#### What is in this article?:

- What is bulk modulus, and 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.

## Predicting bulk modulus

Several sources are available for predicting the bulk modulus of hydraulic fluids^{2,3}.

**Volume lost in pumps and actuators** — The output of a pump or the positional relationship of master and slave cylinders varies with the bulk modulus of the fluid. In the case of pumps, the percentage volume loss in the output is seen as a loss of horsepower. For master-slave cylinders, the volume loss is seen as a reduced stroke from the slave.

**Stopping a moving load** — If a cylinder moves a load at a uniform velocity (that is, constant flow to the cylinder), the cylinder has momentum that the fluid and the system must absorb when a valve controlling upstream and downstream flow is suddenly closed. The downstream fluid pressure will rise from some nominal value to some peak pressure as energy is absorbed. Assuming the cylinder and hydraulic lines to be rigid, and a linear rise in pressure, the fluid's bulk modulus will determine peak pressure. Thus, for a specific maximum pressure, the stiffer the fluid, the less energy is absorbed and the less overshoot. Fluids with higher values of bulk modulus have less energy absorption and less piston overshoot, which translates to better position accuracy.

**Fast load reversals** — Because most fluids are compressible, the fluid in an actuator must be compressed before the cylinder or piston will move a load. In other words, an amount of fluid equal to the compressed volume must be added to an actuator before a load will move. Because this process does not do useful work, it is lost work:

W_{L} = f × d where W_{L} = lost work

f = force

d = distance

Distance refers to an increment of cylinder stroke, so:

W_{L} = ∆P × ∆V_{0}

where ∆P = change in pressure

∆V = change in volume (increment of stroke × piston area)

But ∆V = ∆P × (V_{0} ÷ B), so:

W_{L} = (∆P^{2} × V_{0} ) ÷ B

To calculate lost power, divide by time:

W_{L} = (∆P^{2} × V_{0}) ÷ (B_{t} × 6600)

Because power loss can be significant at higher pressure ranges, let us examine a typical 3000 psig system, that is, ∆P = 3000 psi.

hp_{l} = (1363 V_{0}) ÷ (B × t)

It is now possible to plot lost horsepower versus time for 1 in.^{3} of cylinder volume for various bulk moduli, Figure 2. Lost power increases as cylinder size increases and response time decreases.

Figure 3 illustrates lost power versus response rate for various bulk moduli. The loss in power may look relatively small until we consider an average cylinder. If we assume a bulk modulus of 200,000 psi, a response of 100 Hz, and a stroke of 10 in., the power loss is 6.75 hp / in.^{2} of ram area. Figure 4 relates power loss to total system power available. For example, a 3000-psi, 3.8-gpm system that can supply 6.75 hp cannot move a load at 100 Hz with a 1-in.^{2} piston because all the power is used in compressing the fluid.

### Resonance of hydraulic systems

The natural frequency of a springmass combination is:

ƒ = (1 ÷ 2π) × (kg)^{1/2} ÷ W

Where: ƒ = frequency, Hz

W = weight, lb

k = spring rate, lb/in., and

g = acceleration due to gravity, 32.2 ft/sec^{2}.

To equate this to a hydraulic system, we only need to substitute bulk modulus for spring rate. Thus, a low modulus also lowers the natural frequency of a system. For example, if 1% air content changes the bulk modulus by 50%, its natural frequency decreases by 30%. This greatly reduces the stability of the system.

### Why bulk modulus is important

We can conclude, then, that the absolute value of the bulk modulus of a fluid can seriously affect system performance in relation to position, power level, response time, and stability. Two factors that figure prominently in the control of bulk modulus are fluid temperature and entrained air content. For example, Table 2 shows that raising the temperature of commercial hydraulic fluid by 100° F alone reduces its bulk modulus to 61% of its room-temperature value. Table 2 also indicates that introducing 1% air by volume reduces the bulk modulus to 55% of its room temperature value. If these two conditions occur simultaneously, the net effect is to reduce the bulk modulus by 67%.

In view of today's requirements for higher power and response time, it is more important than ever to pay attention to bulk modulus.

*References*:

*1. ASTM D6793 "*Standard Test Method for Determination of Isothermal Secant and Tangent Bulk Modulus,*" ASTM International, West Conshohocken, Pa. 2. *Handbook of Hydraulic Fluid Technolog

*y, edited by George E. Totten, Marcel Dekker, Inc., 2000.*

3.Hydraulic Fluid Power — Petroleum Fluids — Prediction of Bulk Moduli

3.

*, ANSI.NFPA T2.13.7 R1-1997 (R2005) National Fluid Power Association, Milwaukee.*