Fi ck law

NEWTON: F=m a
LAPLACE:
Nous devons donc envisager l'état présent de l'universe comme
l'effet de son état antérieur et comme la cause de delui qui va suivre.
Une intelligence qui, pour un instant donné, connaîtrait toutes les
forces dont la nature est animée et la situation respective des êtres
qui las composent, si d'ailleurs elle était assez vaste pour soumettre
ces données à l'Analyse, embrasserait dans la même formule les
mouvements des plus grands corps de l'univers et ceux du plus lèger
atome : rien ne serait incertain pour elle, et l'avenir, comme le passé,
serait présent à ses yeux.
“Translation” In principle “Yes”.
Provided that we know the position, velocity and
interaction of all molecules, then the future behavior is
predictable,…BUT

Molecular Simulation
Background
Macroscopic diffusion equations
Fick’s laws

Why Simulation?
1. Predicting properties of (new) materials
2. Understanding phenomena on a
molecular scale.

THE question:
“Can we predict the macroscopic
properties of (classical) many-body
systems?”

…. There are so many molecules.
This is why, before the advent of the computer, it was
impossible to predict the properties of real materials.
What was the alternative?
1. Smart tricks (“theory”)
Only works in special cases
2. Constructing model (“molecular lego”)…

J.D. Bernal’s “ball-bearing
model” of an atomic liquid…

J.D. Bernal constructs a
model of a liquid… (around
1950)..
I took a number of rubber balls
and stuck them together with rods
of a selection of different
lengths ranging from 2.75 to 4
in. I tried to do this in the
first place as casually as
possible, working in my own
office,
being interrupted every five
minutes or so and not remembering
what I had done before the
interruption. However, ...

The computer age (1953…)
Mary-Ann Mansigh
Berni Alder
Tom Wainwright
With computers we can follow the behavior of hundreds to
hundreds of millions of molecules.

A brief summary of:
Entropy, temperature, Boltzmann distributions
and the Second Law of Thermodynamics

The basics:
1. Nature is quantum-mechanical
2. Consequence:
Systems have discrete quantum states.
For finite “closed” systems, the number of
states is finite (but usually very large)
3. Hypothesis: In a closed system, every
state is equally likely to be observed.
4. C o n s e q u e n c e : ALL of equilibrium
Statistical Mechanics and Thermodynamics

First: Simpler example (standard statistics)
Draw N balls from an infinite vessel that contains an
equal number of red and blue balls
¥ ¥

Now consider two systems with total energy E.
This function is very sharply peaked (for macroscopic systems)

Now, allow energy exchange
between 1 and 2.

This is the condition for thermal equilibrium (“no
spontaneous heat flow between 1 and 2”)
Normally, thermal equilibrium means: equal
temperatures…

Let us define:
Then, thermal equilibrium is
equivalent to:
This suggests that b is a function of T.
Relation to classical thermodynamics:

Conjecture: ln W = S
Almost right.
Good features:
•Extensivity
•Third law of thermodynamics comes for
free
Bad feature:
•It assumes that entropy is dimensionless
but (for unfortunate, historical reasons, it
is not…)

We have to live with the past, therefore
With kB= 1.380662 10-23 J/K
In thermodynamics, the absolute (Kelvin)
temperature scale was defined such that
But we found (defined):

And this gives the “statistical” definition of temperature:
In short:
Entropy and temperature are both related to
the fact that we can COUNT states.

How large is W?
For macroscopic systems, super-astronomically large.
For instance, for a glass of water at room temperature:
Macroscopic deviations from the second law of
thermodynamics are not forbidden, but they are
extremely unlikely.

Consider a “small” system
(a molecule, a virus, a
mountain) in thermal
contact with a much larger
system (“bath”).
The total energy is fixed. The higher the energy of
the small system, the lower the energy of the bath.
What happens to the total number of
accessible states?

But, as b=1/kBT :
The probability that the small system is in a given (“labeled”)
state with energy ei is

This is the Boltzmann distribution:
“Low energies are more likely than high
energies”

The probability to find the system in state I is:
Hence, the average energy is

Therefore
This can be compared to the thermodynamic
relation

This suggests that the partition sum
is related to the Helmholtz free energy through

Remarks
The derivation is not difficult
but it takes a few minutes…
We have assumed quantum mechanics (discrete states) but
often we are interested in the classical limit
ìï é ùïü - ® í- ê + úý
2
( E ) p U ( r
) exp 1 d d exp
å òò p r å
i i 3
i
N N i N
h N ! 2
m
i
b b
îï ë ûïþ
1
h
® Volume of phase space
3
1
N!
® Particles are indistinguishable
Integration over the momenta can be carried out for most systems:
3 3 2 2 2 2 d exp dp exp
ìï é p ùïü é ì í- b = í- b p üù æ p
m
ö ê úý ê ýú = ç ¸
îï ë 2 m ûïþ ë î 2
m
þû è ø
N N
N i
i
i
b
ò p å ò

Remarks
Define de Broglie wave length:
1
æ ö
h
2 b
2
2
p
m
L º ç ¸
è ø
Partition function:
, , 1 d exp
( ) ( ) 3
= éë-b ùû L ò r
!
N N
N Q N V T U r
N

Check: ideal gas
Thermo recall (3)
Helmholtz Free energy:
, , 1 d exp
( ) ( ) 3
= éë-b ùû L ò r
!
N N
N Q N V T U r
N
1 d 1
dF = -SdT - pdV
= L ò r
N
=
L V
N N
3 3
N
N N
! !
Pressure
Free energy:
Pressure:
F P
V
æ ¶ ö = - çè ¶ ø¸
æ
L
N
3 - L » ÷ ÷ø
ln 3
æ ¶ ö æ ¶ ö ç ¸= ç ¸= è ¶ ø è ¶ ø
F T F E
T
P F N
= -æ ¶ ö = çè ¶ ø¸
V bV
T
Energy:
(ln 1)
= æ ¶ ö = ¶L = ç ¶ ¸ L ¶ è ø
E b F 3 N 3
Nk T
2 B
b b
!
ö
ç çè
b = - N r
N
F T
VN
1
b
b
Energy:

Relating macroscopic observables to
microscopic quantities
Example:
Heat capacity
Pressure
Diffusion coefficient

Fluctuation expression for heat capacity.
Recall:
with

Then the heat capacity is
Using our expression for E:

Both the numerator and denominator depend on b.
And, finally:

Introduce “scaled” coordinates:

For pairwise additive forces:
Then

i and j are dummy variable hence:
And we can write

But as action equals reaction (Newton’s 3rd law):
And hence
Inserting this in our expression for the pressure, we get:
Where

What to do if you cannot use the virial expression?

Other ensembles?
COURSE:
MD and MC
In the thermodynamic limit the thermodynamic properties are
independent of the ensemble: so buy a bigger computer …
different ensembles
However, it is most of the times much better to think and to carefully
select an appropriate ensemble.
For this it is important to know how to simulate in the various
ensembles.
But for doing this wee need to know the Statistical Thermodynamics
of the various ensembles.

Example (1):
vapour-liquid equilibrium mixture
Measure the composition of the
coexisting vapour and liquid
phases if we start with a
homogeneous liquid of two
different compositions:
– How to mimic this with the N,V,T
ensemble?
– What is a better ensemble?
composition
T
L
V
L+V

Example (2):
swelling of clays
Deep in the earth clay layers can
swell upon adsorption of water:
– How to mimic this in the N,V,T
ensemble?
– What is a better ensemble to use?

Ensembles
• Micro-canonical ensemble: E,V,N
• Canonical ensemble: T,V,N
• Constant pressure ensemble: T,P,N
• Grand-canonical ensemble: T,V,μ

Constant pressure simulations:
N,P,T ensemble
-
- p/kBT Thermo recall (4)
First law of thermodynamics
Consider a small system that can exchange
volume and energy with a large reservoir
E E
V V
W - - = W - æ ¶ W ö - æ ¶ W ö + çè ¶ ø¸ èç ¶ ø¸
ln ln , ln ln i i i i
( ) ( ) ,
V V E E V E E V
E V
V E
L
( )
( )
æ ¶ ö
çè ¶ ø¸
E E ,
V V E pV
ln
i i i i
E ,
V k T k T
B B
W - -
= - -
W
1/kBT
S p
V T
æ ¶ ö = çè ¶ ø¸
Hence, the probability to find Ei,Vi:
( ) ( )
W E - E , V - V exp
E + pV
= =
éë- ùû ( )
( )
( )
å å
W - - éë- + ùû
, ,
( )
,
, exp
m exp
éë- + ùû
b
b
i i i i
i i
j k j k j k j k
i i
P E V
E E V V E pV
E pV
b
, i i V E , i
i
d d d + d i i i E = T S - p V å m N
Hence
T ,N
,
1 =
V N
S
T E
and

Grand-canonical simulations:
μ,V,T ensemble
-
- -μ/kBT Thermo recall (5)
First law of thermodynamics
Consider a small system that can exchange
particles and energy with a large reservoir
W - - = W - æ ¶ W ö - æ ¶ W ö + çè ¶ ø¸ çè ¶ ø¸
ln ln , ln ln i i i i
( ) ( ) ,
N N E E N E E N
E N
N E
L
( )
æ ¶ ö
çè ¶ ø¸
E E N N E N
W - ,
- m
i i i i i
( )
ln
= - +
E ,
N k T k T
B B
W
1/kBT
æ ¶ S
ö ç = - m è ¶ N ¸ ø
T
Hence, the probability to find Ei,Ni:
( ) ( )
W E - E , N - N exp
E - N
= =
éë- ùû ( )
( )
( )
å å
W - - éë- - ùû
, ,
( )
,
, exp
m exp
éë- - ùû
b m
b m
i i i i i
i i
j k j k j k j k k
i i i
P E N
E E N N E N
E N
b m
, i i N E , i
i
E E
N N
d d d + d i i i E = T S - p V å m N
Hence
,
i
i T V
,
1 =
V N
S
T E
and

Computing transport coefficients from an
EQUILIBRIUM simulation.
How?
Use linear response theory (i.e. study decay of fluctuations in
an equilibrium system)
Linear response theory in 3 slides:

Consider the response of an observable A due to an
external field fB that couples to an observable B:
For simplicity, assume that
For small fB we can linearize:

Hence
Now consider a weak field that is switched off at t=0.
fB
DA
0
t

Using exactly the same reasoning as in the static case, we
find:

Average total displacement:
Mean squared displacement:

Macroscopic diffusion equations
Fick’s laws:
(conservation law)
(constitutive law)

62
HHWW 1144
More on Moderators
Calculate the moderating power and ratio for pure
D2O as well as for D2O contaminated with a) 0.25%
and b) 1% H2O.
Comment on the results.
In CANDU systems there is a need for heavy water
upgradors.

slowing down in hydrogeneous
material
63
More on Moderators
slowing down in large mass
number material
u u
7x
6x
5x
4x
3x
2x
x
3x
continuous slowing-down model
2x
x
0 1 2 3 4 5 6 7 n 0 1 2 3 n

64
More on Moderators
A
= + - - úû
ln 1
1
u E
ln 1 ( 1)
2
2
ù
+
D = = é
êë
A
A
A
E
av
z
CCoonnttiinnuuoouuss sslloowwiinngg ddoowwnn mmooddeell oorr FFeerrmmii mmooddeell..
• The scattering of neutrons is isotropic in the CM
system, thus z is independent on neutron energy. z
also represents the average increase in lethargy per
collision, i.e. after n collisions the neutron lethargy
will be increased by nz units.
• Materials of low mass number  z is large  Fermi
model is inapplicable.

65
More on Moderators
MMooddeerraattoorr--ttoo--ffuueell rraattiioo º Nm/Nu.
• Ratio leakage ¯ Sa of the moderator f ¯.
• Ratio ¯ slowing down time p ¯ leakage .
• Water
moderated
reactors, for
example, should
be under
moderated.
• T ratio ¯ (why).

66
One-Speed Interactions
• Particular  general.
Recall:
• Neutrons don’t have a chance to interact with each
other (review test!)  Simultaneous beams, different
intensities, same energy:
Ft = St
(IA + IB + IC + …) = St
(nA + nB + nC + …)v
• In a reactor, if neutrons are moving in all directions
n = nA + nB + nC + …

Rt = St
nv = St
f

w 
One-Speed Interactions
   
R r F r dF r vn r d v n r r
67
n 
w ( r 
, ) d W r
dW
º Neutrons per cm3 at r
whose velocity vector
lies within dW about w.
n(r  ) = ò n(r  , w

)d W
4
p
• Same argument as before 
dI (r  ,w ) = n(r  ,w
)vdW ( , ) ( , )
= = = å W = å = å
( ) ( ) ( , ) ( , ) ( ) ( )
4
dF r dI r
t t t
t
       
w w f
w w
w p
= å
ò ò
   where
(r ) = òvn(r , )dW
f w
p
4

n r n r E w d dE   
Scalar
68
Multiple Energy Interactions
• Generalize to include energy
n 
w ( r 
, E , ) dEd W º Neutrons per cm3 at r with energy
interval (E, E+dE) whose velocity
vector lies within dW about w.
n(r  , E)dE = ò n(r  , E,  )d W
dE ò ò
p
w
4
¥
= W
( ) ( , , )
p
0 4
R r E dE E n r E v E dE E r E dE t t ( , ) ( ) ( , ) ( ) ( ) ( , )    = å = å f
¥
= å
  f
ò
R(r ) (E) (r , E)dE t
0
Thus knowing the material properties St and the neutron flux f as a
function of space and energy, we can calculate the interaction rate
throughout the reactor.

dI (r,w) = n(r ,w)vdW     dI r = n r vdW      
69
Neutron Current
¥
= å
  f
ò
• Similarly R r S () E) r E)dE S
((, and so on …
0
Scalar
• Redefine as
( ,w) ( ,w)
   
   J = òvn(r , )dW
(r ) = òvn(r , )dW
f w
p
4
4
w
p
NNeeuuttrroonn ccuurrrreenntt ddeennssiittyy
• From larger flux to smaller flux! 
• Neutrons are not pushed!
J
• More scattering in one direction
than in the other.

x J · xˆ = J

Net flow of neutrons per second per unit area normal
to the x direction:
( , ) ( , ) ( ) ( , ) ( , ) ˆ       f
a n r t d S r t d r r t d J r t ndA
t
70
  
· = = ò W
J xˆ J n(r , )v cos d x x
p
w q
4

In general: n J · nˆ = J
EEqquuaattiioonn ooff CCoonnttiinnuuiittyy
¶
ò " = ¶
ò "-òå "-ò ·
" " " A
Rate of change in
neutron density
Production
rate
Absorption
rate
“Leakage
in/out” rate
Volume Source
distribution
function
Surface
area
bounding "
Normal
to A
Equation of Continuity

B dA Bd 3r    
( , ) ( , ) ( ) ( , ) ( , ) ˆ       f
a n r t d S r t d r r t d J r t ndA
t
71
Using Gauss’ Divergence Theorem ò · = òÑ·
S V
( , ) ˆ ( , )     
ò · = òÑ· "
J r t ndA J r t d
A
"
¶
ò " = ò "-òå "-ò ·
¶
" " " A

       = -å -Ñ·
¶ f f
1 (r ,t) S(r ,t) (r ) (r ,t) J (r ,t)
v ¶
t a
EEqquuaattiioonn ooff CCoonnttiinnuuiittyy

72
For steady state operation
     
Ñ· J (r ) + å (r ) f
(r ) - S(r ) = 0 a
For non-spacial dependence
¶
n(t) S(t) (t)
t a= -å f
¶
Delayed sources?

73
Fick’s Law
Assumptions:
1.The medium is infinite.
2.The medium is uniform
å not å
(r  ) 3.There are no neutron sources in the medium.
4.Scattering is isotropic in the lab. coordinate system.
5.The neutron flux is a slowly varying function of
position.
6.The neutron flux is not a function of time.
Restrictive! Applicability??

74
Fick’s Law
Current Jx
x
dC/dx
x
f(x)
Negative Flux Gradient
Current Jx
High flux
More collisions
Low flux
Less collisions
• Diffusion: random walk of
an ensemble of particles
from region of high
“concentration” to region of
small “concentration”.
• Flow is proportional to the
negative gradient of the
“concentration”.

Number of neutrons ssccaatttteerreedd per
second from d" at rr and going
through dAz
r dAz r
Removed
(assuming no
buildup)
75
x
y
z
r
dAz
Fick’s Law
q
f
f q 
( ) cos
å e-S d"
r
s
t
4 p
2
 å å
not (r ) s s
Slowly varying
Isotropic

z z J dA dA r e t drd d 
1
»
3
S
77
Fick’s Law
¥
- = S -S
ò ò ò [ ]
= =
s z r
=
p
f
p
q
f q q q f
p
2
0
/ 2
0 0
( ) cos sin
4 r
HHWW 1155
0
æ
S
= + - - = - å
s
z z z ¶
3 2
ö
ö çè
÷ø
æ
÷ ÷ø
ç çè
z
J J J
t
¶f
+ = ?
z z J dA
and show that
= - Ñf = å
J D D s
and generalize 3 S
2t

s
D
DDiiffffuussiioonn
ccooeeffffiicciieenntt
Fick’s law
The current density is proportional to the negative of the gradient
of the neutron flux.

Combine:
Initial condition:
Solve:

Integrating the left-hand side by parts:

Or:
This is how Einstein proposed to measure the
diffusion coefficient of Brownian particles

Other examples: shear viscosity

Other examples: thermal conductivity

Other examples: electrical conductivity

Chapter 4 Fluid Flow, Heat Transfer, and Mass Transfer:
Similarities and Coupling
4.1 Similarities among different types of transport
4.1.1 Basic laws
The transfer of momentum, heat , and species A occurs in the direction of
decreasing vz, T, and wA, as summarized in Fig. 4.1-1. according to Eqs. [1.1-2],
[2.1-1], and [3.1-1]
[4.1-1] also
[1.1-2]
[4.1-2] also
[2.1-2]
[4.1-3] also
[3.1-1]
z
yz
d
dy
t = -m n
q k dT
y
dy
= -
A
j D dw
Ay A
dy
= -r
Newton’s law of viscosity
Fourier’s law of conduction
Fick’s law of diffution

The three basic laws share the same form as follows:
Or
[4.1-4]
[4.1-5]
Flux of gradient of
æ ö æ ö
ç ¸ æ proportonal
transport ö ¸=-ç ¸´ç ç transport
¸ ç ¸ è ç ¸ è cons t
ø ø è ç ø
¸ j d
f f dy
The three-dimensional forms of these basic laws are summarized in Table 4.1-1.
For constant physical properties, Eqs. [4.1-1] through [4.1-3] can be written
as follows:
[4.1-6]
tan
property property
y
=-G f
d v
dy
t = -n r
( ) yz z

[4.1-7]
[4.1-8]
q = -a d r
C T
( ) y v
dy
j D d
=- r
( ) Ay A A
dy
These equations share the same form listed as follows:
[4.1-9]
Flux of diffusivity gradient of
transport of transport transport property
property property concentration
æ ö æ ö æ ö
ç ¸=-ç ¸´ç ¸ ç ¸ ç ¸ ç ¸
çè ø¸ èç ø¸ èç ø¸
In other words, n, a, and DA are the diffusivities of momentum, heat, and mass,
respectively, and rvZ, rCvT, and rA are the concentration of z momentum, thermal
energy, and species mass, respectively.
4.1.2 Coefficients of Transfer
Fig. 4.1-2 shows the transfer of z momentum, heat, and species A from an
interface, where they are more abundant, to an adjacent fluid, and from an adjacent
fluid, where they are more abundant, to an interface. The coefficients of transfer,
according to Eqs. [1.1-35], [2.1-14], and [3.1-21], are defined as follows:

t - m ¶ ¶
= =
' 0 0 [4.1-10]
= =
0
( / )
0
yz y z y
f
- -
z z
v y
C
v v v
¥ ¥
(momentum
transfer coefficient)

[4.1-11]
[4.1-12]
q - k ( ¶ T / ¶
y
)
h = y y = =
y =
( heat transfer coefficient
) T - T T -
T
¥ ¥
j - D ¶ w ¶
y
( / )
Ay y = A A y
=
(mass transfer coefficient) = =
- -
As mentioned in Sec. 3.1.6, Eq. [4.1-12] is for low solubility of species A in the fluid.
These coefficients share the same form listed as follows:
[4.1-13]
[4.1-14]
[4.1-15]
or
Coefficient flux at the
of transfer difference in transport property
æ ö æ ö
ç ¸= ç ¸
è ø è ø
j -G ( ¶ f
/ ¶
y
)
k = f y = =
f
y =
(mass transfer coefficient) f
f - f f -
f
¥ ¥
It is common to divide Cf by rv/2 to make denominator appear in the form of the
kinetic energy rv2
∞/2. As shown in Eq. [1.1-36], the so-called friction coefficient is
defined by
0 0
0 0
0 0
0 0
m
A A A A
k
r w rw w w
¥ ¥
interface
0 0
0 0
'
0
f yz y
=
= =
1 1 2
2 2
f
C
C
t
rn rn
¥ ¥

4.1.3 The Chilton-Colburn Analogy
The analogous behavior of momentum, heat, and mass transfer is apparent from
Examples 1.4-6, 2.2-5, and 3.2-4, where laminar flow over a flat plate was considered.
From Eqs. [1.4-67], [2.2-71], and [3.2-56], at a distance z from the leading edge of the
plate,
C = -
0.323Re (1 2)
fz
2
z
0.323Pr1 3 Re 1 2 z
hz
k
=
k z Sc
D
m 0.323 1 3 Re 1 2
z
A
=
[4.1-16]
[4.1-17]
[4.1-18]
where
Rez
zru
m
= ¥ (local Reynolds number)
Pr p v C
m
= = (Prandtl number)
k
a
Sc v
m
r
= = (Schmidt number)
D D
A A
[4.1-19]
[4.1-20]
[4.1-21]
and υ∞ is the velocity of the fluid approaching the flat plate.

Equations [4.1-16] through [4.1-18] can be rearranged as follows
C fz
= 0.323Re -
(1 2)
2
z
1 Pr2 3 0.323Re (1 2)
PrRe z
hz
k
= -
k z Sc
D Sc
1 2 3 0.323Re 1 2
Re
m
z
A z
=
[4.1-22]
[4.1-23]
[4.1-24]
Since these equations have the same RHS, we see
C hz k z Sc
1 Pr2 3 1 2 3
fz m
k D Sc
2 PrRe Re
A z
= =
Substituting Eqs. [4.1-19] through [4.1-21] into Eq. [4.1-25], we obtain
C fz h k m
Sc
Pr2 3 2 3
2
= =
u rC u ¥ ¥
p
[4.1-25]
[4.1-26]

This equation, known as the Chilton-Colburn analogy,1 is ofen written as follows
fz
2
H D
C
= j = j [4.1-27]
where the j factor for heat transfer
Pr2/3 H
p
j h
v rC ¥
=
[4.1-28]
And the j factor for mass transfer
j k Sc
m 2/3
D
= [4.1-29]
v¥

The Chilton –Colburn analogy for momentum, heat and mass transfer has been
derived here on the basis of laminar flow over a flat plate. However, it has been
observed to be a reasonable approximation in laminar and turbulent flow in
systems of other geometries provided no form drag is present . From drag, which
has no counterpart in heat and mass transfer, makes Cf/2 greater than jH and jD, for
example, in flow around (normal to) cylinders. However, when form drag is
present, the Chilton –Colburn analogy between heat and mass transfer can still be
valid, that is,
jH = jD
[4.1-30]
or
h k Sc
v rC v ¥ ¥
Pr2/3 m 2/3
p
= [4.1-31]
These equations are considered valid for liquid and gases within the ranges
0.6 < Sc < 2500 and 0.6 < Pr < 100 . They have been observed to be a reasonable
approximation for various geometries, such as flow over flat plates, flow around
cylinders, and flow in pipes.

The Chilton –Colburn analogy is useful in that it allows one unknown transfer
coefficient to be evaluated from another transfer coefficient which is known or
measured in the same geometry. For example, by use Eq. [4.1-26] the mass transfer
coefficient km(for low solubility of species A in the fluid) can be estimated from a heat
transfer coefficient h already measured for the same geometry.
It is worth mentioning that for the limiting case of Pr=1, we see that from Eq.[4.1-
26]
fz
2
= [4.1-32]
p
C h
u rC ¥
Which is known as the Reynolds analogy , in honor of Reynolds’ first
recognition of the analogous behavior of momentum and heat transfer in 1874.

4.1.4 Integral-Balance Equations
The integral-balance equations governing momentum, heat , and species
transfer, according to Eq. [1.4-3], [2.2-6], and [3.2-4], respectively, are as follows
¶ òòò vd W = - òò ( vv ´ n ) dA - òò ´ ndA + òòò ( f -Ñ p ) d
W
[4.1-33]
¶ A A b
t
r r t
W W
(momentum transfer)
¶ òòò C Td W = - òò ( vC T ) ´ ndA - òò q ´ ndA + ¶ v v òòò
sd
W
A A
t
r r
W W
(heat transfer)
¶ òòò w W = - òò ( vw ) ´ ndA - òò j ´ ndA + òòò
r d
W
¶ A A A A A A
t
r r
W W
(species transfer)
[4.1-34]
[4.1-35]

In Eq. [4.1-33] the pressure term has been converted from a surface integral to
a volume integral using a Gauss divergence type theorem (i.e., Eq. [A.4-2]).
Furthermore, the body force fand pressure gradient b Ñp
can be considered as
the rate of momentum generation due to these force. In Eq. [4.1-34] the kinetic
and potential energy, and the pressure, viscous, and shaft work are not included
since they are either negligible or irrelevant in most materials processing
problems. In Eq. [4.1-35] ρw= ρ.
AA These integral balance equations share the same form as follows:
rate of net rate of
æ ö æ ö
Rate of rate of
æ ö ç ¸= ç ç inf
low by ¸ ç other
¸ æ ö accumulation ¸+ ç ¸+ ç ¸ è ø è ç convection ¸ ø è ç generation
net inf
low
è ø ø
¸ [4.1-36]
or
¶ W = - ¶ òòò òò v ´ ndA - òò j ´ ndA + òòò s d
W
[4.1-37]
rf ( r f
)
t W A A
f W
f

These equations are summarized in Table 4.1-2. The following integral mass-balance
equation ,Eq.[1.2-4], is also included in the table:
¶ òòò d Ωv =- nòò ( )
´
dA
[4.1-38]
¶ A
t
r r
W

4.1.5 Overall Balance Equations
The overall balance equations for momentum, heat, and species transfer
according to Eqs.[1.4-9], [2.2-8], and [3.2-7], respectively, are as follows
P = v - v +F + F +F
d m m
dt
( ) ( ) ( ) in out v p b
d E T = ( mC T ) - ( mC T ) + Q +
S
dt
v in v out
(momentum transfer) [4.1.-39]
dM A W = ( mw ) - ( mw )
+ J +
R
dt
A in A out A A
(heat transfer) [4.1-
40]
(species transfer) [4.1-41]
These overall balance equations share the same form as follows
Rate of rate of inflow rate of outflow
accumulation by convection by convection
æ ö æ ö æ ö
ç ¸= ç ¸-ç ¸
è ø è ø è ø
rate of other net inflow rate of
+ +
from surroundings generation
æ ö æ ö
ç ¸ ç ¸
è ø è ø
[4.1-42]
or

d F =(m f ) -(m f
) +J +S
dt in out
f f [4.1-43]
Where the total momentum, thermal energy, or species A in the control
volume Ω is
F = òòò rfd
W
W
[4.1-44]
In Eq.[4.1-39] the viscous force Fv at the wall can be considered as the rate
of momentum transfer through the wall by molecular diffusion. The pressure
force Fp and the body force Fb , on the other hand, can be considered as the
rate of momentum generation due to the action of these forces. In Eq.[4.1-40 ]
Q is by conduction, which is similar to diffusion.
The above equations are summarized in Table 4.1-3. The following overall
mass balance equation (i.e. Eq [1.2-6]), is also included in the table
dM m m
dt
- [4.1-
=( ) ( ) in out
45]

4.1.6 Differential Balance Equations
The differential balance equations governing momentum, heat, and species
transfer, according to Eqs. [1.5-6], [2.3-5] and [3.3-5], respectively, are as
follows:
¶ r v -Ñ´r vv -Ñ´t f
-Ñ
¶
b p ( )= ( ) + ( )
t
¶ r = -Ñ´r v -Ñ´ q
+
¶
( ) ( ) v v C T C T s
t
¶ r = -Ñ´r v -Ñ´ j
+
¶
( ) ( ) A A A A w w r
t
(momentum transfer) [4.1-46]
(heat transfer) [4.1-47]
(species transfer) [4.1-48]
In Eq. [4.1-47] the viscous dissipation is neglected and in Eq. [4.1-48] ρwA =ρA
These differential balance equations share the same form as follows:
Rate of rate of net inflow rate of other rate of
æ ö æ ö æ ö æ ö
ç ¸= ç ¸ + ç ¸ +
accumulation ç ¸
è ø è by convection ø è net inflow ø è generation
ø [4.1-49]
or

( ) ( ) s
t f f ¶ rf = -Ñ´r f -Ñ´ +
¶
v j
¶ r = -Ñ´r
¶
v
( ) ( )
t
[4.1-
These equations are summarized in Table 4.1-4. The following equation5 0o]f
continuity, Eq. [1.3-4], is also included in the table:
[4.1-
51]
Table 4.1-5 summarizes these equations for incompressible fluids.

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