Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

Dr. R. Nagarajan

Professor

Dept of Chemical Engineering

IIT Madras

Advanced Transport PhenomenaModule 2 Lecture 7

Conservation Equations: Alternative Formulations

Conservation Equations: Alternative Formulations

EQUIVALENCE BETWEEN LAGRANGIAN & EULERIAN FORMULATIONS

To express this relation, we introduce here the

notion that each field quantity f(x, t) ( including

vectors) possesses a local spatial gradient defined

such that the projection of the vector grad f in any

direction gives the spatial derivative of that scalar f

in that direction; thus,1

( ). .( ,....)

ff

h

grad e3

EQUIVALENCE BETWEEN LAGRANGIAN & EULERIAN FORMULATIONS CONTD…

Where is the unit vector in the direction of

increasing coordinate is the length

increment associated with an increment in the

coordinate

e

and h( ,....)

4


If we define the local material derivative of f in the following reasonably way:

and expand in terms of f(x, t) using a Taylor series about x, t, that is

f ( t ,t t ) x v

0

( . ) ( , )t

Df f t t t f tLim

Dt t

x v x

.( . ) ( , ) ( ) . ...,

xx,

x v x grad vt

t

ff t t t f t f t t

t

5

This valuable kinematic interrelation17 now allows

each of the above-mentioned primitive

conservation equations to be re-expressed in an

equivalent Eulerian form.

6



Then it follows from equation

that an observer moving in that local fluid velocity

v(x, t) will record:.

Df ff

Dt t

v grad

7

0

( . ) ( , )t

Df f t t t f tLim

Dt t

x v x


In particular, each of the local species mass and

element. Mass balance equations

can now be expressed:''

( ) ''( ) ( )

''' ( 1, 2,...., )

( 1,2,...., ),

v.grad j

v.grad j

ii i i

kk k elem

div r i Nt

div k Nt

8

( ) '''( )

''' ( 1, 2,...., )

( 1,2,...., )

ij

j

ii

kk elem

Ddiv r i N

DtD

div k NDt


Note that there is a nonzero species i mass

convective term only when the vectors and

are both perpendicular, there is no

convective contribution to the species i mass

balances despite the presence of total mass

convection

igrad

v

9

The local material derivative also provides a

convenient “shorthand” for making changes in

dependent variables, as shown below. The

distributive property of differentiation makes it clear

that if we derive an equation for and

subtract it from the equation for we

can construct an equation for .

2 2 D v Dt

2 2 D e v Dt

De Dt

ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE

10

The latter could then be used to generate an

equation for by the addition of ,

etc.

If the linear- momentum conservation (balance) Eq. is

multiplied, term by term, by v (scalar product ), we

obtain the following equation for

Dh Dt D Dt

2 2 D v Dt

ALTERNATIVE FORMS OF ENERGY CONSERVATION PDE CONTD…

v . vD Dt

11

Subtracting this from the equation of energy conservation ,

Eqn.

permits us to write

where we have introduced the short hand notation

2

12 iv . Π v . gN

ii

D vdiv

Dt

1

iq''+q''' Π:grad v j ''.gN

ii

Dediv

Dt

Π:grad v (Π.v) - v . Πdiv div


12

2

1

' .2 iq''+q'' Π v ''.g

N

ii

D ve div div m

Dt

Equation therefore governs the rate of change of the

specific internal energy of a fluid parcel

By adding

to both the

RHS and lhs of this equation , we obtain an equation

governing the rate of change of specific enthalpy

of a fluid parcel: h e p

(D p Dt Dp Dt p Dp Dt

vDp Dt p div


13

This equation can be simplified by rewriting it in

terms of that part of the contact stress(T) left after

subtracting the local thermodynamics pressure

1

iq''+q''' v Π:grad v j ''.g

N

ii

Dh Dpdiv pdiv

Dt Dt


14

In terms of this so–called “extra stress,” Eq.

becomes:

1

. iq''+ ''' T grad v j ''.g

N

ii

Dh Dpdiv q

Dt Dt


15

1

iq''+q''' v Π:grad v j ''.g

N

ii

Dh Dpdiv pdiv

Dt Dt

At this point we reiterate that the specific enthalpy,

, of the mixture includes chemical

(bond-energy) contributions, and must be calculated

from a constitutive reaction of the general form:

where the values are the partial specific

enthalpies in the prevailing mixture.

h e p

1 2.1

. , , , ,.... ,N

i ii

h h T p

ih


16

For a mixture of ideal gases this relation simplifies considerably

to :

Alternatively, in terms of mole fractions

1 1

( ). ( ) .

N Ni

i i ii i i

H Th h T

M

1

1

.( )N

i ii

N

i ii

y H TH

hM

y M


17

Here the are the “absolute” molar enthalpies of the pure constituents, that is,

Where is the molar “heat of formation’’ of species i; that is,

1,2,...,iH i N

, ,

( ) ( )

( ) ( ) ,ref

i i ref

T

i f i ref p i

T

H T H T

H T H T C dT

,f i refH T


18

the enthalpy change across the stoichiometric in

which one mole of species i as formed from its

constituent chemical elements in some (arbitrarily

chosen) reference states (e.g. H2(g), O2(g) and

C (graphite) at Tref = 298 K.


19

Because of the (implicit) inclusion in of the “heat of

formation’’ in h, the local energy addition term

appearing on the RHS of the PDE, (Eq.)

is not associated with chemical reactions (this would

give rise to a “double-counting” error).

q


20

1

.N

iit

Dh Dpdiv

Dt D

iq +q T grad v j .g

An explicit chemical-energy generation term enters

energy equations only when expressed in terms of a

“sensible-” (or thermal-) energy density dependent

variable, such as

(or T itself).

, ,ref

T

p mix

T

C dT


21

Using Eqs

and

a PDE for is readily derived

(Eq. ),

,p mixc DT Dt


22

1

N

ii

Dh Dpdiv pdiv

Dt Dt

iq +q v Π:grad v j .g

1 1

( ). ( ) .

N Ni

i i ii i i

H Th h T

M

'' 2

0( ) sin

2w w wq q d d

and its RHS indeed contains (in addition to ) the

explicit “chemical”-energy source term:

Where

q

,

N

i=1i ir h

i i ih T H T M


23

Finally, addition of the equation for

allows us to construct the following PDE for the

“total” enthalpy

2 2D v Dt

2 2oh e p v

0

1

N

ii

Dh pdiv q div

Dt t

iq + T . v m .g


24

MACROSCOPIC MECHANICAL-ENERGY EQUATION (GENERALIZED BERNOULLI

EQUATION)

A widely used macroscopic “mechanical” energy

balance can be derived from our equation for

(Eq. ) by combining

term-by-term volume integration, Gauss’ theorem,

and the rule for differentiating products.

2 2D v Dt

25

2

1

.2 iv Π v.g

N

ii

D vdiv

Dt


EQUATION) CONTD…

We state the result for the important special case

of incompressible flow subject to “gravity” as the

only body force, being expressible in terms of the

spatial gradient of a time independent potential

function , that is

x

g = -grad

26


EQUATION) CONTD…

Then for any fixed macroscopic CV:

The corresponding results for a variable-density fluid

(flow) (Bird, et al. (1960)) are rather more complicated

than Eq. above, and not, exclusively, “mechanical” in

nature.

2 2

2 2

.( )

V S

V S

v p vdV dA

t

dV dA

v. n

T.grad v v T.n

27


EQUATION) CONTD… In contrast to Eq.

note that Eq. (from 59)

makes no reference to changes in thermodynamic internal

energy, nor surface or volume heat addition

hence the name “mechanical” energy equation.28

2 2

''

1

''.2 2

''. . .

V S s

N

i iV S Vi

v ve dV e dA dA

t

q dV dA dV

v. n q n

Π.n v m g

. ,q n dA q dV

Common applications of Eq.

are to the cases of:

a.Passive steady-flow component (pipe length,

elbow, valve, etc.) on the control surfaces of

which the work done by the extra stress can be

neglected.


EQUATION) CONTD…

29

2 2

2 2

.( )

v. n

T.grad v T.n

V S

V S

v p vdV dA

t

dV v dA

Then there must be a net inflow of

to compensate for the

volume integral of T : grad v, a positive quantity

shown in Section to be the local irreversible

dissipation rate of mechanical energy ( into heat).

2 2p v


EQUATION) CONTD…

30

b.Steady-flow liquid pumps, fans, and turbines,

relating the work required for unit mass flow to the

net outflow of . 2 2p v


EQUATION) CONTD…

31

For (b), in cases with a single inlet and single outlet

Eq. may be rewritten in the “ engineering form”.

2 2

1 2

2

2 2

. Re,2

shaft

exit

all fluid filledeachportionsection

W p v p v

m

vK shape


EQUATION) CONTD…

32


EQUATION) CONTD…

Here the indicating sum (RHS) accounts for all

viscous dissipation losses in fluid-containing portions

of the system ( other than those contained in the

“excluded” “pumping-device” shown in Figure), and

, given by:

is the rate at which the mechanical work is done on the fluid by the indicated pumping Device.

shaftW

portion of fixedCS enveloping thepumping device

. pI . dA shaftW v T n

33

Note that the work requirement per-unit-mass-

flow is the sum of that required to change

and that required to overcome

the prevailing viscous dissipation losses

throughout the system. With a suitable change in

signs, this equation can clearly also be used to

predict the output of a turbine system for power

extraction from the fluid.

[ p

2 2v


EQUATION) CONTD…

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Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

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