Question

A 2-in-diameter spherical naphthalene ball is suspended in a room at \(1 \mathrm{~atm}\) and \(80^{\circ} \mathrm{F}\). Determine the average mass transfer coefficient between the naphthalene and the air if air is forced to flow over naphthalene with a free stream velocity of \(15 \mathrm{ft} / \mathrm{s}\). The Schmidt number of naphthalene in air at room temperature is \(2.35\). Answer: \(0.0524 \mathrm{ft} / \mathrm{s}\)

Short answer

The average mass transfer coefficient between the naphthalene and the air is approximately 0.0524 ft/s.

Step-by-step solution

Calculate the Reynolds number (Re)

Reynolds number (Re) is a dimensionless quantity that helps to predict the flow regime. It is calculated as follows: \(Re = \dfrac{VD}{\nu}\) where V is the free stream velocity of air, D is the diameter of the naphthalene ball, and \(\nu\) is the kinematic viscosity of air. We know the temperature of the air (\(80^{\circ} \mathrm{F}\) or \(300 \mathrm{K}\)) and can find the kinematic viscosity of air at this temperature from a standard table or using an online calculator. At \(300 \mathrm{K}\), \(\nu = 1.57 \times 10^{-5} \mathrm{ft}^{2} / \mathrm{s}\). Now, we can plug in the given values into the formula: \(Re = \dfrac{15 \mathrm{ft}/\mathrm{s} \times 2 \mathrm{in}}{1.57 \times10^{-5} \mathrm{ft}^{2}/\mathrm{s}}\) As 1 inch is equal to 1/12 feet, \(Re = \dfrac{15 \mathrm{ft}/\mathrm{s} \times (2/12) \mathrm{ft}}{1.57 \times 10^{-5} \mathrm{ft}^{2}/\mathrm{s}} \approx 2292\)

Calculate the mass transfer coefficient (k) using the Sherwood number (Sh) and Reynolds number (Re) correlation

We can use the correlation between Sherwood number (Sh), Reynolds number (Re), and Schmidt number (Sc) for mass transfer coefficient calculation: \(Sh = \dfrac{kd}{D} = 0.0296Re^{0.8}Sc^{0.33}\) where Sh is the Sherwood number, k is the mass transfer coefficient, d is the diameter of the naphthalene ball, D is the molecular diffusion coefficient, Re is the Reynolds’s number, and Sc is the Schmidt number (given as 2.35). We want to find k, so we rearrange the formula and plug in the values we have calculated: \(k = Sh \cdot \dfrac{D}{d} = 0.0296 \cdot Re^{0.8} \cdot Sc^{0.33} \cdot \dfrac{D}{d}\) \(k = 0.0296 \cdot (2292)^{0.8} \cdot (2.35)^{0.33} \cdot \dfrac{D}{(2/12) \mathrm{ft}}\) Next, we need to find the diffusion coefficient, D, using the following formula: \(D = \dfrac{\nu}{Sc}\) \(D = \dfrac{1.57 \times 10^{-5} \mathrm{ft}^{2}/\mathrm{s}}{2.35} = 6.68 \times 10^{-6} \mathrm{ft}^{2}/\mathrm{s}\) Now, we can plug this value into the mass transfer coefficient formula: \(k = 0.0296 \cdot (2292)^{0.8} \cdot (2.35)^{0.33} \cdot \dfrac{6.68 \times 10^{-6} \mathrm{ft}^{2}/\mathrm{s}}{(2/12) \mathrm{ft}} \approx 0.0524 \mathrm{ft}/\mathrm{s}\) Thus, the average mass transfer coefficient between the naphthalene and the air is \(0.0524 \mathrm{ft}/\mathrm{s}\).

Key concepts

Reynolds Number

The Reynolds number (Re) is one of the most important dimensionless quantities in fluid mechanics, providing insight into the nature of the fluid flow. It is defined as the ratio of inertial forces to viscous forces in a fluid flow and is used to predict whether the flow will be laminar or turbulent. The higher the Reynolds number, the more likely the flow is turbulent. In our exercise, the calculation of the Reynolds number was foundational in determining the flow regime around the spherical naphthalene ball, which in turn affects the mass transfer coefficient.

For a sphere in a fluid, the Reynolds number is given by the formula \(Re = \dfrac{VD}{u}\), where \(V\) is the velocity of the fluid, \(D\) is the diameter of the sphere, and \(u\) represents the kinematic viscosity of the fluid.

Sherwood Number

The Sherwood number (Sh) is a dimensionless number used in mass transfer operations. It represents the ratio of convective mass transfer to diffusive mass transfer. In the context of our problem, we used it to calculate the mass transfer coefficient (k). The Sherwood number helps us understand how efficiently a substance like naphthalene is transferred between phases due to both molecular diffusion and convection.

The Sherwood number is related to both the Reynolds number and the Schmidt number and is often represented in correlations that allow the calculation of the mass transfer coefficient, as seen in our exercise with the formula \(Sh = 0.0296Re^{0.8}Sc^{0.33}\).

Schmidt Number

The Schmidt number (Sc) is a dimensionless number that compares the kinematic viscosity of a fluid to its molecular diffusivity. It is defined as \(Sc = \dfrac{u}{D}\) where \(u\) is the kinematic viscosity and \(D\) is the molecular diffusion coefficient. The Schmidt number is crucial for characterizing fluid flows where there is simultaneous momentum and mass transfer. In our calculation, the given Schmidt number was an essential parameter in determining the correct mass transfer coefficient.

Dimensionless Quantities in Heat and Mass Transfer

Dimensionless quantities, such as the Reynolds, Sherwood, and Schmidt numbers, are pivotal in heat and mass transfer analysis. They allow engineers and scientists to compare different physical situations and scale up processes from laboratory to industrial scale. These quantities encapsulate the properties and flow conditions into non-dimensional forms, making the mathematical analysis of complex phenomena more manageable.

Understanding the interplay between these dimensionless numbers can provide insights into the transport mechanisms involved and allow for the development of correlations to calculate variables such as the mass transfer coefficient. In this way, we can generalize findings and apply them to a variety of systems under similar dimensionless conditions.

Diffusion Coefficient Calculation

The diffusion coefficient (D) quantifies the rate at which a substance diffuses in a medium. It is a crucial parameter in mass transfer operations and is featured in the calculation of both the Sherwood and Schmidt numbers.

In the exercise, we calculated the diffusion coefficient for naphthalene in air using the established relationship between the kinematic viscosity and the Schmidt number: \(D = \dfrac{u}{Sc}\). The diffusion coefficient found was instrumental in determining the final mass transfer coefficient. This kind of calculation is vital for designing and modeling systems where diffusion plays a significant role, such as in chemical reactors or environmental engineering scenarios.