Hello sir Muttalibi is a step solution in detailing mathematics the s ame as an existing step solution EXAMPLE 6-1 Momentum-Flux Correction Factor for Laminar Pipe Flow Vavg Consider laminar flow through a very long straight section of round pipe. It is shown in Chap. 8 that the velocity profile through a cross-sectional area of the pipe is parabolic (Fig. 6-15), with the axial velocity component given by r4 V R V = 2Vavg (1) R2 where R is the radius of the inner wall of the pipe and Vag is the average velocity. Calculate the momentum-flux correction factor through a cross sec- tion of the pipe for the case in which the pipe flow represents an outlet of the control volume, as sketched in Fig. 6-15. Assumptions 1 The flow is incompressible and steady. 2 The control volume slices through the pipe normal to the pipe axis, as sketched in Fig. 6-15. Analysis We substitute the given velocity profile for V in Eq. 6-24 and inte- grate, noting that dA, = 2mr dr, FIGURE 6-15 4 dA, = TR2 Velocity profile over a cross section of a pipe in which the flow is fully developed and laminar. B = 1- 2nr dr (2) Defining a new integration variable y = 1- r2/R? and thus dy = -2r drlR2 (also, y = 1 atr 0, and y = 0 at r = R) and performing the integra- For turbulent flow ßmay have tion, the momentum-flux correction factor for fully developed laminar flow an insignificant effect at inlets becomes and outlets, but for laminar A = -4 [r dy = -- flow Bmay be important and (3) should not be neglected. It is wise to include Bin all Laminar flow: %3D Discussion We have calculated B for an outlet, but the same result would have been obtained if we had considered the cross section of the pipe as an inlet to the control volume. momentum control volume problems.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

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Transcribed Image Text:Hello sir Muttalibi is a step solution in detailing mathematics the same as an existing step solution EXAMPLE 6-1 Momentum-Flux Correction Factor for Laminar Pipe Flow CV Vavg Consider laminar flow through a very long straight section of round pipe. It is shown in Chap. 8 that the velocity profile through a cross-sectional area of the pipe is parabolic (Fig. 6-15), with the axial velocity component given by r4 V R V = 2V 1 avg R2 (1) where R is the radius of the inner wall of the pipe and Vavg is the average velocity. Calculate the momentum-flux correction factor through a cross sec- tion of the pipe for the case in which the pipe flow represents an outlet of the control volume, as sketched in Fig. 6-15. Assumptions 1 The flow is incompressible and steady. 2 The control volume slices through the pipe normal to the pipe axis, as sketched in Fig. 6-15. Analysis We substitute the given velocity profile for V in Eq. 6-24 and inte- grate, noting that dA, = 2ar dr, FIGURE 6–15 %3D Velocity profile over a cross section of a pipe in which the flow is fully developed and laminar. V 4 2rr dr R² B = (2) V avg TR² Defining a new integration variable y = 1 - r2/R? and thus dy = -2r dr/R2 (also, y = 1 at r = 0, and y = 0 at r = R) and performing the integra- For turbulent flow B may have tion, the momentum-flux correction factor for fully developed laminar flow an insignificant effect at inlets and outlets, but for laminar flow Bmay be important and (3) should not be neglected. It is wise to include Bin all becomes 10 4. Laminar flow: B = -4 y² dy = -4 Discussion We have calculated B for an outlet, but the same result would have been obtained if we had considered the cross section of the pipe as an inlet to the control volume. momentum control volume problems.

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