Question

What is the name of the relation \(f 72=\mathrm{St} \operatorname{Pr}^{2 / 3}=\) \(\mathrm{St}_{\text {mass }} \mathrm{Sc}^{2 / 3}\), and what are the names of the variables in it? Under what conditions is it valid? What is the importance of it in engineering?

Short answer

Answer: The given relation is the Law of the Wall or the Von Kármán-Pohlhausen integral relation, which is crucial in the field of engineering. It is used to describe the velocity profile near a solid boundary, essential when studying boundary layers. This relation helps engineers understand the flow dynamics near solid boundaries, estimate heat and mass transfer, and reduce drag in vehicles and fluid transport-related systems, ultimately enhancing efficiency and performance.

Step-by-step solution

Name of the relation

The given relation is known as the Law of the Wall or the Von Kármán-Pohlhausen integral relation. This relation is widely used in fluid mechanics and describes the velocity profile near a solid boundary, which is essential when studying boundary layers.

Names of the variables in the relation

The variables in the Von Kármán-Pohlhausen integral relation are as follows: 1. \(f\): This represents the friction coefficient. 2. \(St\): This represents the Stanton number, a dimensionless number that describes the ratio of heat transferred into a fluid to the thermal capacity of the fluid. 3. \(Pr\): This represents the Prandtl number, a dimensionless number that represents the ratio of viscous diffusion to thermal diffusion in a fluid. 4. \(St_{mass}\): This represents the mass transfer Stanton number, which is similar to the Stanton number, but instead of heat transfer, it deals with mass transfer. 5. \(Sc\): This represents the Schmidt number, a dimensionless number that describes the ratio of momentum diffusivity to mass diffusivity in a fluid.

Conditions for the relation to be valid

The Von Kármán-Pohlhausen integral relation is valid under the following conditions: 1. The flow is steady, incompressible, and laminar. 2. There is a smooth and flat solid boundary. 3. The fluid properties are constant. 4. There are no external forces acting on the fluid.

Importance in engineering

The Law of the Wall has significant importance in engineering, especially in the field of fluid mechanics. Some of the key applications and importance are: 1. It provides a better understanding of the flow dynamics and behavior near solid boundaries, which is essential when designing machines, vehicles, and structures that interact with fluids, such as aircraft wings, submarine hulls, or turbines. 2. It's an essential tool to estimate heat and mass transfer along with fluid flow near solid boundaries, which is crucial for designing efficient heat exchangers, cooling systems, and mass transfer processes, like distillation columns. 3. Engineers use this relation to optimize designs to reduce the drag in vehicles and other fluid-transport-related systems to ensure energy savings and enhance performance.