Sedimentation Law 33 distributions in soils can be represented by a power-law distribution, consistent with a fractal fragmentation model, but with scale invariance valid only over a limited domain range. Three domains—clay, silt and sand—were identified in which power-law scaling was applicable (Bittelliet al., 1999). The boundaries between the domains were relatively constant for different soil types, but did not coincide with the traditional boundaries between clay, silt and sand.
34 Basic Physical Properties of Soil
When the particle is settling into a laminar flow region, the drag force is Fd=πr2ρl
v2
2Cd (2.43)
whereπr2is the orthographic projection of the sphere onto a plane perpendicular to the direction of movement andCdis the drag coefficient.
Dimensional analysis of the problem shows that the drag coefficient depends on the particle size and the flow velocity, as described by the Reynolds number
Re= ρlv2r
η (2.44)
whereηis the dynamic viscosity of the fluid andvis the terminal velocity. Stokes found that for small Reynolds numbers (Re <0.3), the relationship between drag coefficient and Reynolds number is (Allen, 1981):
Cd= 24
Re (2.45)
Substituting eqns (2.45) and (2.44) into eqn (2.43) yields
Fd = 6πrηv (2.46)
This equation is valid only for low Reynolds numbers (Re<0.3), which also deter- mines a critical radius (or diameter) above which Stokes’ equation should not be used (Allen, 1981).
By inserting eqns (2.39)–(2.46) into eqn (2.42), we get ρs
4πr3 3
g–ρl
4πr3 3
g– 6πrηv= 0 (2.47)
Rearranging and simplifying leads to
g(ρs–ρl)4r2 3
1
6η =v (2.48)
which can be further simplified to
v=g(ρs–ρl)2 9
r2
η (2.49)
Often the particle size is expressed as equivalent particle diameter (d= 2r). Thus eqn (2.49) is written as
v=g(ρs–ρl) d2
18η (2.50)
Sedimentation Law 35 If the particle’s falling distanceS is known, then it is possible to determine the falling timet:
t= 18ηS
g(ρs–ρl)d2 (2.51)
These equations are valid only under certain assumptions:
1. The particles are smooth, spherical and rigid.
2. The particles are small enough not to interact with each other.
3. The particle density is known and constant.
4. The particles are uniformly distributed within the liquid.
5. The liquid flow is laminar, no turbulent flow is present and the fall of the particles is unhindered by the proximity of the container walls.
6. The particles are large in comparison with the liquid molecules, so that Brownian movements do not affect the fall.
7. The liquid viscosity is constant during the fall; therefore temperature is also constant, since the liquid viscosity is affected by the liquid temperature.
The program PSP_sedimentation simulates the falling of a soil particle in a soil–water suspension, for different temperatures, concentrations of sodium hexameta- phosphate (HMP) and particle diameters. Themain function imports data saved in the filewaterDensity.dat. HMP is a dispersing agent used for particle size analysis.
The data provide different values of the density and viscosity of water at temperatures ranging from 16 to 26◦C. These values are used in the program to compute the changes in density and viscosity as functions of HMP concentration. The HMP densityρhmpas a function of concentration is given by (Gee and Bauder, 1986)
ρhmp=ρl
1 + 0.63 c 1000
(2.52) whereρlis the density of water andcis the HMP concentration [g L–1].
The HMP viscosity ηhmp is given as a function of concentration by (Gee and Bauder, 1986)
ηhmp=ηl
1 + 4.25 c 1000
(2.53) whereηlis the viscosity of water [kg m–1s–1].
A visual program is also included where the falling of a particle (of variable ra- dius) in a cylinder is simulated. The program imports the visual module and a file calledPSP_readDataFile, which is used to read data from an ASCII file. This file will be used many times throughout the book, and it is described in Appendix A. In the main( ) function of the program, the function readDataFile, written in the file PSP_readDataFile, is used to open and read the file. The temperature, concentration
36 Basic Physical Properties of Soil
and particle size are input by the user after being prompted by the program. The settling particle is then visualized by employing the instructions of the program Visual Python, and the total number of seconds is printed.
The functioncomputeSedimentationTimecalculates the sedimentation time using Stokes’ law, while the function computeSedimentationDepth calculates the pos- ition (depth) of the settling particle at different times. The computation of solution density and viscosity as functions of HMP concentration is written in the functions getSolutionDensityandgetSolutionViscosity, as described in eqns (2.52) and (2.53).
#PSP_sedimentation
from _ _future_ _ import print_function, division import visual
from PSP_readDataFile import readDataFile particleDensity = 2580
g = 9.80665
def getSedimentationTime(solutionDensity, solutionViscosity, particleDiameter, z):
numer = 18 * solutionViscosity * z
denom = (g * (particleDensity - solutionDensity)
* (particleDiameter * 0.000001)**2) return (numer / denom)
def getSedimentationDepth(solutionDensity, solutionViscosity,
particleDiameter, time):
numer = (time * g * (particleDensity - solutionDensity)
* (particleDiameter * 0.000001)**2) denom = 18 * solutionViscosity
return (numer / denom)
def getSolutionDensity(liquidDensity, concentration):
return (liquidDensity * (1 + 0.63*(concentration / 1000))) def getSolutionViscosity(liquidViscosity, concentration):
return (liquidViscosity * (1 + 4.25*(concentration / 1000))) def initializeScene(heightCylinder, radiusParticle):
scene = visual.display(width = 600, height = 600, exit=True) scene.background = visual.color.white
scene.center = (0, 0, -heightCylinder/2) scene.forward = (0, 1, -0.5)
cylinder = visual.cylinder(pos = (0,0,0), radius = 0.05) cylinder.axis = (0,0,-heightCylinder)
cylinder.color = visual.color.white cylinder.opacity = 0.2
Sedimentation Law 37 particle = visual.sphere(pos = (0,0,0), color = visual.color.black) particle.radius = min(radiusParticle * 0.001, 0.02)
return(particle) def main():
A, isFileOk = readDataFile("waterDensity.dat", 2, ’\t’, False) if ((not isFileOk) or (len(A[0]) < 3)):
print("Wrong file!\nMissing data or wrong delimiter in line:", A+1)
return()
temperature = A[:,0]
density = A[:,1]
viscosity = A[:,2]
print ("Available temperature:", temperature) isGoodChoice = False
while not isGoodChoice:
t = float(input("\nInput temperature: ")) for i in range(len(temperature)):
if (temperature[i] == t):
isGoodChoice = True
liquidDensity = density[i]
liquidViscosity = viscosity[i]
if not isGoodChoice:
print ("Warning: not available value")
concentration = float(input("Input concentration [g/l]: "))
diameter = float(input("Input particle diameter [micrometers]: ")) solutionDensity = getSolutionDensity(liquidDensity, concentration) solutionViscosity = getSolutionViscosity(liquidViscosity,
concentration) heightCylinder = 0.1
particle = initializeScene(heightCylinder, diameter/2.0) time = 0
depth = 0
while (depth < heightCylinder):
visual.rate(1000)
depth = getSedimentationDepth(solutionDensity,
solutionViscosity, diameter, time) particle.pos.z = -depth
time += 1
print ("seconds:", time) main()
38 Basic Physical Properties of Soil
Figure 2.13 Visualization of a particle settling in a fluid according to Stokes’ law as simulated by Python and visualized with Visual Python.
Figure 2.13 shows a visualization, using Visual Python, of a 10µm particle settling in a fluid (particle size is not to scale, but was magnified for visualization purposes).
...
2.14 E X E R C I S E S
2.1. Use the program PSP_boxCountingby runningmain. Set different values for thethresholdvariable and observe the evolution of the fractal dimension during refinement. Change the colour of the pore space in the output image generated by the program to red.
2.2. A soil sample collected in the field with a brass cylinder (8 cm diameter, 6 cm hight) has mass of 45 g. The wet sample (including the brass cylinder) weighed 445 g. After oven drying for 24 h at 105◦C, the dry sample (including the cylin- der) weighed 375 g. Assuming that the soil particle density ρs is 2650 kg m–3 and the density of waterρlis 1000 kg m–3, compute the mass basis water contentw, the dry bulk densityρb, the volumetric water contentθ and the total porosityφf. For the constantπ, use four decimal places.
2.3. Use the programPSP_basicProperties.py. Input the values of bulk density and mass basis water content obtained from Exercise 2.2 to calculate the total por- osity, the void ratio, the volumetric water content, the gas porosity and the degree of saturation. Try different values of bulk density and mass basis water content to check the output variable changes. Modify the program to enter the particle density chosen by the user. Pass the variable as an argument to the function and use the new particle density for computing the porosity.
Exercises 39 2.4. Assume that the soil pedon is described by a right prism of 1.5 m depth. Calculate the equivalent water heighthwin millimetres, by knowing that the volumetric soil water contentθis 0.25 m3m–3.
2.5. Compute the geometric mean diameter and the geometric standard deviation for a silt loam soil having textural fractions of silt = 60%, sand = 20% and clay = 20%.
2.6. Derive the dimensions of the dynamic viscosityη, by using eqn (2.47).
2.7. Using Stokes’ law, compute the settling time from the liquid surface at a depth of 0.2 m, for particles of 15µm in diameter. Assume that the density of the solid phase is 2650 kg m–3, the temperature is 20◦C and the liquid is pure water. Repeat the computation for the same particle diameter, but settling in a solution of sodium hexamataposphate. For pure liquid water, use a liquid density of 998.23 kg m–3and a viscosity η = 0.001002 Pa s–1. For the sodium hexamataposphate solution, the liquid density is 998.5 kg m–3and the viscosity is 0.001023 Pa s–1.
2.8. Based on Stokes’ law and usingPSP_sedimentation.py, compute the sedi- mentation times required for spherical particles of sizes 100, 50 and 2µm to settle in pure water, at 20◦C, down to a depth of 0.3 m. Repeat the computation for a solution of hexametaphosphate at concentrations of 5 and 10 g L–1. The required variables, such as liquid and solid density, as well as liquid viscosity, are provided in the program. Check the effect on the settling times by changing the particle diameter and the concentrations of hexametaphosphate.
3
Soil Gas Phase and Gas Diffusion
The gas-phase composition of soil is similar to that of the atmosphere in terms of mol- ecules characterizing the gas phase, but molecules are present in different concentrations to those in the atmosphere. Oxygen is consumed and carbon dioxide is generated by the metabolic activity of microorganisms and plant roots in the soil. Oxygen, carbon dioxide and water vapour all diffuse into and out of the soil. Soil air is typically around 79 % N2. The remaining 21 % is mostly CO2and O2, which vary reciprocally. In well-aerated soil, the CO2 content of the air is around 0.25 % and the O2 content 20.73 %. In poorly aerated soil, O2 levels can approach zero. The absorption of nutrients by roots and the beneficial activity of microorganisms depend on an adequate supply of oxygen. Soils should therefore be managed to provide adequate aeration.
The gas phase is characterized by a high compressibility; therefore, as a first approxi- mation, we can consider the gas phase to have a constant volume. It has low viscosity (about 1/50 of the water viscosity); therefore, computation of energy losses during con- vective movement of gas are commonly neglected. The soil gas phase has a low heat cap- acity; hence it is often ignored in computation of thermal dynamics and heat transport. It also has a very low density; therefore its contribution is usually neglected in the compu- tation of the total soil mass or in relationships between masses of the different soil phases.
The rate of respiration of living organisms in the soil depends primarily on the avail- ability of oxygen and carbon in the soil, and on soil temperature and soil moisture.
Oxygen consumption is greatest when organic matter is incorporated into moist, warm soil. Respiration rates in winter are suppressed because of low temperature and some- times low oxygen concentration. In the summer, low soil water potential may inhibit respiration. Respiration rates are usually highest in spring months, when roots are grow- ing rapidly, root exudates are plentiful for microorganism growth, and moisture and temperature conditions are favourable.
In this chapter, we will first present simple models for gas exchange for individual roots or microorganism colonies, as well as gas exchange between the soil profile and the atmosphere. To introduce the first concepts of numerical solutions for mass exchanges in soils, only steady-state exchange will be considered. Useful results relating to soil aeration can be obtained at steady state, and the simpler steady-state solutions will allow us to start developing numerical tools. The transient flow problems in later chapters should provide clear examples of how the transient gas flow problem can be solved if such solutions are needed.
Soil Physics with Python. First Edition. Marco Bittelli, Gaylon S. Campbell and Fausto Tomei.
© Marco Bittelli, Gaylon S. Campbell and Fausto Tomei 2015. Published in 2015 by Oxford University Press.
Transport Equations 41 The solution of gas flow by diffusion was selected as a first example for presenting a numerical solution of mass flow because of its simplicity in the numerical scheme.
The term ‘diffusion’ refers to the transport of a material within a single phase by ran- dom molecular motion. Diffusion can result from gradients in pressure, temperature, external forces (forced diffusion) or concentration. Here we discuss only diffusion due to concentration gradients. We assume an isobaric and isothermal system, which means pressure and temperature are constant in time and space. Detailed discussions of dif- fusion fluxes and coefficients have been provided by Birdet al.(1960), Cussler (1997) and Polinget al.(2000).