# Bulk Modulus

When analyzing a fluid it is important to consider how easily the fluids volume will change due to a pressure change.  The reason why is because as that fluids volume changes its density will change.  To determine the compressibility of a fluid you will need to determine what the bulk modulus is.  The bulk modulus is found by finding the ratio of the change in pressure over the volume differential.

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

Also, since the density of a fluid will change if its volume changes, the density differential can also be used to determine the bulk modulus.

(Eq 2)  $E_v=-\frac{dP}{dρ/ρ}$

The resulting dimensions of the bulk modulus will be $FL^{-2}$.  For BG units this will be $lb/in^2$ (psi).  While for SI units it will be $N/m^2$ (Pa).  Notice that these are the same units that are used for the modulus of elasticity for a solid.

When the bulk modulus has a large value you can normally consider the fluid to be incompressible.  This is because it will take a large pressure differential to cause a relatively small change in fluid volume or density.  Liquids are normally always considered incompressible due to their large bulk modulus.  The table below shows the bulk modulus of some select liquid.  From this table you can see that a large pressure change will be required to create a volume or density change in these liquids.

 Liquid Temperature Bulk Modulus Fahrenheit Celsius psi Pa Ethyl Alcohol 68 20 1.54e5 1.06e9 Gasoline 60 15.6 1.60e5 1.30e9 Mercury 68 20 4.14e6 2.85e10 SAE 30 oil 60 15.6 2.20e5 1.50e9 Sea Water 60 15.6 3.39e5 2.34e9 Water 60 15.6 3.12e5 2.15e9

### Compression and Expansion of a Gas

The relationship between a gas’s pressure and density when the gas expands or contracts is dependent on the type of process.

##### Isothermal Process

The first process that i’m going to talk about is the isothermal process.  For this process to take place the temperature must remain constant.  This will result in the following pressure and density relationship.

(Eq 3) $\frac{P}{ρ}=constant$

##### Isentropic Process

The next process that I would like to mention is the isentropic process.  For the isentripic process to occur heat cannot be exchanged with the surroundings.  The pressure and density relationship would be almost the same as the isothermal process, except you will need to consider the ratio of the specific heat at constant pressure $c_p$ and the specific heat at constant volume $c_V$.

(Eq 4) $k=\frac{c_p}{c_V}$

Note the two specific heat values can be used to find the ideal gas constant of the gas.

(Eq 5) $R= c_p-c_V$

Finally, once the ratio between the two specific heat values is determined you can now calculate the relationship between the pressure and density.

(Eq 6) $\frac{P}{ρ^k}=constant$

##### Bulk Modulus of a Gas

Unlike a liquid, a gas is considered compressible, since a relatively low pressure change will noticeably change the volume that a gas occupies.  Due to this fact the bulk modulus will change directly with pressure.  The bulk modulus can be found by taking the derivative $\frac{dp}{dρ}$.

The resulting bulk modulus for the isothermal process will be as following.

(Eq 7) $E_v=P$

The resulting bulk modulus for the isentropic process will be as follows.

(Eq 8) $E_v=kP$

### Example

If a gas was being compressed from one state to another determine the following equations.  Write the an equation that can be used to find the final pressure of the gas if it were being compressed isothermally. Write the an equation that will determine the final pressure of the gas if it were compressed isentropically.

### Solution

Isothermal Process

$\frac{p_i}{ρ_i}=\frac{p_f}{ρ_f}$

$p_f = p_i\left(\frac{ρ_f}{ρ_i}\right)$

Isentropic Process

$\frac{p_i}{{ρ_i}^k}=\frac{p_f}{{ρ_f}^k}$

$p_f = p_i\left(\frac{ρ_f}{ρ_i}\right)^k$