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Two pipes of diameters d1 and d2 converge to form a pipe of diameter 2d. If the liquid flows with a velocity of v1 and v2 in the two pipes, what will be the flow velocity in the third pipe?
- a)v1 + v2
- b)v1 + v2/2
- c)v1 + v2/4
- d)2(v1 + v2)
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Two pipes of diameters d1 and d2 converge to form a pipe of diameter 2...
Explanation:
When two pipes of diameters d1 and d2 converge to form a pipe of diameter 2d, we can use the principle of continuity to determine the flow velocity in the third pipe.
Principle of Continuity:
According to the principle of continuity, the mass flow rate of a fluid remains constant along a streamline. In other words, the product of the fluid velocity and the cross-sectional area of the pipe remains constant.
Mathematical Formulation:
Applying the principle of continuity, we can write the equation as:
A1v1 = A2v2 = A3v3
Where A1, A2, and A3 are the cross-sectional areas of the pipes, and v1, v2, and v3 are the flow velocities in the respective pipes.
Relationship between Area and Diameter:
The cross-sectional area of a pipe is directly proportional to the square of its diameter. Therefore, we can write the following relationships:
A1 = π(d1/2)^2
A2 = π(d2/2)^2
A3 = π(d/2)^2
Substituting the Area Equations:
Substituting the area equations into the continuity equation:
π(d1/2)^2v1 = π(d2/2)^2v2 = π(d/2)^2v3
Canceling the π/2^2 terms:
(d1/2)^2v1 = (d2/2)^2v2 = (d/2)^2v3
Simplifying:
(d1/2)^2v1 = (d2/2)^2v2
(d1/2)^2 = (d2/2)^2v2/v1
(d1/2)^2 = (d2/2)^2(v2/v1)
(d1/2)^2/(d2/2)^2 = v2/v1
(d1/d2)^2 = v2/v1
Taking the square root of both sides:
d1/d2 = √(v2/v1)
Simplifying:
d1/d2 = √(v2/v1)
d1 = √(v2/v1)d2
Flow Velocity in the Third Pipe:
Since the diameter of the third pipe is 2d, we can substitute d = 2d into the equation:
d1 = √(v2/v1)d2
d1 = √(v2/v1)(2d)
Simplifying:
d1 = 2√(v2/v1)d
v3 = v2/v1 * (d1/d)^2
v3 = v2/v1 * (2√(v2/v1))^2
v3 = v2/v1 * 4(v2/v1)
v3 = 4(v2/v1)^2
Therefore, the flow velocity in the third pipe is 4 times the square of the ratio of the velocities in the two pipes. Hence, the correct answer is option 'C' - v1 v2/4.
When two pipes of diameters d1 and d2 converge to form a pipe of diameter 2d, we can use the principle of continuity to determine the flow velocity in the third pipe.
Principle of Continuity:
According to the principle of continuity, the mass flow rate of a fluid remains constant along a streamline. In other words, the product of the fluid velocity and the cross-sectional area of the pipe remains constant.
Mathematical Formulation:
Applying the principle of continuity, we can write the equation as:
A1v1 = A2v2 = A3v3
Where A1, A2, and A3 are the cross-sectional areas of the pipes, and v1, v2, and v3 are the flow velocities in the respective pipes.
Relationship between Area and Diameter:
The cross-sectional area of a pipe is directly proportional to the square of its diameter. Therefore, we can write the following relationships:
A1 = π(d1/2)^2
A2 = π(d2/2)^2
A3 = π(d/2)^2
Substituting the Area Equations:
Substituting the area equations into the continuity equation:
π(d1/2)^2v1 = π(d2/2)^2v2 = π(d/2)^2v3
Canceling the π/2^2 terms:
(d1/2)^2v1 = (d2/2)^2v2 = (d/2)^2v3
Simplifying:
(d1/2)^2v1 = (d2/2)^2v2
(d1/2)^2 = (d2/2)^2v2/v1
(d1/2)^2 = (d2/2)^2(v2/v1)
(d1/2)^2/(d2/2)^2 = v2/v1
(d1/d2)^2 = v2/v1
Taking the square root of both sides:
d1/d2 = √(v2/v1)
Simplifying:
d1/d2 = √(v2/v1)
d1 = √(v2/v1)d2
Flow Velocity in the Third Pipe:
Since the diameter of the third pipe is 2d, we can substitute d = 2d into the equation:
d1 = √(v2/v1)d2
d1 = √(v2/v1)(2d)
Simplifying:
d1 = 2√(v2/v1)d
v3 = v2/v1 * (d1/d)^2
v3 = v2/v1 * (2√(v2/v1))^2
v3 = v2/v1 * 4(v2/v1)
v3 = 4(v2/v1)^2
Therefore, the flow velocity in the third pipe is 4 times the square of the ratio of the velocities in the two pipes. Hence, the correct answer is option 'C' - v1 v2/4.
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Community Answer
Two pipes of diameters d1 and d2 converge to form a pipe of diameter 2...
According to the Continuity Equation,
where a represents flow area, v represents flow velocity, i is for inlet conditions and o is for outlet conditions. Thus,
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