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A circular pipe of radius R carries a laminar flow of a fluid. The average velocity is indicated as the local velocity at what radial distance, measured from the centre?
- a)0.50R
- b)0.71R
- c)0.67R
- d)0.29R
Correct answer is option 'B'. Can you explain this answer?
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A circular pipe of radius R carries a laminar flow of a fluid. The ave...
For laminar flow in a circular pipe, the local velocity may be given as,
For laminar flow in a circular pipe, the average velocity may be given as,
......(ii)
For laminar flow in a circular pipe, the average velocity may be given as,
......(ii)Equating (i) and (ii), we get,
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A circular pipe of radius R carries a laminar flow of a fluid. The ave...
To determine the radial distance at which the average velocity of a laminar flow in a circular pipe is indicated, we need to consider the velocity profile of the flow. In laminar flow, the velocity distribution across a pipe cross-section follows a parabolic profile, with the maximum velocity occurring at the center and decreasing as we move towards the pipe wall.
In this case, we are looking for the radial distance at which the average velocity is indicated. The average velocity is defined as the total volumetric flow rate divided by the cross-sectional area of the pipe. Since the flow is laminar and the velocity profile is parabolic, the average velocity is typically taken as the local velocity at the radial distance where the flow velocity is half of the maximum velocity.
Let's consider the velocity profile equation for laminar flow in a circular pipe:
v(r) = (2 * V_avg / R^2) * (R^2 - r^2)
where v(r) is the velocity at a radial distance r from the center of the pipe, V_avg is the average velocity, and R is the radius of the pipe.
To find the radial distance where the velocity is half of the maximum velocity, we can set v(r) equal to half of the maximum velocity, which occurs at r = 0:
(2 * V_avg / R^2) * (R^2 - r^2) = V_max / 2
Simplifying the equation, we get:
R^2 - r^2 = (R^2) / 2
r^2 = (R^2) / 2
Taking the square root of both sides, we find:
r = R / √2
Simplifying further, we get:
r = 0.707 R
So, the radial distance at which the average velocity is indicated is approximately 0.707 times the radius of the pipe.
Therefore, the correct answer is option 'B' - 0.71R.
In this case, we are looking for the radial distance at which the average velocity is indicated. The average velocity is defined as the total volumetric flow rate divided by the cross-sectional area of the pipe. Since the flow is laminar and the velocity profile is parabolic, the average velocity is typically taken as the local velocity at the radial distance where the flow velocity is half of the maximum velocity.
Let's consider the velocity profile equation for laminar flow in a circular pipe:
v(r) = (2 * V_avg / R^2) * (R^2 - r^2)
where v(r) is the velocity at a radial distance r from the center of the pipe, V_avg is the average velocity, and R is the radius of the pipe.
To find the radial distance where the velocity is half of the maximum velocity, we can set v(r) equal to half of the maximum velocity, which occurs at r = 0:
(2 * V_avg / R^2) * (R^2 - r^2) = V_max / 2
Simplifying the equation, we get:
R^2 - r^2 = (R^2) / 2
r^2 = (R^2) / 2
Taking the square root of both sides, we find:
r = R / √2
Simplifying further, we get:
r = 0.707 R
So, the radial distance at which the average velocity is indicated is approximately 0.707 times the radius of the pipe.
Therefore, the correct answer is option 'B' - 0.71R.
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A circular pipe of radius R carries a laminar flow of a fluid. The average velocity is indicated as the local velocity at what radial distance, measured from the centre?a)0.50Rb)0.71Rc)0.67Rd)0.29RCorrect answer is option 'B'. Can you explain this answer?
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