Molecular Transport Equations
Outline
1.Molecular Transport Equations
2.Viscosity of Fluids
3.Fluid Flow
Molecular Transport
“Each molecule of a system has a certain
quantity of mass, thermal energy, and
momentum associated with it.” – Foust
1. What happens when a difference in the
concentration of these properties occur from
one region to another?
2. How is transport different in a solid, gas, and
a liquid?
Molecular Transport
We need a simple physical model to describe
molecular transport - one that does not take
into account the structural differences of the
three states.
driving force
rate of transport =
resistance
Molecular Transport
driving force
rate of transport =
resistance
A driving force is needed to overcome
resistance in order to transport a
property.
Recall: Ohm’s Law from Physics 72
Molecular Transport
Flux
Define: FLUX
: amount of property  being transferred per
unit time through a cross-sectional area
Mathematically,
d
 Z  flux  
dz
Is the equation
dimensionally consistent?
What are the units of:
ψz?
δ?
Γ?
Flux
d
 Z  
dz
Flux in the z-direction: amount of property
transferred per unit time per cross-sectional area
perpendicular to the z-direction of flow
δ: diffusivity, proportionality constant
Flux
d
 Z  
dz
If the transport process is at steady state, what
happens to the flux?
General Property Balance
If the transport
process is at
steady state,
what happens to
the flux?
 rate of
  rate of




property
in
property
out
0
0

 

 rate of generation   rate of accumulation 



 of property
  of property

Flux at Steady State
d
 Z  
dz
z2
2
z1
1
 Z  dz    d
 Z  z2  z1     2  1 
Z 
  1  2 
z2  z1
At steady-state:
Z 
  1  2 
z2  z1
Flux
d
 Z  
dz
What happens when you have an unsteadystate transport process?
General Property Balance
Assume:
1. Transport occurs in the zdirection only.
2. Volume element has a
unit cross-sectional area.
3. R = rate of generation of
property (concentration
per unit time)
 rate of
  rate of




property
in
property
out

 

 rate of generation

 of property
  rate of accumulation 


  of property

General Property Balance
Assume:
1. Transport occurs in the zdirection only.
2. Volume element has a
unit cross-sectional area.
3. R = rate of generation of
property (amount per unit
time per unit volume)
rate of property in   z|z   (area)
rate of property out   z|z z   (area)
WHY?
General Property Balance
Assume:
1. Transport occurs in the zdirection only.
2. Volume element has a
unit cross-sectional area.
3. R = rate of generation of
property (amount per unit
time per unit volume)
rate of generation of property  R   z  
WHY?
General Property Balance
Assume:
1. Transport occurs in the zdirection only.
2. Volume element has a
unit cross-sectional area.
3. R = rate of generation of
property (amount per unit
time per unit volume)
rate of accumulation of property
d

  z   
dt
WHY?
General Property Balance
 rate of
  rate of




property
in
property
out

 

 rate of generation

 of property
  rate of accumulation 


of
property
 

d
 z|z      z|z z      R   z    dt   z   
Dividing by  z    :
 z|z  z|z z
z
d
R 
dt
General Property Balance
 z|z  z|z z
z
d
R 
dt
Taking the limit as z  0 :
But:
 z  
d
dz
d 
d
 2 R 
dz
dt
2
d z
d

R 
dz
dt
General equation for momentum,
energy, and mass conservation
(molecular transport mechanism
only)
Momentum Transport
• Imagine two parallel
plates, with area A,
separated by a
distance Y, with a
fluid in between.
• Imagine the fluid
made up of many
layers – like a stack
of cards.
Momentum Transport
Driving Force – change
in velocity
d
 Z  
dz
Momentum Transport
d
 Z  
dz
d(v x  )
 yx  
dy
Flux of x-directed
momentum in the
y-direction
Momentum Transport
d(v x  )
 yx  
dy
but since:
  
dv x
 yx  
dy
Heat Transport
• Imagine two
parallel plates,
with area A,
separated by a
distance Y, with a
slab of solid in
between.
• What will happen
if it was a fluid
instead of a solid
slab?
Heat Transport
Driving Force –
change in
temperature
d
 Z  
dz
Heat Transport
d
 Z  
dz
qy
A
 
d(  c p T)
Heat flux in the
y-direction
dy
Heat Transport
qy
A
 
d(  cp T)
dy
but since: k  cp
qy
dT
 k
A
dy
Mass Transport
• Imagine a slab of
fused silica, with
thickness Y and
area A.
• Imagine the slab
is covered with
pure air on both
surfaces.
Mass Transport
Driving Force –
change in
concentration
d
 Z  
dz
Mass Transport
d
 Z  
dz
dcA
J  DAB
dy
*
Ay
Mass flux in the
y-direction
Analogy
d(v x  )
 yx  
dy
MOMENTUM
qy
A
 
d(  c p T)
HEAT
dy
dcA
J  DAB
dy
*
Ay
MASS
Assignment
• Compute the steady-state momentum flux τyx
in lbf/ft2 when the lower plate velocity V is 1 ft/s
in the positive x- direction, the plate separation
Y is 0.001 ft, and the fluid viscosity µ is 0.7 cp.
Assignment
• Compute the steady-state momentum flux τyx
in lbf/ft2 when the lower plate velocity V is 1 ft/s
in the positive x- direction, the plate separation
Y is 0.001 ft, and the fluid viscosity µ is 0.7 cp.
ANS: 1.46 x 10-2 lbf/ft2
Assignment
• A plastic panel of area A = 1 ft2 and thickness
Y = 0.252 in. was found to conduct heat at a
rate of 3.0 W at steady state with
temperatures To = 24.00°C and T1 = 26.00°C
imposed on the two main surfaces. What is
the thermal conductivity of the plastic in
cal/cm-s-K at 25°C?
Assignment
• A plastic panel of area A = 1 ft2 and thickness
Y = 0.252 in. was found to conduct heat at a
rate of 3.0 W at steady state with
temperatures To = 24.00°C and T1 = 26.00°C
imposed on the two main surfaces. What is
the thermal conductivity of the plastic in
cal/cm-s-K at 25°C?
ANS: 2.47 x 10-4 cal/cm-s-K
Assignment
• Calculate the steady-state mass flux jAy of
helium for the system at 500°C. The partial
pressure of helium is 1 atm at y = 0 and zero at
the upper surface of the plate. The thickness Y
of the Pyrex plate is 10-2 mm, and its density
ρ(B) is 2.6 g/cm3. The solubility and diffusivity
of helium in pyrex are reported as 0.0084
volumes of gaseous helium per volume of
glass, and DAB = 0.2  10-7 cm2/s, respectively.
Assignment
ANS: 1.05 x 10-11 g/cm2-s