Question

Air flows in a 4-cm-diameter wet pipe at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) with an average velocity of \(4 \mathrm{~m} / \mathrm{s}\) in order to dry the surface. The Nusselt number in this case can be determined from \(\mathrm{Nu}=0.023 \mathrm{Re}^{0.8} \mathrm{Pr}^{0.4}\) where \(\mathrm{Re}=10,550\) and \(\operatorname{Pr}=0.731\). Also, the diffusion coefficient of water vapor in air is \(2.42 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). Using the analogy between heat and mass transfer, the mass transfer coefficient inside the pipe for fully developed flow becomes (a) \(0.0918 \mathrm{~m} / \mathrm{s}\) (b) \(0.0408 \mathrm{~m} / \mathrm{s}\) (c) \(0.0366 \mathrm{~m} / \mathrm{s}\) (d) \(0.0203 \mathrm{~m} / \mathrm{s}\) (e) \(0.0022 \mathrm{~m} / \mathrm{s}\)

Short answer

Answer: The mass transfer coefficient inside the pipe for fully developed flow is approximately 0.0366 m/s.

Step-by-step solution

1: Calculate Nusselt number

To calculate the Nusselt number, use the given formula Nu = 0.023 Re^0.8 Pr^0.4, and substitute the values of Re=10,550 and Pr=0.731: Nu = 0.023 * (10,550)^0.8 * (0.731)^0.4

2: Calculate Sherwood number

As the given problem is based on an analogy between heat and mass transfer, the Sherwood number would be equal to the Nusselt number: Sh = Nu

3: Compute the mass transfer coefficient

Now we will find the mass transfer coefficient, km, by using the equation: \(k_{\mathrm{m}}=\frac{\mathrm{Sh} \cdot \mathrm{D_{wv}}}{\mathrm{d}}\) We need to substitute the given values: Sh (which is equal to Nu), \(D_{wv} = 2.42 \times 10^{-5} \, \mathrm{m^2} \!/\! \mathrm{s}\), and d = 4 cm (0.04 m): \(k_{\mathrm{m}}=\frac{\mathrm{Nu} \cdot 2.42 \times 10^{-5}\mathrm{~m}^{2} /\mathrm{s}}{0.04\mathrm{~m}}\)

4: Evaluate and compare

Evaluate the expression for \(k_{\mathrm{m}}\) and compare the result with the given options: \(k_{\mathrm{m}} \approx 0.0366 \, \mathrm{m} \!/\! \mathrm{s}\) This value matches option (c). Therefore, the mass transfer coefficient inside the pipe for fully developed flow is \(0.0366 \mathrm{~m} / \mathrm{s}\).

Key concepts

Nusselt Number

The Nusselt number, denoted as Nu, is a dimensionless number crucial in the study of convective heat transfer. It compares the rate of heat transfer via conduction within a fluid to that of convection across the fluid's surface. The general formula used for calculating the Nusselt number given in your problem is:\[\text{Nu} = 0.023 \cdot \text{Re}^{0.8} \cdot \text{Pr}^{0.4}\]Here's how it works:

• The number indicates how efficiently heat is being transferred from the wall of the pipe to the flowing fluid.
• A high Nusselt number suggests enhanced convection.
• It's analogous to enhancing the heat transport capacity of the system.

For the air flowing in the pipe, this concept is essential, allowing you to determine how effective the drying process is, as influenced by convection heat transfer.

Sherwood Number

The Sherwood number is the mass transfer equivalent of the Nusselt number in heat transfer. It represents a dimensionless measure for mass transfer similar to Nu for heat transfer. The Sherwood number, presented as Sh, is calculated using an analogy from the Nusselt number, especially when analyzing combined heat and mass transfer processes. In this context:

• The analogy suggests that the Nusselt number and Sherwood number are equal for the system.
• This means a high Sherwood number would indicate a more significant mass transfer rate.
• An understanding of this equivalence helps in simplifying the calculations of mass transfer problems just by knowing the conditions of heat transfer.

This equivalency is particularly useful when evaluating drying or other processes involving simultaneous heat and mass transfer.

Mass Transfer Coefficient

The mass transfer coefficient, usually denoted as \(k_m\), quantifies the rate of mass transfer across a unit area. It effectively translates the dimensionless Sherwood number into a real-world value that engineers use to design and analyze systems.Calculation for \(k_m\) uses:\[k_m = \frac{\text{Sh} \cdot D_{wv}}{d}\]Where:

• \(D_{wv}\) is the diffusion coefficient of water vapor in the air.
• \(d\) is the characteristic length (like the pipe diameter).
• Sh, the Sherwood number, comes from its equivalency to the Nusselt number in this context.

Learning how these factors interplay helps in predicting the performance of systems where mass needs to be transferred effectively, such as in drying operations.

Reynolds Number

The Reynolds number, abbreviated as Re, is critical in evaluating different flow regimes within a fluid system, such as whether the flow is laminar or turbulent. This dimensionless number is calculated as:\[\text{Re} = \frac{\rho u L}{\mu}\]Where:

• \(\rho\) is the fluid density.
• \(u\) is the flow velocity.
• \(L\) is a characteristic length, like pipe diameter.
• \(\mu\) is the fluid's dynamic viscosity.

For air flowing through a pipe, the Reynolds number gives insight into the behavior of air within that pipe:

• Values less than 2300 indicate laminar flow.
• Values greater than 4000 indicate turbulent flow.
• Knowing the flow regime is crucial for accurate calculations in heat and mass transfer operations.

Prandtl Number

The Prandtl number, symbolized as Pr, is another dimensionless number important in fluid dynamics and heat transfer. It bridges the momentum diffusivity (kinematic viscosity) and thermal diffusivity of the fluid.It is calculated by the formula:\[\text{Pr} = \frac{u}{\alpha}\]Here,

• \(u\) represents the kinematic viscosity.
• \(\alpha\) is the thermal diffusivity.

In simpler terms:

• A low Prandtl number suggests that thermal diffusion dominates, making heat transfer slower through the fluid compared to momentum transfer.
• A higher number means the reverse, where heat spreads more efficiently than momentum.

For engineers, Prandtl numbers highlight how heat will spread through a fluid, influencing the design and assessment of systems dealing with thermal processes.