01
-
Upload
tareq-abdussalam-khamllag -
Category
Documents
-
view
218 -
download
0
description
Transcript of 01
![Page 1: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/1.jpg)
Molecular Transport Equations
![Page 2: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/2.jpg)
Outline
1.Molecular Transport Equations
2.Viscosity of Fluids
3.Fluid Flow
![Page 3: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/3.jpg)
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?
![Page 4: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/4.jpg)
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
![Page 5: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/5.jpg)
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
![Page 6: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/6.jpg)
Molecular Transport
![Page 7: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/7.jpg)
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?
![Page 8: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/8.jpg)
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
![Page 9: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/9.jpg)
Flux
If the transport process is at steady state, what happens to the flux?
Z
ddz
![Page 10: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/10.jpg)
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
![Page 11: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/11.jpg)
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
![Page 12: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/12.jpg)
Flux
What happens when you have an unsteady-state transport process?
Z
ddz
![Page 13: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/13.jpg)
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)
![Page 14: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/14.jpg)
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?
![Page 15: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/15.jpg)
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?
![Page 16: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/16.jpg)
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?
![Page 17: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/17.jpg)
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
![Page 18: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/18.jpg)
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)
![Page 19: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/19.jpg)
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.
![Page 20: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/20.jpg)
Momentum Transport
Driving Force – change in velocity
Z
ddz
![Page 21: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/21.jpg)
Momentum Transport
Z
ddz
( )xyx
d vdy
Flux of x-directed momentum in the y-direction
![Page 22: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/22.jpg)
Momentum Transport
( )xyx
d vdy
but since:
xyx
dvdy
![Page 23: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/23.jpg)
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?
![Page 24: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/24.jpg)
Heat Transport
Driving Force – change in temperature
Z
ddz
![Page 25: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/25.jpg)
Heat Transport
Z
ddz
( c T)y pq dA dy
Heat flux in the y-direction
![Page 26: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/26.jpg)
Heat Transport
( T)y pq d cA dy
but since: k pc
yq dTkA dy
![Page 27: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/27.jpg)
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.
![Page 28: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/28.jpg)
Mass Transport
Driving Force – change in concentration
Z
ddz
![Page 29: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/29.jpg)
Mass Transport
Z
ddz
* AAy AB
dcJ Ddy
Mass flux in the y-direction
![Page 30: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/30.jpg)
Analogy
( )xyx
d vdy
( c T)y pq d
A dy
* AAy AB
dcJ Ddy
MOMENTUM HEAT MASS
![Page 31: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/31.jpg)
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.
![Page 32: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/32.jpg)
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
![Page 33: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/33.jpg)
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?
![Page 34: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/34.jpg)
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
![Page 35: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/35.jpg)
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.
![Page 36: 01](https://reader030.fdocuments.us/reader030/viewer/2022033104/577c7c561a28abe0549a30c6/html5/thumbnails/36.jpg)
Assignment
ANS: 1.05 x 10-11 g/cm2-s