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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).
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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)
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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
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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
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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
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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
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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
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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
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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:
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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
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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
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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
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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)
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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
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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)
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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
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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
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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