Bulk modulus and compressibility
for calculating the performance of hydraulic systems

The basis of hydraulic pressure distribution is the incompressibility of liquids. However, the strict term “incompressible” is only relative even for liquids. First, all liquids contain a certain amount of dissolved gases, which are compressible. Second, the value of compressibility always depends on the material to which the pressure is applied. Thus, water has a different compressibility than mercury or oil. Although compressibility is extremely low for all liquids, all other components in which pressure is distributed in the hydraulic system have a certain compressibility. This affects the efficiency of the hydraulic system and should therefore be taken into account for complex installations.

Bulk modulus and compressibility

The bulk modulus (not neuter!) is a parameter (K) that provides information about a material’s elasticity. The simplest example is the behavior of a cube of steel under pressure compared with a cube of rubber. The different deformation and recovery behavior of the two bodies is due to their material and thus its bulk modulus. The bulk modulus is applied to solids.

Compressibility is the reciprocal of the bulk modulus (1/K). This parameter is used to determine the elasticity of liquids and gases. For liquids, compressibility is extremely low and can generally be neglected. However, this changes abruptly as soon as gases occur in the liquid. This can happen when the temperature increases (e.g., overheating brakes when the water content in the brake fluid is too high) or when pressure drops (e.g., ascending too quickly to the surface during dives).

Formula for calculating the bulk modulus and compressibility

To calculate the bulk modulus K, the material-specific values of Young’s modulus E, shear modulus G, and Poisson’s ratio v are required. These are determined through tests.

The formula for calculating the bulk modulus is then

K = (2G(1+v))/(3(-2v))

Compressibility is obtained as the reciprocal.

Relevance of the bulk modulus for hydraulics

The bulk modulus K is required for the design of all components to which hydraulic pressure is applied. This applies in particular to the lines. Hoses and pipes that are too elastic reduce the system’s effective pressure and thus lower efficiency. For this reason, hydraulic flexible lines are reinforced with a steel braid that does not expand under pressure. A rubber line would simply stretch and would not transmit the pressure to the desired point. This makes it easy to identify defective hydraulic hoses: In addition to leakage, the typical damage pattern is a bulge that has formed at the weak point. Such a line can only be replaced and disposed of.