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Today, we'll apply Bernoulli's equation to solve a hydraulic problem. Can anyone remind us what Bernoulli’s principle states?
It states that in a flowing fluid, an increase in velocity occurs simultaneously with a decrease in pressure or potential energy.
Correct! Now, let's consider a tapered pipe where water flows. We need to find the deflection of a manometer with mercury. When we disregard friction, which equation do we apply?
Bernoulli's equation!
Exactly! This helps us to simplify calculations. Remember, we relate the pressure differences using the density of mercury, which is based on the relative density. Can anyone tell me how pressure is tied to flow speed in this theory?
Lower pressure corresponds with higher speeds according to Bernoulli's principle.
Yes! By applying this, we can derive the needed values for h. We'll do the math together to see how we find 17.5 cm for h. Now, what is the implication of neglecting friction here?
It allows us to directly apply Bernoulli's equation without additional losses.
Great summary! We'll delve more into fluid dynamics next.
Now on to our new topic: the Reynolds transport theorem. Why do we find this theorem crucial in fluid mechanics?
It helps in relating the changes in extensive properties of fluids to control volumes.
Correct! We denote extensive properties by capital B. Can anyone explain the difference between extensive and intensive properties?
Extensive properties depend on the size or mass of the system, while intensive properties are independent of the system's size.
Good! In our example, how do we represent the system's mass in relation to its extensive properties?
It is the integral of the extensive property 'b' over the mass density of the fluid.
Exactly! The Reynolds transport theorem ties these ideas together. Can someone summarize the essence of this theorem?
It describes how extensive properties change over time in a control volume, relating flow rates across control surfaces.
Well articulated! In our next class, we will delve into applications where we can utilize this theorem.
Let’s derive the Reynolds transport theorem together! We start by considering a fixed control volume. Who can describe the setup we'll use?
We’ll analyze the fluid that flows within the volume over time, focusing on both inflow and outflow.
Perfect! We’ll look at flow across various surfaces as it changes. Can anyone recall what formulas we use for inflow and outflow?
For inflow, we use the negative component of the flow rate, and for outflow, it’s the positive component.
Exactly right! By combining these components, we'll obtain the net change in extensive properties within the control volume. What do we expect the end result to reveal?
It should express how properties like mass or momentum evolve through a control volume.
Indeed! This makes it a powerful tool in fluid dynamics. Let’s walk through the calculations next.
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The section primarily covers the application of Bernoulli's equation in solving specific hydraulic problems, notably involving a tapered pipe and flow rates. It further introduces the Reynolds transport theorem, emphasizing its significance in relating extensive properties to control volumes and fluid dynamics. The connection between fluid statics, kinematics, and dynamics is also highlighted.
In this section, we revisit concepts from fluid mechanics, focusing on solving a hydraulic problem that involves the flow of water through a tapered pipe. We apply Bernoulli's equation to determine properties like the deflection of a mercury manometer as water flows at a certain discharge rate.
The problem is set up by neglecting friction to enable the application of Bernoulli's equation effectively. The solution follows from applying both the continuity equation and Bernoulli’s equation, leading to a relationship that allows us to solve for the deflection height (h) of mercury in the manometer. This deflection is calculated with given parameters such as diameter, flow rate, and elevation.
Following this practical example, we transition into the more theoretical aspects of fluid dynamics, specifically the Reynolds transport theorem, which is integral to fluid motion laws. We establish relationships between extensive properties of fluids, symbolizing total parameters like momentum and mass. The section discusses the derivation of Reynolds transport theorem, explaining how it relates to control volumes, and emphasizes that it can apply across various fluid properties. Finally, the section prepares for further exploration into how these concepts can be utilized in fluid dynamics applications.
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Welcome back, we are going to start this lecture by solving the practice problem, which where we ended our last lecture.
In this section, the professor is introducing the lecture by revisiting a practice problem discussed in the previous class. This sets the stage for applying theoretical principles to a real-world scenario, emphasizing continuity in learning.
Think of this like a sports coach reviewing last week's game footage before the next match to ensure the team continues to improve through practice.
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So, the problem goes like this, the water flows up a tapered pipe as shown in the figure below. Find the magnitude and direction of the deflection h of the differential mercury manometer corresponding to a discharge of 120 liters per second.
In this problem, the goal is to calculate the deflection (
h) in a mercury manometer due to water flowing through a tapered pipe with a specific discharge rate of 120 liters per second. This requires applying fluid mechanics principles, specifically Bernoulli's equation, to relate flow conditions to pressure differences.
Imagine trying to predict how much water in a garden hose will lift a water gauge based on how quickly the water is flowing. Here, the faster the flow (discharge), the greater the pressure difference and subsequent gauge reading.
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The friction in the pipe can be completely neglected. The reason of neglecting the friction completely is, so that, we are able to apply, you know what, Bernoulli’s equation.
Friction losses in the pipe can complicate the calculations involved in fluid dynamics. By neglecting friction, the professor simplifies the problem to focus on the principles of Bernoulli's equation, which relates pressure, velocity, and height in fluid flow.
This is like ignoring air resistance when calculating how far a ball will roll down a hill. By simplifying the scenario, you can focus solely on the effects of gravity and incline without additional complicating factors.
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So, we can write the Bernoulli equation as p1/gamma + x + h = p2/gamma + 0.8 + x + Sh.
To derive the values of pressure (p1, p2) at different points in the pipe, the professor applies Bernoulli's equation. This equation helps derive the pressure differences based on velocity and height changes along the flow path.
Think of Bernoulli's principle as a seesaw: as one side goes up (height), the other side must come down (pressure) to maintain balance.
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So, by continuity criterion Q is going to be :04:11
The continuity equation expresses the principle of conservation of mass in fluid flow. The equation relates the flow rate (Q) to the cross-sectional area and fluid velocity, allowing determination of one variable if the other is known.
Imagine squeezing a tube of toothpaste: the same amount of toothpaste must exit the tube whether it’s thick or thin. That means if the opening narrows (area decreases), the paste must shoot out quicker (velocity increases).
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Thus, we find h is equal to 17.5 centimeters.
After using Bernoulli's equation and the continuity equation, the final value of the deflection h in the mercury manometer is calculated to be 17.5 centimeters, signifying the pressure difference caused by the fluid flow.
This final measurement can be visualized like checking how high water rises in a measuring cup when you pour it in at a steady rate. The height reflects the flow's effect on pressure.
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So, this completes our Bernoulli equation however, we are going to continue with the final topic of the basics of fluid mechanics that is called fluid dynamics.
The discussion wraps up with the application of Bernoulli’s equation, transitioning to the next major topic: fluid dynamics, which deals with the movement of fluids and the forces acting on them.
Think of this transition as moving from understanding how water behaves in a stationary tank to exploring how it flows through rivers and pipes.
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
Bernoulli's Equation: Relates the velocity of fluid flow to pressure and height.
Reynolds Transport Theorem: Connects extensive properties of system to changes in control volume.
See how the concepts apply in real-world scenarios to understand their practical implications.
In a tapered pipe system, calculating the deflection of a mercury manometer linked to flow rates exemplifies the application of Bernoulli's equation.
Using Reynolds transport theorem helps assess how fluid properties change over time in a given control volume.
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
In a pipe that's wide, fluid flows with pride, less pressure in sight, when the speed takes flight.
Imagine a knight who must balance speeding waters, avoiding frictional foes, using his Bernoulli sword to ensure steady flow.
B.E.P. for Bernoulli's Equation Principle - 'Pressure Goes Down When Speed Goes Up.'
Review key concepts with flashcards.
Review the Definitions for terms.
Term: Bernoulli's Equation
Definition:
An equation that relates the pressure, velocity, and height in a flowing fluid, indicating conservation of energy.
Term: Extensive Properties
Definition:
Properties that depend on the amount of material in a system, such as mass or volume.
Term: Intensive Properties
Definition:
Properties that do not depend on the amount of material in a system, like temperature or pressure.
Term: Reynolds Transport Theorem
Definition:
A theorem that connects the change of extensive properties within a control volume to the flow of those properties across a control surface.