Đường ống dài 2L, đường kính d, nối hai bình có độ chênh H. Nước chảy tầng, bỏ qua tổn thất cục bộ. Nếu ta nối từ giữa ống 4 nhánh song song có chiều dài tương đương L, đường kính d thì khi đó lưu lượng nước chảy trong ống sẽ tăng lên:
Trả lời:
Đáp án đúng: C
Let Q1 be the initial flow rate without adding any pipes, and Q2 be the flow rate after adding 4 parallel branches.
* **Case 1: Original Pipe (length 2L)**
Applying the Darcy-Weisbach formula for head loss:
ΔH = (λ * (2L) * V₁²)/(d * 2g)
Where V₁ is the flow velocity in the original pipe. Flow rate Q₁ = A * V₁ = (πd²/4) * V₁
=> V₁ = Q₁/(πd²/4)
=> ΔH = (λ * 2L * Q₁²)/(d * 2g * (πd²/4)² ) = (16λL * Q₁²)/(π²gd⁵)
* **Case 2: After adding 4 parallel branches**
When 4 parallel branches are added, the length of each branch is L. The head loss across each branch is:
ΔH = (λ * L * V₂²)/(d * 2g) where V₂ is the flow velocity in each branch.
The flow rate through each branch is Q₂/4 = (πd²/4) * V₂
=> V₂ = Q₂/(4*πd²/4) = Q₂/(πd²)
=> ΔH = (λ * L * Q₂²)/(d * 2g * (πd²)²) = (λL * Q₂²)/(2π²gd⁵)
Since the pressure difference H is the same in both cases, we have:
(16λL * Q₁²)/(π²gd⁵) = (λL * Q₂²)/(2π²gd⁵)
=> 16Q₁² = Q₂²/2
=> Q₂² = 32Q₁²
=> Q₂ = √32 * Q₁ = 4√2 * Q₁ ≈ 5.66Q₁
Therefore, the flow rate increases by a factor of approximately 5.66 when 4 parallel branches are added.
