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Chapter 4: Problem 4

The average velocities of water flowing steadily through the nozzle are indicated. If the nozzle is connected onto the end of the hose, outline the control volume to be the entire nozzle and the water inside it. Also, select another control volume to be just the water inside the nozzle. In each case, indicate the open control surfaces, and show the positive direction of their areas. Specify the direction of the velocities through these surfaces. Identify the local and convective changes that occur. Assume water to be incompressible.

Short Answer

Expert verified
The control volumes are the entire nozzle with the water inside it, and just the water inside the nozzle. In both cases, the positive direction of areas and velocities is from the hose into the nozzle, and then out of the nozzle. The primary change is convective in nature, as water moves from the hose through the nozzle.

Step by step solution

01

Identify the Control Volumes

Start by viewing the entire system. There can be two control volumes as indicated in the problem - one that includes both the nozzle and the water inside it, and another that consists only of the water inside the nozzle.
02

Demarcate Open Control Surfaces

For the first control volume that includes both the nozzle and the water, the open control surfaces would be the opening of the hose where water enters the nozzle, and the nozzle exit where water goes out. However, for the second control volume that is only the water, the open control surfaces are again the area where water enters from the hose and the area where water exits the nozzle.
03

Determine the Direction of Areas

The positive direction of the areas would be the direction in which the water is moving on those surfaces. In this case, since water is moving inside the nozzle from the hose, the positive direction at the hose end is into the nozzle. Similarly, as water exits the nozzle, the positive direction at the nozzle end is out of the nozzle.
04

Identify the Velocities

The velocity direction, in both cases, is from the hose into the nozzle, and then out of the nozzle.
05

Identify Local and Convective Changes

As water flows from the hose to the nozzle and then exits, a convective change occurs. The convective change refers to the change in fluid properties due to fluid motion. Since the fluid is incompressible, there won't be any significant local changes, which means changes at a specific point in space, in fluid properties.

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Key Concepts

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

Incompressible Flow
In the realm of fluid mechanics, incompressible flow refers to a fluid whose density does not change significantly when pressure or flow conditions change. Water, the fluid in our exercise, is commonly assumed to be incompressible because its density remains nearly constant under normal conditions. This assumption simplifies the analysis of fluid flow because it means that volume flow rate, not just velocity, is conserved from one cross-section of a nozzle to another.
Convective Changes in Fluid Flow
Convective changes in fluid flow describe the variations of fluid properties, such as velocity and pressure, that are due to the movement of the fluid itself. In the case of the water flowing through the nozzle, the convective change is apparent in the change of velocity as the water moves from the broader hose into the narrower nozzle. The water accelerates as it enters the constricted space of the nozzle, demonstrating a convective change in the fluid's velocity.
Positive Direction of Control Surface Area
In control volume analysis, the positive direction of a control surface area is typically defined in the direction of flow out of the control volume. For our exercise, when water flows into the nozzle from the hose, the positive direction at the hose end is considered into the nozzle. As the water exits the nozzle, the positive direction at the nozzle end is out of the nozzle. This convention helps in applying the principle of conservation of mass and other fundamental equations in fluid mechanics.
Velocity Distribution in Nozzle Flow
Velocity distribution refers to the variation in fluid velocity across different points in a flow cross-section. In nozzle flow, one typically observes an increase in flow velocity as the nozzle narrows because of the conservation of mass for incompressible fluids. However, this ideal increase in velocity assumes a uniform flow without turbulence or viscous losses. In real-world applications, the velocity distribution might be more complex due to these additional factors.

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Most popular questions from this chapter

The human heart has an average discharge of \(0.1\left(10^{-3}\right) \mathrm{m}^{3} / \mathrm{s},\) determined from the volume of blood pumped per beat and the rate of beating. Careful measurements have shown that blood cells pass through the capillaries at about \(0.5 \mathrm{~mm} / \mathrm{s}\). If the average diameter of a capillary is \(6 \mu \mathrm{m}\), estimate the number of capillaries that must be in the human body.

The radius of the circular duct varies as \(r=\left(0.05 e^{-3 x}\right) \mathrm{m},\) where \(x\) is in meters. The flow of a fluid at \(A\) is \(Q=0.004 \mathrm{~m}^{3} / \mathrm{s}\) at \(t=0,\) and it is increasing at \(d Q / d t=0.002 \mathrm{~m}^{3} / \mathrm{s}^{2} .\) If a fluid particle is originally located at \(x=0\) when \(t=0\), determine the time for this particle to arrive at \(x=100 \mathrm{~mm}\).

The cylindrical pressure vessel contains methane at an initial absolute pressure of \(2 \mathrm{MPa}\). If the nozzle is opened, the mass flow depends upon the absolute pressure and is \(\dot{m}=3.5\left(10^{-6}\right) p \mathrm{~kg} / \mathrm{s},\) where \(p\) is in pascals. Assuming the temperature remains constant at \(20^{\circ} \mathrm{C}\), determine the time required for the pressure to drop to \(1.5 \mathrm{MPa}\).

The toy balloon is filled with air and is released. At the instant shown, the air is escaping at an average velocity of \(4 \mathrm{~m} / \mathrm{s}\), measured relative to the balloon, while the balloon is accelerating. For an analysis, why is it best to consider the control volume to be moving? Select this control volume so that it contains the air in the balloon. Indicate the open control surface, and show the positive direction of its area. Also, indicate the direction of the velocity through this surface. Identify the local and convective changes that occur. Assume the air to be incompressible.

In a mixing chamber, Carbon dioxide flows into the tank at \(A\) at \(V_{A}=12 \mathrm{~m} / \mathrm{s}\), and nitrogen flows in at \(B\) at \(V_{B}=6 \mathrm{~m} / \mathrm{s}\). Both enter at a gage pressure of \(310 \mathrm{kPa}\) and a temperature of \(260^{\circ} \mathrm{C}\). Determine the average velocity of the mixed gas at \(C\). The mixture has a density of \(\rho=1.546 \mathrm{~kg} / \mathrm{m}^{3}\).
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