Question

When the ___ is unity, one can expect the momentum and mass transfer by diffusion to be the same. (a) Grashof (b) Reynolds (c) Lewis (d) Schmidt (e) Sherwood

Short answer

Question: Identify the dimensionless number that, when equal to unity, implies that momentum and mass transfer by diffusion are the same. Choose from the options below: (a) Grashof number (b) Reynolds number (c) Lewis number (d) Schmidt number (e) Sherwood number Answer: (d) Schmidt number

Step-by-step solution

Understand the given statement

We are given a statement that when a certain dimensionless number is unity (equal to 1), the momentum and mass transfer by diffusion will be the same. Our task is to identify this dimensionless number from the provided options.

Define the dimensionless numbers in the choices

To do this, we need to understand the meaning of each dimensionless number given in the options: (a) Grashof number (Gr) - a dimensionless number that describes the ratio of buoyancy forces to viscous forces (b) Reynolds number (Re) - a dimensionless number that represents the ratio of inertial forces to viscous forces (c) Lewis number (Le) - a dimensionless number that characterizes the relative importance of thermal and mass diffusivities (d) Schmidt number (Sc) - a dimensionless number that represents the ratio of momentum diffusivity to mass diffusivity (e) Sherwood number (Sh) - a dimensionless number that expresses the ratio of convective mass transfer to diffusive mass transport

Identify the correct choice

After defining the dimensionless numbers, we can now assess which one matches the condition given in the problem statement. Since we are looking for a dimensionless number that becomes unity when the momentum and mass transfer by diffusion are the same, the correct choice should represent this relationship. From the definitions in Step 2, we can see that the Schmidt number (Sc) represents the ratio of momentum diffusivity to mass diffusivity. When the Schmidt number is unity (Sc = 1), this implies that momentum and mass transfer by diffusion are equal. Therefore, the correct answer is (d) Schmidt.

Key concepts

Grashof Number (Gr)

The Grashof number (Gr) is a dimensionless number in fluid dynamics that is used to predict the pattern of fluid flow under the influence of buoyancy forces. Essentially, it provides insights into the ratio of the buoyant to viscous forces present in the fluid. When a fluid is heated, it expands and becomes less dense. This less dense fluid rises while the cooler, denser fluid sinks, creating a convective current.

The Grashof number becomes particularly important in the study of natural convection, which occurs without any external forced movement, such as pumping or wind. It is calculated using the formula: \begin{align*}Gr = \frac{g \beta \Delta T L^3}{u^2}\end{align*}where:

• \( g \) is the acceleration due to gravity,
• \( \beta \) is the thermal expansion coefficient,
• \( \Delta T \) is the temperature difference causing buoyancy,
• \( L \) is the characteristic length, and
• \( u \) is the kinematic viscosity of the fluid.

In cases where the Grashof number is significantly large, it indicates that buoyancy effects dominate over viscous effects, meaning that the flow is likely to be turbulent due to the strong natural convection currents.

Reynolds Number (Re)

The Reynolds number (Re) is arguably one of the most renowned dimensionless numbers in fluid mechanics. It plays a crucial role in predicting the flow characteristics of a fluid. The Re correlates the inertial forces (those which cause fluids to continue to flow) to the viscous forces (which cause fluids to resist flow).Generally, Reynolds number helps determine whether the fluid flow is laminar or turbulent. A low Reynolds number, typically less than 2000, suggests smooth, orderly flow—known as laminar flow. Conversely, a high Reynolds number, usually greater than 4000, is indicative of chaotic and irregular movement, characteristic of turbulent flow. The transitional regime between laminar and turbulent flow occurs within the intermediate range of Reynolds numbers.

The formula to calculate the Reynolds number is as follows:\begin{align*}Re = \frac{\rho u L}{\mu}\end{align*}where:

• \( \rho \) represents the fluid density,
• \( u \) is the velocity of the fluid flow,
• \( L \) stands for the characteristic length, and
• \( \mu \) is the dynamic viscosity of the fluid.

Understanding the Reynolds number is fundamental for designing systems involving fluid flow, such as pipelines, aircraft, and process plants, ensuring the efficiency and effectiveness of these systems.

Lewis Number (Le)

The Lewis number (Le) is a dimensionless quantity used in the domain of heat and mass transfer. It signifies the relationship between the thermal diffusivity and mass diffusivity of a substance and is instrumental in analyzing combined heat and mass transfer processes, like drying or energy storage.The Lewis number is defined as:\begin{align*}Le = \frac{\alpha}{D}\end{align*}where:

• \( \alpha \) is the thermal diffusivity, and
• \( D \) is the mass diffusivity.

A Lewis number of unity (\( Le = 1 \)) implies that the rates of heat and mass transfer by diffusion are equal. This condition simplifies the analysis of simultaneous transfer processes, as it allows an analogy between heat and mass transfer, helping to predict the behavior of one based on the known properties of the other. However, in many real-world scenarios, the Lewis number is not equal to one, which demands a more comprehensive analysis to understand the respective rates of transfer.

Schmidt Number (Sc)

The Schmidt number (Sc) serves as an indicator of the kinematic viscosity versus mass diffusivity and plays a key role in characterizing fluid flow when mass transfer by diffusion is involved. It finds its use predominantly in the study of concentration boundary layers in mass transfer operations.Mathematically, the Schmidt number is given by:\begin{align*}Sc = \frac{u}{D}\end{align*}where:

• \( u \) is the kinematic viscosity of the fluid, and
• \( D \) is the mass diffusivity of the solute in the fluid.

When the Schmidt number is equal to one (\( Sc = 1 \)), it indicates that the momentum and mass diffusivities are identical. Therefore, the fluid's resistance to momentum change is the same as its resistance to mass transfer by diffusion. This relationship is essential for simplifying the analysis of certain fluid flows, making it easier to predict both momentum and mass transfer rates within the same system.

Sherwood Number (Sh)

In the transfer of mass, the Sherwood number (Sh) plays an analogous role to the Nusselt number in heat transfer, indicating the significance of convective over diffusive mass transfer. It is especially useful in evaluating the performance of equipment such as scrubbers, dryers, and absorbers.The Sherwood number can be expressed as:\begin{align*}Sh = \frac{k L}{D}\end{align*}where:

• \( k \) is the convective mass transfer coefficient,
• \( L \) is the characteristic length, and
• \( D \) is the mass diffusivity.

When the Sherwood number is high, it implies a dominant convective mass transfer relative to the diffusive transfer. The exact value of the Sherwood number depends on the flow conditions, geometry of the system, and properties of the fluid and mass being transferred. A higher Sherwood number often means improved efficiency in the mass transport process, making it an essential consideration in the design and optimization of mass transfer equipment.