Indian Indian Institute Institute of Sp Spa ace Scienc Science e & Techn Technolo ology gy
Aerosp Aerospace ace Vehicle Vehicle Design-Team Design-Team 6
Hypersonic Cruise Vehicle Design
Authors:
Kush Sreen Maneesh Kumar Yadav Rohit Potla Abhishek Singh Sanjeev Yadav
Contents 1 Introduction
4
2 Design Ob jectives
5
3 Literature Survey
6
3.1 3.2 3.3 3.4 3.5
Introduction . . . . . . . Weight and Sizing . . . . Aero dynamics . . . . . . Propulsion . . . . . . . . Structures and Materials
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4 Aerodynamics
4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
9
Introduction . . . . . . . . . . . . . . . Conical Sho cks . . . . . . . . . . . . . Hyper personic Small Disturbance Theory Waverider Construction . . . . . . . . Aero dynamic Coefficients . . . . . . . 3-D CFD simulation of Body . . . . . . Vertical Fin . . . . . . . . . . . . . . . Aero dynamic Heating . . . . . . . . . .
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5 Propulsion
5.1 5.2 5.3 5.4
Combustion Chamber . . . . . Nozzle design . . . . . . . . . Offset design and Conclusion . CFD Results of inlet . . . . .
9 9 10 11 12 14 17 18 21
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6 Weight and Sizing
7 Structures and Material Selection
29 32
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Material Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Structural and Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 8 Fuel Selection
Introduction . . . . . . . . . . . . . Hydrogen Fuel of the Future? . . . Range in Cruise . . . . . . . . . . . Fuel as Thermal Protection System
23 25 25 27 29
6.1 The HASA technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1 8.2 8.3 8.4
6 6 6 7 8
32 32 33 36
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9 References
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36 36 37 37 38
1
List of Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Waverider and its Sho ck Cone. . . . . . . . . . . . . . . . . . . . . . . . . . Ramjet Engine and Inlet Ramps . . . . . . . . . . . . . . . . . . . . . . . . Supersonic flow past a Cone . . . . . . . . . . . . . . . . . . . . . . . . . . Generating Cone and Waverider . . . . . . . . . . . . . . . . . . . . . . . . Final Waverider Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesh of the Waverider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mach Number ber Contours of the compression Surface . . . . . . . . . . . . . Caret wing generated from a wedge . . . . . . . . . . . . . . . . . . . . . . CFD Resu Result ltss of temp tempeeratur aturee on comp comprressi ession on surf surfac acee . . . . . . . . . . . . . sket sketcch of of the the wh whol olee inl inlet et syst system em with with on desi design gn shoc shock k pos posit itio ions ns . . . . . . . Sket Sketcch of the the wh whol olee inle inlett syst system em with with on desi design gn shoc shock k posit positio ions ns . . . . . . Table able show showin ingg ramp ramp angl anglee and and corr corres espon pondi ding ng pres pressu sure re reco recov very ery . . . . . . . Prel Prelim imin inar ary y inl inlet et desi design gn value alue (in (incl clud udin ingg wed wedge ge shoc shock k and and fric fricti tion on effec effectt ) Thrust vs Area ratio for nozzle . . . . . . . . . . . . . . . . . . . . . . . . Ideal Ramjet thrust and fuel consumption . . . . . . . . . . . . . . . . . . Perf erforma ormanc ncee compa omparrison ison at off-s off-seet desi design gn ope operati ration on . . . . . . . . . . . . . Inlet 2-D unstructured mesh . . . . . . . . . . . . . . . . . . . . . . . . . Inlet CFD results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumed Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometrical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vehicle cross section geometry . . . . . . . . . . . . . . . . . . . . . . . . . Temp erature Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atmos Atmosphe pheric ric Propert Properties ies variati ariation on with with Altitu Altitude de (yell (yellow ow is design design altitu altitude) de) Iso surface of 0 z velo city . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAD Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAD Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation with Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation with Payload Weight . . . . . . . . . . . . . . . . . . . . . . . . Variation with Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacement of Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 7 9 11 12 15 16 17 19 21 23 24 25 26 27 27 28 28 29 30 33 34 34 35 40 41 55 56 57 58 59 61
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11 12 18 32 36 37 60
List of Tables 1 2 3 4 5 6 7
Cone Parameters for generating Waverider . . . . . . . . . . . . . . Geometrical Parameters of generating cone . . . . . . . . . . . . . Fin Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prope Propert rtie iess of Mate Materi rial alss cons consid ider ered ed for for appl applic icat atio ion n in Airc Aircra raft ft Body Body . Various Fuels and their proper perties . . . . . . . . . . . . . . . . . . . Rang Rangee comp compar aris ison on of JP-1 JP-100 and and LH 2 . . . . . . . . . . . . . . . . . Vehicle Geometry Parameters (in SI Units) . . . . . . . . . . . . . . 2
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Division of Work Each team member was allotted a set of topics and sub-systems to cover and some team members were responsible for sub system integration.
• Abhishek Singh : Propulsion,Inlet Design and report integration. • Maneesh Kumar : Weight and Sizing Analysis, Sensitivity Analysis, Waverider Body Construction in CATIA , CAD Modelling
• Kush Sreen : Aerodynamics,Waverider Generation (Conceptual) , Inlet Design (Coding),CFD Analysis, Structural Analysis and report integration.
• Rohit Potla : Fuel Selection and Thermal Protection System Design. • Sanjeev Yadav : Materials Selection.
3
1
Introduction
Since the Wright Brothers took to the sky in 1903, mankind has endeavoured to fly higher and faster. This ambition was nurtured through the two world wars and a major epoch in manned flight took place when Chuck Yeager in the Bell X-1 prototype became the first man to break the sound barrier. The proverbial frontier of flight was pushed even further. The next major challenge in atmospheric flight was of course to fly at Hypersonic speeds. This has indeed proved a major challenge to aerospace designers. The extreme aero-thermodynamic loads coupled with inefficient air breathing engines at Mach 6 and higher have made this dream elusive. But since the 1990’s with the advent of new materials, progress in the development of a Scram-jet engine and computational techniques progress in Hypersonic flight has been substantial. A flying prototype the X-43 has already proved its mettle. This design project envisages to do the same but with a goal of creating a Hypersonic Cruise vehicle capable of carrying large payloads.A payload capacity of 10 tons with a flight range of 10,000 km is set as an objective. A complete preliminary design analysis of the proposed vehicle is carried out. The final vehicle configuration obtained is thorough aerodynamic, structural and propulsion analysis. In this project we will look at the major sub-systems present in a hypersonic vehicle. Of-course all systems play crucial role in the flight of any aircraft but certain system have that criticality. Due to lack of available data and the limited scope of this project we will try an address those crucial sub systems with the aim of developing a realisable prototype. So you may feel some very critical sub-systems or analysis may be missing but his is done to have paid more emphasis on the sub systems already being developed.
4
2
Design Objectives
The primary objectives set for the hypersonic vehicle are-
• Fly at a cruise velocity of Mach 5. • Cruising altitude of 30 Km. • A payload of 10,000 Kg. • Cruising range of 10,000 Km. These design objectives are set as the final goal which the final vehicle configuration must satisfy. During the design process all the above parameters are taken as fixed inputs. With a high load carrying capacity and long range this vehicle is intended as a military vehicle capable of long range missions. It can carry listening equipment to get high resolution data from the enemy territory or carry out quick response and long range bombing. The high cruise altitude and velocity make it impossible for enemy anti-air defences to bring down the vehicle. A major challenge in the design of the vehicle was to ensure a satisfactory L/D ratio. Normal airfoils don’t produce lift in the hypersonic regime. The whole body of the aircraft must be used to produce lift. To ensure high efficiency of lift production flow from the bottom surface must not leak on to the top surface. This method of lift generation is called compression lift and a Waverider geometry is employed. This method of lift generation leads to one major problem of excessive drag due to wave drag and skin friction drag owing to the large wetted area of the body. A propulsion system capable of delivering enough thrust to keep the aircraft in steady flight has to be made. A ramjet engine is used to develop the thrust required. The selection of fuel can either be LH 2 or Synthetic fuel JP-10. The final consideration was the structural and thermal loads experienced by the aircraft frame. Modern materials like Ti-Al alloys to withstand the loads at temperatures in the range 800-1000 K are used in the outer skin of the aircraft. Active Thermal Protection Systems(TPS) is used to cool the skin using fuel. Vehicle Specifications are listed in Appendix VIII
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3 3.1
Literature Survey Introduction
This section will be broken up into literature reviews done for each of the subsystems of the entire design. Most of the information surveyed is through research papers and doctoral thesis. These have been duly credited in the reference section. The aim of the literature survey was to understand the core concepts related to each subsystem and the integration with the other systems. The design of the NASA X-43 aircraft served as a guide for our project. John D. Anderson’s Hypersonic and High Temperature Gas Dynamics was a rich source of information on how to deal with hypersonic flows and was extensively used especially in the aerodynamic design phase.
3.2
Weight and Sizing
HASA-Hypersonic Aerospace Sizing Analysis for the Preliminary Design of Aerospace Vehicles, a NASA report is used in the initial weight and sizing analysis for the vehicle. The report contains detailed methodologies to obtain the mass and volume estimates of the various components of the aircraft. The method employed is an iterative one and it requires some unknown parameters to initiate the process. The report has various aircraft configurations listed to help in this regard. After determining the final mass/volume of the aircraft the next next is to determine the individual structural mass components. Empirical relations exist to determine these parameters.
3.3
Aerodynamics
Anderson’s Hypersonic and High Temperature Gas Dynamics was used to fundamentals of hypersonic flow. The hypersonic small perturbation theory which is essential to derive the empirical relations of lift and drag coefficients of the waverider is presented in great detail. The Mach independence condition at hypersonic speeds is also covered by it. To generate a waverider geometry a basic methodology is proposed using the wedge and conical shock method. Using the conical shock method requires understanding of the TaylorMaccol numerical solution to conical shocks. The doctoral thesis of Mr Graham L. Feltham is used as a reference while building a numerical solution to the conical flow. The viscous optimization process for waverider geometry is presented again in Anderson’s book.
Figure 1: Waverider and its Shock Cone.
6
The aerodynamic coefficients are evaluated using the methodology developed in Experimental results of a Mach 10 conical-flow derived waverider to 14-X hypersonic aerospace vehicle . This method relies purely on the geometrical parameters of the waverider geometry which have already been obtained. This allows us to compare results of the analytical method with a CFD analysis done later. The design of the vertical tail is done using a plane wedge formulation as devised in Integrated Design of Hypersonic Waveriders Including Inlets and Tailfins . A caret shaped vertical tail is designed using this approach and its aerodynamic coefficients computed. Aerodynamic Heating and shear force are computed using the reference temperature method as mentioned in Anderson. These methods help to design a waverider geometry along a conical shock wave the designed geometry is such that the conical shock wave will sit on the leading edge. This is particularly advantageous as no flow leakage from the compression surface will take place thus L dramatically increasing the D ratio of the entire vehicle.
3.4
Propulsion
The propulsion design was initiated from Sadraey(caption). The working of ramjet engine and the essential upstream parameters required for a ramjet engine to operate were decided. From this survey it was established that a ramjet engine is capable of working at Mach 5 under the given conditions. After the ramjet upstream parameters had been fixed next was the inlet design.Preliminary
Figure 2: Ramjet Engine and Inlet Ramps Design of a 2D Supersonic Inlet to Maximize Total Pressure Recovery(caption) was used to design the inlet ramps. The paper discussed the number of inlet ramps required to achieve the required inlet conditions. Also by increasing the number of ramps increased the pressure recovery ratio but made the fluid flow more complicated so an optimal number of ramps was selected.
7
The nozzle design is done using the Method of Characteristics(MOC). Zuckrow(caption) is used to design and implement the MOC code. The nozzle contour and propulsion parameters ( thrust, exit Mach) are obtained from the MOC code.
3.5
Structures and Materials
The various material properties were collected from different data sources like wikipedia , engineer-materials toolbox and other related sources. Papers on hypersonic structural analysis were used to carry out the loading tests on the vehicle geometry.
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4
Aerodynamics
4.1
Introduction
One of the main challenge of flying at hypersonic speeds is to generate lift efficiently. Conventional airfoils and even supersonic wedge airfoils don’t produce enough lift. The entire aircraft body has to be used as a lifting surface to generate sufficient lift.In such geometry it has to be ensured that the flow from the bottom compression surface does not leak to L the upper surface which has low pressure , this is essential to produce a suitable D ratio. Waverider geometries are used , in which the shock wave remains attached to the leading edge, to generate lift at hypersonic velocities. Aerodynamic Heating becomes significant as Mach number become higher. Due to the large wetted area of the waverider configuration surface temperatures of the skin can exceed 1,000 K. By estimating the heat flux through the aircraft skin an active TPS can be devised to maintain temperatures at serviceable conditions.
4.2
Conical Shocks
Figure 3: Supersonic flow past a Cone Supersonic flow past a cone creates a conical shock wave. Numerical solutions like the Taylor-Maccoll analysis can be used to find the flow field.
γ − 1 1 − V r − 2 2
dV dV dV dV d V dV d V − dθ V dθ + dθ dθ = 0 2V + cotθ + dθ dθ dθ 2
2
r
r
2
r
r
2
V θ =
dV r dθ
r
r
r
r
r
2
(1)
(2)
The Taylor-Maccoll analysis proceeds as follows. With a known free stream Mach number M and known cone angle c, assume a value for the oblique shock angle s. The 9
oblique shock relations provides the values of change in flow variables across the shock, and the deflection a of the flow through the shock. Determine the velocity components V r and V θ immediately downstream of the shock. The differential equation can be evaluated until the value of V θ is equal to zero. V θ = 0 is the normal velocity condition at the surface of the cone. In the same manner an inverse solution can also be found where the shock angle is known and the cone angle is found out.
4.3
Hypersonic Small Disturbance Theory
A frequently employed approach in aerodynamics, instead of sing the flow velocity as a dependent variable we deal with the change in velocity with respect to the free stream , the perturbation velocity. These are a special approximate form of the Euler equations applicable to hypersonic slender bodies. The velocity resolved in x and y directions is
u = V + u
(3)
∞
v = v
(4)
The ’’ represents the perturbation velocities which are very small when compared to the freestream velocity. The Hypersonic small disturbance equations are ∂ρ ∂ (ρv ) ∂ (ρw ) + + =0 ∂x ∂y ∂z
(5)
∂u ∂u ∂u ∂ρ + ρv + ρw =− ρ ∂x ∂y ∂z ∂x
(6)
∂v ∂v ∂v ∂ρ + ρv + ρw =− ∂x ∂y ∂z ∂y
(7)
∂w ∂w ∂w ∂ρ + ρv + ρw =− ρ ∂x ∂y ∂z ∂z
(8)
ρ
∂ ∂x
p ργ
∂ + v ∂y
p ργ
10
∂ + w ∂x
p ργ
=0
(9)
The hypersonic small disturbance equations are a set of coupled, non-linear differential equations for which no analytical solutions have been found out.
4.4
Waverider Construction
Now as we have established the importance of a waverider geometry this section will deal with the actual creation of the waverider geometry using the methods discussed above. From literature (caitation of anderson) the shock angle (β ) to construct an optimized geometry for M=5 is β =12o . Using the Taylor-Maccoll analysis for β =12o the cone specifications are Declination of Base(θ) 6.804
Shock Angle(β ) 12
Cone Angle(δ ) 4.265
Arc on Cone Base(φ) 54.377
Length of cone(l) length of vehicle(lw ) 91.4 80
Table 1: Cone Parameters for generating Waverider
Figure 4: Generating Cone and Waverider Any geometry whose leading edge forms part of the shock cone is a valid waverider. Infinite designs are possible. Based on the initial sizing analysis the dimensions of the waverider are fixed. The waverider is constructed using this data. The waverider obtained is constructed using an inviscid formulation and due to large wetted area has large skin friction drag to minimize skin friction drag viscous optimization of the waverider is done. This involves 11
creating an expansion fan on the upper surface of the waverider. A 2 o taper is provided from the tip to the base of the waverider on the upper surface.
Figure 5: Final Waverider Geometry
4.5
Aerodynamic Coefficients
The lift and drag coefficients are used are obtained from the hypersonic small disturbance theory is
Planform Area(S plan ) Wetted Area(S wet ) (σ ) 1351.831 2892.83 2.8135
δ θcb Arc on Cone Base(φl ) 54.377 4.265 6.804
Table 2: Geometrical Parameters of generating cone
l2 δ 2 σ3 C L = S plan σ 2 − 1
1 − R cos φdφ φl
0
where 12
cb
σ
(10)
Rcb =
θcb σ
(11)
C L = 0.0388
(12)
The drag computed is both skin friction and wave drag the empirical relations for both are
C D = C Dp + C f
C Dp
l2 δ 4σ 2 = S plan σ2 − 1
S wet S plan
(13)
2
2
1 − R − ln R dφ φl
cb
cb
σ
0
σ
−3
(14)
C Dp = 3.738 × 10
(15)
0.664F 0 F 1 √ Re
(16)
C f =
1
∞
1
R 1− dφ
4l2 δ lw F 0 = S l
φl
2
cb
2
σ
0
(17)
1
S + (1 + γM 2 δ 2) S wc F 1 = S wet 2
∞
−4
C f = 6.49071 × 10 C D = 4.7627 × 10
13
−3
(18)
(19)
(20)
The
L D
ratio obtained is L = 8.146 D
The upper bound for the
L D
(21)
ratio for hypersonic flight is given by the relation
L D
=
max
6(M + 2) M ∞
(22)
∞
For a freestream mach number of M =5 ∞
L D
= 8.4
(23)
max
L We can see that the D ratio is quite near the maximum value achievable thus the selected geometry is an optimized one.
4.6
3-D CFD simulation of Body
A full body 3-D CFD simulation was carried out to verify the analytical results for the waverider geometry. The meshing was done in pointwise software and a unstructured mesh with 6.5 million cells was created. The solver utilized was CFD++ and a 2 equation K- turbulence model was used.The solution showed good convergence after 1500 iterations after which the solution was stopped. L Post processing was done in CFD-Post software. Apart from the D ratio it was of interest L to see if the conical shock actually remain attached to the geometry. The D ratio obtained from the CFD is
L D
= 7.97
CF D
14
(24)
Figure 6: Mesh of the Waverider We can see this is a good match with the analytical results obtained with a deviation of 2.1 percent. In figure 7 you can see the contours of Mach number on the compression surface. The cyan and light green represent the conical shock and it is evident from the figure that the conical shock is attached to the leading edge of the waverider. This also validates the Taylor-Maccoll Conical flow solution which predicted a Mach number of M=4.771 downstream of the conical shock.
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Figure 7: Mach Number Contours of the compression Surface
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4.7
Vertical Fin
The design of the vertical fin is based on the same principle as the waverider , to keep the shock attached to the leading edge , with one difference the generating surface for a fin is a wedge instead of a cone . This results in caret shape of the tail fins. The wedge angle is δ f in =2o and W f in =1.5m. From the hypersonic small disturbance theory
Figure 8: Caret wing generated from a wedge
sin β γ + 1 γ + 1 1 = + + sin δ 4 4 M sin δ 2
2
(25)
2
∞
f in
β f in = 12.828o
(26)
The fin parameters and aerodynamic coefficients are
lf in =
C Lfin
W f in 2sin β f in
tan φfin
(27)
2
4W f in lf in (M 2 sin2 β f in − 1) = M 2 (γ + 1) ∞
(28)
∞
C Dfin = C Lfin tan δ f in
17
(29)
W f in 1.5
φf in 60
β f in lf in C Lf in C Df in 12.828 5.85 0.1359 0.00474 Table 3: Fin Parameters
4.8
Aerodynamic Heating
To estimate the heat flux through the skin of the aircraft the reference temperature method is used. The incompressible boundary layer equations are used but the fluid properties are evaluated at a reference temperature T . The wall temperature is taken as T wall =1200K. This value is obtained from the CFD result. The coefficient of heating is ∗
√ 0.332(P r ) C H = 3 √ ∗
Rex
−2 3
∗
(30)
ρ ue x = µ ∗
Rex
∗
(31)
∗
µ C p Pr = k ∗
∗
∗
(32)
∗
The velocity is taken as 1510 m/s the free stream velocity and rest of the values are computed at 30 km altitude (see appendix). The reference temperature is calculated as T T wall = 1 + 0.032M 2 + 0.58 T T ∗
∞
∞
∞
∗
T = 972.9K
∗
ρ =
∞
(35)
∗
T ∗
T ref
18
(33)
(34)
p = 0.00435kg/m3 RT
µ = µref ∗
−1
3 2
T ref + 110 T + 110 ∗
(36)
Figure 9: CFD Results of temperature on compression surface µ = 4.08 × 10 5 kg/ms ∗
−
(37)
Rex = 6.435 × 106 ∗
(38)
∗
P r = 0.4822
(39)
−4
∗
C H = 3.686 × 10
√
Pr = ∗
T aw − T T stag − T
∞
∞
⇒ T aw = 1015.14K
∗
q w = ρ u C H C p (T w − T aw ) ∞
19
(40)
(41)
(42)
q w = 449.3451W/m2
(43)
The heat flux through the skin of the aircraft is 449.3451 W/m2 .This has to be managed through a TPS which is discussed in a later section.
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5
Pro Propuls pulsiion
To ram means means to force in. In ramjet ramjet air is forced forced into into the engine engine air intak intakee by the sheer sheer drive drive of the speed of flight. flight. Ramjet, Ramjet, in principle, principle, can work at subsonic speed but it can be practical only at higher supersonic speed. In a ramjet ramjet,, air undergoes undergoes compress compression ion in the diffuser diffuser,, then then fuel fuel is add added ed and burnt burnt in the burner, burner, and then the combustion combustion products products expand through the nozzle. nozzle. It is helpful helpful to consid consider er first a simpli simplified fied model of an ideal ramjet ramjet.. For ideal ramjet ramjet it is assumed assumed that that compression and expansion processes are reversible and adiabatic, that combustion occurs at constant pressure, that the air/combustion products properties (specific heat ratio γ and the gas constant R) are constant throughout the engine, and, although this is not necessary, that the outlet pressure is equal to the ambient pressure, in other words, that the nozzle is in the design regime. The prel The prelim imin inar ary y desi design gn of the the inle inlett is divi divide ded d into into the the foll folloowing wing subt subtas asks ks.. Th Thee first first subtas subtask k is the selectio selection n of the inlet inlet configu configurat ration ion includ including ing selec selectio tion n of the cross cross secsectional tional shape shape of the supersoni supersonicc part, part, select selection ion of compre compressi ssion on method, method, select selection ion of the number of ramps or oblique shocks, and selection of the subsonic diffuser. The second subtask is the determination of the optimization criteria to maximize the total pressure recovery. For a hypers hypersoni onicc inlet inlet,, the free stream stream is decele decelerat rated ed to the subsonic subsonic engine engine face face entry entry speed through a suitable shock system (including oblique shocks and a normal shock) and a subson subsonic ic diffuse diffuser. r. The shock system system will will decele decelerat ratee the flow to a subsoni subsonicc nu numbe mber, r, and the diffuser diffuser will furthe furtherr reduce reduce the flow flow speed speed to the engine engine face face entry entry speed. speed. The design design criterion is to maximize total pressure recovery.
Figure 10: sketch of the whole inlet system with on design shock positions 21
So have tried out with different possibility such as one ramp followed by normal shock , 2 ramp ( i.e two oblique shock followed by normal shock ) and 3 ramp ( i.e 3 oblique shock foll folloowed by normal normal shoc shock k ). We coul could d hav have gone gone with with more more ramp rampss bu butt it would ould made made design more complex and at off design conditions and a lot of spillage may occur. Therefore we decide decided d to go with with 3 ramps ramps (3 obliqu obliquee shocks) shocks) and normal normal shocks. shocks. Next Next task task was to determine ramp angles which will lead maximum pressure recovery. The inlet is to be designed at the cruise conditions of flight Mach number 5.0 and flight altitude altitude 40,000 ft. At the on-design point, the oblique shock waves waves from the two external external ramps intersect at the cowl leading edge, and the third oblique shock reflects upward to intersect the junction of the final ramp and the throat section. Given flight altitude H , the free stream air specific heats ratio γ can γ can be determined. Given the free stream Mach number , and specifying the normal shock up-stream Mach number , there are 8 basic equations for the 3 oblique shocks.
2cot θ1 (M 02 sin 2 θ1 − 1) tan δ 1 = 2 + M + M 02 (γ + + 1 − 2sin2 θ1 )
(γ + + 1) 2 M 04 sin 2 θ1 − 4(M 4(M 02 sin 2 θ1 − 1)(γM 1)(γM 04 sin 2 θ1 + 1) M 1 = 2γM 02 sin 2 θ1 − (γ + + 1)((γ 1)((γ − 1)M 1)M 02 sin 2 θ1 + 2) 2cot θ2 (M 12 sin 2 θ2 − 1) tan δ 2 = 2 + M + M 12 (γ + + 1 − 2sin2 θ2 )
(44)
(γ + + 1) 2 M 14 sin 2 θ2 − 4(M 4(M 12 sin 2 θ2 − 1)(γM 1)(γM 14 sin 2 θ2 + 1) M 2 = 2γM 12 sin 2 θ2 − (γ + + 1)((γ 1)((γ − 1)M 1)M 12 sin 2 θ2 + 2) 2cot θ3 (M 22 sin 2 θ3 − 1) tan δ 3 = 2 + M + M 22 (γ + + 1 − 2sin2 θ3 )
(46)
(γ + + 1) 2 M 24 sin 2 θ3 − 4(M 4(M 02 sin 2 θ3 − 1)(γM 1)(γM 24 sin 2 θ3 + 1) M 3 = 2γM 22 sin 2 θ3 − (γ + + 1)((γ 1)((γ − 1)M 1)M 22 sin 2 θ3 + 2)
(45)
(47)
(48)
(49)
= M 1 sin θ2 M 0 sin θ1 = M
(50)
= M 2 sin θ3 M 1 sin θ2 = M
(51)
22
Figure 11: Sketch of the whole inlet system with on design shock positions Using oblique and normal shock initially we performed iterations to find out what all ramp angles would be optimum which would lead to maximum pressure recovery. Inlet Mach no. is assumed to be 5 (effect of wedge shock and frictions are not taken in to consideration) Once we got basic estimate about the ramp design we have further modified our design by incorporating wedge shock and friction effect in to our consideration [1], code of which is attached in the appendix.
5.1
Com Combusti bustion on Cham Chamber ber
Since Since flow flow enter entering ing the combus combustio tion n camber camber is subson subsonic ic ( M = 0.6), 6), so we desi design gned ed a subsonic combustion chamber[2], we kept the maximum combustion chamber temperature to be 2400K 2400K .The .The total pressure loss in a combustor is predominantly due to two sources. The first is due to the frictional loss in the viscous layers where we can lump in the mixing loss loss of the fuel and air streams streams prior to chemi chemical cal interac interactio tion. n. The second second source source is due to heat release in an exothermic chemical reaction in the burner. The first is usually modeled as the “cold” flow loss and the second is the “hot” flow loss.For a preliminary level design it is safe to assume stagnation pressure loss in nozzel to be around 5 percentage.
F = (m˙ a + m˙ b )V e − m˙ a V i
P 0a γ − 1 2 = 1+ M a P o γ
23
(52)
γ γ −1
(53)
Figure 12: Table showing ramp angle and corresponding pressure recovery
P 06 γ − 1 2 = 1+ M a P 6 γ
γ γ −1
(54)
(m˙ a + m˙ b )h04 − m˙ a h02 = m˙ a Qf
(55)
m˙ a ((1 + f )h04 − h02) = m˙ f Qf
(56)
f =
T 04 T 0a Qf C p T 04
24
−1 − T T
04
0a
(57)
Figure 13: Preliminary inlet design value (including wedge shock and friction effect )
F = (1 + f )V e − V i m˙ a
F T −1 = M γRT 1 + f 04
a
m˙ a
T 0a
F T γ − 1 = M γRT 1 + f (1 + M ) − 1
m˙ a
5.2
a
04
2
T 0a
a
2
(58)
(59)
(60)
Nozzle design
Nozzle design was done using the MOC analysis. The Matlab code is attached in the Appendix. The nozzle exit Mach number is 4.26 and the nozzle area ratio is 9.86 .
5.3
Offset design and Conclusion
Performance the engine went down as we have operated it on off-design condition i.e away from Machno. = 5, we choose 2 different off-design condition ,one was for Machno.4and other was for Machno.6. Pressure recovery improved for Machno.4, it went up to 0.652 for same inlet and maximum chamber pressure. While in case of Machno.6 total pressure 25
Figure 14: Thrust vs Area ratio for nozzle recovery went down to 0.3487.But since our mission requirement was to cruise at Machno. = 5 we went ahead with current engine design. Mass flow rate was found out to be 765kg/s and max thrust value to be 550KN fuel flow rate in case of liquid hydrogen was mound out be around 7 kg/s. Specific thrust value = 0.718KNs/kg which is also corresponds to graph below for Ideal Ramjet thrust and fuel consumption.
26
Figure 15: Ideal Ramjet thrust and fuel consumption
Figure 16: Performance comparison at off-set design operation
5.4
CFD Results of inlet
A CFD simulation was carried out to check if the inlet shocks did fall on the cowl and that the design was operational. From the images it is clear that the inlet design produces the 27
required results.
Figure 17: Inlet 2-D unstructured mesh
Figure 18: Inlet CFD results
28
6 6.1
Weight and Sizing The HASA technique
For initial weight and sizing analysis of the suggested aircraft configuration, a literature review of existing sizing techniques was undertaken. The technique found to be most suitable for the proposed vehicle was the HASA (Hypersonic Aerospace Sizing Analysis) technique for the preliminary design of Aerospace vehicles. It was developed by NASA and has been implemented in a number of hypersonic vehicle designs. For the aforementioned analysis, some initial geometrical parameters had to be sourced from relevant references while the mission conditions set other parameters like payload and flight velocity (at 30 km of altitude). The following were the assumed initial parameters: From these values, flight parameters like
Figure 19: Assumed Parameters weight of fuel carried, fuel ratio, initial estimate of total weight, volume and equivalent body diameter etc. were calculated. These calculations provided an initial estimate of the vehicle size and weight. For arriving at actual parameters suited to the design requirements, following non-dimensional constants were utilized according to the HASA technique:
29
Figure 20: Geometrical Parameters
30
Once these constants were determined, the actual values of flight parameters confirming to the designed mission requirements were determined using the formulas given below:
31
7
Structures and Material Selection
7.1
Introduction
With the high aero-thermodynamic loads the vehicle experiences at high Mach numbers the structural integrity of the vehicle becomes a major issue. Conventional aircraft materials can’t be used due to the high skin temperatures and high temperature alloys or carbon composites have to be looked into. The aircraft structure must also be able to withstand the bending moments and a cross-section for the aircraft is identified which can handle these loads.
7.2
Material Selection
Different Materials are considered for application in the aircraft design and their properties are listed Material
Den sity(kg/m3 ) Maximum Service Temperature(K) Ultimate Tensile Strength(MPa) Thermal Conductivity(W/mK) Coefficient of Expansion(K 1) −
Inconel X-750
8280
923
1250
12
12.6×10
Ti-Zr-Mo Alloy
10220
1873
685
126
5.3×10
Rene 41 super alloy
8240
1243
1241
80
8.1×10
Ti-Al Alloy
44200
1000
950
7.2
9.2×10
Hafnium Diboride
11900
2300
500
60
7.6×10
Carbon Silicon Carbide
2100
1850
310
15
1.3×10
Carbon-Carbon Composites
1700
1600
240
40
4×10
−6
−6
−6
−6
−6
Table 4: Properties of Materials considered for application in Aircraft Body Inconel X-750 was chosen for outer skin of X-15.Throughout the life of X-15 Program, Inconel X-750 has demonstrated its reliability and strength where speeds up to Mach no 6.70. Modern day use of Inconel X-750 has been restricted due to its high weight and cost. Ti-Zr-Mo alloy this form of molybdenum alloys possesses high recrystallisation temperature up to 1800 K and higher strength compared to many current super alloys. Low thermal expansion coefficient so it can be used in high heat exposure area without incorporating of large number of joints to prevent buckling due to thermal stresses.Manufacturing of this super alloys consume lots of money as well as time. Coverage of large surface area of vehicle is not possible to this alloys due to possible weight penalties incurred. Rene 41 super alloy a cooling system involving the use of lithium in lieu of conventional coolant encased in a Rene 41 honeycomb matrix was proposed by Boeing to solve thermal management problems in sustained hypersonic flight vehicles. Due to high weight penalties of this alloy its use is severely limited. Ti-Al alloy has great potential to achieve weight reduction in scram jet/ramjets by up to 25-30 percent. The high strength offered also makes this a goo d prospective material but due to lower service temperature a active thermal system must be in place while sing this 32
−6
−6
material. Hf Br 2 and ZrBr 2 are both relatively new ceramic compounds that hold great potential to replacing traditional TPS in hypersonic atmospheric vehicles.Their low ablation rates at high temp(first sign of ablation at 3000 K ) allow a reusable hypersonic vehicles to be created. But due to high weight their functionality is reduced. C/SiC carbide consists of carbon fibre reinforced silicon carbide produced using liquid polymer infiltration techniques.Despite the good retention in strength to weight ratio at high temperatures, C/SiC composite is significantly much lower in load bearing properties compared to several materials. This is due to micro cracking during complex component fabrication and microstructural voids (empty spaces) generated during carbon fibre weaving. Due to this reason C/Sic carbide are not used in hypersonic flight. C-C composites are extensively used in hypersonic and reusable vehicles, it has good retention at high temperatures up to 1600K and light weight capability for aero structure integration.Several applications of overcoat sealers are required to be applied on reinforced carbon-carbon (RCC) surface to prevent severe oxidation damage.The coating consisted of SiC substrate, a second chemical vapour deposited Sic layer followed by a last layer of Hafnium carbide. After considering the cost and weight penalties the choice of material for the aircraft skin is Ti-Al alloy due to its low weight and high strength but due to its low service temperature C-C composites are preferred at the nose tip of the vehicle.
7.3
Structural and Thermal Analysis
The cross section bulkhead of the aircraft was subjected to bending stresses which which appear during flight to check the structural integrity of the vehicle. Simultaneously a thermal load study was conducted so a coupled structural-thermal analysis was preformed to check the effect temperature may have on the properties of the material.
Figure 21: Vehicle cross section geometry
33
The material used is Ti-Al alloy with a thickness of 3mm. Assuming this to be uniform the gross weight of the outer skin of the aircraft is 41,000 Kg. The thermal analysis is preformed by applying constant wall temperatures and considering effects of only convection and no radiation. The temperature distribution obtained is shown in figure. This temperature distribution is used to apply the temperature specific properties at different locations during the structural analysis. The structural analysis is carried out
Figure 22: Temperature Distribution by applying a bending moment of the entire geometry. This bending moment simulates the aerodynamic loads which appear of the body during flight. The bending moment calculated is 3000 Nm.
Figure 23: Stress Distribution
34
The maximum stresses developed are at the corners of the geometry and are around 790 MPa. This is below the ultimate stress of Ti-Al alloy of 1000 MPa. This gives and overall factor of safety of 1.25 . The maximum displacement observed is 14 mm (attached in Appendix)
Figure 24: Strain Distribution
35
8
Fuel Selection
8.1
Introduction
Fuel selection plays a crucial role in determining the final weight as well as performance of the aircraft. The fuel selection also plays an important role in determining the the parameters of the propulsion system. This section deals with the selection between different type of fuels mainly hydrogen and other organic based synthetic fuels.
8.2
Hydrogen Fuel of the Future?
As a preliminary various fuels were analysed they are listed below. It is clearly visible the
Fuel Density(kg/m3 ) Heating Value(KJ/Kg) Liquid Hydrogen 89.88 121 JP-10 940 42.14 CH 4 (liquefied) 422.36 55.53 JP-4 751 42.8 Table 5: Various Fuels and their properties great advantage of LH 2 over other hydrocarbon based fuel is its high calorific value which is three times the others. Its inherent disadvantage is its low density which causes larger tank volumes and also the fact that it has to be maintained at cryogenic temperatures. Liquefied CH 4 is fuel which is under development but given its relatively low density compared to other hydrocarbons and not much improvement in calorific value we reject it as a candidate. The last two fuels JP-4 and JP-10 are both hydrocarbon based synthetic fuels and are similar in nature but due to better density characteristics we select JP-10. Hydrogen can be used as the primary fuel in an internal combustion engine or in a fuel cell. A hydrogen internal combustion engine is similar to that of a gasoline engine, where hydrogen combusts with oxygen in the air and produces expanding hot gases that directly move the physical parts of an engine. The only emissions are water vapor and insignificant amounts of nitrous oxides. The efficiency is small, around 20 percent. Disadvantages of hydrogen include fuel High Cost,Low Temperatures needed escape can cause fire or asphyxiation. It will explode at concentrations ranging from 4-75 percent by volume in the presence of sunlight, a flame, or a spark. On the other hand JP-10 is a gas turbine fuel or synthetic fuel.It contains a mixture of (in decreasing order) endo-tetrahydrodicyclopentadiene, exo tetrahydrodicyclopentadiene, and adamantane. It is produced by catalytic hydrogenation of dicyclopentadiene.It is high-energy density hydrocarbon fuel. It is less expensive to synthesize than other fuels.JP-10 is speciality fuel that has been developed for demanding applications.The required properties are: maximum volumetric energy content, clean burning, and good low-temperature performance. To achieve these properties, the fuels are formulated with 36
high-density naphthenes in nearly pure form.
8.3
Range in Cruise
The total mass of the propellant is fixed at 50,000 Kg. The mass flow rate of air which has to be heated from 1192 K to 2400 K for the propulsion system is 765 Kg/ m3 . The cruise velocity is 5400 km/hr. It is quite clear that LH 2 will clearly give much higher range in
Fuel Fuel Flow Rate(kg/s) Range(Km) Liquid Hydrogen 7.66 9900 JP-10 22.03 3500 Table 6: Range comparison of JP-10 and LH 2 cruise for the aircraft. It also satisfies the mission requirement range of 10,000 Km.
8.4
Fuel as Thermal Protection System
An internal forced convection model (caption) is used to calculated the convective heat transfer coefficient of each fuel. Nu = 4.82 + 0.0185(RE D P r)0.827
(61)
where Nu is Nusselt number the ratio of convective to conductive heat transfer across (normal to) the boundary. Nu =
hX k
(62)
The convective heat transfer coefficients are JP-10 : hJP
−10
=77.69 W/K-m2
Liquid Hydrogen : hLH =0.6988 W/K-m2 2
Clearly Liquid hydrogen posses very poor convective properties and cant be used directly in an TPS system some other refrigerant/coolant has to be used in this case. Based on all the above properties we select Liquid Hydrogen as our fuel of choice as it provides excellent heating value and is the only fuel which can fulfil our design range.
37
9
References 1. HASA-Hypersonic Aerospace Sizing Analysis for the Preliminary Design of Aerospace Vehicles, NASA-Contractor Report 182226 2. Randall T. Voland, Lawrence D. Huebner, Charles R. McClinton, X-43A Hypersonic vehicle technology development. 3. Mary Kae Lockwood, Dennis H. Petley, John G. Martin, James L. Hunt, Airbreathing hypersonic vehicle design and analysis methods and interactions. 4. Kevin D. Jones, Helmut Sobieczky, A. Richard Seebass , F. Carroll Dougherty, Waverider Design for Generalized Shock Geometries. 5. Kashif H. Javaid and Varnavas C. Serghides, AN INITIAL DESIGN METHODOLOGY FOR OPERATIONAL HYPERSONIC MILITARY AIRCRAFT. 6. Frederick Ferguson, Nastassja Dasque and Mookesh Dhanasar, Waverider Design and Analysis. 7. John D. Anderson Jr, Hypersonic and High Temperature Gas Dynamics 8. John J. Bertin, Russell M. Cummings, Fifty years of hypersonics: where we’ve been, where we’re going. 9. SHEAM-CHYUN LIN and MING-CHIOU SHEN, FLIGHT SIMULATION OF A WAVERIDER-BASED HYPERSONIC VEHICLE
10. Jeffrey S. Robinson, An Overview of NASA’s Integrated Design and Engineering Analysis (IDEA) Environment 11. H. Alkamhawi, T. Greiner, G. Fuerst, S. Luich, B. Stonebraker, and T. Wray,Hypersonic Aircraft Design 12. N. C. Bissinger , N. A. Blagoveshchensky, A. A. Gubanov , V. N. Gusev , V. P. Starukhin , N. V. Voevodenko , S. M. Zadonsky, Improvement of forebody/inlet intention for hypersonic vehicle 13. Mohammed H Sadraey, Aircraft Design 14. Hongjun Ran and Dimitri Mavris, Preliminary Design of a 2D Supersonic Inlet to Maximize Total Pressure Recovery 15. Aircraft propulsion –Saeed Farokhi 16. BASHETTY SRIKANTH, K.DURGA RAO, Dr.S.SRINIVAS PRASAD, Structural Analysis on Unconventional Section of Air-Breathing Cruise Vehicle 17. http://www.dtic.mil/dtic/tr/fulltext/u2/607169.pdf
38
18. Tiago Cavalcanti Rolim,Paulo Gilberto de Paula Toro, Marco Antonio Sala Minucci, Antˆonio de Carlos de Oliveira, Roberto da Cunha Follador, Experimental results of a Mach 10 conical-flow derived waverider to 14-X hypersonic aerospace vehicle 19. Sheam-Chyun Lin and Yu-Shan Integrated Design of Hypersonic Waveriders Including Inlets and Tailfins
39
APPENDIX I Atmospheric Data taken from US standard atmospheric data.
Figure 25: Atmospheric Properties variation with Altitude (yellow is design altitude)
40
APPENDIX II CFD results for velocity in z direction.The iso surface represents zone of 0 vertical velocity. Notice the iso surface attached to the leading edge indicating no flow movement from bottom to top surface.
Figure 26: Iso surface of 0 z velocity
41
APPENDIX III Inlet-Design Matlab Code % ramp d e s ig n b as ed on 3 o b l i q u e s h oc ks and 1 no rm al s ho ck % c od e w r i t t e n by Kush S r ee n c le ar a l l clc m0=4.77 ; %fr ee stream mach number gamma=1.4; m4 up = 1 . 7 ; %p r e s s u r e r e c o v e r y v a l m3=10; t h et a 1 =0; %i n t i a l g u es s w h i l e a b s ( m3−m4 up)>=0.01 ku=(gamma+1) ∗ (gamma+1) ∗ (m0ˆ4) ∗ ( ( s i n d ( t h e t a 1 ) ) ˆ 2 ) ; a= −4∗((m0ˆ2) ∗ (( sin d ( the ta1 ))ˆ2) − 1 ) ; b=(gamma ∗ (m0ˆ2) ∗ ( ( s i n d ( t h e t a 1 ) ) ˆ 2 ) + 1 ) ; c=2∗gamma ∗ (m0ˆ2) ∗ (( sin d ( the ta 1 ))ˆ2) − (gamma − 1); d=(gamma− 1) ∗ (m0ˆ2) ∗ (( sind ( thet a1 ))ˆ 2)+ 2; e=(ku+a ∗ b ) / ( c ∗ d ) ; %a=(((gamma+1)ˆ2) ∗ ((m0)ˆ4) ∗ (( si nd ( thet a1 ))ˆ2) − 4 ∗ (m0∗m0∗ (( sind ( the ta1 ))ˆ2) − 1 ) ∗ (gamma∗m0∗m0 ∗ ( ( s i n d ( t h e t a 1 ) ) ˆ 2 ) + 1 ) ) / ( ( 2 ∗ gamma∗m0∗m0∗ (( si nd ( the ta 1 ))ˆ2) − (gam ((gamma− 1) ∗m0∗m0∗ ( ( s i n d ( t h e t a 1 ) ) ˆ 2 ) + 2 ) ) ; m1=e ˆ0 . 5 ; thet a2=asi nd (( (m0∗ s i n d ( t h et a 1 ) ) /m1 ) ) ;
%o p t i m i za t i o n c o n d i t i o n
ku=(gamma+1) ∗ (gamma+1) ∗ (m1ˆ4) ∗ ( ( s i n d ( t h e t a 2 ) ) ˆ 2 ) ; a= −4∗((m1ˆ2) ∗ (( sin d ( the ta2 ))ˆ2) − 1 ) ; b=(gamma ∗ (m1ˆ2) ∗ ( ( s i n d ( t h e t a 2 ) ) ˆ 2 ) + 1 ) ; c=2∗gamma ∗ (m1ˆ2) ∗ (( sin d ( the ta 2 ))ˆ2) − (gamma − 1); d=(gamma− 1) ∗ (m1ˆ2) ∗ (( sind ( thet a2 ))ˆ 2)+ 2; e=(ku+a ∗ b ) / ( c ∗ d ) ; 42
%b=(((gamma+1)ˆ2)∗ ((m1)ˆ4) ∗ (( si nd ( thet a2 ))ˆ2) − 4 ∗ (m1∗m1∗ (( sin d ( th eta 2 ))ˆ2) − 1)
∗
(gamma∗m1∗m1∗ (( sind ( the ta2 ))ˆ2)+ 1))/ ( ( 2 ∗ gamma∗m1∗m1 ∗ (( sin d ( the ta 2 ))ˆ2) − (gamma+1)) ∗ ((gamma− 1) ∗m1∗m1∗ (( sin d ( th eta 2 )
m2=e ˆ0 . 5 ; thet a3=asi nd (( (m1∗ s i n d ( t h e t a 2 ) ) / m2 ) ) ; %c=(((gamma+1)ˆ2) ∗ ((m2)ˆ4) ∗ ( ( s i n d ( t h e t a 3 ) ) ˆ 2 ) −4∗(m2∗m2∗ (( sin d ( thet a3 ))ˆ2) − 1) ∗ (gamma∗m2∗m2∗ ( ( s i n d ( t h e t a 3 ) ) ˆ 2 ) + 1 ) ) / ( ( 2 ∗ gamma∗m2∗m2∗ (( sin d ( thet a3 ))ˆ2) − (gamma+1)) ∗ ((gamma ku=(gamma+1) ∗ (gamma+1) ∗ (m2ˆ4) ∗ ( ( s i n d ( t h e t a 3 ) ) ˆ 2 ) ; a= −4∗((m2ˆ2) ∗ (( sin d ( the ta3 ))ˆ2) − 1 ) ; b=(gamma ∗ (m2ˆ2) ∗ ( ( s i n d ( t h e t a 3 ) ) ˆ 2 ) + 1 ) ; c=2∗gamma ∗ (m2ˆ2) ∗ (( sin d ( the ta 3 ))ˆ2) − (gamma − 1); d=(gamma− 1) ∗ (m2ˆ2) ∗ (( sind ( thet a3 ))ˆ 2)+ 2; e=(ku+a ∗ b ) / ( c ∗ d ) ; m3=e ˆ0 . 5 ; d=((gamma− 1) ∗ ((m4 up)ˆ2)+2)/(2 ∗ gamma ∗ (( m4 up)ˆ2) − (gamma − 1 ) ) ; m4=dˆ0.5; i f a bs ( m3−m4 up)>=0.01 t h e t a 1 =t h e t a 1 + 0 . 0 1 ; en d en d %angles a=(2 ∗ c o t d ( t h e t a 1 ) ∗ ((m0)ˆ2 ∗ ( si nd ( the ta 1 ))ˆ2 − 1))/(2+(m0)ˆ2 ∗ (gamma +1 −2∗( s i n d ( t h e t a 1 ) ) ˆ 2 ) ) ; del 1=atand( a ) ; a=(2 ∗ c o t d ( t h e t a 2 ) ∗ ((m1)ˆ2 ∗ ( si nd ( the ta 2 ))ˆ2 − 1))/(2+(m1)ˆ2 ∗ (gamma +1 −2∗( s i n d ( t h e t a 2 ) ) ˆ 2 ) ) ; 43
del 2=atand( a ) ; a=(2 ∗ c o t d ( t h e t a 3 ) ∗ ((m2)ˆ2 ∗ ( si nd ( the ta 3 ))ˆ2 − 1))/(2+(m2)ˆ2 ∗ (gamma +1 −2∗( s i n d ( t h e t a 3 ) ) ˆ 2 ) ) ; del 3=atand( a ) ; %p r e ss u r e r ec o ve ry r a t i o s %a c r o s s o b l i q u e s h oc ks a=(((gamma+1) ∗ (m0)ˆ2 ∗ ( si nd ( the ta 1 )) ˆ 2 ) / ( (gamma− 1) ∗m0ˆ2 ∗ ( sin d ( thet a1 ))ˆ2+2 ))ˆ( b=((gamma+1)/(2 ∗ gamma∗m0ˆ2 ∗ ( sin d ( thet a1 ))ˆ2 − (gamma − 1)))ˆ(1/(gamma − 1 ) ) ; pr1=a ∗ b ; a=(((gamma+1) ∗ (m1)ˆ2 ∗ ( si nd ( the ta 2 )) ˆ 2 ) / ( (gamma− 1) ∗m1ˆ2 ∗ ( sin d ( thet a2 ))ˆ2+2 ))ˆ( b=((gamma+1)/(2 ∗ gamma∗m1ˆ2 ∗ ( sin d ( thet a2 ))ˆ2 − (gamma − 1)))ˆ(1/(gamma − 1 ) ) ; pr2=a ∗ b ; a=(((gamma+1) ∗ (m2)ˆ2 ∗ ( si nd ( the ta 3 )) ˆ 2 ) / ( (gamma− 1) ∗m2ˆ2 ∗ ( sin d ( thet a3 ))ˆ2+2 ))ˆ( b=((gamma+1)/(2 ∗ gamma∗m2ˆ2 ∗ ( sin d ( thet a3 ))ˆ2 − (gamma − 1)))ˆ(1/(gamma − 1 ) ) ; pr3=a ∗ b ; %a c r o s s n or ma l s h o ck a=(((gamma+1) ∗ (m4 upˆ2))/((gamma− 1) ∗ m4 upˆ2+2))ˆ(gamma/(gamma− 1 ) ) ; b=((gamma+1)/(2 ∗ gamma ∗ (m4 up)ˆ2 − (gamma − 1)))ˆ(1/(gamma − 1 ) ) ; pr4=a ∗ b ;
p r t o t = p r 1 ∗ pr2 ∗ pr3 ∗ pr4
44
APPENDIX IV Nozzle Design MOC Matlab Code % % % %
B ri tt o n J e f f r e y Olson Ph .D. Ca nd ida te S ta nf o rd U ni ve ri st y D epa rtm ent o f Aero / A s tr o
%%%%%%%% I n t r o d u c t i o n and Backgro und %%%%%%%% % Th is program g i v e s t he i d e a l n o z zl e geo metry u si n g t he m ethod o f % c h a r a c t e r i s t i c s f o r a Quasi −2D D i v e r g i n g N o z z l e . Assume g a s i s % e x ha u st i ng from a c om bu st io n c hamber t h at h as no m ass f l o w r a t e i n . % Using 2D n oz zl e f lo w r e l a ti o n s , an o pt im al t hr oa t a re a i s found t ha t % w i l l p ro du ce t he max amount o f t h ru s t f o r t he g iv en ambient p r e ss u r e % and c o mb us ti on c hamber p ar am et er s . T hi s Area i s a u t o m a ti c a l l y s e t and % f ed i nt o th e method o f c h a r a c t e r t i s t i c s p or ti on o f t h at code . The % method o f c h a r a c t e r i s t i c s a l s o u s es t he e x i t Mach number t ha t % c or re sp on ds t o t he i d e a l e x it a re a . %%%%%%%% D ir ec ti on s fo r runn ing the program %%%%%%%% % Run d e f a u l t o r m od if y t he p ro bl em p ar am et er s i n n o z z l e .m and n o z c f d .m % 1 ) Run th e n o z zl e .m program − This w i l l d es ig n t he n o zz le % 2 ) Run noz mesh .m− This w i l l s e t up a g r id f o r t he newly d es ig ne d % nozzle % 3 ) Run no z c f d .m− This w i l l s o lv e t h e 2−D E ul er e q u at i o n s on t he % n oz zl e mesh and p lo t th e r e s u l t . % ∗∗∗ Note ∗∗∗ % % You must run them i n t h i s o rd er wi th o ut c l e a r i n g v a r i a bl e s . % Each s c r i p t depends on t h e one be fo re i t . Also , s o l v e r ha s t he % f u n c t i o n s u sed by n o z c fd .m and n eed s t o be i n th e same d i r ec t or y . % % F i g ur e ( 1 ) : S t a t i c t hr us t a s f un ct io n o f e x it a re a % F ig u re ( 2 ) : N oz zl e d e s ig n and p l o t s o f Mach number & P r e ss u r e v s L eng th % F ig ur e ( 3 ) : CFD s im u l at i o n o f d es i gn ed n o zz l e
c l ea r a l l ; cl c ; % Problem p a ra me te rs T c = 2 4 0 0; % Temp era tu re i n t he c om bu st io n chamber (K) 45
P c = 2 86 7 4 9. 7 5 ; % P r es s ur e i n t he co mbu sti on c hamber ( Pa ) P amb = 13 30 .1 9 ; % Ambient p r e s s u r e ( Pa ) T amb = 2 3 2 . 9 ; % Ambient t em pe ra tu re (K) gamma = 1 . 4 ; % Ra t i o o f S p e c i f i c H ea t s Cp/Cv ( Gamma) W = 2 5 .4 ; % M o l ec u l a r wei g h t of gas ( kg / kmol ) width = 1 5; % N o zz l e width ( m eters ) h th = 0 . 23 ; % Throat h e ig ht ( m eter s ) % Method o f C h a r a c t er i s t i c s num = 1 5; % Number o f C h a r a c t e r i s t i c l i n e s t he ta i = . 0 3; % I n i t i a l step in theta p lo tt er = 1 ; % Se t t o ’ 1 ’ t o p lo t n o z z l e dh = h t h / 1 0 0 ; max iter = 10000; R = 831 4/W; %
Part A
%f i n d w he re P b ec om es u h(1) = h th ; A s t ar = h t h ∗ width ; M =1; dM1 = . 1 ; f o r i =1: m a x i t er h ( i ) = h ( 1 ) + ( i − 1) ∗ dh ; Ae( i ) = h( i ) ∗ width ; A Asq = (Ae( i )/ A sta r )ˆ 2; A r a t i o ( i )= s q r t ( A A sq ) ; %Newton Rhapson on Eq. 5. 20 res = 1; if i > 1 M = Ma( i − 1); en d
− An d er so n
text
w hi l e r e s > . 0 0 1 M2 = M + dM1; f u n a 1 = −A Asq + (1/Mˆ2) ∗ ((2/(gamma+1)) ∗ (1+(gamma− 1) ∗Mˆ2/2))ˆ((gamma f u n a 2 = −A Asq + (1/M2ˆ2) ∗ ((2/(gamma+1)) ∗ (1+(gamma− 1) ∗M2ˆ2/2))ˆ((gam d v d m = ( f u n a 2 −funa1)/dM1; M = M − funa1/dv dm; r e s = a bs ( f un a 1 ) ;
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en d Ma( i ) = M; % Fi nd P r e s s u r e P ( i ) = P c ∗ (1+(gamma− 1) ∗Ma( i )ˆ2/2 )ˆ( − gamma/(gamma − 1 ) ) ; % Find t h ru s t f o r ea ch p o in t Te( i ) = T c /(1+(gamma− 1) ∗Ma( i )ˆ 2/ 2) ; Tt( i ) = T c /(1+(gamma − 1 ) / 2 ) ; Ve( i ) = Ma( i ) ∗ sq rt (Te( i ) ∗ gamma∗R ) ; Vt( i ) = sq rt (Tt( i ) ∗ gamma∗R ) ; rho t ( i ) = P( i )/ (R∗ Te( i ) ) ; mdot( i ) = rho t ( i ) ∗ Ve( i ) ∗ Ae( i ) ; TT( i ) = mdot( i ) ∗ Ve( i ) + (P( i ) − P amb) ∗ Ae( i ) ; i f P ( i ) < P amb %break %C al cu la te t he p r es s ur e i f sh ock wave e x i s t s a t t he e x i t p la ne P e x i t = P ( i ) ∗ (1+(gamma ∗ 2/(gamma+1)) ∗ (Ma( i ) ˆ 2 − 1 ) ) ; i f P e xi t <= P amb P( i ) = P exit ; break else en d else en d en d f i g u r e (2 ) pl o t (Ae ,TT) t i t l e ( ’ T hr us t c u rv e ’ ) x l a b e l ( ’ E x i t A re a (mˆ 2 ) ’ ) yl ab el ( ’ Thrust (N) ’) % % %
Part B Determine th e nominal e x i t a re a o f th e n o z zl e t o maximize th ru s t
[ a , b]=max(TT) ; % Over o r U nderexpand t h e n o z z l e b = b; A max = Ae(b); 47
M a x t h r u s t = TT( b ) ; h ol d on ; pl ot (A max , Max thrust , ’ r ∗ ’ ) le ge nd ( ’ Thrust Curve ’ , ’Max Thrust ’ ) % %
Part C Method o f C h a r a c t er i s t i c s
M e = Ma( b ) ;
%Mach number a t i d e a l e x i t
%Fi nd t he ta ma x by u s i n g e q u a ti o n 1 1 . 3 3 t h e ta m a x = ( 1 8 0 / p i ) ∗ ( s q r t ( (gamma+1 )/(gamma− 1)) ∗ ata n (( s q rt ( (gamma − 1) ∗ (M eˆ2 − % D th eta f o r each ch a r l i n e d e l t h e t a = ( t h e t a m a x − t h e t a i ) / ( num − 1); % Find f o r i =1:num % I n i t i a l i z e mach numeber f o r j = 1:num i f i ==1 %Theta f o r each l i n e ( f i r s t l i n e s ) t h et a ( i , j ) = t h e t a i + d e l t h e t a ∗ ( j − 1); nu( i , j ) = th et a ( i , j ) ; K m( i , j ) = th et a ( i , j ) + nu( i , j ) ; K p ( i , j ) = th et a ( i , j ) − nu( i , j ); e l se i f i > 1 K p ( i , j ) = −K m(1 , i ) ; % F i nd T h et a s i f j >= i t h e t a ( i , j ) = d e l t h e t a ∗ ( j −i ) ; else %the ta ( i , j ) = the ta ( j , i − 1); the ta ( i , j ) = thet a ( j , i ); en d nu( i , j ) = th et a ( i , j ) − K p ( i , j ) ; K m( i , j ) = th et a ( i , j ) + nu( i , j ) ; en d 48
% P r a n d t l −Mey er f u n c t i o n ( u s i n g Newton R ha ps on ) dM = . 1 ; % Lea ve a t a bo ut . 1 i f j == 1 M ex ( i , j ) = 1 . 0 0 ; else M ex( i , j ) = M ex( i , j − 1); en d M = M ex( i , j ) ; res = 1; w hi le r e s > . 0 1 M2 = M + dM; f u n v 1 = (−nu( i , j ) ∗ ( pi /18 0)+ ( sq r t ( (gamma+1)/(gamma− 1)) ∗ atan (( sq rt ( (gam f u n v 2 = (−nu( i , j ) ∗ ( pi /18 0)+ ( sq r t ( (gamma+1)/(gamma− 1)) ∗ atan (( sq rt ( (gam dv dm = ( f u nv 2−funv1)/dM; M = M − funv1/dv dm; r e s = a bs ( f un v1 ) ; en d M ex ( i , j ) = M; % Fi nd t h e a n g l e mu mu( i , j ) = (18 0/ pi ) ∗ as in (1/ M ex( i , j ) ) ; en d % Add l a s t p oi nt to ch ar l i n e th et a ( i , num+1) = th et a ( i , num ) ; nu( i ,num+1) = nu( i ,num); K m( i , num+1) = K m( i , num ) ; K p ( i , num+1) = K p ( i , num ) ; en d char = zeros (num,num+1 ,2); f o r i =1:num f o r j =1:num+1 % Draw p o i nt s o f i n t e r s e c t i o n % Po in t 1 of a l l c ha r l i n e s i f j == 1 c h ar ( i , j , 1 ) = 0 ; char( i , j ,2 ) = h th /2; 49
en d % Where f i r s t l i n e h i t s t he symmetry l i n e i f i == 1 & j ==2 c h a r ( i , j , 1 ) = ( − h th /2)/ tan (( pi /180 ) ∗ ( th et a (1 , j −1)−mu( 1 , j − 1 ) ) ) ; c h ar ( i , j , 2 ) = 0 ; en d % Where a l l o th er l i n e s h i t th e symmetry l i n e i f j == i +1 & j >2 c h a r ( i , j , 1 ) = − c h a r ( i − 1,j , 2) / tan (( pi /1 80 ) ∗ ( . 5 ∗ th et a ( i , j − 2) − .5 ∗ ( c ha r ( i , j , 2 ) = 0 ; t e s t ( i , j ) = ( th et a ( i , j − 2) − .5 ∗ (mu( i , j −2)+mu( i , j − 1 ) ) ) ; t e s t p t y ( i , j ) = c h a r ( i − 1 ,j , 2 ) ; t e s t p t x ( i , j ) = c h a r ( i − 1 ,j , 1 ) ; en d % A l l o t h e r data p oi nt s f o r ch a r 1 c a l cu l a t e d i f i ==1 & j >2 & j ˜= i +1 C p = t a n ( ( p i / 1 8 0 ) ∗ ( . 5 ∗ ( the ta ( i , j −2)+t he ta ( i , j − 1))+.5 ∗ (mu( i , j −2)+ C m = t a n ( ( p i / 1 8 0 ) ∗ ( . 5 ∗ ( t h e t a ( j − 1,1)+ th et a ( i , j − 1)) − .5 ∗ (mu( j − 1,1)+ A = [ 1 , − C m;1 , − C p ] ; B = [ c h a r ( 1 , 1 , 2 ) − char (1 ,1 ,1 ) ∗ C m ; cha r (1 , j − 1 , 2 ) − cha r (1 , j − 1 , 1 ) ∗ C p ] ; ite rm (1 ,: )= inv (A) ∗ B ; char( i , j ,1) = iterm (1 ,2) ; char( i , j ,2) = iterm (1 ,1) ; en d % A l l o t h e r p oi nt s f o r a l l ch a r l i n e s c al cu l a te d i f i > 1 & j ˜ = i + 1 & j >2 C p = t a n ( ( p i / 1 8 0 ) ∗ ( . 5 ∗ ( the ta ( i , j −2)+t he ta ( i , j − 1))+.5 ∗ (mu( i , j −2)+ C m = t a n ( ( p i / 1 8 0 ) ∗ ( . 5 ∗ ( the ta ( i − 1, j −1)+t he ta ( i , j − 1)) − .5 ∗ (mu( i − 1, j A = [ 1 , − C m;1 , − C p ] ; B = [ c h a r ( i − 1 , j , 2 ) − c h a r ( i − 1 , j , 1 ) ∗ C m ; c h a r ( i , j − 1 , 2 ) − char ( i , j i t e r m ( 1 , : ) = i n v (A) ∗ B ; char( i , j ,1) = iterm (1 ,2) ; char( i , j ,2) = iterm (1 ,1) ; en d en d en d
%
F i l l i n s i m i l a r p oi nt s ( where ch a r l i n e s s ha re p oi nt s ) 50
f o r i = 2 : num f o r j = 2:num c h ar ( j , i , 1 ) = c ha r ( i − 1, j +1 ,1); c h ar ( j , i , 2 ) = c ha r ( i − 1, j +1 ,2); en d en d % ∗∗∗∗∗∗ Make t he n o z zl e s ha pe and e xt en d t he c ha r l i n e s t o w a ll ∗∗∗∗∗∗ % I n i t i a l s t a r t p o i n t o f t h e n oz z l e ( a t t h r o a t ) no z ( 1 , 1 ) = 0 ; n oz ( 1 , 2 ) = h t h / 2 ; % Find a l l th e p o i nt s o f t h e n o z zl e f o r i = 2 : num % Find d i f f e r e n t s l o p e s and p o i nt s t o i n t e r s e c t m1 = tan (( pi /18 0) ∗ ( the ta ( i − 1,num)+mu( i − 1,num))); i f i ==2 m2 = ( pi /180 ) ∗ theta max ; else m2 = (( pi /18 0) ∗ ( the ta ( i − 1,num+1))); en d m3 = (( pi /18 0) ∗ ( the ta ( i − 1,num))); m4 = tan ( (m2+m3) / 2 ) ; A = [ 1 , −m4 ; 1 ,−m1 ] ; B = [ n o z ( i − 1 , 2 ) − noz ( i − 1 , 1 ) ∗ m4 ; c h a r ( i − 1,num+1,2)
−
c h a r ( i − 1,num+1,1)∗ m1
i t e r m ( 1 , : ) = i n v (A) ∗ B ; n oz ( i , 1 ) = i t e r m ( 1 , 2 ) ; n oz ( i , 2 ) = i t e r m ( 1 , 1 ) ;
en d
% Extend ch a r l i n e s t o w al l c h a r ( i − 1,num+2,1)= noz ( i , 1 ) ; c h a r ( i − 1,num+2,2)= noz ( i , 2 ) ;
%L as t l i n e m1 = tan (( pi /18 0) ∗ ( th et a (num, num)+ mu(num, num ) ) ) ; m2 = (( pi /18 0) ∗ ( th et a (num− 1,num))); m3 = (( pi /18 0) ∗ ( th et a (num, num+ 1) )) ; m4 = tan ( (m2+m3) / 2 ) ; A = [ 1 , −m4 ; 1 , −m1 ] ; B = [ noz(num, 2 ) − noz(num,1) ∗ m4; char (num, num+1,2) 51
− cha r (num, num+1 ,1) ∗m1 ] ;
i t e r m ( 1 , : ) = i n v (A) ∗ B ; n o z ( num+ 1 , 1 ) = i t e r m ( 1 , 2 ) ; n o z ( num+ 1 , 2 ) = i t e r m ( 1 , 1 ) ;
% Extend ch a r l i n e s to w al l cha r (num, num+2,1)= noz (num+1 ,1 ); cha r (num, num+2,2)= noz (num+1 ,2 ); i f p l o t t e r ==1 % P l o t t h e n o z z le s hape figu re (1) ; cl f ; subplot (2 ,1 ,1) ; pl ot (noz ( : , 1 ) , noz ( : , 2 ) , ’ k ’ , ’ LineWidth ’ ,3 ) h ol d on ; [ a , b ] = max ( n o z ) ; pl ot (a (1 ) ,A max/width/ 2 , ’ g ∗ ’ ) % P l o t f o r lo o p f o r cha r l i n e s f o r i = 1 : num f i g u r e (1 ) h ol d on ; plot ( char( i , : ,1 ) , char( i , : ,2 ) ) h ol d on ; p l o t ( c h a r ( i , : , 1 ) , − c ha r ( i , : , 2 ) ) en d % P lo t th e n o zz l e shape ( bottom s i d e ) f i g u r e (1 ) subpl ot (2 ,1 ,1) h ol d on ; p l o t ( n o z ( : , 1 ) , − noz ( : ,2 ) , ’k ’ , ’ LineWidth ’ , 3 ) h ol d on ; p l o t ( a ( 1 ) , − A max/width /2 , ’ g ∗ ’ ) t i t l e ( ’ Max T h ru s t ( minimum l e n g t h ) N o z z l e D e si g n ’ ) x l a b e l ( ’ N o z z le l e n g t h (m) ’ ) y l a b e l ( ’ N o z z le h e i g h t (m) ’ ) l e g e n d ( ’ N o z z le s ha pe ’ , ’ A r e a e x i t ( p r e d i c t e d ) ’ , ’ Char . L i n es ’ ) else en d % Find % e r r o r s i n A/A∗ and Mexit e r r o r A r e a = 1 0 0 ∗ (width ∗ 2 ∗ noz(num,2) 52
− A max)/(A max)
error Mach = 100∗(M e
− M ex(num,num))/M e
% P lo t Mach Number and p r e s s u r e t hr ou gh n o z z l e u s in g t he q ua si −1D % a re a r e l a t i o n s . ( I s e n t ro p i c ex pa ns io n thro ugh n o zz le ) Mnoz ( 1 ) = 1 . 0 ; % Choked Flow M = Mnoz ( 1 ) ; f o r i =1 : s i z e ( n oz , 1 ) Ae( i ) = 2 ∗ n o z ( i , 2 ) ∗ width ; A Asq = (Ae( i )/ A sta r )ˆ 2; A r a t i o ( i )= s q r t ( A A sq ) ; %Newton Rhapson on Eq. 5. 20 res = 1; if i > 1 M = Mnoz( i − 1);
− An d er so n
text
w hi l e r e s > . 0 0 1 M2 = M + dM1; f u n a 1 = −A Asq + (1/Mˆ2) ∗ ((2/(gamma+1)) ∗ (1+(gamma− 1) ∗Mˆ2/2))ˆ((gamma f u n a 2 = −A Asq + (1/M2ˆ2) ∗ ((2/(gamma+1)) ∗ (1+(gamma− 1) ∗M2ˆ2/2))ˆ((gam d v d m = ( f u n a 2 −funa1)/dM1; M = M − funa1/dv dm; r e s = a bs ( f un a 1 ) ;
en d
en d Mnoz( i ) = M; en d % Fi nd P r e s s u r e Pnoz( i ) = P c ∗ (1+(gamma− 1) ∗ Mnoz( i )ˆ2 /2) ˆ( − gamma/(gamma − 1 ) ) ;
figure (1); subpl ot (2 ,1 ,2) pl ot (noz ( : , 1 ) , Mnoz, ’ r ∗ ’ ) h ol d on ; pl ot ( noz ( : ,1 ) , Pnoz/P amb , ’b ∗ ’ ) h ol d on ; plo t ( noz( s iz e (noz ,1 ) ,1 ) ,M e , ’ go ’ ) h ol d on ; plo t ( noz( s iz e (noz ,1 ) ,1) ,1 , ’go ’ ) x l a b e l ( ’ N o z z le l e n g t h (m) ’ ) yl ab el ( ’Mach number and P/P c ’ ) 53
leg end ( ’Mach Number ’ , ’P/P a m b ’ , ’ M e x i t ( pr ed ic te d ) ’ , ’ P a m b/P a m b ’ )
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APPENDIX V CAD drawing of the Geometry
Figure 27: CAD Drawing
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APPENDIX VI Final Geometry of the Vehicle
Figure 28: CAD Drawing
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APPENDIX VII Sensitivity Studies
Figure 29: Variation with Range
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variation with payload weight
Figure 30: Variation with Payload Weight
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variation with geometrical parameters
Figure 31: Variation with Geometry
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APPENDIX VIII Vehicle Geometrical Specifications are Length(m) Breadth(m) Width(m) Structural Volume(m3 ) Wetted Area(m2 ) Total Volume(m3 ) 80 5.7 6.23 1129 1177 1628
Table 7: Vehicle Geometry Parameters (in SI Units)
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