lecture 3 transport phenomena

7/27/2019 Lecture 3 Transport Phenomena

1/18

Buoyant force

Vg

BF

The volume of gas (liquid) is

at rest with respect to itssurrounding gas (liquid): the

force of gravity is balanced

by the buoyant forceAPPF topbottomB )(

bottom

P

topP

P

h

bottomPtopP

Note that the buoyant force does not care whats

ins ide this volume (a brick, a gas, or vacuum): it

depends only on the volume and the density of the

outs idegas (liquid).


2/18

Lecture 3. Transport Phenomena (Ch.1)

Lecture 2 various processes in macro systems near the state of

equilibriumcan be described by a handful of macro parameters. Quasi-

static processessufficiently slow processes, at any moment the systemis a lmostin equilibrium.

It is important to know how much time it takes for a system to approach an

equilibrium state. A system is not in equilibrium when the macroscopic

parameters (T, P, etc.) are not constant throughout the system. To

approach equilibrium, these non-uniformities have to be ironed out through

the transport of energy, momentum, and mass from one part of thesystem to another. The mechanism of transport is molecular collisions. Our

goal - to estimate the characteristic rates of approaching equilibrium, and,

thus, to impose limitations on the rates of quasi-staticprocesses.

1. Transfers of Q (Heat Conduction)

2. Transfers of Mass (Diffusion)

One-dimensional (1D) case:

x

n(x,t)

T(x,t)


3/18

The Mean Free Path of Molecules

Transports energy, momentum, massdue to random

thermal motion of molecules in gases and liquids.

An est imate: one molecule is moving with a constant

speed v, the other molecules are fixed. Model of hard

spheres, the radius of molecule r~ 110-10

m.

The av. distance traversed by a molecule until the 1st collision is the

distance in which the av. # of molecules in this cylinder is 1.

12 2 V

Nlr

nN

V

rl

1

4

12

n

l

2

1Maxwell :

The mean free pathl- the average distance traveled

by a molecule btw two successive collisions.

The average time interval between successive collisions - the collision time:

v

l - the most probable speed of a moleculev

n= N/V

the density of molecules = 4r2the cross section


4/18

Some Numbers:

326

5

23

m104Pa10

K300J/K1038.11

P

Tk

N

V

n

Bair at norm. conditions:

P = 105 Pa: l ~ 10-7 m - 30 times greater than d

P = 10-2 Pa (10-4mbar): l ~ 1m (size of a typical vacuum chamber)

- at this P, there are still ~2.5 1012

molecule/cm3

(!)

m103~

93

N

V

d

3/23/2 Pnd

l

The collision time at norm. conditions: ~ 10-7m / 500m/s = 210-10 s

For H2gas in interstellar space, where the density is ~ 1 molecule/ cm3,

l ~ 1013 m - ~ 100 times greater than the Sun-Earth distance (1.5 1011 m)

for an ideal gas: TnkPTNkPV BB

P

T

n

l 1

nl

1

the intermol. distance


5/18

Transport in Gases (Liquids)

Box 1 Box 2

l

Simpl i f ied approach: consider the ballistic

molecule exchange between two boxeswithin the

gas (thickness of each box should be comparable to

the mean free path of molecules, l). During theaverage time between molecular collisions, ,roughly half the molecules in Box 1 will move to the

right in Box 2, while roughly half the molecules in

Box 2 will move to the left in Box 1.

Each molecule carries some quantity (mass, kin. energy, etc.), withineach box - = N= A ln. E.g., the flux of the number of moleculesacross the border per unit area of the border, Jx:

x

nD

x

nlv

x

nlvlxnlxnv

tA

NJn

3

12

6

1

6

1

x=-l

in a 3D case, on average 1/6 of the

molecules have a velocity along +xorx

-- if n /xis negative, the flux is in the positive xdirection(the current flows from high density to low density)

x=0 x=l

x

n(x,t) Jx

In a 3D case, TKJnDJ thUn

the diffusion constant


6/18

Diffusion

Diffusion the flow of randomly moving particles

caused by variations of the concentration ofparticles. Example: a mixture of two gases, the total

concentration n = n1+n2=const over the volume (P

= const).

J

J

Ficks Law:

n1 n2

x

nD

x

nvlJx

3

1 vlD3

1

- the dif fus ion coeff ic ient

(numerical pre-factor depends on the dimensionality: 3D1/3; 2D1/2)

vlD

3

1 its dimensions [L]2 [t]-1, its units m2 s-1

Typically, at normal conditions, l ~10-7 m, v ~300 m/s D~ 10-5 m2 s-1

(in liquids, Dis much smaller, ~10-10 m2 s-1)

For electrons in well-ordered semiconductor heterostructures at low T:

l ~10-5 m, v ~105m/s D~ 1m2 s-1


7/18

Diffusion Coefficient of an Ideal Gas ( Pr. 1.70)

for an ideal gas:PTk

nl B1

from the equipartition theorem: 2/1Tv P

T

P

TTD

2/32/1

therefore, at a const. temperature:P

D 1

and at a const. pressure:2/3TD


8/18

The Diffusion Equation

n(x ,t) tAxJx

flow in

tAxxJx

flow out

x x+x

xJ

tn x

change of ninside:

x

nDJx

combining with

well get the equation that describes one-dimensional diffusion:

2

2

x

nD

t

n

the solution which corresponds to an initial condition

that all particles are atx =0 at t =0:

Dt

x

Dt

Ctxn

4exp,

2

C is a normalization factor

the rmsdisplacement of particles: Dtx 2

the di f fus ion equat ion

t1=0

x =0

t2=t


9/18

Brownian Motion (self-diffusion)

The experiment by the botanist R. Brown concerning thedrifting of tiny (~ 1m) specks in liquids and gases, had been

known since 1827.

Brownian motion was in focus of the struggle for and against

the atomic structure of matter, which went on during the

second half of the 19thcentury and involved many prominent

physicists.

Ernst Mach: Ifthe belief in the existence of atoms is so crucial in your eyes, I

hereby withdraw from the physicistsway of thought...

Albert Einstein explained the phenomenon on the basis of the kinetic theory

(1905), connected in a quantitative manner the Brownian motion and suchmacroscopic quantities as the coefficients of mobility and viscosity and

brought the debate to a conclusion in a short time.

Observing the Brownian motion under a microscope, Jean Perrin measured

the Boltzman constant and Avogadro number in 1908 (Nobel 1926).

Histor ical backgroun d:


10/18

Brownian

Motion

(cont.) tDx

t

2

a 1D random walkof a drunk

t

xGaussian

distribution

For air at normal conditions , it takes)/sm107.1m/s500m10( 257 Dvl

for a molecule to diffuse over 1m: odor spreads by convections10~ 52

D

Lt

the rms displacement

For electrons in metals at 300K , it takes)/sm103m/s10m10( 2267 Dvl

to diffuse over 1m. For the electron gas in metals, convection can be ignored:

the electron velocities are randomized by impurity/phonon scattering.

s30~2

D

Lt

A body that participates in a random walk, or a subject of random collisions with the

gas molecules. Its average displacement is zero. However, the average square

distance grows l inear ly with t ime:

after Nsteps, the position is nlRR NN

111NR n

- a randomly

oriented unit

vector

01 NR

NNNN RnllRnlRR

22222

1

after averaging ( ): 2221 lRR NN

tlNRN 22

Dt

x

Dt

Ctxn

4exp,

2


11/18

Static Energy Flow by Heat Conduction

T1 T2

x

area A Heat condu ct ion ( static heat flow, T = const)

x

TKJ

x

T

C

K

x

J

Ct

TthU

thU

2

21In general, the energy transport dueto molecular motion is described by

the equation of heat conduction:

Thus, in principle, if you know the initial conditions, e.g. T(x,t=t0), you can

describe the process by solving the equation. Often, you are asked to consider a

different situation: a staticflow of energy from a hotobject to a coldone. (At

what rate the internal energy is transferred between two systems with T1

T2

or

between parts of a non-equilibrium system (if one can introduce Ti) ?) The

temperature distribution in this formulation is time-independent, and we need to

calculate the flux of thermal energy JU due to the heat conduct ion

(diffusion/intermixing of particles with different energies, interactions between the

particles that vibrate but do not move translationally).

T(x)

T1

T2

x

JU


12/18

Fourier Heat Conduction Law

T1 T2

x

AKGTGJ thU

,power

Gthe thermal conductivity [W/K]

R =1/G the thermal resistivity

Electricity Thermal Physics

Charge Q Th. Energy, Q

Currant dQ/dt Power Q/dt

What flows

Flux

Driving forceEl.-stat. pot.

difference

Temperature

difference

Resistance El. resistance R Th. resistance R

Connect ion in ser ies (Pr. 1.57):

Rto t= R1+ R2

Conn ect ion in paral le l :

Rto t-1= R1

-1+ R2-1

T1 T2

T1 T2

G

A

x

tTQ

Kth [W/Km]the thermal conductivity (material-specific)

x

TAK

t

QJ thU

For a window glass (Kth=0.8W/mK, 3 mm thick, A=1m2) and T = 20K:

Wm

KmKW/m 5300

003.0

20)1)(8.0( 2

t

Q

- - if Tincreases from left to

right, energy flows from right to left

~ 10 times greater than in reality, a thin

layer of still air

must contribute to thermal insulation.

Pr. 1.56


13/18

Relaxation Time due to Thermal Conductivity

(a rough estimate)

U = CT1G

environment

T2

GCTTT

TGTC

dtQU

121/

~

Problem 1.60: A frying pan is quickly heated on the stovetop to 2000C. It has

an iron handle that is 20 cm long. Estimate how much time should pass before

the end of the handle is too hot to grab (the density of iron = 7.9 g/cm3, its

specific heat c= 0.45 J/gK, the thermal conductivity Kth=80 W/mK).

x

AKG

th

s

KmsJ

mKkgJmkg

K

Lc

L

AK

LAc

G

C

thth

400~80

1.04507900111

21132

the thermal

conductivity

the thermal conductivity

the heat capacity (specific heat)


14/18

Thermal Conductivity of an Ideal Gas

Box 1 Box 2

T

dx

dTvC

dx

dTlCTTCUUQV

VV

2

1

2

1

2

1

2

1 2121

l

x

TAK

Qth

vl

V

C

lA

vlC

A

vCK VVVth

2

1

2

1

2

1

B

BV

V knf

V

Nkf

V

Cc

2

2

vlknf

K Bth

4

Energy flow, t ~ :

Km

W0.02500m/smJ/Km

723325 101038.1104.2

4

5

4vlkn

fK Bth

the specific

heat capacity

(exp. value0.026 W/mK)

The thermal conductivity of air at norm. conditions:

T1 T2

the time between two

consecutive collisions v

l


15/18

Thermal Conductivity of Gases (cont.)

2.Thermal conductivity of an ideal gas

is independent of the gas density!

This conclusion holds only if L >> l .

ForL < l , Kthn

Dewar

mT

Km

T

vnlvlkn

f

K thBht

,4

1

1. mKth /1

- an argon-filled window helps to reduce Q

(at higher densities, more

molecules participate in the

energy transfer, but they

carry the energy over ashorter distance)


16/18

Sate-of-the-art Bolometers

(direct detectors of e.-m. radiation)

Ti

Nb

0.1 1

104

105

106

107

108

109

HEDDs

meanders

G,

W/Km3

T, K

Ge-ph= Ce/e-phG= Cph/es

electrons

phonons

heat sink

Te

T

Tph

phe

phe GG

~

VTc Geph

1

10

100

1000

10 100 1000

BT85-4

BT87-1

BT100-1

BT121-1

BT121-4

Timeconstan

t(s)

Temperature (mK)

phephephe TTTGTTGRIdt

Q

2power

specific heat of electrons


17/18

Momentum Transfer, Viscosity

uxz

Dragtransfer of the momentum in the

direction perpendicular to velocity.

Laminar flow of a gas (fluid) between two

surfaces moving with respect to each other.z

bottom,top,

xxx

x uuA

Ft

p

zud

AF xx

Fxthe viscous drag force, - the coefficient of viscosity

Fx/A shear stress

Visco sity o f an idealgas( Pr. 1.66 ):

Box 2

Box 1z~

ux(z2)ux(z1)

xxxz uNmzNmuzNmup 21)(

21)(

21

21

zd

dulv

A

F

lA

vuNmp

AA

F xxxzx

2

1

2

1lv

2

1 T1/2


18/18

Effusion of an Ideal Gas

- the process of a gas escaping through a small hole (a>l) the hydrodynamic regime.

The number of molecules that escape through

a hole of area A in 1 sec, Nh, in terms of P(t ), T

(how is T changing in the process?)

At

vmN

At

pNP

x

hh

121

x

h

vm

tAPN

2

m

TkvvTkvmv BxxBxx

22,

2

1

2

1

Nh = - N, whereNis the total # of molecules in a system

m

Tk

V

tAN

Tk

m

m

tA

V

TNk

Tk

m

m

tAPN B

B

B

B 222

NN

m

Tk

V

A

t

N B

1

2

Tk

m

A

VtNtN

B

2,exp)0()(

Depressurizing of a space ship,V- 50m3, A of a hole in a wall10-4m2

(clearly, a

lecture 3 transport phenomena

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