Kinematic Viscosity - The ratio of coefficient of dynamic viscosity to fluid density is called the coefficient of kinematic viscosity

(in²/sec)     (3)

Density – The density, ρ, of a fluid is the mass per unit volume

(lbm/ft³) or (lbf - sec² / in⁴)     (4)

Density is a function of both pressure and temperature

Bulk Modulus - Expanding the above equation for density in a Taylor Series (for 2 variables)

        (5)

Mass density increases as pressure increases and decreases as temperature increases as the sign of is positive and the sign of is negative.

Assuming constant temperature so that = 0, then equation (5) becomes

    (6)

The quantity

(psi)     (7)

is the change in pressure divided by a fractional change in volume at constant temperature. The minus sign indicates a volume decrease with pressure increase. is called the isothermal bulk modulus (or simply bulk modulus) since it was derived assuming constant fluid temperature. The bulk modulus represents fluid compressibility and has a significant effect on the performance of hydraulic systems. Effects of pressure on bulk modulus are large and effects of temperature are usually negligible.

Effective Bulk Modulus - Both entrained air in the fluid or mechanical compliance of tubing/hoses can substantially lower the bulk modulus. Effects of both are additive.

Effects of Entrained Air:

For liquid-air mixtures, an empirical formula for the effective bulk modulus, β’, is

    (8)

where

β_isen isentropic bulk modulus of the fluid w/o entrained air

V_G0 volume of gas entrained in the liquid at atmospheric pressure

V_L0 volume of the liquid at atmospheric pressure

p₀ atmospheric pressure

p fluid pressure

k isentropic exponent (normally, k=1.4)

r_v air to liquid volume ratio

Using equation (8), the effects of entrained air are shown in Figure 1 below.

Figure 1 Effects of Entrained Air on Bulk Modulus

Effects of Mechanical Compliance:

For cylindrical pipelines, the effective bulk modulus, β’, can computed using

    (9)

β_p is the bulk modulus of the pipeline (available in Materials or Engineering Handbooks) and w is given by

for (thick walls)     (10)

for (thin walls)     (11)

where

d_o outer pipe diameter

d_i inner pipe diameter

ν Poisson’s number (0.3 for steel)

S pipe wall thickness

Fluid Properties Example: For a steel pipe with β_isen = 200000 psi, 0.5” O.D., S = 0.012”, compute the effects of mechanical compliance on bulk modulus.

μ_p = ratio of normal stress (on all faces of a cube) to a change in volume

∴ thin walled

Using equation (9)

In this example, the effects of thin walled pipe reduce bulk modulus by 25%.

Combined Effects of Air and Mechanical Compliance:

Effects of entrained air and mechanical compliance combine together like springs in series, i.e.,

    (12)

where

β’ effective bulk modulus (psi)

β_l bulk modulus of the hydraulic fluid

β_MC bulk modulus for mechanical compliance

β_g bulk modulus of air

V_g volume of air in the fluid

V_total total volume of fluid and air

Note the effective bulk modulus will be less than any of the individual bulk modulus (β_l, β_MC or V_tβ_g/Vg). For air, the bulk modulus is computed using

Basic Effects of Fluid Properties

The effects of temperature and pressure on hydraulic system fluid properties and flow characteristics are listed below.

Density – Effects orifice and valve volume flow rates. As density increases, orifice and valve flow rates will decrease (see orifice flow equations).

• Increasing pressure increases density

• Increasing temperature decreases density

Kinematic Viscosity – Effects pipe (tube) volumetric flow rate. As viscosity increases, pipe flow rate will decrease (see orifice flow equations). Kinematic viscosity increases with increased pressure and decreasing temperature.

• Increasing pressure increases kinematic viscosity

• Increasing temperature decreases kinematic viscosity

Bulk Modulus – Effects compressibility of fluid and system response time (see pressure derivative equation). As bulk modulus decreases, the pressure derivative will decrease leading to slower response times. Compressibility will affect the performance of actuators, motors and pumps because the stiffness of the fluid is less as bulk modulus is reduced.

• Increasing pressure increases bulk modulus

• Increasing temperature decreases bulk modulus

• Entrained air and compliance of hoses/tubes/parts decreases bulk modulus

Fluid Properties for Standard Hydraulic Fluids

Mil-Prf-5606

Bulk Modulus at 40 ^oC and 4000 psi: 200,000 psi (minimum)

Specific Gravitity at 60 ^oF Approx 0.88*

Nominal Density: 50 lb_m/ft³

Pour Point: ≤ -60 ^oC

Flash Point: 82 ^oC

Coefficient of Thermal Expansion: 8.6E-04 cm³/ (cm^{3 o}C)

* Not specified in MIL-PRF-5606

Mil-Prf-87257

Bulk Modulus at 40 ^oC and 4000 psi: 200,000 psi (minimum)

Specific Gravitity at 60 ^oF Approx 0.88*

Nominal Density: 49 lb_m/ft³

Pour Point: ≤ -60 ^oC

Flash Point: 160 ^oC minimum

Coefficient of Thermal Expansion: 8.2E-04 cm³/ (cm^{3 o}C)

* Not specified in MIL-PRF-87257

Mil-Prf-83282

Bulk Modulus at 60 ^oC and up to 10000 psi: 200,000 psi (minimum)

Specific Gravitity at 60 ^oF Approx 0.85*

Nominal Density: 49 lb_m/ft³

Pour Point: ≤ -55 ^oC

Flash Point: 205 ^oC

Coefficient of Thermal Expansion: 8.2E-04 cm³/ (cm^{3 o}C)

AS1241 Type V (Phosphate Ester)

Bulk Modulus at 40 ^oC and 4000 psi: 200,000 psi

Density at 25 ^oC 0.97 – 1.02 g/mL

Nominal Density: 63 lb_m/ft³

Pour Point: ≤ -62 ^oC

Flash Point: 149 ^oC

Coefficient of Thermal Expansion: 1.0E-03 in³/ (in^{3 o}F)