# Compressibility of a Fluid

### Bulk Modulus

The Bulk Modulus is used to determine the compressibility of a fluid. It relates the change in pressure over the change in the fluids volume. Refer to equation 1.

(Eq 1) $E_v=-\frac{dP}{dV/V}$

Density can also be used to calculate the bulk modulus since density is a product of mass and volume and as the volume changes the density will change proportionally. Refer to equation 2.

(Eq 2)  $E_v=-\frac{dP}{dρ/ρ}$

### Compression and Expnasion of a Gas

When a gas expands or contracts temperature needs to be considered. The way the temperature behaves due to the compression or expansion of a gas will affect the relationship between pressure and density. If the temperature remains the same during the compression or expansion process, then it is an isothermal process. The relationship between density and pressure for and isothermal process can be seen in equation 3.

(Eq 3) $\frac{P}{ρ}=constant$

Another process to consider is an isentropic process. An isentropic process means that there is no heat loss during the compression or expansion process of a gas. Refer to equation 4 to view the relationship between pressure and density during an isentropic process.

(Eq 4) $\frac{P}{ρ^k}=constant$

(Eq 5) $k=\frac{c_p}{c_v}$

$c_p=specific~heat~at~constant~pressure$

$c_v=specific~heat~at~constant~volume$

Now depending on the type of process, the bulk modulus will be affected. The bulk modulus for an isothermal process can be seen in equation 6, while the bulk modulus for a isentropic process can be seen in equation 7.

(Eq 6) $E_v=P$

(Eq 7) $E_v=kP$