9.2.3. Differential Manometers

Now there is another type of manometer which is called the differential manometers. As the name says that here, we do not measure a particular point pressure measurements where you consider two points. Because many of our applications, we are not worried about absolute pressure. Mostly we are looking it at the two points like I have the pipe and I have the point A and B. So this is a pipe flow, I have the point A and B. At this A and B locations, I want to know it what is the pressure difference between these two point of A and B in this pipe flow. I am not worried about the absolute pressure at the A or the absolute pressure B.

A differential manometer is a device used to measure pressure differences between two points in a fluid system. Unlike other manometers that may measure the absolute pressure at a single point, a differential manometer focuses on the difference in pressure between two specified points. This is particularly useful in fluid dynamics where knowing the pressure difference drives flow is more important than knowing the absolute pressures themselves. For example, in a pipe with fluid flowing, understanding the pressure difference at points A and B informs us how much pressure is available to move the fluid from A to B.

Consider two water tanks at different heights connected by a pipe. If we want to determine how strong the flow is from one tank to the other, we wouldn't need to know the exact pressure in each tank; we would only need to know how the pressure in Tank A compares to Tank B. This is similar to what differential manometers do.

I want to looking for a relative pressure between these two difference between which is driving the flow from one side to the other sides. I am just looking it what is the difference pressure between that. So that means we use differential manometers which measure the difference of pressure between two points in a fluid system.

The working principle of a differential manometer utilizes U-tube geometry where each side of the U-tube is connected to two different points in the system. When there is a pressure difference between these two points, it causes a difference in liquid levels within the U-tube. By measuring this difference in height (h1 and h2) and knowing the densities of the liquids involved, we can calculate the pressure difference. This allows engineers and technicians to assess how much force is acting to move the fluid through the pipes.

Imagine two balloons filled with air at different pressures. The balloon at higher pressure can be likened to point A in our differential manometer, while the balloon with lower pressure represents point B. If we connect a tube to both balloons and observe what happens; air will rush from the higher pressure balloon to the lower pressure one, similar to how fluid flows in a pipe from high to low pressure. The greater the difference in balloon sizes represents the greater pressure difference that the manometer is designed to measure.

If this is the arrangement and consider these tubes are carrying the same the different liquids okay. They are carrying the different liquid having the different density like ρ1 and ρ2 and ρ3 is the manometric density of the manometric liquid okay. If this is the arrangement and measure the distance from this flow systems and this manometer becomes equilibrium after this certain flow that.

In a typical setup of a differential manometer where different liquids are used, the density of each liquid plays a crucial role in determining the pressure difference. Let's assume we have fluid A with density ρA and fluid B with density ρB flowing into the manometer filled with a manometric liquid (say mercury, density ρM). The heights of the liquid columns (h1 and h2) correlate to the pressures at points A and B. By establishing an equilibrium condition in the manometer, you can apply hydrostatic equations to derive the relation between these heights and the respective fluid pressures. The overall goal is to determine how much pressure difference drives flow through the pipe system.

Think of a balanced seesaw with kids of different weights sitting on each side. The heavier child on one side balances against the lighter child on the other side. Similarly, in a differential manometer, the pressures exerted by different liquids with varying densities balance against each other, allowing us to find the pressure difference between two points. The height difference between liquid columns can help us 'weigh' these pressures accurately.

If I look it at this point the pressure which is connected to A part will be the P pressure at that point then because of the fluid what is carried by these conduit, you have a h is the height which will be ρ * g * h. Then you have a manometric height liquid the difference which will be a ρ * g * h.

To express the pressures mathematically, we set the pressure at point A (PA) as a function of the liquid heights in the manometer. If h1 is the height of the liquid on one side and h2 is its counterpart on the other, the relation derived from hydrostatic pressure equations will look like: PA - PB = ρ1 * g * h1 - ρ2 * g * h2. This equation allows us to balance the pressures at points A and B influenced by the densities of the fluids and gravitational acceleration. The final equation helps solve for unknown variables as needed.

Imagine you are trying to weigh two fruit baskets using a balance scale. The heavier basket pushes down more on its side, and you can measure how much lower it goes compared to the lighter basket. In the context of the differential manometer, the pressure difference (similar to the weight difference) affects how high or low the liquid stands on either side of the manometer, giving us the means to calculate the actual pressures indirectly.

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