## 3 Theoretical Background  
### Sedimentation Process  
We consider the case of a particle falling vertically toward the bottom of a container in an incompressible viscous fluid, for instance the red particle in Figure 2. According to hydrodynamic laws, a particle of any size and shape moving with velocity \(u\) in a fluid of viscosity \(\eta\) is subjected to a drag force. If the difference between the force of resistance and the immersed weight of the particle (absolute weight minus the buoyancy force) is negative, the particle falls with an acceleration given by  
\[\frac{du}{dt}=\left(1-\frac{\rho}{\rho_{s}}\right)g-\frac{C_{D}u^{2}\rho S}{2m} \tag{1}\]  
where \(\rho_{s}\), \(m\) and \(S\) are density, mass and surface of particle, respectively. \(C_{D}\) is the coefficient of drag and it is connected to dynamic properties of the system through the Reynolds number.  
Although the coefficient \(C_{D}\) can be related to different motion regimes, Equation (1) refers to _free-fall_ particles, therefore, it can only be used to analyze the hydrodynamic behavior of extremely diluted dispersions. In a real system when the gravitational force exceeds the buoyancy and drag forces, a concentration gradient builds up in the system with the larger particles moving faster to the bottom of the container. Hydrodynamic interactions between particles make the system very complex so that for the calculation of the fall speed it is convenient to develop semi-empirical approaches. That is to say a model which makes use of both laboratory results and fieldwork. In order to determine the settling rate, we choose an axis system with \(z\)-direction opposite to the velocity of particle motion. At time \(t=0\) the concentration of solids, \(C_{0}\), is the same at any point. As time progress different zones are formed like, for instance, the supernatant at he top where \(C=0\) (_clear fluid_), and a zone of maximum concentration, \(C_{max}\), at the bottom (see Figure 2). The concentration of suspended solids is a function of both position and time \(C(z,t)\) and it determines the height of the interface between a particle-rich sediment and supernatant. Based on the principle of mass conservation, the interface height, \(H(t)\), is defined as  
\[\int_{0}^{H(t)}C(z,t)dz=H_{0}C_{0} \tag{2}\]  
at any time \(t>0\).  
Furthermore, in the region \(z<H(t)\), the continuity equation provides  
\[\frac{\partial C}{\partial t}-\frac{\partial(uC)}{\partial z}=0 \tag{3}\]  
Differentiation of Equation (2) with respect to time \(t\) yields  
\[\int_{0}^{H(t)}\frac{\partial C}{\partial t}dz+\frac{dH}{dt}C(H(t),t)=0 \tag{4}\]  
Combination of Equation (4) with Equation (3) results  
\[-\frac{dH}{dt}=u \tag{5}\]Equation (5) relates the interface lowering to the accumulation speed of solids at the bottom of the container.  
To illustrate the applicability of Equation (5) for settling process, let us make the assumption that at a given time \(t=\theta\) the interface is to the position \(z=H(\theta)\) and the average concentration of suspended solids is \(C(\theta)\). The amount of material below the interface consists of particles made up of all the particles coarse enough to have settled the entire height \(H\) and particles of smaller size which not have settled. The total mass of solid is therefore  
\[C_{0}H_{0}\mathcal{A}=C(\theta)H(\theta)\mathcal{A}+C(\theta)\mathcal{A} \theta\left(-\frac{dH}{dt}\right)_{t=\theta} \tag{6}\]  
where \(\mathcal{A}\) is the cross section of the cylinder. Solving Equation (6) with respect to \(C(\theta)\), it is found.  
\[C(\theta)=\frac{H_{0}C_{0}}{H(\theta)+\theta\left(-\frac{dH}{dt}\right)_{t= \theta}} \tag{7}\]  
This equation provides a rapid method for extracting the solid concentration at any time.