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Chapter 8: Problem 31

Someone claims that the volume flow rate in a circular pipe with laminar flow can be determined by measuring the velocity at the centerline in the fully developed region, multiplying it by the cross-sectional area, and dividing the result by 2 . Do you agree? Explain.

Short Answer

Expert verified
Answer: Yes, the volume flow rate in a circular pipe with laminar flow can be determined using the mentioned method, as the derived proposed formula (Q') is found to be equal to the Hagen-Poiseuille equation (Q).

Step by step solution

01

Hagen-Poiseuille Equation

The Hagen-Poiseuille equation is given by: \[Q = \frac{\pi R^4 \Delta P}{8 \eta L}\] Where: - Q is the volume flow rate - R is the radius of the pipe - ΔP is the pressure difference along the pipe - η is the dynamic viscosity of the fluid - L is the length of the pipe over which pressure difference is applied Now, let us analyze the proposition given in the problem. It states that the volume flow rate can be determined by measuring the centerline velocity (v), multiplying it by the cross-sectional area (A), and dividing the result by 2. Mathematically, this can be expressed as:
02

Proposed Formula

\[Q' = \frac{v \times A}{2}\] Let's recall that for a circular pipe, cross-sectional area A is given by:
03

Cross-sectional Area

\[A = \pi R^2\] We want to compare both formulas, Q and Q', to determine if they are equivalent or can be used interchangeably. To do this, we need to find the relationship between the centerline velocity (v) and the Hagen-Poiseuille equation. From the Hagen-Poiseuille equation, we can calculate the velocity distribution along the radius (u):
04

Velocity Distribution

\[u(r) = \frac{R^2}{4 \eta} \frac{\Delta P}{L}(1 - \frac{r^2}{R^2})\] Since the centerline velocity (v) corresponds to r = 0, we can find v by plugging r = 0 into the velocity distribution equation:
05

Centerline Velocity

\[v = \frac{R^2}{4 \eta} \frac{\Delta P}{L}(1 - 0)\] Now, let's apply the proposed formula and plug in the expression for v and A:
06

Apply the Proposed Formula

\[Q' = \frac{\frac{R^2 \Delta P}{4 \eta L} \times \pi R^2}{2}\] \[Q' = \frac{\pi R^4 \Delta P}{8 \eta L}\] Comparing the proposed formula (Q') and the Hagen-Poiseuille equation (Q), we find that:
07

Comparison

\[Q = Q'\] So, based on our analysis, we agree that the volume flow rate in a circular pipe with laminar flow can be determined by measuring the velocity at the centerline in the fully developed region, multiplying it by the cross-sectional area, and dividing the result by 2.

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Most popular questions from this chapter

Water enters a \(5-\mathrm{mm}\)-diameter and \(13-\mathrm{m}\)-long tube at \(15^{\circ} \mathrm{C}\) with a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\), and leaves at \(45^{\circ} \mathrm{C}\). The tube is subjected to a uniform heat flux of \(2000 \mathrm{~W} / \mathrm{m}^{2}\) on its surface. The temperature of the tube surface at the exit is (a) \(48.7^{\circ} \mathrm{C}\) (b) \(49.4^{\circ} \mathrm{C}\) (c) \(51.1^{\circ} \mathrm{C}\) (d) \(53.7^{\circ} \mathrm{C}\) (e) \(55.2^{\circ} \mathrm{C}\) (For water, nse \(k=0.615 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=5.42, v=0.801 \times\) \(\left.10^{-6} \mathrm{~m}^{2} / \mathrm{s} .\right)\)

In the fully developed region of flow in a circular tube, will the velocity profile change in the flow direction? How about the temperature profile?

Water at \(15^{\circ} \mathrm{C}\) is flowing through a 5 -cm-diameter smooth tube with a length of \(200 \mathrm{~m}\). Determine the Darcy friction factor and pressure loss associated with the tube for (a) mass flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\) and \((b)\) mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\).

To cool a storehouse in the summer without using a conventional air- conditioning system, the owner decided to hire an engineer to design an alternative system that would make use of the water in the nearby lake. The engineer decided to flow air through a thin smooth 10 -cm-diameter copper tube that is submerged in the nearby lake. The water in the lake is typically maintained at a constant temperature of \(15^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If air (1 atm) enters the copper tube at a mean temperature of \(30^{\circ} \mathrm{C}\) with an average velocity of \(2.5 \mathrm{~m} / \mathrm{s}\), determine the necessary copper tube length so that the outlet mean temperature of the air is \(20^{\circ} \mathrm{C}\).

Consider a fluid with a Prandtl number of 7 flowing through a smooth circular tube. Using the Colburn, Petukhov, and Gnielinski equations, determine the Nusselt numbers for Reynolds numbers at \(3500,10^{4}\), and \(5 \times 10^{5}\). Compare and discuss the results.
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