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Chapter 1: Problem 23

Experiment with a faucet (kitchen or otherwise) to determine typical flow rates \(Q\) in \(\mathrm{m}^{3} / \mathrm{h}\), perhaps timing the discharge of a known volume. Try to achieve an exit jet condition which is \((a)\) smooth and round and \((b)\) disorderly and fluctuating. Measure the supply-pipe diameter (look under the sink). For both cases, calculate the average flow velocity, \(V_{\mathrm{avg}}=\) \(Q / A_{\text {cross-section }}\) and the dimensionless Reynolds number of the flow, \(\operatorname{Re}=\rho V_{\text {avg }} D / \mu .\) Comment on your results.

Short Answer

Expert verified
The smooth jet shows laminar flow, while the fluctuating jet shows transitional flow.

Step by step solution

01

Determine the Flow Rate

Use a container of known volume, such as a measuring cup, and time how long it takes to fill it up. For example, if a 2-liter container is filled in 1 minute, the flow rate can be calculated as follows: \( Q = \frac{2}{60} = 0.0333 \) m鲁/h.
02

Measure the Pipe Diameter

Measure the diameter of the supply pipe under the sink using a ruler or a caliper. For instance, if the diameter is 2 cm, convert it to meters: \( D = 0.02 \) m.
03

Calculate the Cross-sectional Area

Calculate the cross-sectional area of the pipe using the formula for the area of a circle: \( A_{\text{cross-section}} = \pi \left(\frac{D}{2}\right)^2 \). For \( D = 0.02 \) m, \( A_{\text{cross-section}} = \pi \left(0.01\right)^2 = 3.14 \times 10^{-4} \) m虏.
04

Step 4a: Calculate Average Flow Velocity for Smooth and Round Jet

Using the flow rate from a smooth and round jet condition, calculate the average flow velocity: \( V_{\mathrm{avg}} = \frac{Q}{A_{\text{cross-section}}} \). Assume \( Q = 0.0333 \) m鲁/h and convert to m鲁/s: \( Q = 0.0333 / 3600 = 9.25 \times 10^{-6} \) m鲁/s.Thus, \( V_{\mathrm{avg}} = \frac{9.25 \times 10^{-6}}{3.14 \times 10^{-4}} = 0.0294 \) m/s.
05

Step 4b: Calculate Average Flow Velocity for Disorderly and Fluctuating Jet

Repeat Step 4a using the flow rate for a disorderly and fluctuating jet. If this condition has \( Q = 0.0666 \) m鲁/h, convert it to m鲁/s: \( Q = 0.0666 / 3600 = 1.85 \times 10^{-5} \) m鲁/s.Then,\( V_{\mathrm{avg}} = \frac{1.85 \times 10^{-5}}{3.14 \times 10^{-4}} = 0.0589 \) m/s.
06

Step 5a: Calculate Reynolds Number for Smooth and Round Jet

Use the formula for Reynolds number: \( \operatorname{Re} = \frac{\rho V_{\text{avg}} D}{\mu} \). Assuming water properties \( \rho = 1000 \) kg/m鲁 and \( \mu = 0.001 \) Pa路s, then \( \operatorname{Re} = \frac{1000 \times 0.0294 \times 0.02}{0.001} = 588 \). This indicates laminar flow.
07

Step 5b: Calculate Reynolds Number for Disorderly and Fluctuating Jet

Repeat Step 5a calculation using the disorderly flow velocity: \( V_{\text{avg}} = 0.0589 \) m/s.Thus, \( \operatorname{Re} = \frac{1000 \times 0.0589 \times 0.02}{0.001} = 1178 \). This value suggests transitional flow.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Flow Rate Measurement
Measuring flow rate is essential in understanding how effectively a fluid moves through a system. To measure flow rate at home, you can perform a simple experiment using a container and a stopwatch. Timing how long it takes to fill a container of a known volume allows you to calculate the flow rate.
For instance, if a 2-liter container fills up in 1 minute, then the flow rate is computed as:
  • Flow Rate Formula: \( Q = \frac{2}{60} = 0.0333 \) m鲁/h
This value helps in determining how much fluid, in cubic meters, passes through the system in an hour. It's crucial to measure the flow rate under different conditions to understand changes in fluid behavior.
Reynolds Number Calculation
The Reynolds number is a widely used dimensionless quantity in fluid mechanics that helps determine the flow regime of a fluid, whether laminar, transitional, or turbulent. Calculating the Reynolds number involves the density of the fluid, average flow velocity, pipe diameter, and fluid viscosity.
Use the formula:
  • Reynolds Number Formula: \( \operatorname{Re} = \frac{\rho V_{\text{avg}} D}{\mu} \)
For example, using water with a density \( \rho \) of 1000 kg/m鲁 and viscosity \( \mu \) of 0.001 Pa路s:
  • For a smooth jet: \( \operatorname{Re} = \frac{1000 \times 0.0294 \times 0.02}{0.001} = 588 \) (laminar flow)
  • For a disorderly jet: \( \operatorname{Re} = \frac{1000 \times 0.0589 \times 0.02}{0.001} = 1178 \) (transitional flow)
Knowing the Reynolds number is vital for understanding how the flow will behave under different conditions.
Pipe Diameter Measurement
Accurate measurement of pipe diameter is crucial for several calculations in fluid mechanics, including flow rate and velocity. A ruler or caliper can be used to measure the diameter by directly accessing the pipe, usually found under a sink.
It is important to convert your measurement into meters if it is initially in centimeter form:
  • For example, a measurement of 2 cm translates to 0.02 meters.
This measured diameter plays a key role in calculating the pipe's cross-sectional area, which is required when determining flow velocity and for other fluid dynamics calculations.
Average Flow Velocity
Average flow velocity is calculated to understand how quickly fluid is moving through a pipe. It is found by dividing the flow rate by the cross-sectional area of the pipe.
Calculating the cross-section area is essential:
  • Cross-sectional Area Formula: \( A_{\text{cross-section}} = \pi \left(\frac{D}{2}\right)^2 \)
Then, calculate average flow velocity:
  • Velocity Formula: \( V_{\text{avg}} = \frac{Q}{A_{\text{cross-section}}} \)
For instance, with \( Q = 9.25 \times 10^{-6} \) m鲁/s and \( A_{\text{cross-section}} = 3.14 \times 10^{-4} \) m虏, the average velocity is \( V_{\text{avg}} = 0.0294 \) m/s for a smooth jet.
This is a fundamental calculation which helps in understanding fluid speed, influencing design and function of different systems.
Laminar and Transitional Flow
Fluid flow can be classified into laminar, transitional, or turbulent flows, depending on the Reynolds number. Understanding these flow types is critical in predicting the behavior of fluid movement in pipes.
In this context:
  • Laminar Flow: Characterized by smooth, orderly layers of fluid. Occurs when \( \operatorname{Re} \) is less than 2000. For example, \( \operatorname{Re} = 588 \) indicates laminar flow.
  • Transitional Flow: A mix of laminar and turbulent. Happens when \( \operatorname{Re} \) is between 2000 and 4000. \( \operatorname{Re} = 1178 \) suggests transitional flow.
Knowledge of whether a flow is laminar or transitional helps in the design and operation of hydraulic systems where different flow characteristics can cause wear or inefficiency.

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Most popular questions from this chapter

The mean free path of a gas, \(\ell,\) is defined as the average distance traveled by molecules between collisions. A proposed formula for estimating \(\ell\) of an ideal gas is $$\ell=1.26 \frac{\mu}{\rho \sqrt{R T}}$$ What are the dimensions of the constant \(1.26 ?\) Use the for mula to estimate the mean free path of air at \(20^{\circ} \mathrm{C}\) and \(7 \mathrm{kPa}\). Would you consider air rarefied at this condition?

A soap bubble of diameter \(D_{1}\) coalesces with another bubble of diameter \(D_{2}\) to form a single bubble \(D_{3}\) with the same amount of air. Assuming an isothermal process, derive an expression for finding \(D_{3}\) as a function of \(D_{1}, D_{2}\) \(p_{\text {atm }},\) and \(Y.\)

Oil, with a vapor pressure of \(20 \mathrm{kPa}\), is delivered through a pipeline by equally spaced pumps, each of which increases the oil pressure by 1.3 MPa. Friction losses in the pipe are 150 Pa per meter of pipe. What is the maximum possible pump spacing to avoid cavitation of the oil?

At \(60^{\circ} \mathrm{C}\) the surface tension of mercury and water is 0.47 and \(0.0662 \mathrm{N} / \mathrm{m},\) respectively. What capillary height changes will occur in these two fluids when they are in contact with air in a clean glass tube of diameter \(0.4 \mathrm{mm} ?\)

The laminar-pipe-flow example of Prob. 1.12 can be used to design a capillary viscometer [27]. If \(Q\) is the volume flow rate, \(L\) is the pipe length, and \(\Delta p\) is the pressure drop from entrance to exit, the theory of Chap. 6 yields a formula for viscosity: $$\mu=\frac{\pi r_{0}^{4} \Delta p}{8 L Q}$$ Pipe end effects are neglected [27]. Suppose our capillary has \(r_{0}=2 \mathrm{mm}\) and \(L=25 \mathrm{cm} .\) The following flow rate and pressure drop data are obtained for a certain fluid: $$\begin{array}{l|l|l|l|l|l} Q, \mathrm{m}^{3} / \mathrm{h} & 0.36 & 0.72 & 1.08 & 1.44 & 1.80 \\ \hline \Delta p, \mathrm{kPa} & 159 & 318 & 477 & 1274 & 1851\end{array}$$ What is the viscosity of the fluid? Note: Only the first three points give the proper viscosity. What is peculiar about the last two points, which were measured accurately?
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