What is "gauge pressure" and how does a manometer work?

In summary, gauge pressure refers to the pressure of a fluid relative to the atmospheric pressure. It is measured using a manometer, which is a device that compares the pressure of a fluid to the ambient atmospheric pressure. A typical manometer consists of a U-shaped tube filled with liquid, where one side is open to the atmosphere and the other side is connected to the pressure source. The difference in liquid height between the two sides indicates the gauge pressure, allowing for easy measurement of pressure variations.
  • #1
zenterix
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Homework Statement
The following is from the book "Thermodynamics for Chemical Engineers" by Kenneth Hall.
Relevant Equations
Consider the following figure
1726343315962.png


Manometers measure differential pressures.

The differential equation that expresses pressure is

$$\frac{\partial P}{\partial z}=-\rho g\tag{1}$$

where ##\rho## is the density of the fluid and ##g## is the local acceleration of gravity.

If the density is constant, the integration of (1) is

$$P_2-P_1=-\rho g\Delta z=-\rho g(z_2-z_1)=-\rho g h\tag{2}$$

$$P_1=P_2+\rho gh\tag{3}$$

This indicates that the hydraulic pressure (gauge pressure) is ##\rho gh##.

First of all, where does (1) come from?

What is an intuitive explanation for the meaning of ##-\rho g##?

##\partial P/\partial z## is the rate of change of pressure relative to position ##z##. Since we have a negative sign on the rhs, it seems that ##z## is being measured from bottom to top.

##\rho g## is thus a rate of change of pressure and so ##\rho g h## is change in pressure.

(3) says that the pressure at position 1 is the sum of the pressure at position 2 plus the term ##\rho gh##.

Note that ##\rho g h## is also the pressure that a vertical column of fluid exerts at its base due to gravity.

$$P=\frac{F}{A}=\frac{mg}{A}=\frac{\rho Vg}{A}=\frac{\rho Ahg}{A}=\rho g h$$

In terms of dimensions, we have ##\frac{[m]}{[L]^3}[a][L]=\frac{[F]}{[L]^2}=[P]##. I am not sure if this is correct in terms of notation. Can I write ##[a]## like this as a dimension?

Next, consider the following snippet

Atmospheric pressure ##P_{atm}## is the pressure caused by the weight of the Earth's atmosphere on an object. We might find this pressure called "barometric" pressure.

Absolute pressure ##P_{abs}## is the total pressure, and absolute pressure of zero is perfect vacuum. All thermodynamic calculations must use absolute pressure.

Gauge pressure or manometric pressure ##P_{man}## is the pressure relative to atmospheric pressure.

$$P_{man}=P_{abs}-P_{atm}\tag{4}$$

$$P_{abs}=P_{atm}+P_{man}\tag{5}$$

I would like to understand what gauge pressure is exactly.

It seems that (3) is in the form of (5) but what allows us to conclude this?

As for the U-tube glass manometer depicted in the picture above, it seems that the opening at 2 is open to the atmosphere and the opening at 1 is subjected to some pressure we would like to measure.

We know the density of the fluid (at the temperature of the fluid being used).

The difference in the pressure at 1 relative to the pressure at 2 is the pressure exerted by the column of fluid on the right with height ##h## (which is the height difference between 1 and 2).
 
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  • #2
zenterix said:
What is an intuitive explanation for the meaning of ##-\rho g##?
P1>P2

zenterix said:
I would like to understand what gauge pressure is exactly.
It is a relative value respect to the atmospheric pressure at that specific altitude above sea level and temperature of the air.

That is not the value of all the actual pressure acting on a substance, which is analyzed thermodynamically.
 

FAQ: What is "gauge pressure" and how does a manometer work?

What is gauge pressure?

Gauge pressure is the pressure measurement relative to atmospheric pressure. It indicates how much pressure is above or below the ambient atmospheric pressure. Gauge pressure is commonly used in various applications, such as tire pressure readings and fluid systems, where the atmospheric pressure is not a factor in the operation.

How is gauge pressure different from absolute pressure?

Gauge pressure differs from absolute pressure in that absolute pressure measures the total pressure exerted, including atmospheric pressure, while gauge pressure only measures the pressure above atmospheric pressure. Absolute pressure is calculated by adding atmospheric pressure to gauge pressure.

What is a manometer?

A manometer is a device used to measure the pressure of a gas or liquid. It typically consists of a U-shaped tube filled with a liquid, such as mercury or water, where the difference in liquid height between the two arms of the tube indicates the pressure of the fluid being measured.

How does a manometer work?

A manometer works by balancing the pressure of the fluid against the weight of the liquid column in the tube. When pressure is applied to one side of the manometer, it causes the liquid to move, creating a height difference between the two columns. This height difference can be converted into pressure units, allowing for the measurement of gauge pressure.

What are the types of manometers?

There are several types of manometers, including open-tube manometers, closed-tube manometers, and digital manometers. Open-tube manometers are used for measuring gauge pressure in open systems, while closed-tube manometers are used for measuring pressure in closed systems. Digital manometers provide electronic readings and are often more precise and easier to read than traditional liquid-filled manometers.

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