Even if the concepts described in this book do not involve many considerations about fluid compressibility, it can be interesting to note how the bulk modulus of a fluid is in direct relation with the speed of sound within the fluid.
In fact, from the basic definitions of thermodynamics, the speed of sound c is defined as [13]
(2.8)
Considering the definition of Eq. (2.4), after noticing that ΔV/V = Δρ/ρ (the definition of fluid density, ρ, will be given in Section 2.5), the speed of sound results in
(2.9)
2.5 Fluid Density
Density is an important intensive property of a fluid, and it can be defined as the ratio between the mass and the volume at a given state. Considering reference condition as (p0, T0), the density ρ0 is given by
(2.10)
Typical values of density of different fluids at room temperature and atmospheric conditions are given in Table 2.3.
Following the same line of reasoning presented in Section 2.4, density variations can also be related to pressure and temperature. The isothermal bulk modulus and the isobaric cubic expansion coefficient defined in Section 2.4 can be used to quantify the dependence of density on pressure and temperature.
The variation with respect to pressure (assuming constant temperature, T0) is shown below:
However, the variation with respect to temperature (assuming constant pressure, p0) is as follows:
The chart of Figure 2.4 shows the typical variation of fluid density for three different oils. The plot results from Eq. (2.11) using the following values for the parameters of Eq. (2.12) (values taken from [14]):
From Figure 2.4, one can notice that the change in density over a large pressure variation is not negligible. In particular, for a mineral oil, the same mass of fluid is exposed to a volume reduction (or a density increase) of about 0.7% every 100 bar of pressure variation. Using the same parameters from Table 2.4, the plot of Figure 2.5 can be derived from Eq. (2.12) to represent the variation of fluid density with temperature.
Figure 2.4 Density variation with pressure for three oils (parameters listed in Table 2.4).
Figure 2.5 Density variation with temperature for three oils (parameters listed in Table 2.4).
Table 2.3 Typical values for fluid density.
| Fluid | Density [kg/m3] |
|---|---|
| Mineral oil | 870–900 |
| Water | 1000 |
| Water/glycol | 1060 |
| Water/oil emulsion | 920–940 |
| Vegetable oil | 930 |
| Chlorinated hydrocarbons | 1400 |
| Phosphoric esters | 1150 |
| Silicon‐based fluid | 930–1030 |
Table 2.4 Fluid parameters used for the plot of Figure 2.4.
| Fluid | ρ0 [kg/m3] |
|
γ [K−1] |
|---|---|---|---|
| Mineral oil | 870 | 0.70 · 10−4 | 0.65 · 10−3 |
| HFC |
1050
|