Problem Statement
Determine the bulk modulus of elasticity of a fluid which is compressed in a cylinder from a volume of 0.009 m³ at 70 N/cm² pressure to a volume of 0.0085 m³ at 270 N/cm² pressure.
Given Data
- Initial Volume, \(V_1 = 0.009 \, \text{m}^3\)
- Initial Pressure, \(P_1 = 70 \, \text{N/cm}^2\)
- Final Volume, \(V_2 = 0.0085 \, \text{m}^3\)
- Final Pressure, \(P_2 = 270 \, \text{N/cm}^2\)
Solution
1. Calculate the Change in Pressure (dP) and Volume (dV)
First, we find the difference between the final and initial states for both pressure and volume.
2. Apply the Bulk Modulus Formula
The bulk modulus of elasticity (K) is defined as the ratio of the change in pressure to the fractional change in volume. The negative sign indicates that as pressure increases, volume decreases.
3. Convert Result to Standard SI Units (N/m²)
For consistency with standard engineering and physics notation, we convert the final result to Pascals (N/m²).
The bulk modulus of elasticity of the fluid is \( K = 3.6 \times 10^7 \, \text{N/m}^2 \) (or 36 MPa).
Explanation of Bulk Modulus
The Bulk Modulus of Elasticity (K) is a measure of a substance’s resistance to being compressed. It quantifies how much pressure is needed to cause a given fractional decrease in volume. A higher bulk modulus indicates that a fluid is less compressible.
- High K value: The fluid is difficult to compress (like water or oil). A large change in pressure is required to produce a small change in volume.
- Low K value: The fluid is easy tocompress (like a gas). A small change in pressure can produce a large change in volume.
Physical Meaning
The calculated bulk modulus of 3.6 x 10⁷ N/m² (or 36 MPa) tells us the inherent stiffness of the fluid. It means that for every 36 million Pascals of pressure applied, the fluid’s volume will decrease by 100% (if it were perfectly linear, which it isn’t over large ranges).
This value is relatively low for a liquid, suggesting it is more compressible than typical hydraulic fluids or water (which has a bulk modulus of about 2.2 x 10⁹ N/m²). This property is critical in applications like:
- Hydraulic Systems: In hydraulics, a high bulk modulus is essential. If the fluid compresses too much, the system will feel “spongy” and will not transmit force efficiently.
- Acoustics: The speed of sound in a fluid is directly related to its bulk modulus and density. This calculation is a step towards determining how fast sound would travel through this particular fluid.
