Question

Consider an unconsolidated (uncemented) layer of soil completely saturated with groundwater; the water table is coincident with the surface. Show that of d the upward Darcy velocity \(|v|\) required to fluidize the bed is $$ |v|=\frac{(1-\phi) k g\left(\rho_{s}-\rho_{w}\right)}{\mu} $$ where \(\phi\) is the porosity, \(\rho_{s}\) is the density of the soil particles, and \(\rho_{w}\) is the water density. The condition of a fluidized bed occurs when the pressure at depth in the soil is sufficient to completely support the weight of the overburden. If the pressure exceeds this critical value, the flow can lift the soil layer.

Short answer

The upward Darcy velocity required to fluidize the soil bed is \(|v| = \frac{(1-\phi) kg(\rho_s - \rho_w)}{\mu}\).

Step-by-step solution

Understand the Problem

We need to derive the equation for the upward Darcy velocity \(|v|\) sufficient to fluidize a soil bed. This involves balancing the upward flow-induced force with the weight of the soil particles submerged in water.

Review Darcy's Law

Darcy's Law for fluid flow is given by: \[v = -\frac{k}{\mu} \frac{dP}{dz}\] where \(v\) is the Darcy velocity, \(k\) is the permeability, \(\mu\) is the dynamic viscosity, and \(\frac{dP}{dz}\) is the pressure gradient.

Calculate the Pressure Gradient

To fluidize the bed, the upward pressure gradient must equal the weight of the soil particles per unit volume. The pressure gradient is:\[\frac{dP}{dz} = g(\rho_s - \rho_w)\]This represents the difference in weight between the soil particles and the water.

Substitute into Darcy’s Law

Substitute the calculated pressure gradient into Darcy's Law:\[|v| = \frac{(1-\phi)k}{\mu} g (\rho_s - \rho_w)\]here, the factor \((1-\phi)\) is included to account for the porosity, representing the solid part of the volume.

Finalize the Formula

The derived formula for the Darcy velocity necessary to lift the soil particles and fluidize the bed is:\[|v| = \frac{(1-\phi) kg(\rho_s - \rho_w)}{\mu}\]This equation shows the relationship between the porosity, permeability, density differences, and fluid viscosity in fluidizing a soil layer.

Key concepts

Fluidization

Fluidization is an interesting concept, essential in understanding the behavior of particulate materials immersed in a fluid. Imagine a jar filled with sand, then water is added to it from the bottom. When the water flows upward with enough velocity, it makes the sand particles "float," almost like a liquid. This is fluidization.
Fluidization occurs when the force of the flowing fluid counteracts the gravitational force on the particles. Therefore, fluidization requires a specific fluid velocity, called the "minimum fluidization velocity," to occur. At this point, the upward pressure exerted by the fluid overcomes the weight of the solid particles.

This principle is fundamental in various industrial processes, like fluidized bed reactors used in chemical engineering, where solid particles are made to behave like liquids to improve mixing and chemical reactions.

Porosity

Porosity is a key concept when discussing the void spaces within a material. It is defined as the fraction of the volume of voids over the total volume. For example, in soil or rock, porosity represents the amount of empty space between solid particles.

• Volume of Voids: The space not occupied by solid material. This is where fluids can reside.
• Total Volume: The combined volume of the solid material and the voids.

Porosity, denoted by the symbol \(\phi\), directly influences the ability of fluids to move through a material. A higher porosity means more space for fluids to flow through, thus affecting the fluidization process. For instance, in Darcy's law application, the factor (1-\(\phi\)) is used to account for the volume that is actually occupied by the solid part.

Pressure Gradient

The pressure gradient is the change in pressure over a certain distance within a fluid. It is a driving force in fluid dynamics, influencing how and where fluids flow. In the context of fluidization, understanding the pressure gradient is essential because it determines whether the soil particles will lift or remain stationary.
A pressure gradient can be thought of as a "slope of pressure." In the fluidization of soil, the pressure gradient must be such that it counterbalances the gravitational pull on the soil particles.

• Equation: In Darcy's law, the pressure gradient is represented as \(\frac{dP}{dz}\), capturing how pressure changes with depth.
• Importance: A sufficient upward pressure gradient is required to achieve fluidization.

Without it, the fluid cannot effectively "push" against the forces holding the solid particles down.

Darcy Velocity

Darcy velocity is crucial in analyzing fluid flow through porous media. Named after Henry Darcy, this concept helps us understand how fast a fluid is moving within a porous structure. Contrary to common velocity, Darcy velocity considers the entire cross-sectional area of the porous medium, not just the area occupied by fluid.
According to Darcy's law, the Darcy velocity \(v\) is given by \(v = -\frac{k}{\mu} \frac{dP}{dz}\). Here:

• \(k\): The permeability of the material, which indicates how easy it is for fluids to pass through.
• \(\mu\): The fluid's dynamic viscosity, describing the fluid's resistance to flow.
• \(\frac{dP}{dz}\): The pressure gradient, reflecting the change in pressure over a distance.

The Darcy velocity can predict when fluidization will occur by providing the necessary conditions for the upward fluid force to balance the force of gravity on the soil particles. It is a critical concept in hydrogeology and engineering applications for predicting fluid flow behavior.