Question

A 1-in-diameter Stefan tube is used to measure the binary diffusion coefficient of water vapor in air at \(80^{\circ} \mathrm{F}\) and \(13.8\) psia. The tube is partially filled with water with a distance from the water surface to the open end of the tube of \(10 \mathrm{in}\). Dry air is blown over the open end of the tube so that water vapor rising to the top is removed immediately and the concentration of vapor at the top of the tube is zero. During 10 days of continuous operation at constant pressure and temperature, the amount of water that has evaporated is measured to be \(0.0025 \mathrm{lbm}\). Determine the diffusion coefficient of water vapor in air at \(80^{\circ} \mathrm{F}\) and \(13.8 \mathrm{psia}\).

Short answer

The diffusion coefficient of water vapor in air at 80°F and 13.8 psia is approximately 7.21 x 10^-5 ft^2/s.

Step-by-step solution

1. Conversion of units

Start with converting the given data to appropriate units: 1. Temperature: Convert \(80^{\circ} \mathrm{F}\) to Rankine (R): $$ T (\mathrm{R}) = 80 + 459.67 = 539.67\,\mathrm{R} $$ 2. Pressure: Given in absolute units, no conversion needed. 3. Convert length measurements to meters: $$ 1 \,\mathrm{in} = 0.0254 \,\mathrm{m} $$ Now we have: - Diameter of the Stefan tube: \(1\times 0.0254=0.0254\,\mathrm{m}\) - Distance from the water surface to the open end of the tube: \(10\times 0.0254=0.254\,\mathrm{m}\)

2. Calculate the mass diffusion rate based on evaporated water

Given that the amount of evaporated water during 10 days is \(0.0025\,\mathrm{lbm}\). To find the mass diffusion rate, we need to divide the mass of evaporated water by the time elapsed: $$ \text{Mass diffusion rate} = \frac{0.0025\,\mathrm{lbm}}{10\,\text{days}\times 24\,\text{hours/day}\times 3600\,\text{seconds/hour}} $$ Let's calculate the mass diffusion rate: $$ \text{Mass diffusion rate} = \frac{0.0025}{10\times 24\times 3600} = 2.89\times10^{-6}\,\mathrm{lbm/s} $$

3. Apply Stefan's Law of diffusion

Using the Stefan's Law of diffusion, the mass diffusion rate can be expressed as: $$ \text{Mass diffusion rate}=\frac{D_{AB}A\Delta\rho}{L}, $$ where \(D_{AB}\) is the diffusion coefficient we want to determine, \(A\) is the cross-sectional area of the tube, \(\Delta\rho\) is the difference of water vapor density between the surface and the top of the tube, and \(L\) is the distance from the water surface to the open end of the tube. The cross-sectional area of the tube can be obtained from the given diameter: $$ A = \frac{\pi d^2}{4} = \frac{\pi (0.0254)^2}{4}= 5.067\times10^{-4}\,\mathrm{m^2} $$ The density of water vapor, \(\rho_{H_2O}\), at the given temperature and pressure can be found in engineering tables. For the given temperature (\(80^{\circ} \mathrm{F}\)) and pressure (\(13.8\,\mathrm{psia}\)), we have \(\rho_{H_2O} = 0.00206\,\mathrm{lbm/ft^3}\). The difference between the water vapor density at the surface and at the top of the tube is \(\Delta\rho=\rho_{H_2O}-0\), as the concentration at the top of the tube is zero. Now we can rewrite Stefan's law: $$ 2.89\times10^{-6}\,\mathrm{lbm/s} = \frac{D_{AB}(5.067\times10^{-4}\,\mathrm{m^2 })(0.00206\,\mathrm{lbm/ft^3})}{0.254\,\mathrm{m}} $$

4. Calculate the diffusion coefficient

Solve the equation for \(D_{AB}\): $$ D_{AB} = \frac{2.89\times10^{-6}\,\mathrm{lbm/s} \times 0.254\,\mathrm{m}}{(5.067\times10^{-4}\,\mathrm{m^2})(0.00206\,\mathrm{lbm/ft^3})} $$ $$ D_{AB} = 7.21\times10^{-5}\,\mathrm{ft^2/s} $$ The diffusion coefficient of water vapor in air at \(80^{\circ}\mathrm{F}\) and \(13.8\,\mathrm{psia}\) is \(7.21\times 10^{-5}\,\mathrm{ft^2/s}\).

Key concepts

Introduction to the Stefan Tube Method

The Stefan tube method is a useful experimental technique for measuring the diffusion coefficients of gases in binary mixtures, such as water vapor in air. In this method, a tube, commonly referred to as a Stefan tube, is partially filled with a liquid (water in this case) and the other end is left open to the atmosphere. Dry air is blown over the open end to maintain a zero concentration of the diffusing substance, ensuring unidirectional diffusion from the liquid surface to the opening.

It's important to maintain constant temperature and pressure throughout the experiment as these factors can significantly impact the diffusion rate. By measuring the amount of substance that evaporates over a known period, and with the geometry of the tube known, one can use the laws governing diffusion to calculate the diffusion coefficient. This method is named after the Slovene physicist Jožef Stefan, known for his work on evaporation and related phenomena.

Binary Diffusion

Binary diffusion occurs when two different types of molecules, or species, mix through random molecular motion. In the context of the Stefan tube experimentation, water vapor and air molecules are the two components involved in the diffusion process. Many industrial and natural processes involve binary diffusion, such as the drying of crops, the evaporation of water from ponds, and even the distribution of pollutants in the atmosphere.

Understanding binary diffusion is pivotal because the behavior of the two species and how they interact in different conditions can significantly affect the rate of diffusion. Specific factors include temperature, pressure, the concentration gradient, and the unique properties of the diffusing substances.

Mass Diffusion Rate

Mass diffusion rate is a fundamental parameter that describes the quantity of mass transferring from one region to another per unit time due to diffusion. In our Stefan tube example, it represents how much water vapor moves through a cross-sectional area of the tube each second.

Mathematically, the mass diffusion rate can be determined by dividing the total mass exchanged by the time over which the exchange occurs. It is critical to use consistent and appropriate units when calculating this rate to ensure the accurate application of Stefan's law. Understanding the mass diffusion rate is essential in various engineering and scientific disciplines, including chemical engineering, environmental studies, and materials science.

Stefan's Law of Diffusion

Stefan's Law of diffusion, not to be confused with Stefan-Boltzmann's Law from thermodynamics, is a physical law that relates the mass diffusion rate to the concentration gradient over a certain distance. In the context of the Stefan tube method, it provides a means to compute the diffusion coefficient, which is a measure of how easily molecules of one substance travel through another substance.

Stefan's equation for diffusion is represented as \( \text{Mass diffusion rate}=\frac{D_{AB}A\Delta\rho}{L} \), where \(D_{AB}\) is the diffusion coefficient, \(A\) is the cross-sectional area through which diffusion occurs, \(\Delta\rho\) is the concentration difference, and \(L\) is the diffusive path length. This law enables engineers and scientists to predict diffusion behavior and design systems that either enhance or retard diffusion, depending on the desired outcome.