Ifomula kaBernoulli: Izimiso kanye Nezicelo
Isimiso sikaBernoulli siwumqondo oyisisekelo ku-fluid mechanics, owatholwa nguDaniel Bernoulli, usosayensi waseSwitzerland, ngekhulu le-18. Lesi simiso sichaza ubudlelwano phakathi kwejubane lokugeleza koketshezi kanye nokucindezela okungaphakathi kwalokho kugeleza. Lesi sihloko sizoxoxa ngokujulile ngefomula kaBernoulli, imiqondo yayo eyisisekelo, imvelaphi yezibalo, izibonelo zokubala, kanye nokusetshenziswa kwayo empilweni yansuku zonke.
Ukuqonda Ifomula KaBernoulli
Ifomula kaBernoulli ithi ekugelezeni koketshezi okufanele (ngaphandle kokungqubuzana), inani lamandla omshini (amandla angaba khona, amandla e-kinetic, kanye nengcindezi) ngeyunithi ngayinye lihlala linjalo kulo lonke ukugeleza. Le fomula ichazwa kanje:
\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]
Kuphi:
– \( P \) ukucindezela koketshezi (kuma-pascals, Pa),
– \( \rho \) ubuningi boketshezi (ngamakhilogremu ngemitha yesikwele, kg/m³),
– \( v \) ijubane lokugeleza koketshezi (ngamamitha ngomzuzwana, m/s),
– \( g \) ukusheshisa okubangelwa amandla adonsela phansi (ngamamitha ngomzuzwana oyisikwele, m/s²),
– \( h \) ukuphakama koketshezi ngaphezu kwereferensi (ngamamitha, m).
Isimiso sikaBernoulli
Isimiso sikaBernoulli sisekelwe emthethweni wokulondolozwa kwamandla. Ekugelezeni koketshezi, amandla aphelele ohlelweni kumele ahlale engaguquki uma kungekho mandla angeziwe noma asuswa ohlelweni. Amandla aphelele ngeyunithi ngayinye aqukethe:
1. Amandla Okucindezela (\( P \)): Amandla abangelwa ukucindezela koketshezi.
2. Amandla e-Kinetic (\( \frac{1}{2} \rho v^2 \)): Amandla abangelwa ukunyakaza koketshezi.
3. Amandla Angase Adonselwe Ezandleni Zamandla Adonsela Phansi (\( \rho gh \)): Amandla abangelwa indawo yoketshezi ensimini yamandla adonsela phansi.
Ukutholwa Kwezibalo Kwefomula KaBernoulli
Ukuze siqonde ukuthi i-Formula kaBernoulli ivelaphi, sicabangela into encane yoketshezi ehamba ngokugeleza ngesivinini \(v \) kusukela ephuzwini 1 kuya ephuzwini 2. Uma sicabanga ukuthi akukho ukulahleka kwamandla ngenxa yokungqubuzana noma ukushisa, amandla aphelele kumaphuzu 1 no-2 kumele afane.
Amandla aphelele ephuzwini lesi-1 (\( E_1 \)) angu:
\[ E_1 = P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 \]
Amandla aphelele ephuzwini lesi-2 (\( E_2 \)) angu:
\[ E_2 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \]
Njengoba amandla aphelele kumele ahlale njalo:
\[ P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \]
Lokhu yilokho okubizwa ngokuthi iFomula kaBernoulli.
Isibonelo Sokubala Ukusebenzisa Ifomula KaBernoulli
Ake sithi sinepayipi elivundlile elinamanzi ageleza esuka endaweni A aye endaweni B. Endaweni A, ijubane lamanzi lingu-2 m/s kanti ingcindezi ingu-150,000 Pa. Endaweni B, ijubane lamanzi lingu-4 m/s. Ubuningi bamanzi bungu-1000 kg/m³. Bala ingcindezi endaweni B.
Sebenzisa ifomula kaBernoulli:
\[ P_A + \frac{1}{2} \rho v_A^2 + \rho gh_A = P_B + \frac{1}{2} \rho v_B^2 + \rho gh_B \]
Njengoba ipayipi livundlile, \( h_A = h_B \), ngakho \( \rho gh \) kuzo zombili izindawo zinganakwa:
\[ P_A + \frac{1}{2} \rho v_A^2 = P_B + \frac{1}{2} \rho v_B^2 \]
Faka amanani aziwayo esikhundleni sawo:
\[ 150,000 + \frac{1}{2} \times 1000 \times (2)^2 = P_B + \frac{1}{2} \times 1000 \times (4)^2 \]
Bala amandla e-kinetic kuzo zombili izindawo:
\[ 150,000 + 2000 = P_B + 8000 \]
\[ 152,000 = P_B + 8000 \]
\[P_B = 152,000 – 8000 \]
\[ P_B = 144,000 \, \umbhalo{Pa} \]
Ngakho-ke, ingcindezi endaweni B ingu-144,000 Pa.
Ukusetshenziswa Kwesimiso SikaBernoulli
1. Izindiza: Isimiso sikaBernoulli sisetshenziswa ekwakhiweni kwamaphiko ezindiza. Ukuma kwephiko lendiza kwaklanywa ngendlela yokuthi isivinini somoya ngaphezu kwephiko sibe sikhulu kunesingaphansi kwephiko. Lokhu kuholela ekucindezelekeni okuphansi ngaphezu kwephiko kanye nokucindezela okuphezulu ngaphansi kwephiko, okudala ukuphakanyiswa okuvumela indiza ukuthi indize.
2. Izimoto Zomjaho: Umklamo we-aerodynamic wezimoto zomjaho usebenzisa i-Bernoulli's Principle ukwandisa isivinini kanye nokuqina. Ama-spoiler kanye nama-diffuser asetshenziswa ukulawula ukuhamba komoya, okwandisa amandla okwehla, okuthuthukisa ukubambelela kwemoto emgwaqweni.
3. Ukugeleza Kwegazi Emzimbeni: Isimiso sikaBernoulli sisebenza nasekugelezeni kwegazi emithanjeni yegazi. Umehluko ngesivinini sokugeleza kwegazi kanye nokucindezela ezingxenyeni ezahlukene zomzimba ungafundwa ukuze kutholakale izimo zezokwelapha.
4. I-Venturi Meter: Idivayisi esetshenziselwa ukukala izinga lokugeleza koketshezi epayipini. I-Venturi meter inesigaba esinciphile, esibangela ukwanda kwejubane lokugeleza koketshezi kanye nokwehla kwengcindezi ngokuvumelana neSimiso sikaBernoulli. Umehluko wengcindezi phakathi kwezingxenye ezibanzi nezincane usetshenziselwa ukubala izinga lokugeleza.
5. Izifutho Zokutshiza Amakha: Izifutho zokutshiza amakha noma izifutho zoketshezi zisebenzisa isimiso sikaBernoulli. Uma umoya uvunguzwa ngepayipi elincane, umfutho womoya ozungeze ukuphela kwepayipi uyancipha, udonse uketshezi lwephunga elimnandi ngepayipi bese ulifutha njengenkungu encane.
Izici Ezithonya Ukusetshenziswa Kwesimiso SikaBernoulli
1. Ukungqubuzana: Eqinisweni, uketshezi alufaneleki futhi kuhlala kukhona ukungqubuzana kwangaphakathi (ukujiya) okubangela ukulahlekelwa amandla. Lokhu kwenza ukusetshenziswa kweSimiso sikaBernoulli kube nzima kakhulu.
2. Ukucindezelwa Koketshezi: Isimiso sikaBernoulli kulula ukusisebenzisa eziketshezini ezingacindezelwa (njengamanzi). Kumagesi noma eziketshezini ezingacindezelwa, i-equation kumele ilungiswe.
3. Ukuxokozela: Isimiso sikaBernoulli sisebenza ekuxokozeleni okujwayelekile (okuvamile). Ekuxokozeleni okuguquguqukayo (okungajwayelekile), ukuhlaziywa kuba nzima kakhulu.
Ucwaningo Lwesibonelo: Ukusetshenziswa Kwesimiso SikaBernoulli Ekuklanyweni Kwamaphiko Ezindiza
Umklamo wephiko lendiza uyisibonelo esivelele sokusebenza kwe-Bernoulli's Principle. Isimo sephiko esigobile nesiphansi esiyisicaba sivumela umoya ukuthi ugeleze ngokushesha phezu kwephiko kunangaphansi kwalo. Ngokusho kwe-Bernoulli's Principle, ijubane lokugeleza komoya eliphezulu phezu kwephiko likhiqiza ingcindezi ephansi kunangaphansi kwephiko. Lo mehluko wengcindezi udala amandla okuphakamisa avumela indiza ukuthi indize.
Ake sithi indiza inendawo yamaphiko engama-25 m². Isivinini somoya ngaphezu kwephiko singama-70 m/s, kanti ngaphansi kwephiko singama-50 m/s. Ubuningi bomoya buyi-1.225 kg/m³. Bala ukuphakama okuphumayo.
1. Bala umehluko wokucindezela (\( \Delta P \)) usebenzisa iFomula kaBernoulli:
\[ P_{bottom} – P_{top} = \frac{1}{2} \rho (v_{top}^2 – v_{bottom}^2) \]
Faka amanani aziwayo esikhundleni sawo:
\[ \Delta P = \frac{1}{2} \times 1.225 \times (70^2 – 50^2) \]
\[ \Delta P = \frac{1}{2} \times 1.225 \times (4900 – 2500) \]
\[ \Delta P = \frac{1}{2} \times 1.225 \times 2400 \]
\[ \Delta P = 1.225 \izikhathi ezingu-1200 \]
\[ \Delta P = 1470 \, \text{Pa} \]
2. Bala amandla okuphakamisa (\( F_{lift} \)):
\[ F_{lift} = \Delta P \times A \]
\[ F_{lift} = 1470 \, \text{Pa} \times 25 \, \text{m}^2 \]
\[ F_{lift} = 36,750 \, \text{
N} \]
Ngakho-ke, amandla okuphakamisa aphumelayo angama-newton angu-36,750.
Isiphetho
Isimiso sikaBernoulli singenye yezisekelo ze-fluid mechanics, sichaza ubudlelwano phakathi kwejubane lokugeleza koketshezi kanye nokucindezela. Ngokuqonda nokusebenzisa iFomula kaBernoulli, singahlaziya futhi siklame izinhlelo ezahlukahlukene ezihilela ukugeleza koketshezi, kusukela endizeni kuya kumamitha okugeleza. Nakuba empeleni sivame ukucabangela izici ezifana nokungqubuzana kanye nokuxokozela, isimiso sikaBernoulli sisalokhu siyithuluzi elibalulekile ekuqondeni amandla oketshezi kanye nokuklama izixazululo zobunjiniyela ezisebenzayo.
