1.2. Calculating Local Atmospheric Pressure

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.

A barometer is a device used to measure atmospheric pressure. To find the local atmospheric pressure, we can use the height of a mercury column (R), which is commonly measured in millimeters of mercury (mmHg). In this context, when R is specified as 750 mmHg, it means that this height of mercury column is used to calculate atmospheric pressure. The barometer relies on the fact that atmospheric pressure can support a certain height of mercury, with 13.6 being the density of mercury compared to water.

Imagine a tall glass filled with water. When you stick a straw in it and cover one end with your thumb, the water stays in the straw due to the air pressure pushing down on the surface of the water in the glass. Similarly, the barometer works by balancing the weight of the mercury in the tube with the atmospheric pressure pushing down on the surface of the mercury in the reservoir.

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. p 2 atmospheric pressure so, atmospheric pressure we assume gauge pressure that can be assumed 0. So, pressure at point one will be we have seen was R if you just go back here this P 2 - p 1 z2 - z 1 this is point 2. So, this = R. So, going to, can be written as a S * .. So, pressure will be P= S * R, on calculation it is going to give almost hundred 100,000 Pascal.

In a fluid system, the piezometric head is a measure of the fluid pressure at different points. By applying the principle of hydrostatics between two points (point 1 and point 2), we can derive pressures at these points. The equation can be set up as P2 - P1 = ρg(z2 - z1), where P is the pressure, ρ is the fluid density, g is gravitational acceleration, and z is the elevation. When we rearrange for P1 (the pressure at point 1), we can make calculations knowing that R is equivalent to the mercury height. This ultimately allows us to find that the pressure could be approximately 100,000 Pascal, indicative of standard atmospheric pressure.

Think of water flowing through a pipe with varying heights. The fluid pressure is greater at lower points in the pipe due to the weight of the water above. When using a barometer, we are essentially measuring how much pressure the atmosphere exerts on the mercury, which is equivalent to the weight of the mercury column in the barometer tube.

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. Secondly perfect gas with constant temperature gradient.

In fluid mechanics, we need to consider how gases behave differently than liquids. Gases are compressible, meaning their density and pressure can change significantly with temperature. In this context, we discuss two scenarios: one where the gas behaves ideally at a constant temperature (isothermal process) and another where the gas encounters a gradual change in temperature. Both scenarios have different equations governing their behavior, based on the relationships between pressure (P), volume (V), and temperature (T). Understanding these variations is crucial for applications involving gas behaviors.

Consider a balloon filled with air. If you keep it at a constant temperature while squeezing it, you can feel the increase in pressure inside the balloon. This scenario exemplifies an isothermal process. In contrast, if you heat the balloon, the air inside expands, leading to a change in pressure and volume, demonstrating how temperatures affect gas behavior.

Now, we are going to do perfect gas at constant temperature that is isothermal process, we start with the same old equation. And where gamma is where the is a function of P. So, because in gas it is PV = nRT that is the thumb rule for isothermal.

The Ideal Gas Law, represented by the equation PV = nRT, establishes a relationship between the pressure (P), volume (V), and temperature (T) of a gas, where n is the number of moles and R is the universal gas constant. In an isothermal process, temperature remains constant, which means that as the volume of the gas increases, its pressure decreases, or vice versa. This relationship is crucial for solving problems related to gas behavior under varying conditions.

Consider how an inflatable pool toy behaves when you adjust its volume. If you squeeze the sides, its volume decreases, which increases the pressure inside it. Conversely, if you relieve the pressure by letting air out, the volume increases, and the internal pressure drops. This showcases the relationship described by the Ideal Gas Law in real-life terms.

So, dp will be, so this is . Correct? Because we already seen here, we just multiplied the density with G here. 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 (Z 2 - Z 1).

To find the pressure variation with height (z), we begin with the small change in pressure (dp) as the gas expands. By dividing this change by the current pressure (p), we integrate to uncover the relationship between pressures at two different heights. The integration results in a logarithmic relationship, expressed as ln(P2/P1), demonstrating how pressure changes with height in an isothermal process. This mathematical approach is crucial for deriving the pressure at a given height.

Imagine measuring the height of a mountain: the higher you go, the less air pressure is felt. As you scale up, you can track these low-pressure zones using logarithmic scales, much like how we calculated the pressure difference between two elevations in a gas column.

Pressure measurement devices, we have discussed one already barometers, there are manometers a standard manometer or a differential manometer and the pressure transducers.

In fluid mechanics, there are several devices used to measure pressure. Barometers are specifically for measuring atmospheric pressure, while manometers can measure pressure relative to the atmosphere or between two points (differential). Pressure transducers convert pressure readings into measurable electrical signals for more precise readings in experimental setups. Understanding these tools helps in accurately measuring pressure in various systems, which is essential in fields like engineering and meteorology.

Think of a barometer as your everyday weather app: it tells you the atmospheric pressure outside, helping you prepare for your day. A manometer can be compared to a local weather station that can measure pressure fluctuations within a specific area—vital for predicting how weather behaves in your surroundings.

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