Dimension of Bulk Modulus

The Bulk Modulus of a fluid ascertains how compressible it is. In other words, bulk modulus states how the density of a fluid varies when the fluid is subjected to pressure. It is defined as:

EV = \[\frac{dpd∀}{∀}\] = \[\frac{dpdp}{p}\]

In the above formula, 

Dp is the change in pressure that is required to change the volume by d∀, ∀, by being the initial volume. The negative symbol represents that an increase in pressure goes along with a decrease in volume.

The dimension of the bulk modulus is FL⁻², and the unit of the dimension of the bulk modulus is \[\frac{N}{m^{2}}\].

A high value of bulk modulus represents that it is difficult to compress the fluid. For example, let us take water, its bulk modulus is 2.15 × 10⁹ \[\frac{N}{m^{2}}\] . It implies that it requires significant pressure to change the volume of water by a small amount. Practically, the water along with other liquids is treated as being incompressible.

Bulk Modulus Measurement

The powder diffraction under applied pressure can be used to determine the bulk modulus. It is a fluid attribute that indicates the fluid's ability to change the volume under pressure.

Following are the empirical testing apparatuses' examples for the bulk modulus:

• Normal Impedance and Flow Ripple Apparatus,

• Electronic Speckle Pattern Interferometer,

• Holographic Interferometer,

• Doppler Interferometer,

• Acoustic Coupler,

• Vibration Tester.

There are numerous theoretical formulas that have been derived for calculating the bulk modulus and, despite their differences, the results show good agreement. When compared to experimentally generated values, however, there is a little noticeable difference.

Bulk Modulus: Applications

The bulk modulus is a significant consideration in hydraulic systems. Trapped gas and temperature are the two key factors that influence a fluid's bulk modulus. The bulk modulus of a fluid decreases with an increasing temperature, hence the lower the temperature of a fluid, the harder it is to compress. The bulk modulus of fluids is drastically affected by trapped gas, primarily the air. The lower the bulk modulus, the more entrained or trapped gas there is in a fluid. The only exception is when the bulk modulus of the fluid is lower than that of air, in that case, the reverse is true.

Importance of Bulk Modulus

The absolute value of a fluid's bulk modulus can have an impact on the system's performance in terms of power level, stability, position, and response time. The entrained air content and the fluid temperature are the two important elements in controlling bulk modulus.

Therefore, in short, it is more crucial than ever to pay attention to bulk modulus in light of today's requirements for higher power and response time.

Effect of the Air on the Bulk Modulus

When hydraulic fluids are used, they usually become aerated. Because air is substantially more compressible than oil, aeration has a considerable impact on bulk modulus. Estimating the effects of air in oil on compressibility and bulk modulus is discussed by George Totten.

Also, keep in mind that the solubility of air in fluids rises as pressure rises. As air is dissolved in a fluid at high pressure, bubbles occur when the pressure drops, causing cavitation.

Bulk Modulus Formula

The bulk modulus is defined as the relative change in the volume of a body obtained by a unit compressive or tensile stress substituting throughout the stress uniformly.

The bulk modulus states how a substance reacts when it has been flattened uniformly. 

Certainly, when the external forces are perpendicular to the surface, the bulk modulus is distributed uniformly over the surface of the object. This may also appear when an object is immersed in a fluid and encounters changes in volume without any change in shape.

The δ P represents the volume stress and we state this as the ratio of the magnitude of the change in the amount of force δ F to the surface area. The bulk modulus of a liquid is an estimation of its compressibility. We calculate it as the pressure required to cause a unit change in its volume.

Hence, the bulk modulus formula is given by:

K = V \[\times\] \[\frac{\Delta P}{\Delta V}\]

In the above bulk modulus formula,

K stands for Bulk Modulus

 ΔV  stands for Change in Volume

ΔP stands for Change in Pressure

V stands for Original Volume.

The unit of the bulk modulus is given by Pa. KPa and MPa are the higher units. The bulk modulus is represented by ‘K’. Its dimension is force per unit area. We express it in the unit of Newton per square meter ( \[\frac{N}{m^{2}}\] ) in the metric system.

Dimension Formula of Bulk Modulus

The dimension formula of the bulk modulus is given by,

M¹L⁻¹T⁻²

In the above dimension formula of bulk modulus

M = Mass

L = Length

T = Time.

Dimension Formula of Bulk Modulus Derivation

Bulk Modulus (k) = Bulk Stress × BulkStrain⁻¹……….(1)

As Bulk Stress = Force × Area⁻¹……..(2)

The dimension formula of force =  M¹L¹T⁻² ……(3)

The dimension formula of the area is = M⁰L²T⁰. .....(4)

Substituting the equation (3) and (4) in equation (2) we get,

Bulk Stress = M¹L¹T⁻² ×M⁰L²T⁰⁻¹ 

Hence, the dimension of bulk stress = M¹L⁻¹T⁻²........(5) 

= ΔV/V =  M⁰L⁰T⁰ = Dimensionless Quantity………(6)

Substituting the equation (5) and (6) in equation (1) we get,

Bulk Modulus = Bulk Stress × BulkStrain⁻¹

Or,

K = M¹L⁻¹T⁻² * M⁰L⁰T⁰ = M¹L⁻¹T⁻².

Hence, the bulk modulus is dimensionally represented as M¹L⁻¹T⁻².

Solved Example

1. Calculate the change in temperature if the atmospheric pressure of 0.1 MPa of metal is minimized to zero when this unit is placed in a vacuum. The bulk modulus of the material is 12000 MPa.

Solution:

Given,

Bulk Modulus (K) = 12000 MPa

Change in Temperature, δ P = 0.0 - 0.1

Atmospheric Pressure = 0.1 Mpa

Using the Bulk Modulus Formula,

K = \[\frac{(\Delta V \times \Delta P)}{\Delta V}\]

= \[\frac{\Delta V}{V}\] = -\[\frac{\Delta P}{K}\]

= \[\frac{\Delta V}{V}\] = -\[\frac{-0.1}{120000}\]

= \[\frac{\Delta V}{V}\] = 8.3 × 10⁻⁷

Hence, the fractional change in volume is 8.3 × 10⁻⁷.