Question

Define the following terms: mass-average velocity, diffusion velocity, stationary medium, and moving medium.

Short answer

Define the following terms used in fluid mechanics and mass transport: 1) Mass-average velocity: Mass-average velocity is the weighted average of individual velocities of particles in a fluid mixture, considering their respective masses. It is calculated using the formula: V_m = (Σ m_i v_i)/(Σ m_i), where m_i is the mass of the particle i, v_i is the velocity of particle i, and n is the total number of particles in the mixture. 2) Diffusion velocity: Diffusion velocity is the velocity of a particle or species in a mixture due to concentration gradients, causing particles to move from high concentration areas to low concentration areas. Fick's law is used to determine the diffusion velocity: V_d = -D (dC/dx), where D is the diffusion coefficient, C is the concentration, and x represents the distance in the direction of diffusion. 3) Stationary medium: A stationary medium is a fluid or mixture of particles that are not in motion and have an average velocity of zero. In a stationary medium, properties like pressure, temperature, and concentration are constant, and the concentration gradients drive the movement of particles or species. 4) Moving medium: A moving medium is a fluid or mixture of particles with a non-zero average velocity. In a moving medium, mass transport occurs due to both advection (bulk movement of particles in the flow direction) and diffusion (movement due to concentration gradients). The combined effect of advection and diffusion determines the overall mass transport in a moving medium.

Step-by-step solution

Mass-Average Velocity

Mass-average velocity is the average velocity of a fluid mixture or a mixture of particles with different velocities. It takes into account the individual velocities of each particle and their respective masses. Mathematically, the mass-average velocity (V_m) can be calculated using the following formula: V_m = \frac{\sum_{i=1}^n m_i v_i}{\sum_{i=1}^n m_i} where m_i is the mass of the particle i and v_i is the velocity of particle i, and n is the total number of particles in the mixture. The mass-average velocity represents a weighted average of the individual velocities, where the weights are the masses of the respective particles.

Diffusion Velocity

Diffusion velocity is the velocity of a particle or a species in a mixture due to concentration gradients. This movement is driven by the natural tendency of particles to move from areas of higher concentration to areas of lower concentration until a uniform concentration is achieved. The diffusion velocity (V_d) can be determined using Fick's law: V_d = -D \frac{dC}{dx} where D is the diffusion coefficient, C is the concentration of the particle or species, and x represents the distance in the direction of diffusion.

Stationary Medium

A stationary medium is a fluid or a mixture of particles that are not in motion, or their average velocity is zero. In a stationary medium, no mass transport occurs, and properties such as pressure, temperature, and concentration are constant throughout the medium or remain the same over time. In the context of mass transport, stationary medium refers to the environment where the concentration gradients are the only driving force for the movement of particles or species.

Moving Medium

A moving medium is a fluid or a mixture of particles where the particles are in motion or have a non-zero average velocity. In a moving medium, mass transport occurs due to both advection and diffusion. Advection refers to the bulk movement of particles or species in the same direction as the flow of the medium, while diffusion occurs due to concentration gradients. The combined effect of advection and diffusion determines the overall mass transport in a moving medium.

Key concepts

Mass-Average Velocity

Mass-average velocity is all about understanding how particles in a mixture move, based on their masses and velocities. Imagine you have a bunch of different balls rolling down a hill. Some are bigger and heavier, like bowling balls, while others are smaller, like tennis balls. Each ball moves at its own pace, but the mass-average velocity tells us the overall motion of all the balls combined.
To find this mass-average velocity, you use a special formula:

• This formula is: \[V_m = \frac{\sum_{i=1}^n m_i v_i}{\sum_{i=1}^n m_i}\]
• Here, \(m_i\) represents the mass of each ball,
• \(v_i\) is the velocity of each one,
• and \(n\) is the total number of balls (or particles).

This gives you a weighted average, meaning bigger balls have more influence on the speed we calculate. Mass-average velocity helps us understand where the whole group of particles is heading, making it super useful in science, especially when studying fluids or gases.

Diffusion Velocity

Diffusion velocity explains how particles move because of concentration differences; it's like a perfume spreading out in a room. Imagine you open a bottle of perfume in one corner, and soon enough, the smell reaches the other side of the room. This happens because particles want to spread out evenly.
We use Fick's law to understand this process mathematically. The formula goes like this:

• \(V_d = -D \frac{dC}{dx}\)
• \(V_d\) is the diffusion velocity,
• \(D\) is the diffusion coefficient, telling us how easily particles spread out,
• and \(\frac{dC}{dx}\) represents the concentration gradient, or how steep the difference in particle number is between areas.

The negative sign indicates particles move from high to low concentration areas. This natural spread aims to achieve equilibrium, where the concentration is uniform.

Stationary Medium

A stationary medium is what you picture when everything is calm and still, like a calm pond. In this scenario, all particles in the fluid or gas are at rest, meaning the average velocity is zero. Picture a cup of still water; no currents, no movement.
What characterizes a stationary medium is that there's no mass transport because everything is perfectly stable. Other conditions like pressure, temperature, and concentration remain constant with time and space. Any movement that does occur is purely due to concentration gradients, as particles try to even out any differences. Stationary mediums are crucial for understanding processes where we don't want external forces affecting particle movements.

Moving Medium

In a moving medium, particles are in constant motion, like a river flowing downhill. This means there's always a non-zero average velocity, signifying dynamic mass transport. Such movement combines two main processes: advection and diffusion.

• **Advection** is straightforward; it's the movement of particles along with the flow of the whole medium, similar to leaves floating downstream.
• **Diffusion**, on the other hand, occurs because of concentration differences, as particles spread out over areas of varying concentration.

Together, advection and diffusion define how particles and substances get distributed in a moving medium, influencing how things mix and interact. Whether it's air currents or ocean currents, understanding these dynamics is crucial for fields like meteorology, oceanography, and chemical engineering.