VisCosity, odel tests run in a laboratory using water in a 0.2-ft-diameter pipe yield the Ty vs. Q data shown in Figure b. Perform a dimensional analysis and use model data to predict the wall shear stress in a 0.3-ft- ameter pipe through which water flows at the rate of 1.5 ft/s. 0.7 I 0.6 2 0.5 Diameter D 0.4 Model data (a) 0.3 0.2 0.1 1.5 Flowrate, Q. ft'is 0.5 2 (b) Ib/ft2 Wall shear stress, r, Ibft?
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- 5.13 The torque due to the frictional resistance of the oil film between a rotating shaft and its bearing is found to be dependent on the force F normal to the shaft, the speed of rotation N of the shaft, the dynamic viscosity of the oil, and the shaft diameter D. Establish a correlation among these variables by using dimensional analysis.: The discharge pressure (P) of a gear pump (Fig. 3) is a function of flow rate (Q), gear diameter (D), fluid viscosity (µ) and gear angular speed (w). P = f (Q, D, H, 0). Use the pi theorem to rewrite this function in terms of dimensionless parameters. Suction Discharge Fig. 3: Gear pump P, QThe viscous torque T produced on a disc rotating in a liquid depends upon the characteristic dimension D, the rotational speed N, the density pand the dynamic viscosity u. a) Show that there are two non-dimensional parameters written as: T and a, PND? b) In order to predict the torque on a disc of 0.5 m of diameter which rotates in oil at 200 rpm, a model is made to a scale of 1/5. The model is rotated in water. Calculate the speed of rotation of the model necessary to simulate the rotation of the real disc. c) When the model is tested at 18.75 rpm, the torque was 0.02 N.m. Predict the torque on the full size disc at 200 rpm. Notes: For the oil: the density is 750kg/m² and the dynamic viscosity is 0.2 N.s/m². For water: the density is 1000 kg/ m² and the dynamic viscosity is 0.001 N.s/m². kg.m IN =1
- A tiny aerosol particle of density pp and characteristic diameter Dp falls in air of density p and viscosity u . If the particle is small enough, the creeping flow approximation is valid, and the terminal settling speed of the particle V depends only on Dp, µ, gravitational constant g, and the density difference (pp - p). Use dimensional analysis to generate a relationship for Vas a function of the independent variables. Name any established dimensionless parameters that appear in your analysis.Example: The pressure difference (Ap) between two point in a pipe due to turbulent flow depends on the velocity (V), diameter (D), dynamic viscosity (µ), density (p), roughness size (e), and distance between the points (L). using dimensional analysis determine the general form of the expression (use MLT system).The power P generated by a certain windmill design depends upon its diameter D, the air density p, the wind velocity V, the rotation rate 0, and the number of blades n. (a) Write this relationship in dimensionless form. A model windmill, of diameter 50 cm, develops 2.7 kW at sea level when V= 40 m/s and when rotating at 4800 r/min. (b) What power will be developed by a geometrically and dynamically similar prototype, of diameter 5 m, in winds of 12 m/s at 2000 m standard altitude? (c) What is the appropriate rotation rate of the prototype?
- Speed is usually a function of density, gravitational acceleration, diameter, height difference, viscosity, and length. Using the repetitive variables method and taking density, gravitational acceleration, and diameter as repetitive variables, find the required dimensionless parameters. V = f(p, g, D, Az, u, L)For the drag force of an Aerobee rocket, D=0.00056v (Ib], obtain a linear model near v = 800 ft/sec and find the velocity below 800 ft/sec that the model prediction error becomes greater than 2%. Enter an answer in a whole number in ft/sec.Please solve this problem, Thank you very much! Figure is attached 1. liquids in rotating cylinders rotates as a rigid body and considered at rest. The elevation difference h between the center of the liquid surface and the rim of the liquid surface is a function of angular velocity ?, fluid density ?, gravitational acceleration ?, and radius ?. Use the method of repeating variables to find a dimensionless relationship between the parameters. Show all the steps.
- Buckingham Pi. A mechanical stirrer is used to mix chemicals in a large tank. The required shaft power P is a function of liquid density p, viscosity μ, stirrer blade diameter D, and angular speed w of the spinning blades. (a) use repeating variables p, D, u to find a relation between dimensionless power (1) and w (m2); (b,c) a small 1/3 scale model is used in water to predict the actual required power in a viscous liquid with SG =2 and μ = 12μwater. Find (b) the ratio of speeds, wwater/ wactual, necessary for dynamic similarity and then (c) the predicted ratio of powers Pwater/ Pactual. expecting unit : (a) π1 ~ D ; π^2 ~ D^2; (b) wwater/ wactual: 10^0; (c) Pwater/ Pactual 10^-3 SI constant Patm = 10^5 Pa; pwater - 1000 kg/m^3; pair ~ 1.2kg/m^3; µwater ~ 10^-3 N•s/m^2; pair - 2 x 10^-5 N•s/m^2 ; g = 9.8 m/s^2 =Q1) Under laminar conditions, the volume flow rate Q through a small triangular-section pore of side length (b) and length (L) is a function of viscosity (u), pressure drop per unit length (AP/L), and (b). Using dimensional analysis to rewrite this relation. How does the volume flow changes if the pore size (b) is doubled?1. The thrust of a marine propeller Fr depends on water density p, propeller diameter D, speed of advance through the water V, acceleration due to gravity g, the angular speed of the propeller w, the water pressure 2, and the water viscosity . You want to find a set of dimensionless variables on which the thrust coefficient depends. In other words CT = Fr pV2D² = fen (T₁, T₂, ...Tk) What is k? Explain. Find the 's on the right-hand-side of equation 1 if one of them HAS to be a Froude number gD/V.