Applications — Pitot tube, Venturi meter, orifice flow: Visual walkthrough

2.2.16 · D2 · Physics › Fluid Mechanics › Applications — Pitot tube, Venturi meter, orifice flow

Is page par EK result pictures mein derive kiya gaya hai: Venturi flow rate, yaani woh formula jo ek pinched pipe ko yeh batane deta hai ki usme se kitna paani beh raha hai. Hum ek pipe mein paani ki picture se shuru karte hain aur ek boxed equation par khatam karte hain, ek chhota step ek waqt. Har symbol ko use karne se pehle draw kiya gaya hai.

Parent: Applications — Pitot / Venturi / Orifice.

Step 1 — Pipe draw karo aur jo dikhta hai usse naam do

KYA. Ek horizontal pipe ki picture banao jo baayein taraf chaudi hai aur beech mein pinched hai. Paani left se right ki taraf flow karta hai. Hum ek imaginary line ke saath do spots mark karte hain jise paani follow karta hai — us line ko streamline kaho (woh path jo ek paani ka particle trace karta hai).

Spot 1 chaude hisse mein hai. Uski cross-section area $A_{1}$ hai (wahan pipe ka circle kitna bada hai, square metres mein) aur wahan paani speed $v_{1}$ se chalta hai (metres per second).
Spot 2 narrow throat mein hai. Uski area $A_{2}$ aur speed $v_{2}$ hai.

KYUN. Kisi bhi equation se pehle, hume yeh jaanna zaroori hai ki har letter real picture mein kya cheez point kar raha hai. $A$ ek area hai, $v$ ek speed hai, chota number batata hai kaunsa spot. Poori derivation mein aur kuch nahi aata.

PICTURE. Neeche dono dashed circles dekho: bada wala (blue) $A_{1}$ hai, chota wala (yellow) throat $A_{2}$ hai. Arrow flow direction dikhata hai.

Figure — Applications — Pitot tube, Venturi meter, orifice flow

Step 2 — Narrow part ka ZAROOR zyada tez rehna (Continuity)

KYA. Utna hi paani jo chaude hisse mein per second ghusta hai, throat se per second nikalna chahiye — paani create ya destroy nahi hota aur (hamare liye) squash bhi nahi hota. Ek spot pe per second cross karne wali matra $area \times speed$ hoti hai. Ise $Q$ kaho, yaani volume flow rate (cubic metres per second). To:

$wide area A_{1} wide speed v_{1} = throat area A_{2} throat speed v_{2} = same for all spots Q$

KYUN. Yeh Continuity Equation hai. Hum ise isliye use karte hain kyunki yeh woh ek rule hai jo dono unknown speeds ko pipe geometry se connect karta hai. Iske bina, $v_{1}$ aur $v_{2}$ do alag alag mystery hote.

Kyunki $A_{2} < A_{1}$ (throat chota hai) aur unka product same rehna chahiye, throat speed $v_{2}$ zyada badi hone par majboor hai. Ek chaudi dheemi nadi ka ek narrow tez rapid mein squeeze hona socho.

PICTURE. Per second utna hi paani ek chote window se — toh zyada tez jaana padega. Chota mota arrow (slow, wide) ek lamba patla arrow (fast, narrow) ban jaata hai.

Step 3 — Tez paani kam push karta hai: Bernoulli ka balance

KYA. Us same streamline ke saath, Bernoulli ka rule kehta hai ek certain sum constant rehta hai. Kyunki pipe horizontal hai, dono spots par height same hai, isliye height term dono sides par identical hai aur simply cancel ho jaata hai. Jo bachta hai:

$pressure at spot 1 P_{1} + "motion pressure" at spot 1 \frac{1}{2} ρ v_{1}^{2} = pressure at spot 2 P_{2} + "motion pressure" at spot 2 \frac{1}{2} ρ v_{2}^{2}$

Yahan $P$ pressure hai (push per unit area, pascals mein) aur $ρ$ (Greek "rho") fluid ki density hai (kilograms per cubic metre — fluid ka ek box kitna bhaari hai). Term $\frac{1}{2} ρ v^{2}$ dynamic pressure hai: push ka woh hissa jo purely motion se aata hai.

KYUN. Hum Bernoulli's Equation isliye use karte hain kyunki yeh woh ek law hai jo speed ko pressure ke against trade karta hai. Step 2 ne bataya ki throat faster hai; Bernoulli ab "faster" ko "lower pressure" mein convert karta hai. Humne dekha ki throat par motion upar jaata hai, isliye sum fixed rakhne ke liye pressure neeche aana chahiye — dekho Dynamic vs Static vs Stagnation Pressure.

PICTURE. Neeche bar chart: har spot par, ek green pressure bar aur ek red motion bar same total height tak pahunchte hain. Jahan red (motion) bar tall hai (throat), wahan green (pressure) bar short hai.

Step 4 — Pressure difference isolate karne ke liye rearrange karo

KYA. Dono pressures ko ek side aur dono motion terms ko doosri side le jao:

$the drop we can measure P_{1} - P_{2} = extra motion gained at throat \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2})$

KYUN. Left side, $P_{1} - P_{2}$ , exactly wahi hai jo ek instrument read kar sakta hai — chaude hisse mein pressure zyada hai aur throat mein kam, aur ek manometer woh difference directly report karta hai. Hum equation ko is tarah organize kar rahe hain ki measurable cheez akele baithe.

Dhyan do $v_{2} > v_{1}$ Step 2 se, isliye $v_{2}^{2} - v_{1}^{2} > 0$ , isliye $P_{1} - P_{2} > 0$ : chauda hissa sach mein zyada pressure par hai. Sign automatically sahi aata hai.

PICTURE. Ek see-saw: left pan mein pressure gap baitha hai; right pan mein motion gain baitha hai. Woh balance karte hain.

Step 5 — Continuity se ek unknown khatam karo

KYA. Abhi bhi do speeds hain. Step 2 se, $A_{1} v_{1} = A_{2} v_{2}$ , isliye

$v_{1} = \frac{A _{2}}{A _{1}} v_{2} .$

Ise $v_{2}^{2} - v_{1}^{2}$ mein substitute karo:

$v_{2}^{2} - v_{1}^{2} = v_{2}^{2} - (\frac{A _{2}}{A _{1}})^{2} v_{2}^{2} = v_{2}^{2} (1 - \frac{A _{2}^{2}}{A _{1}^{2}}) .$

To Step 4 ban jaata hai:

$P_{1} - P_{2} = \frac{1}{2} ρ v_{2}^{2} shrink factor 1 - \frac{A _{2}^{2}}{A _{1}^{2}}$

KYUN. Do unknowns ( $v_{1}$ aur $v_{2}$ ) wali equation solve nahi ho sakti. Continuity hume $v_{1}$ ko $v_{2}$ ke terms mein likhne deta hai, sirf ek unknown $v_{2}$ bacha ke. Isliye hum continuity doosri baar use karte hain — speeds nikaalane ke liye nahi, balki unhe eliminate karne ke liye.

PICTURE. Do speed-boxes ( $v_{1}$ , $v_{2}$ ) ek mein merge ho jaate hain, chota area-ratio $A_{2} / A_{1}$ ek shrinking dial ki tarah kaam karta hai.

Recall

$1 - A_{2}^{2} / A_{1}^{2}$ 0 aur 1 ke beech kyun hai? Kyunki $A_{2} < A_{1}$ , fraction $A_{2}^{2} / A_{1}^{2}$ ek positive number hai jo 1 se kam hai, isliye ise 1 se ghatane par ek positive number milta hai jo 1 se kam hai. Yeh kabhi negative nahi hota. Shrink factor ka matlab ::: throat ki kitni motion actually ek readable pressure drop ke roop mein dikhti hai.

Step 6 — Throat speed solve karo

KYA. Dono sides ko $\frac{1}{2} ρ (1 - A_{2}^{2} / A_{1}^{2})$ se divide karo aur positive square root lo (speed positive hoti hai):

$v_{2} = \frac{2 ( P _{1} - P _{2} )}{ρ ( 1 - \frac{A _{2}^{2}}{A _{1}^{2}} )}$

KYUN. Square root isliye aata hai kyunki speed dynamic-pressure term $\frac{1}{2} ρ v^{2}$ mein squared thi. "Squared" ko undo karne ka matlab square root lena hai — yeh wahi ek tool hai jo jawaab deta hai "kaunsa number, squared karke, yeh deta hai?" Hum positive root rakhte hain kyunki throat mein paani ki ek real forward speed hai, negative nahi.

PICTURE. $v_{2}^{2}$ ko wapas $v_{2}$ mein baadalna: parabola-to-line "undo squaring" arrow.

Step 7 — Speed ko flow rate $Q$ mein badlo

KYA. Jo quantity hum actually chahte hain woh $Q$ hai, yaani paani ke litres-per-second. Step 2 se, $Q = A_{2} v_{2}$ . Step-6 ki speed ko $A_{2}$ se multiply karo aur algebra theek karo (root ke andar top aur bottom dono ko $A_{1}^{2}$ se multiply karo taaki messy fraction gayab ho jaye):

$Q = A_{2} v_{2} = A_{1} A_{2} \frac{2 ( P _{1} - P _{2} )}{ρ ( A _{1}^{2} - A _{2}^{2} )}$

Term by term:

$A_{1} A_{2}$ — dono areas overall scale set karte hain (badi pipe, zyada flow).
$2 (P_{1} - P_{2})$ — measured pressure drop; ise double karne par $Q$ sirf $2$ se multiply hota hai.
$ρ$ — ek bhaari fluid same push ke liye slower flow karta hai.
$A_{1}^{2} - A_{2}^{2}$ — squeeze ki geometry; agar pipe thodi hi narrow ho ( $A_{2} \to A_{1}$ ) toh yeh gayab ho jaata hai.

KYUN. Pressure differences woh hain jo hum measure karte hain; flow rate woh hai jo hum chahte hain. Yeh aakhri $A_{2}$ se multiply karna dono ke beech bridge ka kaam karta hai.

PICTURE. Poora formula tree — measured $Δ P$ andar jaata hai, $Q$ bahar aata hai.

Step 8 — Degenerate cases (reader ko kabhi akela mat chodo)

KYA & KYUN, case by case:

Koi squeeze nahi ( $A_{2} = A_{1}$ ). Tab $A_{1}^{2} - A_{2}^{2} = 0$ , denominator zero hai, aur formula blow up karta hai — jab tak $P_{1} - P_{2}$ bhi zero na ho. Constant width wali pipe mein koi pressure drop nahi hota aur koi reading nahi milti: kuch narrow nahi hota, kuch speed up nahi hota. Meter ko ek real pinch chahiye.
Zero flow ( $Q = 0$ ). Tab $P_{1} - P_{2} = 0$ : ruka hua paani dono spots par equal pressure dikhata hai. Sahi — koi motion nahi, koi dynamic dip nahi.
Bahut chota throat ( $A_{2} ≪ A_{1}$ ). Tab $A_{1}^{2} - A_{2}^{2} \approx A_{1}^{2}$ , aur $Q \approx A_{2} 2 (P_{1} - P_{2}) / ρ$ — wide side ki speed negligible ho jaati hai, to yeh throat jet ki plain Pitot reading jaisi lagti hai.
Sign check. $P_{1} - P_{2}$ hamesha positive hota hai (wide side zyada), isliye root ke neeche ka number positive hai aur $Q$ real hai. Agar kisi manometer ne kabhi $P_{2} > P_{1}$ kaha, toh tumne ports swap kar diye hain.

PICTURE. Teen mini-pipes: no-pinch (flat pressure), normal pinch (dip), extreme pinch (deep dip). Flat wale mein padhne ke liye koi dip nahi.

Ek picture mein summary

Ek frame mein poori chain: Continuity throat ko speed up hone par majboor karti hai → Bernoulli us speed-up ko pressure dip mein badalta hai → algebra readable pressure drop ko flow rate $Q$ ke liye swap karta hai.

Recall Feynman retelling — kisi dost ko bolke batao

Paani ek moti pipe mein beh raha hai jiske beech mein ek pinch hai. Kyunki utna hi paani-per-second patli jagah se bhi guzarna hai, wahan use bhagna padhta hai — jaise gaadiyaan ek lane mein merge ho rahi hon. Lekin bhaagta hua paani pipe ki walls par kam dabata hai, isliye pinched spot par lower pressure hoti hai bade hisse se. Hum do pressure gauges lagate hain, dekhte hain pressure kitna gira, aur ulta chalaate hain: bada drop matlab paani zyada zor se beh raha tha, yaani zyada paani per second. Humne woh "ulta" recipe ek boxed formula ke roop mein likhi, $Q = A_{1} A_{2} 2Δ P / [ρ (A_{1}^{2} - A_{2}^{2})]$ . Aur humne weird cases check kiye: jo pipe kabhi narrow nahi hoti woh koi drop nahi dikhati aur kuch measure nahi karti, jabki bilkul ruka hua paani har jagah equal pressure dikhata hai — bilkul jaise common sense maangta hai.

Connections

Bernoulli's Equation — Steps 3–4, speed-for-pressure trade.
Continuity Equation — Steps 2 aur 5, dono speeds ko link karta hai.
Dynamic vs Static vs Stagnation Pressure — $\frac{1}{2} ρ v^{2}$ ka matlab.
Manometers and Pressure Measurement — $P_{1} - P_{2}$ actually kaise padha jaata hai.
Torricelli's Law — ek draining tank ke liye wahi energy idea.
Projectile Motion — orifice jet ka companion.