3Lecture Conservation Eqs

8/3/2019 3Lecture Conservation Eqs

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Indian Institute of Technology, Kharagpur, IndiaIndian Institute of Technology, Kharagpur, India -- 721302721302

Fundamental Considerations ofFundamental Considerations of

Continuum ConservationContinuum Conservation

January 2011

Suman Chakraborty

Professor

Mechanical Engineering Department

Indian Institute of Technology (IIT) Kharagpur, India

E-mail: [email protected]

http://www.stanford.edu/~sumancha/
mailto:[email protected]:[email protected]


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General Philosophy of ConservationGeneral Philosophy of Conservation

Balance: IN OUT + GENERATED = CHANGE


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Reynolds Transport Theorem (RTT)Reynolds Transport Theorem (RTT)

Control mass system Control volume concept

Eulerian approach

Applicable for any property :-

mass, momentum, energy etc

Can be vectors or Scalars

Lagrangian approach


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I

II

IIIN=extensive property of a system

n= N per unit mass

Nt

= (NI

)t

+ (NII

)tNt+t = (NII

)t+t + (NIII

)t+t

t

NNLt

t

NNLt

dt

dN tIIttIIt

ttt

tsys

)()(

00

Rate of outflow -inflow

cv

ndt

cs

r dAVn ).(

cs

r

cvsys

dAVnndtdt

dN ).( RTT

where Vr

is the velocity of fluid relative to the CV


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Conservation of MassConservation of MassN=m (mass)n=1

Assuming stationary CV : Vr

=V

( . )sys cv cs

dm

d V dAdt t

=0

0).( dVdt

cvcv

Non-deformable CVUsing Gauss divergence theorem

0).(

V

t

For any arbitrary CV


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Conservation of Linear MomentumConservation of Linear Momentum

Assuming stationary CV :Vr

=V

N mV

n V

( )( . )

cv cssys

d mVVd V V dA

dt t

( )sys

sys

d mVFdt

Newtons second law

cvsys FFas t0

Surface forces

Body forces


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Surface forces: Concept of StressSurface forces: Concept of Stress

Definition of StressThe stress field is the distribution of internal "tractions" that balance a given set

of external tractions.Traction T

represents the force per unit area acting at

a given location on the body's surface. Traction T is abound vector, which means T cannot slide along its

line of action or translate to another location and

keep the same meaning. In other words, a traction

vector cannot be fully described unless both the force

and the surface where the force acts on has beenspecified. Given both Fand s, the traction Tcan be defined as

n

iTComponents ofT

are

designated by 2 indices

Direction normal to

the chosen area

Direction of action of the force component


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Stress tensor components are denoted by where iis the direction normal to

the face on which it is acting andjrepresents the direction of action of the

concerned force component

23

2221

33

3231

1312

11

3x

2x1x

Stress Tensor Components

ij

Our objective now is to express the traction vector at a point on any

arbitrary plane in terms of the stress tensor components at that

point


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1 1 2 2 3 3

n

i i i i ji jT n n n n

O

A

B

C

x1

x2

x3

Consider surfaces:

S1

: OAB with surface normal along x3S2

: OBC with surface normal along x2S3

: OAC with surface normal along x1& S: ABC with surface normal along

knjninn 321

Applying force balance:

where b

is the body force per unit volume

h is the perpendicular distance from O to ABC

Noting Si

=Sni

and taking Limit as h0

Generalised

to => Cauchys theorem

Traction Vector on an arbitrary surfaceTraction Vector on an arbitrary surface

1 11 11 1 21 2 31 3 1 1 13 3

nF S S S T S Shb Sha

3312211111 nnnTn


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Conservation of angular momentumConservation of angular momentum

Net moment w.r.t

centroidal

axis (OO) of the element=Io

as y, x0

valid when couple stresses (body

couples) are absent.

jiij

)( 222112 yxyxKyxxy

x21

x22

x22x21

y12

y11y

11

y12


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bodysurfcv FFF

cv

i

cs

icvidbdAnF .

cv

i

cv

i

cv

i

cv

icvidVudu

tdbdF ).()(.

assuming non-deformable control volume

(From RTT)

Since choice of CV is arbitrary,

( ) .( ) .i i i iu u V bt

Can be represented in tensor notation as:

i

j

ij

ji

j

i bx

uux

ut

)()(

Naviers

equation of

equillibrium

NavierNavierss equation of equilibriumequation of equilibrium1 2 3

i i i ii j k

where


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Substituting the Constitutive RelationshipSubstituting the Constitutive Relationship

where p is the thermodynamic pressure

(satisfies Eq. of state)

For a Newtonian fluid, is a linear function of eij

dev

ij ijkl klC e

dev

ij ij ijp

}]{}[{21

i

j

j

i

i

j

j

i

j

i

xu

xu

xu

xu

xu

a function of the rate of deformation

Symmetric part

Rate of deformation eij

Anti-Symmetric part

ij is a function ofeij

only

ij

A 4th

order tensor that maps a 2nd

order tensor onto a 2nd

order tensor

Note that


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lkjiijkl DCBACs

Thus, for isotropic fluid,

jkiljlikklijijklC

jiij

(Volume dilation coefficient)(Viscosity coefficient)

)(i

j

j

iij

k

kijij

x

u

x

u

x

up

Finally the stress tensor yields the form

Define

isotropic

scalar ).)(.().)(.().)(.( CBDADBCADCBA For isotropy

jjiijjiijjii CBDADBCADCBA

)( jkiljlikklijlkji DCBA

Using we can show

Noting

2dev k

ij ij ij

k

ue

x

Special case: Homogeneous + Isotropic Fluid

Position independent for homogeneous


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11 22 33

3

mp

k

km

x

upp

)(

32

Stokes hypothesis: pm

=p 32

automatically satisfied for incompressible fluid as:

and also for monoatomic

gas

0k

k

u

x

time scale of change needs to be larger compared to

the molecular relaxation time

The thermodynamic pressure (p) accounts for translational + rotational +

vibrational

modes of energy of the molecules

The mechanical pressure considers only the translational mode

32 known as the bulk viscosity of fluid

Towards the Stokes Hypothesis


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NavierNavier--Stokes equationStokes equation

i

j

ij

ji

j

i bx

uux

ut

)()( Navier

equation

LHS= ][)]([)()(j

ij

ij

j

iji

j

ixuu

tuu

xtuuu

xu

t

0 (from continuity)

Substituting the expression for stress in RHS we obtain:

i

k

k

ij

i

jij

ij

i bx

u

xx

u

xx

P

x

uu

t

u

]

3[][][

=0for incompressible

fluid

Navier-Stokes equation


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Conservation of EnergyConservation of EnergyN=E (Total energy )

= Internal energy (mi)+Kinetic

Energy (mv2/2)+Potential Energy (mgz)

n=e

Assuming stationary CV : Vr

=V

( . )sys cv cs

dEed e V dA

dt t

cv j

jd

x

eu

t

e]

)()([

using non deformable control volume and Gauss divergence theorem

cvcv j

j

j

j

sysdDt

De

dx

u

tex

e

ut

e

dt

dE

)}

)(

()({

=0


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First law of ThermodynamicsFirst law of Thermodynamics

cvcv

sys

WQdt

dE

dAnqdQQcscv

cv.

'''''

'''[ ]jjcv

qQ d

x

( )

n

cv i i i i

cv cs

ii i i

cv cs

W b u d T u dA

b u d u ndA

cv j

iij

ii dx

uub ])([

Q

is the rate of heat generation

q is the heat flux


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j

iij

ii

i

i

x

u

ubx

qQDt

De

)('''

For any arbitrary choice of control volume, combining the above two results we get,

Above is the statement of total energy conservation, however we are interested in

thermal energy only and hence we would subtract the mechanical energy from the

above equation, to get an equation in terms of specific internal

energy (i)

Multiplying ui

with the Naviers

equation:

iji

i i ij

Duu u b

Dt x

Mechanical Energy

And then subtracting we obtain,

j

i

ijj

j

x

u

x

q

QDt

Di

''' })(

{j

j

j

j

x

u

t

i

x

iu

t

i

Dt

Di

j

j

x

iu

t

i

)()(

j

iij

j

j

j

j

x

u

x

qQ

x

iu

t

i

''')()(


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Vpx

u

j

iij .

We need to find

j

iij

x

u

where

])()()[(

])()()[(

2

2

3

3

22

1

3

3

12

1

2

2

1

2

3

3

2

22

3

3

1

12

2

2

1

1

32

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

x

u

It is important to note that >0 (Viscous dissipation)

Using the stress tensor for the Newtonian and Stokesian

fluid we get,

Using the expression for and noting that

j

iij

x

u

we get, ''' j

j

q Dh DpQ

Dt Dt x

Generalised

thermal energy

conservation equation

ph i

Viscous heating because of the energy dissipation

due to work done against viscous shear


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h=h(T, p) for a simple compressible pure substance & assume no phase change

dPP

hdTCdP

P

hdT

T

hdh

T

p

Tp

again [ ] [ ] p T p P

s s s vdh TdS vdP T dT T v dP T dT T v dP

T P T T

andpdT

dv

v

1

gives(1 )p

Dh DT DPC v T

Dt Dt Dt

Substituting the above relation in''' j

j

q Dh DpQ

Dt Dt x

and assuming Fouriers law of heat conduction to be valid, we obtain:

).(''' TkQDt

DpT

Dt

DTCp

Express enthalpy gradient in terms of pressureExpress enthalpy gradient in terms of pressure

and temperature gradientsand temperature gradients

Governing differential equation

for T


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Species Conservation. . . .

, , , , (all for species )i in i out i gen i stored im m m m

.ii i iu r

t

ir

This principle can be applied to a differential control volume in a

manner very similar to the continuity relation. The only new

consideration is that the mass of speciesican be created or destroyedvia chemical reactions, so that :

Where Is the generation rate ofiper unit volume

with units of3

kg of i

s m

Species velocity = velocity of i w.r.t. ground (this

velocity is a continuum average that includes the

effects of bulk flow and diffusion)

iu


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There is a constraint on these variables, namely :

1

for N speciesN

i

i

Next, we define several variables. The mass averaged velocity

of the mixture is

1 mass averaged velocity

N

i i

i

u

u

The quantity

1

N

i i

i

u u

the local rate at which mass passes through a unitcross sectional area perpendicular to .u


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the drift velocity of species i causes a flux which can be

expressed as :

drift component of species i (w.r.t. )i i i J u u u

Therefore ,

i i i iu J u

iJ

.i i i i J u r t

Substituting this into the expression for species conservation :


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From Ficks law for binary systems where the drift velocity is

due only to diffusion, we can write :

. .i i ij i iu D C r t

. . .i i i iu J rt

or

i i J D C


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Diffusive

SV

t

2).()(

General form of the Conservation EquationsGeneral form of the Conservation Equations

General conservative form

Unsteady Source

S Eqn

1 0 0 continuity

u x-mom

v y-mom

w z-mom

T k/Cp energy

C D r species

xbx

V

x

P

).(

3

yby

VyP

).(

3

zbz

V

z

P

).(

3

p

gen

pp Cq

CDtDpT

C

Advective

3Lecture Conservation Eqs

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