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Assignment Number 1 1. Show that by using Buckingham’s pi theorem pressure drop depends on Reynolds number. 2. The time period T of water surface waves is known to depend on the wave length ?, depth of flow D, density of the fluid ?, acceleration due to gravity g and surface tension ?. Obtain the expression for time period by using both Rayelign method and Buckingham’s pi theorem. 3. In a tidal model, the horizontal scale ratio is 1/500. The vertical scale is 1/50. What model period would correspond to a prototype period of 12 hours. 4. A model boat, 1/100 size of its prototype has 0.12N of resistance when the simulating a speed of 5m/sec of the prototype. Water is the fluid in the both the cases. What is the corresponding resistance of the prototype? 5. A pipe of diameter 1.5m is required to transport an oil of specific gravity 0.9 and kinematic viscosity 0.03 stoke at the rate of 3 m³/sec. If a 15cm diameter pipe with water is 0.01 stoke is used to model the above flow , find the velocity and discharge in the model. 6. A proposed model of the river stretch of 15kM is to have a horizontal scale of 1/200 and a vertical scale of 1/40. If the normal depth, wide and discharge are 2m, 90m and 152 m³/sec. Estimate the corresponding model quantities. What values of the Mannings roughness N is to be provided in the model to represented a prototype roughness value of 0.25? 7. The efficiency ? of a fan depends upon the viscosity of the fluid, the angular velocity ?, diameter D of the rotor and the discharge Q Express ? in terms of dimensionless parameters using Buckingham Pi theorem as ? = ?[?D²/?, Q/?D³] 8. The drag on the airplane cruising at 240 mph in standard air is to be determined from tests on a 1 10 scale model placed in a pressurized wind tunnel To minimize compressibility effects, the airspeed in the wind tunnel is also to be 240 mph Determine ( the required air pressure in the tunnel (assuming the same air temperature for model and prototype) and ( the drag on the prototype corresponding to a measured force of 1 lb on the model. 9. An agitator of diameter D requires power to rotate at a constant speed N in a liquid density ? and viscosity ? i show with the help of Pi theorem that P=?N³D?F(?ND²/?).

Assignment Number 1
1. Show that by using Buckingham’s pi theorem pressure drop depends on Reynolds number.
2. The time period T of water surface waves is known to depend on the wave length ?, depth of flow D, density of the fluid ?, acceleration due to gravity g and surface tension ?. Obtain the expression for time period by using both Rayelign method and Buckingham’s pi theorem.
3. In a tidal model, the horizontal scale ratio is 1/500. The vertical scale is 1/50. What model period would correspond to a prototype period of 12 hours.
4. A model boat, 1/100 size of its prototype has 0.12N of resistance when the simulating a speed of 5m/sec of the prototype. Water is the fluid in the both the cases. What is the corresponding resistance of the prototype?
5. A pipe of diameter 1.5m is required to transport an oil of specific gravity 0.9 and kinematic viscosity 0.03 stoke at the rate of 3 m³/sec. If a 15cm diameter pipe with water is 0.01 stoke is used to model the above flow , find the velocity and discharge in the model.
6. A proposed model of the river stretch of 15kM is to have a horizontal scale of 1/200 and a vertical scale of 1/40. If the normal depth, wide and discharge are 2m, 90m and 152 m³/sec. Estimate the corresponding model quantities. What values of the Mannings roughness N is to be provided in the model to represented a prototype roughness value of 0.25?
7. The efficiency ? of a fan depends upon the viscosity of the fluid, the angular velocity ?, diameter D of the rotor and the discharge Q Express ? in terms of dimensionless parameters using Buckingham Pi theorem as ? = ?[?D²/?, Q/?D³]
8. The drag on the airplane cruising at 240 mph in standard air is to be determined from tests on a 1 10 scale model placed in a pressurized wind tunnel To minimize compressibility effects, the airspeed in the wind tunnel is also to be 240 mph Determine ( the required air pressure in the tunnel (assuming the same air temperature for model and prototype) and ( the drag on the prototype corresponding to a measured force of 1 lb on the model.
9. An agitator of diameter D requires power to rotate at a constant speed N in a liquid density ? and viscosity ? i show with the help of Pi theorem that P=?N³D?F(?ND²/?).

Added by William B.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Supreeta N

11/15/2023

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For the first question, we need to show that the pressure drop depends on the Reynolds number using Buckingham's pi theorem. Buckingham's pi theorem states that if there are n variables involved in a problem and these variables have m fundamental dimensions, then Show more…

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Assignment Number 1 1. Show that by using Buckingham’s pi theorem pressure drop depends on Reynolds number. 2. The time period T of water surface waves is known to depend on the wave length ́λ, depth of flow D, density of the fluid ρ, acceleration due to gravity g and surface tension σ. Obtain the expression for time period by using both Rayleigh method and Buckingham’s pi theorem. 3. In a tidal model, the horizontal scale ratio is 1/500. The vertical scale is 1/50. What model period would correspond to a prototype period of 12 hours. 4. A model boat, 1/100 size of its prototype has 0.12N of resistance when the simulating a speed of 5m/sec of the prototype. Water is the fluid in the both the cases. What is the corresponding resistance of the prototype? 5. A pipe of diameter 1.5m is required to transport an oil of specific gravity 0.9 and kinematic viscosity 0.03 stoke at the rate of 3 m³/sec. If a 15cm diameter pipe with water is 0.01 stoke is used to model the above flow , find the velocity and discharge in the model. 6. A proposed model of the river stretch of 15kM is to have a horizontal scale of 1/200 and a vertical scale of 1/40. If the normal depth, wide and discharge are 2m, 90m and 152 m³/sec. Estimate the corresponding model quantities. What values of the Mannings roughness N is to be provided in the model to represented a prototype roughness value of 0.25? 7. The efficiency η of a fan depends upon the viscosity of the fluid, the angular velocity ω, diameter D of the rotor and the discharge Q Express η in terms of dimensionless parameters using Buckingham Pi theorem as η = Φ[ωD²/ν, Q/ωD³] 8. The drag on the airplane cruising at 240 mph in standard air is to be determined from tests on a 1 10 scale model placed in a pressurized wind tunnel To minimize compressibility effects, the airspeed in the wind tunnel is also to be 240 mph Determine ( the required air pressure in the tunnel (assuming the same air temperature for model and prototype) and ( the drag on the prototype corresponding to a measured force of 1 lb on the model. 9. An agitator of diameter D requires power to rotate at a constant speed N in a liquid density ρ and viscosity μ i show with the help of Pi theorem that P=ρN³D⁵F(ρND²/μ).

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00:01 Hello students this question is related to buckingham's pi theorem in this question we are asked to express or prove that the pressure drop is a function of reynolds number that is re number for that first let's look into the buckingham's pi theorem that is according to buckingham's pi theorem the number of dimensionless groups let it be represented as items that can be formed as equal to total number of variables variables let be represented by the letter n that is n be the total number of variables and minus the number of reference dimensions let it be represented by k.

02:20 So in the case of pressure drop in pipe the parameters involved are pressure drop which is represented by the letter delta p in the next parameter this density of time sorry density of fluid.

02:57 Sorry, which is represented by the letter rho for the the diameter of the pipe let it be represented by the letter d and the velocity of fluid that is represented by the letter capital v and the viscosity of fluid that is mu here the total number of variables that is the number of variables n is equal to 5 as this pressure drop density of fluid diameter of the pipe velocity and viscosity therefore n is equal to 5 and these can be represented in terms of dimension that is m raised to minus 1 t raised to minus 2 the density can be represented as ml raised to minus 3 then diameter plus represented as l velocity can be represented as l t raised to minus 1 and the viscosity can be represented as m raised to minus 1 t raised to minus 1 yeah ml and t are known dimensions there are they are mass length and temperature so according to buckingham's theorem we can take the reference dimensions that is k as 3 therefore according to the theorem the total number of dimensionless group that can be formed is equal to n minus k that is total number of variable that is 5 and known dimension which is equal to 3 therefore we can make 2 dimensionless group let's make that let pi 1 be the fourth dimensional group and pi 2 be the second dimensional group we can form pi 1 such that by multiplying by multiplying the pressure drop by a function of variables that are dimensionless that are dimensionless that is pi 1 can be represented as pi 1 is equal to delta p when we divide this delta p divided by rho into the velocity square the entire dimension the entire dimension of p1 becomes dimensionless there will be no dimension for pi 1 dimensionless we are multiplying such that p1 becomes dimensionless therefore it's the same way let's create pi 2 such that we are multiplying viscosity that is mu by that the function of variables that are dimensionless that is pi 2 can be represented as pi 2 is equal to mu divided by rho into v multiplied by d...

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