#### What is in this article?:

- Bulk Modulus: What is it? When is it Important?
- Predicting bulk modulus

You should consider bulk modulus — the measure of a fluid's resistance to compression —of a hydraulic fluid if position, response time, and stability are critical.

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Despite the frequent assumption that hydraulic fluid is incompressible, the fact remains: All fluids have some degree of compressibility. Granted, fluid compressibility may be neglected in systems that do not require tight control of response and where operating pressure and fluid volume are moderate. However, when applying high pressure to a large volume of fluid, a significant amount of energy can be expended to compress the fluid — essentially squeezing the fluid's molecules closer together.

The result can be delayed response — a loaded actuator may not move until upstream fluid has been compressed, and the energy stored in the fluid may cause the actuator to continue moving after its control valve has closed. *Bulk modulus* is a property that indicates the compressibility of a fluid. With many of today's hydraulic systems operating at pressures 5000 psi and higher, ignoring bulk modulus can compromise response time of a system.

Applied pressure should directly affect the action of the system rather than compress the fluid. This is why it is so important to design systems with as little fluid as possible beween the control valve and the actuator.

### What is bulk modulus?

Most substances diminish in volume when exposed to a uniform, externally applied pressure. A typical plot of volume, V, versus pressure, P, is shown in Figure 1. The curve shows that volume of the fluid is a function of applied pressure, compressibility of the fluid, k, and initial volume of the fluid, V_{0}:

V = f (P, V_{0}, k)

V_{0} = initial volume, in, l, or m^{3}

P = pressure, psig, Pa, or bar

k = compressibility, usually negative, in.^{2}/lb

(V - V_{0}) ÷ V = specific volume, commonly used for x-axis

The term bulk modulus usually means *the reciprocal of compressibility* and defines the slope of the curve when plotted against specific volume, Figure 1. Because specific volume is dimensionless, units of bulk modulus are the same as pressure — psig (bar, Pa, N/m^{2}). Thus, the bulk modulus is a measure of resistance to compressibility of a fluid. A flat slope signifies a fairly compressible fluid having a low bulk modulus. A steep slope indicates a stiff, or only slightly compressible fluid.

### Defining bulk modulus

The plot in Figure 1 is not a straight line, so its slope changes from point to point. Two common methods are used to define the slope, or bulk modulus^{1}:

*Secant bulk modulus* is the product of the original fluid volume and the slope of the line drawn from the origin to any specified point on the plot of pressure versus specific volume (the slope of the secant line to the point).

Mathematically, secant bulk modulus, B_{S}, is:

B_{S} = (V_{0} × P) ÷ (V_{0} –V)

*Tangent bulk modulus* is the product of fluid volume at any specified pressure and the derivative of fluid pressure with respect to volume at that point (the slope of the tangent line to the point). Mathematically, tangent bulk modulus, B_{T}, is:

B_{T} = V_{0} (*dP*/*dV*)

Before giving some typical values for bulk moduli, we must take one more variable into consideration, namely, temperature.

### Temperature and bulk modulus

Temperature is important because a fluid compresses as its temperature rises. As the temperature rises, the fluid attempts to expand, which, in turn, creates additional pressure. This can occur rapidly or slowly. Compressing the fluid very slowly allows generated heat to dissipate. This bulk modulus is called *isothermal* (constant temperature) bulk modulus. *Adiabatic* or *isentropic* bulk modulus occurs by compressing the fluid rapidly and measuring the pressure — even though it results from both compression and thermal expansion.

Because we are concerned with rapidly moving, tightly controlled systems, most hydraulic applications are considered isentropic. Therefore, most of the bulk moduli discussed here are isentropic. Table 1 shows values of isentropic secant modulus for some typical hydraulic fluids at a fixed pressure and temperature.

### Effect of air on bulk modulus

Designers should be cautious before using published bulk modulus values. The values usually are determined by laboratory methods that take special precautions to degas the fluid before it is trapped and compressed.

However, hydraulic fluids typically become aerated in use. Aeration has a significant effect on bulk modulus because air is much more compressible than oil. George Totten^{2} discusses estimating the effects of air in oil on compressibility and bulk modulus. Also, realize that the solubility of air in fluids increases with pressure. Air dissolved in a fluid at high pressure can form bubbles when pressure drops — a phenomenon that can cause cavitation.