Concept: Dimensional Analysis and Dynamic Scaling Context ...
Transcript of Concept: Dimensional Analysis and Dynamic Scaling Context ...
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Concept: Dimensional Analysis and Dynamic Scaling
Context: Universal Scaling of a Pendulum
Defining Question: Why do you need not fear gigantic ants or gargantuan mosquitos?
Read Chapter 5 pp. 431-436Dynamics of Walking and RunningLecture 34 will follow the textbook.
EngrD 2190 – Lecture 33
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Include yourself
Due Friday in lecture.
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Calculation Sessions this week - Tours of Unit Operations LabNo exposed toes, exposed arms or exposed legs.Each team must submit an analysis of the distillation column,
to be graded as homework.
Prelim 3: Tuesday 11/23, 7:30-9:30 p.m., 245 Olin Hall.Graphical Modeling for Mass Balances - Part 2:
translating thermodynamic maps: T-(x,y) to x-y, for example.operating lines for multistage absorbers and strippersoperating lines for multistage distillationphase maps for pure substances - separating condensible vapors
from non-condensible vapors.design using single-stage and countercurrent multi-stage units -
may include use of T-(x,y), P-(x,y), and H-(x,y) diagrams.Open book, open notes, open exercise solutions.Bring a calculator and a ruler. Graphing calculators are allowed.
Laptops allowed as for Prelims 1 and 2.
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Dimensional Analysisand Dynamic Scaling
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A ‘Water Bear’ (aka a Tardigrade) - Nature’s Toughest Animal
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Dynamic Scaling Example 1 - Motion Through a Fluid
Swimmer stops swimming - swimmer glides for 1-3 body lengths
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Dynamic Scaling Example 1 - Motion Through a Fluid
Propellers stop - ship glides for ~100 ship lengths
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Dynamic Scaling Example 1 - Motion Through a Fluid
Propulsion stops - paramecium glides for ~0 body lengths
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Dynamic Scaling Example 1 - Motion Through a Fluid
Key ratio for motion through fluids:forces frictional
forces inertial
Ratio is negligible for microfluidics
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Dynamic Scaling Example 1 - Motion Through a Fluid
Key ratio for motion through fluids:forces frictional
forces inertial
Ratio is large for commercial chemical processes
Reactor cascade connected by 20” pipes.
Flow is 40 tons/minute
Partial oxidation of cyclohexane to cyclohexanol.First step in process to synthesize Nylon.
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Flixborough England 1974
Explosion killed 28 and caused $450,000,000 in damages.
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Dynamic Scaling Example 2 - Distillation Column Scale-UpMcCabe-Thiele analysis: 10 equilibrium stages
bench scalemodel
total height: 60 cm
tray diameter: 10 cm
vapor holes in sieve tray: 3 mm
Liquid depth on tray: 2 cm
actualcolumn (100)
60 m
10 m
30 cm (~14 inches)
2 m (~6½ feet)
The bench-scale model worked well. The commercial-scale unit failed. Why?
Flow behavior depends on viscosity, density, and surface tension,which cannot be scaled.
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reactionrate
1
10
100
1,000
10,000
100,000
temperature
desired reaction
undesired reaction
Dynamic Scaling Example 3 - Chemical Reactor Scale-Up
Why?
CH2=CH2 + O2
heat
CH2
OH2C
kJ/mol105(O)CHCH½OCHCH :reaction desired 22222
kJ/mol1320OH2CO2O2CHCH :reaction undesired 22222
reactortemperature
bench scalemodel
reactor diameter: 1 cm
reactor length: 10 cm
workedfine
actualreactor (100)
1 m
10 m
blowedup
rate energy generated mass in reactor reactor volume = L(r2)
rate energy removed reactor surface area = L(2r)
So how do we scale-up the reactor?
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Dynamic Scaling Example 3 - Chemical Reactor Scale-Up
CH2=CH2 + O2
heat
CH2
OH2C
So how do we scale-up the reactor?
Many small reactors!
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Chemical Process Modeling and Analysis
Mathematical Modelingprocess flowsheet equations
Graphical Modelingprocess flowsheet paths on phase mapsprocess unit operating lines
Dimensional Analysisbench-scale unit commercial-scale unit scale upbench-scale unit micro-scale unit scale down
moderate time interval (minutes) long time interval (years)
moderate time interval (minutes) short time interval (msec)
In general: convenient size, duration, or cost inconvenient or inaccessible
How? What are the rules?
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Dimensional Analysis Example 1: A Pendulum
How does period change with pendulum length? Mass? Angle?
100
Dimensional Analysis and Dynamic Scaling!
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Dimensional Analysis Example 1: A Pendulum
will always be given
(none)LL][
radiuslength arc
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We seek an equation of the form ),,,( gmt p
has dimensions of time must also have dimensions of time
The function cannot contain m. Why? Because no other parameter has dimensions of mass.
Can the function contain ? Yes, because we can cancel ’s dimensions with g.
If is in the function, it is present as the ratio /g.
Furthermore, the ratio must be present as its square root.
2
22 T
TLL
TL][L,][
gg
TT1/22
1/2
g
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Dimensional analysis yields
The pendulum period of oscillation scales as the square root of the length.
)(1/2
gt p
For our model pendulum ( = 10 cm) we measure tp 0.64 sec.
For large pendulum ( 10 m) we predict tp 100½ 0.64 sec 6.4 sec.
Without dynamic scaling one would naively predict tp 100 0.64 sec 64 sec.
This prediction is valid only for the same angle in model and large pendulum.
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Dimensional analysis yields )(1/2
gt p
(/g)1/2
tp = /8
= /6
= /4
The pendulum period depends on and only.
To find the function we must conductexperiments, but our experimental agendais shortened by dimensional analysis;vary only and .
1/2
2
gt p
Experiments show the period is independent of .
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Prelim 2 2021 Statistics
Mean: 74 / 120 (62%)Std. Deviation: 16
A - L: Lucy (Front of room)M - Z: Kelsey (Back of room)
Solution is posted.
Problem 1: 22 7 / 35 (63%)Problem 2: 13 5 / 20 (64%)Problem 3: 15 4 / 20 (75%)Problem 4: 23 10 / 45 (52%)