The pressure drop AP in flow through pipes per unit length is found to depend on the average velocity u, diameter d, density of the fluid p, and viscosity u. Using MLT set of dimensions evaluate the dimensionless parameters correlating this phenomenon. *
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- When fluid in a pipe is accelerated linearly from rest, it begins as laminar flow andthen undergoes transition to turbulence at a time ttr which depends upon the pipe diameter D,fluid acceleration a, density ρ, and viscosity µ. Arrange this into a dimensionless relationbetween ttr and D. (Fluid Mechanics)When fluid in a pipe is accelerated linearly from rest, it begins as laminar flow andthen undergoes transition to turbulence at a time ttr which depends upon the pipe diameter D,fluid acceleration a, density ρ, and viscosity µ. Arrange this into a dimensionless relationbetween ttr and D.Using II-Theorem method to Express (n) in terms of dimensionless groups.The efficiency (n) of a fan depends upon density (p), and dynamic viscosity (u), of the fluid, angular velocity (@), diameter of the rotator (D), and discharge (Q). Q3/ A petroleum crude oil having a kinematics viscosity 0.0001 m?/s is flowing through the piping arrangement shown in the below Figure,The total mass flow rate is equal 10 kg/s entering in pipe (A) . The flow divides to three pipes ( B, C, D). The steel pipes are schedule 40 pipe. note that the dynamic viscosity 0.088 kg/m.s. Calculate the following using SI units: 1- The type of flow in pipe (A). 2- The mass velocity in pipe (B) GB. 3- The velocity in pipe (D) Up. 4- The Volumetric flow rate in pipe (D) QD. 5- The Volumetric flow rate in pipe (C) Qc. Og = 2o mm Ug = 2UA Perolenm crude oIL A ma = 1o Kg/s O = 5o mm mic = ? Go = 7000 k9/m.s Nate that!- O, = 30 mm. D:0iameter. U:velocity G mass velocity mimass How vate
- MLT By dimensional analysis, obtain an expression for the drag force (F) on a partially submerged body moving with a relative velocity (u) in a fluid; the other variables being the linear dimension (L), surface roughness (e), fluid density (p), and gravitational acceleration (g).Taylor number (Ta) is used here to describe the ratio between the inertia effect and the viscous effect. By applying Buckingham Pi's Theorem, determine an equation for Ta as a function of the radius of inner cylinder (r), cylinder tangential velocity (v), fluid dynamic viscosity (u), gap distance (L) and fluid density (p). Q4During World War II, Sir Geoffrey Taylor, a British fl uiddynamicist, used dimensional analysis to estimate theenergy released by an atomic bomb explosion. He assumedthat the energy released E , was a function of blast waveradius R , air density ρ, and time t . Arrange these variablesinto a single dimensionless group, which we may term theblast wave number .
- There are many common dimensionless numbers used to describedifferent physical effects. Such dimensionless numbers are usually a ratioof specific properties, for example, the Reynolds number represents theratio of inertial to viscous forces.Choose three of the following dimensionless numbers and describe whatproperties the ratio represents and the context in which these numbers areimportant: Froude Number, Euler Number, Mach Number, Weber Number,Nusselt Number, Prandtl Number.In the study of turbulent flow, turbulent viscous dissipation rate ? (rate of energy loss per unit mass) is known to be a function of length scale l and velocity scale u′ of the large-scale turbulent eddies. Using dimensional analysis (Buckingham pi and the method of repeating variables) and showing all of your work, generate an expression for ? as a function of l and u′.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:
- Suppose we know little about the strength of materials butare told that the bending stress σ in a beam is proportionalto the beam half-thickness y and also depends on thebending moment M and the beam area moment of inertiaI . We also learn that, for the particular case M = 2900in ∙ lbf, y = 1.5 in, and I = 0.4 in4 , the predicted stressis 75 MPa. Using this information and dimensional reasoningonly, find, to three significant figures, the onlypossible dimensionally homogeneous formula σ=y f ( M , I ).Qus A fluid flow situatton depends the Velocity (V), the on density several lineor dimension, tsh, L2, pressure drop (PP) » gravity (o),Viscosity Ms Susface tension (), and bulk mo dulus of elasticity k. Apply dimen sional analysis. to these variables ed MKEATaylor number (Ta) is used here to describe the ratio between the inertia effect andthe viscous effect. By applying Buckingham Pi’s Theorem, determine an equation forTa as a function of the radius of inner cylinder (r), cylinder tangential velocity (v),fluid dynamic viscosity (μ), gap distance (L) and fluid density (ρ).