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1 列年 Journal of ronautcs Astronautcs and Aaton Srs B Vol.4 No. pp (009) 77 * 立路理論利力漏利率拉雷諾量力率行理論 / 數易參數力狀 / 廉率良理論度 R& of Nol Gun/Watrjt Rockt Launchng Systm Mng-Fang Lu Hsng-Jun L Chh-Luong Chang and Shu-Hao Chuang** partmnt of Mchancal Engnrng Natonal Chung-Hsng Unrsty No. 50 Kuo Kuang Rd. Tachung 40 Tawan R.O.C. partmnt of Occupaton Safty and Halth Chung Hwa Unrsty of Mdcal Tchnology No.89 Wnhwa st St. Rnd Shang Tanan 77 Tawan R.O.C. ABSTRACT Th rsarch s focusd on gun/watrjt rockt and ts launchr wth flash-collapsbl lockng mchansm. To mak us of th adancd frst gun scond rockt propulson thory w slct PU as matral for launchr bas and assocatd xtrudng md-rod matchng wll wth nozzl as th launchng gud to gan som ntal rockt locty by outr-shll ffct. In ths mannr w can dramatcally mpro th propulson ffcncy of conntonal watrjt rockt and sol ts many nhrnt * 97 年年 9 ** To whom corrspondnc should b addrssd E-mal: shachuang@mal.hwa.du.tw

asly adjustabl ky paramtrs such as ntrnal prssur ar/watr rato nozzl sz shap/ar rsstanc launchr condtons and tc.

2 78 呂李 troublsom problms. Moror wth th ad of our arlr Lagrangan Rynolds transport quaton momntum quaton and rockt gnralzd total kntc powr quaton w analyz and tst th propulson prformanc of ths nol launchng systm ncludng a fw asly adjustabl ky paramtrs such as ntrnal prssur ar/watr rato nozzl sz shap/ar rsstanc launchr condtons and tc. u to smplcty low-cost and abundanc of thsasly adjustabl paramtrs ths launchng systm shows hgh alu for acadmc propulson rsarch and prod mportant rlatons for futurstc arospac nnoat launch mods. Kywords: Flash-collapsbl launchr Gun/watrjt rockt Analyss and xprmnt () 力來量力易廉樂年來行 Mazu t al.(996) RINA(999) 利 (PET Bottl) 行力 [] 行力行力力力流流率量 [3] 率不 000 C.C. 0~50 m 易漏漏 PET 不數累漏連了利數 () 000 C.C. PET 050 m 3 4 Braun(967) Krrbrock(99) [45] 理論參數率率率度流更良參 [67] 拉雷諾念量力率 [7-0] 念量參數行力行理論數理量利不 0.5 m 率度流不流率力 C.C.

砲式氣壓水箭新型發射系統之研發 79 ΣF = ΣFxtsurf + P M (t ) g (t ) = (V ) ρ d t = V M + V M t t K K ρ d CScomong ρ da t () = MV ρ d m t 上式中因為瞬間之系統質量不隨時間改變 M (t ) = 0 故 t 可得圖 4 含自動開傘系統砲式氣壓水箭發射情形

Transport Equaton [6-9] Lagrangan 在此意謂著追蹤火箭外殼 ( Bsyst ) t = K K βρ d + CScomong βρ da t () 其中 B 為某一特定物理量 β 為單位質量之某特定物理量為流體相對於火箭之控制體積的相對速度 CS 後面加上 comong 表示隨著火箭一起移動我們並採用最

而火箭推進系統之通化總動力功率 (TKP Gnralzd Total Kntc Powr)即真正由系統本身所發出的功率包含系統本身動能每秒的變化及火箭推進系統對外在環境做出的功率 W ( Ek syst ) t + W = (V ) ρ d + W.

3 砲式氣壓水箭新型發射系統之研發 79 ΣF = ΣFxtsurf + P M (t ) g (t ) = (V ) ρ d t = V M + V M t t K K ρ d CScomong ρ da t () = MV ρ d m t 上式中因為瞬間之系統質量不隨時間改變 M (t ) = 0 故 t 可得圖 4 含自動開傘系統砲式氣壓水箭發射情形 Fxt surf + P M (t ) g (t ) + m = M (t ) V ρ d t 二火箭相關推進動力公式通常分析火箭推進系統(圖 5)之推進性能均採傳統雷諾輸送公式 Rynolds Transport Equaton 處理但其並非最直接而自然者故本文採用拉氏雷諾輸送公式 LRTE Lagrangan Rynolds Transport Equaton [6-9] Lagrangan 在此意謂著追蹤火箭外殼 ( Bsyst ) t = K K βρ d + CScomong βρ da t () 其中 B 為某一特定物理量 β 為單位質量之某特定物理量為流體相對於火箭之控制體積的相對速度 CS 後面加上 comong 表示隨著火箭一起移動我們並採用最先進的交織輸送法(Intrwad Transport raton)可高效率且更清晰地推導火箭之動量方程式及相關的重要通化方程式[8] 其過程如下 (3) (3)式中 V 為火箭外殼(亦即控制體積)速度 Fxt surf 為作用在火箭外表面的力 (如風壓阻力黏滯力等 ) P 代表火箭出口的壓力與面積 g(t)為重力加速度 m 為出口質流率而火箭推進系統之通化總動力功率 (TKP Gnralzd Total Kntc Powr)即真正由系統本身所發出的功率包含系統本身動能每秒的變化及火箭推進系統對外在環境做出的功率 W ( Ek syst ) t + W = (V ) ρ d + W. t (4) 進一步分析可得 ( Ek syst ) t + + W = V ρ d Vc ρ d t t ρ d + W t = V [V M (t )] + Vc ρ d V ρ d t t V ρ d + ( ) ρ d + W. t t (5) 接著利用 LRTE 將(5)式處理如下 + W = V V M (t ) ρ d t t ( ) ρ d V ρ d + (6) t ( Ek syst ) V m + m + W 圖 5 火箭的推進系統概圖上式中之 W 可表示為

4 80 呂李 W = ( F ) V P A ( V ) xt surf c c ()( ) + g t V ρ d = ( F ) V PA ( V ) + V g() t M () t g() t ρ d (3)(7)(6) 理 TKP surf c c E ( ksyst ) + W = PA g() t ρ d t V d + d + m ρ ( ) ρ t (7) (8) + ( ) ρd + m t comong + m ( V ) ρd t comong PA ( V) m ( V) [( ) ( )] () = V PA + m g t ρd V ρd + t ( ) ( ) V d t ρ 率 () TKP 不參不 (8) TKP 立 V 不論 TKP 量 TKP 力若來更參了利力率 (TKP) 來利流流來率 (APP Gnralzd Aalabl Propulson Powr)[8] ( ) ( ρ ) t APP = TKP V d (9) 流量 ( ρ ) 流量流率 t (8)(9) APP[8] E ( ksyst ) ( ) APP = + W V d t ρ t = PA g t ρ d V ρ d () + ( ) ρ d + m t ( V ) ( ρ d ) t (9) (0) 力率 (TP Gnralzd Thrust Powr) APP 率流 TP = APP PA ( V) m ( V) = PA g( t) ρ d V ρ d APP PA ( V) m ( V) TP η P = = APP APP () =. (0) () 利量 TKP 來力 (3) F xt surf 力 [89] Fxt surf = ρ arv ACd (3) ρ ar 度 A C d 力數流不流流流利量律 = (4) A A 度量律流理 t ρ = ρ d d t + ρ d t t A = m m A t A (5) m 量 (3)(5)(3)

5 8 ρarv AC + PA M() t g() t A = M () tv m + ρa ρa A t A (6) 力率 (8) 率 E ( ksyst ) + W = PA (7) t P 力力力率利量念 ((4)) 理力率 (8) 理 A g() t ρd = g() t ( m ) = g() t m A (8) (8) V ρd = V ( m ) = V m A (9) 3 A A 3 ρd = m ρ t A t A comng (0) 力率 (8) 理 A A PA = PA g() t m V m A A m ρ + ρa A t A. () (6)() 兩參數了力 P g(t) 9.8 m/s P 兩 V ν V t 數兩數兩數數了聯立度 V 度 t 兩數度數便 V ν 便行利量力率聯立來力率 () 度 V V A m PA PA A t = g() t + ( m ) m A A ρ AA R ρ A ( ma R ) ( ma R ) () () 量 (6) 流度 = { ρarv AC + PA M( t) g( t) t nomnator A 3 ( ) 3 ρ ρ A PA PA A Mt () gt () + A A m ( m ) ( m ) A A A + ρa } ( ρaa) R ( ma R ) ( ma R ) ( mar) nomnator = M ( t). (3) V Δt 理量度流度 V ( t + Δ t) = V ( t) + V Δt (4) ( t+ Δ t) = ( t) + Δt t (5) 理 ()(3) 度流度行行度數 St ( + Δ t) = St () + V () tδ t. (6) 4. 若若度 [80] 度理論來

8 呂李利數力行度 0.5 m 滑行省行易漏量力 (PU Polyurthan) 6 7 度.

0638 m 8 448 kpa 6 9 448 kpa 0 ( ) k Px ( ) = P (7) x 0 ( )

6 8 呂李利數力行度 0.5 m 滑行省行易漏量力 (PU Polyurthan) 6 7 度.5 m m 0.4 mm m kpa kpa 0 ( ) k Px ( ) = P (7) x 0 ( ) 量.9 kg 9 不例行利理力行降流流率力度降力狀理列列 P 0 0 P(x) ( x) 離 x 離 ( x) = 0 + Ax k.4 x +Δx x x+δ x x+δx PxAdx ( ) = Mgx s x + MV ( ) s x x (8) M s 理 V ( x +Δx) k k+ x +Δx = P0 0 ( x) x M ( sgδ x+ MsV x) Ms k + (9)

7 83 利數離 Δt 量 ( 流 ) Δ ( t+ Δ t) = A ( t) Δt ( t+δt) k k P() t () t = P( t +Δt) ( t+δ t). (30) 4.3 行度若降漏洞流不不力行不量數力數.8 量.7 kg m m 0.5 m 度.5 m 量 0 不例不力了 kpa 力 0. kg 0 量 m 3 降量力 448 kpa 0 m (Kpa) (Kg) (m) 理論率拉雷諾量力率 [8] 行理論 / 數參數力量度度狀 / 廉參數率良理論度利力漏率易漏漏數 448 kpa m 3 不離度 40 m/s 易 36 m 易 50% 來念參量來力若滑車 465 m/s [0] 力車利 [0] 度率 [8] 力力力不更量不論行度度 [80] 理度率 0

8 84 呂李 [] Mazu B K. Makng of PET-Bottl Rockt st d. Ech Publcaton Co. Tokyo 996. (n Japans) [] Th Royal Insttuton of Naal Archtcts Watrjt Propulson Th Naal Archtct 999. [3] Ly W. Rockts Mssls and Mn n Spac nd d Vkng Prss Nw York 968. [4] Braun W. V. and Frdrck I. O. Hstory of Rocktry & Spac Tral Thomas Y. Crowll 967. [5] Krrbrock J. L. Arcraft Engns and Gas Turbns nd d. Th MIT Prss Cambrdg MA. 99. [6] L H. J. and Huang S. C. On thraton Procss of Rynolds Transport quaton Th Intrnatonal Journal of Mchancal Engnrng Educaton Vol. No. 993 pp [7] L H. J. and L H. W. rng th Gnralzd Rockt Kntc Powr Equatons and Assocatd Propulson Indxs Th Japan Socty of Mchancal Engnrs Intrnatonal Journal Srs B Vol. 4 No. Fbruary 999 pp [8] L H. J. and Chang C. L. rng th Gnralzd Powr and Effcncy Equaton for Jt Propulson Systm Th Japan Socty of Mchancal Engnrs Intrnatonal Journal Intrnatonal Journal Srs B Vol. 44 No pp [9] L H. J. Chu C. H. and Hsa W. K. Intgratd Enrgy Mthod for Propulson ynamcs Analyss of Ar-Prssurzd Watrjt Rockt Th Japan Socty for ronautcal and Spac Scncs Vol. 44. No pp. -7. [0] Yuan K. L. L H. J. and Ln H. Propulson Suprorty Analyss of U-turn Launch Mod for Satllt Rockt Th Japan Socty for ronautcal and Spac Scncs Vol. 5 No pp

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