Momentum-Flux Correction Factor for Laminar Pipe Flow 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

Question

Momentum-Flux Correction Factor for Laminar Pipe Flow

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

[latex]V=2 V_{ ext {avg }}\left(1-\frac{r^{2}}{R^{2}} ight)[/latex] (1)

where R is the radius of the inner wall of the pipe and [latex]V_{ ext {avg }}[/latex] is the average velocity. Calculate the momentum-flux correction factor through a cross section of the pipe for the case in which the pipe flow represents an outlet of the control volume, as sketched in Fig. 6–15.

Accepted Answer

SOLUTION For a given velocity distribution we are to calculate the momentum-flux correction factor.

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 integrate, noting that [latex]d A_{c}=2 \pi r d r[/latex],

[latex]\beta=\frac{1}{A_{c}} \int_{A_{c}}\left(\frac{V}{V_{ avg }} ight)^{2} d A_{c}=\frac{4}{\pi R^{2}} \int_{0}^{R}\left(1-\frac{r^{2}}{R^{2}} ight)^{2} 2 \pi r d r[/latex] (2)

Defining a new integration variable [latex]y=1-r^{2} / R^{2} ext { and thus } d y=-2 r d r / R^{2}[/latex] (also, y = 1 at r = 0, and y = 0 at r = R) and performing the integration, the momentumflux correction factor for fully developed laminar flow becomes

Laminar flow: [latex]\beta=-4 \int_{1}^{0} y^{2} d y=-4\left[\frac{y^{3}}{3} ight]_{1}^{0}=\frac{4}{3}[/latex] (3)

Discussion We have calculated 𝛽 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.

Question 6.1: Momentum-Flux Correction Factor for Laminar Pipe Flow Consid...

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