Absolute Pressure and Gauge Pressure

Whenever you have a regular health checkup at your doctor's office, they will likely measure your blood pressure. In that case, a medical tool called the sphygmomanometer is inflated around your arm to temporarily stop the blood flow in an artery.

Fig. 1 - Blood pressure measurements are made using a sphygmomanometer, which ignores the air pressure around it. It's an example of gauge pressure.

As the pressure from the cuff is slowly released, the artery expands and blood rushes back into the arm. This allows the medical professional to quantify the force exerted by the blood flow against the artery walls, as the heart pumps the blood through the body. This procedure takes no longer than a couple of minutes, yet requires a thorough understanding of fluid mechanics, types of pressure, and their measurements. In this article, we'll focus on gauge pressure (e.g., blood pressure) and absolute pressure, their differences, and applications!

Earth's atmosphere is filled with air which constantly exerts a pressure on all objects and surfaces, as it experiences the gravitational pull toward the planet's surface. This pressure is known as atmospheric pressure.

Sometimes it's also referred to as the barometric or air pressure and has been assigned a special unit of the standard atmosphere (\(\mathrm{atm}\)), where \(1 \, \mathrm{atm}\) describes the average atmospheric pressure at sea level at \(15\,^\circ \mathrm{C}\).

Although we tend to assume atmospheric pressure to be a constant value, it differs across the planet's surface with altitude and temperature. These fluctuations are very minor near the Earth's surface, where most of the measurements we deal with occur, hence the valid assumption for a constant atmospheric pressure value. However, if the calculations deal with sensitive weather patterns or the pressure measured at the top of a mountain or deep in the ocean, the atmospheric pressure value should be adjusted accordingly.

So, we can use the fact that atmospheric pressure is a constant value at sea level to define two types of pressure: one that accounts for this additional atmospheric pressure (absolute pressure), and one that ignores it (gauge pressure).

In other words, atmospheric pressure is set as the reference point for the gauge pressure. If the absolute pressure is greater than atmospheric pressure, the gauge pressure will have a positive value. On the other hand, if the absolute pressure is smaller than atmospheric pressure, the gauge pressure will be negative.

The absolute pressure is the total pressure exerted by a fluid and includes the atmospheric pressure in its measurements.

Now that we have established the basic definitions for different kinds of pressure measurements, let's identify the key differences between gauge pressure and absolute pressure.

Difference between Gauge Pressure and Absolute Pressure

Returning to the blood pressure example from earlier, the sphygmomanometer used by the doctor will have a scale ranging from \(0 \, \mathrm{mmHg}\) to around \(300 \, \mathrm{mmHg}\). Considering that humans experience atmospheric pressure, shouldn't the scale go up to a minimum of \(1060 \, \mathrm{mmHg}\), as before taking any measurements on a patient, the sphygmomanometer will already display a value of roughly \(760 \, \mathrm{mmHg}\)?

Atmospheric pressure is always present on Earth and is experienced by the sphygmomanometer and the blood flowing through the patient, so keeping a record of this constant measurement doesn't make sense. The interesting value is the blood pressure compared to its surroundings because that value is what the arteries will feel as a net force per unit of artery area. So we are interested in the blood's gauge pressure. Thus, for convenience purposes, it's common to adapt these kinds of instruments to read zero at the atmospheric pressure, which means utilizing the gauge pressure.

On the other hand, if we have a completely sealed system where the atmospheric pressure has an effect on the events occurring, we use absolute pressure. It's commonly used in laboratory settings as well as manufacturing, where even the slightest fluctuation in the setup can alter the results. A great example is vacuum-packaged food, where the longevity of the food depends on the quality of the vacuum seal, or the lack of any pressure, including atmospheric pressure.

Fig. 2 - When food is placed into a vacuum packing chamber, an absolute pressure gauge is used to measure the pressure drop way below the atmospheric pressure.

Relationship between Absolute Pressure and Gauge Pressure

Just based on the definitions of both values, it's clear that absolute pressure and gauge pressure are closely related. A visual reference for connecting the two pressures is visible in Figure 3.

Fig. 3 - The final measurement of pressure in a system can be made relative to zero (absolute pressure) or to the atmospheric pressure (gauge pressure).

We can then use this diagram to come up with a mathematical expression connecting the two types of pressures.

Formula Describing Absolute and Gauge Pressure

Based on the explanation provided above, the simple version of the relationship between absolute and gauge pressure is

$$P=P_0+P_\mathrm{G}, $$

where \(P\) is the absolute pressure, \(P_0\) is the atmospheric pressure, and \(P_\mathrm{G}\) is the gauge pressure. Considering \(P_0\) is a constant value, we only need to elaborate on the gauge pressure, so let's explain it in more detail.

Gauge Pressure

Pressure in a fluid occurs as the upper layers press their weight onto the lower layers. So let's imagine a closed container filled with a liquid, meaning we can ignore the atmospheric pressure. The pressure \(P\) exerted on the bottom of this container by a vertical, rectangular column of liquid can be expressed mathematically as

$$ P=\frac{F}{A}, $$

where \(F\) is the force the fluid exerts and \(A\) is the area over which the fluid exerts this force. The force in this case is simply the force of gravity acting on the liquid of mass \(m\) with the gravitational acceleration \(g\). Mass can be re-expressed in terms of liquid density \(\rho\) and volume \(V\) as

$$ m=\rho V,$$

where volume is that of the rectangular column, meaning it's equal to the surface area \(A\) multiplied by the column's height \(h\). Now we simply insert all of these values into the pressure equation,

$$ P=\frac{mg}{A}=\frac{\rho gV}{A}= \frac{\rho g \bcancel{A}h}{\bcancel{A}}, $$

to obtain the expression for hydrostatic pressure:

$$ P_\text{fluid}=\rho h g. $$

This means that a fluid that isn't moving will exert a pressure that only depends on the density of the fluid and its depth.

Absolute Pressure

We can use the hydrostatic pressure equation to complete the absolute pressure formula mentioned earlier to obtain

$$P=P_0+\rho gh. $$

This equation simply tells us that the total pressure exerted on the bottom of a column of fluid that is at sea level is the hydrostatic pressure (which is the gauge pressure) plus the atmospheric pressure. This is logical because both fluid (hydrostatic pressure) and air (atmospheric pressure) exert a force on the bottom of the column of fluid.

Example of Absolute and Gauge Pressure

Let's look at an example problem applying the absolute and gauge pressure formula!