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

[zenterix]

Homework Statement

The following is from the book "Thermodynamics for Chemical Engineers" by Kenneth Hall.

Relevant Equations

Consider the following figure

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).

 

[Lnewqban]

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

P₁>P₂

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.