Compressibility & bulk modulus

Oil feels solid — but it does squeeze, just a little. That tiny springiness sets how stiff a hydraulic system is, and a few air bubbles can wreck it.

Source: Rabie, Fluid Power Engineering, Ch. 2.

Before you start

What you need first

Pressure in Pa (Topic 2) and volume (Topic 8).

What you'll be able to do

Use the bulk modulus $B=-\dfrac{\Delta p}{\Delta V/V}$.
Find the fractional volume change under pressure.
Explain why trapped air destroys stiffness.

Start here · the idea

Oil compresses — just a little

Squeeze trapped oil hard and its volume shrinks by about 1% per 150 bar. Tiny — which is exactly why hydraulics works (the oil passes the push along almost rigidly) — but not zero, and that small "give" is what we measure here.

The property that says how hard the oil resists being squeezed is its bulk modulus $B$: a big $B$ means very stiff (hard to compress).

Bulk modulus

$$B = -\dfrac{\Delta p}{\Delta V/V} \qquad\Longleftrightarrow\qquad \dfrac{\Delta V}{V} = -\dfrac{\Delta p}{B}$$

where:
Symbol	Meaning	SI unit
B	bulk modulus (stiffness of the oil)	Pa
$\Delta p$	change in pressure	Pa
$\Delta V$	change in volume	m³
V	original volume	m³

For hydraulic oil, $B \approx 1.5\!-\!2.0~\text{GPa}$ (a handy value is $1.7~\text{GPa}$). The minus sign just says volume shrinks as pressure rises.

Oil behaves like a very stiff spring

Because it gives a little under load and pushes back, a trapped column of oil acts like a spring — a very stiff one. Push harder (more pressure) and it compresses a touch more.

This springiness is good and bad: it lets the system absorb shock, but it also means a loaded actuator "sags" slightly and can bounce — the higher the $B$, the stiffer and more precise the machine.

Stiffness rises with $B$: a bigger bulk modulus → a stiffer "spring" → less sag under load.

Trapped air wrecks the stiffness

Air is roughly ten thousand times more compressible than oil. So even a tiny bubble of trapped air dominates the "give": the mixture's effective bulk modulus $B_e$ collapses far below the oil's own $B$.

A system with a little entrained air (air bubbles trapped and carried along in the oil) feels spongy — the actuator bounces, responds slowly, and holds position poorly. This is why bleeding air out and using anti-foam oil (Topic 6) matter so much, and it connects straight to aeration (Topic 12).

⚠️ The diesel effect. Worse still: when oil carrying air bubbles is suddenly compressed (for example at the pump outlet), each bubble is squeezed so fast that it heats up like the air in a diesel engine's cylinder — to 1000 °C or more. That can char the surrounding oil and burn the seals — one more reason to keep air out of the system.

✏️ Try it yourself — no numbers needed

Two identical actuators hold the same load. One has clean oil; the other has a small amount of trapped air. Which one "sags" and bounces more under the load, and why — even though the oil's own bulk modulus is the same in both?

The aerated one sags and bounces more. Why: the air bubble is enormously more compressible than the oil, so the mixture's effective bulk modulus $B_e$ drops far below the pure-oil value. A lower $B_e$ means more "give" — a softer, springier column — for the same load.

Common mistakes to avoid

Mistake	Fix
Mixing units of $B$ and $\Delta p$	Both in Pa (or both in bar) — $B\approx1.7~\text{GPa}=1.7\times10^9~\text{Pa}$.
Treating oil as perfectly incompressible	It compresses ~1% per 150 bar — small but it sets stiffness.
Ignoring trapped air	A tiny air fraction can cut the effective $B$ several-fold.

Recap — the whole topic on one screen

$$B=-\dfrac{\Delta p}{\Delta V/V} \qquad \dfrac{\Delta V}{V}=-\dfrac{\Delta p}{B}\qquad B_{\text{oil}}\approx 1.7~\text{GPa}$$

Idea	What you own now
Bulk modulus	Stiffness of the oil; big $B$ = hard to compress
Volume change	$\Delta V/V=-\Delta p/B$ — ~1% per 150 bar
Trapped air	Collapses the effective $B_e$ → spongy system

Next topic

Thermal expansion & trapped-oil pressure

Bulk modulus also explains a hidden danger: heat trapped oil and, because it cannot expand, the pressure rockets. Next we combine expansion with $B$ to see how far.

→ Thermal expansion & trapped-oil pressure

Symbol	Meaning	SI unit
B	bulk modulus (stiffness of the oil)	Pa
\(\Delta p\)	change in pressure	Pa
\(\Delta V\)	change in volume	m³
V	original volume	m³

Mistake	Fix
Mixing units of \(B\) and \(\Delta p\)	Both in Pa (or both in bar) — \(B\approx1.7~\text{GPa}=1.7\times10^9~\text{Pa}\).
Treating oil as perfectly incompressible	It compresses ~1% per 150 bar — small but it sets stiffness.
Ignoring trapped air	A tiny air fraction can cut the effective \(B\) several-fold.

Idea	What you own now
Bulk modulus	Stiffness of the oil; big \(B\) = hard to compress
Volume change	\(\Delta V/V=-\Delta p/B\) — ~1% per 150 bar
Trapped air	Collapses the effective \(B_e\) → spongy system