Te Tātai a Bernoulli: Ngā Mātāpono me ngā Whakamahinga
Ko te mātāpono a Bernoulli he ariā taketake i roto i te miihini wai, i kitea e Daniel Bernoulli, he kaipūtaiao Huiterangi, i te rautau 18. E whakamārama ana tēnei mātāpono i te whanaungatanga i waenga i te tere o te rere o te wai me te pēhanga i roto i taua rere. Ka matapakihia e tēnei tuhinga te tātai a Bernoulli, ōna ariā taketake, ngā putanga pāngarau, ngā tauira tātai, me ōna whakamahinga i roto i te oranga o ia rā.
Te Mārama ki te Tātai a Bernoulli
E ai ki te tātai a Bernoulli, i roto i te rere wai pai (me te kore waku), he pumau te nui o te pūngao miihini (pūngao pūmanawa, pūngao nekeneke, me te pēhanga) mō ia rōrahi puta noa i te rere. E ai ki tēnei tātai:
\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{pūmau} \]
kāore i te mana:
– Ko te pēhanga wai (i roto i ngā pascal, Pa) te \( P \)
– Ko te mātotoru o te wai (i roto i ngā kirokaramu mō ia mita pūtoru, kg/m³),
– Ko te tere rere o te wai (i roto i ngā mita ia hekona, m/s) te \( v \),
– Ko te whakaterenga nā te kaha ā-papatipu (i roto i ngā mita ia hekona tapawhā, m/s²),
– Ko te \( h \) te teitei o te wai i runga ake i te tohutoro (i roto i ngā mita, m).
Te Mātāpono a Bernoulli
Ko te mātāpono a Bernoulli e ahu mai ana i te ture tiaki pūngao. I roto i te rere o te wai, me noho pūmau te pūngao katoa i roto i tētahi pūnaha mēnā kāore he pūngao e tāpirihia, e tangohia rānei i te pūnaha. Ko te pūngao katoa mō ia rōrahi kotahi ko:
1. Pūngao Pēhanga (\( P \)): Pūngao i puta mai i te pēhanga wai.
2. Pūngao Kinetic (\( \frac{1}{2} \rho v^2 \)): Pūngao i puta mai i te nekehanga wai.
3. Pūngao Pūmanawa Ā-Toi (\( \rho gh \)): Pūngao e puta mai ana i te tūranga o te wai i roto i te papa ā-tōi.
Te Whakaputa Pāngarau o te Tātai a Bernoulli
Hei mārama ki te pūtake o te Tātai a Bernoulli, ka whakaarohia he huānga iti o te wai e neke ana i roto i te rere me te tere \( v \) mai i te pūwāhi 1 ki te pūwāhi 2. Mēnā ka whakaarohia kāore he ngaronga pūngao nā te waku, nā te wera rānei, me rite tonu te pūngao katoa i ngā pūwāhi 1 me te 2.
Ko te pūngao katoa i te pūwāhi 1 (\( E_1 \)) ko:
\[ E_1 = P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 \]
Ko te pūngao katoa i te pūwāhi 2 (\( E_2 \)) ko:
\[ E_2 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \]
Nā te mea me pumau tonu te pūngao katoa:
\[ P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \]
Koinei te mea e kiia nei ko te Tātai a Bernoulli.
Tauira Tātaitanga mā te whakamahi i te Tātai a Bernoulli
Me kī he paipa whakapae tā tātou e rere ana te wai mai i te pūwāhi A ki te pūwāhi B. I te pūwāhi A, ko te tere o te wai he 2 m/s, ā, ko te pēhanga he 150,000 Pa. I te pūwāhi B, ko te tere o te wai he 4 m/s. Ko te mātotoru o te wai he 1000 kg/m³. Tātaihia te pēhanga i te pūwāhi B.
Whakamahia te Tātai a Bernoulli:
\[ P_A + \frac{1}{2} \rho v_A^2 + \rho gh_A = P_B + \frac{1}{2} \rho v_B^2 + \rho gh_B \]
Nā te mea he whakapae te paipa, \( h_A = h_B \), nō reira ka taea te wareware i \( \rho gh \) i ngā pūwāhi e rua:
\[ P_A + \frac{1}{2} \rho v_A^2 = P_B + \frac{1}{2} \rho v_B^2 \]
Whakakapia ngā uara e mōhiotia ana:
\[ 150,000 + \frac{1}{2} \times 1000 \times (2)^2 = P_B + \frac{1}{2} \times 1000 \times (4)^2 \]
Tātaihia te pūngao nekeneke i ngā pūwāhi e rua:
\[ 150,000 + 2000 = P_B + 8000 \]
\[ 152,000 = P_B + 8000 \]
\[ P_B = 152,000 – 8000 \]
\[ P_B = 144,000 \, \text{Pa} \]
Nō reira, ko te pēhanga i te pūwāhi B ko 144,000 Pa.
Te Whakamahinga o te Mātāpono a Bernoulli
1. Ngā Rererangi: Ka whakamahia te mātāpono a Bernoulli i roto i te hoahoa o ngā parirau rererangi. Ko te āhua o te parirau rererangi kua hangaia kia nui ake te tere o te hau i runga ake i te parirau i raro iho i te parirau. Ko te hua o tēnei he iti ake te pēhanga i runga ake i te parirau, he nui ake te pēhanga i raro iho i te parirau, ka hangaia te ara hiki e taea ai e te rererangi te rere.
2. Ngā Waka Reihi: Ko te hoahoa ā-rangi o ngā waka reihi e whakamahi ana i te Mātāpono a Bernoulli hei whakanui ake i te tere me te pumau. Ka whakamahia ngā mea whakahāwea me ngā mea whakarara hei whakahaere i te rere o te hau, hei whakanui ake i te kaha o te heke, hei whakapai ake i te kukume o te waka i runga i te rori.
3. Te Rere o te Toto i roto i te Tinana: Ka pā anō hoki te Mātāpono a Bernoulli ki te rere o te toto i roto i ngā uaua toto. Ka taea te ako i ngā rerekētanga o te tere me te pēhanga o te rere o te toto i ngā wāhanga rerekē o te tinana hei tātaritanga i ngā mate hauora.
4. Mita Venturi: He taputapu e whakamahia ana hei ine i te tere rere o te wai i roto i te paipa. He wāhanga whaiti kei te mita Venturi, e hua ake ai te tere rere o te wai me te heke o te pēhanga e ai ki te Mātāpono a Bernoulli. Ko te rerekētanga pēhanga i waenga i ngā wāhanga whānui me ngā wāhanga whaiti ka whakamahia hei tatau i te tere rere.
5. Ngā Pūwero Hinu Kakara: Ka whakamahia e ngā pūwero hinu kakara, ngā pūwero wai rānei te mātāpono a Bernoulli. Ina pupuhihia te hau mā roto i tētahi ngongo whaiti, ka heke te pēhanga hau i te pito o te ngongo, ka kukume i te wai hinu kakara ki runga mā roto i te ngongo, ka pūwerohia hei kohu angiangi.
Ngā Take e Pā Ana ki te Whakamahinga o te Mātāpono a Bernoulli
1. Te waku: Ko te mea pono, ehara ngā wai i te mea tino pai, ā, kei reira tonu te waku o roto (te matotoru) e mate ai te pūngao. Nā tēnei ka uaua ake te whakamahi i te Mātāpono a Bernoulli.
2. Te Āheinga ki te Whakakōpeke i te Wai: He māmā ake te whakamahi i te mātāpono a Bernoulli ki ngā wai kore e taea te kōpeke (pērā i te wai). Mō ngā hau, mō ngā wai rānei e taea te kōpeke, me whakarerekē te whārite.
3. Te Ārai: Ka pā te mātāpono a Bernoulli ki te rere laminar (tūturu). I roto i te rere ārai (tūturu), ka uaua ake te tātari.
Akoranga Take: Te Whakamahinga o te Mātāpono a Bernoulli i roto i te Hoahoa Parirau Rererangi
Ko te hoahoa o te parirau o te waka rererangi he tauira matarohia o te Mātāpono a Bernoulli e mahi ana. Mā te āhua piko o runga me te papatahi o raro o te parirau ka tere ake te rere o te hau i runga i te parirau i tō raro iho. E ai ki te Mātāpono a Bernoulli, ko te tere o te rere o te hau i runga i te parirau ka iti ake te pēhanga i tō raro iho. Nā tēnei rerekētanga pēhanga ka hangaia te kaha hiki e taea ai e te waka rererangi te rere.
Me kī he 25 m² te whānui o te parirau o tētahi waka rererangi. Ko te tere o te hau i runga ake i te parirau he 70 m/s, ko te tere o raro i te parirau he 50 m/s. Ko te mātotoru o te hau he 1.225 kg/m³. Tātaihia te hua o te hiki.
1. Tātaihia te rerekētanga pēhanga (\( \Delta P \)) mā te whakamahi i te Tātai a Bernoulli:
\[ P_{raro} – P_{runga} = \frac{1}{2} \rho (v_{runga}^2 – v_{raro}^2) \]
Whakakapia ngā uara e mōhiotia ana:
\[ \Delta P = \frac{1}{2} \times 1.225 \times (70^2 – 50^2) \]
\[ \Delta P = \frac{1}{2} \times 1.225 \times (4900 – 2500) \]
\[ \Delta P = \frac{1}{2} \times 1.225 \times 2400 \]
\[ \Delta P = 1.225 \whakanuia ki te 1200 \]
\[ \Delta P = 1470 \, \text{Pa} \]
2. Tātaihia te kaha hiki (\( F_{hiki} \)):
\[ F_{hikinga} = \Delta P \times A \]
\[ F_{hikinga} = 1470 \, \text{Pa} \times 25 \, \text{m}^2 \]
\[ F_{hikinga} = 36,750 \, \kuputuhi{
N} \]
Nō reira, ko te kaha hiki e hua ake ana ko 36,750 newtons.
Whakamutunga
Ko te Mātāpono a Bernoulli tētahi o ngā kaupapa matua o te miihini wai, e whakamārama ana i te whanaungatanga i waenga i te tere o te rere o te wai me te pēhanga. Mā te mārama me te whakamahi i te Tātai a Bernoulli, ka taea e tātou te tātari me te hoahoa i ngā momo pūnaha e uru ana ki te rere o te wai, mai i te waka rererangi ki ngā mita rere. Ahakoa i roto i te mahi he maha ngā wā me whakaaro tātou ki ngā āhuatanga pēnei i te waku me te aueue, he taputapu nui tonu te Mātāpono a Bernoulli ki te mārama ki ngā hihiri wai me te hoahoa i ngā otinga hangarau whai hua.
