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The naphthalene sublimation technique involves the use of a mass transfer experiment coupled with an analysis based on the heat and mass transfer analogy to obtain local or average convection heat transfer coefficients for complex surface geometries. A coating of naphthalene, which is a volatile solid at room temperature, is applied to the surface and is then subjected to airflow in a wind tunnel. Alternatively, solid objects may be cast from liquid naphthalene. Over a designated time interval, \(\Delta t\), there is a discernible loss of naphthalene due to sublimation, and by measuring the surface recession at locations of interest or the mass loss of the sample, local or average mass transfer coefficients may be determined. Consider a rectangular rod of naphthalene exposed to air in cross flow at \(V=10 \mathrm{~m} / \mathrm{s}, T_{\mathrm{s}}=300 \mathrm{~K}\), as in Problem 6.10, except now \(c=10 \mathrm{~mm}\) and \(d=30 \mathrm{~mm}\). Determine the change in mass of the \(L=500\)-mm-long rod over a time period of \(\Delta t=30 \mathrm{~min}\). Naphthalene has a molecular weight of \(M_{\mathrm{A}}=128.16 \mathrm{~kg} / \mathrm{kmol}\), and its solid-vapor saturation pressure at \(27^{\circ} \mathrm{C}\) and \(1 \mathrm{ltm}\) is \(p_{\text {A, } a t}=1.33 \times 10^{-4}\) bar.

Step by step solution

01

Determine the Reynolds Number

To determine the mass transfer coefficient, we need to first calculate the Reynolds number of the flow over the rectangular rod. Based on the given velocity, \(V\), and the dimensions of the rod, we can determine the Reynolds number using the relation: \( Re = \frac{\rho V d}{\mu} \) where \(Re\) is the Reynolds number, \(\rho\) is the air density, \(V\) is the air velocity, \(d\) is the characteristic length, and \(\mu\) is the dynamic viscosity of air. We are given \(V = 10 \: m/s\) and \(d = 30 \: mm = 0.03 \: m\). We need to look up the air density and viscosity at the given temperature, \(T_s = 300 \: K\). According to standard air properties at this temperature: \( \rho = 1.16 \: kg/m^3 \) \( \mu = 1.85 \times 10^{-5} \: kg/(m \cdot s) \) Now we can calculate the Reynolds number: \( Re = \frac{(1.16)(10)(0.03)}{1.85 \times 10^{-5}} = 18802 \)

02

Determine the Nusselt Number

Based on the obtained Reynolds number, we'll use the analogy between heat and mass transfer to determine the Nusselt number, Nu. For a rectangular rod in cross-flow, we can use the relation: \( Nu = 0.27 Re^{0.6} Pr^{1/3} \) where \(Pr\) is the Prandtl number of air. We need to look up the Prandtl number for air at the given temperature, \(T_s = 300 \: K\). According to standard air properties at this temperature: \( Pr = 0.7 \) Now we can calculate the Nusselt number: \( Nu = 0.27 (18802)^{0.6} (0.7)^{1/3} = 101.54 \)

03

Determine the Mass Transfer Coefficient

The mass transfer coefficient, \(Sh\), can be determined from the Nusselt number using the heat and mass transfer analogy: \( Sh = Nu \) Therefore, the mass transfer coefficient is: \( Sh = 101.54 \)

04

Calculate the Mass Loss

We know that the mass loss rate over a surface, \(\Delta m / \Delta t\), can be determined from the mass transfer coefficient and the saturation pressure of naphthalene, as: \( \frac{\Delta m}{\Delta t} = Sh \cdot C_p \cdot A \cdot \frac{p_{A \atop}}{M_A} \) where \(C_p\) is the heat capacity of air, \(A\) is the surface area of the naphthalene rod, \(p_{A \atop}\) is the saturation pressure of naphthalene, and \(M_A\) is the molecular weight of naphthalene. We need to look up the heat capacity of air at the given temperature, \(T_s = 300 \: K\): \( C_p = 1006 \: J/(kg \cdot K) \) The surface area of the rod can be calculated as: \( A = 2(L \cdot c + L \cdot d) \) where \(L = 500 \: mm = 0.5\: m\), \(c = 10 \: mm = 0.01 \: m\), and \(d=30\:mm = 0.03\: m\): \( A = 2(0.5\cdot 0.01 + 0.5 \cdot 0.03) = 0.04 \: m^2 \) We are given the saturation pressure of naphthalene, \(p_{A \atop} = 1.33 \times 10^{-4} \: bar = 13.3\: Pa\), and the molecular weight, \(M_A = 128.16 \: kg/kmol\). Now we can calculate the mass loss over the given time period: \( \Delta m = (101.54)(1006)(0.04) \frac{13.3}{128.16} (30\cdot60) \: kg \) \( \Delta m = 0.190 \: kg \) So, the change in mass of the naphthalene rod over the 30-minute time period is approximately \(0.190 \: kg\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Naphthalene Sublimation

Naphthalene sublimation is a fascinating process used widely in both educational and industrial experiments to analyze mass transfer. This process involves the transition of naphthalene, a volatile crystalline compound, from a solid state directly into a vapor without passing through a liquid phase. Sublimation primarily occurs at room temperature, making naphthalene an ideal substance for such experiments. In a typical setup, naphthalene is applied as a coating on a surface or cast into a sample shape, like a rod, which is then exposed to airflow.

By exposing the naphthalene-coated surface to a controlled airflow, sublimation occurs, and the lost mass due to the sublimation process can be measured. This loss allows scientists and engineers to evaluate the rate of mass transfer. The sublimation process helps in determining the mass transfer coefficient, which is crucial for designing or analyzing equipment involving mass transfer systems such as dryers and sublimation reactors.

Mass Transfer Coefficient

The mass transfer coefficient is a vital concept when studying how substances move and transfer between phases, like from solid to gas in sublimation. It quantifies the rate at which mass is transferred per unit area and concentration difference.

In the context of naphthalene sublimation, the mass transfer coefficient is crucial because it determines how quickly naphthalene vaporizes into the airflow. The coefficient is influenced by several factors such as:

• Flow velocity
• Surface geometry
• Temperature

The higher the mass transfer coefficient, the more efficient the sublimation and hence, the loss of mass. It's often calculated using analogies from heat transfer, employing parameters like the Sherwood or Nusselt numbers based on empirical correlations.

Reynolds Number

The Reynolds Number, denoted as \( Re \), is a dimensionless value that helps determine the flow regime around an object, such as laminar or turbulent flow. Calculating the Reynolds number is a crucial step when performing analyses involving fluid flow over objects.

For a rectangular rod of naphthalene in cross-flow, the Reynolds number can be calculated using the formula:\[Re = \frac{\rho V d}{\mu}\]where:

• \( \rho \) is the density of the air
• \( V \) is the velocity of the air
• \( d \) is a characteristic dimension (e.g., the width or height of an object)
• \( \mu \) is the dynamic viscosity of the air

Understanding the Reynolds number is vital because it impacts the mass transfer processes. A higher Reynolds number indicates turbulent flow, which usually enhances mass transfer due to increased mixing.

Nusselt Number

The Nusselt Number, symbolized as \( Nu \), is an essential dimensionless quantity in heat and mass transfer processes, symbolizing the enhancement of heat transfer through a fluid layer as compared to pure conduction. In our problem context, it directly applies to evaluating convection heat and mass transfer.

It is calculated with experimental correlations. For a rectangular rod, in particular, the Nusselt number can be computed using:\[Nu = 0.27 Re^{0.6} Pr^{1/3}\]Here, \( Pr \) is the Prandtl number, which is specific to the fluid's properties, relating momentum diffusivity (viscosity) and thermal diffusivity.

The Nusselt number is akin to the Sherwood number in mass transfer scenarios, offering a pathway to calculate the mass transfer coefficient. By knowing the Nusselt number, one can infer how well equipment or a system will perform in terms of heat and mass transfer, aiding in engineering designs and efficiency calculations.