Molecular Transport Equations
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
Molecular Transport
Flux
Define: FLUX
: amount of property Γ being transferred per unit
time through a cross-sectional area
Mathematically,
dΓ
ψ Z = flux = −δ
dz
Is the equation
dimensionally consistent?
What are the units of:
δ?
Γ?
ψz?
Flux
dΓ
ψ Z = −δ
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
dΓ
ψ Z = −δ
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
Flux at Steady State
Flux at Steady State
Flux at Steady State
dΓ
ψ Z = −δ
dz
z2
Γ2
z1
Γ1
ψ Z ∫ dz =
−δ ∫ dΓ
ψ Z ( z2 − z1 ) = −δ ( Γ2 − Γ1 )
ψZ =
δ ( Γ1 − Γ 2 )
z2 − z1
At steady-state:
ψZ =
δ ( Γ1 − Γ 2 )
z2 − z1
Flux
dΓ
ψ Z = −δ
dz
What happens when you have an unsteady-state
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   rate of accumulation 
+
=


of
property
of
property

 

General Property Balance
General Property Balance
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 ψ z|z ⋅ Α (area)
rate of property=
out ψ z|z +∆z ⋅ Α (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 ⋅ Α )
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
dΓ
=
⋅ ( ∆z ⋅ Α )
dt
WHY?
General Property Balance
 rate of
  rate of


−

property
in
property
out

 

 rate of generation   rate of accumulation 
+
=


of
property
of
property

 

dΓ
⋅ ( ∆z ⋅ Α )
])
(ψ z|z ⋅ Α ) − (ψ z|z +∆z ⋅ Α ) + ( R ⋅ [ ∆z ⋅ Α=
dt
Dividing by ( ∆z ⋅ Α ) :
ψ z|z −ψ z|z +∆z
∆z
dΓ
+R =
dt
General Property Balance
ψ z|z −ψ z|z +∆z
∆z
dΓ
+R =
dt
Taking the limit as ∆z → 0 :
dψ z
dΓ
−
+R =
dz
dt
dΓ
But: ψ z = −δ
dz
d Γ
dΓ
δ 2 +R =
dz
dt
2
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
dΓ
ψ Z = −δ
dz
Momentum Transport
dΓ
ψ Z = −δ
dz
d(v x ρ )
τ yx = −ν
dy
Flux of x-directed
momentum in the
y-direction
Momentum Transport
d(v x ρ )
τ yx = −ν
dy
but since:
µ = νρ
dv x
τ yx = − µ
dy
Momentum Transport
dv x
τ yx = − µ
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
dΓ
ψ Z = −δ
dz
Heat Transport
dΓ
ψ Z = −δ
dz
qy
A
= −α
d( ρ c p T)
Heat flux in the
y-direction
dy
Heat Transport
qy
A
= −α
d( ρ cp T)
dy
but since: k = αρ cp
qy
dT
= −k
A
dy
Heat Transport
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
dΓ
ψ Z = −δ
dz
Mass Transport
dΓ
ψ Z = −δ
dz
dc A
J = −DAB
dy
*
Ay
Mass flux in the
y-direction
Mass Transport
dc A
J = −DAB
dy
*
Ay
Analogy
d(v x ρ )
τ yx = −ν
dy
MOMENTUM
qy
A
= −α
d( ρ c p T)
HEAT
dy
dc A
J = −DAB
dy
*
Ay
MASS