Homework # cm. 11 cm cm. 100 cm. MAE Propulsion Systems, II

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1 Homework #3.1 Nitrous Oxide HTPB Hybrid Rocket design Desired Thrust of 8 knt Operate near optimal mixture ratio (based on C*) Nozzle A/A * =16.4, exit diameter = cm, Nozzle Exit Divergence angle = 10 Single Circular Grain Port, Initial Diameter 11 cm Grain length 100 cm Ambient pressure 60 kpa Assume Isentropic Flow in Nozzle D throat = cm cm 11 cm cm 100 cm 81

2 2.1 (2) Homework #3.1 82

3 Homework #3.1 83

4 Homework #3.1 84

5 CEA Mean Combustion Property Calculations, P 0 ~3000 kpa N20/HTPB Ideal combustion properties O/F Ratio T0 g Mol.Wt K kg/kg-mol

6 86

7 N 2 O Injector / Oxidizer properties 50 injector ports, 2.0 mm in diameter Assume each port has a discharge coefficient C d =0.81 Assume Super-Saturation Liquid N 2 O density 892 kg/m 3 87

8 Gaseous N 2 O Viscosity properties Use Southerland s Law (semi-empirical fit) for Viscosity terms T µ(t 0 ) = µ 0 s T s 3/2 T s + C s T 0 + C s C s = 240 o K T s = 300 o K µ s = pa sec T 0 = actual flame temperature Assume Boundary Layer Prandtl Number of 0.5 Assume combustor efficiency (C * actual/c * ideal) =

9 HTPB Properties Solid propellant density of 930 kg/m 3 Latent heat of vaporization (h n ), 1.8 MJ/kg Solid Grain temperature (assume constant) 300 K 89

10 Build Dynamic Response Model.. Assume averaged properties are constant as a function of distance along the propellant grain length Use Isentropic analysis for Nozzle $ & M O /F (t) = P 2/3 r & % h νsolid fuel Δh flame fuel ' ) ) ( 0.23 $ m & ox % µ e L ' ) ( 1/5 $ & & % D L t 0 r ( τ ) dτ ' ) ) ( 3/5 M O /F +1 M O /F Use integrator of your choice 90

11 Homework #6 (cont d) Select Injector pressure to give an approximate steady burn Thrust of 8 knt Choose optimal propellant masses to give a 10 second burn time Plot predicted linear regression rate as a function of N 2 O mass velocity and total mass velocity!r vs ρ e U e =! m ox A c ρ t U t =! m total A c Repeat Analysis to calculate the motor length that gives the optimal (or very close) O/F Ratio 6

12 State Vector 91

13 Calculation Summary State r Vector X (t) = (t ) State Equations X! (t ) = t!x (t ) = F X (t ) P 0 ( t ) M ox ( t ) M Fuel ( t ) Chamber pressure Mean fuel port radius = Current oxidizer mass Current fuel mass P 0 A burn r fuel ρ V fuel R g T 0 P 0 P A * 0 c r c p T 0 T fuel = P 2/3 r ρ fuel h ν fuel M ox M fuel 2 γr g T 0 V c γ γ+1 ( γ 1) A ox C dox 2ρ A ox p ox P 0 c A ox C dox 2ρ ox ( p ox P 0 ) ρ fuel A burn r + R T g 0 A ox C dox 2ρ ox ( p ox P 0 ) V c ( ) 4 5 µ ox L 1 5 = F X (t ) 92

14 Calculation Summary (2)!X (t ) = F X Integrate Trapezoidal Rule (t ) sample interval =Δt X k+1 = X k + Δt F X k X k+1 = X k + Δt F X k + F X k+1 2 D port(t ) = 1 2 r (t ) A burn( t) = π D port(t ) L port A c(t ) = π 4 D 2 port(t ) Geometry Calculations V c(t ) = A c(t ) L port A ox = N inj π 4 D 2 inj A * = π 4 D throat G ox (t) = t M ox (t ) A c(t ) =!m ox (t) A c(t ) 93

15 Calculation Summary (3)!m nozzle (t) =!m exit (t) = P 0 (t) A * γ R g T 0actual 2 γ +1 γ+1 γ 1 De Laval Nozzle Calculations M exit A exit A = 1 2 γ 1 1+ * M exit γ +1 2 M exit p exit (t) = P 0 (t) T exit = T 0actual F vac (t) = λ exit!m exit (t) M exit 1+ γ 1 2 M exit 1+ γ 1 2 M exit 2 2 γ γ 1 2 1γ+1 2 γ 1 γ R g T exit + A exit p exit (t) λ = 1 ( 2 1+ cosθ exit ) F SL (t) = F vac (t) A exit p SL I spvac (t) = F vac (t) g 0!m exit (t) I sp SL (t) = F SL (t) g 0!m exit (t) I sp mean (T ) = 0 T F vac (t) dt g 0 M ox (T )+ M fuel (T 94 )

16 Calculation Summary (4) End of Frame Calculations O / F (t ) = M ox (t ) M fuel (t ) { γ, M W,C p,c v, R } g CEA( P (t ) 0 (t),o / F (t ),η * ) η * = T 0 act T 0ideal Enhancement Option Total Flux Mean Regression Rate r = P 2/3 r ρ fiel Δh flame surface h v 0.23 G 1/5 ox /3 P r Δh flame surface h v 0.23 µ L 1/5 L D port 95 4 µ L 1/5

17 C* max ~ M O/F = 6.0 Initial Conditions. Start with this value Pick P 0 (o) ~ 2 P amb Pick T 0 (o), g (o), MR (0) corresponding to MR = 6.0 (Table Lookup) compute initial g, c p, µ, etc. [ T 0,R g,m W,g ] = f(m R ) 96

18 INITIAL CONDITIONS: MR~6, P0 ~ 2 pamb (enough to choke nozzle) Table Lookup Combustion/ Ox Gas Properties η Startup Initial Geometry / Mass Properties { Rport, Mox, Mfuel } 97

19 Time Loop, once per sample interval { Rport, Mox, Mfuel } Homework #7 (Project 2) (cont d) Δt { Ρfuel, hvfuel, Tfue} { A*, Lchamber, Aox, CdOx } Integrate Chamber Pressure Dynamics Equations Use Variable Time step Be sure to Capture Initial Startup Transient { A*, Aexit } Isentropic Nozzle Equations Evaluate New Gas properties at End of each frame η Combustion/ Ox Gas Properties Table Lookup use new props. and state vector For each new frame 98

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