2.2 - Derivation of Pressure Variation
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Introduction to Barometers
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Let's start our exploration with barometers, which measure atmospheric pressure using a column of mercury. Can anyone tell me what happens to the height of mercury in a barometer with changes in atmospheric pressure?
I believe that if the atmospheric pressure increases, the mercury will rise higher in the tube?
Exactly! The height R of the mercury column reflects the local atmospheric pressure. We can calculate it using the formula P = S * R. Remember, S is the density of mercury! What unit do we get for P when we do this?
Oh, that would be Pascals, right?
Yes, great job! So, if R is 750 mm, what is your calculated atmospheric pressure?
It should be roughly 100,000 Pascals.
Correct! This concept of barometric pressure is fundamental in fluid mechanics.
Understanding Incompressible Fluids
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Now, moving on to incompressible fluids! Who can explain how we calculate pressure between two points in a fluid using the piezometric head?
I think we compare the height differences and use the hydrostatic pressure equation?
Exactly! The principle states that pressure at point 1 can be derived from the piezometric head between points 1 and 2. Can someone express this mathematically?
Is it something like P2 - P1 = R?
Almost! Remember we can also factor in z1 and z2. The final relationship will help you visualize pressure variations with height!
Exploring Compressible Fluids and Isothermal Processes
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Now let’s consider compressible fluids. When we discuss isothermal processes—who can remind me what that means?
It means the temperature remains constant while pressure and volume can change?
Exactly! The ideal gas law states PV=nRT, and we can derive pressures under these conditions. What equation describes how pressure changes with height for a perfect gas?
Uh, is it p = p2 e^(-[Mg/(RTs)](Z2-Z1))?
Yes, brilliant! That's a key equation for understanding how pressure behaves in a gas as its height changes!
Pressure Measurement Devices
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Let’s dive into pressure measurement devices! New to us are manometers. Can anyone explain the difference between a standard and a differential manometer?
A standard manometer compares pressure in a system to atmospheric pressure, while a differential manometer measures pressure differences between two points, right?
Exactly! And what are practical applications for using these devices?
They're used in various systems, especially where pressure changes frequently, like in water distribution systems!
That’s correct! Pressure measurement is crucial in engineering applications.
Applications and Examples
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Finally, let's consider some applications. Why do you think we typically use mercury in barometers and manometers?
Because mercury is denser than water, right? So it requires less height for a preset pressure!
Correct! Given standard atmospheres, how high would a column of water need to be compared to mercury?
It would need to be much taller since the density of mercury is around 13.6 times more than water!
You're all doing wonderfully! Understanding the practical applications of these theories solidifies our knowledge of fluid mechanics.
Introduction & Overview
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Quick Overview
Standard
The section delves into how atmospheric pressure is calculated using barometers, illustrates pressure variations in incompressible fluids, and discusses the behavior of perfect gases under isothermal conditions. It also touches upon practical applications of pressure measurement devices such as manometers.
Detailed
Derivation of Pressure Variation
This section covers the fundamental principles involved in measuring atmospheric pressure using barometers and illustrates how pressure varies in both incompressible and compressible fluids. The barometer works on the principle of hydrostatic pressure, where the height of mercury (R) is linked to the local atmospheric pressure. The relationship can be mathematically expressed, allowing for the calculation of pressure at varying heights in a fluid column. Additionally, the section explains the isothermal behavior of perfect gases, using the equations of state to derive pressure variations as a function of height in a constant temperature condition.
- Barometric Pressure Calculation: It illustrates how to derive pressure (P1) from a piezometric head and explains how gauge pressure can be assumed as zero under specific conditions. A derivation demonstrates that pressure (P) can be written as a product of density (S) and the height of mercury (R), giving a specific pressure in Pascals.
- Pressure Variations in Compressible Fluids: The section discusses isothermal processes where pressure varies with height, governed by the equation of state, displaying pressure decrease with elevation.
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The derivation leads to an exponential relationship for pressure as:
$$ p = p_2 e^{- rac{Mg}{RT_s}(Z_2 - Z_1)} $$
showcasing pressure loss with increasing height in stable temperature conditions. - Measurement Devices: It briefly describes different pressure measuring devices, such as standard and differential manometers, highlighting their respective uses in measuring gauges and pressure differences.
These principles are crucial not only in understanding fluid mechanics but also have practical implications in engineering applications involving pressure measurement and control.
Audio Book
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Understanding Barometers
Chapter 1 of 5
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Chapter Content
So, this is the barometer. This is the principle of working of barometers. Now, a simple question is, what is the local atmospheric pressure when R is 750 millimeters of Hg. This is R we are just going to see we have given the =13.6 then we have assumed incompressible fluid constant, we are going to see how it works. So, this is point 1 here and this is point 2 here.
Detailed Explanation
A barometer is a device used to measure atmospheric pressure. In this context, 'R' represents the height of a mercury column, specifically 750 millimeters in this case. The section highlights the incompressibility of the fluid (mercury), which is essential for determining pressure based on the height of the liquid column. Essentially, atmospheric pressure is calculated using the height of the mercury in the barometer and adjusting for the density of the liquid and gravitational force.
Examples & Analogies
Imagine a tall glass filled with water and how the height of the water column can tell you about pressure. If the water is calm and the glass is sealed, changes in the atmosphere can cause the water level to rise or fall, just like mercury in a barometer reacts to atmospheric pressure changes.
Calculating Pressure Using Piezometric Head
Chapter 2 of 5
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Chapter Content
So, we have seen this equation piezometric head is constant. So, we can also write let me erase this one. P1 is . We are putting this piezometric head equation between point 1 and point 2 and we want to calculate the pressure at p1. P2 atmospheric pressure so, atmospheric pressure we assume gauge pressure that can be assumed 0.
Detailed Explanation
The piezometric head is an important concept in fluid mechanics which states that the height of the fluid column can be used to determine the pressure at various points within the fluid system. Here, P1 represents the pressure at point 1, and P2 is referenced to atmospheric pressure. By utilizing the difference in height between two points in a fluid column, we can calculate the pressure at point 1.
Examples & Analogies
Think of a water fountain. The height to which the water shoots up from the fountain is directly related to the pressure in the water pipes below. If the fountain is high, it means there’s a lot of pressure pushing the water up. Just like measuring the height of water allows us to know pressure in a piezometric scenario.
Isothermal Process in Perfect Gases
Chapter 3 of 5
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Chapter Content
Now, we must also be a little aware about the pressure variations in a compressible fluid. So, there are 2 processes, one is perfect gas at constant temperature isothermal that we have been seeing till now.
Detailed Explanation
In this section, pressure variations in compressible fluids like gases are discussed, particularly focusing on the 'isothermal' process, where the temperature remains constant. When we consider an ideal gas, we use the gas law (PV = nRT) to derive relationships regarding pressure and density changes at constant temperature. These relationships are different from those observed in liquids due to compressibility effects.
Examples & Analogies
Consider a balloon filled with air. If you apply pressure by squeezing it, the volume decreases, and the air inside gets compressed, leading to a rise in pressure. However, if the temperature remains the same while applying pressure, it illustrates the isothermal process in action.
Integrating Pressure Changes
Chapter 4 of 5
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Chapter Content
So, now we start simply integrating from P1 to P2, and z1 to z2 simple integration, and if you integrate this, we are going to get because it is dp / p. So, that becomes ln (p2/p1) and dz is a simple integration yielding (Z2 - Z1).
Detailed Explanation
This chunk describes the mathematical integration necessary to derive the relationship between pressures at different heights for gases. The integral transforms the differential pressure change over height into a logarithmic relationship between the pressures at two points, linking them through the height difference. This method illustrates how pressure decreases with increasing height in a column of gas or liquid under constant temperature.
Examples & Analogies
Imagine hiking up a mountain—each step higher means you are farther from the 'pressure' of the sea level air. If you could measure the air pressure at every step, you'd notice it goes down logarithmically as you ascend. This is akin to how we derive pressure changes mathematically through integration.
Practical Applications of Pressure Measurement
Chapter 5 of 5
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Chapter Content
Pressure measurement devices, we have discussed one already barometers, there are manometers a standard manometer or a differential manometer and the pressure transducers. This is out of the scope, but you must have already done it in your fluid mechanics class.
Detailed Explanation
This chunk introduces different types of pressure measurement devices such as barometers, standard manometers, and differential manometers. These tools are critical in various engineering applications to monitor and measure pressure, ensuring safe and efficient operations. Understanding how they work and their applications can significantly enhance one's knowledge of fluid mechanics and hydraulics.
Examples & Analogies
Think about the pressure gauge in your car's tires. It helps you know whether your tires are properly inflated. Barometers function in a similar manner for atmosphere, measuring pressure changes to forecast weather conditions.
Key Concepts
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Piezometric Head: The height of a fluid column indicating pressure at a point.
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Gauge Pressure: Pressure measurement not including atmospheric pressure.
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Isothermal Process: A process in which temperature remains constant while pressure changes.
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Manometer: A device to measure fluid pressure relative to another reference point.
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Barometer: A device used to measure atmospheric pressure using a fluid column.
Examples & Applications
Example: Calculating the atmospheric pressure using a barometer where R is 750 mm, resulting in approximately 100,000 Pascals.
Example: Pressure variation calculation in a manometer connected to a pipe with water at 500 kPa resulting in a height of 51 meters.
Memory Aids
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Rhymes
Pressure high, mercury's shy; down it goes when the winds blow by.
Stories
Imagine a giant barometer in a kingdom, telling the villagers when to prepare for storms by the mercury rising—this height kept them dry and safe.
Memory Tools
Remember: 'B' for Barometer measures 'B' for Barometric pressure, and 'M' for Manometer measures 'M' for comparison.
Acronyms
P.G.B. - Piezometric head, Gauge pressure, Barometer – key concepts to remember!
Flash Cards
Glossary
- Piezometric Head
The height of a fluid column corresponding to the pressure at a specific point in a fluid.
- Gauge Pressure
The pressure relative to the surrounding atmosphere, typically taken as zero.
- Isothermal Process
A process during which the temperature remains constant.
- Manometer
A device for measuring the pressure of a fluid by balancing it against a column of liquid.
- Barometer
An instrument that measures atmospheric pressure.
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