Nkhani yokhudza Equation of continuity
Yesani kutsegula pompo yamadzi pang'onopang'ono mukuyang'anitsitsa liwiro la madzi omwe akutuluka mkamwa mwa pompo. Tsopano yerekezerani, kodi liwiro la madzi limathamanga bwanji, pamene pakamwa pa pompo patsekedwa kapena ayi? Chifukwa chiyani zili choncho? Kuti mumvetse izi, chonde phunzirani za continuity equation.
Mtsinje
Mu mtsinje wokhazikika, liwiro la tinthu tamadzimadzi pa mfundo imodzi, mwachitsanzo mfundo A nthawi zonse limakhala lofanana. Podutsa mfundo B, liwiro la tinthu tamadzimadzi likhoza kusintha. Komabe, pofika pa mfundo B, tinthu tamadzimadzi tomwe timatsatira kuchokera kumbuyo timayenda pa liwiro lomwelo ndi tinthu tamadzimadzi tomwe timatsogolera. Momwemonso pofika pa mfundo C ndi zina zotero. Mzere Woyenda ndi curve yomwe imalumikiza mfundo A, B, ndi C.
Chubu choyenda
Tikhoza kufotokoza kusinthasintha kulikonse kudzera mu mfundo iliyonse ya kuyenda kwa madzi. Ngati tiwona kuyenda kwa madzi kokhazikika, kusinthasintha kwina komwe kumadutsa ngodya inayake pamalo ongopeka kumapanga chubu choyenda. Palibe tinthu tamadzi tomwe timakumana koma nthawi zonse timakhala ofanana, ndipo chubu choyenda chidzafanana ndi chitoliro chomwe mawonekedwe ake akadali ofanana. Madzi olowa kumapeto kwa chubucho adzatuluka mu chubucho kumapeto kwina.
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Debit
Kutuluka kwa madzi kumatanthauza kuchuluka kwa madzi omwe akuyenda kudutsa gawo linalake panthawi inayake. Mwa masamu, zitha kunenedwa:
Kutuluka kwa madzi = Kuchuluka kwa madzi / nthawi yopuma
Q = V / t
Kuti muwonjezere kumvetsetsa kwanu, tikugwiritsa ntchito chitsanzo. Mwachitsanzo, madzi amayenda kudzera mu chitoliro. Mapaipi nthawi zambiri amakhala ozungulira ndipo amakhala ndi malo enaake opingasa. Chitolirocho chilinso ndi kutalika kwake.
Madzi akamayenda mu chitoliro mpaka L, voliyumu ya madzi mu chitoliro ndi V = AL (V = voliyumu ya madzi, A = malo opingasa ndi L = kutalika kwa chitoliro). Chifukwa bola ngati akuyenda mu chitoliro motsatira L, madziwo amafunika nthawi inayake, ndiye kuti tinganene kuti kuchuluka kwa madzi ndi:
Chifukwa v = s / t = L / t —> L = vt, equation yomwe ili pamwambapa yasinthidwa kukhala:
Motero, madzi akamayenda kudzera mu chitoliro chomwe chili ndi malo ndi kutalika kwa gawo linalake, kwa nthawi inayake, kuchuluka kwa madzi otuluka (Q) kumakhala kofanana ndi malo otuluka (A) ochulukitsidwa ndi liwiro la madzi otuluka (v).
Kulinganiza kwa kupitiriza
Unikani momwe madzi amayendera mu chitoliro chomwe chili ndi mainchesi osiyana, monga momwe chithunzi chili pansipa chikusonyezera.
Chithunzichi chikuwonetsa kuyenda kwa madzi kuchokera kumanzere kupita kumanja (madzi akuyenda kuchokera ku chitoliro chachikulu cha mainchesi kupita ku mainchesi ang'onoang'ono). Mzere wodulidwa ndi wowongoka.
A 1 = malo opingasa mapaipi akuluakulu, A 2 = malo opingasa ang'onoang'ono, v 1 = liwiro la kuyenda kwa madzi mu chitoliro chachikulu, v 2 = liwiro la kuyenda kwa madzi mu chitoliro chaching'ono, L = mtunda woyenda ndi madzi.
Mu mtsinje wokhazikika, liwiro la kuyenda kwa tinthu tamadzimadzi pamalo amodzi limafanana ndi liwiro la kuyenda kwa tinthu tina tamadzimadzi tomwe timadutsa pamalo amenewo. Kuyenda kwa madzimadzi sikumadutsananso (mizere ndi yofanana). Chifukwa chake kuchuluka kwa madzimadzi omwe amalowa kumapeto kwa chitolirocho kuyenera kukhala kofanana ndi kuchuluka kwa madzimadzi omwe amatuluka kumapeto ena. Ngati madzimadziwo ali ndi kuchuluka kwina komwe kumabwera ku chubu chokhala ndi mainchesi akulu, ndiye kuti madziwo adzatuluka pa chitoliro chaching'ono chokhala ndi mainchesi ofanana.
Tsopano ganizirani chithunzi cha chitoliro chomwe chili pamwambapa. Onaninso chitoliro chachikulu cha m'mimba mwake ndi chitoliro chaching'ono cha m'mimba mwake.
Mu nthawi inayake, madzi ena amadutsa mu gawo la chitoliro ndi mainchesi akulu (A 1 ) mpaka L 1 (L 1 = v 1 t). Kuchuluka kwa madzi oyenda ndi V 1 = A 1 L 1 = A 1 v 1 t. Mu nthawi yomweyo, madzi ena amadutsa mu gawo laling'ono la chitoliro (A 2 ) mpaka L 2 (L 2 = v 2 t). Kuchuluka kwa madzi oyenda ndi V 2 = A 2 L 2 = A 2 v 2 t.
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Kuyerekeza kwa kupitiriza kwa madzi osagwedezeka
Mu madzi osagwedezeka, kuchuluka kwa madzi nthawi zonse kumakhala kofanana pamalo aliwonse omwe amadutsa. Kuchuluka kwa madzi omwe amatuluka mu chitoliro chomwe chili ndi dera lopingasa la A1 (lalikulu la chitoliro) panthawi inayake ndi:
Momwemonso, kuchuluka kwa madzi omwe akuyenda mu chitoliro chomwe chili ndi gawo la A 2 (m'mimba mwake wa chitoliro chaching'ono) panthawi inayake ndi:
Mu madzi oyenda bwino, kuchuluka kwa madzi otuluka kumakhala kofanana ndi kuchuluka kwa madzi otuluka, kotero:
m 1 = m 2
ρ A 1 v 1 t = ρ A 2 v 2 t
A 1 v 1 = A 2 v 2
Kotero, equation ya continuity equation mu incompressible fluids:
A 1 v 1 = A 2 v 2 — Equation 1
A 1 = dera lopingasa 1, A 2 = dera lopingasa 2, v 1 = liwiro la kuyenda kwa madzi mu gawo 1, v 2 = liwiro la kuyenda kwa madzi mu gawo 2, A v = liwiro la kuyenda kwa voliyumu (V / t) = kutuluka.
Chiyerekezo 1 chikuwonetsa kuti kutuluka kwa madzi nthawi zonse kumakhala kofanana pamalo aliwonse omwe ali m'mphepete mwa chitoliro kapena chubu choyendera madzi. Pamene gawo lodutsa chitoliro likuchepa, kuchuluka kwa madzi kumawonjezeka, pomwe pamene gawo lodutsa chitoliro likukulirakulira, kuchuluka kwa madzi kumachepa.
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Gawo la pakamwa pa pompo likakhala lotsekedwa, madzi amatuluka molemera kuposa gawo la pakamwa pa pompo silinatsekedwe. Izi zili choncho chifukwa gawo lodutsa la pompo limakhala laling'ono gawo la pakamwa pa pompo likakhala lotsekedwa kotero kuti liwiro la madzi limakwera. Koma ndikofunikira kudziwa kuti liwiro la madzi kapena kutuluka kwa madzi nthawi zonse kumakhala kofanana pamalo aliwonse oyenda madzi, kaya gawo la pakamwa pa pompo lathu latsekedwa kapena ayi. Ndiye kodi kusintha kwa madzi kumakhudza bwanji?
Nanga bwanji za kuyenda kwa madzi mumtsinje? Gawo la mtsinje wozama lili ndi gawo lalikulu kuposa gawo losaya la mtsinje, kotero liwiro la kuyenda kwa madzi mumtsinje wozama ndi lochepa kuposa liwiro la kuyenda kwa madzi mu gawo losaya la mtsinje.
Kuyerekeza kwa kupitiriza kwa madzi opanikizika
Pa madzi opanikizika, kuchuluka kwa madzi sikofanana nthawi zonse. Mwa kuyankhula kwina, kuchuluka kwa madzi kumasintha akapanikizika. Ngati madziwo sanakanikizidwe, kuchuluka kwa madziwo kumachotsedwa mu equation, kotero pankhaniyi, kuchuluka kwa madziwo kumaphatikizidwa mu equation. Ponena za equation yomwe idachokera kale, tiyeni tipeze equation ya madzi opanikizika.
Unyinji wa madzi omwe amalowa ndi wofanana ndi unyinji wa madzi omwe amatuluka, kotero:
m 1 = m 2
ρ A 1 v 1 t = ρ A 2 v 2 t
Nthawi yoyenda kwa madzi ndi yofanana kotero kuti imatha kuchotsedwa. Ma equation amasinthidwa kukhala:
ρ A 1 v 1 = ρ A 2 v 2 → Equation 2
Iyi ndi equation ya madzi opanikizika. Kusiyana kuli mu kuchuluka kwa madzi. Ngati madziwo akakamizidwa, kuchuluka kwa madzi kumasintha. Mosiyana ndi zimenezi, ngati madziwo sanakanikizidwe, kuchuluka kwa madzi nthawi zonse kumakhala kofanana kotero kuti kungathe kuchotsedwa mu equation.
Chitsanzo cha vuto 1:
Madzi amayenda kudzera mu chitoliro chokhala ndi mainchesi 10 cm pa liwiro la 2 m/s. Kodi madziwo amatuluka bwanji?
Zodziwika:
Chitoliro cha mapaipi = 10 cm
Utali wa chitoliro (r) = 5 cm = 0.05 m
Liwiro la madzi (v) = 2 m/s
Akufuna: Debit
Yankho:
Q = A v
Q = (π r 2 ) (2 m/s)
Q = (3.14)(0.05 m) 2 (2 m/s)
Q = (3.14)(0.0025 m2 ) (2 m/s)
Q = 0.0157 m 3 /s
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Chitsanzo cha vuto 2:
Chitoliro chokhala ndi mainchesi 20 cm chimalumikizidwa ku chitoliro china chokhala ndi mainchesi 10. Ngati liwiro la madzi mu chitoliro chokhala ndi mainchesi 20 cm ndi 4 m/s, kodi liwiro la madzi mu chitoliro chokhala ndi mainchesi 10 ndi lotani?
Zodziwika:
Chidutswa 1 = 20 cm (r 1 = 10 cm = 0.1 m)
v 1 = 4 m/s
Chidutswa 2 = 10 cm (r 2 = 5 cm = 0.05 m)
Akufuna: v 1
Yankho:
Funso 1 = Funso 2
A 1 v 1 = A 2 v 2
( π r 12 ) (4 m/s) = (π r 22 ) (v 2 )
(0.1 m) 2 (4 m/s) = (0.05 m) 2 (v 2 )
(0.01 m2 ) (4 m/s) = (0.0025 m2 ) (v 2 )
(0.04 m 3 /s) = (0.0025 m 2 )(v 2 )
v 2 = 16 m/s
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