Dimension of Bulk Modulus

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


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.

FAQs on Dimension of Bulk Modulus

1. Explain the relationship between Compressibility and Bulk Modulus.

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?

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. 


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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.

3. Distinguish between the bulk modulus and an elastic modulus.

The bulk modulus is a measurement of a solid's incompressibility; the higher the bulk modulus, the more force is required to compress the solid by a certain amount. While Elastic modulus measures the resistance to stretching of the material and the material with higher elastic modulus is harder to strain. The bulk modulus is linked to dilatational strain, which happens without a change in shape but with a change in volume, therefore yielding does not occur in this condition of stress. An elastic modulus includes a distortional part which can cause yielding.

4. What is the effect of a temperature on the bulk modulus?

The ratio of the infinitesimal pressure rise to the consequent proportional reduction in volume is the bulk modulus of a substance. And when the temperature of the material increases, the volume of material increase as given:

B = -F \[\frac{V}{A}\] dV where, A is the area, F is force and V is volume


The volume of the molecules expands as they absorb the energy and become kinetically charged, raising the bulk modulus value. As a result, we can see how temperature can affect bulk modulus.

5. Why do liquids have a bulk modulus?

Both the liquids and gases known as the fluids have a bulk modulus but no shear modulus. And having no shear stress means that even an arbitrarily small shear force can cause a fluid to flow. The bulk modulus of a liquid is proportional to its compressibility and is defined as the pressure needed to alter the volume of a liquid by one unit. Since liquids are practically incompressible, they require very large pressures to cause any significant volume change.

6. What are the materials with the highest bulk modulus?

The bulk modulus scales with the valence electron density in most of the cases. It is commonly known that under very high pressures the elements tend toward the same bulk modulus value. And this can also be explained in the terms of the valence electron density in that the hydrostatic pressure bearing down on these elements raises the valence electron density. The further compression will be hard as the electron density rises, resulting in a bulk modulus limiting value.

7. What is the bulk modulus for a perfectly elastic rubber?

The bulk modulus of perfectly elastic rubber is infinite. And it means that the perfectly elastic rubber is incompressible. And an incompressible means the volume does not change during the distortion. So,


Bulk modulus = change in pressure/volumetric strain


Also,


Bulk modulus = Young's modulus/3(1–2*poisson's ratio)


It means that if the Poisson's ratio is 0.5, which is for perfectly elastic rubber, then the bulk modulus is infinity (if we put 0.5 in place of the Poisson's ratio in the 2nd formulae given above). To learn more, Students can visit Vedantu and can also download the study materials in a PDF format for free.

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