Water flows through a horizontal tube of a variable cross-section (figure 13-E7). The area of the cross-section at A and B are 4 mm² and 2 mm² respectively. If 1 cc of water enters per second through A, find (a) the speed of the water at A, (b) the speed of the water at B and (c) the pressure difference Pₐ-Pᵦ.
Answers
Given,
Volume rate of the flow of water = 1 cc/sec.
= 1 cm³/sec.
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Since, Volume rate of the flow of water is constant throughout the Horizontal pipe, therefore,
(a). At A,
Area × Velocity = 1 cm³/s.
4/100 cm² × Velocity = 1
∴ Velocity = 25 cm/seconds.
(b). At B,
Area × Velocity = 1 cm³/s.
2/100 × Velocity = 1
Velocity = 50 cm/seconds.
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(c).
Using the Bernoulli's Equation,
P₁ + h₁ρg + 1/2 ρv₁² = P₂ + h₂ρg + 1/2 ρv₂²
∴ P₁ - P₂ = 1/2 ρ(v₂² - v₁²) + ρg(h₂ - h₁)
∴ P₁ - P₂ = 1/2 ρ(v₂² - v₁²) [ ∵ In case of the Horizontal tube, Height difference is zero.]
∴ P₁ - P₂ = 1/2 × 1 × (50² - 25²)
∴ P₁ - P₂ = 937.5 dyne/cm²
∴ P₁ - P₂ = 93.75 N/m² or 93.75 Pa.
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Hope it helps.
Answer
Since, Volume rate of the flow of water is constant throughout the Horizontal pipe, therefore,
(a). At A
Area × Velocity = 1 cm³/s.
4/100 cm² × Velocity = 1
∴ Velocity = 25 cm/seconds.
(b). At B,
Area × Velocity = 1 cm³/s.
2/100 × Velocity = 1