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Thursday, 25 January 2018

DERIVATION OF INCLINED SINGLE COLUMN MANOMETER

We were discussing the basic definition and significance of Pascal’s Law along with its derivation , Vapour pressure and cavitationAbsolute pressure, Gauge pressure, Atmospheric pressure and Vacuum pressurepressure measurementPiezometer and also the basic concept of  U-tube manometer in our previous posts.

Today we will understand here the basic concept of inclined single column manometer to measure the pressure at a point in fluid, in the subject of fluid mechanics, with the help of this post.

Single column manometer

Single column manometer is one modified form of U-Tube manometer. There will be one reservoir with large cross-sectional area about 100 times as compared to the area of glass tube. One limb (let us say left) of the glass tube will be connected with the reservoir and another limb (right) of glass tube will be open to atmosphere as displayed here in following figure.

This complete set-up will be termed as single column manometer. Pressure will be measured at a point in the fluid by connecting the single column manometer with the container filled with liquid whose pressure needs to be measured. Rise of liquid in right limb of glass tube will provide the pressure head.

There are basically two types of single column manometers, on the basis of right limb of manometer, as mentioned here.
Inclined single column manometer

Inclined single column manometer

In case of inclined single column manometer, other limb of glass tube will be inclined as displayed here in following figure.

Let us consider we have one container filled with a liquid and we need to measure the pressure of liquid at point A in the container. Let us consider that we are using the pressure measuring device “Inclined single column manometer” here to measure the pressure of liquid at point A as displayed here in following figure.
Let us consider the following terms from above figure.

XX is the datum line between in the reservoir and in the inclined limb of manometer
P = Pressure at point A and we need to measure this pressure or we need to find the expression for pressure at this point.

Let us consider that container, filled with a liquid whose pressure is to be measured, is connected now with inclined single column manometer. 

Once inclined single column manometer will be connected with container, heavy liquid in reservoir will move downward due to the high pressure of liquid at point A in the container. Therefore heavy liquid will rise in inclined limb of the manometer.

Inclined single column manometer will be quite sensitive manometer as distance moved by heavy liquid in the right limb will be more due to inclination of right limb of the manometer.

L = Length of heavy liquid moved in the inclined limb from XX
h1 = Height of lower specific gravity liquid above the datum line
h2 = Height of higher specific gravity liquid above the datum line
h2 = L x Sin θ
Δh = Fall of heavy liquid in the reservoir
S1 = Specific gravity of the light liquid i.e. specific gravity of liquid in container
S2 = Specific gravity of the heavy liquid i.e. specific gravity of liquid in reservoir and in right limb of the manometer
ρ1 = Density of the light liquid = 1000 x S1
ρ2 = Density of the heavy liquid = 1000 x S2
YY = Datum line after connecting the manometer with container

Level of heavy liquid in the reservoir will be dropped and therefore there will be respective rise in the level of heavy liquid in the right limb.

Pressure in the left limb above the datum line YY = P + ρ1g (Δh + h1)
Pressure in the right column above the datum line YY = ρ2g (Δh + h2)

As pressure is same for the horizontal surface and therefore we will have following equation as mentioned here
P + ρ1g (Δh + h1) = ρ2g (Δh + h2)
P = ρ2g (Δh + h2) - ρ1g (Δh + h1)
P = Δh (ρ2g - ρ1g) + ρ2g h- ρ1g h1

As, value of Δh will be quite small and therefore we may neglect the term Δh (ρ2g - ρ1g) and we will have following equation.

P = L x Sin θ x ρ2g - ρ1g h1

Do you have any suggestions? Please write in comment box.

Reference:

Fluid mechanics, By R. K. Bansal
Image Courtesy: Google

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