Lecture 4 Transport of mass, momentum and heat. Bai Transport of mass, momentum and heat Energy transfer • in species transport equations, there is a source term due to …

X.S. Bai Transport of mass, momentum and heat

Lecture 4

Transport of mass, momentum and heat


Flow and molecular motion

turbulent jetflame


Mean speed of molecules: microscopic scale

1/ 28 B

ii

k Tsmπ

⎛ ⎞= ⎜ ⎟⎝ ⎠

kB is Boltzmann’s constant, mi is the mass of molecule i.


Mean speed of molecules: microscopic scale

1 1

1

1

N N

i i i i Ni i

i iNi

ii

v vv Y v

ρ ρ

ρρ

= =

=

=

= = =∑ ∑

∑∑

Define a mass averaged mean velocity


Mass transfer

• A mixture containing N species, each species has different velocity

ii Vvv +=

Velocityof

species iMass averages

velocity ofmixture

Diffusionvelocity ofspecies i

01

1

=

=

∑

∑

=

=N

iii

N

iii

VY

vYv


Mass transfer


Mass transfer

•

– diffusion velocity due to concentration gradient– diffusion velo. due to temperature gradient (Soret)– diffusion velocity due to pressure gradient

• the last two effects can be neglected• diffusion by concentration gradient is modeled by Fick’s Law

iV

iiNii YDVY ∇−=

Binary diffusivityNegative sign indicate flux fromhigh concentration to low


A control volume CV for mass conservation

: reaction rate of i: mass fraction of i

density of the mixture: velocity of i

i

i

i

Y

v

ω

ρ :

A control volume CV for mass conservation: the boundary is open,species can enter or leave the CV; it can also be formed or consumed in the CV


Law of mass conservation

Law of mass conservation says that the increase of the total mass of species i in the CV is equal to the net mass flow into the CV + the formation rate of the species in the CV.


Mass transfer

• Transport of species i in a mixture of N species, in a control volume ...

iiii vY

tY ωρρ

+⋅−∇=∂∂

Change of mass ofspecies i

= Net massflux in

+Formation

rate ofspecies i

Known fromchemical kinetics


Mass transfer

• Using Fick’s Law one has the transport equation for species i

• This equation is not closed without reaction kinetics rate information !

iiiNii YDvY

tY ωρρρ

+∇⋅∇=⋅∇+∂∂

Convectivetransfer

diffusivetransfer


Mass transfer

• Summation over all species we have

∑∑∑

==

= +⋅−∇=∂

∂ N

ii

N

iii

N

ii

vYt

Y

11

1 ωρ

ρ

=1v =0

Law ofmassaction

0=⋅∇+∂∂ v

tρρ Hi, I know it !

It’s the continuityequation !


Momentum conservation


Momentum conservation

We consider the momentum conservation of the mixture.

The law of momentum conservation is the Newton’s second law applied to Fluid Mechanics, which states that the change of the total momentum in the CV (Fig.4.1) + the exchange of momentum through the boundary of the CV is equal to the forces acting on the boundary + the body forces on the CV.


Momentum transfer

( )

( )

, or ( 1 2 3)

or , or

i j iji

j i j

iji ij

j i j

ρu u τρuv pvv pI i , ,t t x x x

τu uv pv v pI ρ ρut t x x x

ρ ρ τ

ρ ρ τ

∂ ∂∂∂ ∂+∇ ⋅ = ∇ ⋅ + + = − + =

∂ ∂ ∂ ∂ ∂

∂∂ ∂∂ ∂+ ⋅∇ = ∇ ⋅ + + = − +

∂ ∂ ∂ ∂ ∂

23

ji kij ij

j i k

uu uτ μ δ μx x x

⎛ ⎞∂∂ ∂= − −⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠


Energy conservation


A control volume CV for energy conservation

Energy flow outWork done to the

environment

Heat transfer from the environment

: enthalpy of i: mass fraction of i

density of the mixture: velocity of i

i

i

i

hY

vρ :

Energy flow in

the boundary is open, species can enter or leave the CV; it can also be formed or consumed in the CV


Total energy change per unit mass

2

2

1

2

1

2

12

12

12

12

total

N

i iiN

i ii

e e v

Y e v

pY h v

ph v

ρ

ρ

=

=

= +

= +

= − +

⎛ ⎞= − +⎜ ⎟⎝ ⎠

∑

∑


The energy flux at an open boundary

• Energy flux due to mass flow of species through the CV boundaries

• Energy flux due to temperature gradient through the CV boundaries

• Energy flux due to concentration gradient at the boundaries (Dufour effect, often negligible)


Energy flux due to flow motion

( ) ( )2 21 12 2

1 1

2 21 12 2

1 1

2 21 12 2

1 1 1 1

212

N N

i i i i i ii i

N N

i i i i ii i

N N N N

i i i i i i i ii i i i

pv e v Y v V h v

p pv Y h v YV h v

p pv Y h v Y v YV h YV v

pvh v v

ρ ρρ

ρ ρρ ρ

ρ ρ ρ ρρ ρ

ρ ρρ

= =

= =

= = = =

⎛ ⎞+ = + − +⎜ ⎟

⎝ ⎠⎛ ⎞ ⎛ ⎞

= − + + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= + − + + + − +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎛

= + − +⎝

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

1

N

i i ii

YV hρ=

⎞+⎜ ⎟

⎠∑


Energy flux due to temperature gradients

q Tλ= − ∇

heat conductivity


Total energy flux

• Through the boundary of a control volume

212

1

N

total i i ii

pq q vh v v YV hρ ρ ρρ =

⎛ ⎞= + + − + +⎜ ⎟

⎝ ⎠∑

conduction convection diffusion


work per unit area done to the environment

( )pI vτ− + ⋅


First law of thermodynamics

The first law of thermodynamics states that the energy increase in a system is equal to the heat transfer from the environment to the system - work done to the system. In differential equation form one can write

( ) ( )total total re q Q pI vtρ τ∂

+∇ ⋅ = +∇ ⋅ − + ⋅⎡ ⎤⎣ ⎦∂


Energy transport equation – conventional forms

( )

( )( )

212

212

1

N

i i ii

r

h p vt

q vh v p v YV h

Q pI v

ρ ρ

ρ ρ ρ

τ=

∂− +

∂⎛ ⎞

+∇ ⋅ + + − + +⎜ ⎟⎝ ⎠

= + ∇ ⋅ − + ⋅⎡ ⎤⎣ ⎦

∑



( )( )

( ) ( )

( ) ( ) ( )

2122 1

2

2122 1

2

:

:

vv v v v pItv v v pI v pI vtv vv pI v pI v

t

ρ ρ τ

ρ ρ τ τ

ρρ τ τ

∂⎛ ⎞⋅ + ⋅∇ = ⋅ ∇ ⋅ − +⎜ ⎟∂⎝ ⎠∂

⇒ + ⋅∇ = ∇⋅ − + ⋅ − − + ∇⎡ ⎤⎣ ⎦∂∂

⇒ +∇⋅ = ∇ ⋅ − + ⋅ − − + ∇⎡ ⎤⎣ ⎦∂



( ) ( )

( )1

:

N

i i ii

r

h p q vh v p YV ht

Q pI v

ρ ρ ρ

τ=

∂ ⎛ ⎞− + ∇ ⋅ + + − +⎜ ⎟∂ ⎝ ⎠

= + − + ∇

∑

jj

D v uDt t t x

∂ ∂ ∂= + ⋅∇ ≡ +∂ ∂ ∂

1:

N

i i i ri

Dh Dp q YV h Q vDt Dt

ρ ρ τ=

⎛ ⎞− = −∇ ⋅ + + + ∇⎜ ⎟

⎝ ⎠∑



1

11 :N

i i ri i

Dh Dp ρα h ρα h Y Q vDt Dt Le

ρ τ=

⎛ ⎞⎛ ⎞− = ∇ ⋅ ∇ − − ∇ + + ∇⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑

ii p i

α λLeD ρc D

≡ =

Lewis number



1

11 :N

i i ri i

Dh Dp ρα h ρα h Y Q vDt Dt Le

ρ τ=

⎛ ⎞⎛ ⎞− = ∇ ⋅ ∇ − − ∇ + + ∇⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑

( )Dh ρα hDt

ρ =∇⋅ ∇



( ) ( )1

p

N

i pi i i i ri

DT DpcDt Dt

T D c Y T h Q v

ρ

λ ρ ω τ=

−

= ∇ ⋅ ∇ + ∇ ⋅∇ − + + : ∇∑

( ) ( )1

N

p i pi i i ii

DTc T Dc Y T hDt

ρ λ ρ ω=

=∇⋅ ∇ + ∇ ⋅∇ −∑


Energy transfer

• in species transport equations, there is a source term due to chemical reaction, why there is no source term due to chemical reactions in the energy equation?

• chemical energy is released to thermal energy, however the system’s energy is not changed due to chemical reactions !

• If energy equation is expressed as temperature transport equation, one should have the reaction source term


Summary of governing equations

• Unkowns– 3 velocity

components– pressure– temperature or

enthalpy– N species– density

• total unknowns– N+6

• Governing equations– continuity 1– N-S equations 3– temperature or

enthalpy1

– equation of state 1– species transport N

• total equations– N+6


Momentum:

0xu

j

j =+∂ρ∂

∂ρ∂t

j

ij

ij

jiixx

px

uutu

∂

∂+

∂∂

−=∂

∂+

∂∂ τρρ

Pri

ri j j

u hh h Qt x x x

ρρ μ⎛ ⎞∂∂ ∂ ∂+ = +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

ij

i

jj

ijixY

ScxxYu

tY ω

∂∂μ

∂∂

∂ρ∂

∂ρ∂

+⎟⎟

⎠

⎞⎜⎜

⎝

⎛=+

∑=i i

iWYTRp ρ0

Mass:

Energy:

Species:

Equation of state:

Simplified governing equations (unity Lewis number)


Chemical reaction rates

( )

∏ ∏

∑

= =

=

−=

−=≡

N

i

N

iijbijfj

j

L

jijij

ii

ijij CkCkq

qdt

dC

1 1

1

'''

''' νν

ννω

LjMM i

N

iiji

N

iij ,...,1

1

''

1

' =⇔∑∑==

ννReactions:

Reactionrates:


A comparison of mass diffusivity, heat conductivity and viscosity

Using kinetic theory it can be shown (cf. Turns book):

12/32

2/1

3

3

32

31 −∝⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛== PT

PT

mTkcD

ij

Bij

σπξ

Mean diffusion coefficient for species i in the mixture

12/3

,1

/

1 −

≠=

∝−

=

∑PT

DX

YD N

ijjijj

ii



Using kinetic theory it can be shown (cf. Turns book):

2/12/1

43

3

21 T

mTkc

Vnkk B

B ∝⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎠

⎞⎜⎝

⎛=σπ

ξ

Thermal diffusion coefficientpc

kρ

α ≡

Lewis numberi

iiip

i LeD

DDckLe αα

ρ=∴=≡ ,



Using kinetic theory it can be shown:

2/12/1

4332

31 TTmkc B ∝⎟⎟

⎠

⎞⎜⎜⎝

⎛==

σπξρμ

Prandtl numberαν

ραμ

=≡Pr

Schmidt numberi

i DSc μ

≡

Lecture 4 Transport of mass, momentum and heat. Bai Transport of mass, momentum and heat Energy transfer • in species transport equations, there is a source term due to …

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