section 2: lecture 1 integral form of the conservation...

1MAE 5420 - Compressible Fluid Flow

Section 2: Lecture 1

Integral Form of the Conservation Equations

for Compressible Flow

Anderson: Chapter 2 pp. 41-54


Section 1 Review

• Equation of State:

• Relationship of Rg to specific heats,

• Internal Energy and Enthalpy

p = !RgT"> Rg =RuM

w

- Ru = 8314.4126 J/°K-(kg-mole)

- Rg (air) = 287.056 J/°K-(kg-mole)

cp= c

v+ R

g

h = e + Pv cv=

de

dT

!"#

$%&v

cp =dh

dT

!"#

$%&p

! =cp

cv


Chapter 1 Review (cont’d)

• First Law of Thermodynamics, reversible process

• First Law of Thermodynamics, isentropic process(adiabatic, reversible)

de= !pdv

de= dq ! pdv dh = dq+ vdp

dh = vdp


Chapter 1 Review (cont’d)

• Second Law of Thermodynamics, nonadiabatic,

reversible process

• Second Law of Thermodynamics, isentropic process(adiabatic, reversible)

s2! s

1= cp ln2

T2

T1

"

#$

%

&' ! Rg ln

p2

p1

"

#$

%

&'

p2

p1

=T2

T1

!

"#

$

%&

'' (1 p

2

p1

!

"#

$

%& =

'2

'1

!

"#

$

%&

(


Chapter 1 Review (concluded)

• Speed of Sound for calorically Perfect gas

• Mathematic definition of Mach Number

c = ! RgT

M =V

! RgT


Concept of a Control Volume

• Arbitrary Volume (C.V), fixed in space, with fluid flux across its boundaries

• Outer surface of C.V is known as Control Surface (C.S.)

• System defined as the Mass

Within the control volume at

Some Instant in time t

• Mass within control volume

Must OBEY Laws of mass

And momentum conservation

mout

•

min

•


Conservation of Mass• At any instant in time, the mass contained within the C.V. is conserved

• Within an elemental piece dv of the control volume

• And the total Contained mass is

!

!tm

C .V( ) = m

•

C .V .

"""in

# m

•

C .V .

"""out

dmC .V

= !dv

moutmout

•

min

•

dvmC .V

= !dvc.v.

"""


Conservation of Mass (continued)

• The mass flow out of the C.V. across an elemental piece of

the control surface, ds, is

• And the incremental

mass-flow into the C.V. is:

• Integrating over the entire

control surface

d m

•

out

!"

#$ = %V

outcos &( )ds

moutmout

•

min

•

dv

n

v

!

dS

m

•

C .V . = ! "V!>

• ds

!>#$

%&

C .S .

''

d m

•

in

!"

#$ = %&V

incos '( )ds

! "V!>

• ds

!>#$

%&

C .S .

'' =((t

"dvc.v.

'''#

$)%

&*


Conservation of Mass (concluded)

• One Dimensional Steady Flow (mass flow in = mass flow out)

!!t

"dvc.v.

###$

%&'

()= 0 = * "V

*>

• ds

*>$%

'(

C .S .

## + "1V1A1= "

2V2A2

“continuity equation”


Conservation of Momentum• Newton’s second law applied to control volume

• Momentum flow across control surface

• But from earlier analysis, across an incremental

Surface area

F

!>

" =d

dtmV

!>#$

%& =

Momentum flow across C.S. +

Rate of Change of Momentum within C.V.

vn

V

n

V

m

•!"

#$

C .S .

%% V

&>

m

•

= !V">

• ds

">

# m

•$%

&'

C .S .

(( V

">

= !V">

• ds

">$%

&'

C .S .

(( V

">


Conservation of Momentum (cont’d)

• Rate of Momentum Change within Control Volume

• But from earlier (continuity) analysis

For an incremental volume, dv

• And total net rate of momentum change within the control volume is

dVn

V

n

V

!

!t(mV

">

)

m = !dv

!!t(mV

">

)C .V .

=!!t

#dvV">$

%&'

C .V .

((( =!!t

#V">$

%&' dv

C .V .

(((



• Then Newton’s second law becomes

• Resolution of Forces Acting on Control Volume

- Body Forces (electo-magnetic, gravitational, buoyancy etc.)

- Pressure Forces

- Viscous or frictional Forces

F

!>

" =d

dtmV

!>#$

%& = 'V

!>

• ds

!>#$

%&

C .S .

(( V

!>

+))t

'V!>#

$%& dv

C .V .

(((

F!>

"#$%&body

= ' fb!>

dvC .V .

(((

F!>

"#$%&pressure

= ! p( )C .S .

'' dS!> (Minus sign

Because pressure

Acts inward)

F!>

"#$%&fric

= | f!>

' dS!>#

$%&

C .S .

(( |dS

SC .S .

!>


Conservation of Momentum (continued)

• Collected terms for conservation of Momentum

• Steady, In-viscous

Flow, Body Forces negligible

! fb">

dvC .V .

### " p( )C .S .

## dS">

+ | f">

$ dS">%

&'(

C .S .

## |dS

SC .S .

">

=

!V">

• ds">%

&'(

C .S .

## V">

+))t

!V">%

&'( dv

C .V .

###

! fb">

dvC .V .

### " p( )C .S .

## dS">

+ | f">

$ dS">%

&'(

C .S .

## |dS

SC .S .

">

=

!V">

• ds">%

&'(

C .S .

## V">

+))t

!V">%

&'( dv

C .V .

### ! p( )C .S .

"" dS!>

= #V!>

• ds!>$

%&'

C .S .

"" V!>



• One-Dimensional Flow

! p( )C .S .

"" dS!>

= #V!>

• ds!>$

%&'

C .S .

"" V!>

p1

p2

A2

A2

V1

V2

• No flow across solid boundary

! p2A2! p

1A1( ) i_

x = "2V2A2V2i_

x! "1V1A1V1i_

x

# p1+ "

1V1

2( )A1 = p2+ "

2V2

2$% &'A2


Conservation of Momentum (concluded)

• One-Dimensional Flow

! p( )C .S .

"" dS!>

= #V!>

• ds!>$

%&'

C .S .

"" V!>

p1

p2

A2

A2

V1

V2

• No flow across solid boundary

p1+ !

1V1

2( )A1 = p2+ !

2V2

2"# $%A2 &

! j =pj

RgTj

=' pj' RgTj

=' pjcj2& p

11+ '

V1

2

c1

2

(

)*+

,-A1= p

21+ '

V2

2

c2

2

(

)*+

,-A2

p2

p1

=A1

A2

1+ 'M1

2( )1+ 'M

2

2( )


Conservation of Energy

• From First law of thermodynamics (reversible process)

(corresponding to inviscid fluid)

Rate of change of fluid energy as it flows thru C.V.

Rate of external heat addition to C.V.

Rate of work done to fluid inside of C.V.

dE

dt=dQ

dt+dW

dtdE

dt!

dQ

dt!

dW

dt!


Conservation of Energy (cont’d)

Rate of change of fluid energy as it flows thru C.V. •

Earlier thermodynamic discussions were for static fluid… flow thru C.V. is not static

… Kinetic Energy terms must also Be captured … following process similar to

Momentum flow through C.V.

dE

dt!

dE

dt=

!!t

" e +V2

2

#

$%

&

'( dv

#

$%%

&

'((

C .V .

))) + "V*>

• d S

*>

e +V2

2

#

$%

&

'(

C .S .

))

Rate of change of energy with C.V Energy Flow Across C.S.



Rate of heat addition to C.V. •dQ

dt! ! q

•"#

$% dv

"#&

$%'

C .V .

(((

• dW

dt! Rate of work done to fluid inside of C.V.

dW

dt!

d

dt(F • dx) = F •V ! dx = direction " of " motion

Consider: 1) Pressure Forces, 2) Body Forces



1) Pressure Forces

2) Body Forces

! (pd S!>

) •V!>

C .S .

""

(! f">

dv) •V">

C .V .

###

MAE 5420 - Compressible Fluid Flow 20


• Collected Energy EquationScalar equation


Conservation of Energy (concluded)

• Steady, Inviscid quasi 1-D Flow, Body Forces negligible

1-D eqn.



• Steady, Inviscid quasi-1-D Flow, Body Forces negligible



• Steady, Inviscid quasi 1-D Flow, Body Forces negligible, collected

• Divide through by massflow



• but from continuity, equation of state



• but from continuity, equation of state

h = e + Pv = e +RgT

Combine internal energy and thermal energy into enthalpy


Summary

• Continuity (conservation of mass)

• Steady quasi One-dimensional Flow


Summary (continued)

• Newton’s Second law-- Time rate of change of momentumEquals integral of external forces



Summary (concluded)

• Conservation of Energy--


section 2: lecture 1 integral form of the conservation...

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