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Two pipes, each of diameter d, converge to form a pipe of diameter D. What should be the relation between d and D such that the flow velocity in the third pipe becomes double of that in each of the two pipes?
- a)D = d
- b)D = 2d
- c)D = 3d
- d)D = 4d
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Two pipes, each of diameter d, converge to form a pipe of diameter D. ...
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,
A1v1 + A2v2 = Av
d2v + d2v = D2v
D = d.
where a represents flow area, v represents flow velocity, i is for inlet conditions and o is for outlet conditions. Thus,
A1v1 + A2v2 = Av
d2v + d2v = D2v
D = d.
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Two pipes, each of diameter d, converge to form a pipe of diameter D. ...
Explanation:
When two pipes of diameter d converge to form a pipe of diameter D, the flow velocity in the third pipe can be determined using the principle of continuity. According to the principle of continuity, the mass flow rate through each pipe is constant.
The equation for mass flow rate:
The equation for mass flow rate (Q) is given by:
Q = A1 * V1 = A2 * V2 = A3 * V3
Where:
- Q is the mass flow rate
- A1, A2, and A3 are the cross-sectional areas of the respective pipes
- V1, V2, and V3 are the flow velocities in the respective pipes
Relation between diameters and cross-sectional areas:
The cross-sectional area of a pipe is directly proportional to the square of its diameter. So, the relation between the diameters and cross-sectional areas can be expressed as:
A1 = (π/4) * d^2
A2 = (π/4) * d^2
A3 = (π/4) * D^2
Relation between flow velocities:
To find the relation between the flow velocities, we can rearrange the equation for mass flow rate as follows:
V3 = (A1/A3) * V1 = (A2/A3) * V2
Since we want the flow velocity in the third pipe to become double of that in each of the two pipes, we can set up the following equation:
2V1 = (A1/A3) * V1
Simplifying the equation, we get:
2 = (A1/A3)
Substituting the values of cross-sectional areas:
Substituting the values of A1 and A3 in terms of diameters, we have:
2 = ((π/4) * d^2) / ((π/4) * D^2)
Simplifying the equation further, we get:
2 = (d^2) / (D^2)
Taking the square root of both sides, we get:
√2 = d / D
Conclusion:
Therefore, the relation between d and D should be D = d for the flow velocity in the third pipe to become double of that in each of the two pipes. So, option A is the correct answer.
When two pipes of diameter d converge to form a pipe of diameter D, the flow velocity in the third pipe can be determined using the principle of continuity. According to the principle of continuity, the mass flow rate through each pipe is constant.
The equation for mass flow rate:
The equation for mass flow rate (Q) is given by:
Q = A1 * V1 = A2 * V2 = A3 * V3
Where:
- Q is the mass flow rate
- A1, A2, and A3 are the cross-sectional areas of the respective pipes
- V1, V2, and V3 are the flow velocities in the respective pipes
Relation between diameters and cross-sectional areas:
The cross-sectional area of a pipe is directly proportional to the square of its diameter. So, the relation between the diameters and cross-sectional areas can be expressed as:
A1 = (π/4) * d^2
A2 = (π/4) * d^2
A3 = (π/4) * D^2
Relation between flow velocities:
To find the relation between the flow velocities, we can rearrange the equation for mass flow rate as follows:
V3 = (A1/A3) * V1 = (A2/A3) * V2
Since we want the flow velocity in the third pipe to become double of that in each of the two pipes, we can set up the following equation:
2V1 = (A1/A3) * V1
Simplifying the equation, we get:
2 = (A1/A3)
Substituting the values of cross-sectional areas:
Substituting the values of A1 and A3 in terms of diameters, we have:
2 = ((π/4) * d^2) / ((π/4) * D^2)
Simplifying the equation further, we get:
2 = (d^2) / (D^2)
Taking the square root of both sides, we get:
√2 = d / D
Conclusion:
Therefore, the relation between d and D should be D = d for the flow velocity in the third pipe to become double of that in each of the two pipes. So, option A is the correct answer.
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Two pipes, each of diameter d, converge to form a pipe of diameter D. What should be the relation between d and D such that the flow velocity in the third pipe becomes double of that in each of the two pipes?a)D = db)D = 2dc)D = 3dd)D = 4dCorrect answer is option 'A'. Can you explain this answer?
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