Dimension of Bulk Modulus

What is the Bulk Modulus(Eᵥ)?

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 = \[-\frac{dp}{d∀ / ∀}\] = \[\frac{dp}{dp / p}\]

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 N/m².

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 N/m². 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 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 ×  ΔP /  ΔV

In the above bulk modulus formula,

K stands for Bulk Modulus

Δ V  stands for Change in Volume

Δ P 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 (N/m²) in the metric system.


Dimension Formula of Bulk Modulus

The dimension formula of the bulk modulus is given by,

[M¹ L⁻¹ T⁻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 × [ Bulk Strain]⁻¹……….(1)

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

The dimension formula of force =  [M¹ L¹ T⁻²] ……(3)

The dimension formula of the area is = [M⁰ L² T ⁰]. .....(4)

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

Bulk Stress =  [M¹ L¹ T⁻²] × [M⁰ L² T ⁰]⁻¹ 

Hence, the dimension of bulk stress = [M¹ L⁻¹ T⁻²]........(5) 

= Δ V/V =  [M⁰ L⁰ T ⁰] = Dimensionless Quantity………(6)

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

Bulk Modulus = Bulk Stress × [ Bulk Strain]⁻¹

Or,

K = [M¹ L⁻¹ T⁻²] * [M⁰ L⁰ T ⁰] = [M¹ L⁻¹ T⁻²].

Hence, the bulk modulus is dimensionally represented as [M¹ L⁻¹ T⁻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 = (ΔV × ΔP) /ΔV)

= Δ V/ V = -Δ P / K

= Δ V/ V = -0.1/120000

= Δ V/ V = 8.3 × 10⁻⁷

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

FAQs (Frequently Asked Questions)

1. Explain the relationship between Compressibility and Bulk Modulus.

Answer: The relationship between compressibility and bulk modulus states that the bulk modulus is the inverse of compressibility.

  • Bulk modulus is defined as the ratio between the increase in pressure and decrease in the volume of a material. It is represented by the letter K.

  • Compressibility is stated as the ratio of change in volume to change in pressure. It is represented by letter B. In the case of fluids, compressibility relies on either the adiabatic process or the thermal process.

2. Why is Bulk Modulus important?

Answer: We can state that the absolute value of the bulk modulus of fluid can seriously affect the system performance with reference to power level, position, stability, and response time.

Two predominant factors of bulk modulus are fluid temperature and entrained air content. 

[Image will be uploaded soon]

For example, the above table shows that increasing the temperature of commercial hydraulic fluid by 100°F alone reduces its bulk modulus by 61% of its room temperature value. 

The above table also denotes that the bulk modulus reduces to 55% of its room temperature by introducing 1% of air by volume. If these two situations occur simultaneously,  the net effect is to minimize the bulk modulus by 67%.

Considering present needs, the requirements for high power and response time is more important than ever to focus on bulk modulus.