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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 ȹν,
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.
 Fall '11
 KUO

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