Warm greetings to all🤩 in the 15th post on #Day14 ! Today, we will complete Fluid Statics part 2. Here we go🦥,
Manometer🦦
A standard technique for measuring pressure involves the use of liquid columns in vertical or inclined tubes. Pressure-measuring devices based on this technique are called manometers. The mercury barometer is an example of one type of manometer, but there are many other configurations possible depending on the particular application. Three common types of manometers include the piezometer tube, the U-tube manometer, and the inclined-tube manometer.
Piezometer Tube 📌
The simplest type of manometer consists of a vertical tube, open at the top, and attached to the container in which the pressure is desired, as illustrated in Fig. 2.9.
It is a sphygmomanometer, the traditional instrument used to measure blood pressure. Since manometers involve columns of fluids at rest, the fundamental equation describing their use is below:
which gives the pressure at any elevation within a homogeneous fluid in terms of a reference pressure and the vertical distance h between p and p(0). Remember that in a fluid at rest pressure will increase as we move downward and will decrease as we move upward. Application of this equation to the piezometer tube of Fig. 2.9 indicates that the pressure can be determined by a measurement of through the relationship:
Note that since the tube is open at the top, the pressure can be set equal to zero (we are now using gauge pressure), with the height h1 measured from the meniscus at the upper surface to point (1). Since point (1) and point A within the container are at the same elevation, p(A)=p(1). Although the piezometer tube is a very simple and accurate pressure-measuring device, it has several disadvantages. It is suitable only if the pressure in the container is greater than atmospheric pressure (otherwise air would be sucked into the system), and the pressure to be measured must be relatively small so the required height of the column is reasonable. Also the fluid in the container in which the pressure is to be measured must be a liquid rather than a gas.
U-Tube Manometer 📌
To overcome the difficulties noted previously, another type of manometer which is widely used consists of a tube formed into the shape of a U, as is shown in Fig. 2.10.
The fluid in the manometer is called the gauge fluid. To find the pressure p(A) in terms of the various column heights, we start at one end of the system and work our way around to the other end, simply utilizing the equation above. Thus, for the U-tube manometer shown in Fig. 2.10, we will start at point A and work around to the open end. The pressure at points A and (1) are the same, and as we move from point (1) to (2) the pressure will increase by y1*h1. The pressure at point (2) is equal to the pressure at point (3), since the pressures at equal elevations in a continuous mass of fluid at rest must be the same. Note that we could not simply “jump across” from point (1) to a point at the same elevation in the right-hand tube since these would not be points within the same continuous mass of fluid. With the pressure at point (3) specified, we now move to the open end where the pressure is zero. As we move vertically upward the pressure decreases by an amount y2*h2. In equation form these various steps can be expressed as
and, therefore, the pressure can be written in terms of the column heights as
A major advantage of the U-tube manometer lies in the fact that the gauge fluid can be different from the fluid in the container in which the pressure is to be determined. For example, the fluid in A in Fig. 2.10 can be either a liquid or a gas. If A does contain a gas, the contribution of the gas column,y1*h1, is almost always negligible so that p (A)=p (2), and in this instance becomes:
p(A)==y2*h2
Thus, for a given pressure the height h2, is governed by the specific weight y2, of the gauge fluid used in the manometer. If the pressure p (A) is large, then a heavy gauge fluid, such as mercury, can be used and a reasonable column height (not too long) can still be maintained. Alternatively, if the pressure p (A) is small, a lighter gauge fluid, such as water, can be used so that a relatively large column height (which is easily read) can be achieved.
The U-tube manometer is also widely used to measure the difference is pressure between two containers or 2 points is a given system. Consider a manometer connected between containers A and B is shown in Fig. 2.11. The difference is pressure between A and B can be found
by again starting at one end of the system and working around to the other end. For example, at A the pressure is p (A), which is equal to p(1) and as we move to point (2) the pressure increases by y1*h1. The pressure at p(2) is equal to p(3), and as we move upward to point (4) the pressure decreases by y2*h2. Similarly, as we continue to move upward from point (4) to (5) the pressure decreases by y3*h3. Finally, p(5)=p (B), since they are at equal elevations. Thus,
Or, as indicated in the figure in the margin, we could start at B and work our way around to A to obtain the same result. In either case, the pressure difference is
When the time comes to substitute in numbers, be sure to use a consistent system of units!
Inclined Manometer📌
To measure small pressure changes, a manometer of the type shown in Fig. 2.12 is frequently used.
One leg of the manometer is inclined at an angle O (teta) and the differential reading l(2) is measured along the inclined tube. The difference in pressure p (A)-p (B) can be expressed as
where it is noted to be that (1) and (2) is due to the vertical distance between the points, which can be expressed as l(2)*sinO(teta). Thus, for relatively small angles the differential reading along the inclined tube can be made large even for small pressure differences. The inclined-tube manometer is often used to measure small differences in gas pressures so that if pipes A and B contain a gas, then
where the contributions of the gas columns h1 and h3 have been neglected. Equation above shows that the differential reading l2 (for a given pressure difference) of the inclined-tube manometer can be increased over that obtained with a conventional U-tube manometer by the factor 1/sinO (teta). Recall that sinO-0 as u teta(O) is going to 0.
References📖
Fundamentals of Fluid Mechanics by Bruce R. Munson 7th edition
Introduction to Fluid mechanics/Edward J. Shaughnessy, Ira M. Katz, James P. Schaffer.
Additional Resources 🧪🔑
You can get deep insight about Process/Chemical Engineering from these sources😉:
https://t.me/ebookstorage/178- Engineering Heat Transfer
https://t.me/ebookstorage/159- Fundamentals of Engineering Thermodynamics (9th Edition) (2018)
https://t.me/ebookstorage/183-Heat Transfer applications and principles
https://t.me/ebookstorage/171-Design and Operation of Heat Exchangers and their Networks (2020)
https://t.me/ebookstorage/215- Industrial Separation Processes (book)
https://t.me/ebookstorage/214- Advanced Process Engineering Control (book)
https://t.me/OilAndGas- Information about Oil&Gas (mainly arabic lang)
https://t.me/chemical_worlds- Chemical Engineering Books, Quizzes and GATE Study Group
https://t.me/chemical_environmental- Discussion group related to Chemical Engineering Problems
https://t.me/chemicalengineeringworld_cew- Everything related to Chemical Engineering
https://t.me/ebookgate- Chemical Engineering E-books (Telegram Channel)
https://www.youtube.com/channel/UCqioh32NOJc8P7cPo3jHrbg- Piping Analysis
https://www.youtube.com/channel/UCQfMyugsjrVUWU0v_ZxQs2Q -Mechanics of engineered devices
http://chemicalengineeringguy.com/- suggests a wide range of courses in Chemical engineering (you can find free courses on topic of Aspen HYSYS, Aspen Plus)
https://www.youtube.com/user/LearnEngineeringTeam- suggests working principles of every engineered devices, equipment and etch.
https://www.youtube.com/channel/UCR0EfsRZIwA5TVDaQbTqwEQ- suggests great information about pumps, compressors with animation.
Today we have already completed learning about Fluid Statics in 2 parts. Now, time to say goodbye👋🏻 until tomorrow and Stay tuned for more content 😉🌝✨!
✏️Note: If you need one of those books or links, you can contact me via my email or LinkedIn profile.
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