Introduction of compressible flow - FKM

79 downloads 499 Views 552KB Size Report
pressure and temperature. In general, gases are highly compressible and liquids have a very low compressibility. Part one : Introduction of Compressible Flow. 1 ...
COMPRESSIBLE FLOW

COMPRESSIBLE FLOW Introduction The compressibility of a fluid is, basically, a measure of the change in density that will be produced in the fluid by a specific change in pressure and temperature. In general, gases are highly compressible and liquids have a very low compressibility.

Part one : Introduction of Compressible Flow

1

COMPRESSIBLE FLOW

Application ; Aircraft design Gas and steam turbines Reciprocating engines Natural gas transmission lines Combustion chambers Compressibility effect ; Supersonic – the flow velocity is relatively high compared to the speed of sound in the gas. Subsonic

Part one : Introduction of Compressible Flow

2

COMPRESSIBLE FLOW

Fundamental assumptions 1. The gas is continuous. 2. The gas is perfect (obeys the perfect gas law) 3. Gravitational effects on the flow field are negligible. 4. Magnetic and electrical effects are negligible. 5. The effects of viscosity are negligible. Applied principles 1. Conservation of mass (continuity equation) 2. Conservation of momentum (Newton’s law) 3. Conservation

of

energy

(first

law

of

thermodynamics) 4. Equation of state

Part one : Introduction of Compressible Flow

3

COMPRESSIBLE FLOW

Perfect gas law :

P

ρ

= RT

P : Pressure

ρ : Density

R : Universal gas constant

Rair = 287.04( kgJ⋅ K ) T : Temperature

Part one : Introduction of Compressible Flow

4

COMPRESSIBLE FLOW

Conservation laws : Conservation of mass Rate of increase of mass of fluid in control volume

=

Rate mass enters control volume

_

Rate mass leaves control volume

Conservation of momentum : Net force on gas in control volume in direction considered

=

Rate of increase of momentum in direction considered of fluid in control l

_

Part one : Introduction of Compressible Flow

+

Rate momentum leaves control volume in direction considered

Rate momentum leaves control volume in direction considered

5

COMPRESSIBLE FLOW

Conservation of energy : Rate of increase in internal energy and kinetic energy of gas in control volume

=

+

Rate enthalpy and kinetic energy leave control volume

Rate heat is transferred into control volume

Part one : Introduction of Compressible Flow

_

_

Rate enthalpy and kinetic energy enter control volume

Rate work is done by gas in control volume

6

COMPRESSIBLE FLOW

Definition : A control volume is a volume in space (geometric entity, independent of mass) through which fluid may flow Enthalpy H, is the sum of internal energy U and the product of pressure P and volume V appears.

H = U + PV

Part one : Introduction of Compressible Flow

7

COMPRESSIBLE FLOW

COMPRESSIBLE FLOW Introduction

Many of the compressible flows that occur in engineering practice can be adequately modeled as a flow through a duct or streamtube whose cross-sectional area is changing relatively slowly in the flow direction.

A duct is a solid walled channel, whereas a streamtube is defined by considering a closed curve drawn in a fluid flow.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

8

COMPRESSIBLE FLOW

Quasi-one-dimensional flow is flows in which the flow area is changing but in which the flow at any section can be treated as one-dimensional.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

9

COMPRESSIBLE FLOW

CONTINUITY EQUATION

The continuity equation is obtained by applying the principle of conservation of mass to flow through a control volume.

One-dimensional flow is being considered.

There is no mass transfer across the control volume. The only mass transfer occurs through the ends of the control volume.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

10

COMPRESSIBLE FLOW

Mass enters through the left hand face of the control volume be equal to the rate at which mass leaves through the right hand face of the control volume.

m& 1 = m& 2 We know that

m& = ρVA

We considered ;

ρ1V1 A1 = ρ 2V2 A2 For the differentially short control volume indicated,

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

11

COMPRESSIBLE FLOW

above equation gives ;

ρVA = ( ρ + dρ )(V + dV )( A + dA) Neglecting higher order terms, we found ;

VAdρ + ρAdV + ρVdA = 0 Dividing this equation by

ρVA

then gives ;



dV dA + + =0 V A ρ

This equation relates the fractional changes in density, velocity and area over a short length of the control volume.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

12

COMPRESSIBLE FLOW

MOMENTUM EQUATION (Euler’s equation)

The flow is steady flow.

Gravitational forces are being neglected.

The only forces acting on the control volume are the pressure forces and the frictional force exerted on the surface of the control volume.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

13

COMPRESSIBLE FLOW

The net force on the control volume in the x-direction is ; PA − ( p + dP )( A + dA) + 12 [( P + ( P + dP )][( A + dA) − A] − dFµ

Note : dx is too small, dPdA have been neglected. Mean pressure on the curved surface can be taken as the average of the pressures acting on the two end surfaces. dFµ is the frictional force.

Rearranging above equation, we found the net force on the control volume in the x-direction is ;

− AdP − dFµ

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

14

COMPRESSIBLE FLOW

Since the rate at which momentum crosses any section of the duct is equal to

m& V , we found that ;

ρVA[(V + dV ) − V ] = ρVAdV

The above equation can be written as ;

− Adρ − dFµ = ρVAdV Frictional force is assumed to be negligible. The Euler’s equation for steady flow through a duct becomes;



dP

ρ

= VdV

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

15

COMPRESSIBLE FLOW

Integrating Euler’s equation ;

V2 dP +∫ =C 2 ρ

(For compressible)

And if density can be assumed constant, Euler’s equation become ;

V2 P + =C 2 ρ

(For incompressible)

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

16

COMPRESSIBLE FLOW

STEADY FLOW ENERGY EQUATION

For flow through the type of control volume considered as before, we found ;

V22 V12 h2 + = h1 + +q−w 2 2 h = enthalpy per mass V = velocity q = heat transferred into the control volume per unit mass of fluid flowing through it w = work done by the fluid per unit mass

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

17

COMPRESSIBLE FLOW

Assumption ; No work is done, w=0 Perfect gases is considered,

h = c pT

Steady flow energy equation ;

V22 V12 c pT2 + = c pT1 + +q 2 2 Applying this equation to the flow through the differentially short control volume gives ;

V2 (V + dV ) 2 c pT + + dq = c p (T + dT ) + 2 2

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

18

COMPRESSIBLE FLOW

Neglecting higher order terms gives ;

c p dT + VdV = dq This equation indicates that in compressible flows, changes in velocity will, in general, induce changes in temperature and that heat addition can

cause

velocity

changes

as

well

as

temperature changes. If the flow is adiabatic i.e., if there is no heat transfer to of from the flow, it gives ;

V22 V12 = c pT1 + c pT2 + 2 2

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

19

COMPRESSIBLE FLOW

Steady flow energy equation for adiabatic flow becomes ;

c p dT + VdV = 0 This equation shows that in adiabatic flow, an increase in velocity is always accompanied by a decrease in temperature.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

20

COMPRESSIBLE FLOW

EQUATION OF STATE

When applied between any two points in the flow ;

P1 P2 = ρ1T1 ρ 2T2 When applied between the inlet and the exit of a differentially short control volume, this equation becomes ;

P P + dP = ρT ( ρ + dρ )(T + dT )

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

21

COMPRESSIBLE FLOW

Higher order terms are neglected and it gives ;

P P (1 + = ρT ρT

dP P

)(1 − dρρ )(1 − dTT )

dP dρ dT − − =0 P T ρ This equation shows how the changes in pressure, density and temperature are interrelated in compressible flow.

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

22

COMPRESSIBLE FLOW

ENTROPY CONSIDERATIONS

In studying compressible flows, another variable, the entropy, s, has to be introduced. The entropy basically places limitations on which flow processes are physically possible and which are physically excluded. The entropy change between any two points in the flow is given by ;

s2 − s1 = c p ln

Since

[ ]− R ln[ ] T2 T1

P2 P1

(1)

R = c p − cv , this equation can be written;

⎡ s2 − s1 = ln ⎢ cp ⎣

( )( ) T2 T1

P2 P1



γ −1 γ

⎤ ⎥ ⎦

If there is no change in entropy, i.e., if the flow is

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

23

COMPRESSIBLE FLOW

isentropic, this equation requires that :

T2 ⎛ P2 ⎞ = ⎜⎜ ⎟⎟ T1 ⎝ P1 ⎠

γ −1 γ

hence, since the perfect gas law gives ;

T2 P2 ρ1 = T1 P1 ρ 2 it follows that in isentropic flow :

P2 ⎛ ρ 2 ⎞ = ⎜⎜ ⎟⎟ P1 ⎝ ρ1 ⎠

γ

in isentropic flows, then

P

ρ γ is a constant.

If equation (1) is applied between the inlet and the exit

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

24

COMPRESSIBLE FLOW

of a differentially short control volume, it gives ;

( s + ds ) − s = c p ln[T +TdT ] − R ln[ P +PdP ]

neglecting small value, the above equation gives;

ds = c p

dT T

− R dPP

(2)

which can be written as ;

ds dT ⎛ γ − 1 ⎞ dP ⎟⎟ = − ⎜⎜ cp T ⎝ γ ⎠ P lastly, it is noted that in an isentropic flow, equation (2) gives;

RT c p dT = dP P using the perfect gas law ;

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

25

COMPRESSIBLE FLOW

c p dT =

dP

ρ

(3)

but the energy equation for isentropic flow, i.e., for flow with no heat transfer, it gives ;

c p dT + VdV = 0 which using equation (3) gives ;

dP

ρ

+ VdV = 0

Part two : The Equation of Steady One-Dimensional Compressible Fluid Flow

26