Upload
randolf-paul
View
241
Download
10
Embed Size (px)
Citation preview
ME 475/675 Introduction to
CombustionLecture 11
Announcements
• Midterm 1• September 29, 2014• Review Friday, September 26
• HW 5 Due Friday, September 26, 2014
Chapter 3 Introduction to Mass Transfer
• Consider two species, x and o• Concentration of “x” is larger on the left, of “o” is larger on the right• Species diffuse through each other
• they move from regions of high to low concentrations • Think of perfume in a room• Mass flux is driven by concentration difference• Analogously, heat transfer is driven by temperature differences
• There may also be bulk motion of the mixture (advection, like wind)• Total rate of mass flux: (sum of component mass flux)
x x
xx
x x
x
xx x
x x
xx
x x
x
xx xx x
xx
x x
x
xx xx x
xx
x x
x
xx x
o o
ooo
ooo ooo
oo o ooo o
ooo ooo
oo o oo
MassFraction
Yx
YoYx�̇� ¿1-
0-
Chapter 3 Introduction to Mass Transfer
• Rate of mass flux of “x” in the direction
Advection (Bulk Motion) Diffusion (due to concentration gradient)
• Diffusion coefficient of x through o • Units • Appendix D, pp. 707-9
• For gases, book shows that
x x
xx
x x
x
xx x
x x
xx
x x
x
xx xx x
xx
x x
x
xx xx x
xx
x x
x
xx x
o o
ooo
ooo ooo
oo o ooo o
ooo ooo
oo o oo
MassFraction
Y x
YoYx �̇� ¿
Stefan Problem (no reaction)
• One dimensional tube (Cartesian)• Gas B is stationary: • Gas A moves upward • Want to find this
• ; • but treat as constant
Y
x
YB
YA
L-𝑌 𝐴 , ∞
𝑌 𝐴 , 𝑖
A
B+A
Mass Flux of evaporating liquid A
• For • (dimensionless)• increases slowly for small • Then very rapidly for > 0.95
• What is the shape of the versus x profile?
0 0.2 0.4 0.6 0.80
2
4
6
86.908
0
m Y( )
10 Y𝑌 𝐴 , 𝑖
�̇�𝐴}} over {{ { } rsub { }} over { }𝜌 𝒟 𝐴𝐵 𝐿 ¿¿
Profile Shape• but
• Ratio: ;
• For
• Large profiles exhibit a boundary layer near exit (large advection near interface)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
10.99
0
YA x .05( )
YA x .1( )
YA x .5( )
YA x .9( )
YA x .99( )
10 x𝑥𝐿
=0.99
=0.9
=0.5
=0.1
=0.05
Liquid-Vapor Interface Boundary Condition
• At interface need
• So
• Saturation pressure at temperature T• For water, tables in thermodynamics textbook• Or use Clausius-Slapeyron Equation (page 18 eqn. 2.19)
A+BVapor
𝑌 𝐴 , 𝑖
LiquidA
Clausius-Clapeyron Equation (page 18)• Relates saturation pressure at a given temperature to the saturation
conditions at another temperature and pressure
• If given , we can use this to find • Page 701, Table B: , at P = 1 atm
Problem 3.9
• Consider liquid n-hexane in a 50-mm-diameter graduated cylinder. Air blows across the top of the cylinder. The distance from the liquid-air interface to the open end of the cylinder is 20 cm. Assume the diffusivity of n-hexane is 8.8x10-6 m2/s. The liquid n-hexane is at 25C. Estimate the evaporation rate of the n-hexane. (Hint: review the Clausius-Clapeyron relation a applied in Example 3.1)
Stefan Problem (no reaction)
• One dimensional tube (Cartesian)• Gas B is stationary • but has a concentration gradient
• Diffusion of B down = advection up
• ; • ; =
Y
x
YBYA
L-𝑌 𝐴 , ∞
YA,i