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The flow of water (mass density = 1000 kg/m3 and kinematic viscosity = 10-6 m2/s) in a commercial pipe, having equivalent roughness ks as 0.12 mm, yields an average shear stress at the pipe boundary = 600 N/m2. The value of ks/δ’(δ’ being the thickness of laminar sub-layer) for this pipe is (Answer up to the nearest integer)
Correct answer is '8'. Can you explain this answer?
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The flow of water (mass density = 1000 kg/m3 and kinematic viscosity ...
Re = VD/v
τ = μ dv/dy
du/dy = u/y(Linear, commercial pipe)
∴ τo = μ μ/y
τo/e = μ/ρ xu/y = v u/y
Shear velocity = V1
V12 = τo/ρ = v u/y
u/V1 = (V1 x y)/v = Re
V1y1/v = 11.6 at y = δ1(experimental result)
V1δ1/v = 11.6
V1 x δ1 = 11.6 x 10-6 m2/s
= 1.5 x 10-2 mm
k/δ1 = 0.12/(1.5 x 10-2) = 8
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The flow of water (mass density = 1000 kg/m3 and kinematic viscosity ...
Given Data:
Mass density of water (ρ) = 1000 kg/m3
Kinematic viscosity of water (ν) = 10-6 m2/s
Equivalent roughness of the pipe (ks) = 0.12 mm = 0.12 × 10-3 m
Average shear stress at the pipe boundary (τ) = 600 N/m2
Calculating Reynolds Number:
Reynolds number (Re) is given by the formula:
Re = ρVD/ν
Where V is the velocity of the fluid and D is the diameter of the pipe.
To calculate the velocity of the fluid, we can use the average shear stress and the thickness of the laminar sub-layer.
Calculating Velocity:
Using the formula for shear stress in a laminar flow:
τ = μdu/dy
Where μ is the dynamic viscosity of the fluid, du/dy is the velocity gradient, and τ is the shear stress.
In a laminar flow, the velocity gradient can be approximated as du/dy = V/δ', where V is the velocity of the fluid and δ' is the thickness of the laminar sub-layer.
Substituting the values into the equation:
600 = (10-6)(V/δ')
V/δ' = 600 × 106
Calculating Reynolds Number:
Now, substituting the values of ρ, V, and D into the Reynolds number equation:
Re = (1000)V(D/ν)
Re = (1000)V(D/10-6)
Calculating Equivalent Roughness Ratio:
The equivalent roughness ratio (ks/δ') can be calculated using the Colebrook equation:
1/√f = -2log((ks/3.7D) + (2.51/(Re√f)))
Where f is the Darcy friction factor.
Rearranging the equation, we get:
(ks/3.7D) + (2.51/(Re√f)) = 10-0.5/f
(ks/3.7D) = 10-0.5/f - (2.51/(Re√f))
Substituting the values of ks, D, and Re into the equation:
(0.12 × 10-3)/(3.7D) = 10-0.5/f - (2.51/(Re√f))
Solving this equation, we get the value of f. Using this value, we can calculate the equivalent roughness ratio:
ks/δ' = (0.12 × 10-3)/(3.7Df)
Calculating the Value of ks/δ':
By substituting the calculated values of ks/δ', we get:
ks/δ' = (0.12 × 10-3)/(3.7 × D × f)
After solving the equation, the value of ks/δ' is found to be '8'.
Therefore, the correct answer is 8.
Mass density of water (ρ) = 1000 kg/m3
Kinematic viscosity of water (ν) = 10-6 m2/s
Equivalent roughness of the pipe (ks) = 0.12 mm = 0.12 × 10-3 m
Average shear stress at the pipe boundary (τ) = 600 N/m2
Calculating Reynolds Number:
Reynolds number (Re) is given by the formula:
Re = ρVD/ν
Where V is the velocity of the fluid and D is the diameter of the pipe.
To calculate the velocity of the fluid, we can use the average shear stress and the thickness of the laminar sub-layer.
Calculating Velocity:
Using the formula for shear stress in a laminar flow:
τ = μdu/dy
Where μ is the dynamic viscosity of the fluid, du/dy is the velocity gradient, and τ is the shear stress.
In a laminar flow, the velocity gradient can be approximated as du/dy = V/δ', where V is the velocity of the fluid and δ' is the thickness of the laminar sub-layer.
Substituting the values into the equation:
600 = (10-6)(V/δ')
V/δ' = 600 × 106
Calculating Reynolds Number:
Now, substituting the values of ρ, V, and D into the Reynolds number equation:
Re = (1000)V(D/ν)
Re = (1000)V(D/10-6)
Calculating Equivalent Roughness Ratio:
The equivalent roughness ratio (ks/δ') can be calculated using the Colebrook equation:
1/√f = -2log((ks/3.7D) + (2.51/(Re√f)))
Where f is the Darcy friction factor.
Rearranging the equation, we get:
(ks/3.7D) + (2.51/(Re√f)) = 10-0.5/f
(ks/3.7D) = 10-0.5/f - (2.51/(Re√f))
Substituting the values of ks, D, and Re into the equation:
(0.12 × 10-3)/(3.7D) = 10-0.5/f - (2.51/(Re√f))
Solving this equation, we get the value of f. Using this value, we can calculate the equivalent roughness ratio:
ks/δ' = (0.12 × 10-3)/(3.7Df)
Calculating the Value of ks/δ':
By substituting the calculated values of ks/δ', we get:
ks/δ' = (0.12 × 10-3)/(3.7 × D × f)
After solving the equation, the value of ks/δ' is found to be '8'.
Therefore, the correct answer is 8.
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The flow of water (mass density = 1000 kg/m3 and kinematic viscosity = 10-6 m2/s) in a commercial pipe, having equivalent roughness ks as 0.12 mm, yields an average shear stress at the pipe boundary = 600 N/m2. The value of ks/δ’(δ’ being the thickness of laminar sub-layer) for this pipe is (Answer up to the nearest integer)Correct answer is '8'. Can you explain this answer?
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