Problem 1: The discharge pressure (P) of a centrifugal pump shown below is a function of flow rate (Q), impeller diameter (D), fluid density (p), and impeller angular speed (2). P = f (Q. D, p. 2). Use the Buckingham pi technique to rewrite this function in terms of dimensionless parameters, ₁ g (1₂).
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- : 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, QProblem 5 s): The discharge pressure (P) of a screw pump (Fig. 5) is a function of flow rate (Q), screw diameter (D), fluid viscosity (u) and screw angular speed (w). P = f (Q, D, μ, w). Use the pi theorem to rewrite this function in terms of dimensionless parameters, ₁ g (T₂). Choose Q, D, and u as repeating variables. Screw Fig. 5: Screw pumpThe velocity V of propagation of ripples on the surface of a shallow liquid depends on the gravitational acceleration g and the liquid depth h. If Buckingham's Theorem is used to identify the salient dimensionless group(s), how many dimensionless group(s) will be obtained? Number of dimensionless group(s) = 1. {1} (Enter your answer as a number.)
- 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?A football, meant to be thrown at 60 mi/h in sea-level air( ρ = 1.22 kg/m 3 , μ = 1.78 E-5 N . m 2 ), is to be testedusing a one-quarter scale model in a water tunnel ( ρ =998 kg/m 3 , μ = 0.0010 N . s/m 2 ). For dynamic similarity,what is the ratio of prototype force to model force?( a ) 3.86 : 1, ( b ) 16 : 1, ( c ) 32 : 1, ( d ) 56 : 1, ( e ) 64 : 1Q4: Use dimensional analysis to show that in a problem involving shallow water waves (Figure 1), both the Froude number (Fr T) and the Reynolds number %3| Vgh (Re = pch are relevant dimensionless parameters Fr = f (Re). 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 u. P.u Figure 1
- The differential equation for small-amplitude vibrations y(r, f) of a simple beam is given by a*y + E = 0 ax pA where p = beam material density A = cross-sectional area I= area moment of inertia E = Young's modulus Use only the quantities p, E, and A to nondimensionalize y, x, and t, and rewrite the differential equation in dimensionless form. Do any parameters remain? Could they be removed by further manipulation of the variables?When 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.Q4: Use dimensional analysis to show that in a problem involving shallow water waves (Figure 1), both the Froude number (Fr = and the Reynolds number (Re pch. are relevant dimensionless parameters Fr = f (Re). The wave speed c of %3D waves on the surface of a liquid is a function of depth h, gravitational acceleration g, fluid density p, and fluid viscosity u. P. u Figure 1
- 5. The drag force, fd, on a sphere located in a pipe through which a fluid is flowing is to be determined experimentally. Assume that the drag is a function of the sphere diameter, d, the pipe diameter, D, the fluid velocity, V, and the fluid density, p. (a) What dimensionless parameters would you use for this prob- lem? (b) Some experiments using water indicate that for d = 0.2in., D = 0.5in, and V = 2ft/s, the drag is 1.5 x 10-3lb. Estimate the drag on a sphere located in a 2 - ft-diameter pipe through which water is flowing with a velocity of 6ft/s. The sphere diameter is such that geometric similarity is maintained. (Ans: 31.11b) SphereProblem 4: The power P developed by a wind turbine is a function of diameter D, air density p, wind speed V, and rotational rate @. Viscous effect is negligible. (4a) Rewrite the above relationship in a dimensionless form; (4b) In a wind tunnel, a small model with a diameter of 90cm, rotating at 1200 RPM (revolution per minute), delivered 200 watts when the wind speed is 12m/s. The data are to be used for a prototype of diameter of 50m and wind speed of 8 m/s. For dynamic similarity, what will be (i) the rotational speed of the prototype turbine? (ii) the power delivered by the prototype turbine? Assume air has sea-level density.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.