Page 1
Fusee a eau. (Compressed air over water rockets) msc Performance parameters Created by John Gwynn September 2002 © All rights reserved . Launcher Physical characteristics Nozzle Diameter mm d 2 (π)r2 Nozzle Area m Na Length of rocket body from tip to nozzle(L) mm Diameter of rocket body (D) mm Aerodynamic ratio L/D Type of bottle used for main reservoir Volume of bottle ml Type of bottle used for payload capsule/ Coiffe Height of liquid level Hydraulic head (h) mm Rocket launch pressure Bar. Pe bar 2 Launch pressure in Pa or N/m Density of liquid Kgs/m3 Dn (ρ) 3 Da Density of air/gas Kgs/m 2 Acceleration due to gravity (g) 9.81 m/s Discharge coefficient 0,97 > 0,998 W Coefficient of contraction 0,52 > 0,99 X 2*g*h 2*Pa/Dn 1 Jet velocity V1 from a pressurised reservoir 0.5 Note to power 0.5 V1=W *(2gh+ 2Pa/Dn) Velocity of water jet discharged V1 m/s
*
Comparison Table showing how parametrs change when you vary one parameter at a time> In this case Launch pressure in Bar Badoit 0,33L
Exp. 22 0,000380133 520 80 6,5 Badoit 1L 1000 Aix les Bains 230 7 709275 1000 1,1614 9,81 0,97 0,8 4,5126 1418,55 37,72350196
Α1
ρ
ψ χ
36,5917969
Air jet 2*Pa/2 Da Air jet V1=W *( Pa/Dn)0.5 Note to power 0.5 3.Tsiolkovski Parameter V2=V1*LN(M1/M2) m/s Mass initial gms M1 Mass final gms M2 Ratio M1/M2 Ln M1/M2 3 Volume of liquid ml cm Volume of air ml cm3 Final velocity of fusee/rocket V2 Tsiolkovski formula V2=V1*LN(M1/M2) m/s F= Pe*At*Cf Cf divergent expansion nozzle coef.1.6>2.0 4.DischargeFlow Rate Qs m3/s Qs= X*A*V1 m3/s Qs Equivalent ml/s Mass flow rate m* Kgs/s Tjet Duration of jet stream Qs/B23 Seconds Air Qs= X*A*V1 m3/s Air Qs Equivalent ml/s Air Mass flow rate m* Kgs/s
22 22 22 0,000380133 0,00038013 0,000380133 520 520 287 80 80 58 6,5 6,5 4,95 Badoit 1L Badoit 1L Badoit 0,33L 1000 1000 350 Aix les Bains Aix les Bains 230 230 125 7 9 9 709275 911925 911925 1000 1000 1000 1,1614 1,1614 1,1614 9,81 9,81 9,81 0,97 0,97 0,97 0,8 0,8 0,8 4,5126 4,5126 2,4525 1418,55 1823,85 1823,85 37,72350196 42,7593569 42,73526062
22 22 22 22 22 22 22 22 22 22 0,000380133 0,00038013 0,000380133 0,000380133 0,000380133 0,000380133 0,000380133 0,000380133 0,000380133 0,000380133 520 520 520 520 520 520 520 520 520 520 80 80 80 80 80 80 80 80 80 80 6,5 6,5 6,5 6,5 6,5 6,5 6,5 6,5 6,5 6,5 Badoit 1L Badoit 1L Badoit 1L Badoit 1L Badoit 1L Badoit 1L Badoit 1L Badoit 1L Badoit 1L Badoit 1L 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 Aix Aix Aix Aix Aix Aix Aix Aix Aix Aix 230 230 230 230 230 230 230 230 230 230 2 3 4 5 6 7 8 9 10 15 202650 303975 405300 506625 607950 709275 810600 911925 1013250 1519875 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 9,81 0,97 0,8 4,5126 405,3 20,24382869
9,81 9,81 9,81 9,81 9,81 9,81 9,81 9,81 0,97 0,97 0,97 0,97 0,97 0,97 0,97 0,97 0,8 0,8 0,8 0,8 0,8 0,8 0,8 0,8 4,5126 4,5126 4,5126 4,5126 4,5126 4,5126 4,5126 4,5126 607,95 810,6 1013,25 1215,9 1418,55 1621,2 1823,85 2026,5 24,7479817 28,55017688 31,90239176 34,93440425 37,72350196 40,32012649 42,75935687 45,06675715
9,81 0,97 0,8 4,5126 3039,75 55,17483666
19,63651383
24,0055423 27,69367158 30,94532001 33,88637212
36,5917969 39,11052269 41,47657616 43,71475444
53,51959156
36,5917969
41,4765762
41,4532028
305353,4527 305353,4527 552,5879593 552,5879593 536,0103205 536,0103205
392597,296 626,57585 607,778575
392597,2964 626,5758504 607,7785749
600 750 100 118 6 6,355932203 1,791759469 1,849388582 500 632 500 368
750 118 6,3559322 1,84938858 632 368
245 70 3,5 1,252762968 175 175
600 100 6 1,791759469 500 500
600 600 600 600 600 600 600 600 100 100 100 100 100 100 100 100 6 6 6 6 6 6 6 6 1,79175947 1,791759469 1,791759469 1,791759469 1,791759469 1,791759469 1,791759469 1,791759469 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500 500
600 100 6 1,791759469 500 500
65,56369859 67,67245138
43,0121577 49,62039828 55,44657015 60,71622812 65,56369859 70,07664938 74,31604809
95,89423496
76,7063064
51,93103739
35,18390959
1
1
1
1
1
1
1
0,011127791 11127,79117 11,12779117 0,044932547
0,011127791 11127,79117 11,12779117 0,056794739
0,01261328 12613,2827 12,6132827 0,05010591
0,012606175 12606,17469 12,60617469 0,013882086
0,005971585 5971,58499 5,97158499 0,083729864
0,00730023 7300,2335 7,3002335 0,06849096
0,008421816 8421,816365 8,421816365 0,059369616
0,163004045 0,163004045 163004,045 163004,045 0,195930862 0,195930862 0,006134817 0,004515225
0,18482921 184829,214 0,22216472 0,00398205
0,184829214 184829,2139 0,222164715 0,00189364
37142,85714 37142,85714
1
1
1
1
1
78,3263252 1
1
0,009410663 0,010305055 0,011127791 0,011893751 0,012613283 0,013293926 9410,662712 10305,0548 11127,79117 11893,75122 12613,28268 13293,92649 9,410662712 10,3050548 11,12779117 11,89375122 12,61328268 13,29392649 0,053131221 0,048519878 0,044932547 0,042038882 0,039640751 0,03761116
0,016275638 16275,63795 16,27563795 0,030720762
37142,8571 37142,85714 37142,85714 37142,85714 37142,85714 37142,85714 37142,85714 37142,85714
37142,85714
Created by John Gwynn for Alex and Gabriel
[email protected] © All rights reserved Kinematic viscosity of air at 200m Vis 14* 10-6 m2/s
γ
L/Vis ρ
Density of air (de) at 200m Kg/m3 Coeficient of drag Cd 0,05 Equiv L/D=6 Cd varies from 0,4 > 0,05 Cd depends on aerodynamic pure form shape Cross section area A m2 V2 Velocity of fusee/rocket. m/s Vair Head on wind velocity m/s Reynolds NumberRe =(V2+Vair) L/Vis m/s Reynolds NumberRe ~72000(V2+Vair) L m/s 2
(V2+Vair) 5.Aerodynamic drag Force (Fd) Fd = 0,5(de)(Cd)A(V2+Vair) 2
N
6.Thrust Force F= q* V1 + ( Pe-Pa)*Se N Mass flow from nozzle/ Debit d'gas q Kgs/s q*V1 N q*V2 N Velocity of fluid /gas V1 F= q* V1 + ( PI-Pa)*Se N Note: For ideal nozzle Pl=Pa F= q* V2 + ( PI-Pa)*Se N 1/g 7. Specific Impulse Specific Impulse Isp= F/g.m* Sec Specific impulse 2 for water jet period t F(t) /(g*m) Specific impulse due to comp.air Ariane5 F=Thrust force N, m* =mass flowrate Kgs/s Acceleration a F=m*a Acceleration a1 =F/ M1 m/s2 2 F=m*a a3 =F/ M1 m/s F=m*a a2 =F/ M2 m/s2 Multiple of g 9.81 m/s2
ιω ια
37142,8571
20500
37142,85714
1,202
1,202
1,202
1,202
1,202
1,202
1,202
1,202
1,202
1,202
1,202
1,202
1,202
0,05
0,05
0,05
0,07
0,05
0,05
0,05
0,05
0,05
0,05
0,05
0,05
0,05
0,05
0,005026548 0,005026548 65,56369859 67,67245138 0 0 2435223,091 2513548,194 2454704,875 2533656,58
0,00502655 76,7063064 0 2849091,38 2871884,11
0,002642079 51,93103739 0 1064586,267 1073102,957
0,005026548 35,18390959 0 1306830,928 1317285,575
0,00502655 43,0121577 0 1597594,43 1610375,18
0,005026548 49,62039828 0 1843043,365 1857787,712
0,005026548 55,44657015 0 2059444,034 2075919,586
0,005026548 60,71622812 0 2255174,187 2273215,581
0,005026548 65,56369859 0 2435223,091 2454704,875
0,005026548 70,07664938 0 2602846,977 2623669,753
0,005026548 0,005026548 74,31604809 78,3263252 0 0 2760310,358 2909263,508 2782392,841 2932537,616
0,005026548 95,89423496 0 3561785,87 3590280,157
4298,598573 4579,560676
5883,85744
2696,832645
1237,907494
1850,04571 2462,183926 3074,322142 3686,460357 4298,598573 4910,736789 5522,875004
6135,01322
9195,704299
0,649293749
0,69173245
0,88874357
0,299759101
0,186983172
0,27944529 0,371907403 0,464369518 0,556831634 0,649293749 0,741755864
0,83421798 0,926680095
1,388990672
11,12779117 11,12779117 407,1858743 407,1858743 729,5791461 753,0449068
12,6132827 523,15578 967,52
12,60617469 522,566316 654,6517292
5,97158499 117,2611112 210,1037064
7,3002335 8,421816365 9,410662712 10,3050548 11,12779117 11,89375122 12,61328268 13,29392649 175,246064 233,2310165 291,2159691 349,2009217 407,1858743 465,1708269 523,1557796 581,1407322 313,998794 417,8938823 521,7889702 625,6840581 729,5791461 833,474234 937,3693219 1041,26441
16,27563795 871,0654953 1560,739849
407,1858743 407,1858743
523,15578
522,566316
117,2611112
175,246064 233,2310165 291,2159691 349,2009217 407,1858743 465,1708269 523,1557796 581,1407322
871,0654953
0,101936799 0,101936799
523,15578 0,1019368
522,566316 0,101936799
0,101936799
0,1019368 0,101936799 0,101936799 0,101936799 0,101936799 0,101936799 0,101936799 0,101936799
0,101936799
3,730050652 3,730050652 7,460101305 7,460101305
4,22798942 8,45597883
4,225606809 8,451213618
2,001683367 0,167600676
2,44704814 2,823004238 3,154466871 3,45426831 3,730050652 3,986801498 4,227989415 4,456142144 0,16760068 0,167600676 0,167600676 0,167600676 0,167600676 0,167600676 0,167600676 0,167600676
5,455615857 0,167600676
1,202
431
1214,883087 4071,858743 7289,298523 123,8412933 743,0477597
1003,137566 3450,727749 6375,874359 102,2566326 649,9362241
1288,83944 4433,52356 8191,77612 131,380167 835,043437
2670,824368 7465,233085 9347,885287 272,2552872 952,8935053
350,1728441 1172,611112 2101,037064 35,69549889 214,1729933
523,331324 1752,46064 3139,98794 53,3467201 320,08032
696,4898038 2332,310165 4178,938823 70,99794127 425,9876476
869,6482837 2912,159691 5217,889702 88,64916246 531,8949747
1042,806764 3492,009217 6256,840581 106,3003836 637,8023019
1215,965243 1389,123723 1562,282203 1735,440683 4071,858743 4651,708269 5231,557796 5811,407322 7295,791461 8334,74234 9373,693219 10412,6441 123,9516048 141,602826 159,2540472 176,9052684 743,709629 849,6169562 955,5242833 1061,43161
2601,233082 8710,654953 15607,39849 265,1613744 1590,968246
407,1858743 407,1858743 485,3135316 431,3898058 539,2372573 5392,372573
407,1858743 407,1858743 485,3135316 431,3898058 539,2372573 4569,807265
523,15578 523,15578 623,974541 554,644 693,305045 5875,46648 49,469639 91,4885855
522,566316 522,566316 623,9745406 554,6440 693,3050451 9904,357787 49,49753243 62,00867566
117,2611112 1 138,661009 123,2542302 154,0677878 1540,677878
175,246064 1 207,991514 184,881345 231,101682 2311,01682
233,2310165 1 277,322018 246,5084605 308,1355756 3081,355756
291,2159691 1 346,6525226 308,1355756 385,1694695 3851,694695
349,2009217 1 415,9830271 369,7626907 462,2033634 4622,033634
407,1858743 1 485,3135316 431,3898058 539,2372573 5392,372573
465,1708269 1 554,6440361 493,016921 616,2711512 6162,711512
871,0654953 1 1039,957568 924,4067268 1155,508409 11555,08409
0,040
0,040
0,040
0,040
0,040
0,040
0,040
Refer to specific duct formula developed for modelling output before adding expansion nozzle. 8.Potential Increase in jet efficiency by using expansion nozzle Launch Force (F) N without Tuyau/ divergent nozzle Cf>1 Cf divergent expansion nozzle coef.1.6>2.0 F= Pe*At*Cf Assume Cf=1.8 N
1,0 1,8 1,6 2,0 2,0 1,8
Potential acceleration at launch with divergent nozzle/tuyau m/s2 Modified V1 for expansion nozzle V1=F/q
Measured Velocity at decollage Launch pressure Pe=7 bar No divergent tuyau /nozzle Cf>1 Test Data Phase2 Velocity 3,18m at 0,04s > m/s Phase1 Velocity m/s Ln M1/M2 Calculated V2 using Tsiolkovski formula > 2 Measured acceleration (h1)2.24m >(h2)5,59 t0.04s m/s Measured acceleration (h0)0m >(h2)5,59 t0.08s m/s2 Rateof change of acceleration per second. Flight time to apogee tap sec Impulse time ι Altitude 250>270m
27.10.02 83,75 56,15 1,84938858 103,843169 693,75 1046,875 13085,9375 3,280 4,52 0,040 0,05010591
523,1557796 581,1407322 1 1 623,9745406 693,3050451 554,6440361 616,2711512 693,3050451 770,338939 6933,050451 7703,38939
79,500 79,500 44,370 42,987 1,791759469 1,849388582 79,5
3,280 0,040
0,040
0,040
0,040
Created by John Gwynn for Alex and Gabriel
[email protected] © John Gwynn 2003 All rights reserved World Record.323,092m (1060 ft.) Approximations>Research Launch pressure Pe bar Tsiolkovski formula V2=V1*LN(M1/M2) V2 m/s Fd Drag force N Thrust Force F N Tjet s Height of jet Hjet m (Constant Kh )
7
7
9
9
2
3
4
5
6
7
8
9
10
15
35,184 0,187 117,261 0,084 1,644
43,012 0,279 175,246 0,068 1,644
49,620 0,372 233,231 0,059 1,644
55,447 0,464 291,216 0,053 1,644
60,716 0,557 349,201 0,049 1,644
65,564 0,649 407,186 0,045 1,644
70,077 0,742 465,171 0,042 1,644
74,316 0,834 523,156 0,040 1,644
78,326 0,927 581,141 0,038 1,644
95,8942 1,3890 871,0655 0,0307 1,644
35,695
53,347
70,998
88,649
106,300
123,952
141,603
159,254
176,905
265,1614
84,04148258
125,599518 167,1575529 208,7155881 250,2736233 291,8316584 333,3896936 374,9477288 416,5057639
624,2959398
21,83210941
32,6279633 43,42381729 54,21967123 65,01552517
75,8113791 86,60723304 97,40308698 108,1989409
162,1782106
91,489
κ
Acceleration in ( g ) Mulitple of g ( 9.81m/s2 ) Height of flight approximation (Apogee) Assume body thrown vertically upwards after 3m No aero drag 2 1. Basic approximation h2 = (V2) /2g m. 2. Observations Gwynn Specific impluse factor a2*ι > h3 m Measured acceln./jet Impulse time. > V1 2 3. Gwynn parameter a2 /g > v2 Modified Aerodynamic drag Force for higher V2 (Fd) 2 (V2+Vair) 2 Fd= 0,5(de)(Cd)A(V2+Vair) 2 h = V2.t + 0.5.a.t 4.Neglecting air resistance h = v1t - 0.5(9.81)t2 Normally t=V2/g Try tapogee V2*t 2 JG Exp.value>1 h = t* a2 /g > Approx. V2 (Value130) * t 0.5(9.81)t2 4.Neglecting air resistance h=v1t - 0.5(9.81)t2
m
0,045 1,644
0,057 2,078
0,050 2,078
0,014 0,575
322,1330275 322,1330275 291,5719409 255,0349743 75,74391027 66,25241836
426,613725 410,456393 34,7609747 85,1216551
195,9773628 0 0 97,1349139
5737,139943 4389,382939 0,866582222 0,663006526
7245,69617 1,09444628
9435,191498 1,048743059
1237,907494 0,186983172
1850,04571 2462,183926 3074,322142 3686,460357 4298,598573 4910,736789 5522,875004 6135,01322 0,27944529 0,371907403 0,464369518 0,556831634 0,649293749 0,741755864 0,83421798 0,926680095
9195,704299 1,388990672
8,103975535 8,103975535 3,280 3,280 260,76 260,76 248,4400257 217,3079322 52,769952 52,769952 207,990048 207,990048
8,53720693 4,250 355,9375 361,767034 88,5965625 267,340938
0 4,250 0 412,8233841 88,5965625 -88,5965625
3,280 115,4032235 71,60931886 52,769952 62,63327147
3,280 3,280 3,280 3,280 3,280 3,280 3,280 3,280 141,079877 162,7549064 181,8647501 199,1492282 215,0489314 229,85141 243,7566377 256,9103467 107,01972 142,4301207 177,8405216 213,2509225 248,6613235 284,0717244 319,4821253 354,8925262 52,769952 52,769952 52,769952 52,769952 52,769952 52,769952 52,769952 52,769952 88,3099253 109,9849544 129,0947981 146,3792762 162,2789794 177,081458 190,9866857 204,1403947
3,280 314,5330907 531,9445308 52,769952 261,7631387
67,67245138 20 6,56 3,048780488 9,295062463 0,548408685 0,663006526
83,75 10 10,56 0,9469697 0,89675161 0,05290834 1,09444628
83,75 10 10,56 0,946969697 0,896751607 0,031386306 1,048743059
33,83622569 3,28 110,9828203 73,58233406 1,058438532 0,731512514 384,7043185 268,2955612 220,0023602
41,875 41,875 3,28 3,28 137,35 137,35 150,322197 144,0448592 1,05138244 1,013978889 0,73151251 -0,552590906 497,588469 515,3621257 299,947618 540,681346 245,957047 443,3587037
35,18390959 20 6,56 3,048780488 9,295062463 0,464753123 0,186983172 0,186983172 17,5919548 3,28 57,70161174 10,78923042 1,087335126 0,497231218 107,842659 98,45940417 80,73671142
43,0121577 20 6,56 3,04878049 9,29506246 0,46475312 0,27944529 0,27944529 21,5060789 3,28 70,5399386 19,7120534 1,07089094 0,49723122 163,645108 146,246994 119,922535
95,89423496 20 6,56 3,048780488 9,295062463 0,464753123 1,388990672 1,388990672 47,94711748 3,28 157,2665453 218,4417645 1,031197514 0,497231218 844,7125631 637,9266519 523,0998546
2
5.FLs = 0.5(M2)( VH) + M2.g.h + FD.Vm.tv Vinitial Horizontal Deviation in flight Max> 20m Flight time secs VH Note: at apogee Vv > 0 Horz velocity comp ( VH)2 0.5(M2)( VH)2 Aerodynamic drag force FD e Vm t Vm*t Fd*Vm*t e jets LN(Hjet) FLs Substituting and resolving for height ( h ) m Jet Efficiency expansion 100% when Pe=Pa
Possibly>
© All rights reserved by John Gwynn 2003 msc
[email protected]
0,82 ?Constant. K
79,5 20 6,56 3,048780488 9,295062463 0,464753123 0,866582222 2,718282 39,75 3,28 130,38 112,9849901 1,045957305 0,497231218 389,2949287 h 281,1877528 0,82 230,5739573
49,62039828 20 6,56 3,048780488 9,295062463 0,464753123 0,371907403 0,371907403 24,81019914 3,28 81,37745318 30,26487729 1,061167392 0,497231218 219,7872062 192,7192414 158,0297779
55,44657015 20 6,56 3,048780488 9,295062463 0,464753123 0,464369518 0,464369518 27,72328508 3,28 90,93237505 42,22622321 1,054568017 0,497231218 276,1471657 237,977767 195,1417689
60,71622812 20 6,56 3,048780488 9,295062463 0,464753123 0,556831634 0,556831634 30,35811406 3,28 99,57461412 55,44629506 1,049716238 0,497231218 332,6622083 282,1112743 231,3312449
65,56369859 20 6,56 3,048780488 9,295062463 0,464753123 0,649293749 0,649293749 32,7818493 3,28 107,5244657 69,81496345 1,045957305 0,497231218 389,2949287 325,1938962 266,6589949
70,07664938 20 6,56 3,048780488 9,295062463 0,464753123 0,741755864 0,741755864 35,03832469 3,28 114,925705 85,24681565 1,042935029 0,497231218 446,0209064 367,2878059 301,1760009
74,31604809 20 6,56 3,048780488 9,295062463 0,464753123 0,83421798 0,83421798 37,15802405 3,28 121,8783189 101,673085 1,040436931 0,497231218 502,8231542 408,4457861 334,9255446
78,3263252 20 6,56 3,048780488 9,295062463 0,464753123 0,926680095 0,926680095 39,1631626 3,28 128,4551733 119,0368523 1,038327411 0,497231218 559,6892905 448,7132366 367,944854
