Soil Texture Analysis and Soil Density

Texture Analysis Using Hydrometer and Stokes' Law

• Hydrometer analysis is used for texture analysis to calculate the settling speed of soil particles.

• The calculation of settling speed is based on Stokes' Law.

• Stokes' Law Principle: At terminal velocity, the forces acting on a settling particle are in balance. These forces include:

  • Gravity (downward pull).

  • Frictional drag (resistance from the fluid).

  • Buoyancy (upward force due to displaced fluid).

• Stokes' Law Equation for Terminal Velocity ($v$): The general form is: 
  where $\'eta$ is the dynamic viscosity of the fluid. As presented in the lecture, the velocity calculation uses a more explicit form or combines constants: $v_{t}=\frac{2R^2\left(p_{s}-p_{f}\right)g}{9\eta}$
  (The lecture often simplifies by grouping constants, effectively making water density part of the overall constant).

  • In this equation:

  • $r$: Radius of the particle (in meters). Textbooks might use diameter ($d$), where $r = d/2$. The answer remains the same in magnitude but requires using the correct variable.

  • $g$: Gravitational acceleration (approximately $9.8 \text{ m/s}^2$).

  • $\rho_p$: Density of the particle (e.g., quartz).

  • $\rho_w$: Density of water.

  • $\eta$: Dynamic viscosity of water ($\text{Pa} \cdot \text{s}$ or $\text{kg}/(\text{m} \cdot \text{s})$).

For all measurements in this context, particle density, water density, gravitational force ($g$), and viscosity of water ( $\eta$ )are constant. The only variable changing is the particle radius ($r$).

Importance of Units in Calculations

• Unit Consistency: Always ensure correct units are used in all equations (e.g., kilogram per meter cube ($\text{kg/m}^3$) or gram per centimeter cube ($\text{g/cm}^3$) for density).

• Unit Cancellation: Verify that units cancel out correctly to yield the desired output unit. For velocity, the final unit should be meters per second ($\text{m/s}$). For example, if density is $\text{kg/m}^3$, gravitational force is $\text{m/s}^2$, radius squared is $\text{m}^2$, and viscosity is $\text{Pa} \cdot \text{s}$ ($\text{kg}/(\text{m} \cdot \text{s})$), the units should simplify to $\text{m/s}$. Always double-check your units.

Applying Stokes' Law with Constants

• Particle Density: For quartz, which is generally assumed for soil particles, the density is $2650 \text{ kg/m}^3$.

• Water Density: The density of water is $1000 \text{ kg/m}^3$.

• Viscosity of Water: A typical value for the dynamic viscosity of water is $0.001 \text{ Pa} \cdot \text{s}$.

• After substituting these constant values along with $g = 9.8 \text{ m/s}^2$ into Stokes' Law, the equation simplifies to:
  $v = (\text{Constant}) \times r^2$

  Where the constant value is approximately $3,597,000 \text{ m}^{-1}\text{s}^{-1}$. Therefore, $v = 3,597,000 \times r^2$

• Example Particle Sizes and Calculations:

  • Clay Particle: Upper limit of clay size has a diameter of $2 \text{ microns}$ ($2 \times 10^{-6} \text{ m}$), so the radius ($r$) is $1 \text{ micron}$ ($1 \times 10^{-6} \text{ m}$).

  • Sand Particle: Lower limit of sand size has a diameter of $0.05 \text{ mm}$ ($5 \times 10^{-5} \text{ m}$), so the radius ($r$) is $0.025 \text{ mm}$ ($2.5 \times 10^{-5} \text{ m}$).

Calculating Time for Particle Settling

• To determine the time required to wait before taking a hydrometer reading at a specific depth, we use the relationship:
  $\text{velocity} (v) = \frac{\text{distance} (d)}{\text{time} (t)}$

• Therefore, the time to wait ($t$) is: $t = \frac{\text{distance} (d)}{v}$

• Measurement Distance: The hydrometer typically measures the density of the suspension at a distance of $0.1 \text{ m}$ ($10 \text{ cm}$) from the top of the solution.

• Application: This calculation helps determine how long to wait for larger particles (e.g., sand) to settle out of the suspension so that only specific particle sizes (e.g., silt or clay) remain at the measurement depth. For example, one might wait two hours for silt particles (diameter $2 \text{ microns}$) to settle below the measurement depth.

Porosity and Density of Soil: Introduction

• Density: Defined as mass per unit volume.

  • Notation: Often represented by an uppercase D or the Greek letter $\rho$ (rho).

• Soil Volume Considerations: The total volume of soil includes both solid particles and pore spaces. These pore spaces can be filled with liquid (water) or gas (air).

Types of Soil Density

• There are two primary types of density used in soil analysis:

  1. Particle Density

  2. Bulk Density

Particle Density ($\rho_p$)

• Definition: The mass of the soil particles divided by the volume of the particles only (excluding pore spaces).

• Constant Value: For practical purposes in this class and generally, particle density is considered a constant.

• Standard Value: The particle density of soil is typically assumed to be that of quartz:

  • $2.65 \text{ g/cm}^3$

  • $2650 \text{ kg/m}^3$

• This number is crucial and should be remembered.

• Particle density measures the density of the solid mineral components, without accounting for any pore volume.

Bulk Density ($\rho_b$)

• Definition: The mass of dry soil divided by the total volume of the soil sample (including both solid particles and all pore spaces).