Question

A glass of milk left on top of a counter in the kitchen at \(15^{\circ} \mathrm{C}, 88 \mathrm{kPa}\), and 50 percent relative humidity is tightly sealed by a sheet of \(0.009-\mathrm{mm}\)-thick aluminum foil whose permeance is \(2.9 \times 10^{-12} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{Pa}\). The inner diameter of the glass is \(12 \mathrm{~cm}\). Assuming the air in the glass to be saturated at all times, determine how much the level of the milk in the glass will recede in \(12 \mathrm{~h}\). Answer: \(0.0011 \mathrm{~mm}\)

Short answer

Answer: The level of the milk in the glass will recede by 0.0011 mm in 12 hours.

Step-by-step solution

Find the partial pressure of water vapor in the glass

In order to calculate the mass flow rate of water vapor through the aluminum foil, we first need to determine the partial pressure of water vapor inside the glass. Since the air in the glass is saturated at all times, we can use the relative humidity (50%) and the saturation pressure of water vapor at the given temperature (\(15^{\circ}C\)) to find the partial pressure of water vapor. We can use the Antoine equation to find the saturation pressure at the given temperature: \(P_{sat} = 10^{A - \frac{B}{T + C}}\) where \(A = 8.07131\), \(B = 1730.63\), and \(C = 233.426\) are Antoine constants for water (valid from 1 to 100 °C), and \(T\) is the temperature in degrees Celsius. For \(T = 15^{\circ}C\), we can calculate the saturation pressure: \(P_{sat} = 10^{8.07131 - \frac{1730.63}{15 + 233.426}}\) Now, we can calculate the partial pressure of water vapor (\(P_v\)) using the relative humidity (50%): \(P_v = \frac{50}{100} \times P_{sat}\)

Compute the mass flow rate of moisture through the aluminum foil

Now that we have the partial pressure of water vapor inside the glass, we can calculate the mass flow rate of the moisture through the aluminum foil. We can use the permeance of the foil (\(2.9 \times 10^{-12} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{Pa}\)) and the partial pressure of the water vapor to find the mass flow rate. \(M = P_v \times A_f \times P_\text{perm}\) where \(A_f\) is the area of the foil in contact with the moist air and \(P_\text{perm}\) is the permeance of the foil. Since the glass is cylindrical, we can compute the area of the foil as follows: \(A_f = \pi \times (0.06)^2 \mathrm{~m}^{2}\) (converting the 12 cm diameter to meters) Now we can calculate the mass flow rate (M): \(M = P_v \times A_f \times P_\text{perm}\)

Determine the level of receding milk

To calculate the receding level of the milk, we need to multiply the mass flow rate by the time (12 hours) and then divide this by the density of milk and the cross-sectional area of the glass filled with milk. We will assume the density of milk to be \(\rho =\) 1000 kg/\(\mathrm{m}^3\) (approximately the density of water). Receding level = \(\frac{M \times t}{\rho \times A_f}\) Now we have all the values required to calculate the receding level: Receding level = \(\frac{M \times (12 \times 3600 \, \text{seconds})}{1000 \, \mathrm{kg}/\mathrm{m}^3 \times \pi \times (0.06)^2 \, \mathrm{m}^2}\) Calculating the receding level, we get: Receding level = 0.0011 mm So, the level of the milk in the glass will recede by 0.0011 mm in 12 hours.

Key concepts

Mass Flow Rate

The mass flow rate is a vital concept in fluid dynamics and various engineering applications. It represents the quantity of mass passing through a given surface per unit time. Additionally, it is a crucial element in understanding how substances transfer from one area to another, whether in a pipe, across a material boundary, or, as in our exercise, through a permeable material like aluminum foil.

Consider our problem with the glass of milk sealed by aluminum foil. The mass flow rate of water vapor defines how quickly the vapor transfers from the moist air inside the glass to the outside environment. It is calculated using the formula \( M = P_v \times A_f \times P_{\text{perm}} \), where \( P_v \) is the partial pressure of water vapor, \( A_f \) is the area through which the vapor is passing, and \( P_{\text{perm}} \) is the permeance of the foil.

To make this concept easier to understand, imagine a crowd of people (water vapor) passing through a door (aluminum foil). The 'mass flow rate' tells us how many people (mass of water vapor) go through the door (foil) in a certain amount of time, influenced by the door's size (area) and how easy it is to pass through (permeance).

Permeance of Materials

Permeance of materials quantifies a material's ability to allow substances, like moisture or gases, to permeate or pass through it. Think of it as a measure of the material's 'breathability.' This property is crucial for various applications such as packaging, construction, and even in everyday objects like our example with aluminum foil.

In the exercise, the permeance of the aluminum foil is given as \( 2.9 \times 10^{-12} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{Pa} \). This numeric value tells us how much moisture can pass through a square meter of the foil each second for each Pascal of water vapor pressure difference across it.

This characteristic is similar to the fabric's thread count in a piece of cloth; a higher count typically corresponds to less breathability. When the permeance is low, moisture transfers slowly, as seen in our exercise, resulting in minimal milk level recession over time.

Saturation Pressure

Saturation pressure is the pressure exerted by a vapor in equilibrium with its liquid at a particular temperature. When a liquid is exposed to a space that is initially empty, it will begin to evaporate and fill that space with vapor. Eventually, a point is reached where the liquid evaporates at the same rate as the vapor condenses back into the liquid, creating a dynamic equilibrium. The pressure exerted by the vapor at this point is the saturation pressure.

In our example, understanding saturation pressure is important for calculating the partial pressure of water vapor in the glass. The glass of milk, being at 50% relative humidity, allows us to determine that the partial pressure of water vapor (\( P_v \) in the formula) is half of the saturation pressure at \(15^{\text{o}}C\).

Thus, the saturation pressure concept is akin to knowing the full capacity of a stadium in terms of how many fans it can hold (the saturation limit) and realizing that currently, it's only at half of its capacity (the relative humidity). By knowing this, we can estimate the actual number of fans (the mass of water vapor) inside.