concept: dimensional analysis and dynamic scaling context: … · 2020. 12. 14. · e = 28 kilotons...
TRANSCRIPT
![Page 1: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/1.jpg)
Concept: Dimensional Analysis and Dynamic Scaling
Context: Designing a Dynamically Similar Model
Defining Question: Restaurants display models of food items.Why don’t fashion boutiques display clothes on dolls?
EngrD 2190 – Lecture 37
![Page 2: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/2.jpg)
1. Derive a set of dimensionless groups given a list of parameters.
Lectures 33, 34, and 35 last week:
Pendulums Swinging, People Walking, and Spheres Falling.
Practice Exercises: 5.8, 5.15, 5.20, and 5.22.
2. Use a Universal Correlation of Dimensionless Groups.
Lecture 36.
Practice Exercises: 5.28, 5.29, and 5.30.
3. Design a Dynamically Similar Model by Scaling.
Lectures 37 and 38
Practice Exercises: 5.33, 5.34, and 5.39.
Solutions to practice exercises are posted at the EngrD 2190 homepage.
Three skills in dimensional analysis / dynamic scaling
![Page 3: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/3.jpg)
Designing a Dynamically Similar Model
Assume the system is described by n groups: 1, 2, …, n
1. Set (i)model = (i)actual for i = 1 to n1.
2. Use the model to measure n.
3. Use n to calculate parameters for the actual system.
![Page 4: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/4.jpg)
blood0.010 m/sec
10 m
1 mm I.D.P = ?
Example 1: Design a model to predict the pressure drop in an artificial kidney.
Table 5.8. The parameters of fluid flow through a pipe.
our goal
Use to design model
use 10 in model
need 10 in model
![Page 5: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/5.jpg)
blood0.010 m/sec
10 m
1 mm I.D.P = ?
Example 1: Design a model to predict the pressure drop in an artificial kidney.
Table 5.8. The parameters of fluid flow through a pipe.
~10 Use vegetable oil in the model.
0.050930
vegetable oil
first guessUse 2 to calculate
Use 3 to calculate
![Page 6: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/6.jpg)
Example 1: Design a model to predict the pressure drop in an artificial kidney.
Use 2 to calculate the pipe length (d) in our model.
tubeof m100m01.0m001.0
m10model
kidneymodel
kidneymodel
dd
dd
Use Re (3) to calculate the fluid velocity (v) in our model.
cm/sec4.1m/sec014.0
01.093005.0
004.0001.001.01025
model
modelfluidkidney
fluidmodel
kidney
fluid
model
fluid
v
dvdv
vdvd
reasonable
Re = 2.6
If we used water in the model, vmodel = 0.03 cm/sec. Too slow. If glycerin, vmodel = 25 cm/sec. Too fast.
![Page 7: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/7.jpg)
blood0.010 m/sec
10 m
1 mm I.D.P = ?
Example 1: Design a model to predict the pressure drop in an artificial kidney.
Table 5.8. The parameters of fluid flow through a pipe.
0.050930
vegetable oil
first guessfrom 2
from Re
100 m
0.014 m/sec
We build the model and measure P = 9.0 105 Pa = 8.9 atm = 130 psi
Use 1 to calculate P in our artificial kidney.
psi74atm0.5Pa100.9)(
1025)01.0930)014.0(
100.9)(
5kidney
22
5
kidneyfluid2
modelfluid2kidney
modelfluid2
kidneyfluid2
P
vv
PP
vP
vP
![Page 8: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/8.jpg)
blood0.010 m/sec
10 m
1 mm I.D.P = ?
Example 1: Design a model to predict the pressure drop in an artificial kidney.
Blood pressure is measured in units of mm Hg. 80/160 ~ 1.5 psi/3.1 psi.
psi74atm0.5Pa100.9)( 5kidney P
Pressure drop is too large. What to do?
Use a hundred 100-cm tubes in parallel. Total length = 10 m, P = 0.74 psi
![Page 9: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/9.jpg)
Example 2: Hot microspheres descending and cooling in a liquid.
Cu
v = 5 cm/sec
q = ?
0.1 mm water
We need the sphere’s heat transfer coefficient, h.
Design a more convenient model with larger spheres.
))(area surface sphere(
))(area(
fluidsphere TThq
Tqh
![Page 10: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/10.jpg)
Example 2: Hot microspheres descending and cooling in a liquid.
Cu
v = 5 cm/sec
q = ?
0.1 mm water
our goal
Use to design model
The Prandtl number comprises fluid properties only.
Dynamic similarity requirement (Pr)model = (Pr)actual demands model system and actual system use same fluid: water.
![Page 11: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/11.jpg)
Example 2: Hot microspheres descending and cooling in a liquid.
Calculate vmodel.
use Prandtl number;model must use water
use Reynolds number
measure
m/sec105m01.0
m10m/sec05.01
ReRe
44
modelemicrosphermodel
modelfluidemicrospher
fluidmodel
emicrospher
fluid
model
fluid
emicrosphermodel
dvdv
dvdv
vdvd
vmodel = 0.5 mm/sec = 30 cm/min Too slow for a free-falling 1-cm copper sphere in water.
![Page 12: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/12.jpg)
Example 2: Hot microspheres descending and cooling in a liquid.
vmodel = 0.5 mm/sec = 30 cm/min
Cu
Use capstan toslowly feed line.
Line also servesas thermcouple.
30 cm/min
Conduct experiments, measure hmodel = 240 J/(m2ꞏsecꞏK)
Calculate the Nusselt Number -
Calculate hmicrosphere
1.4K)secJ/(m59.0
)m01.0K))(secJ/(m240(Nu2
khd
K)secJ/(m104.2m0001.0
K)secJ/(m59.01.4
1.4
1.4
NuNu
24emicrospher
emicrospheremicrospher
emicrospher
modelemicrospher
h
dkh
khd
![Page 13: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/13.jpg)
recap on Designing a Dynamically Similar Model by ScalingAssume the system is described by n groups: 1, 2, …, n
1. Set (i)model = (i)actual for i = 1 to n1.
2. Use the model to measure n.
3. Use n to calculate parameters for the actual system.
If your model has a different length and/or mass dimension -your model is smaller or less massive - your model must alsochange something so there is a different time dimension.
Usually, use a different fluid for the model.
Fluid parameters: viscosity:
surface tension:
LTM][
heat capacity:
2TM][
thermal conductivity:
2
2
P TL][C
3T
ML][k
![Page 14: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/14.jpg)
Example 3: A Universal Correlation for an Explosive Shock Wave.
![Page 15: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/15.jpg)
Example 3: A Universal Correlation for an Explosive Shock Wave.
![Page 16: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/16.jpg)
![Page 17: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/17.jpg)
Example 3: A Universal Correlation for an Explosive Shock Wave.
4 parameters – 3 dimensions= 1 group
dcba trE
cadbadad
cba
dcba trE
232
32 TLMLMTL
TLM Set a = 1
From experiments, = 11/5
5
2
rEt
1/52Thus
Etr
From the declassified photos, Geoffrey Taylor calculated the energy of the Trinity explosion:
E = 28 kilotons of TNT
![Page 18: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/18.jpg)
Dimensional Analysisand Dynamic Scaling
Trucks, Power Shovels, and Barbie Dolls
![Page 19: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/19.jpg)
Dynamic Similarity in Trucks
![Page 20: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/20.jpg)
Dynamic Similarity in Trucks
![Page 21: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/21.jpg)
Dynamic Similarity in Trucks
![Page 22: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/22.jpg)
Dynamic Similarity in Trucks
Taylor Dunn ET-3000 Terex TR-100
Height to Load Bed
3.5 ft 15 ft×4.3
![Page 23: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/23.jpg)
Dynamic Similarity in Trucks
Taylor Dunn ET-3000 Terex TR-100
Vehicle Weight
0.95 ton 76 ton×(4.3)3
![Page 24: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/24.jpg)
Dynamic Similarity in Trucks
Taylor Dunn ET-3000 Terex TR-100
Load Capacity
3000 lb = 1.5 ton 100 ton×(4.1)3
![Page 25: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/25.jpg)
Dynamic Similarity in Trucks
Taylor Dunn ET-3000 Terex TR-100
Engine Power
15 hp 1050 hp×(4.1)3
![Page 26: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/26.jpg)
Dynamic Similarity in Trucks
Taylor Dunn ET-3000 Terex TR-100
Tires
20.5×8×1020 inches = 1.7 ft
59/80R6314 ft
×(4.3)3/2
![Page 27: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/27.jpg)
How Tires Carry the Load
Tire Contact Area Scales as Height Squared
![Page 28: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/28.jpg)
How to Scale Tire Height?Weight scales as (Height)3
Increase Tire Height as (Height)1: Tire Area increases as (Height)2
Not Enough
Increase Tire Height as (Height)3/2: Tire Area increases as (Height3/2)2 = (Height)3
Enough!
![Page 29: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/29.jpg)
Dynamic Similarity in Trucks
Taylor Dunn ET-3000 Terex TR-100
Tires
20.5×8×1020 inches = 1.4 ft
59/80R6314 ft
×(4.3)3/2
![Page 30: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/30.jpg)
![Page 31: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/31.jpg)
![Page 32: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/32.jpg)
![Page 33: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/33.jpg)
![Page 34: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/34.jpg)
![Page 35: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/35.jpg)
![Page 36: Concept: Dimensional Analysis and Dynamic Scaling Context: … · 2020. 12. 14. · E = 28 kilotons of TNT. Dimensional Analysis and Dynamic Scaling Trucks, Power Shovels, and Barbie](https://reader035.vdocuments.site/reader035/viewer/2022071516/6138d5bf0ad5d20676498161/html5/thumbnails/36.jpg)
Why does this garmentdisplay poorly?
What scaling wasignored?
Thread count!
The cloth used for thisgarment is the samecloth used at full scale.
The scaled equivalent isburlap at full scale, whichhas a thread count ofabout 10-12 per inch.
Burlap garmentshang poorly.