A vacuum cleaner creates a pressure drop Ap across its fan. Relate this pressure drop to the impeller diameter D and width h, its rotational speed o, the air density p, and the inlet and outlet diameters di and do. Write the final form in terms of dimensionless parameters using Buckingham n theorem.
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- Evaluate the dimensionless groups hcD/k,UD/, and cp/k for water, n-butyl alcohol, mercury, hydrogen, air, and saturated steam at a temperature of 100C. Let D=1m,U=1m/sec, and hc=1W/m2K.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.— Required information The speed of propagation C of a capillary (very small) wave in deep water is known to be a function only of density p, wavelength A, and surface tension Y. Find the proper functional relationship, completing it with a dimensionless constant.
- The time t d to drain a liquid from a hole in the bottom of atank is a function of the hole diameter d , the initial fluidvolume y 0 , the initial liquid depth h 0 , and the density ρ andviscosity μ of the fluid. Rewrite this relation as a dimensionlessfunction, using Ipsen’s method.The 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 thrust F of a propeller is generally thought to be afunction of its diameter D and angular velocity V , the forwardspeed V , and the density ρ and viscosity μ of the fl uid.Rewrite this relationship as a dimensionless function.
- 3- Consider laminar flow over a flat plate. The boundary layer thickness & grows with distance x down the plate and is also a function of free-stream velocity U, fluid viscosity u, and fluid density p. Find the dimensionless parameters for this problem, being sure to rearrange if necessary to agree with the standard dimensionless groups in fluid mechanics.Q1: Consider laminar flow over a flat plate. The boundary layer thickness o grows with distance x down the plate and is also a function of free-stream velocity U, fluid viscosity u, and fluid density p. Find the dimensionless parameters for this problem, being sure to rearrange if neessary to agree with the standard dimensionless groups in fluid mechanics. Answer: Q2: The power input P to a centrifugal pump is assumed to be a function of the volume flow Q, impeller diameter D, rotational rate 2, and the density p and viscosity u of the fluid. Rewrite these variables as a dimensionless relationship. Hint: Take 2, p, and D as repeating variables. P e paD? = f( Answer:A liquid of density ? and viscosity ? is pumped at volume flow rate V· through a pump of diameter D. The blades of the pump rotate at angular velocity ? . The pump supplies a pressure rise ΔP to the liquid. Using dimensional analysis, generate a dimensionless relationship for ΔP as a function of the other parameters in the problem. Identify any established nondimensional parameters that appear in your result. Hint: For consistency (and whenever possible), it is wise to choose a length, a density, and a velocity (or angular velocity) as repeating variables.
- In an experimental investigation, it is found that the discharge of oil through a pipeline relates to the pressure drop per unit length of the pipeline P, the radius of the pipe r, the density of oil p, the tapering angle 0 and the viscosity of oil µ. Derive the non-dimensional parameters related to this problem (you may use Buckingham's PI theorem)During World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional analysis to estimate theenergy released by an atomic bomb explosion. He assumed that the energy released E, was a function of blastwave radius R, air density ρ, and time t. Arrange these variables into single dimensionless group, which we mayterm the blast wave number.Consider a boundary layer growing along a thin flat plate. The boundary layer thickness & at a downstream distance x is a function of x, the fluid density p, dynamic viscosity, and free stream velocity V. Use Buckingham's theorem with p, x and V as repeating variables, to obtain the relationship between dimensionless parameters Is. Figure 3.2