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When water flows at a rate Q through a capillary tube of radius r that is placed horizontally, a pressure difference p develops across the ends of the tube. If the radius of the tube is doubled and the rate of flow halved, the pressure difference becomes
- a)p/32
- b)p/8
- c)p
- d)8p
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
When water flows at a rate Q through a capillary tube of radius r that...
If the length of capillary tube is l, the pressure difference is given by
where η is the coefficient of viscosity of water. If r becomes 2r and Q becomes Q/2, we have
Hence the correct choice is (a).
where η is the coefficient of viscosity of water. If r becomes 2r and Q becomes Q/2, we have
Hence the correct choice is (a).
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When water flows at a rate Q through a capillary tube of radius r that...
Answer:
To understand why the pressure difference becomes p/32 when the radius of the capillary tube is doubled and the rate of flow is halved, we need to apply the principles of fluid mechanics and the Hagen-Poiseuille equation.
Hagen-Poiseuille Equation:
The Hagen-Poiseuille equation describes the flow of an incompressible and viscous fluid through a cylindrical pipe or tube. It states that the flow rate (Q) through a tube is directly proportional to the fourth power of the radius (r) and the pressure difference (ΔP), and inversely proportional to the viscosity of the fluid (η) and the length of the tube (L).
Mathematically, the Hagen-Poiseuille equation is given by:
Q = (π * r^4 * ΔP) / (8 * η * L)
where:
Q = flow rate
r = radius of the tube
ΔP = pressure difference across the ends of the tube
η = viscosity of the fluid
L = length of the tube
Effect of Doubling the Radius:
When the radius of the capillary tube is doubled, the flow rate (Q) is directly proportional to the fourth power of the radius (r^4). Therefore, the flow rate becomes 2^4 = 16 times the original value.
Effect of Halving the Flow Rate:
When the rate of flow is halved, the flow rate (Q) becomes 1/2 times the original value.
Pressure Difference Calculation:
From the Hagen-Poiseuille equation, we can express the pressure difference (ΔP) in terms of the flow rate (Q) and other variables:
ΔP = (8 * η * L * Q) / (π * r^4)
Now, let's calculate the pressure difference for the original scenario and the new scenario.
Original Scenario:
Q_original = Q
ΔP_original = (8 * η * L * Q_original) / (π * r^4)
New Scenario:
Q_new = Q_original / 2
r_new = 2 * r
ΔP_new = (8 * η * L * Q_new) / (π * (r_new)^4)
= (8 * η * L * (Q_original / 2)) / (π * (2r)^4)
= (8 * η * L * Q_original) / (π * 16r^4)
= (1/2) * (8 * η * L * Q_original) / (π * r^4)
= (1/2) * ΔP_original
The pressure difference in the new scenario is half of the original pressure difference. Therefore, the correct answer is option 'A': p/32.
To understand why the pressure difference becomes p/32 when the radius of the capillary tube is doubled and the rate of flow is halved, we need to apply the principles of fluid mechanics and the Hagen-Poiseuille equation.
Hagen-Poiseuille Equation:
The Hagen-Poiseuille equation describes the flow of an incompressible and viscous fluid through a cylindrical pipe or tube. It states that the flow rate (Q) through a tube is directly proportional to the fourth power of the radius (r) and the pressure difference (ΔP), and inversely proportional to the viscosity of the fluid (η) and the length of the tube (L).
Mathematically, the Hagen-Poiseuille equation is given by:
Q = (π * r^4 * ΔP) / (8 * η * L)
where:
Q = flow rate
r = radius of the tube
ΔP = pressure difference across the ends of the tube
η = viscosity of the fluid
L = length of the tube
Effect of Doubling the Radius:
When the radius of the capillary tube is doubled, the flow rate (Q) is directly proportional to the fourth power of the radius (r^4). Therefore, the flow rate becomes 2^4 = 16 times the original value.
Effect of Halving the Flow Rate:
When the rate of flow is halved, the flow rate (Q) becomes 1/2 times the original value.
Pressure Difference Calculation:
From the Hagen-Poiseuille equation, we can express the pressure difference (ΔP) in terms of the flow rate (Q) and other variables:
ΔP = (8 * η * L * Q) / (π * r^4)
Now, let's calculate the pressure difference for the original scenario and the new scenario.
Original Scenario:
Q_original = Q
ΔP_original = (8 * η * L * Q_original) / (π * r^4)
New Scenario:
Q_new = Q_original / 2
r_new = 2 * r
ΔP_new = (8 * η * L * Q_new) / (π * (r_new)^4)
= (8 * η * L * (Q_original / 2)) / (π * (2r)^4)
= (8 * η * L * Q_original) / (π * 16r^4)
= (1/2) * (8 * η * L * Q_original) / (π * r^4)
= (1/2) * ΔP_original
The pressure difference in the new scenario is half of the original pressure difference. Therefore, the correct answer is option 'A': p/32.
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