Dimensional Analysis

Keeping track of units.

"Dimensions" are different from "units".

dimensions M – mass L – length T – time

units M – grams, kg, ton, slug, amu, ... L – meters, km, feet, miles, ... T – second, minute, hour, day, ...

$100 Million error

Mars Lander

Can’t mix apples and oranges.

Only like-dimensioned quantities may be added, subtracted, compared, or equated.

The dimensions are:

An expression may be dimensionally correct but have mixed units:

Convert one unit to another using a conversion factor

221 attvdd oo

22

TT

LT

T

LLL

inches0.02mm15.8cm0.8d

inch

cm54.21cm54.2inch1

Are these equations correct?

2

2

r

vmF

atd2

1

A

vLP 8

Use dimensions to determine the units of viscosity .

Dimensionless units.

What are the dimensions of G, the gravitational constant?

Relative error is dimensionless.

It makes it easy to determine which term is important.

1.0

02.02

5.3

03.0

15

02.0

F

F

221

r

mmGF

r

r

m

m

m

m

F

F

2

2

2

1

1

Dimensional analysis helps us understand physical dependences.

Look at the period (time) for the oscillation of a mass on a spring.

What does the period depend on? Mass Spring constant (M/T2) Gravity (L/T2)

T m k g Dimensions:

The period can’t depend on g because there is no L on left side.

Correct dependence is:

22 T

L

T

MMT

k

mT

Scaling

Examine how different physical magnitudes depend on the size of a system (defined by a length parameter L)

Material properties (e.g., strengths, moduli, densities, coefficients of friction) are held constant.

If you scale the water jug up to a town tank, what problems arise? If you scale up a chemistry lab reaction to industrial plant size, what

problems arise?

Dimensionless systems simplify analysis of some problems.

Look at Aerodynamics example Independent variables

• lift force F,

• wing area A,

• velocity v,

• air density • air viscosity

Can do experiments varying one variable while holding all others constant.

Takes a long time (& money).

Buckingham Pi Theorem

This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.

If problem has n independent variables And m "primitive dimensions" (L, M, T) Then you can simplify to n-m dimensionless variables. Not all solutions are physically meaningful

Look at Aerodynamics example Independent variables

• F, A, v, , Dimensionless variables = n-m = 5-3 = 2

Aerodynamics

The most general power-law expression would be

Dimensionally this look like

To be dimensionless, the powers must all be 0.

edcba vAFf

edcb

a

LT

M

L

M

T

LL

T

MLf

32

2

ecaedcbaeda TLMf 232

02

032

0

eca

edcba

eda

Aeronautics (continued)

Now add some insight to get physically meaningful solution. We want to get a variable with the lift force F, so a = 1. We only have three equations so must eliminate one variable.

Set e = 0. The exponents become

The solution is

2002

100321

1001

cc

bdcb

dd

t)coefficien(lift 201211

Av

FvAFf

Aeronautics (continued)

A second physically meaningful solution has a=0, c=1 The exponents become

The solution is

These two dimensionless variables are to test aerodynamic shapes in wind tunnels.

1010

01320

1010

21

cc

bdcb

dd

number) (Reynolds21

21 1110

AvvAFf

DiffusionDiffusion

What do these processes have in common?

1) Hydrogen embrittlement of pressure vessels in nuclearpower plants

2) Flow of electrons through conductors

3) Dispersion of pollutants from smoke stacks

4) Transdermal drug delivery

5) Influenza epidemics

6) Chemical reactions

7) Absorption of oxygen into the bloodstream

They all depend on

Diffusion (conduction)

What is diffusion? The transport of material--atomsor molecules--by random motion

What is conduction? The transport of heat or electronsby random motion.

Brownian motion causes the ink particles to move erraticallyin all directions. A concentration of ink particles willdisperse.

Place a drop of ink into a glass of water. What Place a drop of ink into a glass of water. What happens?happens?

Because there are more ways for the particles to drift apartthan there are for the particles to drift closer together.

Why does random motion cause spreading of a Why does random motion cause spreading of a concentrationconcentrationof particles?of particles?

We can also explain the spreading of a We can also explain the spreading of a concentration by entropy.concentration by entropy.

The second law of thermodynamics says that systems tend towards maximum entropy – or maximum disorder.

Area of high concentration and low/zero concentration is an ordered state and the mixed state is the disordered state!

Other examples?Other examples?

Why do metal cooking spoons have plastic handles?

Other examples?Other examples?

What happens if someone across the room sprays perfume?

Perfume diffusion simulation

Diffusion Progression Rates:Gas: 10 cm/min

Liquids: 0.05 cm/min

Solids: 0.00001 cm/min

After adding milk and sugar, why do we stir our After adding milk and sugar, why do we stir our coffee?coffee?

Diffusion is slow!

Agitation (or stirring) can move fluids much larger distances in the same amount of time, which can accelerate the diffusion process.

In each of these examples, molecules (or heat) are In each of these examples, molecules (or heat) are moving down a gradient!moving down a gradient!

(From an area of high concentration to an area of low concentration)

dz

dcDJ i

i Fick’s Law:

Ji is called the flux. It has units of ))(( 2 tl

diffused material of amount

D is called the diffusion coefficient. It has units oft

l 2

Temperature Diffusivity

(°C) (cm2/s)

CO2-N20 0 0.096

Ar-O2 20 0.2Ethanol(5%)-Water 25 1.13E-05

Water(13%)-Butanol 30 1.24E-05H2-Ni 85 1.16E-08Al-Cu 20 1.30E-30

(gas)

(liquid)

(solid)

Diffusivity ValuesDiffusivity Values

Greater the diffusivity, greater the flux!

Do our definitions of flux make sense?

N2

CO2

(constant T & P)

C(*)

capillary area time

removed gas of amount flux) dioxide carbon(

• If double area of capillary, expect the amount of gas transported to double.

• Want flux independent of apparatus – normalize by area.

lengthcapillary

difference ionconcentrat dioxide carbon flux) dioxide carbon(

• Flux is proportional to the concentration gradient – steeper the gradient, more material transported.

• Flux is inversely proportional to capillary length – increasing the distance to travel will decrease the flux.

2lengthtime

massJ

dx

dcDJ i

i

Steady diffusion across a thin filmSteady diffusion across a thin film

Now let’s use our diffusion equation to predict the concentration profile of a material diffusing across a thin film!

Well-mixed dilute solution with concentration ci,l

Well-mixed dilute solution with concentration ci,0

Thin film

If we are at steady-state (the concentration profile has no time dependence, or in other words, there is no accumulation of i in the film), we have a linear concentration profile.

ci,0

ci,l

l

Concentration-dependent diffusionConcentration-dependent diffusion

z=0 z=zc z=l

ci,0

ci,c

ci,l

D1

D2

Which diffusivity is greater? How do you know?

Consider two neighboring thin films with a separation at ci,c:

Unsteady state diffusionUnsteady state diffusion

Back to a drop of ink in a glass of water…

If consider diffusion in the z-direction only:

z=0

How does the concentration profile change with time?

t = 0(add ink drop – all ink

located at z = 0)

t

z

Heat TransferHeat Transfer

Occurs by three means:

1. Conduction:

• Occurs between two static objects

• Heat flows from the hotter to the cooler object

• For example, holding a cup of hot coffee

2. Convection:

• Transport of heat via a fluid medium

• Currents caused by hot air rising, fan circulating air

3. Radiation:

• Transport of energy as electromagnetic waves; the receiving body absorbs the waves and is warmed

• For example, warmth of a fire

Heat moves down a temperature gradient!Heat moves down a temperature gradient!

(From an area of high temperature to an area of low temperature)

dz

dTkqz Fourier’s Law:

qz is called the heat flux. It has units of ))(( 2 tl

energy

k is called the thermal conductivity. It has units of ))()(( Ttl

energy

α is called the thermal diffusivity. It is defined as)ˆ)(( pC

k

and has units oft

l 2

T k(°C) (cal/cm s C)

H2 27 4.23E-04

O2 27 6.35E-05Benzene 23 3.78E-04

Water 60 1.56E-03Steel 100 9.08E-01Wood -- 9.00E-05

(gas)

(liquid)

(solid)

Thermal Conductivity ValuesThermal Conductivity Values

Greater the thermal conductivity, greater the heat flux!

Consider a two-paneled door:

metalwood

What will the steady-state temperature profile look like? Why?

Tc

TH

Heat ConductionHeat Conduction

z

kmetal > kwood

Here’s a heat-conducting bar with a fixed temperature T at each end:T(t,0)=0; T(t,100)=100. 2k1 = k2 .

z=0 z=100T(t,0)=0 T(t,100)=100

κ1 κ2

At steady-state:

21

21 .kinkin dz

dCkconst

dz

dCk

Therefore, the ratios of the temperature gradients in each sectionmust equal the inverse ratios of the k’s.

(Constant flux)

2. Heat transfer—Fourier’s Law

heat flux in z-direction q

A

d c T

dzz p

( ); is thermal diffusivity,

is density, cp is heat capacity, T is thermal energy (heat).

3. Mass transfer—Fick’s Law

mass flux of A in z-direction J Ddc

dzAz ABA ; D is molecular

diffusivity of A in B, CA is the concentration of A.

1. Momentum transfer—Newton’s Law

flux of x-momentum in z direction

zx

xdv

dz

( ), vx is velocity

in x-direction, is density, is viscosity.

Gradient transport Gradient transport summarysummary

Chemical Engineering Labs Data Analysis

Chromatography Lab

Write-Up

First, describe what happened, taking care to differentiate between different colors and different solvents.

For the methanol and 2-propanol experiments, calculate flow rates from the purple ink.

Dye Flow Rates

5.0 cm

2.9 cm

Flow rate of pink = 2.9/5.0 = 0.59

Write-Up Continued

Present a graph showing the flow rate of each component as a function of the proportion of 2-propanol in the solvent (approximately 100% for 2-propanol and 0% for methanol).

Assuming that flow rate changes linearly with 2-propanol concentration, draw a line which represents the expected flow rate for each component in various concentrations. Deduce the expected equations of each of these lines.

Dye Flow Rate vs. Percentage Methanol

y = 0.002x + 0.8

y = -0.0015x + 1

0.7

0.75

0.8

0.85

0.9

0.95

1

0 50 100

Percent Methanol

Re

lati

ve

Flo

w R

ate

Pink Dye FlowRate

Purple Dye FlowRate

Linear (Purple DyeFlow Rate)

Linear (Pink DyeFlow Rate)

Write-Up Continued

Notice that for the purple pen, one ink component flows faster in 2-propanol while the other flows faster in methanol. If we were attempting to separate these two components, and we were really unlucky by choosing the wrong concentration of methanol, we could obtain no separation. Estimate the methanol concentration in 2-propanol of this most undesirable solvent.

Chemical Engineering Labs Data Analysis

Heat Transfer Lab

Heat Transfer Lab

Calculate the total electrical energy input for both experiments (with and without the lid):

Calculate the water's energy increase for both experiments:

seconds wattsaverageE(Joules)

ifoTT

Cg

4.186J masswater E(Joules)

Heat Transfer Lab

Calculate the efficiencies with and without the lid:

What difference did putting a lid on the pot make?

If the efficiency is less than 1.0, where did the remaining energy go? Try to think of all possible “losses”.

How could you improve the efficiencies?

inputenergy electrical total

increaseenergy swater'e

Chemical Engineering Labs Data Analysis

Distillation

Distillation

Use the temperature data and the specific gravity data to determine the weight percent ethanol in each sample.

Distillate #1

T = 24.0 º C

s.g. = 0.920

Information from the Data Table At 20ºC

45% ethanol has specific gravity = 0.92472

46% ethanol has specific gravity = 0.92257

At 25ºC

45% ethanol has specific gravity = 0.92085

46% ethanol has specific gravity = 0.91868

Specfic Gravity vs. Temperature

0.9180

0.9190

0.9200

0.9210

0.9220

0.9230

0.9240

0.9250

0.9260

20 21 22 23 24 25

Temperature (C)

Sp

ecif

ic G

ravi

ty 45% ethanol

46% ethanol

Linear (45% ethanol)

Linear (46% ethanol).

Determine the % Ethanol in Distillate #1 At 24ºC the distance between the 45% and 46% alcohol

lines is 1.2 cm. At 24ºC the distance between the 45% and the data

point is 0.9 cm.

So

And the % ethanol = 45.75% %75.0

%12.1

9.0

x

x

Grams of Ethanol in each solutionMass of Distillate #1

= volume of the solution * specific gravity

Grams of ethanol in Distillate #1

= grams of solution * % alcohol ÷ 100

Moles of ethanol

= grams of ethanol ÷ 46.07 g/mol

Grams of Water in each Solution

Calculate the total moles of water in each sample

moles of H2O= grams of H2O ÷ 18.02grams/mole

Perform a mole balance analysis for ethanol and water to check whether all material is accounted for.