A series of experiments was conducted for a ship model of 3-ft length in a water tank. The relevant variables are the length of the model, L, water density, p, tawing velocity, V, the viscosity, μ, and the gravitational acceleration, g. The experimental result was V (ft/s) 10 D (lbf) 0.022 20 30 40 50 60 0.079 0.169 0.281 0.45 0.618 70 0.731
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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 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 =1A fighter aircraft is to operate at 200 m/s in air at standard conditions (p= 1.2 kg/m³). A model is constructed to 1/20 scale (length ratio) and tested in a wind tunnel at the same air temperature (same air viscosity) to determine the drag force. The drag force Fp is a function of the air density p and viscosity µ, the velocity V and the cross section length of the airplane D. a) Use dimensional analysis to prove that: F, and PV²D² pVD What are these two dimensionless numbers? b) If the model is tested at 50 m/s, what air density should be used in the wind tunnel? Comment the result and give a recommendation. c) If the model drag force is 250 N at 50 m/s, what will be the drag force of the prototype? Lift force 200 m/s Drag force
- 3- Use dimensional analysis to show that in a problem involving shallow water waves, both the Froude number and the Reynolds number are relevant dimensionless parameters. The wave speed c of waves on the surface of a liquid is a function of depth h, gravitational acceleration g. fluid density p, and fluid viscosity μ. Manipulate your's to get the parameters into the following form: Fr= √=f(Re) where Re=pch μ h Too 8 P₂ μIn this lab we will drop a steel ball (Psteel = 8000 kg/m³, rball = 2.18 x10³ m) through the air (Nair = 1.8 x105 Pa.s, Pair = 1.29 kg/m³), and through a viscous soap solution (Nsoap = 9.0 Pa.s [you will measure this number in the lab, but use this number for the prelab], Psoap = 1000 kg/m³). %3D 1 Pa = 1 N/m?. Calculate the terminal speed of the ball falling through the air, assuming that inertial drag is the dominant contributor. (For a sphere CD = and A is the area of the circle of radius r of the ball). Neglecting drag, estimate the time in seconds it takes for the steel ball to reach the terminal speed found in part 2 (i). Neglecting drag, estimate how far the ball would have travelled before reaching its terminal speed and comment on whether it is reasonable to neglect drag when studying the motion of this ball as it falls through the air from rest from the top of a 30 cm tall beaker or even from the ceiling of the room. Neglecting drag it would take 36 m to reach the terminal…b) The variation in the experimental density of water, p, with temperature T, in the range of 20°CThe following table lists temperatures and specific volumes of water vapor at two pressures: p = 1.5 MPa v(m³/kg) p = 1.0 MPa T ("C) v(m³/kg) T ("C) 200 0.2060 200 0.1325 240 280 0.2275 0.2480 240 280 0.1483 0.1627 Data encountered in solving problems often do not fall exactly on the grid of values provided by property tables, and linear interpolation between adjacent table entries becomes necessary. Using the data provided here, estimate i. the specific volume at T= 240 °Č, p = 1.25 MPa, in m/kg ii. the temperature at p = 1.5 MPa, v = 0.1555 m/kg, in °C ii. the specific volume at T = 220 °C, p = 1.4 MPa, in m'/kg1. 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.> | E9 docs.google.com/form تبديل الحساب Questions 7 نقاط Q1/ The power of 6-blade flat blade turbine agitator in a tank is a function of diameter of impeller, number of rotations of the impeller per unit time, viscosity and density of liquid. From a dimensional analysis, obtain a relation between the power and the four variables. 3. صفحة 2 منThe conduction heat transfer in an extended surface, known as a fin, yields the following equation for the temperature T, if the temperature distribution is assumed to be one-dimensional in x, where x is the distance from the base of the fin, as shown in figure: To Fin >X T(x) h,T Heat Loss d²T_hp (T-T) = 0 dx² ΚΑ dT dx = 0 Here, p is the perimeter of the fin, being 2R for a cylindrical fin of radius R; A is the cross-sectional area, being R2 for a cylindrical fin; k is the At x = 0: T = T₁ dT At x=L: :0 dx thermal conductivity of the material; T is the ambient fluid temperature; and h in the convective heat transfer coefficient. The boundary conditions are as follows: where L is the length of the fin. Solve this equation to obtain 7(x) by using Euler's method for R=1cm, h= 20 W/m².K, k = 15 W/m-K, L = 25 cm, T₁ = 80°C, and T = 20°C.PROPAGATION OF ERROR You are tasked to supervise the design of 0.5MW wind turbine to be constructed in a Wind Farm in Pililla, Rizal. Suppose the design criteria are as follows: The air density as surveyed after 2 year period averaged at 1.225 kg / cubic meter. The sweeping diameter of the rotor blades is 125m, and the anemometer reading in the area amounts to 4.2m/s on a 2 year period of survey. What is the theoretical power output of the turbine assuming 100% efficiency of operation? Your answer must be in kW and in two decimal places with correct signs. In the previous problem, if the diameter measurement is off by 0.1m and the anemometer reading is also fluctuating by 0.05m/s and if the error measurement for theoretical power output must be limited to 3%, WHY OR WHY NOT should you accept the design? * Show complete solution in paper.At a sudden contraction in a pipe the diameter changes from D, to D2. The pressure drop, Ap, which develops across the con- traction is a function of D, and Dz, as well as the velocity, V, in the larger pipe, and the fluid density, p, and viscosity, u. Use D, V, and u as repeating variables to determine a suitable set of dimensionless parameters. Why would ti be incorrect to include the velocity in the smaller pipe a sa nadditional variable?SEE MORE QUESTIONS