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 forcerate of transport = resistance

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

Molecular Transport

Flux

Define: FLUX: amount of property being transferred per unit time through a cross-sectional area

Mathematically,

Z

dfluxdz

Is the equation dimensionally consistent?

What are the units of:ψz? δ? Γ?

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

Flux

If the transport process is at steady state, what happens to the flux?

Z

ddz

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

Flux at Steady State

Z

ddz

At steady-state:

2 2

1 1

2 1 2 1

1 2

2 1

zZ z

Z

Z

dz d

z z

z z

1 2

2 1Z z z

Flux

What happens when you have an unsteady-state transport process?

Z

ddz

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)

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

areaarea

WHY?

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?

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 propertyd zdt

WHY?

General Property Balance

rate of rate of property in property out

rate of generation rate of accumulation of property of property

|z |zz z zdR z zdt

|z |zDividing by : z z z dz R

z dt

General Property Balance

|z |zz z z dR

z dt

Taking the limit as 0: zd dz Rdz dt

But: z

ddz

2

2d dRdz 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

Z

ddz

Momentum Transport

Z

ddz

( )xyx

d vdy

Flux of x-directed momentum in the y-direction

Momentum Transport

( )xyx

d vdy

but since:

xyx

dvdy

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

Z

ddz

Heat Transport

Z

ddz

( c T)y pq dA dy

Heat flux in the y-direction

Heat Transport

( T)y pq d cA dy

but since: k pc

yq dTkA 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

Z

ddz

Mass Transport

Z

ddz

* AAy AB

dcJ Ddy

Mass flux in the y-direction

Analogy

( )xyx

d vdy

( c T)y pq d

A dy

* AAy AB

dcJ Ddy

MOMENTUM HEAT 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