To determine Verify the kinetic energy correction factor α for fully developed, laminar flow in a circular tube is 2.0 Explanation Write velocity profile for the non-uniform parabolic velocity profile for the fully developed laminar flow in a circular tube. u = u c ( 1 − r 2 R 2 ) Here, maximum velocity is u c , radius of the pipe is R , and radial distance of the point at which the velocity should be measured. is r . Write the expression for the kinetic energy in terms of average velocity. K E a v e r a g e = 1 2 ρ u ¯ 3 A (I) Here, the density of water is ρ , average velocity is u ¯ , and cross-sectional area of the pipe.is A . Write the expression for cross sectional of the pipe. A = π R 2 Here, radius of the pipe is R . Substitute π R 2 for A in equation (I) K E a v e r a g e = 1 2 ρ u ¯ 3 ( π R 2 ) (II) Write the Equation to calculate the kinetic energy based on the uniform parabolic velocity profile. K E n o n − u n i f o r m = ∫ 0 π 1 2 ρ u 3 ( d A ) (III) Here, the axial velocity is u , and area of the elemental strip.is d A . Substitute 2 π r d r for d A in Equation (III) K E n o n − u n i f o r m = ∫ 0 R ρ u 3 ( 2 π r d r ) (IV) Here, radius of the elemental strip r . Consider ( 1 − r 2 R 2 ) for t , t = ( 1 − r 2 R 2 ) (V) Differentiate the Equation (V) with respect to r d t = − 2 r R 2 d r (VI) Rearrange the Equation (VI) d r = R 2 − 2 r d t Calculate the limits of the integration At r = 0 ; t = 1 − 0 R 3 = 1 At r = R ; t = 1 − R 2 R 2 = 0 Substitute u c ( 1 − r 2 R 2 ) for u in Equation (IV) K E n o n − u n i f o r m = ∫ 0 R ρ ( u c ( 1 − r 2 R 2 ) ) 3 ( 2 π r d r ) (VII) Substitute new limits and ( 1 − r 2 R 2 ) for t in Equation (VII) K E n o n − u n i f o r m = π ρ u c 3 ∫ 1 0 ( t ) 3 ( r ( R 2 d t − 2 r ) ) = − π ρ u c 3 R 2 2 ∫ 1 0 ( t ) 3 d t = − π ρ u c 3 R 2 2 ( t 4 4 ) 1 0 (VIII) Apply the limits of the integration K E n o n − u n i f o r m = − π ρ u c 3 R 2 2 ( 0 4 4 − 1 4 4 ) = − π ρ u c 3 R 2 2 ( − 1 4 ) = π ρ u c 3 R 2 8 (IX) Write the Equation for the mass using the average velocity

8th Edition

ISBN: 9781119571490

Author: GERHART

Publisher: WILEY

See similar textbooks

Videos

Question

Chapter 6.9, Problem 88P

To determine

Verify the kinetic energy correction factor α for fully developed, laminar flow in a circular tube is 2.0

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.9, Problem 87P

Chapter 6.9, Problem 89P

Students have asked these similar questions

Please answer with the sketches.

The beam is made of elastic perfectly plastic material. Determine the shape factor for the cross section of the beam (Figure Q3). [Take σy = 250 MPa, yNA = 110.94 mm, I = 78.08 x 106 mm²] y 25 mm 75 mm I 25 mm 200 mm 25 mm 125 Figure Q3

A beam of the cross section shown in Figure Q3 is made of a steel that is assumed to be elastic- perfectectly plastic material with E = 200 GPa and σy = 240 MPa. Determine: i. The shape factor of the cross section ii. The bending moment at which the plastic zones at the top and bottom of the bar are 30 mm thick. 15 mm 30 mm 15 mm 30 mm 30 mm 30 mm

Chapter 6 Solutions

FUNDAMENTALS OF FLUID MECHANICS

Ch. 6.1 - Prob. 1P Ch. 6.1 - The velocity in a certain flow field is given by...Ch. 6.1 - The flow in the plane two-dimensional channel...Ch. 6.1 - The three components of velocity in a flow field...Ch. 6.1 - Determine an expression for the vorticity of the...Ch. 6.1 - According to Eq. 6.134, the x-velocity in fully...Ch. 6.1 - For a certain incompressible, two-dimensional flow...Ch. 6.1 - An incompressible viscous fluid is placed between...Ch. 6.1 - A viscous fluid is contained in the space between...Ch. 6.1 - ..Air is delivered through a constant-diameter...

Ch. 6.2 - For a certain incompressible flow field it is...Ch. 6.2 - Prob. 12P Ch. 6.2 - Prob. 14P Ch. 6.2 - For each of the following stream functions, with...Ch. 6.2 - The stream function for an incompressible,...Ch. 6.2 - Prob. 17P Ch. 6.2 - Prob. 18P Ch. 6.2 - In a two-dimensional, incompressible flow field,...Ch. 6.2 - The stream function for an incompressible flow...Ch. 6.2 - The stream function for an incompressible,...Ch. 6.2 - Consider the incompressible, two-dimensional flow...Ch. 6.3 - A fluid with a density of 2000 kg/m3 flows...Ch. 6.3 - Prob. 24P Ch. 6.3 - Prob. 25P Ch. 6.4 - The stream function for a given two-dimensional...Ch. 6.4 - Prob. 27P Ch. 6.4 - Prob. 28P Ch. 6.4 - Prob. 29P Ch. 6.4 - The velocity potential for a certain inviscid flow...Ch. 6.4 - Prob. 31P Ch. 6.4 - Prob. 32P Ch. 6.4 - Prob. 33P Ch. 6.4 - Prob. 34P Ch. 6.4 - Prob. 35P Ch. 6.4 - Prob. 36P Ch. 6.4 - Prob. 37P Ch. 6.5 - Prob. 38P Ch. 6.5 - Prob. 39P Ch. 6.5 - Water flows through a two-dimensional diffuser...Ch. 6.5 - Prob. 41P Ch. 6.5 - Prob. 42P Ch. 6.5 - Prob. 43P Ch. 6.5 - Prob. 44P Ch. 6.5 - Prob. 45P Ch. 6.5 - Prob. 46P Ch. 6.5 - Consider the flow of a liquid of viscosity μ and...Ch. 6.5 - Prob. 48P Ch. 6.5 - Show that the circulation of a free vortex for any...Ch. 6.5 - Prob. 50P Ch. 6.6 - Potential flow against a flat plate (Fig. P6.51a)...Ch. 6.6 - Prob. 52P Ch. 6.6 - Prob. 53P Ch. 6.6 - Prob. 54P Ch. 6.6 - Prob. 55P Ch. 6.6 - Prob. 56P Ch. 6.6 - A 15-mph wind flows over a Quonset hut having a...Ch. 6.6 - Prob. 58P Ch. 6.6 - Prob. 59P Ch. 6.6 - Prob. 60P Ch. 6.6 - Prob. 61P Ch. 6.6 - Prob. 62P Ch. 6.6 - The velocity potential for a cylinder (Fig. P6.63)...Ch. 6.6 - (See The Wide World of Fluids article titled “A...Ch. 6.6 - Prob. 65P Ch. 6.6 - Air at 25 °C flows normal to the axis of an...Ch. 6.8 - Determine the shearing stress for an...Ch. 6.8 - Prob. 68P Ch. 6.8 - The velocity of a fluid particle moving along a...Ch. 6.8 - “Stokes’s first problem” involves the...Ch. 6.9 - Oil (SAE 30) at 15.6 °C flows steadily between...Ch. 6.9 - Prob. 72P Ch. 6.9 - Prob. 73P Ch. 6.9 - We will see in Chapter 8 that the pressure drop in...Ch. 6.9 - (See The Wide World of Fluids article titled “10...Ch. 6.9 - The bearing shown in Fig. P6.76 consists of two...Ch. 6.9 - Prob. 77P Ch. 6.9 - Prob. 78P Ch. 6.9 - An incompressible, viscous fluid is placed between...Ch. 6.9 - Two immiscible, incompressible, viscous fluids...Ch. 6.9 - Prob. 81P Ch. 6.9 - A viscous fluid (specific weight = 80 lb/ft3;...Ch. 6.9 - A flat block is pulled along a horizontal flat...Ch. 6.9 - A viscosity motor/pump is shown in Fig. P6.84. The...Ch. 6.9 - A vertical shaft passes through a bearing and is...Ch. 6.9 - A viscous fluid is contained between two long...Ch. 6.9 - Verify that the momentum correction factor β for...Ch. 6.9 - Verify that the kinetic energy correction factor α...Ch. 6.9 - A simple flow system to be used for steady-flow...Ch. 6.9 - (a) Show that for Poiseuille flow in a tube of...Ch. 6.9 - An infinitely long, solid, vertical cylinder of...Ch. 6.9 - We will see in Chapter 8 that the pressure drop in...Ch. 6.9 - A liquid (viscosity = 0.002 N · s/m2; density =...Ch. 6.9 - Fluid with kinematic viscosity ν flows down an...Ch. 6.9 - Blood flows at volume rate Q in a circular tube of...Ch. 6.9 - An incompressible Newtonian fluid flows steadily...Ch. 6.9 - Prob. 97P Ch. 6.9 - Prob. 98P Ch. 6.9 - Prob. 99P Ch. 6.10 - Prob. 101P Ch. 6.10 - Prob. 102P Ch. 6.11 - Prob. 1LLP Ch. 6.11 - Prob. 2LLP Ch. 6.11 - Prob. 3LLP

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.