Problem Statement
Assuming that the bulk modulus of elasticity of water is \(2.07 \times 10^6 \, \text{kN/m}^2\) at standard atmospheric conditions, determine the increase of pressure necessary to produce a 1% reduction in volume at the same temperature.
Given Data
- Bulk Modulus, \(K = 2.07 \times 10^6 \, \text{kN/m}^2\)
- Volume Reduction = 1%
Solution
1. Define Bulk Modulus Formula
The bulk modulus of elasticity (\(K\)) relates the change in pressure (\(dp\)) to the fractional change in volume (\(-\frac{dV}{V}\)). The negative sign indicates that as pressure increases, volume decreases.
2. Determine the Fractional Volume Change
A 1% reduction in volume means the change in volume (\(dV\)) relative to the original volume (\(V\)) is -1/100.
3. Rearrange the Formula and Calculate Pressure Increase (\(dp\))
We rearrange the formula to solve for the increase in pressure, \(dp\).
Now, substitute the given values into the formula.
This can also be expressed using scientific notation.
The increase in pressure necessary is \( dp = 2.07 \times 10^4 \, \text{kN/m}^2 \).
Explanation of Bulk Modulus
Bulk Modulus (\(K\)) is a measure of a substance’s resistance to uniform compression. It describes how much pressure is needed to cause a given fractional decrease in volume. A material with a high bulk modulus (like water or steel) is difficult to compress, while a material with a low bulk modulus (like air) is easily compressed.
The formula essentially states that the required pressure increase (\(dp\)) is directly proportional to the material’s inherent resistance to compression (\(K\)) and the desired fractional change in volume (\(-\frac{dV}{V}\)).
Physical Meaning
The result, \(2.07 \times 10^4 \, \text{kN/m}^2\), is the immense pressure required to compress water by just 1%. This is equivalent to 20,700,000 Pascals (Pa) or about 204 times standard atmospheric pressure.
This calculation highlights why liquids like water are often treated as incompressible in many fluid dynamics problems. While they can be compressed, the pressure needed to do so is enormous. This property is fundamental to hydraulics, where pressure applied to an enclosed fluid is transmitted throughout, allowing for the multiplication of force.
