ME 475/675 Introduction to

CombustionLecture 11

Mass Transfer, Stefan problem, Liquid/vapor interface boundary condition, Example problem 3.9

Announcements

• Midterm 1• Monday, September 28, 2015

• 8-10 AM, PE 104?• Review Friday, September 25

• HW 4 Due now (but I’ll accept it Monday, no penalty)• HW 5 Due Friday, September 25, 2015• Tesla Lecture

• https://nevada.formstack.com/forms/engineering_distinguishedlecture[nevada.formstack.com]

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 (around) 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, wind)• Total mass flux rate: (sum of component mass fluxes)

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-

Quantitative phonenena and expressions

• 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 • Constant, 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

• Increases with and • Decreases with and

• For (strong wind)• (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 ; need

• 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• ; (page 701 for fuels)

• Let and (tabulated for fuels on page 701)• Let that we are tying to find at temperature

• If given , we can use this to find • Page 701, Table B: , at

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 25°C. Estimate the evaporation rate of the n-hexane. (Hint: review the Clausius-Clapeyron relation 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

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