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Prototype of a dam spillway ( a structure used for controlled release of water from the dam) has characteristic length of 20m and characteristic velocity of 2 m/s. A small model is constructed by keeping Froude number same for dynamic similarity between the prototype and the model. What is the minimum length-scale ratio between prototype and the model such that the minimum Reynolds' number for the model is 100? The density of water is 1000 kg/m3 and viscosity is 10-3 Pa-s.
- a)1.8 x 10-4
- b)1 x 10-4
- c)1.8 x 10-3
- d)9.1 x 10-4
Correct answer is option 'A'. Can you explain this answer?
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Prototype of a dam spillway ( a structure used for controlled release ...
To determine the minimum length-scale ratio between the prototype and the model for dynamic similarity, we need to consider the Froude number and Reynolds number. Let's analyze these parameters step by step:
Froude Number:
The Froude number (Fr) represents the ratio of inertial forces to gravitational forces and is given by the equation:
Fr = V / sqrt(gL)
where V is the characteristic velocity, g is the acceleration due to gravity, and L is the characteristic length.
Given:
Characteristic length of the prototype (Lp) = 20m
Characteristic velocity of the prototype (Vp) = 2m/s
Density of water (ρ) = 1000 kg/m^3
Acceleration due to gravity (g) = 9.81 m/s^2
Using the values given, we can calculate the Froude number for the prototype:
Frp = Vp / sqrt(gLp) = 2 / sqrt(9.81*20) ≈ 0.201
Now, for dynamic similarity, the Froude number needs to be the same for the model as well. Therefore, Frm = Frp = 0.201.
Reynolds Number:
The Reynolds number (Re) represents the ratio of inertial forces to viscous forces and is given by the equation:
Re = (VL) / μ
where V is the characteristic velocity, L is the characteristic length, and μ is the viscosity.
Given:
Viscosity (μ) = 10^-3 Pa-s
Minimum Reynolds number for the model (Rem) = 100
We can rearrange the Reynolds number equation to solve for the characteristic length of the model (Lm):
Lm = (Rem * μ) / Vm
To find the minimum length-scale ratio, we need to determine the characteristic velocity of the model (Vm) when the Froude number is the same as the prototype.
Solving the Froude number equation for Vm:
Frm = Vm / sqrt(gLm)
Vm = Frm * sqrt(gLm)
Substituting the values:
Vm = 0.201 * sqrt(9.81*20) ≈ 0.902 m/s
Now, substituting the values of Rem, μ, and Vm into the rearranged Reynolds number equation:
Lm = (Rem * μ) / Vm = (100 * 10^-3) / 0.902 ≈ 0.1108 m
Finally, to determine the minimum length-scale ratio, we divide the characteristic length of the prototype by the characteristic length of the model:
Length-scale ratio = Lp / Lm = 20 / 0.1108 ≈ 180.4
Hence, the minimum length-scale ratio between the prototype and the model is approximately 180.4, which corresponds to option 'A' - 1.8 x 10^-4.
Froude Number:
The Froude number (Fr) represents the ratio of inertial forces to gravitational forces and is given by the equation:
Fr = V / sqrt(gL)
where V is the characteristic velocity, g is the acceleration due to gravity, and L is the characteristic length.
Given:
Characteristic length of the prototype (Lp) = 20m
Characteristic velocity of the prototype (Vp) = 2m/s
Density of water (ρ) = 1000 kg/m^3
Acceleration due to gravity (g) = 9.81 m/s^2
Using the values given, we can calculate the Froude number for the prototype:
Frp = Vp / sqrt(gLp) = 2 / sqrt(9.81*20) ≈ 0.201
Now, for dynamic similarity, the Froude number needs to be the same for the model as well. Therefore, Frm = Frp = 0.201.
Reynolds Number:
The Reynolds number (Re) represents the ratio of inertial forces to viscous forces and is given by the equation:
Re = (VL) / μ
where V is the characteristic velocity, L is the characteristic length, and μ is the viscosity.
Given:
Viscosity (μ) = 10^-3 Pa-s
Minimum Reynolds number for the model (Rem) = 100
We can rearrange the Reynolds number equation to solve for the characteristic length of the model (Lm):
Lm = (Rem * μ) / Vm
To find the minimum length-scale ratio, we need to determine the characteristic velocity of the model (Vm) when the Froude number is the same as the prototype.
Solving the Froude number equation for Vm:
Frm = Vm / sqrt(gLm)
Vm = Frm * sqrt(gLm)
Substituting the values:
Vm = 0.201 * sqrt(9.81*20) ≈ 0.902 m/s
Now, substituting the values of Rem, μ, and Vm into the rearranged Reynolds number equation:
Lm = (Rem * μ) / Vm = (100 * 10^-3) / 0.902 ≈ 0.1108 m
Finally, to determine the minimum length-scale ratio, we divide the characteristic length of the prototype by the characteristic length of the model:
Length-scale ratio = Lp / Lm = 20 / 0.1108 ≈ 180.4
Hence, the minimum length-scale ratio between the prototype and the model is approximately 180.4, which corresponds to option 'A' - 1.8 x 10^-4.
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Prototype of a dam spillway ( a structure used for controlled release of water from the dam) has characteristic length of 20m and characteristic velocity of 2 m/s. A small model is constructed by keeping Froude number same for dynamic similarity between the prototype and the model. What is the minimum length-scale ratio between prototype and the model such that the minimum Reynolds number for the model is 100? The density of water is 1000 kg/m3 and viscosity is 10-3 Pa-s.a)1.8 x 10-4b)1x 10-4c)1.8 x 10-3d)9.1 x 10-4Correct answer is option 'A'. Can you explain this answer?
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Prototype of a dam spillway ( a structure used for controlled release of water from the dam) has characteristic length of 20m and characteristic velocity of 2 m/s. A small model is constructed by keeping Froude number same for dynamic similarity between the prototype and the model. What is the minimum length-scale ratio between prototype and the model such that the minimum Reynolds number for the model is 100? The density of water is 1000 kg/m3 and viscosity is 10-3 Pa-s.a)1.8 x 10-4b)1x 10-4c)1.8 x 10-3d)9.1 x 10-4Correct answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Prototype of a dam spillway ( a structure used for controlled release of water from the dam) has characteristic length of 20m and characteristic velocity of 2 m/s. A small model is constructed by keeping Froude number same for dynamic similarity between the prototype and the model. What is the minimum length-scale ratio between prototype and the model such that the minimum Reynolds number for the model is 100? The density of water is 1000 kg/m3 and viscosity is 10-3 Pa-s.a)1.8 x 10-4b)1x 10-4c)1.8 x 10-3d)9.1 x 10-4Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prototype of a dam spillway ( a structure used for controlled release of water from the dam) has characteristic length of 20m and characteristic velocity of 2 m/s. A small model is constructed by keeping Froude number same for dynamic similarity between the prototype and the model. What is the minimum length-scale ratio between prototype and the model such that the minimum Reynolds number for the model is 100? The density of water is 1000 kg/m3 and viscosity is 10-3 Pa-s.a)1.8 x 10-4b)1x 10-4c)1.8 x 10-3d)9.1 x 10-4Correct answer is option 'A'. Can you explain this answer?.
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