Problem 65 - College Physics

Question

Crude oil with a bulk modulus of \(2.35 \mathrm{GPa}\) is leaking from a deep- sea well \(2250 \mathrm{~m}\) below the surface of the ocean, where the water pressure is \(2.27 \times 10^{7} \mathrm{~Pa}\). Suppose 35,600 barrels of oil leak from the wellhead; assuming all that oil reaches the surface, how many barrels will it be on the surface?

Step-by-Step Solution

Answer

Approximately 35,945 barrels at the surface.

Step 1: Understand the Given Values

We are given the following values:1. Bulk Modulus of crude oil \(K = 2.35 \times 10^9 \text{ Pa}\).2. Initial pressure at depth \(P_i = 2.27 \times 10^7 \text{ Pa}\).3. Change in pressure \(\Delta P = -2.27 \times 10^7 \text{ Pa}\) (since the oil reaches the surface where pressure is essentially zero).4. Initial volume \(V_i = 35,600\) barrels.

Step 2: Use Bulk Modulus to Find Volume Change

The bulk modulus formula is given by:\[K = - \frac{\Delta P}{\Delta V / V_i}\]This can be rearranged to find the change in volume:\[\Delta V = -\frac{\Delta P \times V_i}{K}\].

Step 3: Calculate \(\Delta V\)

Substitute the given values into the formula:\[\Delta V = -\frac{(-2.27 \times 10^7 \text{ Pa}) \times 35,600 \text{ barrels}}{2.35 \times 10^9 \text{ Pa}}\]Calculate \(\Delta V\):\[\Delta V \approx 345.11 \text{ barrels}\].

Step 4: Compute Surface Volume

The volume at the surface, \(V_s\), is given by:\[V_s = V_i + \Delta V\]Substitute the values:\[V_s = 35,600 + 345.11 = 35,945.11 \text{ barrels}\].

Step 5: Summary and Conclusion

The initial volume of crude oil was 35,600 barrels. Due to decompression, the volume expands by approximately 345.11 barrels upon reaching the surface. Therefore, the total volume at the surface becomes 35,945.11 barrels.

Key Concepts

Bulk Modulus

In fluid mechanics, the bulk modulus (K) of a substance is a measure of its resistance to uniform compression. It is a fundamental property that correlates pressure changes with volume changes in materials. The bulk modulus is expressed in Pascals (Pa), and it's defined by the equation: \[ K = - \frac{\Delta P}{\Delta V / V_i} \] where \( \Delta P \)is the change in pressure, \( \Delta V \)is the change in volume, and \( V_i \)is the original volume.
This formula indicates that for any given substance, as the pressure increases, the volume tends to decrease, provided the substance has a positive bulk modulus. However, the magnitude of this decrease is directly related to the value of the bulk modulus. A higher bulk modulus means the substance is less compressible.
Understanding bulk modulus helps in analyzing the behavior of fluids under pressure, which is essential in fields like underwater engineering, where pressure differences are significant.

Pressure Changes

Pressure changes in fluids are a critical aspect of fluid mechanics. Pressure is the force exerted per unit area and, in a liquid at depth, it is often considerably higher than at the surface due to the weight of the fluid above. In our exercise, the pressure at the depth of 2250 meters is \(2.27 \times 10^7 \text{ Pa}\).
When oil leaks from such a deep-sea well and rises to the surface, the pressure exerted on it decreases significantly as the surrounding water pressure decreases. This change in pressure is \(\Delta P = -2.27 \times 10^7 \text{ Pa} \). Since atmospheric pressure at the surface is much less than deep-sea pressure, it is assumed to be negligible in this context.
Thus, understanding how the pressure decreases allows us to predict changes in the fluid's volume, as seen with the crude oil expansion on reaching the surface.

Volume Expansion

Volume expansion refers to the increase in volume that a fluid undergoes when the pressure on it is reduced. In fluids, this is often observed when they are subjected to substantial changes in pressure, such as moving from deep underwater to the surface.
Using the relationship provided by the bulk modulus, the volume expansion of the crude oil can be calculated: \[ \Delta V = -\frac{\Delta P \times V_i}{K} \]. For this problem, we calculated \(\Delta V\) to be approximately 345.11 barrels, indicating the oil expands due to the pressure decrease as it rises.
This expansion needs to be considered when assessing potential storage and environmental impact, especially in the case of oil spills, where the initial and final volumes significantly differ.

Oil Leakage

Oil leakage, particularly from a deep-sea well, involves numerous challenges. These include understanding fluid properties like bulk modulus, as well as managing the considerable pressure differential from deep sea to surface level.
When oil leaks from a deep well, its volume at the surface can be significantly larger than at the depth it was initially stored. This increase is due to reduced pressure, resulting in volumetric expansion. Therefore, while 35,600 barrels may leak at depth, at the surface, this becomes about 35,945.11 barrels due to expansion.
Understanding these changes can help in planning for containment and recovery efforts, which are crucial during environmental assessments and in minimizing potential damage from such leaks.