerful tool is Dimensional Analysis Basics of Dimensional Analysis Dimensional Analysis is a technique for reducing the factors influencing a physical phenomenon to their simplest form for further quanKtaKve analysis. Obvious Features:  ­ Underlying behavior should not depend on choice of units used (metric, SI, etc.)  ­  Units should match on either side of a relaKonship Perhaps Non ­obvious Feature:  ­ We can generally simplify problems by converKng them to dimensionless form. Base Units and Buckingham ­π Buckingham ­π Theorem: When a complete relaKonship between physical quanKKes is expressed in dimensionless form, the number of independent quanKKes that appear is reduced by the maximum number of quanKKes that are dimensionally independent. Dimensionally independent quanKKes will not have units that could be reproduced from the units of other quanKKes. Example: If quanKKes are speed, distance, and Kme, the units of one can be rewrihen in terms of units of the other two. Oddball Example From Textbook by Sonin, 2001: What is the diameter, d, of the circular imprent leN by an elasKc ball covered in dye when it impacts a hard surface (in other words, what is largest flat deformaKon at impact)? Step 1: What quanKKes could influence (assuming surface perfectly rigid)? Quan%ty Variable Name Base Units Ball Diameter D L Ball Density Ρ ML ­3 ElasKc Modulus E MT ­2L ­1 Poisson’s RaKo ν Velocity at impact V LT ­1 Example, ConKnued Independent quanKty Selected base quanKKes d = f ( D,V ,ν, ρ, E ) L LT ­1 Use base quanKKes to find dimensionless output € € ML ­3 d = f (V ,ν, ρ, E ) D ȹ E ȹ d = f ȹ ν, 2 ȹ D ȹ ρV Ⱥ Likewise, use base quanKKes to find dimensionless input “Experimental” Comparison Comparison of paint diameter to ball diameter, as funcKon of independent dimensionless quanKty ρ € Ball materials: Alumina (green) Aluminum (red) Rubber (blue) In ­Class Example d Period of ideal pendulum oscillaKon, Tp: Step 1: What are units of the dependent variable (period) and possible influenKal quanKKes? In ­Class Example Step 2: What can you mulKply/divide our dependent variable (frequency) by to make the output dimensionless? (hint: square ­roots are ok) Tp = f ( m, d, g,θ 0 ) Step 3: What is leN? (hint: radians are considered dimensionless, and mass cannot be canceled, so should it be a€ to affect?) ble Result Tp g L € For small θ0, frequency relaKonship is constant, and this holds for any pendulum Summary So Far •  Dimensional analysis can lend insight into criKcal factors impacKng physical phenomena •  Conversion to dimensionless form can reduce the number of independent variables to consider during experimentaKon This Weeks Lab: Fluid Machinery Fluid machinery adds or extract energy from moving fluids Major types: 1. PosiKve Displacement Devices (fluid confined) –  Compressors, piston systems, many pumps, etc. 2. Turbomachines (fluid parKally or unconfined) –  Windmills, turbines, etc. Turbomachines Further classificaKons: •  Axial flow devices (flow parallel to axis of rotaKon)  ­ Jet engines, propellers, wind turbines •  Radial (or Centrifugal) flow devices (flow enters axially, exits radially, or vice versa) –  Impellers, some pumps, blowers Our System: Impeller Pump Spinning blades pull in air and force it out at elevated pressure/speed. Key funcKonality is to increase energy (ΔE) or equivalently, do work upon, the fluid. P3 − P2 V32 − V22 ΔE = + g( h2 − h1 ) + ρ 2 € PotenKal Factors In the ME395 Impeller system (a blower), potenKal design factors include: Factor Variable Dimensions Impeller diameter D L Inlet pipe diameter Li L Outlet pipe diameter Lo L Flow Rate Q L3T ­1 RotaKonal Speed ω T ­1 Fluid density ρ ML ­3 Fluid viscosity μ ML ­1T ­1 AcceleraKon due to gravity g LT ­2 Dimensionless Analysis To iniKate our dimensionless analysis, we will look at work done (per unit fluid) ˙ W = f ( D, Li , Lo, Q,ω , ρ, µ, g) ˙ ȹ Li Lo Q W µ g ȹ , , 3, 3 5 = f ȹ 2 , 2 ȹ) ρω D ȹ D D ωD ρωD ω D Ⱥ € € Constant Geometry If geometry is kept proporKonal, we can reduce the equaKon to a funcKon of well ­known dimensionless parameters for blowers: Q φ= 3 ωD Flow coefficient ˙ W 3 5 = τ = f (φ , Re, Fr )) ρω D ΔE Head ψ = 2 2 = f 1 (φ , Re, Fr ) coefficient ω D €W ˙ τ= 3 5 = f 2 (φ, Re, Fr ) Power ρω D € Coefficient € QρΔE η = ˙ = f 3 (φ, Re, Fr) W Efficiency Scaling Analysis given Geometric Similarity Now, for a given geometry, we can search for relaKonship between flow coefficient and other coefficients (such as power coefficient) that are funcKons of Re and Fr. Self ­Similar Performance In fluid dynamics, there are many cases where flow behavior does not depend on Re and Fr. If so, curves will collapse onto one master curve, producing self ­similar performance Test Setup Experimental Methods In our experiment, we want to find relaKonships between flow coefficient and hydrostaKc head coefficient, as a funcKon of Reynold’s number and Froude number. What can we change? 1. Flow rate, using a valve (doesn’t affect Re & Fr) 2. Impeller speed (affects Re & Fr) What can we measure? 1. Inlet flow velocity 2. Inlet flow condiKons 3. Pressure condiKons at inlet & outlet (ambient) Tasks 1.  Determine: Does self ­similarity occur for this system? 2.  For a geometrically scaled applicaKon, what impeller speed should give a specified pressure and flow rate? –  Note: change in gas as well as dimensions! –  Results from similarity analysis must sKll hold! Obtaining Coefficients From pressure change and velocity, we must reconstruct Re, Fr, ψ, and φ. To do so, we will use tools from fluid dynamics: •  ConservaKon of Mass •  Bernoulli’s EquaKon Control Volumes We look at various control volumes for fluid flow through our system. Points of interest: 1 – outside blower 2 – before impeller 3 – aNer impeller Measured experimentally: P1, V2, P3 ­P2 V1, P1 1 2 V2, P2 Volume 1: no mechanical work yet V3, P3 3 Volume 2: mechanical work done Flow Rate CalculaKons First, to esKmate flow rate, we must make an assumpKon about flow profiles. Simple version: ȹ πL2 ȹ Q = AV = ȹ in ȹV2 ȹ 4 Ⱥ Velocity before entering impeller Other, more accurate profiles can also be used. € ConservaKon of Mass In enclosed space, at steady ­state, with incompressible flow, flow rate should be constant. Therefore: ȹ πL2 ȹ ȹ πL2 ȹ Q = ȹ in ȹV2 = ȹ out ȹV3 ȹ 4 Ⱥ ȹ 4 Ⱥ Meanwhile, we can assume far enough from inlet, € V1 ≈ 0 Bernoulli’s EquaKon (simplified ConservaKon of Energy) Assuming incompressible, inviscid fluid, with no external work done (as in control volume 1)… Two points on the same streamline must saKsfy: P1 V12 P2 V22 + =+ ρ2 ρ 2 Therefore: € V22 P2 = P1 − ρ 2 And P 3 = P2 + ΔP, with ΔP measured experimentally € € Energy Added Now that we know our states at each point, we can calculate the energy imparted on the fluid P3 − P2 V32 − V22 ΔE = + g( h2 − h1 ) + ρ 2 Which gives us enough informaKon to calculate all of € our dimensionless quanKKes for our experimental testbed. Challenge for Analysis The calculaKons to this point will allow you to plot flow coefficient vs. hydrostaKc head coefficient from your experimental data (various combinaKons of Re and Fr) – Does self ­similarity of flow coefficient vs. head coefficient occur?? To perform the tasks in the leher, you must also: 1. Convert task leher specificaKons (fixed outlet pressure, varied flow rates) to specificaKons in terms of our dependent quanKKes (energy to be added at the various flow rates), given scaled geometry, gas selecKon. Challenge for Analysis 2. Work out rotor speed providing desired energy increase and flow rate, while saKsfying experimental relaKonship between flow coefficient and head coefficient. Q φ= 3 ωD € € ΔE ψ= 2 2 ωD Specified What motor speed produces desired Q and ΔE while also saKsfying experimental relaKonship between φ and ψ? Experimental Issues •  Remember to incorporate uncertainty analysis •  Check pressure transducer and velocity sensor accuracy data •  LimitaKons on assumpKons (flow profile, losses in impeller) Safety •  Safety Glasses •  Powerful motor •  Do not drop anything into blower! •  Do not leave control valve shut ...
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This note was uploaded on 11/22/2011 for the course ME ME360 taught by Professor Kuo during the Fall '11 term at University of Michigan.

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