The variables in a Saybolt viscometer are the time (t) required to empty a certain volume of oil of density (ρ) and viscosity (μ), a length (L) representing the dimensions of the viscometer, and the gravitational acceleration (g). Using the Buckingham Pi-theorem, show that
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The variables in a Saybolt viscometer are the time (t) required to empty a certain volume of oil of density (ρ) and viscosity (μ), a length (L) representing the dimensions of the viscometer, and the gravitational acceleration (g). Using the Buckingham Pi-theorem, show that
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- For a venturi meter given below, the volumetric flow rate is defined in terms of the geometrical parameters, the density of working fluid (p), and density of the manometer liquid (pm) as 4. Q = f(D, D2, A2, g, h, Pmv Pr) %3D Write down the balance equations and show your work to end up with an expression for the volumetric flow rate in terms of the variables defined above.Q2/ By using the power series method make a dimensional analysis for the following variables: The frictional torque of a disc T = f (disk diameter D, rotating speed N, viscosity of fluid µ, and its density p).In the following section, at least 2 to up to 5 answers may be correct. 1) For a fluid, the assumption (simplifying notion) of incompressibility has important consequences: Pascal’s principle: a change of pressure in an enclosed fluid at rest is transmitted undiminished to all points in the fluid. pressure changes are transmitted immediately from one place to another. the speed of sound then is infinite (just within this approximation). pressure becomes unpredictable. none of the above. 2) Archimedes’ principle can be summarized as: an immersed object is buoyed up by a force equal to the weight of the fluid it displaces. a bathtub is fun, and may lead to important physical discoveries regarding the volume of an object and how much water it displaces, and the weight of that amount of water. boats swim because of the work done by sailors. submarines are always doomed. fish swim because they are less heavy than water 3) A…
- 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.2- The resistance (R) experienced by a partially submerged body depends upon the velocity (u), length of the body (L), dynamic viscosity (u) and density (p) of the fluid, and gravitational acceleration (g). Obtain a dimensionless expression for (R). Ans. R=(u²L p) f LgWhen a liquid in a beaker is stired, whirlpool will form and there will be an elevation difference h, between the center of the liquid surface and the rim of the liquid surface. Apply the method of repeating variables to generate a dimensional relationship for elevation difference (h), angular velocity (@) of the whirlpool, fluid density (p). gravitational acceleration (2), and radius (R) of the container. Take o. pand R as the repeating variables.
- 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 =1The pressure difference ∆p produced by a water pump, and the power P required to operate it, each depend on the size of the pump, measured by the diameter D of the impeller, the volume flow rate ˙q, the rate of rotation ω, the water density ρ and dynamic viscosity µ. (a) Express the non-dimensional pressure difference and power as separate functions of the other non-dimensional groups. (b) Tests on a model pump are performed at 0.5 × full scale, at a rotation rate that is 2 × the full-scale value. To achieve dynamic similarity in the model test: (i) what would the volume flow rate of the water need to be in the model test compared to the full-scale? (ii) What would the pressure difference be compared to the full scale? (iii) What would the power consumption be relative to the full scale?1. In Buckingham π theorem, if n is the number of variables and m is the number of basic dimensions, the number of independent dimensionless parameters would be,A. (m – n) B. (n – m) C. (m + n) D. (n/m)
- Q2/ By using the power series method make a dimensional analysis for the following variables; The frictional torque of a disc T- f(disk diameter D. rotating speed N, viscosity of fluid u, and its density p).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)Q1: The drag force (F) of an aircraft during flight depends upon the length of aircraft L, flow velocity V, air viscosity u, air density p, and bulk modulus of air K. (1) Derive the relationship between the above variables and the drag force by using Buckingham's II- theorem. (2) Explain the physical meaning of the obtained dimensionless groups.