molecular transport equations
DESCRIPTION
Molecular Transport Equations. Outline. Molecular Transport Equations Viscosity of Fluids Fluid Flow. Molecular Transport. “Each molecule of a system has a certain quantity of mass, thermal energy, and momentum associated with it.” – Foust - PowerPoint PPT PresentationTRANSCRIPT
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Molecular Transport Equations
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Outline
1.Molecular Transport Equations
2.Viscosity of Fluids
3.Fluid Flow
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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?
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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 forcerate of transport =
resistance
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Molecular Transport
A driving force is needed to overcome resistance in order to transport a property.Recall: Ohm’s Law from Physics 72
driving forcerate of transport =
resistance
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Molecular Transport
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Flux
Define: FLUX: amount of property being transferred per unit time through a cross-sectional area
Mathematically,
Z
dflux
dz
Is the equation dimensionally consistent?
What are the units of:ψz? δ? Γ?
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Flux
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
Z
ddz
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Flux
If the transport process is at steady state, what happens to the flux?
Z
ddz
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General Property Balance
rate of rate of
property in property out
rate of generation rate of accumulation
of property of property
If the transport process is at steady state, what happens to the flux?
0 0
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Flux at Steady State
Z
ddz
At steady-state:
2 2
1 1
2 1 2 1
1 2
2 1
z
Z z
Z
Z
dz d
z z
z z
1 2
2 1Z z z
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Flux
What happens when you have an unsteady-state transport process?
Z
ddz
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General Property Balance
rate of rate of
property in property out
rate of generation rate of accumulation
of property of property
Assume:1. Transport occurs in the z-
direction only.2. Volume element has a
unit cross-sectional area.3. R = rate of generation of
property (concentration per unit time)
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General Property Balance
Assume:1. Transport occurs in the z-
direction 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 ( )
rate of property out ( )z z
z z z
area
area
WHY?
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General Property Balance
Assume:1. Transport occurs in the z-
direction 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?
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General Property Balance
Assume:1. Transport occurs in the z-
direction 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
dz
dtWHY?
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General Property Balance
rate of rate of
property in property out
rate of generation rate of accumulation
of property of property
|z |zz z z
dR z z
dt
|z |zDividing by : z z z d
z Rz dt
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General Property Balance
|z |zz z z d
Rz dt
Taking the limit as 0: zd dz R
dz dt
But: z
ddz
2
2
d dR
dz dt
General equation for momentum, energy, and mass conservation (molecular transport mechanism only)
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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.
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Momentum Transport
Driving Force – change in velocity
Z
ddz
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Momentum Transport
Z
ddz
( )xyx
d vdy
Flux of x-directed momentum in the y-direction
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Momentum Transport
( )xyx
d vdy
but since:
xyx
dvdy
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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?
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Heat Transport
Driving Force – change in temperature
Z
ddz
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Heat Transport
Z
ddz
( c T)y pq d
A dy
Heat flux in the y-direction
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Heat Transport
( T)y pq d c
A dy
but since: k pc
yq dTk
A dy
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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.
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Mass Transport
Driving Force – change in concentration
Z
ddz
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Mass Transport
Z
ddz
* AAy AB
dcJ D
dy
Mass flux in the y-direction
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Analogy
( )xyx
d vdy
( c T)y pq d
A dy
* A
Ay AB
dcJ D
dy
MOMENTUM HEAT MASS
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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.
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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
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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?
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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
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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.
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Assignment
ANS: 1.05 x 10-11 g/cm2-s