Quasi one dimensional flow equation

The equations which describe flow through a combustor within the constraints of a steady one-dimensional approximation will be developed in this section. The equations will also be applied to simpler flow situations in other propulsion subsystems, such as, nozzles, inlets, and turbo machinery cascades. The equations are particularly simple to deal with, yet they are commonly acknowledged to be reasonably accurate, even for rather complex problems. This attributes make the equation set to be developed particularly useful for preliminary design purposes. The analysis accounts for the following effects.

1. Changes in combustor cross-sectional area 2. Chemical reactions

3. Variations in gas molecular weight and specific heat 4. Exchange of heat with the surroundings

5. Drag caused by internal bodies or solid particles in the flow 6. Losses due to friction on the combustor walls

These equations are developed in such a manner that close correspondence with the combustor problems at hand is maintained without significant loss in generality. The format due to Shapiro is followed, where the basic equations are treated sequentially and then a final set of equations are developed from them.

4.1.1 Equation of State

We assume that the fluid flowing through the system behave as a perfect gas R Tu

p W

=

r

(26) The perfect gas law will be useful here in logarithmic form:[1]

dp d dT dW

p T W

r

= r +

-(27) Note that the state equation is often written in the form

0

Z R Tu

p W

= r

(28) Here the function Z is called the compressibility and measures the number of mols formed from an initial quantity of un reacted gas. In that sense it is equal to the ratio of the molecular weight before and after chemical changes take place, Z=W0/W. The logarithmic differential of this equation has a similar form to Equation 28. with W replaced by Z, that is [1].

dp d dT dZ

p T Z

r

= r + +

(29)

4.1.2 Speed of Sound

The speed of sound in a semi-perfect gas is given

( )

2 k u

s

kR T

a p C

r W

r r

æ¶ ö é ¶ ù

=çè¶ ÷ø =êë¶ úû= ⁽³⁰⁾

The quantity k is the isentropic exponent, and when the gas is both thermally and calorifically perfect, k reduces to g the ratio of sensible specific heats In logarithmic form the speed of sound is given as below [1] [33] [34].

1 2

da dk dT dW

a k T W

æ ö

= ç + - ÷

è ø ⁽³¹⁾

4.1.3 Mach Number

The Mach number is the ratio of the local fluid speed to the local sound speed

M V

= a

(32)

It will be convenient to deal with the square of the Mach number, which is given by

2 2

2 2

dM dV dW dk dT

M = V + W - k - T ₍₃₃₎

4.1.4 Conservation of Mass

The integral form of the mass conservation relation for steady flow is

ˆ 0

A

V ndA

r × =

ò

^r ₍₃₄₎

Here Vr

is the velocity vector A is the surface area of the control volume, which is fixed in space, and ˆn is the unit outward normal vector to the control surface.

Considering the fluid to be a continuum we may use Gauss’ divergence theorem to write the equation in differential form. In cylindrical coordinates, the conservation of mass equation may be written in differential form as follows [7].

(

rVr

) (

V

) (

rVz

)

⁰

r r r ^q z r

q

¶ + ¶ + ¶ =

¶ ¶ ¶ ⁽³⁵⁾

The quantities r,q, and z are the radial azimuthal and axial coordinates respectively and the velocity components in those directions are denoted by the corresponding coordinates ,while r denotes the fluid density. Multiplying by drdq and integrating from 0 to rw(q,z) and 0 to 2p, respectively, eliminates the middle term in Equation 35 leaving [1].

( ) ( )

2 2

0 0 0

0

rw

r w z

rV d rV drd

z

p p

r q + ¶ r q =

ò ò ò

¶ ₍₃₆₎

Integrating the second term in Equation 36 over r using Leibniz’rule yields

( ) ( ) ( )

2 2

0 0

0

w

r w z z w

rV rV dr rV dr d

z dz

p p

r r r q

é +ì¶ - üù =

ê í¶ ýú

ê î þú

ë û

ò ò

₍₃₇₎

The boundary condition at the wall for frictionless flow is

, , w r w

z w

dr V dz =V

(38)

Then the integrated mass conservation equation reduces to

( )

2

, ,

0 0 ,

0

rw

w w r w w

z w w r w

w w z w

r V dr

rV dr r V d

z r V dz

p r

r r q

r

é¶ æ öù

+ - =

ê¶ çç ÷÷ú

ê è øú

ë û

ò ò

₍₃₉₎

The second term in the square brackets is zero by Equation 39. and the remaining term is the rate of change of mass flow through the stream tube, so that

( )

2 0 0

0

rw

z

rV drd dm

z dz

p r q

¶ = =

¶

ò ò

₍₄₀₎

Thus the mass flow rate is constant, that is,

( )

2 0 0

rw

m rV drdz const

p

r q

=

ò ò

= ₍₄₁₎

In the quasi-one-dimensional approximation we assume that the variation of the stream tube cross-sectional area [1].

( )

^, ₁

rw z z q

¶ <<

¶ ⁽⁴²⁾

This suggests that it is reasonable to further assume that the flow properties vary little across the stream tube and that we may approximate the mass flow rate as follows

( ) ( ) ( ) ( )

m z =r z V z A z =const ₍₄₃₎

With these constraints in mind we may write the quasi one dimensional mass conservation equation in the following differential form.

dm d dV dA 0

m V A

r

= r + + =

(44)

For most problems concerning combustors, including the case of jet propulsion combustors which form an important part of the present study, the requirement of slowly varying cross-sectional area is closely met [1].

There are some systems, however, that contain relatively large changes in cross-sectional area or abrupt turns in the flow path. In such configurations the quasi-one-dimensional approximation would fail locally for some distance downstream of these flow path disturbances.

One example of such a case is a two-stage magnetohydrodynamics coal combustor in which a vertical gasifier and slag removal stage is mated to a horizontal combustor stage, causing a 90^o change in flow direction in a short distance. In more conventional combustors, like those in jet propulsion engines the flow path is smooth reasonably straight and characterized by small changes in flow area making the application of Equation 44 quite appropriate [1] [37].

4.1.5 Conservation of Energy

Assuming that the quasi-one-dimensional approximation serves to describe all pertinent variables as being constant across the stream tube we consider the conservation of mass to be expressed as follows,

( )

²

k 2 t

m dQ dW m dh dæ V ö mdh

- = ç + ÷=

è ø ⁽⁴⁵⁾

This equation is applied across the ends of a control volume removed from regions in which mixing occurs and in which dQ represents the incremental amount of heat per unit mass added to the stream from external sources by way of conduction, convection or radiation. The term dWk represents the incremental work per unit mass extracted from the stream and dht is the increment in the total enthalpy per unit mass of fluid flowing through the control volume [1] [33] [36].

The mass flow m appearing in Equation 45 is the sum of the oxidant and fuel flow rates and may be described as follows,

o f

m m= +m ₍₄₆₎

4.1.6 Conservation of Momentum

The integral form of the momentum equation for the control volume is

( ^ˆ )

^{b s}

A A

V n VdA F dF dA

u

dA

r × = r +

ò ò ò

r r r r

(47)

For the current quasi-one-dimensional application, the momentum conservation equation reduces to,

w w z

mdV = -Adp-t dA -dF ₍₄₈₎

The shear stress may be expressed in non dimensional form as the skin friction coefficient as defined by the following equation.

1 2

2

w

Cf

V t

= r

(49)

The wetted area may be related to the cross-sectional area by means of the hydraulic diameter D, where D/4 is the ratio of cross-sectional area to the wetted perimeter bounding that area, or [1].

4 dAw dz

A = D

(50)

Substituting these expressions in to the one-dimensional momentum equations yields

2

2 2

2 2

1 1

4 0

2 2 1

2

f z

dF

dp dV dz

kM kM C

p V D kpAM

é ù

ê ú

+ + ê + ú=

ê ú

ë û ⁽⁵¹⁾

4.2 The Equations of Motion in Standard Form

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