Transcript
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
What happens when a difference in the concentration of these properties occur from one region to another?
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.
Molecular Transport
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,
Is the equation dimensionally consistent?
What are the units of:
?z? ?? ??
Flux
Flux in the z-direction: amount of property transferred per unit time per cross-sectional area perpendicular to the z-direction of flow
?: diffusivity, proportionality constant
Flux
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?
0
0
Flux at Steady State
At steady-state:
Flux
What happens when you have an unsteady-state transport process?
General Property Balance
Assume:
Transport occurs in the z-direction only.
Volume element has a unit cross-sectional area.
R = rate of generation of property (concentration per unit time)
General Property Balance
Assume:
Transport occurs in the z-direction only.
Volume element has a unit cross-sectional area.
R = rate of generation of property (amount per unit time per unit volume)
WHY?
General Property Balance
Assume:
Transport occurs in the z-direction only.
Volume element has a unit cross-sectional area.
R = rate of generation of property (amount per unit time per unit volume)
WHY?
General Property Balance
Assume:
Transport occurs in the z-direction only.
Volume element has a unit cross-sectional area.
R = rate of generation of property (amount per unit time per unit volume)
WHY?
General Property Balance
General Property Balance
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
Momentum Transport
Flux of x-directed momentum in the y-direction
Momentum Transport
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
Heat Transport
Heat flux in the
y-direction
Heat Transport
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
Mass Transport
Mass flux in the
y-direction
Analogy
MOMENTUM
HEAT
MASS
Assignment
Compute the steady-state momentum flux ?yx
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 ?yx
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 ?(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