Compare The Hydrostatic Pressure At The Bottom Of A Milk Bot
Compare the hydrostatic pressure at the bottom of a milk bottle before and after cream separation
In the past, milk was delivered in bottles, and the cream would naturally separate and rise to the top due to differences in density, resulting in a layered composition. This process of phase separation impacts the pressure exerted at the bottom of the bottle, a critical factor in understanding fluid statics. The primary objective of this analysis is to determine whether the hydrostatic pressure at the bottom of the bottle remains constant during the phase separation of the cream and to quantify the change in pressure, if any, after the cream has separated.
Initially, the milk in the bottle is well-mixed, and the hydrostatic pressure at its bottom results from the combined weight of the entire fluid column. Over time, as the cream separates and floats atop the milk, the distribution of mass within the bottle changes, which could influence the pressure. To analyze this, key parameters provided include the volumes and densities of the components, alongside the geometric dimensions of the bottle.
Understanding the physical parameters
The given data are as follows:
- Volume of cream, \(V_c = 4.6 \times 10^{-6} \, \mathrm{m}^3\)
- Volume of milk (minus cream), \(V_m = 4.554 \times 10^{-4} \, \mathrm{m}^3\)
- Density of cream, \(\rho_c = 994 \, \mathrm{kg/m}^3\)
- Density of milk, \(\rho_m = 1033 \, \mathrm{kg/m}^3\)
- Diameter of the top section (where cream separates), \(d_t = 26 \, \mathrm{mm} = 0.026 \, \mathrm{m}\)
- Average diameter of the rest of the bottle, \(d_b = 75 \, \mathrm{mm} = 0.075 \, \mathrm{m}\)
Calculating the hydrostatic pressure before separation
When the milk is uniformly mixed, the total volume of the fluid, \(V_{total}\), is the sum of the volumes of cream and milk:
\[
V_{total} = V_c + V_m = 4.6 \times 10^{-6} + 4.554 \times 10^{-4} \approx 4.600 \times 10^{-6} + 4.554 \times 10^{-4} \approx 4.6 \times 10^{-6} + 4.554 \times 10^{-4}
\]
Numerically, this sum is approximately \(4.5986 \times 10^{-4} \, \mathrm{m}^3\).
The hydrostatic pressure at the bottom of the bottle when the mixture is uniform is given by:
\[
P_{initial} = \rho_{avg} \, g \, h_{avg}
\]
where \(\rho_{avg}\) is the average density of the mixture, \(g = 9.81 \, \mathrm{m/s^2}\) is acceleration due to gravity, and \(h_{avg}\) is the average height of the fluid column.
Calculating \(\rho_{avg}\):
\[
\rho_{avg} = \frac{\rho_c V_c + \rho_m V_m}{V_c + V_m}
\]
Plugging in the values:
\[
\rho_{avg} = \frac{994 \times 4.6 \times 10^{-6} + 1033 \times 4.554 \times 10^{-4}}{4.6 \times 10^{-6} + 4.554 \times 10^{-4}}
\]
Compute numerator:
\[
994 \times 4.6 \times 10^{-6} \approx 4.5724 \times 10^{-3}
\]
\[
1033 \times 4.554 \times 10^{-4} \approx 0.470 \, \mathrm{kg}
\]
Sum of numerator:
\[
4.5724 \times 10^{-3} + 0.470 \approx 0.4746 \, \mathrm{kg}
\]
Total volume in denominator:
\[
4.6 \times 10^{-6} + 4.554 \times 10^{-4} \approx 4.5986 \times 10^{-4} \, \mathrm{m}^3
\]
Thus:
\[
\rho_{avg} \approx \frac{0.4746}{4.5986 \times 10^{-4}} \approx 1032 \, \mathrm{kg/m}^3
\]
Using this, and assuming the entire fluid height, \(h_{total}\), is approximately the height of the bottle:
The total volume \(V_{total}\) relates to height via the cross-sectional area of the main section:
\[
A_b = \pi \left(\frac{d_b}{2}\right)^2 = \pi \times (0.0375)^2 \approx 4.42 \times 10^{-3} \, \mathrm{m}^2
\]
Height of the entire fluid column:
\[
h_{total} = \frac{V_{total}}{A_b} \approx \frac{4.5986 \times 10^{-4}}{4.42 \times 10^{-3}} \approx 0.104 \, \mathrm{m}
\]
Thus, initial hydrostatic pressure:
\[
P_{initial} \approx 1032 \times 9.81 \times 0.104 \approx 1051 \, \mathrm{Pa}
\]
Calculating the hydrostatic pressure after phase separation
After the cream rises and separates, the fluid composition changes. The cream is now at the top, occupying a volume \(V_c = 4.6 \times 10^{-6}\, \mathrm{m}^3\), and the milk (minus cream) occupies the lower part, with volume \(V_m = 4.554 \times 10^{-4}\, \mathrm{m}^3\).
The height of the cream layer, \(h_c\), based on its volume and cross-sectional area at the top:
\[
h_c = \frac{V_c}{A_t}
\]
Area at the top:
\[
A_t = \pi \left(\frac{d_t}{2}\right)^2 = \pi \times (0.013)^2 \approx 5.33 \times 10^{-4} \, \mathrm{m}^2
\]
Calculate \(h_c\):
\[
h_c = \frac{4.6 \times 10^{-6}}{5.33 \times 10^{-4}} \approx 0.00863 \, \mathrm{m}
\]
Similarly, height of the milk layer, \(h_m\), is:
\[
h_m = \frac{V_m}{A_b} = \frac{4.554 \times 10^{-4}}{4.42 \times 10^{-3}} \approx 0.103 \, \mathrm{m}
\]
Total height of the fluid is approximately:
\[
h_{total, separated} = h_c + h_m \approx 0.00863 + 0.103 \approx 0.112 \, \mathrm{m}
\]
The pressure at the bottom of the bottle after separation is the sum of the weight of the cream and milk layers above the bottom point.
Using hydrostatic pressure formula:
\[
P_{final} = \rho_c g h_c + \rho_m g h_m
\]
Calculating each component:
\[
\rho_c g h_c = 994 \times 9.81 \times 0.00863 \approx 84.4 \, \mathrm{Pa}
\]
\[
\rho_m g h_m = 1033 \times 9.81 \times 0.103 \approx 1044 \, \mathrm{Pa}
\]
Sum of both contributions:
\[
P_{final} \approx 84.4 + 1044 \approx 1128.4 \, \mathrm{Pa}
\]
Assessing the change in pressure
The initial hydrostatic pressure was approximately 1051 Pa, whereas after the cream separates and rises to the top, the pressure increases to roughly 1128.4 Pa. Therefore, the change in hydrostatic pressure at the bottom of the bottle is:
\[
\Delta P = P_{final} - P_{initial} \approx 1128.4 - 1051 \approx 77.4 \, \mathrm{Pa}
\]
This indicates that the pressure at the bottom of the bottle increases after phase separation due to the redistribution of mass within the fluid column. The decreasing height of the overall fluid column does not compensate for the increased weight of the denser milk layer below the cream, leading to a net increase in hydrostatic pressure.
Conclusion
In summary, the hydrostatic pressure at the bottom of the milk bottle does not remain constant during the phase separation of the cream. Instead, it increases by approximately 77.4 Pa due to the change in the distribution and densities of the layers within the bottle. This analysis highlights the importance of understanding fluid stratification and density variations when assessing hydrostatic forces in real-world scenarios.
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