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In a steady laminar flow of a given discharge of a liquid through a circular pipe of diameter d, the hydraulic gradient varies
- a)directly as square root of diameter
- b)directly as square of diameter
- c)inversely as square of diameter
- d)inversely as fourth power of diameter
Correct answer is option 'D'. Can you explain this answer?
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In a steady laminar flow of a given discharge of a liquid through a ci...
The hydraulic gradient in a steady laminar flow of a liquid through a circular pipe is a measure of the change in pressure per unit length of the pipe. It is determined by the properties of the liquid and the characteristics of the pipe, such as its diameter.
To understand why the hydraulic gradient varies inversely as the fourth power of the diameter, let's consider the factors that influence the pressure drop along the pipe.
1. Flow rate: The discharge of the liquid through the pipe is constant, meaning that the flow rate remains the same regardless of the pipe diameter. This can be expressed as Q = A * V, where Q is the flow rate, A is the cross-sectional area of the pipe, and V is the average velocity of the liquid.
2. Velocity distribution: In laminar flow, the velocity distribution across the pipe is parabolic, with the maximum velocity occurring at the center of the pipe and decreasing towards the walls. The velocity profile can be described using the Hagen-Poiseuille equation, which relates the velocity to the pressure drop.
3. Pressure drop: The pressure drop along the pipe is caused by the frictional resistance between the liquid and the pipe walls. This resistance is influenced by the viscosity of the liquid, the length of the pipe, and the diameter of the pipe.
Now, let's consider how the hydraulic gradient varies with the pipe diameter.
- The cross-sectional area of the pipe is directly proportional to the square of the diameter (A ∝ d^2).
- The average velocity of the liquid is inversely proportional to the diameter (V ∝ 1/d).
- The pressure drop is directly proportional to the length of the pipe and the viscosity of the liquid, but it is also influenced by the velocity distribution.
From the above relationships, we can deduce that the hydraulic gradient is inversely proportional to the fourth power of the diameter.
- The hydraulic gradient (ΔP/L) is proportional to the pressure drop (ΔP) per unit length of the pipe (L).
- The pressure drop (ΔP) is influenced by the velocity distribution, which is inversely proportional to the diameter (V ∝ 1/d).
- Thus, the hydraulic gradient is inversely proportional to the fourth power of the diameter (ΔP/L ∝ 1/d^4).
Therefore, option D is the correct answer: the hydraulic gradient varies inversely as the fourth power of the diameter.
To understand why the hydraulic gradient varies inversely as the fourth power of the diameter, let's consider the factors that influence the pressure drop along the pipe.
1. Flow rate: The discharge of the liquid through the pipe is constant, meaning that the flow rate remains the same regardless of the pipe diameter. This can be expressed as Q = A * V, where Q is the flow rate, A is the cross-sectional area of the pipe, and V is the average velocity of the liquid.
2. Velocity distribution: In laminar flow, the velocity distribution across the pipe is parabolic, with the maximum velocity occurring at the center of the pipe and decreasing towards the walls. The velocity profile can be described using the Hagen-Poiseuille equation, which relates the velocity to the pressure drop.
3. Pressure drop: The pressure drop along the pipe is caused by the frictional resistance between the liquid and the pipe walls. This resistance is influenced by the viscosity of the liquid, the length of the pipe, and the diameter of the pipe.
Now, let's consider how the hydraulic gradient varies with the pipe diameter.
- The cross-sectional area of the pipe is directly proportional to the square of the diameter (A ∝ d^2).
- The average velocity of the liquid is inversely proportional to the diameter (V ∝ 1/d).
- The pressure drop is directly proportional to the length of the pipe and the viscosity of the liquid, but it is also influenced by the velocity distribution.
From the above relationships, we can deduce that the hydraulic gradient is inversely proportional to the fourth power of the diameter.
- The hydraulic gradient (ΔP/L) is proportional to the pressure drop (ΔP) per unit length of the pipe (L).
- The pressure drop (ΔP) is influenced by the velocity distribution, which is inversely proportional to the diameter (V ∝ 1/d).
- Thus, the hydraulic gradient is inversely proportional to the fourth power of the diameter (ΔP/L ∝ 1/d^4).
Therefore, option D is the correct answer: the hydraulic gradient varies inversely as the fourth power of the diameter.
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In a steady laminar flow of a given discharge of a liquid through a circular pipe of diameter d, the hydraulic gradient variesa)directly as square root of diameterb)directly as square of diameterc)inversely as square of diameterd)inversely as fourth power of diameterCorrect answer is option 'D'. Can you explain this answer?
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