Tsarin Bernoulli: Ka'idoji da Amfani
Ka'idar Bernoulli wani muhimmin ra'ayi ne a fannin injinan ruwa, wanda Daniel Bernoulli, wani masanin kimiyya ɗan ƙasar Switzerland, ya gano a ƙarni na 18. Wannan ƙa'ida ta bayyana alaƙar da ke tsakanin saurin kwararar ruwa da matsin lamba a cikin wannan kwararar. Wannan kasida za ta tattauna dalla-dalla game da dabarar Bernoulli, manyan manufofinta, abubuwan da ta samo asali daga lissafi, misalan lissafi, da aikace-aikacenta a rayuwar yau da kullun.
Fahimtar Tsarin Bernoulli
Tsarin Bernoulli ya bayyana cewa a cikin kwararar ruwa mai kyau (ba tare da gogayya ba), adadin kuzarin injiniya (ikon da zai iya aiki, kuzarin motsi, da matsin lamba) a kowace ƙarar raka'a yana nan a duk tsawon kwararar. An bayyana wannan dabarar kamar haka:
\[ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} \]
Ina:
– \( P \) shine matsin lamba na ruwa (a cikin pascals, Pa),
– \( \rho \) shine yawan ruwan (a cikin kilogiram a kowace mita mai siffar cubic, kg/m³),
– \( v \) shine saurin kwararar ruwa (a cikin mita a kowace daƙiƙa, m/s),
– \( g \) shine hanzarin da aka samu sakamakon nauyi (a cikin mita a kowace daƙiƙa murabba'i, m/s²),
– \( h \) shine tsayin ruwan da ke sama da ma'aunin (a cikin mita, m).
Ka'idar Bernoulli
Ka'idar Bernoulli ta dogara ne akan dokar kiyaye makamashi. A cikin kwararar ruwa, jimlar kuzarin da ke cikin tsarin dole ne ya kasance daidai idan ba a ƙara ko cire makamashi daga tsarin ba. Jimlar kuzarin da ke cikin kowace naúrar ya ƙunshi:
1. Makamashin Matsi (\( P \)): Makamashin da matsin ruwa ke haifarwa.
2. Makamashin Kinetic (\( \frac{1}{2} \rho v^2 \)): Makamashin da motsin ruwa ke haifarwa.
3. Makamashin Ƙarfin Nauyi (\( \rho gh \)): Makamashin da aka samu ta hanyar matsayin ruwa a cikin filin nauyi.
Tsarin Lissafi na Bernoulli
Domin fahimtar yadda aka samo Tsarin Bernoulli, za mu yi la'akari da ƙaramin abu na ruwa da ke motsawa a cikin kwararar da ke da saurin gudu \( v \) daga aya ta 1 zuwa aya ta 2. Idan muka ɗauka babu asarar kuzari saboda gogayya ko zafi, jimlar kuzarin da ke maki 1 da 2 dole ne ya zama iri ɗaya.
Jimlar kuzarin da ke wurin 1 (\( E_1 \)) shine:
\[ E_1 = P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 \]
Jimlar kuzarin da ke wurin 2 (\( E_2 \)) shine:
\[ E_2 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \]
Tunda jimlar makamashi dole ne ya kasance mai dorewa:
\[ P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \]
Wannan shine abin da ake kira Tsarin Bernoulli.
Misalin Lissafi Ta Amfani da Tsarin Bernoulli
A ce muna da bututun kwance wanda ruwa ke gudana daga wuri A zuwa wuri B. A wuri A, saurin ruwa shine m2/s kuma matsin lamba shine 150,000 Pa. A wuri B, saurin ruwa shine m4/s. Yawan ruwa shine 1000 kg/m³. Lissafa matsin lamba a wuri B.
Yi amfani da Tsarin Bernoulli:
\[ P_A + \frac{1}{2} \rho v_A^2 + \rho gh_A = P_B + \frac{1}{2} \rho v_B^2 + \rho gh_B \]
Tunda bututun yana kwance, \(h_A = h_B \), don haka \( \rho gh \) a duka wurare biyu za a iya yin watsi da su:
\[ P_A + \frac{1}{2} \rho v_A^2 = P_B + \frac{1}{2} \rho v_B^2 \]
Maye gurbin dabi'un da aka sani:
\[ 150,000 + \frac{1}{2} \sau 1000 \sau (2)^2 = P_B + \frac{1}{2} \sau 1000 \sau (4)^2 \]
Lissafa kuzarin motsi a duka wurare biyu:
\[ 150,000 + 2000 = P_B + 8000 \]
\[ 152,000 = P_B + 8000 \]
\[ P_B = 152,000 – 8000 \]
\[ P_B = 144,000 \, \rubutu{Pa} \]
Don haka, matsin lamba a wurin B shine 144,000 Pa.
Amfani da Ka'idar Bernoulli
1. Jiragen Sama: Ana amfani da ƙa'idar Bernoulli wajen tsara fikafikan jirgin sama. An tsara siffar fikafikan jirgin sama ta yadda saurin iska da ke sama da fikafikan ya fi ƙasa da fikafikan. Wannan yana haifar da ƙarancin matsin lamba a sama da fikafikan da kuma matsin lamba mafi girma a ƙasa da fikafikan, wanda ke haifar da ɗagawa wanda ke ba jirgin damar tashi.
2. Motocin Gudu: Tsarin sararin samaniya na motocin tsere yana amfani da ƙa'idar Bernoulli don ƙara gudu da kwanciyar hankali. Ana amfani da na'urori masu lalata da masu watsawa don daidaita iska, ƙara ƙarfin ƙasa, wanda ke inganta jan hankalin abin hawa a kan hanya.
3. Guduwar Jini a Jiki: Ka'idar Bernoulli kuma ta shafi kwararar jini ta hanyoyin jini. Ana iya nazarin bambance-bambancen da ke tsakanin saurin kwararar jini da matsin lamba a sassa daban-daban na jiki don gano yanayin lafiya.
4. Ma'aunin Venturi: Na'urar da ake amfani da ita don auna yawan kwararar ruwa a cikin bututu. Ma'aunin Venturi yana da sashe mai ƙunci, wanda ke haifar da ƙaruwar saurin kwararar ruwa da raguwar matsin lamba bisa ga ƙa'idar Bernoulli. Ana amfani da bambancin matsin lamba tsakanin sassan faɗi da kunkuntar don ƙididdige yawan kwararar.
5. Feshin Turare: Feshin turare ko feshin ruwa suna amfani da ƙa'idar Bernoulli. Idan aka hura iska ta cikin wani ƙaramin bututu, matsin lamba na iska da ke kewaye da ƙarshen bututun yana raguwa, yana jawo ruwan turaren ta cikin bututun kuma yana fesa shi a matsayin hazo mai laushi.
Abubuwan da ke Tasiri ga Amfani da Ka'idar Bernoulli
1. Rikici: A zahiri, ruwa ba shi da kyau kuma akwai rikici na ciki (danko) wanda ke haifar da asarar makamashi. Wannan yana sa amfani da Ka'idar Bernoulli ya fi rikitarwa.
2. Matsewar Ruwa: Ka'idar Bernoulli ta fi sauƙi a yi amfani da ita ga ruwaye marasa matsewa (kamar ruwa). Ga iskar gas ko ruwaye masu matsewa, dole ne a daidaita daidaiton.
3. Hayaniya: Ka'idar Bernoulli ta shafi kwararar laminar (na yau da kullun). A cikin kwararar da ba ta dace ba (na yau da kullun), binciken ya zama mai rikitarwa.
Nazarin Lamarin: Amfani da Ka'idar Bernoulli a Tsarin Fika-fikan Jiragen Sama
Tsarin fikafikan jirgin sama misali ne na ƙa'idar Bernoulli a aikace. Siffar fikafikan da ke lanƙwasa da kuma ƙasa mai faɗi suna ba da damar iska ta yi sauri ta wuce fikafikan fiye da ƙarƙashinsa. A cewar ƙa'idar Bernoulli, saurin iska mai yawa a kan fikafikan yana haifar da ƙarancin matsin lamba fiye da ƙarƙashin fikafikan. Wannan bambancin matsin lamba yana haifar da ƙarfin ɗagawa wanda ke ba jirgin damar tashi.
A ce jirgin sama yana da yankin fikafikai na 25 m². Saurin iska da ke sama da fikafikan shine 70 m/s, yayin da ƙasa da fikafikan yake 50 m/s. Yawan iska shine 1.225 kg/m³. A ƙididdige ɗagawar da aka samu.
1. Lissafa bambancin matsin lamba (\( \Delta P \)) ta amfani da Tsarin Bernoulli:
\[ P_{kasa} – P_{sama} = \frac{1}{2} \rho (v_{sama}^2 – v_{kasa}^2) \]
Maye gurbin dabi'un da aka sani:
\[ \Delta P = \frac{1}{2} \sau 1.225 \sau (70^2 – 50^2) \]
\[ \Delta P = \frac{1}{2} \sau 1.225 \sau (4900 - 2500) \]
\[ \Delta P = \frac{1}{2} \sau 1.225 \sau 2400 \]
\[ \Delta P = 1.225 \sau 1200 \]
\[ \Delta P = 1470 \, \text{Pa} \]
2. Lissafa ƙarfin ɗagawa (\( F_{lift} \)):
\[ F_{lift} = \Delta P \times A \]
\[ F_{lift} = 1470 \, \text{Pa} \times 25 \, \text{m}^2 \]
\[ F_{lift} = 36,750 \, \rubutu{
N} \]
Don haka, ƙarfin ɗagawa da aka samu shine newtons 36,750.
Kammalawa
Ka'idar Bernoulli ɗaya ce daga cikin muhimman abubuwan da suka shafi tsarin sarrafa ruwa, tana bayyana alaƙar da ke tsakanin saurin kwararar ruwa da matsin lamba. Ta hanyar fahimtar da amfani da Tsarin Bernoulli, za mu iya yin nazari da tsara tsarin da ya shafi kwararar ruwa, daga jiragen sama zuwa mita masu kwarara. Kodayake a aikace sau da yawa dole ne mu yi la'akari da abubuwa kamar gogayya da turmutsitsi, ƙa'idar Bernoulli ta kasance kayan aiki mai mahimmanci wajen fahimtar yanayin ruwa da kuma tsara ingantattun hanyoyin magance matsalolin injiniya.
