2. Differential Manometers

Welcome back to the lecture number 4 of this week. Last week we stopped sorry last lecture we stopped at differential manometers this was the slide that we were going to talk about, we saw some devices that can be used to measure pressures one of them was manometers in which a standard manometer and a differential manometer.

In this part, we are being reintroduced to the concept of differential manometers. Manometers are instruments used to measure fluid pressure by balancing it against a fluid column. The term 'differential manometer' refers to a type of manometer that measures the difference in pressure between two points in a fluid system, often using two different fluids, such as water and mercury.

Imagine using a balance scale to compare the weight of two objects. Just as the scale shows you the difference in weight, a differential manometer estimates the pressure difference between two points in a fluid system, giving insights into system performance.

So we need to find out the drop in pressure between points 1 and 2. So how are we going to approach this problem, so where if you see this is an orifice.

The goal is to calculate the pressure difference (drop) between two points labeled 1 and 2 in a fluid system. The orifice mentioned implies a point where fluid flows through an opening, which plays a crucial role in pressure measurement. Understanding the setup is essential for applying the principles of the differential manometer.

Consider a pipe with a small hole (orifice) through which water flows. By measuring how high the water rises or falls in the manometer connected to different points of the pipe, we can infer if the pressure is higher or lower at each point, just like checking the height of fluid in a straw to understand fluid dynamics.

So firstly P 1 this is P 1 here plus h 1 gamma w minus h 2, so if we travels around this direction and reach this point, so, minus h 2 into gamma Hg again we are traversing across the water, so minus h 3 gamma w should be equal to P 2 very p - p = h ( - g )

To analyze the pressure variation, we use a systematic approach by applying hydrostatic principles. The pressure at point 1 plus the height-adjusted pressure differences due to fluid heights (h1, h2, h3) and the specific weights of the fluids (gamma w for water, gamma Hg for mercury) relate to the pressure at point 2. This equation forms the basis for calculating pressure differences in a differential manometer.

Think of how a plant absorbs water using its roots. The pressure of the water column needs to be sufficient to push water up. Similarly, in our setup, the heights of different fluids give us 'pressure lifts' or drops, allowing us to establish how the pressures compare throughout our system.

So we can write we can 1 2 2 γ Hg - γ-water and this is the pressure difference. Very very simple question on differential manometers, and this is how it works.

The calculations lead us to express the pressure difference between points 1 and 2 as a function of the heights (h2) of the different fluids and their weights (gamma Hg and gamma water). This summarization points out the simplicity of the method: by knowing the weights and heights, we can predict pressure differences.

Imagine balancing two scales with different types of fruits. The weight of each fruit can give an idea of how heavy the total load is. In our case, by looking at how high or low these liquids are, we estimate their 'load' or pressure exerted.

So, what we say we have a 6 meter deep tank, so this is 6 meter, right? And contains 4 meters of water, and 2 meters of oil of relative density 0.88.

In this example, we set up a scenario involving a tank that is 6 meters deep. Inside, there are layers of fluids: 4 meters of water sitting atop 2 meters of oil with a given relative density. This setup is typical for practical applications where multiple fluids are present, and understanding the pressure at the bottom of the tank is vital.

Picture a multi-layered dessert with jelly (oil) sitting on top of custard (water). Just like how the weight of the jelly affects how deep it sinks into the custard, the pressure from the liquid layers in the tank helps determine how the different densities influence overall pressure.

First, determine the pressure while water interface, that is, p2, so p2 is written as p1 plus pressure due to 2 meter of oil, very nice.

To find the pressure at the bottom of the tank (P3), we first need to determine the pressure at the water-oil interface (P2). We calculate P2 as the sum of atmospheric pressure (P1) and the pressure exerted by the 2 meters of oil above it. Understanding how to traverse through the fluid layers in this manner is essential for accurate pressure calculations.

Think of filling a glass with different liquids. The weight of the upper layers pushes down on the lower layers, just like our fluids. If we want to know how much pressure is on the glass bottom, we need to add up how heavy each liquid is above it.

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