Equivalent bulk modulus

The bulk modulus of a liquid can be substantially lowered by gas or vapor entrained in the liquid in the form of bubbles and/or pockets and by the mechanical compliance of a flexible container. The equation which gives the equivalent bulk modulus of liquid-gas mixture in a flexible container is

where the subscripts , , and refer to the gas, liquid, and container, respectively. It is assumed that the initial total volume of the container , and that . Thus the effective bulk modulus will be less than any one of the values , , or . The bulk modulus of the liquid is obtained from manufactures' data. The isothermal bulk modulus of air is equal to the pressure level , the adiabatic value used for air is .

Usually, the major source of mechanical compliance is the hydraulic lines connecting valves and pumps to actuators. The bulk modulus for a thick-walled cylindrical container is

where is the inner diameter, is the outer diameter, is the wall thickness (), is the modulus of elasticity of the wall material, and is the Poisson's ratio for the material. For a thin-walled cylinder such that , and because for metals, (6.31) approximates to the formula

generally used for hydraulic tubing.

Estimates of entrapped air in hydraulic systems run as high as when the fluid is at atmospheric pressure. As pressure is increased, much of this air dissolves into the liquid and does not affect the bulk modulus. In any practical case it is difficult to determine the effective bulk modulus other than by direct measurement.