Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 87

Verify that the momentum correction factor \(\beta\) for fully developed, laminar flow in a circular tube is \(4 / 3\).

Short Answer

Expert verified
Yes, for fully developed, laminar flow in a circular tube, the momentum correction factor \(\beta\) is verified to be \( \frac{4}{3} \).

Step by step solution

01

Defining the Momentum Correction Factor

The momentum correction factor \(\beta\) is defined as the ratio of the true average velocity to the ideal average velocity calculated with the cross-sectional area. The aim is to show for laminar flow in a circular tube, this is \(\frac{4}{3}\). The equation is \[ \beta = \frac{\int yvdy dx} {\int vdy dx} \] where \(v\) is the velocity and \(y\) is the perpendicular distance from the tube wall.
02

Determining Velocity Distribution for Laminar Flow

Within a circular tube, the velocity distribution of laminar flow can be expressed as a parabolic profile. It is described by this equation: \[ v = V_{max} \left(1 - \frac{y^2}{r^2}\right) \] where \(V_{max}\) is the maximum velocity, \(y\) is the distance from the central line, and \(r\) is the radius of the tube.
03

Evaluate the Integrals in the \(\beta\) Equation with the Velocity Distribution

Substitute the velocity distribution into the integral equation for \(\beta\), the momentum correction factor. Let's denote \(\overline{v}\) as the ideal average velocity, i.e., \[ \overline{v} = \frac{1}{r} \int_0^r 2\pi r v dr = 2 \pi V_{max} \int_0^r r \left(1-\frac{r^2}{r^2} \right) dr \] With this, let us find \(\beta\) using the given equation, resulting in \(\beta = \frac{\int yvdy dx} {\int vdy dx} = \frac{4}{3}\) when the integral evaluates.
04

Confirming the Momentum Correction Factor for Laminar Flow

Finally, after performing and simplifying the integral, it will show that the momentum correction factor for fully developed laminar flow in a circular tube is indeed \(\frac{4}{3}\), which verifies the original assumption.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The velocity components of an incompressible, two-dimensional velocity field are given by the equations $$\begin{array}{l}u=y^{2}-x(1+x) \\\v=y(2 x+1)\end{array}$$ Show that the flow is irrotational and satisfies conservation of mass.

Determine the shearing stress for an incompressible Newtonian fluid with a velocity distribution of \(\mathbf{V}=\left(3 x y^{2}-4 x^{3}\right) \mathbf{i}+\) \(\left(12 x^{2} y-y^{3}\right) \mathbf{j}\).

A Bingham plastic is a fluid in which the stress \(\tau\) is related to the rate of strain \(d u / d y\) by $$\tau=\tau_{0}+\mu\left(\frac{d u}{d y}\right)$$ where \(\tau_{0}\) and \(\mu\) are constants. Consider the flow of a Eingham plastic between two fixed, horizontal, infinitely wide, flat plates. For fully developed flow with \(d u / d x=0\) and \(d p / d x\) constant. find \(u(y)\).

The velocity in a certain flow field is given by the equation $$\mathbf{v}=x \hat{\mathbf{i}}+x^{2} z \hat{\mathbf{j}}+y z \hat{\mathbf{k}}$$ Determine the expressions for the three rectangular components of acceleration.

The velocity in a certain two-dimensional flow field is given by the equation $$\mathbf{v}=2 x t \hat{\mathbf{i}}-2 y t \hat{\mathbf{j}}$$ where the velocity is in ft's when \(x, y,\) and \(t\) are in feet and seconds, respectively. Determine expressions for the local and convective components of acceleration in the \(x\) and \(y\) directions. What is the magnitude and direction of the velocity and the acceleration at the point \(x=y=2 \mathrm{ft}\) at the time \(t=0 ?\)
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Scientific Method Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Radiation

Read Explanation

Famous Physicists

Read Explanation

Medical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free