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4.7 REAL NOZZLE

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Ideal Rocket

 Working fluid is homogeneous perfect gas,

 No heat transfer (q=0, adiabatic),

 No frictional loss, no boundary layer loss,

 No shocks,

 Invariant gas composition in nozzle,

 Steady flow,

 One-dimensional flow, i.e., flow is axial and properties are constant across any plane normal to flow,

 Chemical equilibrium in combustion chamber.

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Real Nozzle Effects

 Stagnation pressure loss in the chamber:  Non-isentropic flow, including heat and mass transfer, friction,

 Two-dimensional flow (divergence, varying properties),  Boundary layer (BL) and wall friction:

 Lower velocity in BL: effects include pressure gradient, heat transfer, wall roughness, nozzle geometry.

 Multi-phase flow:  liquid drops and solid particles have higher density (thus lower velocity),  momentum transfer from gas to large drops also slows gas down.

 Unsteady flow  Nozzle flow chemical kinetics:

 Re-association of relatively unstable (high positive heat of formation) molecules as gas cools in the nozzle.

 Throat erosion leading to lower expansion ratio,  Non-uniform properties:

 mixing loss can be a major effect,

 Real gas (not perfect gas) properties,  Non-optimal expansion.

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 Flow in Ideal Rocket Nozzles:  One-dimensional,

 Isentropic.

 Real Nozzles:  Flow is never truly one-dimensional Shape of nozzle walls is important.

 Entire nozzle shape must into account variations in velocity and pressure on surfaces normal to streamlines.

 Other influences on flow:  Friction,

 Heat transfer,

 Composition change,

 Shocks.

 Shape of the supersonic or divergent part of the nozzle will dictate shock formation and performance gain/loss.

Area Ratio is only important geometric variable.

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Nozzle Contours

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CONICAL NOZZLE

 Shape

 Thrust:  Momentum Equation:

 Exit Velocity:

 Exit Area – projected:

 Exit Area – spherical:

 Mass Flow Rate:

F

x∑ = T + pa− p

e( )Ae= ρ (v ⋅ n)v

xdA

CS∫

pa pe

r

CS

T

v

e

α

v ⋅ n = v

e ve,x= v

ecosφ

R

φ dφ

Ae= π r2

dA = 2π R ⋅R sinφ dφ

m = ρ v Asph

= ρ ve⋅2π R2 (1− cosα )

Asph= 2π R2 (1− cosα )

Asph

Ae

= 21+ cosα

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CONICAL NOZZLE

 Thrust:  Conical Nozzle:

 Ideal Nozzle:

 Thrust Loss due to Divergence Loss:

 Small difference between Ae and Asph

 Contribution of pressure term small

 Exit pressure does not have any directional influence as exit velocity

 Area Ratio:

 Nozzle Length:

T

conic= 1+ cosα

2m v

e+ (p

e− p

a)A

sph⎡⎣ ⎤⎦

Tisentr ,1−d= m v

e+ (p

e− p

a)A

e

Tconic,approx= λ m v

e+ (p

e− p

a)A

e

Asph= A

e

Ae

A *= D *+2L tanα

D *

⎛⎝⎜

⎞⎠⎟

2

L = D *2

Ae

A *−1

⎛

⎝⎜

⎞

⎠⎟ ⋅ tan−1 α

ve,conic

visentr,1−d( )

e

= λ =1+ cosα

2

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Conical Nozzle

 General Observations:

 Conical nozzle contour is the most simple contour.

 It has high divergence losses – lower angle will reduce divergence losses, but   longer nozzle (for same expansion) is heavier,

  frictional and boundary layer losses will be greater.

 Effective divergence loss accounted by λ applied to momentum thrust term.

 Serves as reference contour for the length definition of profiled nozzles (profiled nozzle length is typically 75% – 85% of conical length with same ε)

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Perfect Nozzle

 The Perfect Nozzle is shaped in such a manner as to provide uniform parallel flow at the exit plane.   Complete elimination of flow divergence loss!!

 This perfect case is not a practical case, and produces very long nozzles, therefore not used for propulsion application.

 The “Method of Characteristics” is used to analytically determine the contour needed to achieve ideal (uniform parallel) flow conditions at the exit plane.

 The “Method of Characteristics” is widely used to determine nozzles with practical contours.

 Designing a shaped nozzle requires 2–dimensional flow.

 The curvature of the streamlines is significant, so that gradients of velocity and pressure perpendicular to streamlines become important.

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Supersonic Nozzle Design

 Objective of Design:  Development of nearly 1–dimensional flow at nozzle exit while minimizing

pressure loss.

 Design Approach/Philosophy:  To generate a wave–free flow downstream, reflected wave has to be eliminated.

 The design of the opposing wall is such as to “cancel” the incident wave by forming parallel surface to resulting velocity vector.

 Far downstream nozzle contour is a result of the initial expansion just downstream of the throat.

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Oblique Shocks and Expansion Waves

 Concave Corner  Convex Corner

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Method of Characteristics

 Background:  In supersonic flow, the influence of a small pressure disturbance is limited to a

specific region.  Pressure disturbance propagates relative to fluid as a spherical sound wave at local

velocity of sound a.

 Center of sound wave moves downstream with velocity u.

 Changes in fluid properties may be thought of as propagating along Mach lines:   Mach line is straight, if flow upstream is uniform.

  All properties of flow immediately downstream of a Mach line are uniform.

A

Source of small pressure disturbance

B Uniform parallel supersonic flow

Limit of Influence

Mach Angle

Zone of Silence

v = a t

d = u t

u

α = sin−1 at

ut

⎛⎝⎜

⎞⎠⎟= sin−1 1

M

⎛⎝⎜

⎞⎠⎟

α = tan−1 at

ut( )2− at( )2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= tan−1 1

M2 −1

⎛

⎝⎜⎞

⎠⎟

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Method of Characteristics

 Expansion at Infinitesimal Corner:  Velocity change due to an expansion:

 Change in Mach number is related to change in direction of streamline (for isentropic flow):

 Change in Mach number determines temperature, density, and pressure.  See handout about Method of Characteristics.

 Wall curvature controls the flow field downstream of Mach lines.

dUU

= dθ

M2 −1

dM2 =

2M2 1+ 0.5 γ −1( )M2⎡⎣ ⎤⎦M2 −1

dθ

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Method of Characteristics

 Intersection of Mach Lines:  Streamlines upstream of O (blue region) and downstream of O (red region) must be

parallel!!

Mach number must be uniform!

 Knowing M, MM, MM, MM, MM will determine flow immediately downstream of O.

 Conditions:

(i) δθ1−δθ

2= δθ

3−δθ

4

(ii) δM1+δM

2= δM

3+δM

4

(iii) δθ2= m

1δM

2

(iv) δθ4= m

3δM

4

δθ1, δθ3, δM1, δM3

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Method of Characteristics

 The initial expansion occurs inside “1AI5”.

 At “I” the design Mach number is reached.

 The flow downstream of the left running characteristic “IP” is uniform and parallel.

 The contour “AP” is calculated with MoC, such that incoming expansion waves are compensated.

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Contour Design

 Design of a Parabolic Contoured Nozzle  In 1960, G.V.R. Rao proposed a simple optimization method for nozzle design –

provides close approximation to a thrust-optimized contour.

 G.V.R. Rao, “Approximation of Optimum Thrust Nozzle Contour,” ARS Journal, Vol. 30, No. 6, June 1960, p. 561

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Contour Design

 Influence of Nozzle Design on Performance

95

96

97

98

99

100

50 55 60 65 70 75 80 85 90 95 100

Conical nozzle, 15-degree

bell nozzle, eps=10

bell nozzle, eps=20

bell nozzle, eps=30

bell nozzle, eps=40

c

Nozzle length / Length of conical 15o-nozzle · 100

l

Typical length of bell nozzle: 75%-85% of length of conical nozzle

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Why Altitude-Compensating Nozzles?

 Example:  Conventional TCA performance characteristic vs. flight altitude based on ideal gas

analysis.

 ε = 45, p0=100 bar, ϒ = 1.2, MW = 22kg/kmol

1

2

3

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Altitude-Adjusting Nozzles

Plug nozzle (“Aerospike”)

Bell nozzle

Dual-bell nozzle

Extendible nozzle

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Altitude-Adjusting Nozzles

 Extendible nozzles are being used on the RL-10 and Japanese upper stage engines.

 Detailed nozzle design and mechanical design of a reliable deployment mechanism are key.

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Altitude-Adjusting Nozzles

 Altitude adjusting nozzles expand at free surface

Plug Cluster Nozzle

Linear Aerospike Nozzle

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Altitude-Adjusting Nozzles

 Truncated Aerospike nozzles offer improved mission-averaged performance, shorter lengths, TVC, and improved structural efficiency.

 Clustering losses and inter-thruster interactions, end-wall effects, base region flow and heat transfer need development

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Performance Definition

 ‘Four’ types (according to Sutton):  Theoretical performance (based on calculations, loss types specified) at operating

conditions,

 Delivered (actually measured),  Performance at standard conditions:

 p0=1000 psia, optimally expanded at SL or stipulated e in vacuum,

 Propellant combination, not propulsion system, performance,

 Guaranteed minimum performance.

 Associated conditions must be clearly defined:  Chamber and ambient pressures (SL or vacuum),

 Nozzle geometry ( , , etc.),

 Propellants and propellant conditions (T, composition, O/F),

 Type of thermochemical analysis (equilibrium chemistry or invariant composition during nozzle flow).

! !

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Real Nozzle Effects

 Stagnation pressure loss in the chamber:  Non-isentropic flow, including heat and mass transfer, friction,

 Two-dimensional flow (divergence),  Boundary layer and wall friction:

 Lower velocity in BL: effects include pressure gradient, heat transfer, wall roughness, nozzle geometry,

 Multi-phase flow:  liquid drops and solid particles have higher density (thus lower velocity),  momentum transfer from gas to large drops also slows gas down.

 Unsteady flow  Nozzle flow chemical kinetics:

 Re-association of relatively unstable (high positive heat of formation) molecules as gas cools in the nozzle.

 Throat erosion leading to lower expansion ratio,  Non-uniform properties:

 mixing loss can be a major effect,

 Real gas (not perfect gas) properties,  Non-optimal expansion.

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4.8 SUPPLEMENT - TABLES